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Multiple model adaptive control with mixing

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Multiple model adaptive control with mixing

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Content MULTIPLEMODELADAPTIVECONTROLWITHMIXING by MatthewKuipers ADissertationPresentedtothe FACULTYOFTHEGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (ELECTRICALENGINEERING) August2009 Copyright 2009 MatthewKuipers Acknowledgements Although my name alone appears on the author list, I do not pretend to have created this dissertation in a vacuum. There are a number of professors, colleagues, friends, and family members who played a role in developing whatever merits my research may have. It is my pleasure to acknowledge the contributions someofthesepeople. First, I would like to thank my advisor, Professor Petros Ioannou, who played a vital role in the direc- tion and development of my research. I am grateful to had an advisor who gave me tremendous latitude in exploring the topics of adaptive control theory, while guiding me towards meaningful and potentially fruitful research. His enthusiasm for adaptive control and optimism for success gave me reason to pursue research paths that I may have otherwise abandoned. In fact, it was Professor Ioannou who encouraged metocontinuealineofresearchthatbecamethebasisofChapter3and,ultimately,thedissertation. What also makes Professor Ioannou remarkable as an advisor is that he genuinely cares about his students’ best interests. Totheuninitiated, this qualitymayseemtobe aprerequisiteforaresearch advisor, butitis not. Iconsidermyselffortunatetohadanadvisorwhogavemeunbiased,sincereadvisement. Members of my research committees deserve special considerations. I would like to thank Professor MichaelSafonov,ProfessorHenrykFlashner,ProfessorEdmondJonckheere,andProfessorMichaelNeely for providing constructive criticism and insightful suggestions. These comments had a positive impact on thedissertation’sfinalform. It is my hope that my research has contributed to the research area of robust adaptive control theory, especiallywithinitssubfieldmultiplemodeladaptivecontrol. ForthisreasonIwouldliketoacknowledge some of the pioneers of multiple model adaptive control: Professor Michael Athans, Professor Stephen ii Morse, Professor Kumpati Narendra, Professor Jo˜ ao Hespanha, and Professor Michael Safonov, among others. The works of these researchers provided not only inspiration but also a substantial foundation on which my research stands. Special thanks is given to Professor Athans, Dr. Sajjad Fekri, Professor Hesphana, Professor Safonov, and their respective co-workers for their comments and questions, which helpedmefocusmyresearcheffortstowardsspecificissuesandpracticalconcerns. Mostofall,Iwouldliketothankmyfamily,particularlymyparentsPeterandJenniferandmybrother Luke,foralltheirloveandsupportovertheyears. iii iv Table of Contents Acknowledgements List of Tables List of Figures Abstract Chapter 1: Introduction 1.1 The Problem 1.2 Existing Approaches in Literature 1.3 Contribution 1.4 Outline Chapter 2: Adaptive Mixing Control 2.1 Introduction 2.2 Notation and Preliminaries 2.3 A Simple Example 2.3.1 Simulation 2.4 General Problem Formulation 2.5 Conceptual Framework 2.5.1 Multicontroller 2.5.2 Robust Adaptive Supervisor 2.5.3 Stability and Robustness Results 2.6 Conclusions Chapter 3: A Benchmark Example 3.1 Introduction 3.2 The Two-Cart Model 3.3 Robust Adaptive Control Designs 3.4 Simulation Results 3.4.1 Case1: Zero Initial Condition 3.4.2 Case2: Non-zero Initial Condition 3.4.2.1 Satisfied Model Assumptions 3.4.2.2 Violated Model Assumptions 3.5 Conclusions Chapter 4: Adaptive Control of an Airbreathing Hypersonic Flight Vehicle 4.1 Introduction 4.2 Adaptive Mixing Control 4.3 Airbreathing Hypersonic Flight Vehicle (AHFV) Model ii vi vii ix 1 1 3 4 7 8 8 10 16 34 35 37 38 43 46 48 58 58 59 62 66 66 69 72 74 75 76 76 82 88 v 4.4 Adaptive Mixing Control Design of AHFVs 4.4.1 Multicontroller 4.4.2 Robust Adaptive Supervisor 4.5 Simulation Results 4.6 Conclusion Chapter 5: Concluding Remarks and Suggestions for Future Work 5.1 Adaptive Mixing Control With Switching 5.2 Unpredictable Parameter Changes Bibliography 95 96 101 105 107 117 119 122 127 vi List of Tables Table 3.1: Case 1: Assumptions Satisfied Table 3.2: Case 2: Assumptions Satisfied Table 3.3: Case 2: Assumptions Violated Table 4.1: Summary of robust control design 67 71 71 100 vii List of Figures Figure 1.1: Multiple model adaptive control architecture: Based on observed data, the supervisor Σ S selects/blends/mixes candidate controllers Figure 1.2: Adaptive mixing control architecture Figure 2.1: AMC: Based on observed data, the supervisor Σ S mixes candidate controls to apply to the unknown plant P Figure 2.2: Robust adaptive supervisor: The adaptive law Σ T adjusts the estimate θ(t) to make the estimation error ε(t) “small.” By considering the estimate as the true unknown parameter, the mixing signal β(t) activates the candidate controllers Figure 2.3: The mixing signal β Figure 2.4: Block diagram of adaptive mixing control scheme Figure 2.5: Simulation results Figure 2.6: Multicontroller implementation and robust performance formulation Figure 3.1: Mass-spring-dashpot system benchmark example Figure 3.2: The adaptive mixing control architecture Figure 3.3: M is constructed from smooth bump functions such that requirements M1,M2 are satisfied Figure 3.4: Time delay margin: Mixing control (solid); mixed- μ controller (dash) Figure 3.5: Expected output RMS: Adaptive mixing control (solid); mixed- μ controller (dash); switching-based supervisor (inset, dash-dot) Figure 3.6: Comparison of the plant outputs of the Q-blending and output-blending schemes with θ* = 0.325, τ = 0.05, and nominal disturbance model Figure 3.7: Case 1: Comparison of the start-up transients of adaptive mixing control (AMC), RMMAC, and supervisory adaptive control (SAC): θ* = 1.75, τ = 0.05, and nominal disturbance model 2 3 9 9 21 26 35 41 59 60 63 64 64 69 69 viii Figure 3.8: Case 1 simulation results with θ* = 1.02, τ = 0.05, and nominal disturbance model: Adaptive mixing control (AMC), RMMAC, and supervisory adaptive control (SAC) Figure 3.9: Case 1 simulation results with θ* = 1.02, τ = 0.05, and the off-nominal disturbance model with a disturbance power of 100• Θ and a bandwidth of 30• α: Adaptive mixing control (AMC), RMMAC, and supervisory adaptive control (SAC) Figure 3.10: Case 2: Results for θ* = 0.064. AMC (blue), RMMAC (green), and GNARC (red) Figure 4.1: The adaptive mixing control architecture Figure 4.2: Vehicle geometry Figure 4.3: Mixed- μ problem formulation Figure 4.4: Multicontroller implemented by Q-mixing Figure 4.5: Simulation results for the case with real and complex perturbations Figure 4.6: Tracking errors Figure 4.7: Mixing signal β Figure 5.1: Simulation results for AMC with an abrupt parameter jump Figure 5.2: Modified AMC scheme with estimate resetting Figure 5.3: The reset-and-hold algorithm Figure 5.4: Simulation results for AMC with reset and hold, RMMAC, and supervisory switching control with an abrupt parameter jump Figure 5.5: Simulation result for adaptive mixing controller with a conventional adaptive controller 70 72 73 79 89 97 101 104 106 107 123 124 124 125 126 Abstract Despite the remarkable theoretical accomplishments and successful applications of adaptive control, the field is not sufficiently mature to solve challenging control problems requiring strict performance and safetyguarantees. Towardsaddressingtheseissues,anoveldeterministicmultiple-modeladaptivecontrol approachcalledadaptivemixingcontrolisproposed. Inthisapproach,adaptationcomesfromahigh-levelsystemcalledthesupervisorthatmixesintofeed- back a number of candidate controllers, each finely-tuned to a subset of the parameter space. The mixing signal,thesupervisor’soutput,isgeneratedbyestimatingtheunknownparametersand,ateveryinstantof time,calculatingthecontributionlevelofeachcandidatecontrollerbasedoncertaintyequivalence. The proposed architecture provides two characteristics relevant to solving stringent, performance- driven applications. First, the full-suite of linear time invariant control tools is available. A disadvantage of conventional adaptive control is its restriction to utilizing only those control laws whose solutions can be feasibly computed in real-time, such as model reference and pole-placement type controllers. Because itscandidatecontrollersarecomputedoffline,theproposedapproachsuffersnosuchrestriction. Second, the supervisor’s output is smooth and does not necessarily depend on explicit a priori knowledge of the disturbance model. These characteristics can lead to improved performance by avoiding the unnecessary switchingandchatteringbehaviorsassociatedwithsomeothermultipleadaptivecontrolapproaches. Thestabilityandrobustnesspropertiesoftheadaptiveschemeareanalyzed. Itisshownthatthemean- squareregulationerrorisoftheorderofthemodelingerror. Andwhentheparameterestimateconvergesto ix its true value, which is guaranteed if a persistence of excitation condition is satisfied, the adaptive closed- loop system converges exponentially fast to a closed-loop system comprised of the plant and some linear timeinvariantcontrollerthatsatisfiesthecontrolobjective. Three examples are presented. First, a pedagogical example introduces the proposed approach in a tutorial manner. Second, to demonstrate the performance capabilities of an proposed scheme, a bench- mark control example is considered. The third example is a robust adaptive velocity and altitude tracking controllerforamultiple-input/multiple-outputairbreathinghypersonicflightvehiclemodel. x Chapter1 Introduction 1.1 TheProblem The problem that is being investigated is the control of uncertain plants whose level of parametric un- certainty requires some level of “intelligence” to achieve acceptable performance requirements. All real systems are subjected to uncertainty due to unmodeled dynamics, unknown system parameters, distur- bances, and process changes. A practical control design, therefore, must be able to maintain performance and stability robustly in the presence of these uncertainties. When model uncertainties are sufficiently small, modern linear time invariant (LTI) control theories, e.g.,H ∞ andμ-synthesis [75, 65, 77], ensure, when possible, satisfactory closed-loop objectives specified in meaningful engineering terms (frequency weights on the relevant transfer functions) are met. However, changes in operating conditions, failure or degradation of components, or unexpected changes in system dynamics may all violate the assumption of small uncertainty, particularly parametric uncertainty. The impact of such “large” uncertainty is that a single fixed LTI controller may no longer achieve satisfactory closed-loop behavior, let alone stability. What is needed is a controller that is able to monitor the plant dynamics in order to adjust its control law tocompensateforsuchparametricuncertaintyandothermodelingerrors[29,Sec. 1.3]. Adaptive control copes with large parametric uncertainty by tuning controller gains in response to estimated changes in the model. Since in conventional (robust) adaptive control [29, 30] the controller 1 d u y β Figure 1.1: Multiple model adaptive control architecture: Based on observed data, the supervisor Σ S selects/blends/mixescandidatecontrollers. gainsarecalculatedinrealtimebasedontheestimatedplantmodel,thecomplicatedrelationshipbetween plant parameters andH ∞ andμ-synthesis controller gains has precluded the use of conventional adaptive versionsofthesemodernrobustcompensators. By using candidate controllers designed off-line, the multiple model adaptive control (MMAC) archi- tecture,showninFig. 1.1,avoidsreal-timecontrollersynthesisissuesand,therefore,providesanattractive frameworkforcombiningadaptiveandmodernrobusttools. TheMMACarchitecturecomprisestwolevels ofcontrol: (1)alow-levelsystem C(β)calledthemulticontrollerthatiscapableofgeneratingfinely-tuned candidate controls and (2) a high-level system Σ S called the supervisor that influences the controlu by adjustingthemulticontroller,typicallybyselectingorweightingcandidatecontrollersbasedonprocessed plantinput/outputdata. The approach taken is a novel MMAC approach called adaptive mixing control (AMC), shown in Figure 1.2. Each of the p candidate controllers K 1 ,...,K p is tuned to a small subset of the parameter uncertainty, allowing for increased performance. The set of candidate controllers is sufficiently rich such that for every admissible plant there exists at least one controller that achieves the performance objective. By monitoring the plant’s input/output data, the robust adaptive supervisor system “mixes” the candidate controllers. The supervisor comprises two subsystems: the online parameter estimatorE and mixerM. The online parameter estimator generates real-time estimates θ(t) of the unknown parameter vector θ ∗ , andthemixerdeterminestheparticipationleveleachcandidatecontrollerbasedonθ(t). 2 G(s; θ * ) K 1 K N … Multicontroller E Robust Supervisor Mixer Robust Estimator Mixing Strategy M r disturbance and noise β θ u u y Figure1.2: Adaptivemixingcontrolarchitecture 1.2 ExistingApproachesinLiterature The MMAC concept is not new and has been around for quite some time. One recent approach is the so- called supervisory control [46, 47, 25], in which controller selection is made by continuously comparing in real time suitably defined norm-squared estimation error, also referred to as “performance signals”. Here β(t) is a piecewise continuous switching signal that takes on values from the candidate controller index set. As a manifestation of the certainty equivalence principle, the candidate controller associated with the smallest performance signal is placed in the loop according to an appropriate switching logic. Followingtheideaofsupervisorycontrol,logic-basedswitchingandmultiplemodelswerecombinedwith conventional adaptive control[49, 50, 32] with the objective of improving the sometime poor transient performance of conventional adaptive schemes. Also incorporating logic-based switching is the so-called unfalsifiedcontrolapproach[60,71],whichisanonidentifier-baseddeterministicapproach. The advantages of switching type approaches – rapid adaptation and relaxation of the convexity and linear-in-the-parameter assumptions of conventional adaptive control – are claimed in the literature [26]. However,logic-basedswitching,themechanismthatendowsthesedesirablebehaviors,mayalsointroduce performance issues. The first issue is, after a switch, the possibility of poor transient behavior resulting from an improperly initialized controller. This issue is a motivation for the bumpless transfer problem 3 [2, 69, 39, 76] and is an active area of research. The second issue demonstrated by simulations is, if hysteresis is used and the true model is near the boundary of two candidate models, the supervisor may persistently select a controller that, while stabilizing, does not achieve desirable closed-loop behavior. To encourage switching, the hysteresis constant may be reduced or replaced altogether with a dwell time logic,butattheincreasedriskoflong-termintermittentswitchingbetweenmultiplecontrollers. AnotherpromisingMMACapproachisbasedontheso-calledrobustMMAC(RMMAC)methodology thatprovidesguidelinesfordesigningboththecandidatecontrollerset(usingmixed- μsynthesistools)and the supervisor[15, 16, 17]. The RMMAC approach originated from the multiple model adaptive estima- tion/MMAC methods [3] of the 1970s, of which there have been numerous successful applications based on adaptations of these methods [22, 23, 61]. The RMMAC supervisor is based on a dynamic hypoth- esis testing scheme that generates for each candidate the posterior probability that its model is “closest” to the true plant. These probabilities are used to weight the candidate controller outputs, or, as done in the RMMAC/S variant, to switch into the loop the candidate controller associated with the highest pos- terior probability. Given accurate disturbance and noise models that satisfy the standard Kalman filter assumptions,simulationsdemonstraterapidadaptationandsuperiorperformancecomparedtoanonadap- tive mixed- μ compensator [17]. Acknowledged within the same reference, however, is that special care is needed to compensate for an inaccurate stochastic disturbance model. The RMMAC/XI architecture was proposed to handle a range of disturbance powers, at the cost of additional Kalman filters. Still, if the disturbancepowerissignificantlyoutsidetheexpectedrange,poorperformancemayoccur. Andalthough loss of stabilizability is not an issue (because there is no estimated model), it should be noted that no stabilityresultshavebeenpublished. 1.3 Contribution The nature of MMAC schemes with logic-based switching is that if the measured data suggests small deviationsintheplantmodelthenthecourseofactioniseithernochangeoralargechange(i.e.,switch)in 4 thecontrollaw. TheuniquefeatureofAMCwithrespecttootherdeterministicMMACapproachesisthat themulticontrolleristunedcontinuouslybythemixingsignalβ(t),ratherthanswitchedbyadiscontinuous signal,andtherebyside-steppingissuesofswitching;theleveloftuningagreeswithwhatthedatasuggests. BecausestabilityofAMCschemesrequirethatadaptationisslowinsomesense,thereisbuilt-incontroller conditioning that can reduce transient behaviors as the supervisor transitions the multicontroller from one candidate controller to another. While the performance of an AMC scheme can be improved by incorporatingaprioriknowledgeofthedisturbance,theevaluationsof[36]demonstratethatanAMCcan achievesatisfactoryperformancedespiteanumberofunknowndeviationsinthedisturbancemodel. From a conventional adaptive control viewpoint, AMC can be viewed as an indirect adaptive control approach in which the parameterized controller is not necessarily constructed by model reference, pole placement, or linear quadratic methods, but rather by arbitrary methods, including modern robust H ∞ and mixed- μ. Since the underlying control law can be constructed using the full-suite of powerful LTI design tools, the control engineer is capable incorporating relevant performance and robustness specifications into the design. Thisanimportantsteptowardspracticaladaptivecontroldesign. Insummary,themaincontributionsofthisthesisare • The AMC approach is a new MMAC approach that continuously tunes the control law based on robustonlineparameteridentification. • TheAMCapproachextendstheresultsofconventionaladaptivecontrolinthesensethattheparam- eterized controller may be constructed by arbitrary LTI techniques, including modern robust tools suchasH ∞ andμ-synthesis. • The stability and robustness analysis of the AMC approach for single-input/single-output (SISO) plants and a class of multiple-input/multiple-output (MIMO) are presented. The mean-square regu- lation error is of the order of the modeling error provided the unmodeled dynamics satisfy a norm- bound condition. And when the parameter estimate converges to its true value, which is guaranteed 5 ifapersistenceofexcitationconditionissatisfied,theadaptiveclosed-loopsystemconvergesexpo- nentially fast to a closed-loop system comprised of the plant and some LTI controller that satisfies thecontrolobjective. • An AMC solution to a benchmark control problem is presented. This AMC scheme demonstrates satisfactory performance for the ideal case, as well as when a number of model assumptions are violated. • Using the AMC approach, a robust adaptive velocity and altitude controller for an airbreathing hy- personicflightvehicleisdeveloped. SimulationresultssuggestthattheAMCschemeprovidessupe- riorperformanceoverthe“best”nonadaptivecontroller. Thisaircraftmodelismultiple-input/multiple- output(MIMO)withmultipleunknownparameters,unmodeleddynamics,andunmeasurableexoge- nousdisturbances. • To the best of our knowledge, it is for the first time stability and performance preserving controller interpolation via the Youla parameterization is integrated with adaptive control. This interpolation techniquealsoprovidesamethodofincludingunstablecandidatecontrollers. • By using the AMC approach, one can vary the control objective as a function of the unknown plant parameter vector θ ∗ . This would be useful, for example, if extreme values of θ ∗ are a result of damages or failures, while nominal values ofθ ∗ indicate a healthy system. For this scenario, one can design conservative candidate controllers for the damaged models, while constructing more aggressivecandidatecontrollersforthehealthymodels. • By using the adaptive mixing control approach in conjunction with mixed- μ synthesis tools, the designerisabletotakeintoaccountunmodeleddynamics,uncertainparameters,unmeasurableplant disturbances,unmeasurablesensornoise,andexplicitquantitativeperformancespecifications. Thematerialinthisthesisisbasedontheauthor’sworks[9,10,18,27,36,37]. 6 1.4 Outline The following chapters describe the details of the AMC approach. Chapter 2 provides the problem for- mulation, introduces the AMC architecture, and analyzes its stability and robustness properties. First, an AMC scheme is designed for a first-order SISO system to illustrate its design and analysis. Second, the design and analysis is extended to a more generaln th SISO plant. The focus of the presentation is on the sufficientpropertiesoftheindividualcomponentsofAMC.Thispresentationemphasizesthemodularityof thearchitecture,allowingthedesignertoutilizeeitheroff-the-shelforad-hocdesignsforthecomponents. InChapter3,wedesignanAMCschemeforabenchmarkcontrolproblem. Thesystemtobecontrolledis a SISO mass-spring-dashpot (MSD) system with one unknown parameter. We compare the performance of the AMC scheme to a nonadaptive mixed- μ scheme, a RMMAC scheme, and a supervisory adaptive control scheme to evaluate the advantages and disadvantages of the AMC scheme. In Chapter 4, a robust adaptive altitude and velocity tracking controller for an airbreathing hypersonic flight vehicle is designed usinganAMC.ThevehiclemodelisMIMOwiththreeunknownsystemparameters,aswellasunmodeled dynamics. FuturedirectionsarediscussedinChapter5. 7 Chapter2 AdaptiveMixingControl 2.1 Introduction The focus of this chapter is the presentation and analysis of a novel MMAC architecture, adaptive mix- ing control (AMC), that “mixes” the candidate controllers, each designed to meet the control objective on a subset of the parametric uncertainty, in a continuous manner based on a robust adaptive law. The multicontroller is not only capable of generating any of the candidate control laws but also, by controller interpolation, a stable mix of candidate control laws. This allows the multicontroller to evolve from one controllertoanotherinacontinuousmanner. Moreover,providedcertainconditionsontheplantinputare met, the AMC converges exponentially fast to meet the control objective. The supervisor, shown in Fig. 2.2, generates the mixing signal β(t) by processing the estimate θ(t) of the unknown plant parameters θ(t) with a system called the mixer M that determines the level of participation of the candidate con- trollers. Thisdeterminationisamanifestationofcertaintyequivalence: ateveryfixedt≥ 0,thecandidate controllersthatweredesignedforθ ∗ =θ(t)aremixedsuchthatclosed-loopobjectivesaremet. TheimmediatemotivationofAMCistoprovideanadaptivecontrolapproachthatiscapableofincor- poratingthefullsuiteofpowerfulLTItools,whileavoidingsomeoftheperformanceissuesassociatedwith undesirableswitchingphenomenaandanunknownoruncertaindisturbancemodel,offeringanalternative to the existing MMAC approaches for particular applications. The unique feature of AMC with respect 8 d u y β Figure 2.1: AMC: Based on observed data, the supervisor Σ S mixes candidate controls to apply to the unknownplantP u β y y u Figure 2.2: Robust adaptive supervisor: The adaptive law Σ T adjusts the estimateθ(t) to make the esti- mation error 1 (t) “small.” By considering the estimate as the true unknown parameter, the mixing signal β(t)activatesthecandidatecontrollers. to existing MMAC approaches, including RMMAC, is that the intent is not to converge to one controller, but rather a stable mix of candidate controllers. This mixing behavior, together with tuning the mixing signalcontinuously,avoidsswitchingand,inturn,someofitsundesirablebehaviors. Also,whileanAMC scheme’s performance may be improved by incorporating knowledge of the disturbance, the evaluations inChapter3and[36],wherethelatterfocusesonamultipleestimatorvariantoftheapproachpresentedin this paper, demonstrate that adaptive mixing can achieve satisfactory performance despite significant per- turbationsinthedisturbancepowerandbandwidth. Last,utilizingpre-computedcontrollers,AMCavoids computational and existence issues of calculating controller gains when stabilizability of the estimated plantislost. In Section 2.2 we define notation and present a number of useful results used in the stability and robustnessanalysisofAMC.WedesignandanalyzeanAMCschemeforasimplefirst-orderSISOsystem. The aim of this example is to introduce the design and analysis of AMC in a tractable context. The problem formulation and AMC approach are generalized in Sections 2.4 and 2.5 to a n th order SISO 9 process. Special focus is put on sufficient properties of each component in the AMC architecture that ensurestability. Thismodularityisanattractivefeaturebecauseawide-classofimplementations,eitheroff theshelforadhoc,maybeused. TheanalysisofthegeneralizedAMCschemeisfoundintheAppendix. 2.2 NotationandPreliminaries For A ∈ R m×n , the transpose of A is denoted by A T . For the n-vector x, |x| is the Euclidean norm (x T x) 1/2 . TheinducedmatrixnormkAkisdefinedby kAk 4 = sup |x|=1 |Ax| (2.1) andkAkisgivenbykAk = [λ m (A T A)] 1/2 ,whereλ m (A)isthemaximumeigenvalueofA. Ify :R + → R n ,thentheL p normofy isdenotedaskyk p andthetruncatedL 2δ normisdefinedas ky t k 2δ 4 = Z t 0 e −δ(t−τ) y T (τ)y(τ)dτ 1 2 (2.2) whereδ ≥ 0 is a constant, provided that the integral in (2.2) exists. Byky t k 2 we wean thatky t k 2δ with δ = 0, and we say thaty ∈ L 2e ifky t k 2 exists. Consider the functiony ∈ L 2e . If there exists constants c 0 ,c 1 > 0suchthatforanyconstantμ≥ 0thefunctiony satisfies Z t+T t y T (τ)y(τ)dτ ≤c 0 μT +c 1 , ∀t,T ≥ 0 (2.3) then we say that y ∈ S(μ). Similarly, if there exists constants c 0 ,c 1 > 0 such that for any function w : [0, ∞)→R + ,wherew∈L 1 ,thefunctiony satisfies Z t+T t y T (τ)y(τ)dτ ≤c 0 Z t+T t w(τ)dτ +c 1 , ∀t,T ≥ 0 10 thenwesaythaty∈S(w). LetH(s)andh(t)bethetransferfunctionandimpulseresponse,respectively,ofsomeLTIsystem. If H(s)isapropertransferfunctionandanalyticinRe[s]≥−δ/2forsomeδ≥ 0,whereRe[s]denotesthe real part ofs, then theH ∞ system norm is given bykHk ∞ 4 = sup jω |H(jω)|. Thek·k 2δ system norm ofH(s) is defined askHk 2δ 4 = 1 √ 2π n R ∞ −∞ H jω− δ 2 2 dω o1 2 . The inducedL ∞ system norm ofH is given bykHk ∞−gn =khk 1 . Ify = H(s)u andkuk ∞ = u 0 then|y|≤kHk ∞−gn u 0 . Consider another transferfunctionF(s). WesayH =F ifH andF shareacommonminimalrealization. Let X,Y ⊂ R m and f : X → R n . By δX we mean the boundary of the set X; by X −Y we mean the set-theoretic difference. An open ball of radius r centered around pointx 0 ∈ X is denoted as B r (x 0 ) = {x ∈ R m : |x−x 0 | < r}. Let e i ∈ R N denote the i th standard basis vector, i.e., the i th componentofe i isaone;allothercomponentsarezero. Considerthelinearsystem ˙ x(t) =A(t)x(t)+B(t)u(t), x(t 0 ) =x 0 y(t) =C(t)x(t)+D(t)u(t) (2.4) wherex(t)∈R n is the state vector;y(t)∈R r is the measurable output vector;u(t)∈R m is the control input of the system; the elements of the matrices A(t) ∈ R n×n , B(t) ∈ R n×m , C(t) ∈ R l×n , and D(t)∈R l×m are bounded continuous functions of time; andx(t 0 ) denotes the value ofx(t) at the initial timet =t 0 ≥ 0. Thesolutionx(t),y(t)of(2.4)isgivenby x(t) = Φ(t,t 0 )x(t 0 )+ R t t0 Φ(t,τ)B(τ)u(τ)dτ y(t) =C(t)x(t)+D(t)u(t) (2.5) 11 where Φ(t,t 0 ) is the state transition matrix defined as the matrix that satisfies the linear homogeneous matrixequation ∂Φ(t,t 0 ) ∂t =A(t)Φ(t,t 0 ), Φ(t 0 ,t 0 ) =I. (2.6) IfA(t),B(t),C(t),andD(t)donotdependontime,thesystem(2.4)takestheformof ˙ x(t) =Ax(t)+Bu(t), x(t 0 ) =x 0 y(t) =Cx(t)+Du(t) (2.7) whereA,B,C, andD are matrices of the same dimension as in (2.4) but with constant elements. For the LTIsystem(2.7),Φ(t,t 0 )onlydependsonthedifferencet−t 0 ,i.e., Φ(t,t 0 ) =e A(t−t0) (2.8) wheree At isgivenby e At =L −1 [(sI−A) −1 ]. (2.9) whereL −1 [·] denotes the inverse Laplace transform ands is the Laplace variable. Because the definition (2.6)ofΦ(t,t 0 )onlydependsonA(t),bysayingthestatetransitionmatrix(orsimplythetransitionmatrix) Φ(t,τ)ofA(t)wemeanthestatetransitionmatrixofthesystem(2.4). Considerthehomogeneoussystem ˙ x(t) =A(t)x(t), x(t 0 ) =x 0 (2.10) where A(t) has been previously defined. A state x e is said to be an equilibrium state of a dynamical systemif,fortheinitialconditionx 0 =x e ,thesolutionx(t)of(2.10)isgivenbyx(t) =x e ,∀t≥t 0 . The 12 equilibrium statex e is said to be exponentially stable (e.s.) if there exists anα > 0 and for every > 0 thereexistsaδ> 0suchthat |x(t;t 0 ,x 0 )−x e |≤e −α(t−t0) , ∀t≥t 0 (2.11) whenever|x 0 −x e |<δ. LetkΦ(t,t 0 )k be the induced matrix norm of Φ(t,t 0 ) at each timet≥t 0 . The equilibriumstatex e = 0of(2.10)isexponentiallystable(e.s.) ifandonlyifthereexistspositiveconstants α 0 ,λ 0 suchthatthestatetransitionmatrixΦ(t,t 0 )of(2.10)satisfies kΦ(t,t 0 )k≤α 0 e −λ0(t−t0) , ∀t≥t 0 ≥ 0. (2.12) ConsiderA(t)from(2.10). IfthestatetransitionmatrixΦ(t,t 0 )ofA(t)satisfiescondition(2.12),thenwe say thatA(t) is e.s. For the LTI case where the elements ofA(t) are constant, a necessary and sufficient condition for exponential stability ofx e = 0 is that all eigenvalues ofA have negative real parts. If this conditionholds,wesaythatAisastabilitymatrixor,equivalently,thatAisHurwitz. Consider the LTI system (2.7). The pair (A,B) is stabilizable if and only if there exists a matrix F ∈R m×n suchthatA−BF isastabilitymatrix. Thepair(C,A)isdetectableifandonlyifthereexists a matrixL ∈ R n×r such thatA−LC is a stability matrix. Ifu(t) = y(t) = 0,∀t ≥ 0, implies that for any initial conditionx 0 the statex(t)→ 0 ast→∞, then the pair (C,A) is detectable. Output injection isaprocedurebywhichLy(t)isaddedandsubtractedfromthefirstequationof(2.7),yielding ˙ x(t) = (A−LC)x(t)+Ly(t)+(B−LD)u(t). (2.13) Properties of y(t) and u(t) can then be used to establish guarantees on x. This technique is useful in stabilityandrobustnessanalysisofAMCschemes. 13 The following key results are used in the stability and robustness analysis of the AMC scheme. The results are well known, and, unless stated otherwise, their proofs can be found in [30] and the references within. Theorem1. LetΩ⊂R 2n be compact andθ be any constant inΩ. Let the elements of the parameterized detectible pair (C(θ),A(θ)) be continuously differentiable with respect toθ ∈ Ω, whereA(θ) ∈ R n×n andC(θ)∈R l×n . 1. Then there exists an continuously differentiable matrix functionL : Ω→R n×l , such thatA I (θ) 4 = A(θ)−L(θ)C(θ) is a stability matrix uniformly inθ ∈ Ω, i.e.,∃σ > 0 such that∀θ ∈ Ω,A I (θ) satisfies max i Re{λ i [A I (θ)]}<−σ (2.14) whereλ i (A I (θ))isthei th eigenvalueofthematrixA I (θ). 2. If θ(t) ∈ Ω for all t ≥ 0 and ˙ θ ∈ L 2 is satisfied in addition to the conditions in 1), then the equilibriumx e = 0of ˙ x =A I (θ(t))xise.s. 3. Ifθ(t) ∈ Ω for allt ≥ 0 and ˙ θ ∈ S(μ 2 ) is satisfied in addition to the conditions in 1), then there existsaμ ∗ > 0suchthatifμ∈ [0,μ ∗ )theequilibriumx e = 0of ˙ x =A I (θ(t))xise.s. TheproofofTheorem1isacombinationofthewell-knownresultsof[12]andthelineartimevarying (LTV)stabilityresultsfoundin[30]. Thecontinuouslydifferentiabilityconditiononthepair(C(θ),A(θ)) canberelaxedtoaLipschitzrequirement. Lemma 2. Consider the LTV system (2.4) with the initial conditionx(0) = x 0 . If there exists constants α 0 > 0andλ 0 > 0suchthatthestatestatetransitionmatrixΦ(t,τ)of(2.4)satisfies kΦ(t,τ)k≤α 0 e −λ0(t−τ) , ∀t≥τ ≥ 0 (2.15) andu∈L 2e ,thenwehave 14 1. foranyδ∈ [0,δ 1 )where0<δ 1 < 2α 0 isarbitrary,wehave kx t k 2δ ≤ cλ 0 p (δ 1 −δ)(2α 0 −δ) ku t k 2δ + t wherec = sup t kBkand t isanexponentiallydecayingtozerotermbecausex 0 6= 0. 2. u∈L 2 ⇒x∈L 2 ∩L ∞ , ˙ x∈L 2 ,andlim t→∞ |x(t)| = 0 3. u∈S(μ)⇒x∈S(μ)∩L ∞ Lemma 3. Consider the LTI system given by y = H(s)u where H(s) is a strictly proper rational function of s. If H(s) is analytic in Re[s] ≥ −δ/2 for some δ ≥ 0 and u ∈ L 2e then we have |y(t)|≤kH(s)k 2δ ku t k 2δ . ThefollowingBellman-Gronwall(B-G)lemmaisusefulforestablishingboundedness. Lemma4(B-GLemma) . Letc 1 ,c 2 bepositiveconstantsandg(t)beapiece-wisecontinuousfunctionof t. Ifforallt≥t 0 ≥ 0,thefunctiony(t)satisfiestheinequality y(t)≤c 1 +c 2 Z t t0 e −δ(t−τ) g 2 (τ)y(τ)dτ thenforallt≥t 0 ≥ 0 y(t)≤c 1 e −δ(t−t0) e c2 R t t 0 g 2 (τ)dτ +c 1 δ R t t0 e −δ(t−s) e c2 R t s g 2 (τ)dτ ds. 15 2.3 ASimpleExample Considertheuncertainplant y = 1 s−θ ∗ (1+Δ m (s))(u+d), θ ∗ ∈ Ω = [−2.5, 2.5] (2.16) whereθ ∗ is an unknown constant that belongs to the known interval Ω; d is a bounded disturbance, i.e., |d(t)| ≤ d 0 ∀t ≥ 0; and Δ m (s) is a multiplicative plant uncertainty. Δ m (s) is assumed to be a proper rationaltransferfunctionthatisanalyticinRe[s]≥−δ 0 /2forsomeknownδ 0 > 0. Wereferto G 0 (s) = 1 s−θ ∗ as the nominal model, and (2.16) as the overall plant. The control objective is to place the pole of the closed-loopnominalplantintheinterval [−5,−3];guaranteethaty anduarebounded;andwhend 0 = 0, y anduconvergetozeroast→∞. Weconsidercontrollawsoftheform u =−ky (2.17) wherek is to be chosen in a way that the control objective is met for anyθ ∗ ∈ [−2.5, 2.5]. The question is whether a single fixed value ofk will meet the control objective given thatθ ∗ is unknown except that it belongs to the interval [−2.5, 2.5]. If we apply the control (2.17) to the plant (2.16), we obtain the closed-loopsystem y = 1 s−θ ∗ +(1+Δ m (s))k (1+Δ m (s))d. (2.18) 16 WhenΔ m = 0,theclosed-looppoleof(2.18)is s =θ ∗ −k (2.19) and it follows from the bound−2.5 ≤ θ ∗ ≤ 2.5 that a single fixed value of k cannot meet the control objective. Therefore, the control design proceeds following an adaptive control approach. To develop the adaptive controller we first assume thatθ ∗ is known or measurable. Once a control law parameterized by θ ∗ isdeveloped,werelaxthisassumptionandreplaceθ ∗ inthecontrollawwithitsestimateθ. Let us assume that the large parametric uncertainty Ω = [−2.5, 2.5] is divided into three smaller subintervalscalledparametersubsetsanddefinedas Ω 1 = [0.5, 2.5], Ω 2 = [−1, 1], Ω 3 = [−2.5, −0.5]. (2.20) Let us also assume that the following three fixed controllers (candidate controllers) are used based on whichparametersubsetcontainsθ ∗ : u =−k 1 y, k 1 = 5.5, ifθ ∗ ∈ Ω 1 (2.21) u =−k 2 y, k 2 = 4, ifθ ∗ ∈ Ω 2 (2.22) u =−k 3 y, k 3 = 2.5, ifθ ∗ ∈ Ω 3 (2.23) Ifθ ∗ is known a priori then the control objective of placing the nominal closed-loop pole in the interval [−5, −3] is met by selecting the appropriate candidate control law from (2.21)-(2.23) based on which 17 parametersubsetcontainsθ ∗ . Ifθ ∗ belongstomorethanoneparametersubset,thenanyoftheappropriate candidatecontrollawsmaybechosen. Furthermore,overallclosed-loopstabilityisguaranteedif kΔ m k ∞ < 1 k i 1 s−θ ∗ +k i −1 ∞ , ∀θ ∗ ∈ Ω i , i = 1,2,3 (2.24) ≤ min 3 5.5 , 3 4 , 3 2.5 = 3 5.5 . (2.25) Ifθ ∗ is not known a priori and possibly time varying but is measured or estimated online, a strategy needstobedevelopedforconstructingacontrollawthatmeetsthecontrolobjectivebasedonthereal-time knowledgeofθ ∗ . Ourobjectiveistodesignacontrollaw u =−k(θ ∗ )y (2.26) from the candidate control laws (2.21)-(2.23) continuous in θ ∗ to avoid a discontinuous control signal as θ ∗ varies across parameter subsets. Discontinuities in the control signal is a practical concern because it mayleadtocontrolchatteringorpoortransientperformance. Becausethesimplecontrollaw u =−k(θ ∗ )y, k(θ ∗ ) =θ ∗ +3 (2.27) achievesthecontrolobjective,itmaynotbeobviouswhyweseektoconstructtheoverallcontrollerk(θ ∗ ) fromthecandidatecontrollersk 1 ,k 2 ,k 3 andtheircorrespondingparametersubsetsΩ 1 ,Ω 2 ,Ω 3 . Ingeneral, the unknown parameterθ ∗ may not enter into the control law gains in a tractable manner (e.g., as in the case ofH ∞ andμ-synthesis compensators,) making it difficult to parameterize the controller gains with respect to θ ∗ . However, given the candidate controllers and parameter subsets, it is straightforward to assign θ ∗ a stabilizing candidate controller by testing which parameter subsets contain θ ∗ . We follow 18 the latter principle to constructk(θ ∗ ) and develop an approach called control mixing where the candidate controllawsareweightedbasedonthereal-timeinformationon θ ∗ . Theparameteroverlaps Ω 1 ∩Ω 2 = [0.5, 1], Ω 2 ∩Ω 3 = [−1, −0.5]. (2.28) provide a domain in which the controller weights can be varied to continuously transition from one con- troller to another. The method for this simple example is described below. In Section 2.5, we extend the method to a general plant. Examples that demonstrate the advantage of the multiple model approach becauseθ ∗ entersintothecontrollergainsinanintractablemannerareconsideredinChapters3and4. Letusconsideracontrollawoftheform u =−k(θ ∗ )y, k(θ ∗ ) =β 1 (θ ∗ )k 1 +β 2 (θ ∗ )k 2 +β 3 (θ ∗ )k 3 (2.29) wherek 1 = 5.5,k 2 = 4,andk 3 = 2.5arestabilizinggainsifθ ∗ ∈ Ω 1 ,Ω 2 ,Ω 3 ,respectively,andβ 1 ,β 2 ,β 3 areweightstobechosensothatforanyθ ∗ ∈ Ω ifθ ∗ / ∈ Ω i thenβ i (θ ∗ ) = 0, i = 1,2,3 (2.30) β 1 (θ ∗ ),β 2 (θ ∗ ),β 3 (θ ∗ )≥ 0, β 1 (θ ∗ )+β 2 (θ ∗ )+β 3 (θ ∗ ) = 1. (2.31) We now verify that the control law (2.29) with (2.30)-(2.31) guarantees that the control objective is sat- isfied. Ifθ ∗ belongs to the interval (1, 2.5] ⊂ Ω 1 , then it follows from (2.30) thatβ 2 ,β 3 = 0 and from (2.31)thatβ 1 = 1,i.e., u =−k(θ ∗ )y, k(θ ∗ ) =k 1 (2.32) 19 whichhasbeenshowntomeetthecontrolobjective. Similarly,ifθ ∗ belongstotheinterval(0.5, 0.5)⊂ Ω 2 then u =−k(θ ∗ )y, k(θ ∗ ) =k 2 (2.33) meetsthecontrolobjective;ifθ ∗ belongstotheinterval[−2.5, −1)⊂ Ω 3 then u =−k(θ ∗ )y, k(θ ∗ ) =k 3 (2.34) meets the control objective. Now all that remains is to establish that the control objective is met on the model overlaps (2.28). Ifθ ∗ belongs to the model overlap Ω 1 ∩Ω 2 = [0.5, 1] then the control law is of theform u =−ky, k =β 1 k 1 +β 2 k 2 (2.35) wherek 1 = 5.5,k 2 = 4,andβ 1 ,β 2 ≥ 0aresomeconstantsthatsatisfyβ 1 +β 2 = 1. Observefromthese conditionsimposedonβ 1 ,β 2 thatk isboundedby 4≤k≤ 5.5. (2.36) Therefore,theclosed-looppole s =θ ∗ −k satisfies −5≤s≤−3 (2.37) 20 and the control objective is satisfied. Similarly, ifθ ∗ belongs to Ω 2 ∩Ω 3 = [−1, −0.5], then the control lawisoftheform u =−ky, k =β 2 k 2 +β 3 k 3 (2.38) wherek 2 = 4, k 3 = 2.5, andβ 2 ,β 3 ≥ 0 are some constants that satisfyβ 2 +β 3 = 1. Because these conditionsimposedonβ 2 ,β 3 implythat2.5≤k≤ 4,theclosed-looppole(2.19),therefore,satisfies −5≤s≤−3 (2.39) and the control objective is met. Therefore, if the conditions (2.30)-(2.31) on β are satisfied, the mixing controllaw(2.29)satisfiesthecontrolobjectiveforeveryθ ∗ belongingtoΩ = [−2.5, 2.5]. The next step is to define β i for which (2.30)-(2.31) are satisfied. Consider the weights β 1 ,β 2 ,β 3 showninFigure2.3andchosenas β i (θ ∗ ) = ˜ β i (θ ∗ ) ˜ β 1 (θ ∗ )+ ˜ β 2 (θ ∗ )+ ˜ β 3 (θ ∗ ) , i = 1,2,3 (2.40) −2 −1 0 1 2 0 0.2 0.4 0.6 0.8 1 θ * β i β 3 β 1 β 2 Figure2.3: Themixingsignalβ 21 where ˜ β 1 (θ ∗ ) =ψ θ ∗ −1.75 1.25 , ˜ β 2 (θ ∗ ) =ψ(θ ∗ ), ˜ β 3 (θ ∗ ) =ψ θ ∗ +1.75 1.25 (2.41) andψ isthesmoothbumpfunction ψ(x) =        e − 1 1−x 2 , |x|< 1; 0, otherwise. (2.42) Wedefinethemixingsignalasβ = [β 1 β 2 β 3 ] T . Now we establish whether or not the mixing signal (2.40) satisfies the requirements (2.30)-(2.31) for anyθ ∗ ∈ Ω. From(2.42)and(2.41)itfollowsthat ˜ β 1 (θ ∗ )iszeroifθ ∗ / ∈ (0.5, 2.5]. Inturn,from(2.40),it followsthatβ 1 iszeroifθ ∗ / ∈ Ω 1 = (0.5, 2.5]. Similarly,because ˜ β 2 (θ ∗ )iszerowhenθ ∗ doesnotbelong to the interval (−1, 1) and ˜ β 3 (θ ∗ ) is zero outside [−2,5, −0.5), we have thatβ 2 (θ ∗ ) is zero whenθ ∗ does not belong to the interval Ω 2 = (−1, 1) andβ 3 (θ ∗ ) is zero whenθ ∗ does not belong to the interval Ω 3 = [−2.5, −0.5). Therefore, condition (2.30) is satisfied. It follows fromψ(x)≥ 0 and (2.40),(2.41) thatβ 1 ,β 2 ,β 3 ≥ 0,whichsatisfiestheleftconditionof(2.31). From(2.42)and(2.41)itfollowsthat ˜ β 1 (θ ∗ )> 0, forallθ ∗ ∈ (0.5, 2.5]. (2.43) Similarly,wehavethat ˜ β 2 (θ ∗ )> 0, forallθ ∗ ∈ (−1, 1); ˜ β 3 (θ ∗ )> 0, forallθ ∗ ∈ [−2.5, −0.5). (2.44) Therefore,foranyθ ∗ belongingtotheinterval[−2.5, 2.5],wehavethat ˜ β 1 (θ ∗ )+ ˜ β 2 (θ ∗ )+ ˜ β 3 (θ ∗ )> 0 (2.45) 22 and,from(2.40), β 1 (θ ∗ )+β 2 (θ ∗ )+β 3 (θ ∗ ) = 1 (2.46) which satisfies the right condition of (2.31). The mixing signal (2.40) satisfies conditions (2.30)-(2.31) and,therefore,theparameterizedcontrollawgivenby(2.29),(2.40)meetsthecontrolobjective. Wenowanalyzetherobustnessofthemixingcontrolscheme(2.29),(2.40)appliedtothetrueplant G(s) = (1+Δ m (s))G 0 (s). (2.47) ItfollowsfromtheNyquiststabilitycriterionthatasufficientconditionforstabilityis kΔ m k ∞ < 1 k(θ ∗ ) 1 s−θ ∗ +k(θ ∗ ) −1 ∞ . (2.48) The right hand side of (2.48) takes on its minimum value over Ω when θ ∗ = 2.5, k(θ ∗ )| θ ∗ =2.5 = 5.5. Evaluating(2.48)atθ ∗ = 2.5yieldsthesufficientconditionfortheclosed-loopsystemtobestable kΔ m k ∞ < 3 5.5 . (2.49) We now relax the assumption thatθ ∗ is available for direct measurement. The implementation of the mixing control law (2.29) requires thatθ ∗ is known online. In application, this knowledge may come by monitoring certain auxiliary signals, or it may be based on the results of an online parameter estimator, whichistheapproachthatweuse. Becauseweareinterestedinanonlinemethodtogeneratetheestimate θ(t) ofθ ∗ , we seek to develop a dynamical system called the adaptive law whose inputs are measurable 23 signals and output isθ(t). The parameter estimator is designed by following the procedures of [30, 29]. Werewritethenominalplantmodel y = 1 s−θ ∗ u (2.50) as ˙ y−θ ∗ y =u (2.51) orequivalently ¯ z =θ ∗ ¯ φ (2.52) ¯ z = ˙ y−u, ¯ φ =y. (2.53) If ¯ z and ¯ φwereavailableformeasurement,wecouldgeneratetheestimationerror 1 = ¯ z−θ ¯ φ (2.54) given the estimateθ. Because ˙ y is not measurable, (2.54) is not implementable. Therefore, we filter both sidesof(2.52)bythestablefilter F(s) = λ s+λ F η (s), λ> 0 (2.55) 24 whereF η (s) is a stable minimum phase filter an analytic inRe[s] ≥ −δ 0 /2, to develop the linear para- metricmodel(LPM) z =θ ∗ φ (2.56) z =sF(s)y−F(s)u, φ =F(s)y. (2.57) Forthisexample,letuschooseF η (s) = 1. In(2.56),z andφaregeneratedbyfilteringy andu. Giventhe estimateθ,theestimationerrorisgeneratedby 1 =z−θφ (2.58) which will be used to drive the adaptive law, whose task is to make 1 “small.” This LPM can be used to generate a wide-class of adaptive laws for generating the estimate θ ofθ ∗ [30, 29], and we choose the adaptivelawasthegradientalgorithmwithprojectionmodification ˙ θ = Pr Ω (γφ) =        γφ, |θ|< 2.5orγφsgnθ≤ 0 0, otherwise = 1 m 2 = z−θφ m 2 , m 2 = 1+n d , ˙ n d =−δ 0 n d +u 2 +y 2 (2.59) whereγ > 0 is the adaptive gain and Pr Ω is the projection operator that restrictsθ(t) to Ω. We refer to 1 = z−θφ as the unnormalized estimation error to distinguish it from the normalized estimation error = 1 /m 2 .Theadaptivelaw(2.59)guaranteesthat θ∈ Ω (2.60) ,m, ˙ θ∈L 2 ∩L ∞ (2.61) 25 β u y d + + Mixer β 3 β 2 β 1 Ω 3 Ω 2 Ω 1 Multicontroller Adaptive Law Plant θ () 11 2 2 3 3 kk k ββ β −+ + () Pr θ γεφ Ω = S Σ () * 1 1 m s θ +Δ − Figure2.4: Blockdiagramofadaptivemixingcontrolscheme when applied to the nominal plant. In the presence of model uncertainty Δ m (s) and plant disturbanced we are no longer able to express the unknown parameterθ ∗ in the LPM form (2.56), where all signals are measurableandθ ∗ istheonlyunknown. Rather,theLPMoftheoverallplanttakestheform z =θ ∗ φ+η (2.62) η = Δ m F(s)u+(1+Δ m )F(s)d (2.63) where η is an unknown modeling error term, which negatively impacts estimation. When the adaptive law (2.59) is applied to the overall plant, property (2.61) no longer holds and is replaced with the weaker guaranteethat ,m, ˙ θ∈S(η 2 /m 2 )∩L ∞ . (2.64) TocompletetheAMCdesign,wecombinetheadaptivelaw(2.59)withk(θ ∗ )byreplacingθ ∗ withits estimateθ(t),andtheAMCschemeofFigure2.4mayberealizedbythefollowingequations: 26 Filters ˙ x E =A E x E +B E     u y     =     −λ 0 0 −λ     x E +     λ 0 0 λ         u y     (2.65) 1 =z−θφ = −1 −(λ+θ) x E +y (2.66) z = −1 −λ x E +y, φ = 0 1 x E (2.67) Adaptivelaw ˙ θ = Pr Ω (γφ) =        γφ, |θ|< 2.5orγφsgnθ≤ 0 0, otherwise = 1 m 2 = z−θφ m 2 , m 2 = 1+n d , ˙ n d =−δ 0 n d +u 2 +y 2 (2.68) Controllaw u =−(β 1 k 1 +β 2 k 2 +β 3 k 3 )y; β i (θ) = ˜ β i (θ) ˜ β 1 (θ)+ ˜ β 2 (θ)+ ˜ β 3 (θ) , i = 1,2,3 (2.69) where ˜ β 1 , ˜ β 2 , ˜ β 3 aregivenby(2.41). Inthesequel,werefertotheequationsof(2.69)asthemulticontroller andmixer,respectively. Wealsolet ˙ x P =A P x P +B P (u+d), y =C P x P (2.70) be the controllable and observable state-space realization of the plant (2.16), where x P (t)∈R np andA P , B P ,C P areofcompatibledimensions. The AMC’s stability and robustness properties will now be established. Let us define the dynamical system (2.65),(2.66) with input [u y] T and output 1 as the error modelE(θ). The parameterized system Σ(θ)istheclosed-loopsystemcomprisingtheplant,parameterizedcontroller k(θ),anderrormodelE(θ), 27 with the output defined as 1 . It follows from (2.70),(2.65),(2.66), andu =−k(θ)y that the dynamics of Σ(θ)aregivenby ˙ x =A(θ)x+Bd =         A P −B P k(θ)C P 0 0 −λk(θ)C P −λ 0 λC P 0 −λ             x P x E     +     B P 0     d (2.71) 1 =C(θ)x = C P −1 −(λ+θ) x (2.72) wherex = x T P x T E T andθ(t) is tuned by the adaptive law (2.68). The closed-loop system is in a form readily applicable to the tunability analysis approach of [44]. It has been establish in [56] that along the trajectories of closed-loop adaptive system (2.71)-(2.72),(2.68) there exists a unique global solution [x T (t) θ(t)] T ,∀t∈ [0,∞). ThestabilityanalysisofAMCiscarriedoutinfoursteps. InthefirstthreestepsweconsidertheAMC schemeappliedtothenominalplant(Δ m ,d 0 = 0)withtheobjectiveofestablishingthatx→ 0ast→∞, which implies thaty,u→ 0 as well. Then in the fourth step we consider the AMC scheme applied to the truesystem,andweanalyzeitsrobustnessproperties. Forallsteps,assumethatθ ∗ isanyconstantinΩ. Step1: Establishthatforallfixedθ∈ Ω,(C(θ),A(θ))isadetectablepair. SinceΔ m (s),d 0 = 0,theplantsystemmatricesaregivenby A P =θ ∗ , B P = 1, C P = 1 (2.73) andΣ(θ)is,therefore,givenby ˙ x =         θ ∗ −k(θ) 0 0 −λk(θ) −λ 0 λ 0 −λ         x, 1 = 1 −1 −(λ+θ) x (2.74) 28 wherex = y x T E T . To establish that the pair (C(θ),A(θ)) is detectable for all fixedθ ∈ Ω we let 1 ≡ 0 and establish thatx converges to zero ast→∞. Consider the initializationθ(0) =θ 0 , whereθ 0 is any constant in Ω. If we let 1 ≡ 0 then from (2.68) there is no adaptation, i.e.,θ ≡ θ 0 ; therefore, the closed-loopsystemisanLTIsystem. Since 1 ≡ 0,itfollowsfrom(2.58)thatz =θ 0 φand,togetherwith (2.57),wehavethaty andusatisfy sλ s+λ y− λ s+λ u =θ 0 λ s+λ y. (2.75) Sincethereisnoadaptation,theparameterizedcontrollerk(θ 0 )isconstantand,therefore,usatisfies u =−k(θ 0 )y. (2.76) Bysubstituting(2.76)into(2.75),wecanrewrite(2.75)as (s−θ 0 +k(θ 0 ))y = 0 (2.77) andsincek wasconstructedtoensurethe zero of(s−θ 0 +k(θ 0 ))liesintheinterval[−5, −3], we have thaty ∈ L ∞ andy → 0 ast → ∞. In turn, it follows from (2.76) thatu ∈ L ∞ andu → 0 ast → ∞. Because u,y are bounded and converge to zero, it follows from (2.65) that x E ∈ L ∞ and x E → 0 as t→∞. Since 1 ≡ 0impliesx→ 0ast→∞,theparameterizedclosed-loopsystem Σ(θ 0 )isdetectable onΩ. Step 2: Establish that along the solutions of the closed-loop adaptive system (2.74),(2.68) there exists a vector-valued function L : Ω → R 3×1 such thatA I (t) 4 = A(θ(t))−L(θ(t))C(θ(t)) is exponentially stable. Since ˜ β 1 , ˜ β 2 ,and ˜ β 3 aresmoothand,therefore,continuouslydifferentiable,itfollowsfrom(2.69)that k(θ) is continuously differentiable. Therefore, from (2.74) we have thatA(θ) andC(θ) are continuously 29 differentiableinθ. Sincetheadaptivelawguaranteesthatθ(t)∈ Ωandthepair(C(θ),A(θ))isdetectable for every fixedθ(t) ∈ Ω, it follows from result 1) of Theorem 1 that there exists an analytical function L : Ω→R 3×1 suchthatforeachfixedt≥ 0thatA I (θ(t)) 4 =A(θ(t))−L(θ(t))C(θ(t))ise.s. uniformly inθ(t), i.e., there exists a constantσ > 0 that∀t≥ 0 we have thatmax i Re{λ i [A I (θ(t))]}<−σ, where λ i (A) is thei th eigenvalue of the matrixA∈R 3×3 . Observe that sinceL is analytic on the compact set Ω,kLk∈L ∞ . Because the adaptive law additionally guarantees that ˙ θ∈L 2 , it follows from result 2) of Theorem1thatthetransitionmatrixΦ(t,τ)forthedifferentialequation ˙ z(t) =A I (θ(t))z(t) (2.78) satisfies kΦ(t,τ)k≤λ 0 e −α0(t−τ) (2.79) forsomepositiveconstantsλ 0 ,α 0 andt≥τ ≥ 0,i.e.,A I (θ(t))ise.s. Step3: Establishboundednessandconvergenceofx. Letδ∈ [0,δ 1 ),whereδ 1 < min{2α 0 ,δ 0 },andc> 0denotesanyfiniteconstant. Byapplyingoutputinjection,werewrite(2.74)as ˙ x =A I (θ(t))x+L(θ(t)) 1 =A I (θ(t))x+L(θ(t))m 2 (2.80) where in Step 2 we established e.s. of A I (θ(t)). The proof proceeds by establishing that m ∈ L ∞ , which together with kLk ∈ L ∞ and the properties (2.61) guaranteed by the adaptive law establishes Lm 2 ∈L 2 ∩L ∞ . Thisresulttogetherwith(2.80)willestablishboundednessandconvergenceofx. Byapplyingresult1)ofLemma2to(2.80),wehavethat kx t k 2δ ≤ck m 2 t k 2δ +c. (2.81) 30 Sincey isasubvectorofxwehavethat ky t k 2δ ≤kx t k 2δ ≤ck(m 2 ) t k 2δ +c (2.82) andthereforeitfollowsfromu =−k(θ(t))y andk∈L ∞ (from(2.69)and0≤β 1 ,β 2 ,β 3 ≤ 1)that ku t k 2δ ≤cky t k 2δ ≤ck(m 2 ) t k 2δ +c. (2.83) Considerthefictitiousnormalizationsignal m 2 f 4 = 1+ku t k 2 2δ +ky t k 2 2δ . (2.84) Note that becauseδ <δ 0 , it follows from the definitions ofm andm f thatm≤m f . Substituting (2.82), (2.83)into(2.84)yields m 2 f ≤ck(m 2 ) t k 2 2δ +c≤ck(mm f ) t k 2 2δ +c wherethesecondinequalityisobtainedbyusingm≤m f . Fromthedefinitionofk(·) t k 2δ itfollowsthat m 2 f ≤c Z t 0 e −δ(t−τ) ((τ)m(τ)) 2 m 2 f (τ)dτ +c. (2.85) ApplyingtheB-GLemmato(2.85)with g(τ) =(τ)m(τ)yields m 2 f ≤ce −δt e c R t 0 g 2 (τ)dτ +cδ Z t 0 e −δ(t−s) e c R t s g 2 (τ)dτ ds. (2.86) Boundednessofm f ,andinturnm,followsfromg =m∈L 2 ,whichisguaranteedby(2.61). We now turn our attention to the injected system (2.80). The termLm 2 can be viewed as the input ¯ u into the exponentially stable linear system ˙ x =A I (t)x+ ¯ u. BecausekLk,m∈L ∞ andm∈L 2 ∩L ∞ , 31 the input ¯ u = Lm 2 belongs toL 2 ∩L ∞ . Since the equilibrium solutionx e = 0 ofA I (t) is e.s. and Lm 2 ∈L 2 ∩L ∞ ,itfollowsfromresult2)ofLemma2and(2.80)thatx∈L 2 ∩L ∞ , ˙ x∈L 2 ∩L ∞ ,and x→ 0 ast→∞. From the convergence property ofx, and consequentlyL 1 , it follows from (2.80) that ˙ x→ 0ast→ 0. Step4: Establishrobustnessclaims Now suppose that Δ m ,d 0 6= 0. Consequently, the robust adaptive law no longer guarantees that ,m, ˙ θ∈L 2 ,butrather,itguaranteesthat ,m, ˙ θ∈S(η 2 /m 2 ). Nonetheless, the analysis approach for the nominal case can be applied to the robustness analysis with onlyminormodification. ItfollowsfromLemma3,d∈L ∞ ,and(2.55)wehave |η(t)|≤ Δ 1 ku t k 2δ0 +Δ 2 , Δ 1 4 =kΔ m Fk 2δ0 , Δ 2 4 =k(1+Δ m )Fk ∞−gn d 0 and becausem 2 = 1+ku t k 2 2δ0 +ky t k 2 2δ0 ≥ 1, it follows that|η(t)|/m ≤ Δ 1 +Δ 2 . Therefore,,m, ˙ θ∈S(μ 2 ),whereμ 2 4 =c(Δ 2 1 +Δ 2 2 ). Towards establishing detectability of (C(θ),A(θ)) for any constant θ ∈ Ω, y is shown to converge to zero just as in the nominal case. Therefore, the plant’s states converge to zero by the observability of (C P ,A P ). The remainder of showing that (C(θ),A(θ)) is detectable proceeds just as in the nominal case. Observethatdetectabilityof(C(θ),A(θ))isindependentoftheplant,aconsequenceofthecertainty equivalence stabilization theorem: subject to mild conditions, detectability is a property endowed by the factthattheparameterizedcontrollerk(θ)internallystabilizestheerrormodelE(θ)[45]. 32 Following Step 2, there exists an analytic functionL : Ω → R ¯ n×1 , where ¯ n is the dimension ofx, such that the system matrixA I (θ) = A(θ)−L(θ)C(θ) is e.s. for any constantθ ∈ Ω. We apply output injectiontotheparameterizedsystemtorewritethedynamicsofxas ˙ x =A I (θ(t))x+L(θ(t)) 1 +Bd. (2.87) In contrast to the ideal case where ˙ θ ∈ L 2 ∩L ∞ , here the robust adaptive law only guarantees that ˙ θ∈S(μ 2 )∩L ∞ . Itthenfollowsfromresult3)ofTheorem1thatA I (t)ise.s. providedthat c Δ 2 1 +Δ 2 2 <μ ∗ (2.88) forsomeμ ∗ . Condition(2.88)maynotbesatisfied,evenforsmallΔ 1 ,unlessd 0 issufficientlysmall. One waytodealwiththedisturbancetermistodesignthecomponentF η (s)ofF(s)suchthatΔ 2 issufficiently small, saycΔ 2 2 < μ ∗ /2 so that forcΔ 2 1 < μ ∗ /2, condition (2.88) is always satisfied. The constant Δ 2 can be made arbitrarily small by choosingF η (s) = c wherec > 0 is a small design constant. Because this would slow adaptation across all frequencies, a more practical approach would be to shapeF η such that|F η (jω)| is small in the frequency range of the disturbance. We continue with the assumption that condition (2.88) is satisfied. Therefore the homogeneous part of (2.87) is e.s., i.e., the transition matrix Φ(t,τ)ofA I (t)satisfieskΦ(t,τ)k≤λ 0 e −α0(t−τ) forsomepositiveconstantsλ 0 ,α 0 andt≥τ ≥ 0. BecauseA I (t) is e.s. and|d(t)|≤d 0 , it follows from (2.87) and result 1) of Lemma 2 that the bound (2.81)onkx t k 2δ stillholds,andthereforethebound(2.86)onm 2 f stillholds. Becausem∈S(μ 2 )∩L ∞ , wehavethatc R t s ((τ)m(τ)) 2 dτ ≤cμ 2 (t−s),whichwhensubstitutedinto(2.86)yields m 2 f ≤ce −δt e cμ 2 t +cδ Z t 0 e −δ(t−s) e cμ 2 (t−s) ds. (2.89) It follows that for cμ 2 ≤ δ, we have m f ∈ L ∞ . From m f ∈ L ∞ and m ≤ m f , we have that m is bounded. It follows fromm,kLk∈L ∞ andm∈S(μ 2 )∩L ∞ thatLm 2 ∈S(μ 2 )∩L ∞ . Therefore, it 33 follows from (2.87) and the e.s. ofA I (t) thatx∈L ∞ . Furthermore, if lim t→∞ d(t) = 0 then it follows from result 3) of Lemma 2 thatx∈S(μ 2 ), i.e., the mean value ofx is of the order of the modeling error characterizedbyμ 2 . Also,fromx,Lm 2 ∈S(μ 2 )∩L ∞ ,itfollowsfrom(2.87)that ˙ x∈L ∞ ∩S(μ 2 ). Theconditionforstabilityis,therefore, μ 2 <δ ∗ 4 = min{cμ ∗ , √ δ}, 0<δ< min{δ 0 ,2α 0 } (2.90) forsomeconstantc> 0andμ ∗ > 0istheboundforμ 2 forA I (t)tobee.s. Insummary,wehaveshownthatintheabsenceofmultiplicativeuncertaintyΔ m andexogenousinput d that the statesx P ,x E ,θ remain bounded andx P ,x E → 0 ast→∞. When Δ m andd satisfy condition cμ 2 < δ ∗ , then all closed-loop states are bounded and, if d → 0 ast → ∞,x and ˙ x areμ 2 -small in the m.s.s. 2.3.1 Simulation Let us now simulate the AMC scheme (2.65)-(2.69) applied to the plant given by (2.16). For simulation purposes, we use the plant parametersθ ∗ = 2.5 ∈ Ω 1 , Δ m (s) = −2μs 1+μs ,μ = 0.1, andd 0 = 0. We use the control parameters γ = 100, λ = 5, δ 0 = 4, and θ(0) = 0. For comparison, we also simulate an adaptivepoleplacementcontrol(APPC)schemewiththecontrolu =−(θ(t)+3)y andthesameadaptive law and initialization as the AMC scheme. The plant output is shown in Figure 2.5(a). In this example, both the AMC and APPC schemes regulate the plant output to zero, where the AMC scheme has slightly fasterconvergence. SimulationsshowthattheAPPCschemeremainsstableforμ≤ 0.113;AMCremains stable for μ ≤ 0.12; and perfect identification θ(t) ≡ θ ∗ remains stable for μ < 0.125. AMC’s mild improvementinperformanceandrobustnessisfromthefactthatoscillationsinθ,showninFigure2.5(b), caused by exciting Δ m , are not seen in the control whenθ(t) ≥ 0.5. Said differently, outside the model overlaps, k(θ) is constant and therefore the controller/supervisor loop is “open” in the sense that small 34 0 2 4 6 8 10 −8 −6 −4 −2 0 2 4 6 8 10 12 Time, sec y(t) (a) Plant output: Adaptive mixing control (solid); Adap- tivepoleplacementcontrol(dashed) 0 2 4 6 8 10 −1 −0.5 0 0.5 1 1.5 2 2.5 Time, sec θ(t) (b) Estimateθ(t) ofθ ∗ = 2.5: Adaptive mixing control (solid);Adaptivepoleplacementcontrol(dashed) Figure2.5: Simulationresults deviationsintheparameterestimatedonotaffectthecontrol. ThisisnotthecaseinAPPC,andtherefore, theoscillationsinucausedbyθ furtherexcitesΔ m . 2.4 GeneralProblemFormulation ConsidertheSISOLTIplant y =G(s;θ ∗ )u+d, G(s;θ ∗ ) =G 0 (s;θ ∗ )(1+Δ m (s)) (2.91) G 0 (s) = N 0 (s) D 0 (s) = θ ∗T b α n−1 (s) s n +θ ∗T a α n−1 (s) (2.92) y m =y+ν (2.93) whereG 0 (s)representsthenominalplant;thevectorθ ∗ 4 = [θ ∗T b θ ∗T a ] T ∈ Ω⊂R 2n containstheunknown parametersofG 0 (s;θ ∗ );α n−1 (s) 4 = [s n−1 s n−2 ... s 1] T ;y m isthemeasuredvalueofy corruptedby the bounded sensor noiseν, i.e.,|ν(t)| ≤ ν 0 ,∀t ≥ 0; Δ m (s) is an unknown multiplicative perturbation; anddisaboundeddisturbance,i.e.,|d(t)|≤d 0 ,∀t≥ 0;Thecontrolobjectiveistochoosetheplantinput u so that the plant outputy is regulated close to zero. We make the following assumptions on the plant to meetthecontrolobjective: 35 P1. D 0 (s)isamonicpolynomialwhosedegreenisknown. P2. Degree(N 0 )<n. P3. Δ m (s)isproper,rational,andanalyticinRe[s]≥−δ 0 /2forsomeknownδ 0 > 0. P4. θ ∗ ∈ ΩforsomeknowncompactconvexsetΩ⊂R 2n . Considerthestate-spacerealizationof(2.91) ˙ x P =A P x P +B P u, y m =C P x P +d+ν. (2.94) Wemaketheadditionalassumptiontomakecontrolmeaningful: P5. Thepairs(C P ,A P )and(A P ,B P )aredetectableandstabilizable,respectively,onΩ. It should be emphasized that both unstable and nonminimum phase plants are admissible despite require- mentsP1-P5. Given is a family of p candidate controllersC 4 = {K i (s)} i∈I , where K 1 (s),...,K p (s) are rational transferfunctionsandI denotestheindexset{1,...,p},andtheparameterpartitionP 4 = Ω i ⊂R 2n i∈I , where each parameter subset Ω i is compact andP covers Ω, i.e., Ω⊂∪ i∈I Ω i . The candidate controller setC andparameterpartitionP hasbeendevelopedsuchthatforeveryi∈I andeachθ ∗ ∈ Ω i ,thecontrol lawu =−K i (s)y m yields a stable closed-loop system that meets some performance requirements. If Ω i containsθ ∗ andthecontrolischosenasu =−K i (s)y m ,thenasufficientconditionforstabilityoverΩis kΔ m k ∞ ≤ min θ ∗ ∈Ωi K i (s)G(s;θ ∗ ) 1+K i (s)G(s;θ ∗ ) −1 ∞ . (2.95) Let δΩ i and δΩ are the boundaries of the sets Ω i and Ω, respectively. Facilitating control mixing, the partition has an overlapping property: for alli∈I and anyθ ∗ ∈ (δΩ i −δΩ) there exist constantsr > 0 andj 6= i such thatB r (θ ∗ ) ⊂ Ω j . The parameter overlapsO is the set of all pointsθ ∗ ∈ Ω that belong 36 to more than one parameter subset. The candidate controllers and parameter subsets can, for example, be generatedbythemethodof[16],modifiedtogenerateoverlappingparametersubsets. Remark 1. A conventional adaptive controller could be included in the candidate controller set C to accountforthecaseθ ∗ / ∈ Ω. Thisisthetopicoffuturework. Remark 2. As in many MMAC approaches, the complexity of the controller design increases with the number of candidate controllers, which may result in an impractical design. A priori knowledge can be used to decrease the number of unknown parameters and size of Ω, and, in turn, decrease the number of candidatecontrollers. The Problem: The objective of this paper is to propose a provably correct MMAC scheme which is capable of achieving 1) global boundedness of all system signals and 2) regulation of all plant signals in theabsenceofunmodeleddynamics,disturbances,andsensornoise. In order to avoid poor performance during identification, we prohibit prerouted search algorithm that step through candidate controllers in a predetermined order and probing signals that excite the plant for identification purposes. A deterministic approach is pursued because the disturbance is only known to be bounded. The unknown parameter vectorsθ ∗ a ,θ ∗ b enter the plant model linearly and the plant remains detectable and stabilizable on Ω. Therefore, in order to side-step issues associated with discontinuous switching among candidate controllers, we present a deterministic MMAC approach that tunes the multi- controllerinacontinuousmannerbasedontheestimateθ(t)ofθ ∗ . 2.5 ConceptualFramework Theadaptivemixingcontrolarchitecturecomprisestwosystems: themulticontrollerC(β,θ)andtherobust adaptivesupervisorΣ S ,whichinturnconsistsofarobustparameterestimatorandmixerM. Fornotational simplicity, the following definitions are made. If θ ∈ Ω i (for some i ∈ I), then Ω i is said to be an active parameter subset. I θ denotes the index set of all active parameter subsets at θ ∈ Ω, i.e., I θ = 37 {i∈I :θ∈ Ω i }. Wedefinethesetofalladmissiblemixingvaluesatθ∈ ΩasB θ 4 ={β = [β 1 ...β p ] T ∈ [0,1] p : P i∈I β i = 1; β i = 0, i / ∈I θ }. Thesetofalladmissiblemixingvalues∪ θ∈Ω B θ isdenotedbyB. 2.5.1 Multicontroller The multicontrollerC(β,θ), constructed fromC, is a dynamical system capable of generating a mix of candidate control laws. The multicontrollerC(β,θ) is given by the stabilizable and detectable state-space realization ˙ x C =A C (β,θ)x C +B C (β,θ)y m , u =−C C (β,θ)x C (2.96) wherex C (t)∈R nc is the multicontroller state vector; the system matricesA C ,B C ,C C are of compatible dimensions;andthemixingsignalβ(t) 4 = [β 1 (t)...β p ] T andparameterestimateθ(t)aregeneratedbythe supervisor Σ S and tuneC. For fixed values ofβ ∈B andθ∈ Ω, the multicontrolleru =−K(s;β,θ)y m hasthetransferfunction K(s;β,θ) =C C (β,θ)(sI−A C (β,θ)) −1 B C (β,θ) (2.97) = N K (s;β,θ) D K (s;β,θ) . (2.98) ThemulticontrollerC(β,θ)satisfiesthreeproperties: C1. TheelementsofA C (β,θ),B C (β,θ),andC C (β,θ)arecontinuouslydifferentiableinβ andθ. C2. K(s;e i ,θ) =K i (s),whereθ∈ Ω i ande i ∈R p isi th standardbasisvector. C3. Forallθ ∗ ∈ Ωandanyβ ∗ ∈B θ ∗,K(s;β ∗ ,θ ∗ )internallystabilizestheplant. 38 Property C1 ensures that the closed-loop system varies slowly if β andθ are tuned slowly. Property C2 allowsforeachcandidatecontrollertoberecovered. PropertyC3ensuresthatC(β,θ)isastabilizingcer- taintyequivalencecontrollawforanyadmissiblemixingsignal,independentofthemixerimplementation. Themulticontrollercanviewasageneralizationofthemulticontrollerusedinsupervisorycontrol[46]. Construction of the multicontroller involves interpolating the candidate controllers over the parameter overlaps O. Numerous controller interpolation approaches have been proposed in the context of gain scheduling. These methods interpolate controller poles, zeros, and gains [51]; solutions of the Riccati equationsforanH ∞ design[57];state-spacecoefficientmatricesofbalancedcontrollerrealizations[33]; state and observer gains [28]; controller output [7, 34, 52], i.e.,u = P i∈I β i u i , whereu i =−K i (s)y m . Asingainscheduling,theseinterpolationmethodsmaynotsatisfythepoint-wisestabilityrequirementC3 (cf. the counter examples of [67, 53]). Thus, if one of these interpolation methods is used then property C3shouldbeverified. Fortunately,therealsoexisttheoreticallyjustifiedmethods[62,67,53],whichcanbeusedtoconstruct themulticontroller. Thefollowingresultisadaptedfrom[67]: Theorem 5. Consider the nominal plantG 0 (s;θ ∗ ) given by (2.92) and the stable coprime factorization G 0 (s;θ ∗ ) = N(s;θ ∗ )M −1 (s;θ ∗ ) = ˜ M −1 (s;θ ∗ ) ˜ N(s;θ ∗ ) that depends on θ ∗ smoothly. Let the stable transferfunctionsX(s;θ ∗ ),Y(s;θ ∗ ), ˜ X(s;θ ∗ ),and ˜ Y(s;θ ∗ )dependonθ ∗ smoothlyandsatisfythedouble Bezoutidentity(omittingdependenciesonsandθ ∗ )     I 0 0 I     =     ˜ X ˜ Y − ˜ N ˜ M         M −Y N X     =     M −Y N X         ˜ X ˜ Y − ˜ N ˜ M     , ∀θ ∗ ∈ Ω. 39 Suppose θ ∗ ∈ ¯ Ω, where ¯ Ω ⊂ Ω is some compact set, and K 1 (s),...,K ¯ p (s) are ¯ p stabilizing negative feedback controllers for anyθ ∗ ∈ ¯ Ω. Let each controllerK i (s) be given a stable coprime factorization K i (s) = ¯ U i (s) ¯ V −1 i (s),andwedefine Q i (s;θ ∗ ) = ˜ Y(s;θ ∗ ) ¯ V i (s)− ˜ X(s;θ ∗ ) ¯ U i (s) · ˜ N(s;θ ∗ ) ¯ U i (s)+ ˜ M(s;θ ∗ ) ¯ V i (s) −1 (2.99) U i (s;θ ∗ ) =M(s;θ ∗ )Q i (s;θ ∗ )−Y(s;θ ∗ ) (2.100) V i (s;θ ∗ ) =N(s;θ ∗ )Q i (s;θ ∗ )−X(s;θ ∗ ) (2.101) fori = 1,...,¯ p. Ifthecontrollerisgivenby K(s; ¯ β,θ ∗ ) =U(s; ¯ β,θ ∗ )V −1 (s; ¯ β,θ ∗ ), ¯ β = ¯ β 1 ... ¯ β ¯ p T (2.102) whereU(s) = P ¯ p i=1 ¯ β i U i (s;θ ∗ ),V(s) = P ¯ p i=1 ¯ β i V i (s;θ ∗ ), and ¯ β 1 ,..., ¯ β ¯ p are some positive constants that satisfy P ¯ p i=1 ¯ β i = 1, thenK(s) is stabilizing andK(s) =K i (s) if ¯ β j = 1 forj =i and ¯ β j = 0 for j6=i. MotivatedbyTheorem5,considerthemulticontroller K(s;β,θ) =U(s;β,θ)V −1 (s;β,θ) (2.103) U(s;β,θ) = p X i=1 β i U i (s;ϕ Ωi (θ)) (2.104) V(s;β,θ) = p X i=1 β i V i (s;ϕ Ωi (θ)) (2.105) whereQ i ,U i ,andV i aregivenby(2.99),(2.100),and(2.101),respectively,andthefunctionϕ Ωi : Ω→ Ω i denotes a smooth projection function 1 : ϕ Ωi is continuously differentiable andϕ Ωi (θ) =θ ifθ∈ Ω i . The 1 ϕ Ω i is not the same as the projection operatorPr Ω i {·} used in the adaptive law. ϕ Ω i can be constructed, for example, from smoothbumpfunctionsψ. 40 -y m u - + + 1 p ii i QQ β = = ∑ Y − N 1 X − M (a) Multicontrollerstructure. P Δ w e K M(P,K) (b) Generalcontrolconfiguration. Figure2.6: Multicontrollerimplementationandrobustperformanceformulation. functionϕ Ωi is used to ensure that eachQ i is only evaluated on Ω i and, therefore, stable; otherwise,Q i maygenerateanunboundedout-of-the-loopsignalif β i = 0andθ / ∈ Ω i . We now examine if this multicontroller satisfies properties C1-C3. Property C1 follows immediately by inspection of the filters of Theorem 5 that define the multicontroller K(s;β,θ ∗ ) and because ϕ Ωi is smooth. Let us now consider property C3. For fixed constantsθ ∗ ∈ Ω andβ ∗ ∈B θ ∗, we haveβ ∗ i = 0 for i / ∈I θ ∗,andϕ Ωi (θ ∗ ) =θ ∗ fori∈I θ ∗. Thus, K(s;β ∗ ,θ ∗ ) =   X i∈I θ ∗ β ∗ i U i (s;θ ∗ )     X i∈I θ ∗ β ∗ i V i (s,θ ∗ )   −1 . (2.106) BecauseI θ ∗ indexes only stabilizing controllers, it follows from Theorem 5 thatK(s;β ∗ ,θ ∗ ) internally stabilizes the plant, satisfying C3. Moreover, ifβ ∗ =e i for somei∈I θ ∗,K(s,β ∗ ,θ ∗ ) =K i (s), and C2 is satisfied. Fig. 2.6(a) shows an implementation of the multicontroller that avoids unbounded out-of-the- loopsignals[68]. ByQ-blending wemeanthemulticontrollerscheme(2.103),whichisgiveninamultipleinput-multiple output (MIMO) format to emphasize that this approach is suitable for extensions of AMC to the MIMO case. We now consider robust performance, formulated in a standard μ-analysis framework as shown in Fig. 2.6(b), as the control objective. Motivated to work with a normalized uncertainty, we model the multiplicative uncertainty as Δ m (s) = w m (s)Δ(s), where the known stable transfer function w m (s) 41 serves as a frequency dependent weight and Δ(s) is any stable transfer function satisfyingkΔk ∞ ≤ 1. The transfer matrix P(s;θ ∗ ) is assumed to contain the plant G 0 (s;θ ∗ ) and its interconnection with the performance and uncertainty weights that describe the control objective; w(t) ∈ R nw is the exogenous input (for example, say w = [d ν] T ), and e(t) ∈ R ne is the exogenous output. The matrix transfer functionM(P(θ ∗ ),K)isformedbyabsorbingthecontrollerK intoP asshowninFig.2.6(b). Wesaythat a controllerK yields robust performance ifM(P(θ ∗ ),K) is stable and satisfieskM(P(θ ∗ ),K)k ˆ Δ < 1, where ˆ Δ 4 = diag(Δ,Δ P )satisfiesk ˆ Δk ∞ ≤ 1;Δ P isastablen w ×n e transfermatrix;andkMk ˆ Δ denotes sup ω∈R μ ˆ Δ (M(jω)). LetusassumethateachcandidateK i (s)yieldsrobustperformanceforallθ ∗ ∈ Ω i . Asdemonstratedbythefollowingresult,aQ-blendingmulticontrollerpreservesrobustperformance. Lemma 6. If the multicontroller K(β,θ ∗ ) is given by (2.103) and, for some generalized plant P and i = 1,...,p, the candidate controllerK i yields robust performance for allθ ∗ ∈ Ω i , then for allθ ∗ ∈ Ω andβ ∗ ∈B θ ∗ themulticontrollerK(β ∗ ,θ ∗ )yieldsrobustperformancewithrespecttoP. Proof. The transfer matrix M has the form M = M 1 + M 2 QM 3 (cf. [65, pp. 150]), where Q = P p i=1 β ∗ i Q i . For anyθ ∗ ∈ Ω andβ ∗ ∈ B θ ∗, it follows that P i∈I θ ∗ β ∗ i = 1, β ∗ i = 0 fori / ∈ I θ ∗, and kM(P,K i )k ˆ Δ < 1fori∈I θ ∗. Thus, M(P,K(β ∗ ,θ ∗ )) =M 1 +M 2 X i∈I θ ∗ β ∗ i Q i M 3 = X i∈I θ ∗ β ∗ i (M 1 +M 2 Q i M 3 ) = X i∈I θ ∗ β ∗ i M(P,K i ) and,consequently,M(P,K(β ∗ ,θ ∗ ))isstableandkM(P,K(β ∗ ,θ ∗ ))k ˆ Δ < 1. Remark3. The choice of controller interpolation scheme affects the complexity multicontroller. Say, for example, that the candidate controller orders aren 1 ,...,n p . An interpolation approach that schedules 42 the controller gains with respect toθ(t) will result in a multicontroller order of max i {n i }. The output- blendingschemeu = P p i=1 β i u i resultsinamulticontrollerorderof P p i=1 n i . ToanalyzetheQ-blending scheme shown in Fig. 2.6(a), let us assume that the orders of ˜ Y and ˜ X −1 aren 0 . Then the order of the Q-blending multicontroller is 2(n+n 0 )+nn 0 P p i=1 n i . Thus, the a priori theoretical properties of the Q-blendingapproachshouldbecarefullyweighedagainstitscomplexity. 2.5.2 RobustAdaptiveSupervisor The robust adaptive supervisor, shown in Fig. 2.2, is a dynamical system that takes as input the measured plantsignalsu(t),y m (t)andoutputsthemixingsignalsβ(t)that“configures”themulticontrollerC(β,θ). Note that, because we are implementing the supervisor, the multicontroller may access the states of the supervisor,includingtheparameterestimateθ(t). ThemixerM implementsthemappingβ : Ω→ [0,1] p . ThefollowingpropertiesofM areassumed M1. β(θ)iscontinuouslydifferentiable. M2. β(θ)∈B θ , ∀θ∈ Ω. Property M1, together with C1 ensures that if θ is tuned slowly then the closed-loop system will vary slowly. M2, together with C3, ensures that C(β(θ),θ) is a certainty equivalence stabilizing controller, i.e., for any θ ∗ ∈ Ω the controller C(β(θ ∗ ),θ ∗ ) meets the control objective. Thus, if the control law u =−K(s;θ ∗ )y m is applied to the nominal plant (2.92), the closed-loop system is internally stable, and theclosed-loopcharacteristicpolynomial N K (s;θ ∗ )N 0 (s;θ ∗ b )+D K (s;θ ∗ )D 0 (s;θ ∗ a )isHurwitz. Further- more,whenthiscontrollawisappliedtotheoverallplant(2.91),theclosed-loopsystemisstableif kΔ m k ∞ ≤ min θ ∗ ∈Ω K(s;θ ∗ )G(s;θ ∗ ) 1+K(s;θ ∗ )G(s;θ ∗ ) −1 ∞ . (2.107) If C(β,θ) satisfies C3, the design of the mixer M is independent of C(β,θ); otherwise, one must ensure, for all θ ∗ ∈ Ω, that C(β(θ ∗ ),θ ∗ ) meets the control objectives. Assuming requirement C3 is 43 satisfied, the designer has considerable freedom in constructingM, and one such approach, as in Section 2.3andinthesequal,istodefineβ basedonthesmoothbumpfunctionψ giveninSection2.2. Because θ ∗ is unknown, C(β(θ ∗ ),θ ∗ ) cannot be calculated and, therefore, cannot be implemented. Thus, the AMC approach replacesθ ∗ with its estimateθ. The well known counter example of Rohrs et al., [58] demonstrates that for even small modeling errors that an adaptive system may become unstable. Thus,becauseofthepresenceofmultiplicativeuncertaintyΔ M ,disturbanced,andsensornoiseν,weuse arobustonlineparameterestimatortoregainasmuchoftherobustnessoftheknowncaseaspossible. The robust parameter estimator comprises an error model E(θ) and a robust adaptive law Σ T , also referred to as the tuner. The error modelE(θ) is constructed by selecting an appropriate parameterization of the plant model. We proceed with the design of the error model by constructing a LPM using the same technique as in Section 2.3. The interested reader is referred to [30, Sec. 2.4.1] for a detailed description. TheLPMofthenominalsystem(2.92)isgivenbyz =θ ∗T φ,where z =s n F(s)y m (2.108) φ = α T n−1 (s)F(s)u −α T n−1 (s)F(s)y m T (2.109) F = λ n (s+λ) n F η (s), F η (s) = N F (s) D F (s) (2.110) whereλ> 0 is a design constant andF η (s) is a proper stable minimum-phase filter. The estimation error 1 isgeneratedbyregardingθ(t)asthetrueparameterθ ∗ ,i.e., 1 =z−ˆ z =z−θ T φ. Wedefinetheerror modelE(θ) as the dynamical system whose inputs are the observed datay m andu, and its output is 1 . TheerrormodelE(θ)isrealizedfrom(2.108)-(2.110)andhastheform ˙ x E =A E x E +B E u+G E y m (2.111) z =C z x E +D z y m , ˆ z =θ T C E x E , φ =C E x E (2.112) 1 = (C z −θ T C E )x E +D z y m (2.113) 44 whereA E isHurwitzandθ :R + →R 2n istunedbyΣ T . NotethatE(θ)isaffineinθ. When the error modelE(θ) is connected to the true plantG(s) in the presence of the multiplicative perturbationΔ m (s)andboundeddisturbancedandnoiseν,z isgivenbyz =θ ∗T φ+η,where η =N 0 (s)Δ m (s)F(s)u+D 0 (s)F(s)(d+ν) (2.114) isthemodelingerrortermandactsasanestimationdisturbance. ThefilterF(s)canbechosentomitigate thedeleteriouseffectsofη onestimation. Forthispurpose,F(s)willtypicallyhaveabandpassfrequency response to filter out signal bias and disturbances at low frequencies and sensor noise and unmodeled dynamicsathighfrequencies. Forsimplicity,wechooseF(s)tobeanalyticinRe[s]≥−δ 0 /2. TherobustadaptivelawΣ T canbeimplementedbyawide-classofalgorithms[30,Chapter8.5]whose dynamicstakethefairlygeneralform ˙ θ =f θ (θ,x a ,n d , 1 ,x E ) (2.115) ˙ x a =f a (x a ,θ,n d ,x E ) (2.116) ˙ n d =−δ 0 n d +|u| 2 +|y m | 2 (2.117) where x T = [θ T x T a n d ] T is the adaptive law’s state; x a (t) ∈ R na is an auxiliary state required by the tuning algorithm; and n d is the dynamic component of the normalization signal m 2 = 1 +n d that guarantees thatφ/m,η/m∈L ∞ . We choosef θ andf a to be implemented with projection modification [30, 29] in order to constrainθ(t) to Ω. The adaptive law Σ T is implemented by any of the algorithms foundin[30,29]withprojectionmodificationthatguarantees E1. ,m, ˙ θ∈S(η 2 /m 2 )∩L ∞ ,ifΔ m 6= 0,d 0 6= 0, orν 0 6= 0 E2. ,m, ˙ θ∈L 2 ∩L ∞ ,ifΔ m ,d 0 ,ν 0 = 0 E3. θ∈ Ω, x a ∈L ∞ 45 where 4 = 1 /m 2 isthenormalized estimationerror. Additionally,weassumethattherobustadaptivelaw satisfies E4. f θ (·,·,·,0,·) = 0 i.e., adaptation ceases when 1 = 0. This assumption is satisfied by all adaptive laws of [30, 29] except thosewithσ-modification. Remark4. Iftheplantparametersenterthemodelinanonlinearfashion,onecanoftenoverparameterize the plant model. Overparameterization increases the class of admissible plants; thus, there may be some loss of performance, and, as in the example of [46, Sec. X], plant models that are not stabilizable may be introduced. Because the controller gains are computed offline, AMC avoids the computational and existence issues that arise in conventional adaptive control when stabilizability is lost. Furthermore, the adaptive law can be modified to ensure thatθ(t) does not remain in a specified neighborhood about the pointsthatleadtoalossofstabilizability(cf. Sec. 7.6of[30]). Remark 5. We believe that there will be no significant difficulties in extending AMC to the MIMO case, whichiscurrentlyunderinvestigation. 2.5.3 StabilityandRobustnessResults Wenowsummarizethemainresults. Theorem7. Let the unknown plant be given by (2.91) and satisfying the plant assumptions P1-P5. Con- sidertheAMCschemewiththemulticontrollerC(β,θ)givenby(2.96)andsatisfyingassumptionsC1-C3; error model given by (2.111)-(2.113); robust adaptive law Σ T given by (2.115)-(2.117) and satisfying assumptionsE1-E4;andmixer M satisfyingM1-M2. 1. If Δ m ,d,ν = 0 then x, ˙ x → 0 as t → ∞, where x 4 = [x T P x T C x T E ] T . Furthermore, let the multicontroller be given by the Q-blending scheme (2.103), and for some generalized plant P 46 and i = 1,...,p let the candidate controller K i yield robust performance for all θ ∗ ∈ Ω i . If lim t→∞ θ(t) =θ ∗ ,thenx PC 4 = x T P x T C T →x ∗ PC ast→∞,wherex ∗ PC (t)isthesolutionto ˙ x ∗ PC =     A P −B P C ∗ C B ∗ C C P A ∗ C     x ∗ PC =A ∗ PC x ∗ PC (2.118) and the triplet (A ∗ C ,B ∗ C ,C ∗ C ) is the state-space realization of some controller that yields robust performancewithrespecttothegeneralizedplantP. 2. There exists δ ∗ > 0 such that if cΔ 2 1 < δ ∗ , where Δ 1 4 = kN 0 Δ m Fk 2δ0 and c > 0 is a finite constant, then the AMC scheme guarantees that x, ˙ x, ˙ θ,θ,x T ∈ L ∞ . Furthermore, there exists constantsc 0 ,c 1 > 0suchthat Z t 0 |y(τ)| 2 dτ ≤c 0 μ 2 t+c 1 (2.119) whereμ 2 =c(Δ 2 1 +d 2 0 +ν 2 0 ). The proof is given in the Appendix. In result 1), exponential convergence of estimate θ to the true valueθ ∗ can be guaranteed providedφ is persistently exciting (PE). The control input can be augmented with an auxiliary signalu a (t) that forcesu to be sufficiently rich of order 2n, guaranteeingφ is PE if the plant is controllable and observable. For output tracking of controllable, observable plants, the reference signalr may naturally ensure thatφ is PE; otherwise, guaranteeing thatφ is PE can be accomplished by augmentingr orutobesufficientlyrichoforder2n. TheinterestedreaderisreferredtoChapters3and4 of[30]. 47 2.6 Conclusions WehavepresentedtheMMACapproachAMC.Thegeneralframeworkwaspresentedinawaythathigh- lighted the modular aspect of the architecture, where one is concerned that the components satisfy certain properties. A description about how one would go about constructing the components was also provided. TheuniquebehaviorofAMCwhencomparedtootherdeterministicMMACapproachesisthatthecontrol law is continuously tuned based on the real-time data, avoiding practical issues of chattering and possi- bly poor transients due to controller switching. Continuous tuning of the control law is accomplished by estimating the unknown parameter online and, based on which candidate controllers provide the desired closed-loop behavior, adjusting the weights of the multicontroller in a continuous manner to meet control objective. AkeycontributionisthestabilityandanalysisofAMC.Wefoundthatthetunabilityapproachof[44] for analyzing AMC approach useful because the approach is fairly independent of implementation details of the individual components. For the nominal case, it was shown that the AMC scheme drives the plant states to zero. In the presence of unmodeled dynamics and bounded disturbances that satisfy specified bounds, it was shown that the closed-loop state remains bounded and the regulation error is of the order of the modeling error. A simple example was presented, whose role is to introduce the AMC approach. It alsodemonstratedmildperformanceandrobustnessimprovementoverananalogousAPPCscheme. Appendix Proof of Theorem 7: Let us define the parameterized controller Σ C (θ) 4 = C(β(θ),θ) as the mixer- multicontroller interconnection. For any constant θ ∈ Ω, Σ C (θ) has the transfer function (2.98). We 48 also define the parameterized systemΣ(θ) as the plant-parameterized controller-error model interconnec- tion, with its output chosen as 1 . From (2.94),(2.96),(2.111)-(2.113), the parameterized system Σ(θ) can bewrittencompactlyas ˙ x =A(θ)x+B(d+ν), 1 =C(θ)x (2.120) wherex 4 = [x T P x T C x T E ] T andthetriplet(C(θ),A(θ),B)isdefinedintheobviousmanner. Theclosed-loop adaptivesystem(Σ(θ(t)),Σ T )isformedbyreplacingtheparameterθ oftheparameterizedsystemΣwith the tuned estimatesθ(t) generated by the robust adaptive law Σ T . The closed-loop system (2.120) is in a form suitable for analysis using the tunability approach of [44]. Also, it has been establish in [56] that alongthetrajectoriesof(Σ(θ(t)),Σ T )thereexistsauniqueglobalsolution[x T (t) x T T (t)] T ,∀t∈ [0,∞). Step1: Establishthatforallfixedθ∈ Ω,{C(θ),A(θ)}isadetectablepair. Consider the adaptive law initializationθ(0) = θ 0 ∈ Ω, whereθ 0 = [θ T b0 θ T a0 ] T is any fixed constant in Ω. The vectors θ T b0 ,θ T a0 represent the initial estimates of the coefficients of the N 0 (s) and D 0 (s), respectively. Letd ≡ 0,ν ≡ 0, and 1 ≡ 0. Thus, it follows thaty m ≡ y and, because 1 ≡ 0, there is no adaptation, i.e.,θ ≡ θ 0 . Therefore the closed-loop system is an LTI system. Since 1 ≡ 0, we have z =θ T 0 φ. Thus,itfollowsfrom(2.108)and(2.109)thatthesignalsy m andusatisfy F(s)(s n +θ T a0 α n−1 (s) | {z } D0(s;θa 0 ) )y m =F(s)θ T b0 α n−1 (s) | {z } N0(s;θ b 0 ) u. (2.121) Similarly,theparameterizedcontrollerΣ C (θ 0 )isanLTIsystemandy m andualsosatisfy u =−K(s;θ 0 )y m =− N K (s;θ 0 ) D K (s;θ 0 ) y m . (2.122) 49 From(2.121)and(2.122),y m andusatisfy     F(s)D 0 (s;θ a0 ) −F(s)N 0 (s;θ b0 ) N K (s;θ 0 ) D K (s;θ 0 )         y m u     =     0 0     . Thecharacteristicequationoftheabovesystemis F(s)(D 0 (s;θ a0 )D K (s;θ 0 )+N 0 (s;θ b0 )N K (s;θ 0 )) = 0. SinceF(s) is a stable minimum phase filter andD 0 (s;θ a0 )D K (s;θ 0 )+N 0 (s;θ b0 )N K (s;θ 0 ) is Hurwitz by properties C3 and M2, we have that y m ,u → 0 as t → ∞. From the detectability of (C P ,A P ) and (C C (β(θ 0 ),θ 0 ),A C (β(θ 0 ),θ 0 )), together with the convergence of y m and u to zero, it follows that x P ,x C → 0 ast → ∞. SinceA E is a stability matrix, the convergence ofy m andu to zero implies that x E → 0 ast → ∞. Therefore, because it has been shown thatx converges to zero for allθ 0 ∈ Ω when d,n, 1 ≡ 0,theparameterizedpair(C(θ 0 ),A(θ 0 ))isdetectableonΩ. Step 2: Establish that along the solutions of (Σ(θ), Σ T ) there exists a functionL : Ω → R 1×¯ n such thatA I (t) 4 =A(θ(t))−L(θ(t))C(θ(t))isexponentiallystable. Recallthattherobustadaptivelawguaranteesthat,m, ˙ θ∈S(η 2 /m 2 ). Applying[30,Lemma3.3.2] to(2.114),togetherwith|d|≤d 0 and|ν|≤ν 0 ,yields |η(t)|≤ Δ 1 ku t k 2δ0 +Δ 2 (2.123) Δ 1 4 =kN 0 FΔ m k 2δ0 , Δ 2 4 =kD 0 Fk ∞−gn (d 0 +ν 0 ) (2.124) and sincem 2 = 1+ku t k 2 2δ0 +k(y m ) t k 2 2δ0 andm≥ 1, it follows that|η(t)|/m≤ Δ 1 +Δ 2 . Therefore, ,m, ˙ θ∈S(μ 2 ),whereμ 2 4 =c(Δ 2 1 +Δ 2 2 )andcissomeconstant. Because properties C1 and M1 guarantee that C(β(θ),θ) and β(θ), respectively, are continuously differentiable, the parameterized controller Σ C is continuously differentiable with respect toθ. The error 50 modelE(θ)isaffineinθand,therefore,continuouslydifferentiable. Consequently,thepair(C(θ),A(θ))is continuouslydifferentiablewithrespecttoθ. Furthermore,becausetheadaptivelawguaranteesthatθ(t)∈ Ω and ˙ θ ∈ S(μ 2 ), it follows from the detectability result of Step 1 and result 2) of Theorem 1 that there existsacontinuouslydifferentiablefunctionL : Ω→R ¯ n×1 suchthatA I (t) 4 =A(θ(t))−L(θ(t))C(θ(t))is e.s. providedthatμ 2 <μ ∗ forsomeμ ∗ . ForlargeΔ 2 ,i.e.,larged 0 orν 0 ,thisconditionmaynotbesatisfied evenforsmallΔ 1 . Byusingthelengthyanalysisapproachof[30,Section9.9.1]involvingacontradiction argument, boundedness of the closed-loop signals can be proven provided Δ 1 satisfies a bound condition that is independent of Δ 2 . However, for simplicity, we continue with an alternative analysis approach, where we assume the filterF(s) is chosen so that Δ 2 is sufficiently small, saycΔ 2 2 < μ ∗ /2, so that for cΔ 2 1 <μ ∗ /2,theinequalityμ 2 <μ ∗ isalwayssatisfied. Therefore,A I (t)ise.s.,i.e.,thetransitionmatrix Φ(t,τ) ofA I (t) satisfieskΦ(t,τ)k ≤ λ 0 e −α0(t−τ) for some positive constantsλ 0 ,α 0 andt ≥ τ ≥ 0. NotethatifΔ m ,d,ν = 0,theadaptivelawguaranteesthat ˙ θ∈L 2 ,andfromresult1)ofTheorem1,A I (t) ise.s. SinceL(t)iscontinuousandΩiscompact,kLk∈L ∞ ,whereL(t)isaslightabuseofnotationand istakentomeanL(θ(t)). Step3: Establishboundednessandconvergenceofx. Letδ ∈ [0,δ 1 ), whereδ 1 < min{2α 0 ,δ 0 }, andc > 0 denotes any finite constant. Recall thatδ 0 is definedin(2.117). Byapplyingoutputinjection,werewrite(2.120)as ˙ x =A I (t)x+B(d+ν)+L(t) 1 (2.125) whereinStep2weestablishede.s. ofthehomogeneouspartof(2.125). Weestablishthatm∈L ∞ : ByLemma3.3.3of[30]andthee.s. propertyofA I ,wehavethat kx t k 2δ ≤ck( 1 ) t k 2δ +c (2.126) 51 where 1 ∈L 2e because 1 isacontinuousfunctionoftime. ApplyingtheL 2δ normtoy m =C P x P +d+ν and u = −C C (t)x C , wherekC C (t)k is bounded (a consequence of the continuously differentiability of C C (β,β(θ))andthecompactnessofΩ),yields k(y m ) t k 2δ ≤ck(x P ) t k 2δ +c≤ck( 1 ) t k 2δ +c (2.127) ku t k 2δ ≤ck(x C ) t k 2δ ≤ck( 1 ) t k 2δ +c (2.128) where the second inequalities of (2.127) and (2.128) were obtained by first recognizing that x P and x C are subvectors of x and then applying inequality (2.126). Consider the fictitious normalization signal m 2 f 4 = 1+ku t k 2 2δ +k(y m ) t k 2 2δ . Note that becauseδ < δ 0 , it follows from the definitions ofm andm f thatm≤m f . Substituting(2.127),(2.128),and 1 =m 2 intothedefinitionofm f yields m 2 f ≤ck(m 2 ) t k 2 2δ +c≤ck(mm f ) t k 2 2δ +c (2.129) wherethesecondinequalityisobtainedbyusingm≤m f . Fromthedefinitionofk(·) t k 2δ itfollowsthat m 2 f ≤c Z t 0 e −δ(t−τ) ((τ)m(τ)) 2 m 2 f (τ)dτ +c. (2.130) Applying the Bellman-Gronwall Lemma (cf. [30, Lemma 3.3.9]) to (2.130) yields the inequality m 2 f ≤ ce −δt e c R t 0 g 2 (τ)dτ +cδ R t 0 e −δ(t−s) e c R t s g 2 (τ)dτ ds,whereg =m. LetusassumethatF(s)ischosensuch that cΔ 2 2 ≤ δ/2. Because m ∈ S(μ 2 ) implies c R t s ((τ)m(τ)) 2 dτ ≤ cμ 2 (t−s), it follows that for cΔ 2 1 ≤ δ/2, we have m f ∈ L ∞ . Since m ≤ m f , we have that m ∈ L ∞ , and together with m ∈ S(μ 2 )∩L ∞ (property E1) implies that 1 = m 2 ∈ S(μ 2 )∩L ∞ . Moreover, it follows thatn d ∈ L ∞ becausem 2 = 1+n d ,which,togetherwithpropertyE3,impliesx T = θ T x T a n d T ∈L ∞ . Wenowturnourattentiontotheinjectedsystem(2.125). RecallthatA I (t)ise.s.,dandν arebounded, and ¯ u =L 1 is inS(μ 2 )∩L ∞ . Therefore,x∈L ∞ , and, in turn, ˙ x∈L ∞ . Now we examine the mean- square properties of x. From [30, Corollary 3.3.3] it follows that x ∈ S(μ 2 ) and, in turn, y ∈ S(μ 2 ) 52 sincey is a subvector ofx. Thus, (2.119) holds. To summarize, the condition for stability iscΔ 2 1 <δ ∗ 4 = min{μ ∗ /2,δ/2},forsomeconstants0<δ< min{δ 0 ,2α 0 }andc> 0,whereμ ∗ > 0istheboundforμ 2 suchthatA I (t)ise.s. Let us consider that Δ m ,d,ν ≡ 0. For this case, ¯ u =L 1 is aL 2 ∩L ∞ function becausekLk∈L ∞ and 1 ∈L 2 ∩L ∞ (fromE2andm∈L ∞ ). Thus,sinceA I (t)ise.s.,wehavex∈L 2 ∩L ∞ , ˙ x∈L 2 ∩L ∞ , andx → 0 ast → ∞. From the convergence ofx, and consequentlyL 1 , it follows from (2.125) that ˙ x→ 0ast→ 0. Wenowconsiderthecasethatlim t→∞ θ(t) =θ ∗ . Itfollowsfrom(2.94)and(2.96)that ˙ x PC =     A P −B P C C (β(θ),θ) B C (β(θ),θ)C P A C β(θ),θ)     x PC (2.131) =A PC (t)x PC (2.132) wherex PC = [x T P x T C ] T andθ(t)istunedbytheadaptivelaw. LetA ∗ PC 4 = lim t→∞ A PC (t). FromLemma6, the triplet (A ∗ PC ,B ∗ PC ,C ∗ PC ) is the state space realization of a controller that yields robust performance for the generalized plant P. From (2.118) and (2.132), the dynamics of ˜ x = x PC −x ∗ PC is given by ˙ ˜ x = A ∗ PC ˜ x + ˜ u, ˜ u 4 = (A PC (t)−A ∗ PC )x PC . Because A ∗ PC is Hurwitz and lim t→∞ ˜ u → 0, we have lim t→∞ ˜ x(t) = 0. Therefore,x PC →x ∗ PC ast→∞. Thefollowingresultadoptedfrom[12]isusedtoestablishtheresultsofTheorem1. Lemma 8. Let D ⊂ R l×n ×R n×n be the open set of detectable pairs (C,A), where C ∈ R l×n and A∈R n×n . LetS n denote the linear space of all symmetric positive matrices of dimensionn×n. There exists a unique analytic functionP : D → S n whose valueP(A,C) is positive definite and is such that thematrixA−PC T C isstableandsatisfiesthematrixRicattiequation:PA T +AP−PC T CP+I = 0. 53 ProofofTheorem1: Letusproveresult1). LetP betheanalyticalfunctionofLemma8,andP(θ)takes on the valueP(A(θ),C(θ)). BecauseA(θ) andC(θ) are continuously differentiable andP is analytical, thenP(θ)iscontinuouslydifferentiable. Therefore, A I (θ) =A(θ)−P(θ)C T (θ) | {z } L(θ) C(θ) (2.133) iscontinuouslydifferentiable. Since the eigenvalues of the matrix A I depend continuously on its elements, A I (θ) is continuously differentiable,andΩiscompact,itfollowsthat max i {Re[λ i (A I (θ))]}<−σ, ∀θ∈ Ω. (2.134) forsomefiniteconstantσ. ByLemma8,wehavethat−σ< 0,whichestablishesresult1). Now let us prove results 2) and 3). When now letθ vary with time. Taking the time derivative of the i th ,j th elementa ij (θ)ofthematrixA I yields ˙ a ij = ∂a ij (θ) ∂θ ˙ θ. (2.135) Sincetheelement ∂aij ∂θ isboundedbecausea ij (θ)iscontinuouslydifferentiableandΩiscompact,wehave that ˙ θ∈L 2 ⇒k ˙ A T k∈L 2 (2.136) ˙ θ∈S(μ 2 )⇒k ˙ A T k∈S(μ 2 ) (2.137) whereA T (t) =A I (θ(t)). From(2.134),itfollowsthattheLyapunovequation A T T (t)P(t)+P(t)A T (t) =−I (2.138) 54 hasauniqueboundedsolutionP(t)foreachfixedt. WeconsiderthefollowingLyapunovfunction: V(t,x) =x T P(t)x. (2.139) Thenalongthesolution ˙ x =A T (t)xwehave ˙ V =−|x(t)| 2 +x T (t) ˙ P(t)x(t). (2.140) From(2.138), ˙ P satisfies A T (t) T ˙ P(t)+ ˙ P(t)A T (t) =−Q(t), ∀t≥ 0 (2.141) whereQ(t) = ˙ A T T (t)P(t)+P(t) ˙ A T (t). BecauseA T (t) =A I (θ(t))isstable, ˙ P(t)isgivenby ˙ P(t) = Z ∞ 0 e A T T (t)τ Q(t)e A T (t)τ dτ (2.142) whichsatisfies(2.141)foreacht≥ 0,therefore, k ˙ P(t)k≤kQ(t)k Z ∞ 0 ke A T T (t)τ kke A T (t)τ kdτ. (2.143) Because(2.134)impliesthatke A T (t)τ k≤α 1 e −α0τ forsomeα 0 ,α 1 > 0,itfollowsthat k ˙ P(t)k≤ckQ(t)k (2.144) forsomec≥ 0. Then, kQ(t)k≤ 2kP(t)kk ˙ A T (t)k (2.145) 55 togetherwithP ∈L ∞ implythat k ˙ P(t)k≤βk ˙ A T (t)k, ∀t≥ 0 (2.146) for some constant β ≥ 0. Using (2.140) in (2.146) and noting that P satisfies 0 < β 1 ≤ λ min (P) ≤ λ max (P)≤β 2 forsomeβ 1 ,β 2 > 0,wehavethat ˙ V(t)≤−|x(t)| 2 +βk ˙ A T )k|x(t)| 2 (2.147) ≤−β −1 2 V(t)+ββ −1 1 k ˙ A T (t)kV(t) (2.148) therefore, V(t)≤e − R t t 0 (β −1 2 −ββ −1 1 k ˙ A T (τ)k)dτ V(t 0 ) (2.149) ≤e −β −1 2 (t−t0) e ββ −1 1 R t t 0 k ˙ A T (τ)kdτ V(t 0 ). (2.150) Letusproveresult3)first. UsingtheSchwartzinequalityandk ˙ A T k∈S(μ 2 )wehave Z t t0 k ˙ A T (τ)kdτ ≤ Z t t0 k ˙ A T (τ)k 2 dτ 1 2 √ t−t 0 (2.151) ≤ [μ 2 (t−t 0 ) 2 +α 0 (t−t 0 )] 1 2 (2.152) ≤μ(t−t 0 )+ √ α 0 √ t−t 0 . (2.153) Therefore, V(t)≤e −α(t−t0) y(t)V(t 0 ) (2.154) 56 whereα = (1−γ)β −1 2 −ββ −1 1 μ, y(t) = exp −γβ −1 2 (t−t 0 )+ββ −1 1 √ α 0 √ t−t 0 (2.155) = exp   −γβ −1 2 √ t−t 0 − ββ −1 1 √ α 0 2γβ −1 2 ! 2 + α 0 β 2 β 2 4γβ 2 1   (2.156) andγ isanarbitraryconstantthatsatisfies0<γ < 1. Itcanbeshownthat y(t)≤ exp α 0 β 2 β 2 4γβ 2 1 4 =c, ∀t≥t 0 (2.157) hence V(t)≤ce −α(t−t0) V(t 0 ). (2.158) Choosingμ ∗ = β1(1−γ) β2β , we have that∀μ ∈ [0,μ),α > 0 and, therefore,V(t) → 0 exponentially fast, whichimpliesthatthesolutionx e = 0isexponentiallystable. Theproofofresult2)followsfromthatofresult3),becausek ˙ A T k∈L 2 impliesresult3)withμ = 0. 57 Chapter3 ABenchmarkExample 3.1 Introduction In this chapter we investigate the performance capabilities of an AMC scheme for a benchmark control design problem. The plant is the SISO two-cart mass-spring-dashpot (MSD) system shown in Figure 3.1(a)withanuncertainparameterandtimedelay,whicharemajorobstaclesinachievingsuperiordistur- bance rejection. This benchmark example has been investigated in [15, 16]. In these works, an RMMAC scheme was designed and it demonstrated superior performance rejection performance when compared to the “best” nonadaptive compensator. However, since the supervisor utilizes Kalman filters to drive the hypothesis testing, the RMMAC scheme can suffer from poor performance due to either large initial state estimateerrororinaccurateknowledgeofthedisturbance/noisestatistics. Since the RMMAC methodology separates control from identification, any suitably designed supervi- sor may be used. In this chapter we design an AMC supervisory scheme that is used in conjunction with thesamecandidatecontrollersetusedintheRMMACschemewiththeaimofmitigatingtheaboveissues ofRMMAC.TheoverallAMCisshowninFigure3.2. AspresentedinChapter2,theAMCapproachuses onlinerobustparameteridentificationtomixadaptivelythecandidatecontrollaws. AlthoughtheRMMAC candidatecontrollersetcanbeusedinconjunctionwithothersupervisoryschemes,suchastheonesfrom supervisory (switching) adaptive control [1, 25], adaptive control with multiple models [49, 50, 32], and 58 unfalsified control [60, 71], a mixing approach was chosen to avoid poor transient performance caused by unnecessary controller switching. The performances of the AMC, RMMAC, and supervisory adaptive control schemes – as well as a nonadaptive mixed- μ compensator – are compared on the two-cart bench- markproblem. WeassumethereaderisfamiliarwiththeRMMACdesignforthebenchmarkexample(cf. [15,16].) In Section 3.2 the MSD model is presented. The design of the AMC scheme is the focus of Section 3.3. ThesimulationresultsarediscussedinSection3.4. 3.2 TheTwo-CartModel The two-cart mass-spring-damper (MSD) system is used as a testbed for adaptive control design. The MSD system, shown in Figure 3.1(a), is composed of two masses,m 1 andm 2 , interconnected by springs anddashpots. TheMSDsystempossessesimportantpracticalcharacteristicspresentinmanyrealsystems, suchasparametricuncertainty,unmodeleddynamics,plantdisturbance,andmeasurementnoise. The plant disturbanced(t) is a low-frequency, stationary stochastic process that acts on m 2 , which is generatedbyshapingacontinuous-timewhitegaussianprocessasfollows d(s) = a s+a ξ(s) (3.1) m 1 m 2 k 1 b 1 k 2 b 2 e - τs u d u x 1 x 2 d (a) The two-cart system possesses an unmodeled control delay, uncertain spring constant, process disturbance, and measurement noise. 10 −2 10 −1 10 0 10 1 −80 −60 −40 −20 0 20 40 ω, rad/s 20 log |G 0 (jω)| θ * increasing (b) Bodeplotofnominalmodelforvariousvaluesofθ ∗ = k 1 . Figure3.1: Mass-spring-dashpotsystembenchmarkexample. 59 Unknown Plant C 1 C N … Candidate Robust Controllers E 1 Robust Supervisor … Adaptive Control Mixer Robust Estimator θ n ( ) 1 β θ ( ) 4 β θ 4 1 i i β β = ∑ T uu β = Figure3.2: Theadaptivemixingcontrolarchitecture whereξ(t)haszeromeanandunitintensity(Ξ = 1),anda> 0isthedisturbancebandwidth. The only measurement available ism 2 ’s displacement plus additive white gaussian noise, i.e. y(t) = x 2 (t)+n(t),wherethestatisticsofthesensornoisen(t)areknowntobe E{n(t)} = 0, E{n(t)n(τ)} = 10 −6 δ(t−τ). (3.2) Thestate-spacerepresentationofthetwo-cartMSDmodelisgivenas ˙ x(t) = Ax(t)+Bu d (t)+Lξ(t) (3.3) y(t) = Cx(t)+n(t) (3.4) wherethestatevectoris x = x 1 (t) x 2 (t) ˙ x 1 (t) ˙ x 2 (t) d(t) T (3.5) 60 and A =                 0 0 1 0 0 0 0 0 1 0 − θ ∗ m1 θ ∗ m1 − b1 m1 b1 m1 0 θ ∗ m2 − θ ∗ +k2 m2 b1 m2 − b m2 1 m2 0 0 0 0 −a                 (3.6) B = 0 0 1 m1 0 0 T , L = 0 0 0 0 a T (3.7) C = 0 1 0 0 0 (3.8) (3.9) whereθ ∗ 4 =k 1 . Theknownparametersofthemechanicalsystemare m 1 =m 2 = 1, k 2 = 0.15, b 1 =b 2 = 0.1 a = 0.1 (3.10) m 4 =m 1 +m 2 , b 4 =b 1 +b 2 (3.11) andtheunknownconstantparameterk 1 isrestrictedtothecompactset θ ∗ ∈ Ω ={k : 0.25≤k≤ 1.75}. (3.12) Thecontrolu(t)isappliedtom 1 throughacontrolchannelwithanunmodeleddelayτ. Themaximum delaythroughthischannelisknowntobe0.05sec.,i.e. τ ≤ 0.05 sec. (3.13) 61 Theperformancevariablez(t) =x 2 (t)istobekeptsmall. Therefore,thecontrolobjectiveistodesign acontrollawu(t)sothattheeffectofthedisturbanceandnoiseonz(t)isattenuated. 3.3 RobustAdaptiveControlDesigns In this section, the candidate controller set will be described briefly and then the two subsystems of the high-levelsupervisoraredescribed: (1)robuston-lineparameterestimatorsand(2)adaptivecontrolmixer. In[16],thecandidatecontrollersetC ={K i (s)} 4 i=1 ,whereeachK(s)∈C isamixed- μcompensator, andthecorrespondingparameterpartition ˜ Ω 1 = [1.02,1.75], ˜ Ω 2 = [0.64,1.02] ˜ Ω 3 = [0.40,0.64], ˜ Ω 4 = [0.25,0.40] weregeneratedbythe%FNARC method(cf. [16].) Alsogeneratedforcomparisonpurposeswasanon- adaptiverobust(mixed- μ)controllerforthecompleteparameterspaceΩ. Becausetheparameterpartition P = { ˜ Ω i } 4 i=1 does not satisfy the overlapping property (cf. Section 2.4,) the multicontrollerC(β) and monitoringsystemM designsarecoupled. For a fair comparison, the AMC and supervisory control schemes utilize the candidate controller set C developed in the RMMAC design. Let us now consider the design of an AMC scheme. We start the design of the multicontrollerK(β) by enlarging each parameter subset ˜ Ω so that mixing regions are artificially introduced. After expanding the boundaries of ˜ Ω i , i = 1,...,4, by 10%, we have the new parameter partition Ω 1 = [0.92,1.93], Ω 2 = [0.58,1.12], Ω 3 = [0.36,0.70], and Ω 4 = [0.23,0.44]. The multicontroller design is completed by performing output blending, i.e., u(t) = P 4 i=1 β i (t)u i (t). It is clear that requirements C1 and C2 are satisfied. Using standard mixed- μ analysis tools, we have found property C3 holds over Ω. Also, for comparison, an AMC scheme was developed using the Q-blending approachtoconstructthemulticontroller. 62 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 θ β i β 4 β 3 β 2 β 1 Figure3.3:M isconstructedfromsmoothbumpfunctionssuchthatrequirementsM1,M2aresatisfied. The design of the mixing systemM (shown in Fig. 3.3) is accomplished by defining the functions ˜ β i , i = 1,...,4,as ˜ β 1 (θ) =ψ θ−1.422 0.504 , ˜ β 2 (θ) =ψ θ−0.849 0.273 ˜ β 3 (θ) =ψ θ−0.532 0.172 , ˜ β 4 (θ) =ψ θ−0.333 0.108 . whereψisthesmoothbumpfunction. Themixingsignalβ isgeneratedbynormalizing ˜ β = [ ˜ β 1 ... ˜ β 4 ] T , i.e.,β = ˜ β/ P 4 i=1 ˜ β i ,andrequirementsM1andM2aresatisfied. Figure 3.4 shows a plot of delay margin of the controlled system yielded byC(β(θ ∗ )). We see that the certainty equivalence control is robustly stable. Moreover, as shown in Fig 3.5 , the closed loop system achieves superior predicted output RMS performance, assuming perfect estimation, compared to the nonadaptive robust compensator. Of course, estimation is not perfect and actual performance must be analyzed by simulations and experiments. Nonetheless, assuming that estimation is of sufficient quality, theanalysisshowninFigure3.4and3.5provideausefulevaluationtoolforthedesigner. AlsoplottedintheinsetofFigure3.5isthepredictedperformanceofasupervisorschemethatselects onlyonecontrollertoclosetheloopbasedonthepartition{ ˜ Ω i } 4 i=1 . Thisclassofsupervisorsincludesthe 63 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Figure3.4: Timedelaymargin: Mixingcontrol(solid);mixed- μcontroller(dash) Figure3.5: ExpectedoutputRMS:Adaptivemixingcontrol(solid);mixed- μcontroller(dash);switching- basedsupervisor(inset,dash-dot) supervisorsofRMMACandsupervisoryadaptivecontrol. ObservethereductionoftheRMSpeaksforthe mixing scheme compared to the switching-based scheme. If the design objective is to satisfy maximum RMS requirements then the designer, by exploiting the performance behavior on the parameter overlaps, may be able to either 1) reduce the number of candidate controllers or 2) achieve improved performance (lower maximum RMS levels) with a fixed number of candidate controllers. Neither option is pursued in thisexample. 64 The final component of the adaptive mixing control scheme is the robust adaptive law. The LPM z(t) =θ ∗ φ(t)+η(t)isusedtoderivetheadaptivelaw,where z(t) =D K 0 (s)F(s)y m (t)−N K 0 (s)F(s)u(t) (3.14) φ(t) =F(s)u(t)−D U 0 (s)F(s)y m (t), (3.15) F(s) = 5 4 (s+5) 4 F η (s) (3.16) η(t)isthemodelingerror;F η (s)isthebandpassfilter F η =k bp (s 2 + bp s+0.005688)(s 2 + bp s+0.02807) (s 2 +0.08s+0.08)(s 2 +0.543s+0.178) · (s 2 + bp s+80.15)(s 2 + bp s+395.6) (s 2 +4.59s+12.67)(s 2 +1.5s+28.09) wherek bp = 0.010001and bp = 1×10 −5 . ThepassbandisW P = [.3, 5]rad/s,withapeak-to-peak0.25 dB ripple; and the stopband isW S = [.1, 10] rad/s, with a -40 dB attenuation level. These values were chosen to make use of the frequency range that is largely affected byθ ∗ , as shown graphically in Fig. 3.1, whilefilteringouttheremainingfrequenciesthatmaybecomedominatedbyη. Anadaptivelawusingthegradientmethodbasedonintegralcostisused: ˙ θ =                −5(rθ+q), if|θ−1|< 0.75or if5(rθ+q)sgn(θ)≥ 0 0, otherwise ˙ r =−0.1r+ φ 2 m 2 , ˙ q =−0.1q− zφ m 2 , r(0) =q(0) = 0 whereθ(t)istheestimateofθ ∗ andm 2 (t) 4 = 1+n d isthedynamicnormalizationsignal,where ˙ n d =−0.04n d +(F η (s)u) 2 +(F η (s)y) 2 , n d (0) = 0. 65 Thedesignparametersoftheadaptivelawwerechosenbytrialanderror. Remark 6. The AMC supervisor does not require any extra states when an additional candidateK N+1 is added. Contrast this with dynamic hypothesis testing, which requires n (the dimension of the plant) additionalstatesfortheKalmanfilterandonefortheposteriorprobabilityevaluator. Thesupervisoryadaptivecontrolschemeisconstructedbymonitoringtheestimationerrorsignals i = z(t)−θ ∗ i φ(t)fori = 1,2,3,4,whereeachconstantθ ∗ i ischosenasthecenteroftheinterval ˜ Ω i ,andz(t) andφ(t) are generated by (3.14) and (3.15), respectively. Note that the filters that generatez(t) andφ(t) areidenticaltothefiltersin theAMCscheme. Thus, anyeffectattributedtothefilterF(s)isexperienced by both schemes. The supervisory scheme generates the monitoring signalsμ i = R t 0 e −0.04(t−τ) i (τ)dτ (i = 1,2,3,4), and selects the controller corresponding to the smallest monitoring signal subject to a hysteresislogic. Hysteresislogicprohibitsswitchingunlessthesmallestmonitoringsignalisatleast0.1% smaller than the currently selected monitoring signal. The value of the hysteresis constant was chosen by trial and error based on an acceptable compromise between rapid response and reduced likelihood of switchingduetonoiseandmodelingerror. 3.4 SimulationResults WenowpresentsomesimulationresultstoinvestigatetheperformancesoftheAMC RMMAC,andsuper- visory adaptive control (SAC) schems. First, we consider zero initial condition of the plant. Simulations usingthenominalmodelandanoff-nominalmodelwithastrongerdisturbancearepresented. Second,we consider the effects of non-zero initial condition of the plant. These investigations include nominal and off-nominalsimulations. 3.4.1 Case1: ZeroInitialCondition First we investigate compare the performance of the output-blending and Q-blending AMC schemes. Fig. 3.6 shows the results of two different adaptive mixing controllers: an output blending scheme and 66 Table3.1: Case1: AssumptionsSatisfied θ ∗ Short-termRMS Long-termRMS AMC 0.021 0.012 RMMAC 1.75 0.014 0.012 SAC 0.016 0.010 AMC 0.018 0.012 RMMAC 1.385 0.016 0.012 SAC 0.023 0.012 AMC 0.016 0.016 RMMAC 1.02 0.022 0.020 SAC 0.019 0.019 AMC 0.014 0.013 RMMAC 0.830 0.015 0.013 SAC 0.023 0.012 AMC 0.018 0.017 RMMAC 0.640 0.019 0.017 SAC 0.023 0.017 AMC 0.015 0.014 RMMAC 0.520 0.014 0.013 SAC 0.018 0.014 AMC 0.019 0.018 RMMAC 0.400 0.017 0.018 SAC 0.020 0.021 AMC 0.018 0.017 RMMAC 0.325 0.015 0.014 SAC 0.018 0.014 AMC 0.019 0.017 RMMAC 0.25 0.016 0.017 SAC 0.018 0.016 a Q-blending scheme. The output-blending scheme exhibits better transient and long-term performance compared to the Q-blending mixing scheme, which is more susceptible to bursting-type behaviors be- cause the Q-blending multicontroller depends on both β(t) andθ(t). This makes it more susceptible toθ variations. Based on the above results, we chose to use the AMC scheme with output blending to compare with the RMMAC and supervisory adaptive control schemes. Although an extensive evaluation of the AMC, RMMAC,andsupervisoryschemesarebeyondthescopeofthispaper,wepresentsomesimulationresults that illustrate some of the potential benefits of the AMC scheme. Table 3.1 presents short-term ( 0≤ t≤ 67 100) and long-term ( 100≤t≤ 200) output RMS values for various values ofθ ∗ , and withτ = 0.05 and allmodelassumptionssatisfied. TheresultsofTable3.1aretheaverageoffivetrials. While the three schemes achieve remarkable performance, the following observations were made. In general, the RMMAC scheme exhibits the smallest start-up transient if θ ∗ takes on values near the boundary ofΩ (see Fig. 3.7), whereas the adaptive mixing controller typically has the smallest ifθ ∗ takes onvaluesaround1(seeFig.3.8). Thesupervisoryadaptiveschemetendstoperformanumberaswitches before reliable data are observed, often creating a large transient (see Fig. 3.8). These observations are congruent with the short-term RMS values of Table 3.1. Additionally, when θ ∗ takes on a value near the boundaryoftwocandidatemodels,thesupervisoryadaptivecontrollermaycreateadditionaltransientsby switching between controllers. This last observation is illustrated in Fig. 3.8, particularly Figs. 3.8(c) and 3.8(d). Foraclearpresentation,wehaveplottedtheresultspair-wise,althoughtheschemesweresimulated side-by-side. Next we consider the off-nominal case in which the disturbance model’s power and bandwidth are increased to 100 and 3, respectively, and all control designs remain unchanged. The results are shown in Fig 3.9. Because the increased disturbance power and bandwidth, the magnitudes of the residual signals of the RMMAC’s supervisor are larger than expected. In fact, the residual signals become so large that numerical round-off effects cause the controller weights to be computed as their lower limit of 1×10 −6 , as can be seen in Fig. 3.9(b), and this causes the poor performance 1 . The AMC and supervisory adaptive schemes,however,arereasonablyrobusttothisplantdisturbance,asillustratedinFigs.3.9(c)and3.9(d). 1 Inalllikelihood,thisbehaviorcanbefixedbyextendingtheRMMAC/XIschemetocopewithaboundedrangeofainaddition to a range of Ξ. This comes at the cost of complexity. Moreover, there is still a risk of the same undesirable behavior if these parameterstakeonavalueoutsideoftheexpectedranges. 68 0 20 40 60 80 100 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Time, sec Output x 2 (t) Q blending Output blending Figure 3.6: Comparison of the plant outputs of the Q-blending and output-blending schemes with θ ∗ = 0.325,τ = 0.05,andnominaldisturbancemodel. 0 5 10 15 20 25 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Time, sec Output x 2 (t) AMC RMMAC SAC Figure 3.7: Case 1: Comparison of the start-up transients of adaptive mixing control (AMC), RMMAC, andsupervisoryadaptivecontrol(SAC):θ ∗ = 1.75,τ = 0.05,andnominaldisturbancemodel. 3.4.2 Case2: Non-zeroInitialCondition Now we investigate the effects of non-zero initial condition of the plant. In this section we compare the AMC, RMMAC, and GNARC schemes by simulating them them simulated side-by-side. In all simula- tions,theplant’sinitialconditionsaregivenas x 0 = x 1 (0) x 2 (0) ˙ x 1 (0) ˙ x 2 (0) d(0) T ∼ U[−0.1,0.1] U[−0.1,0.1] U[−1,1] U[−1,1] U[−0.1,0.1] T 69 0 50 100 150 200 250 300 350 400 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Time, sec Output x 2 (t) RMMAC AMC (a) Plantoutput: AMCandRMMAC. 0 50 100 150 200 250 300 350 400 0 0.5 1 β 1 (t) 0 50 100 150 200 250 300 350 400 0 0.5 1 β 2 (t) 0 50 100 150 200 250 300 350 400 0 0.5 1 β 3 (t) 0 50 100 150 200 250 300 350 400 0 0.5 1 Time, sec β 4 (t) (b) Controllerweights: AMCandRMMAC. 0 50 100 150 200 250 300 350 400 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Time, sec Output x 2 (t) SAC AMC (c) Plantoutput: AMCandSAC. 0 50 100 150 200 250 300 350 400 0 0.5 1 β 1 (t) 0 50 100 150 200 250 300 350 400 0 0.5 1 β 2 (t) 0 50 100 150 200 250 300 350 400 0 0.5 1 β 3 (t) 0 50 100 150 200 250 300 350 400 0 0.5 1 Time, sec β 4 (t) (d) Controllerweights: AMCandSAC. Figure3.8: Case1simulationresultswithθ ∗ = 1.02,τ = 0.05,andnominaldisturbancemodel: Adaptive mixingcontrol(AMC),RMMAC,andsupervisoryadaptivecontrol(SAC). whereU[a,b] is a uniform distribution on the interval [a,b]. The choice of the intervals were motivated by observations of the approximate range of the plant state when the GNARC is used to control the plant. Thecontrolchanneltime-delayforallsimulationswaschosenas τ = 0.01. Foreachexperiment,werunfiveMonte-Carlosimulations,andtheaveragedresultsaregiveninTables 3.2 and 3.3. Two RMS values are given: (1) transient and (2) long-term. Transient RMS is calculated for the first half of the simulation time and is used to judge learning performance. Long-term RMS is calculated over the final half of the simulation time and is used to predict asymptotic performance. Also, themetric“%RMSincreasefromAMC”isgivenandisdefinedas %I 4 = RMS−RMS AMC RMS AMC 100% (3.17) which quantifies AMC’s improvement or deterioration over the other schemes. Above, “RMS” is either RMMAC’sorGNARC’sRMS,dependingonthecontext. 70 Table3.2: Case2: AssumptionsSatisfied θ ∗ TransientRMS Long-termRMS (%I) (%I) AMC 0.025 0.009 RMMAC 1.385 0.203(701%) 0.009(0%) GNARC 0.039(54%) 0.038(318%) AMC 0.016 0.011 RMMAC 1.020 0.111(600%) 0.017(52%) GNARC 0.038(143%) 0.035(204%) AMC 0.017 0.010 RMMAC 0.830 0.086(400%) 0.010(0%) GNARC 0.035(106%) 0.030(192%) AMC 0.021 0.014 RMMAC 0.640 0.037(77%) 0.015(11%) GNARC 0.031(49%) 0.027(93%) AMC 0.023 0.011 RMMAC 0.520 0.060(159%) 0.011(−7%) GNARC 0.036(56%) 0.026(135%) AMC 0.051 0.015 RMMAC 0.400 0.066(30%) 0.015(3%) GNARC 0.046(−10%) 0.034(130%) AMC 0.044 0.013 RMMAC 0.325 0.039(−11%) 0.012(−8%) GNARC 0.056(27%) 0.044(245%) Table3.3: Case2: AssumptionsViolated Violation TransientRMS Long-termRMS (%I) (%I) AdaptiveMixingControl Dist. Bandwidth 0.185 0.173 RMMAC a = 3rad/s 0.396(114%) 0.358(106%) GNARC θ ∗ = 0.83 0.242(31%) 0.224(29%) AdaptiveMixingControl Dist. Power 0.099 0.098 RMMAC Ξ = 100 0.210(111%) 0.198(103%) GNARC θ ∗ = 0.83 0.287(189%) 0.268(175%) AdaptiveMixingControl TimeVaryingθ ∗ 0.014 0.012 RMMAC Slowsin 0.071(398%) 0.013(10%) GNARC 0.042(195%) 0.039(228%) AdaptiveMixingControl TimeVaryingθ ∗ 0.066 0.067 RMMAC Fastsin 0.187(183%) 0.112(68%) GNARC 0.058(−12%) 0.058(−13%) AdaptiveMixingControl TimeVaryingθ ∗ 0.025 0.013 RMMAC Step 0.022(−12%) 0.014(2%) GNARC 0.037(48%) 0.035(162%) 71 0 20 40 60 80 100 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 x 10 6 Time, sec Output x 2 (t) RMMAC AMC (a) Plantoutput: AMCandRMMAC. 0 20 40 60 80 100 0 0.5 1 β 1 (t) 0 20 40 60 80 100 0 0.5 1 β 2 (t) 0 20 40 60 80 100 0 0.5 1 β 3 (t) 0 20 40 60 80 100 0 0.5 1 Time, sec β 4 (t) (b) Controllerweights: AMCandRMMAC. 0 10 20 30 40 50 60 70 80 90 100 -15 -10 -5 0 5 10 15 Time, sec Output x 2 (t) SAC AMC (c) Plantoutput: AMCandSAC. 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 β 1 (t) 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 β 2 (t) 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 β 3 (t) 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 Time, sec β 4 (t) (d) Controllerweights: AMCandSAC. Figure 3.9: Case 1 simulation results withθ ∗ = 1.02,τ = 0.05, and the off-nominal disturbance model withadisturbancepowerof100·Ξandabandwidthof30·a: Adaptivemixingcontrol(AMC),RMMAC, andsupervisoryadaptivecontrol(SAC). 3.4.2.1 SatisfiedModelAssumptions These experiments test the AMC, RMMAC, and GNARC schemes when the plant matches the model (3.1)-(3.13). Note that the model still contains the control delay, uncertainty in θ ∗ , process disturbance d(t), and measurement noiseθ(t). Table 3.2 shows the results. Both adaptive schemes possess superior disturbance rejection performance relative to the GNARC design. The AMC and RMMAC schemes have comparable long-term performance. In general, we see that Figure 3.5 does an acceptable job predicting long-term RMS output. Observe the improved long-term performance for the cases when θ ∗ takes on a valuenearmodelboundaries,i.e.,forthecaseswhenθ ∗ ∈{1.020, 0.640 0.400}. For illustrative purposes Figure 3.10 shows the averaged output and control weight time histories for θ ∗ = 0.64. Figure3.10(a)and3.10(b)showtheresultswhentheinitialconditionsarechosentobenonzero as described. Figure 3.10(c) and 3.10(d) show the same simulations when the plant initial conditions are 72 0 10 20 30 40 50 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 Time, sec Output x 2 (t) (a) Output,θ ∗ =0.64,nonzeroplantinitialconditions 0 50 100 150 200 0 0.5 1 β 1 (t) 0 50 100 150 200 0 0.5 1 β 2 (t) 0 50 100 150 200 0 0.5 1 β 3 (t) 0 50 100 150 200 0 0.5 1 Time, sec β 4 (t) (b) Control weights,θ ∗ = 0.64, nonzero plant initial con- ditions 0 10 20 30 40 50 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 Time, sec Output x 2 (t) (c) Output,θ ∗ =0.64,zeroplantinitialconditions 0 50 100 150 200 0 0.5 1 β 1 (t) 0 50 100 150 200 0 0.5 1 β 2 (t) 0 50 100 150 200 0 0.5 1 β 3 (t) 0 50 100 150 200 0 0.5 1 Time, sec β 4 (t) (d) Controlweights,θ ∗ =0.64,zeroplantinitialconditions Figure3.10: Case2: Resultsforθ ∗ = 0.64. AMC(blue),RMMAC(green),andGNARC(red) chosen to be zero. These plots show that initial conditions can have a large effect on RMMACs transient performance. This is not a surprising result because the RMMAC scheme uses Kalman filters to drive the dynamichypothesistestingscheme,andonewouldexpectthatiftheKalmanfiltersinitialstateestimation errorislargethenmodelidentificationmaynotbereliableuntilthestateestimationerrorconvergestozero. If the Kalman filters were initialized with true plant states then model identification will converge rapidly tothecorrectmodel. Inthiscase,RMMACstransientperformanceiscomparableto(andsometimesbetter than)AMCstransientperformance. TheseobservationsmotivatetechniquestopreconditiontheRMMACs Kalmanfiltersiftheinitialstateestimationerrorcannotbeguaranteedtobesmall. 73 3.4.2.2 ViolatedModelAssumptions These simulations concern unknown violations of the model assumptions, namely perturbations in the disturbancemodelandatime-varyingparameter. Uncertaintyinthedisturbancemodelisarealengineering issuebecauseinmanyapplicationsthedisturbancemodelispoorlyknownortime-varying. In[15,16],the RMMAC/XI design was proposed to mitigate the ill effects of unexpected increased disturbance power, and,asreportedin[59]butnotimplementedhere,thismodificationregainedtheperformanceoftheknown case. The cost of XI modificationis increased complexity because twicethe number ofKalman filters are required. SincetheXImodificationaddssignificantcomplexitythatmaylimititsapplication,Experiment 2 is intended to investigate if AMC may be a suitable design approach when the disturbance model is uncertain. There are two test cases concerning the disturbance model: first a300% increase in disturbance band- width;andthena100%increaseindisturbancepower. Inbothcases,θ ∗ ischosentobe0.83,whichisthe center of model #2. The first two rows of Table 3.3 show the results. Comparatively, AMC demonstrates excellenttransientandlong-termRMSperformance. Next, the constant parameter assumption is violated asθ ∗ is allowed to vary. The issue of parameter variationsisimportanttoconsiderbecauseitisoneofthemotivationsforadaptivecontrol. The first five Monte Carlo simulations use the slow sinusoid waveform θ ∗ = 1− 0.75sin(0.01t). Theseexperimentsaresimulatedfor1,000secondsandtheresultsareshowninTable3.3. BothAMCand RMMACschemesmaintainlong-termperformancelevelssimilartotheconstant θ ∗ case. 74 Thentheadaptiveschemesarestressedbyrapidlyvaryingθ ∗ . ThenextsetofMonteCarlosimulations usesthefastsinusoidwaveformθ ∗ = 1−0.75sin(0.5t). Thesesimulationsrunfor200seconds. Thefinal fiveMonteCarlosimulationsusethestepwaveform θ ∗ (t) =                        0.325, t∈ [0,100) S [600,700] 0.520, t∈ [100,200) S [500,600) 0.830, t∈ [200,300) S [400,500) 1.385, t∈ [300,400) (3.18) Thesesimulationsrunfor700seconds. Table 3.3 shows the results for the two fast parameter waveforms. Both adaptive schemes perform poorly with the fast sinusoid parameter changes. Although the AMC scheme demonstrates poor transient performance, its long-term RMS is comparable to the GNARCs scheme because, for the most part, AMC keepsC 1 andC 2 in the loop, andθ ∗ spends most of its time in Ω 1 and Ω 2 . The RMMAC scheme, on the other hand, rapidly “switches” among all controllers. For the step waveform, both AMC and RMMAC schemesperformwellbyachievinglong-termRMSvaluesnearnominalmodellevels. 3.5 Conclusions AsolutiontoabenchmarkexamplehasbeendevelopedbyusingtheAMC ˙ TheAMCapproachmixescan- didatecontrollersbymonitoringthereal-timeestimateoftheunknownparameter. Themotivationforthis design was to develop an adaptive control scheme that is capable of maintaining satisfactory performance despiteuncertaintyinthedisturbancemodelwhileavoidingdiscontinuousswitchingamongthecandidate controller set. Monte Carlo simulations were conducted to compare the AMC scheme with the stochastic MMACapproachRMMACandanon-adaptivemixed- μcompensator. Thesimulationresultsdemonstrate that the AMC scheme achieves satisfactory performance despite perturbations in the disturbance model, non-zeroplantinitialconditions,andcertainclassesofparametertimevariations. 75 Chapter4 AdaptiveControlofanAirbreathingHypersonicFlightVehicle 4.1 Introduction Air-breathing hypersonic flight vehicle (AHFV) research, with the ultimate goal of feasible and afford- able atmospheric hypersonic flight using air-breathing propulsion systems, will expand the boundaries of aeronautics and provides for the development of new technologies for space access, high speed civilian transportation, and defense applications [13, 19, 20, 5, 11, 48, 42] . Hypersonic flight technologies have been studied in countries around the world for more than a half of a century since the devolvement of the hydrocarbon-fueled conventional ramjet (CRJ) engine concept [19, 20, 11] . As the most recent develop- ments in the field, in 1996 the Hyper-X program was initiated with an initial aim of developing and flight testing small scale (X-43A, X-43B, X-43C, and X-43D) and one full-scale demonstrator vehicles. The X-43A is “smart scaled” from a 200-foot operational concept. It has been used in three scramjet-powered andun-poweredflighttests,twoatMach7andoneatMach10[42,41]. Subsequent to the successful X-43 flights, NASA Aeronautics Directorate and the Air Force have turnedtheirfocustofoundationalhypersonicresearchanddevelopmentofpredictivecapabilitiesinsupport of the next generation of highly reliable reusable launch vehicles (HRRLVs) program. Until recently a major obstacle For control engineers was the lack of any open source model with sufficient fidelity incorporatingthespecificfeaturesofthisclassofvehiclewhichseparatesthemfromconventionalaircrafts. 76 TworecentmodelsdevelopedinthislightforthelongitudinaldynamicsofagenericAHFVarepresentedin [6,55]and[43,10],respectively. Themodelin[6,55]representsthelongitudinaldynamicsofafullscale vehicle developed based on first principles to give flight control engineers a fundamental understanding of the flight dynamic characteristics of these vehicles. The vehicle model in [43, 10] , CSULA-GHV (California State University Los Angeles - Generic Hypersonic Vehicle), has been developed by starting withasetofmissionrequirementsandusingobliqueshocktheoryforinitialsizingofafull-scalevehicle. Computationalfluiddynamical(CFD)isusedtocalculateaerodynamicforcesandmoments,asimplefinite element model to extract elastic modes, and quasi 1-D flow with heat added to model combustion in the scramjetengine. Sincethesemodelshavebecomeavailable,severalcontroldesignshavebeendeveloped, includingthedesignsof[24,64,54,55,40,21,63]fortheBolenderandDomanmodelandthedesignsof [27,18,38]forthemodeldevelopedbyMirmiranietal. Afterdecadesofresearch,challengesinmanyaspectsofhypersonicflightstillremaintobeaddressed before the full promise of AHFVs is realized. It has been known, however, that control design for these vehiclesposesaspecialchallengeduemainlytotheuniversallyacceptedconfigurationoftightlyintegrated airframe with the scramjet engine. This geometry, which uses the forebody for compression and the aftbody of the vehicle for exhaust expansion of exhaust gases, gives rise to significant coupling between aerodynamics, propulsion, and structure that, unlike conventional aircraft, cannot be neglected in control design. Furthermore, it is known that due to the slender geometry of these vehicles and the light weight requirement the full scale vehicle will possess low fundamental frequencies that will change appreciably with heating during the high speed flight, resulting in thermo-elastic coupling that needs to be accounted forincontroldesign. Becauseafullyoperationalvehiclefliesthroughawiderangeofspeedsandaltitudes in a short span of time, it is subject to fast changing atmospheric conditions. Unlike conventional aircraft there is little or no flight or engine test data available to model the flight or propulsion characteristics and guidethe design. Therefore, forthedesign ofthecontrolsystemsfor thefirstgenerationof these vehicles thecontrolengineermustcontendwithmodelswithsignificantparametricuncertainty. 77 Because of the uncertainties in AHFV dynamics, a practical control design should robustly achieve performanceandstabilityobjectives. Controlrelevantperformanceandrobustnessspecificationsaregiven effectively in terms of frequency dependent weights, and a control approach that explicitly accounts for these specifications is critical for designing a practical control system. Robust multivariable control tech- niques [65, 77], e.g., H ∞ and mixed- μ synthesis, provide quantifiable robust performance and stability guarantees. However, these approaches may fail to meet stringent control requirements when parametric uncertaintyislarge(see,forexample,[70]). Whenparametricuncertaintyisthemajorobstacleinobtaining a satisfactory control design, what is needed is an intelligent approach that is capable of reducing para- metric uncertainty online. Despite its remarkabletheoretical achievements, robustadaptive control theory [29]hasnotreachedalevelofmaturitytohandlecontrolapplicationsinvolvingstringentperformanceand robustness requirements. The complicated and intractable relationship between the plant parameters and thegainsofaH ∞ ormixed- μcompensatorisanimpedimentinthedesignoftraditionaladaptiveversions of robust multivariable controllers. Conversely, designing conventional adaptive control systems using the conventional model reference, pole placement, or optimal linear quadratic methods is challenging for applicationswithstringentfrequencydependentperformanceandrobustnessrequirements. The multiple model adaptive control (MMAC) architecture provides an attractive framework for com- biningtoolsfromadaptivecontrolwithrobustnon-adaptiveschemes. TheMMACarchitecturecomprises two levels of control: (1) a low-level system called the candidate controller set generates finely-tuned controls (with respect to a subset of the uncertainty space); (2) a high-level system called the supervisor determinesthecontrolu,generatedfromthecandidatecontrolsignals,byprocessingtheplantinput/output data. SomeofthepopularMMACschemesintheliteratureincludesupervisory(switching)adaptivecon- trol[1,25],adaptivecontrolwithmultiplemodels[49,50,32],unfalsifiedcontrol[60,71],robustmultiple modeladaptivecontrol[15,16,17](RMMAC),andadaptivemixingcontrol[36,35](AMC). Inthispaper,thecontroldesignisconductedusingtheAMCapproach,showninFigure4.1 1 . Eachof the candidate controllers is designed using mixed- μ synthesis tools [77, 74, 4] to achieve robust-stability 1 The architecture of Figure 4.1 is conceptual; the candidate controllers are not simply run in parallel. The multicontroller imple- mentationisdescribedinSection4.4.4.4.1. 78 G(s;p*) K 1 K N … Multicontroller E Robust Supervisor Mixer Robust Estimator Mixing Strategy M r disturbance and noise u u y Figure4.1: Theadaptivemixingcontrolarchitecture and -performance for a subset of the parametric uncertainty. What makes our design different from other MMAC approaches is that the supervisor of the AMC scheme determines which candidate controllers are allowed to participate in the control of the aircraft based on an online parameter estimate. At any given time, the true parameter vector p ∗ is assumed to take on the value of its online estimate ˆ p(t) and only thosecontrollersdesignedforthatrealizationofp ∗ areeligibletobeplacedintheclosedloop. Themixing strategy determines how multiple controllers are placed in the loop. The strategy considered in this paper is based on interpolating theQ filters of a Youla parameterization [73] in order to preserve stability and performancepropertiesofthecandidatecontrollersduringmixing. Incontrasttologic-basedswitchingschemes,issuesrelatedtoswitchingtransientsandcontrolchatter- ingareavoidedbecausethecontrollawiscontinuouswithrespecttotheobserveddata. UnlikeRMMAC, AMC does not rely on stochastic disturbance models. Thus, the AMC supervisor is likely to be more robust to uncertain disturbances. For the control of AHFVs, these factors are important because exciting the high-frequency unmodeled dynamics should be avoided and the disturbance may be poorly modeled ornon-stationary. The remainder of the paper is structured as follows: In Section 4.2, we describe the AMC approach. In Section 4.3 we describe the AHFV model and characterize its major uncertainties. In Section 4.4, we 79 applytheAMCapproachtodeveloparobustadaptivevelocityandaltitudetrackingcontroller. InSection 4.5, simulation results are presented for hypersonic cruise (Mach 10, 98,425 ft altitude). The paper is concludedwithabriefsummaryandfinalremarksinSection4.6. Notation: LetA be a given complex matrix and let Δ be an arbitrary complex matrix with a known block-diagonal structure and that satisfies the maximum singular value bound ¯ σ(Δ)≤ 1. The structured singular value [65, 14] ofA with respect to the structured Δ is denoted byμ Δ (A). For a vectorx,|x| is theEuclideannorm(x T x) 1/2 andthecorrespondinginducedmatrixnormofAisdenotedaskAk. Ifx(t) is a function of time, then theL p norm ofx is denoted askxk p and the truncatedL 2δ norm is defined as kx t k 2δ 4 = R t 0 e −δ(t−τ) x T (τ)x(τ)dτ 1 2 , whereδ ≥ 0 is a constant, provided that the integral exists. By kx t k 2 we mean thatkx t k 2δ withδ = 0, and we say thatx∈L 2e ifkx t k 2 exists and is finite for at≥ 0. Letx∈L 2e ,andconsidertheset S(μ S ) = ( x : Z t+T t x T (τ)x(τ)dτ ≤c 0 μ S T +c 1 , ∀t,T ≥ 0 ) for a given constantμ S , wherec 0 ,c 1 ≥ 0 are some finite constants and are independent ofμ S . We say that x is μ S -small in the mean square sense (m.s.s.) if x ∈ S(μ S ). Furthermore, consider the signal w : [0,∞)→R + andtheset S(w) = ( x : Z t+T t x T (τ)x(τ)dτ ≤c 0 Z t+T t w(τ)dτ +c 1 , ∀t,T ≥ 0 ) wherec 0 ,c 1 ≥ 0aresomefiniteconstants. Wesaythatxisw-smallinthem.s.s. if x∈S(w). LetX andY be subsets ofR m . ByδX we mean the boundary of the setX; byX−Y we mean the set-theoretic difference. An open ball of radius r centered around pointx 0 ∈ X is denoted asB r (x 0 ) = {x ∈ R m : |x−x 0 | < r}. The vectore i ∈ R k denotes thei th standard basis vector ofR k , i.e., thei th componentofe i isaone;allothercomponentsarezero. 80 LetH(s) andh(t) be the transfer function matrix and impulse response matrix, respectively, of some LTI system. IfH(s) is a proper transfer function and analytic inRe[s] ≥ −δ/2 for someδ ≥ 0, where Re[s] denotes the real part ofs, then theH ∞ system norm is given bykHk ∞ 4 = sup jω ¯ σ(H(jω)). Simi- larly,kHk Δ 4 =sup jω μ Δ (H(jω))istheΔnormofH(s)withrespecttothestructureduncertaintyΔ. The δ-shifted H 2 systemnormofH(s)isdefinedaskHk 2δ 4 = 1 √ 2π n R ∞ −∞ tr H H jω− δ 2 H jω− δ 2 2 dω o1 2 , where (·) H is the conjugate transpose. The inducedL ∞ system norm ofH is denoted bykHk ∞−gn . If y =H(s)uandu∈L ∞ thenkyk ∞ ≤kHk ∞−gn kuk ∞ . Considerthefeedbackinterconnectionofthelinear(generalized)plantP,withtheinput(d,u)andthe output(e,y),andthelinearcontrollerK givenby     e y     =P     d u     =     P 11 P 12 P 11 P 12         d u     , u =Ky (4.1) where u(t) ∈ R nu is the control signal; d(t) ∈ R n d is a signal comprising the exogenous inputs, such as noises, disturbances, and reference signals; e(t) ∈ R ne is a signal generated by the performance and robustness weights (filters); and y(t) ∈ R ny is the measured output signal used in feedback control. The linear fractional transformation (LFT) F l (P,K) denotes the transfer matrix between d and e, i.e. e = N de d and is given by N de = F l (P,K) = P 11 +P 12 K(I−P 22 ) −1 P 21 . Let ˆ Δ be an arbitrary n d ×n e transfer matrix with known block-diagonal structure, which is determined by the robustness and performance specifications, and that satisfies k ˆ Δk ∞ ≤ 1. We say that the controller K yields robust performanceif N de isstable, kN de k ˆ Δ < 1. (4.2) The proposed approach uses a multiple model/controller approach, and a superscript indexi denotes an associationwiththei th candidatemodel/controller. 81 4.2 AdaptiveMixingControl A concise introduction of the design principles and theoretical results of AMC are presented. A thorough descriptionofAMCanditstheoreticalbackgroundarepresentedin[36]. Letusconsidertheuncertainplant ˙ x P = A P (p ∗ )x P +B P (p ∗ )u+B w d w +B Δ d Δ , p ∗ ∈ Ω y = x P +d n e Δ = W Δ (s)(C Δ,1 x P +C Δ,2 d w +C Δ,3 u), e P =W P (s) x T P d T w u T T d Δ = Δ(s)e Δ (4.3) where x P (t) ∈ R n is the plant state vector; u(t) ∈ R nu is the control; d n (t) ∈ R n and d w (t) ∈ R nw are the exogenous, unmeasurable sensor noise and plant disturbance, respectively;y(t)∈R n is the noise corrupted sensor measurement;d Δ (t) ∈ R n d ande Δ (t) ∈ R n δ model the effect of complex uncertainty Δ(s)W Δ (s),whereΔ(s)isanarbitraryproperstabletransfermatrixsatisfyingkΔk ∞ ≤ 1andW Δ (s)isa knownproperstabletransfermatrix;e P (t)∈R n P istheperformancesignal,whereW P (s)isafrequency dependent weight chosen by the designer to specify performance, i.e., loop-shaping designs, disturbance rejection,controlpenalties,etc.;A P (p ∗ ),B P (p ∗ ),B w ,B Δ ,andC Δ,i (i=1,2,3)arematricesofappropriate dimensions, wherep ∗ ∈ Ω⊂R np is a vector comprising the unknown elements ofA P (p ∗ ) andB P (p ∗ ) 2 . Wemakethefollowingassumptions: Assumption1. Thepair(A P ,B P )isstabilizable. Assumption2. Thevectorp ∗ belongstoaknowncompact,convexsetΩ. Assumption3. The exogenous signalsd n (t) andd w (t) are bounded, i.e.,|d n (t)|≤d 0 and|d w (t)|≤d 1 forallt≥ 0andsomeconstantsd 0 ,d 1 ≥ 0. 2 We focus on parametric uncertainty ofA P (p ∗ ) andB P (p ∗ ) because the uncertainty ofp ∗ can be reduced online by estimation techniques by processing the measurable signalsu andy. The matricesBw,B Δ , andC Δ,i (i = 1,...,k), however, may also contain uncertain parameters. The proposed approach can be extended to this more general case by accounting for these additional realuncertaintiesintheoff-linecontrollerdesign. 82 Assumption4. Thereexistsaknownδ 0 > 0suchthatthecomplexuncertaintyΔ(s)W Δ (s)isanalyticin Re[s]≥−δ 0 /2. The plant (4.3) is in a form suitable for robust multivariable control design. If we rewrite (4.3) as the generalizedplantP(s)of(4.1),with e = [e T Δ e T P ] T , d = [d T w d T n ] T (4.4) then the control objective is to find a controller K(s) that yields robust performance requirement (4.2) for all p ∗ ∈ Ω, where ˆ Δ = diag(Δ,Δ P ) and Δ P is an arbitrary (n w +n)×n P complex matrix that satisfieskΔ P k ∞ ≤ 1. WhenΩislargethentheremaynotexistanLTIcontrollerthatsatisfiesthecontrol objective for allp ∗ ∈ Ω. Adaptive control is capable of overcoming parametric uncertainty by adjusting the controller online in response to real-time data. In this paper we adopt a multiple model approach: k candidate controller K 1 (s),...,K k (s) are developed off-line by partitioning the unknown-parameter spaceΩ intok parameter subsetsΩ 1 ,...,Ω k , such that for eachi∈I 4 ={1,...,k} the controllerK i (s) meets the controller objective for all p ∗ ∈ Ω i . The advantage of this approach is that the candidate controllers can be developed using powerful, well-established LTI tools. The problem of partitioning the modelspacehasbeeninvestigatedin[1]and[16]. Letusfurtherassumethefollowingofthepartitioning: Assumption5. Foranyp ∗ ∈ Ωthereexistsani∈I suchthatp ∗ ∈ Ω i . Assumption6. TheparametersubsetsΩ 1 ,...,Ω k satisfyanoverlappingproperty 3 : foralli∈I andany p ∗ ∈ (δΩ i −δΩ)thereexistconstantsr> 0andj6=isuchthatB r (p ∗ )⊂ Ω j . Theparameteroverlapset O 4 = [ i,j∈I,i6=j (Ω i ∩Ω j ) (4.5) 3 Astraightforwardmodificationofthemethodologyin[16]willgenerateanoverlappingpartition. 83 is the set of all pointsp ∗ ∈ Ω that belong to more than one parameter subset. This domain allows one to define a continuous mapping from Ω to the set of stabilizing controllers. Defining this mapping is explainedinwhatfollows. Aconceptualdiagram oftheAMCarchitecture isshowninFig. 4.1. Therobustsupervisor’sroleis to estimateastablemixofcandidatecontrollers,whichiscommunicatedtothemulticontroller u =K(s;β)y (4.6) via the mixing signal β(t) ∈ [0, 1] k . If the i-th component β i (t) of β(t) takes on a nonzero value thenK i (s) participates in the control; otherwise,K i (s) does not. At everyt, the mixer generatesβ(t) by processinganestimate ˆ p(t)ofp ∗ and,basedoncertaintyequivalence,thosecandidatecontrollerK i (s)that weredesignedforp ∗ = ˆ p(t)canparticipateincontrol,whilealsoenforcingtherestriction P k i=1 β i (t) = 1. Thus,forsomeconstantp∈ Ω,wedefinethesetofadmissiblemixingsignalsatpas B p 4 = ( β(t)∈ [0, 1] k : k X i=1 β i = 1, β i = 0ifp / ∈ Ω i (i = 1,...,k) ) . (4.7) Togenerateadmissiblemixingsignalsfortheestimate ˆ p(t),themixercanbeimplementedas β i = f i β (ˆ p) P j∈I f j β (ˆ p) , i = 1,...,k (4.8) wheref i β is a Lipschitz function of ˆ p that is positive if ˆ p∈ Ω i , and zero otherwise. Trapezoidal functions canbeusedtoconstructthemixer,forexample. Therobustparameterestimatorcanbedevelopedusingawide-classofalgorithms[29]. Forexample,a robustadaptivelawbasedonthegradientmethodcanbeconstructedbyconsideringthei-thstateequation ˙ x i = n X j=1 a ∗ ij x j + nu X j=1 b ∗ ij u j (4.9) 84 ofthenominalplant ˙ x P =A P (p ∗ )x P +B P (p ∗ )u,wherex i isthei-thcomponentof x P anda ∗ ij andb ∗ ij are theunknownij-thelementsof A P andB P ,respectively. Westacktheunknownparametersintothevector p ∗ i 4 = [a ∗ i1 ... a ∗ in b ∗ i1 ... b ∗ inu ] T . Therobustadaptivelawthatgeneratestheestimate ˆ p i (t)ofp ∗ i isthen givenby ˙ ˆ p i = Pr Ω {Γφ i }, Γ∈R np×np (4.10) where φ =F(s)[y T u T ] T , z i =sF(s)y i , i = z i − ˆ p T i φ m 2 (4.11) m 2 = 1+n d , ˙ n d =−δ 0 n d +|u| 2 +|y| 2 (4.12) y i isthei-thcomponentof y;Pr Ω istheprojectoroperatorthatrestricts ˆ p i toΩ;Γ> 0istheadaptivegain; δ 0 > 0isaconstant(cf. Assumption4);andF(s)isadesignfiltersatisfyingthefollowingassumption: Assumption 7. F(s) is a stable, minimum-phase filter of relative degree 1 or greater and is analytic in Re[s]≥−δ 0 /2. The multicontrollerK(s;β) is a dynamical system capable of generating a mix of candidate control laws. Conceptually, construction of the multicontroller can be viewed as interpolating the candidate con- trollers over the parameter overlap setO and is equivalent to the controller interpolation problem for gain scheduled controllers. The difference is that gain scheduling interpolates with respect to a measurable signal and AMC interpolates with respect to the estimated parameters. Thus, the AMC scheme is capable ofadjustingtounknownchangesintheplantdynamics. Remark7. There have been numerous ad hoc approaches to the controller interpolation problem. These includeinterpolatingcontrollerpoles,zeros,andgains[51];solutionsoftheRiccatiequationsforanH ∞ design [57]; state-space coefficient matrices of balanced controller realizations [33]; state and observer gains [28]; and candidate controller outputs [7, 34, 52]. Theoretically justified interpolations have also 85 been proposed. One approach is to mix the Q filters of a Youla parameterization [73], as done in the interpolation schemes of [67, 53]. Any of these interpolation methods, including the ad hoc approaches, maybeapplicableforconstructingthemulticontroller. WhentheparametersubsetsΩ 1 ,...,Ω k shareacommonpointp 0 ,thenthemulticontrollercanbecon- structedbythecontrollerinterpolationmethodof[53]. ConsidertheplantG(s) = (sI−A P (p 0 )) −1 B P (p 0 ), arbitrarystabilizingpositivefeedbackcontrollerK 0 (s),andcandidatecontrollersK 1 (s),...,K k (s)given bythecoprimefactorizations G =NM −1 = ˜ M −1 ˜ N, N,M, ˜ N, ˜ M ∈RH ∞ (4.13) K 0 =U 0 V −1 0 = ˜ V −1 0 ˜ U 0 , U 0 ,V 0 , ˜ U 0 , ˜ V 0 ∈RH ∞ (4.14) K i =U i (V i ) −1 = ( ˜ V i ) −1 ˜ U i , U i ,V i , ˜ U i , ˜ V i ∈RH ∞ , i = 1,...,k (4.15) thatsatisfythedoubleBezoutequation     I 0 0 I     =     ˜ V 0 − ˜ U 0 − ˜ N ˜ M         M U 0 N V 0     =     M U 0 N V 0         ˜ V 0 − ˜ U 0 − ˜ N ˜ M     (4.16)     I 0 0 I     =     ˜ V i − ˜ U i − ˜ N ˜ M         M U i N V i     =     M U i N V i         ˜ V i − ˜ U i − ˜ N ˜ M     , i = 1,...,k. (4.17) Considerthemulticontrollergivenby K(s;β) = (U 0 +MQ(β))(V 0 +NQ(β)) −1 (4.18) Q(β) = X β i Q i , Q i = ˜ V i K i −K 0 V 0 . (4.19) Lemma 9. Consider the generalized plant P(s) (4.1),(4.4) for the system (4.3) and the k controllers K 1 (s),...,K k (s) that are stabililizing for allp ∗ belonging to Ω 1 ,...Ω k , respectively. If the multicon- trollerK(s;β) is given by (4.18) thenK(s;e i ) = K i (s) (i = 1,...,k), wheree i ∈ R k is thei-th basis 86 vector. Furthermore,ifthecandidatecontrollerK i (s)foreachi = 1,...,k yieldsrobustperformancefor allp ∗ ∈ Ω i ,thenK(s;β ∗ )yieldsrobustperformancewithrespecttoP(s)forallp ∗ ∈ Ωandanyconstant β ∗ ∈B p ∗. The proof for the first result of Lemma 9 can be found in Theorem 3.5 of [52], and the proof for the second result follows Lemma 3 of [35]. The coefficients ofK(s;β) are continuous inβ. This property andLemma9playcrucialrolesinthestabilityanalysisofAMCschemes. We now summarize the stability results of the AMC scheme. Let the adaptive law’s filters (4.11) and multicontroller(4.18)begivenstabilizableanddetectablerealizationswithstatesx E andx C ,respectively. Thestatesoftheclosed-loopAMCschemeare x 4 = x T P x T C x T E T , ˆ p 4 = ˆ p T 1 ... ˆ p T k T ,andn d . Theorem 10. Consider the unknown plant given by (4.3) under Assumptions 1-6. The AMC scheme comprising the mixer (4.8), the robust online parameter estimators (4.10) with the design Filter F(s) satisfyingAssumption7,andthemulticontroller(4.18)guaranteesthefollowing: 1. IfΔ(s),d n ,d w = 0,thenx→ 0ast→∞. 2. Thereexistsaconstantδ ∗ > 0suchthatif μ 2 S 4 =c(Δ 2 2 +Δ 2 3 )<δ ∗ where c is a finite constant, Δ 2 4 = kFB Δ ΔW Δ C Δ1 k 2δ0 , and Δ 3 4 = kFB Δ ΔW Δ C Δ3 k 2δ0 , then x,ˆ p,n d ∈ L ∞ and R t 0 |x P (τ)| 2 dτ ≤ c 0 (Δ 2 2 + Δ 2 3 + d 2 0 + d 2 1 )t + c 1 for some finite constants c 0 ,c 1 ≥ 0. Proof of Theorem 10 is given in the Appendix. Result 2 of Theorem 10 implies that the plant’s output errorissmallinthemeansquaresenseandisoftheorderofthemodelingerror. 87 4.3 AirbreathingHypersonicFlightVehicle(AHFV)Model It has been mentioned in Section 4.1 that two recent models developed for the longitudinal dynamics of genericAHFVsarepresentedin[6,55]and[43,10],respectively. Betweenthesetwomodels,weconsider the CSULA-GHV model in [43, 10]. CSULA-GHV is a two-dimensional CFD-based model of a full- scale scramjet powered generic AHFV that has been developed at the Multidisciplinary Flight Dynamics and Control Laboratory (MFDCLab), at California State University, Los Angeles [10]. This vehicle’s integratedairframe-propulsionsystemconfiguration,whichwaschosentobegenericwhilepossessingthe unique aero-propulsion coupling characteristics of the X-30 and X-43 vehicles, is shown in Figure 4.2. Thecompleteaero-propulsiondata,generatedoverawideflightenvelopeinthehypersoniccruiseregime, for the vehicle can be found in [10]. The equations of motion, assuming a round non-rotating earth and momentarilyneglectingstructuraldynamicseffects,aregivenby ˙ V = Tcos(α)−D m − μ 0 sin(γ) r 2 , ˙ γ = L+Tsin(α) mV − (μ 0 −V 2 r)cos(γ) Vr 2 (4.20) ˙ h =Vsin(γ), ˙ α =q− ˙ γ, ˙ q = M yy I yy (4.21) L = 1 2 ρV 2 SC L (M,α,δ T ,δ e ), D = 1 2 ρV 2 SC D (M,α,δ T ,δ e ) (4.22) T = 1 2 ρV 2 SC T (M,α,δ T ,δ e ), M yy = 1 2 ρV 2 S¯ cC M (M,α,δ T ,δ e ) (4.23) whereV (ft/s) is the vehicle’s velocity;γ (rad) is the flight-path angle; h (ft) is the altitude;α (rad) is the angle of attack; q (rad/s) is the pitch rate; m (slug) is the vehicle’s mass; I yy (slug·ft 2 ) is the vehicle’s momentofinertia;μ 0 (ft 3 ·slug −1 ·s −2 )istheearth’sgravitationalconstant;andr (ft)isthevehicle’sradial distancetotheearth’scenter. ThenondimensionalcoefficientsofliftC L ,dragC D ,thrustC T ,andpitching momentC M are each a function of Mach numberM, angle of attackα, and the control input comprising the throttle setting δ T (fuel equivalence ratio) and elevon deflection angle δ e (rad), capturing the aero- propulsion coupling and the pitch-up moment caused by the underslung engine position. The interested reader can find the analytical expressions, derived by curve fitting the aero-propulsion data, for C L ,C D , 88 C T ,andC M in[9]. Inthisstudywechosethevehicle’scenterofgravity(c.g.) locationas50%itslength, resulting in a statically unstable system. This selection approximately corresponds to the c.g. location of theX-43a. Dimension Reference Stations Schematic Figure4.2: Vehiclegeometry A five state linearized model of the longitudinal rigid-body dynamics (4.20)–(4.23) trimmed at nom- inal hypersonic cruise flight (M = 10 at 98,425 ft altitude) is used to carry out control studies. After numericallylinearizing(4.20)–(4.23)arounditsstraight-and-levelflighttrimcondition x P0 = 9929 0 98425 0.2 0 T , u 0 = 0.063 −7.67 T (4.24) thenominalrigid-bodydynamicsaregivenby ˙ x P =A P x P +B P u+B w d w (4.25) 89 where x P = ΔV Δγ Δh Δα Δq T , u = Δδ T Δδ e T (4.26) A P =                 −559.58×10 −6 −31.5135 4.7783×10 −6 −73.4472 0 446.556×10 −9 1.3509×10 −9 −110.5899×10 −9 112.7357×10 −3 0 0 9.9293×10 3 0 0 0 −446.556×10 −9 −1.3509×10 −9 110.5899×10 −9 −112.7357×10 −3 1 16.068×10 −6 0 −136.4571×10 −9 6.303 0                 (4.27) B P =                 94.1435 −14.2580 17.0861×10 −3 −10.0777×10 −3 0 0 −17.0861×10 −3 10.0777×10 −3 −5.3916 5.1058                 , B w =                 −5.5958×10 −4 −100.25 3.5127×10 −7 0.1123 0 0 −3.5127×10 −7 −0.1123 1.6068×10 −5 6.303                 (4.28) where x P and u are interpreted as perturbations from trim (x P0 ,u 0 ), the vector d w = [d x d z ] T com- prises the longitudinal and vertical wind gustsd x andd z , respectively, which are modeled by the Dryden turbulencemodel[31,66] d x =F x (s)w x , F x (s) = p 2V 0 σ 2 x /L x 1 s+V 0 /L x , σ x = 10.8, L x = 65574 (4.29) d z =F z (s)w z , F z (s) = p 3V 0 σ 2 z /L z s+V 0 /( √ 3L z ) (s+V 0 /L z ) 2 , σ z = 6.88, L z = 26229 (4.30) and w x , w z are unit-variance, zero-mean white stochastic processes. For compact notation, we use the termsW w = diag(F x , F z )andw w = [w x w z ] T . Weassumethatd x andd z areboundedfunctions: Assumption8. |d x (t)|≤d 1 and|d z (t)|≤d 2 forallt≥ 0andsomeconstantsd 1 ,d 2 ≥ 0. 90 For the purposes of robust controller design and analysis, several sources of uncertainties, motivated by the challenges associated with the control of this class of flight vehicles, are incorporated into the nominalmodel(4.25). Weadoptanuncertaintymodelsimilarto[8]. Thefirstsourceofuncertaintyarises fromthevehicle’sstructuraldynamics. Elasticmodesexcitedbymaneuveringandhighfrequencycontrol actioncouldaffectaero-propulsionperformanceofthevehicleduetothetightlyintegratedairframe-engine configuration. Inthispaperwefocusontheinteractionbetweenfuselagedeformationandaero-propulsion performance. The premise here is that structural dynamics are excited by the force produced by control surface(elevon)deflectionandtheresultingstructuraldeformationmanifestsitselfmainlyaschangesinthe angle of attack and control surface deflection, thereby interacting with the aero-propulsion dynamics [9], possibly resulting in instability from servo interactions. Excitation from other aero-propulsion forces and inertialeffectsarenotmodeled. Nonetheless,theproposedmodeldoescaptureimportantaero-propulsion andstructuralinteractionsandisusefulforcontroldesignandanalysisproposes. The first bending mode is considered. The modal frequency ω sd = 20.3296 rad/s and mode shape functionφ(x)wereobtainedviaaNASTRANfiniteelementmodel. Verticaldeflectionz(x,t)(ft)isgiven by z(x,t) = φ(x)η(t), where, as in [8], the generalized elastic coordinate η(t) (ft) is governed by the secondorderdynamics η = 1 s 2 +2ζ sd ω sd s+ω 2 sd P M sd (4.31) where the damping ratio ζ sd = 0.01 was chosen such that the elastic mode is lightly damped; P = φ(x h )F N,δe (lbf) is the generalized modal force produced by the normal forceF N,δe at the elevon hinge x h astheresultofelevondeflectionandisobtainedfromtheaero-propulsiondata[9];and M sd = 1.2531 (slugs) is the modal mass. Angular deformation of the vehicle body at any pointx p along the vehicle is approximatedby θ(t,x p ) =φ 0 (x p )η(t) (rad). (4.32) 91 Elasticdeflectionsatthenoseofthevehicleperturbsthebowshock,whichsignificantlyimpactsaero- propulsion performance. For the purpose of approximating this effect, the elastic deflection at the nose locationx n istreatedasalocalchangeinangleofattack,i.e.,thequantity α+φ 0 (x n )η, φ 0 (x n ) = 0.0042 (4.33) replaces α in the calculation of the aero-propulsion forces and pitching moment. Similarly, the elastic deflectionattheelevonhingeperturbstheangleofelevondeflection;therefore,thequantity δ e −φ 0 (x h )η, φ 0 (x h ) =−0.005 (4.34) replacesδ e inthecalculationoftheaero-propulsionforcesandpitchingmoment. Theminussignin(4.34) is because of the sign convention that δ e is positive for trailing edge up. Taking into account structural perturbation(4.34),thelinearapproximationof(4.31)isgivenby η =T sd (s)δ e , T sd (s) = k N s 2 +2ζ sd ω sd s+ω 2 sd +k N φ 0 (x h ) . (4.35) where k N = −1802 is the calculated by evaluating ∂P/M sd ∂δe numerically from the CFD data. Because of the high level of uncertainty associated with the structural dynamics, the admissible class of structural dynamicsismodeledas η = Δ 2 (s)W sd (s)δ e , W sd (s) = k u (s+2) s 2 +2ζ u ω u s+ω 2 u (4.36) whereΔ 2 (s) is anarbitrary properstable transferfunction satisfyingkΔ 2 k ∞ ≤ 1; and the coverfunction W sd is specified by the parametersk u = 2150, ζ u = 0.25, andω u = 20 rad/s such that|W sd (jω)| ≥ |T sd (jω)| for allω∈R. Airframe heating has the effect of reducing the natural frequency of the bending 92 mode. Accounting for this thermal effect, we assume that Δ 2 (s) is analytic inRe[s] ≥ −0.6ζ sd ω sd , al- lowingmodalfrequenciesupto60%oftheNASTRANestimatedvalue. Afterincorporatingthestructural dynamicsintotherigid-bodydynamics,theequationsofmotionbecome ˙ x P =A P x P +B P u+B w d w +B sd η (4.37) η = Δ 2 (s)W sd (s)δ e (4.38) whereB sd =a 4 φ 0 (x n )+b 2 φ 0 (x h ),a 4 isthefourthcolumnofA P ,andb 2 isthesecondcolumnofB P . The second source of uncertainty is in the system parameters. Leta ij andb ij denote the (i,j) com- ponents ofA P andB P , respectively. Because of the increased sensitivity to variations in angle of attack resulting from the highly integrated airframe-engine configuration, uncertainty in the stability derivative C Mα is considered, which corresponds to the elementa 54 ofA P . Additionally, uncertainty in control ef- fectiveness is considered. Variations in throttle and elevon authority are taken into account in the form of resultant independent real uncertainties in the control derivativesC T d T andC M de , respectively, which correspondtouncertaintiesintheelementsb 11 andb 52 ofB P ,respectively. Considering the uncertainties in the parametersa 54 , b 11 , andb 52 , the complete uncertainty model is givenby ˙ x P =A P x P +B P u+B w d w +B Δ d Δ (4.39) d Δ = Δe Δ (4.40) e Δ =W Δ C Δ x P +D Δ d T Δ d T w u T T (4.41) 93 where B Δ =                 0 0 0 0 a 54 B sd b 11 0 0 0 0 0 0 0 0 b 52                 , C Δ =             0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0             (4.42) D Δ = D Δ,1 . . . D Δ,2 . . . D Δ,3 =             0 φ 0 (x n ) 0 0 . . . 0 1/V 0 . . . 0 0 0 0 0 0 . . . 0 0 . . . 0 1 0 φ 0 (x h ) 0 0 . . . 0 0 . . . 1 0 0 0 0 0 . . . 0 0 . . . 0 1             (4.43) Δ = diag(δ 1 ,Δ 2 ,δ 3 ,δ 4 ), W Δ = diag(W cmα ,W sd ,W ce ). (4.44) δ 1 ,δ 3 ,δ 4 in (4.44) are real perturbations with magnitude less than one, i.e.,kΔk ∞ ≤ 1; andW cmα and W ce reflect the uncertainty levels for the parametersa 54 ,b 11 ,b 52 . The true valuesa ∗ 54 ,b ∗ 11 , andb ∗ 52 ofa 54 , b 11 ,andb 52 ,respectively,aregivenby a ∗ 54 = (1+W cmα δ 1 )a 54 (4.45)     b ∗ 11 b ∗ 52     =     I 2×2 +W ce     δ 3 δ 4             b 11 b 52     . (4.46) For notational convenience, we define the vector p ∗ 4 = [a ∗ 54 a ∗ 52 b ∗ 11 ] T . In this work, the values of W cmα ,W ce arechosenas W cmα = 0.5, W ce = 0.5I 2×2 (4.47) toreflect50%uncertaintyineachuncertainparameter. 94 For simulation purposes, first order actuator models are used with bandwidths 30 rad/s and 100 rad/s forδ e andδ T ,respectively,i.e., u =G act (s)u c , G act = diag 100 s+100 , 30 s+30 . (4.48) whereu c isthecommandedcontrol. Itshouldbenotedthattheaboveuncertaintymodeldiffersfrom[8]intworegards. First,theproposed uncertainty model (4.39) captures elevon perturbation (4.34) caused by the structural dynamics. Second, inthiswork,largerparametricuncertaintiesinC Mα ,C T δT ,andC M δe areconsidered. 4.4 AdaptiveMixingControlDesignofAHFVs Thecontrolobjectiveisrobustvelocityandaltitudetracking. InSection4.4.4.4.1,weformalizethecontrol objective in terms of aμ-synthesis optimization problem in which the bandwidth of velocity and altitude trackingisemphasized. BecausetheAHFV’suncertaintieslimitachievableperformance,weaimtodesign anadaptiveschemethatincreasesthepotentialperformancebyusingonlinelearningtoreduceparametric uncertainty. Previousadaptivecontrolapproachesforhypersonicvehiclesincludelinearquadratic[27,38] and nonlinear [72] approaches, but none of these approaches quantifies robustness and performance spec- ifications in relevant engineering terms, such as the frequency weights commonly used in robust multi- variablecontroldesignandanalysis. Inconsiderationofthelargeparametricuncertaintiesandunmodeled dynamicsoftheAHFVmodel,aswellasthedesiretospecifyrobust-stabilityand-performanceobjectives, theAMCapproachischosenforcontrollerdesign. 95 4.4.1 Multicontroller The generalcontrol configuration, shown inFigure 4.3(a), is constructed to formulate a standard mixed- μ synthesisproblemfortheplant(4.39)–(4.41). ThegeneralizedplantP ofFigure4.3(a)isdescribedby         e Δ e 0 y         =P(s)         d Δ d 0 u c         , P(s) =         P 11 (s) P 12 (s) P 13 (s) P 21 (s) P 22 (s) P 23 (s) P 31 (s) P 32 (s) P 33 (s)         (4.49) d Δ = Δe Δ , u c =K(s)y (4.50) whered 0 = ˆ r T c w T w w T n T is the exogenous input vector; e 0 = e T P e T u T is the error vector; and the transfermatricesP 11 (s),...,P 33 (s),aswellasthesignals ˆ r c ,e P ,ande u aretobedefinedbyaugmenting the AHFV model with performance and robustness weights. The structure of P(s) is shown in Figure 4.3(b)anditsdescriptionfollows. TosimplifyFigure4.3(b),considertheaugmentedsystem G a (s) =C a (sI−A P ) −1 B a +D a (4.51) C a =     C Δ D −1 e     , B a = [B Δ B w B P D u ], D a =     D Δ 0 5×8     (4.52) wherethescalingconstants D −1 e = diag(0.1, 180/π, 0.1, 180/π, 180/π), D u = diag(0.75, 1800/π) (4.53) 96 P K u c y Δ d Δ e Δ d 0 e 0 (a) Thegeneralrobustcontrolcon- figuration W n W u W P (c P ) R W Δ W w + − + + w n w w d Δ e u e P e Δ d w G a u c u n x P r c y (b) ThegeneralizedplantP Figure4.3: Mixed- μproblemformulation areusedtoimprovethenumericalperformanceofthecontrollersynthesisalgorithm. Theexogenousinput ˆ r c isanormalizedversionofthereferencecommand r c = [V c γ c h c α c q c ] T (4.54) such that each component of ˆ r has unit magnitude or less. Although our objective is velocity and altitude tracking, the real commandsV c andh c are augmented with the fictitious commandsγ c ,α c , andq c , which aresettozeroduringruntime. Thereferenceweightischosenas R = diag(1, 0.018/π, 1, 0.018/π, 0.018/π) (4.55) toemphasizetrackingperformanceinthevelocityandaltitudechannels. Smallnonzeroweightsonthefic- titiousreferencecomponentsγ c ,α c ,andq c specifysomedisturbancerejectionintheγ,α,andq channels withoutsignificantlyimpactingperformanceintherealreferencechannelsV andh. The weightW w (s) = diag(F x (s),F z (s)), whereF x andF z are given by (4.29)-(4.30), characterizes atmosphericdisturbancesandcommunicatesdisturbancerejectionperformancetothecontrollersynthesis 97 algorithm. The weightW Δ = diag(W cmα ,W sd ,W ce ) characterizes model uncertainty and was derived inSection4.3. Thecontrollerweight W u (s) = diag(W δT , W δe ) (4.56) W δT = 1 2 s+0.001 s+100 , W δe = 1 2 s+0.0003 s+30 (4.57) waschosentolimitinputmagnitudesathighfrequenciesandcanbeadjustedtoavoidactuatorsaturation. The weightW n = 10 −4 characterizes sensor noise and, because this choice does not reflect actual noise characteristics,canbereplacedwithanaccuratemodelwhenavailable. Theperformanceweight W P (s,c P ) = diag(W V , W γ , W h , W α , W q ) (4.58) W V = s/M W +ω V s+ω V A W , W h = s/M W +ω h s+ω h A W (4.59) W i = 0.02s (s+0.055)(s+0.06) , i =γ,α,q (4.60) A W = 10 −4 , M W = 1.5, ω V = 0.0625c P , ω h = 0.5c P (4.61) emphasizes tracking and disturbance rejection performance, with a bandwidth ofω V = 0.625c P rad/s in thevelocitychannelandω h = 0.5c P rad/sinthealtitudechannel,wherec P isadesignconstantthatgives thedesignerthecapabilitytoadjustthebandwidthoftheperformanceweights. TheweightsW γ ,W α ,and W q werechosenasbandpassfilterstopenalizepeaksaroundcrossover. 98 Insummaryofthegeneralcontrolproblemformulation,thecomponentsofP(s)aregivenby P 11 =W Δ (C Δ HB Δ +D Δ1 ) P 12 = [0 W Δ (C Δ HB w W w +D Δ2 ) 0] P 13 =W Δ (C Δ HB P +D Δ3 )D u G act P 21 =     −D −1 e HB Δ 0     P 22 =     W P R −D −1 e HB w W w 0 0 0 0     P 23 =     −D −1 e HB P D u G act W u     P 31 =−D −1 e HB Δ P 32 = R −D −1 e HB w W w W n P 33 =−D −1 e HB P D u G act (4.62) whereH(s) = (sI−A P ) −1 . UsingtheD,G-Kiterationalgorithm[74,4],acontroller K(s)issynthesized to yield the robust performance condition (4.2), where ˆ Δ = diag(Δ, Δ P ) and Δ P is a 12×7 complex stabletransferfunctionsatisfyingkΔ P k ∞ ≤ 1. Iftheperformanceobjectiveischosentoodemanding(c p is too large, for example), a controllerK(s) that achieves robust performance will not exist; uncertainty limitsachievableperformance. Whenconsideringthefullrangeparametricuncertainty,i.e., ∀p ∗ ∈ Ω 4 = [a ∗ 54 b ∗ 52 b ∗ 11 ] T ∈R 3 : 0.5a 54 ≤a ∗ 54 ≤ 1.5a 54 , 0.5b 52 ≤b ∗ 52 ≤ 1.5b 52 , 0.5b 11 ≤b ∗ 11 ≤ 1.5b 11 } (4.63) the choice ofc P = 1 yields a global non-adaptive robust controller (GNARC) K GNARC that achieves an upperbound ¯ μ = 1.07onkNk ˆ Δ . ThecontrollerK GNARC representsthe“best”non-adaptivedesignand isusedforcomparisonwiththeadaptivescheme. TheAMCapproachispursuedinanattempttoconstructacontrolschemethatiscapableofsatisfying the more stringent robust performance requirement of c P > 1. The first step of the AMC design is to construct the candidate controller setK 1 ,...,K k , where each controller is designed for a subset of the parameter space Ω. Since each candidate controller is designed with less uncertainty than the complete model,c P iscapableofattainingavaluegreaterthanunitywhileachievingtheupperbound ¯ μ≈ 1. Table 99 Table4.1: Summaryofrobustcontroldesign a ∗ 54 b ∗ 52 b ∗ 11 LB UB LB UB LB UB c P ¯ μ K 1 ,Ω 1 3.15 6.78 2.55 5.49 47.07 101.27 4.00 1.10 K 2 ,Ω 2 3.15 6.78 2.55 5.49 91.63 141.22 4.00 0.98 K 3 ,Ω 3 3.15 6.78 4.97 7.66 47.07 101.27 4.00 0.98 K 4 ,Ω 4 3.15 6.78 4.97 7.66 91.63 141.22 4.00 0.92 K 5 ,Ω 5 6.13 9.45 2.55 5.49 47.07 101.27 4.00 1.25 K 6 ,Ω 6 6.13 9.45 2.55 5.49 91.63 141.22 4.00 0.97 K 7 ,Ω 7 6.13 9.45 4.97 7.66 47.07 101.27 4.00 0.98 K 8 ,Ω 8 6.13 9.45 4.97 7.66 91.63 141.22 4.00 0.90 K GNARC 3.15 9.45 2.55 7.66 47.07 141.22 1.00 1.07 4.1summarizesthecandidatecontrollerdesign. Theparameterspaceforeachofthethreeparametersa ∗ 54 , b ∗ 11 , andb ∗ 52 is partitioned into two intervals so that the maximum value ofc P is same for each parameter subset. By splitting each of the three intervals into two subintervals, there are 2 3 = 8 parameter subsets. The lower bound (LB) and upper bound (UB) of each parameter subset are given in columns 2-4. For i = 1,...,8,theparametersubsetΩ i isthesetofvaluesofp ∗ allowedbymodeli;forexample,following Table4.1,wehavethat Ω 1 = [a ∗ 54 b ∗ 52 b ∗ 11 ] T ∈R 3 : 3.15≤a ∗ 54 ≤ 6.78, 2.55≤b ∗ 52 ≤ 5.49, 47.07≤b ∗ 11 ≤ 101.27 . (4.64) Thevaluec P = 4waschosenforallcandidatemodels. Theupperboundsonμachievedbythecontrollers synthesized by the D,G-K iteration algorithm are given in column 6. Although K 1 , K 5 , andK GNARC violatetherobustperformancecondition(4.2),weconsiderthesecandidatecontrollersacceptablebecause theviolationsareslightandrobust-stabilityisachieved. The multicontroller is constructed using theQ-mixing approach (4.18) presented in Section 4.2. To avoidunboundedout-of-the-loopsignals,themulticontroller K(s;β)isrealizedwiththeinternalstructure showninFigure4.4. 100 y u c - + + + s s 1 s 2 s 4 s 3 r Figure4.4: MulticontrollerimplementedbyQ-mixing 4.4.2 RobustAdaptiveSupervisor We now design the mixerM subsystem to ensure that ˆ p7→β(ˆ p)∈B ˆ p and that this assignment is contin- uousin ˆ p. Tosatisfytheserequirementswedefinethemixingsignalas β i (ˆ p) = f i (ˆ p) P 8 i=1 f i (ˆ p) , f i (ˆ p) = 3 Y j=1 ˜ β i j (ˆ p), i = 1,...,8 (4.65) ˜ β i 1 (ˆ p 1 ) =        1− ˜ u r (ˆ p 1 ;6.13,6.78), i = 1,...,4 ˜ u r (ˆ p 1 ;6.13,6.78), i = 5,...,8 (4.66) ˜ β i 2 (ˆ p 2 ) =        1− ˜ u r (ˆ p 2 ;4.97,5.49), i = 1,2,5,6 ˜ u r (ˆ p 2 ;4.97,5.49), i = 3,4,7,8 (4.67) ˜ β i 3 (ˆ p 3 ) =        1− ˜ u r (ˆ p 3 ;91.63,101.27), i = 1,3,5,7 ˜ u r (ˆ p 3 ;91.63,101.27), i = 2,4,6,8 (4.68) ˜ u r (τ;t 1 ,t 2 ) = r(τ−t 1 ) t 2 −t 1 − r(τ−t 2 ) t 2 −t 1 , r(τ) =        τ, ifτ ≥ 0 0, otherwise . (4.69) We combine the multicontroller K(s;β) and mixer (4.65) with an online parameter estimator, also called the adaptive law. Because of the presence of uncertainty Δ, disturbancew w , and sensor noisew n , weusearobustadaptivelaw[30,29]toregainasmuchoftherobustnessoftheknowncaseaspossible. 101 Byabsorbingp ∗ backintothesystemmatricesA P andB P ,werewritetheplantequation(4.39)as ˙ x P =A P (p ∗ )x P +B P (p ∗ )u+B w d w +B sd T ∗ sd (s)δ e | {z } ˜ η , u =G act (s)u c . (4.70) where T ∗ sd (s) 4 = Δ(s)W sd (s) is the true structural dynamics and ˜ η is the modeling error term. Recall that we assume that the real part of the poles of T ∗ sd attain a maximum value of 60% of−ζ sd ω sd . The developmentoftherobustadaptivelawproceedsbyconsideringthepitch-rate q stateequation ˙ q = X j∈{1,2,3,5} a 5j x j +a ∗ 54 α+b 51 δ T +b ∗ 52 δ e + ˜ η 5 (4.71) ˜ η 5 = [b w,51 b w,52 ]d w +b sd,5 T ∗ sd δ e (4.72) where ˜ η 5 isthefifthcomponentof ˜ η. Alinearparametricmodel(LPM)of(4.71)isgivenby z 12 =p ∗T 12 φ 12 +η 12 , p ∗ 12 = [a ∗ 54 b ∗ 52 ] T (4.73) where z 12 = s Λ(s) q− 1 Λ(s) ( X j∈{1,2,3,5} a 5j x j +b 51 δ T ), φ 12 = 1 Λ(s)     α δ e     (4.74) η 12 = 1 Λ(s) ˜ η 5 , Λ(s) = (s+100) 2 . (4.75) Similarlyforb ∗ 11 ,wedeveloptheLPM z 3 =p ∗ 3 φ 3 +η 3 , p ∗ 3 =b ∗ 11 (4.76) 102 where z 3 = s Λ(s) V − 1 Λ(s) ( 5 X j=1 a 1j x j +b 12 δ e ), φ 3 = 1 Λ(s) δ T (4.77) η 3 = 1 Λ(s) ˜ η 1 , ˜ η 1 = [b w,11 b w,12 ]d w +b sd,1 T ∗ sd δ e (4.78) byconsideringtheV stateequation. A wide-class of robust adaptive laws can be developed using the LPMs (4.73),(4.76). In this study we choosethegradientmethodwithprojection. TherobustadaptivelawusingtheLPM(4.73)isgivenby ˙ ˆ p 12 = Pr ¯ Ω12 {Γ 12 p1 φ 12 }, Γ 12 = diag(10 9 ,10 11 ) (4.79) ˙ n d =−δ 0 n d +δ 2 e , δ 0 = 0.4 (4.80) 12 = z 12 − ˆ p T 12 φ 12 m 2 (4.81) m 2 = 1+n d (4.82) where ¯ Ω 12 ={(a 54 , b 52 )∈R 2 : 3.15≤a 54 ≤ 9.45, 2.55≤b 52 ≤ 7.66} andδ 0 > 0 was chosen such thatT ∗ sd isanalyticinRe[s]≥−δ 0 /2. TherobustadaptivelawusingtheLPM(4.76)isgivenby ˙ ˆ p 3 = Pr ¯ Ω3 {Γ 3 p2 φ 3 }, Γ 3 = 10 9 (4.83) 3 = z 3 − ˆ p T 3 φ 3 m 2 (4.84) where ¯ Ω 3 ={b 11 ∈R : 47.07≤b 11 ≤ 141.22}. The online estimate ˆ p = ˆ p T 12 ˆ p 3 T is combined with themixerandmulticontrollertogeneratetheadaptivecontrollaw u c =K(β(ˆ p))y. (4.85) 103 It should be emphasized that the focus of this section was on illustrating an adaptive control approach that incorporates robustness and performance objectives. When more accurate knowledge of the robust- ness and performance requirements is obtained, the AMC design can be modified straightforwardly for a more practical control design. Moreover, this initial study focuses on robust tracking performance only during the nominal hypersonic cruise phase of the mission. Although hypersonic cruise may constitute a significant portion of the complete mission, a practical design will need to address all phases of the mission. Thisisthefocusoffutureresearch. 0 100 200 300 400 −20 0 20 40 60 80 100 120 t, sec Velocity, ft/s (a) Velocity trajectories: Velocity ΔV(t) (blue) andreferenceVc(t)(green) 0 100 200 300 400 −20 0 20 40 60 80 100 120 t, sec Altitude, ft (b) Altitudetrajectories: AltitudeΔh(t)(blue)and referencehc(t)(green) 0 100 200 300 400 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 t, sec Throttle setting (c) ControlinputΔδ T (t) 0 100 200 300 400 −1.5 −1 −0.5 0 0.5 1 1.5 t, sec Elevon deflection, deg (d) ControlinputΔδe(t) Figure4.5: Simulationresultsforthecasewithrealandcomplexperturbations. 104 4.5 SimulationResults Simulation results are presented to demonstrate the potential performance improvements of the AMC scheme when compared to a non-adaptive robust control design, i.e., mixed- μ synthesis. The AHFV model(4.39)-(4.40)isused,withrealizeduncertainty δ 1 = 0, Δ 2 = 1, δ 3 ,δ 4 = 1 (4.86) andatmosphericturbulencemodel(4.29),(4.30). ForthischoiceofuncertaintyΔ,theunknownparameter vectorp ∗ belongstobothparametersubsetsΩ 4 andΩ 8 . Satisfactoryperformanceisachievediftherobust adaptive supervisor identifies thatp ∗ ∈ Ω 4 ∩ Ω 8 and, thus, mixes the candidate controllersK 4 andK 8 into the loop, while removing all remaining candidate controllers. It should be emphasized that ratio of the mix of K 4 and K 8 is not critical; any mix of these two controllers using the Q-mixing strategy of Section 4.4.4.4.1 will result in satisfactory performance. The velocity and altitude commandsV c (t) and h c (t),respectively,aregeneratedbyfilteringthepulse r c (t) =        100, 0≤t< 200 0, else (4.87) asdescribedby     V c (t) h c (t)     =     F V (s) 0 0 F h (s)     r c (t) (4.88) F V = ω V s+ω V , ω V = 0.05 (4.89) W h = ω 2 h s 2 +2ζ h ω h s+ω 2 h , ζ h = 0.8, ω h = 0.0468. (4.90) Figure 4.5 illustrates the tracking performance of the AMC scheme (4.65),(4.79),(4.83) and (4.85). The AMC scheme achieves tight velocity and altitude tracking, as evident in Figures 4.5(a) and 4.5(b). The 105 0 100 200 300 400 −15 −10 −5 0 5 10 15 t, sec Velocity Tracking Error, ft/s (a) Velocity tracking error: AMC (blue) and GNARC(green) 0 100 200 300 400 −10 −5 0 5 10 t, sec Altitude Tracking Error, ft (b) Altitude tracking error: AMC (blue) and GNARC(green) Figure4.6: Trackingerrors. controlsignalsareshowninFigures4.5(c)and4.5(d). Also,theAMCschemewassimulatedside-by-side with the non-adaptive mixed- μ controllerK GNARC of Section 4.4.4.4.1 to demonstrate the potential for increasedperformanceprovidedbyadaptation. RecallthatthecontrollerK GNARC obtainedbyincreasing the performance design variable c P until μ ≈ 1 represents the best non-adaptive solution. Moreover, the candidate controllers K 1 ,...,K 8 of the AMC scheme and the non-adaptive controller K GNARC were synthesized using the same mixed- μ problem formulation to maintain a fair comparison. Figure 4.6 shows the tracking errors for the AMC and GNARC schemes. Compared to the GNARC scheme, the AMCschemedemonstratesimprovedperformanceintermsofsmallertrackingerrorsinbothvelocityand altitude channels. Figure 4.7 shows the time history of the mixing signalβ(t). Only the first 10 seconds of simulation are shown to depict the transient behavior and to highlight the speed of convergence to the correct controllersK 4 andK 8 . Fort ≥ 10, the mixing signal remained approximately constant. Figure 4.7 demonstrates that the AMC scheme maintains a high-level of performance despite that the robust adaptivesupervisorneverconvergestoonecontroller;rather,controllersK 4 andK 8 aremixedintheloop simultaneously. This behavior is fundamentally different from the MMAC schemes of [46, 16], which were designed to either switch or converge to only one controller. The advantage of the mixing technique in this example is that intermittent switching between controllersK 4 andK 8 , which may be caused by modelmismatchandnoise,isavoided;inturn,switchingtransientsandcontrolchatterarealsoavoided. 106 0 5 10 0 0.5 1 t, sec Mixing Signal β 1 0 5 10 0 0.5 1 t, sec Mixing signal β 2 0 5 10 0 0.5 1 t, sec Mixing signal β 3 0 5 10 0 0.5 1 t, sec Mixing signal β 4 0 5 10 0 0.5 1 t, sec Mixing signal β 5 0 5 10 0 0.5 1 t, sec Mixing signal β 6 0 5 10 0 0.5 1 t, sec Mixing signal β 7 0 5 10 0 0.5 1 t, sec Mixing signal β 8 Figure4.7: Mixingsignalβ 4.6 Conclusion In this paper we have presented the design of an adaptive velocity and altitude tracking controller for an AHFV.BecausetheMIMOaircraftmodelpossessessignificantparametricuncertaintyandunmodeleddy- namicsandthecontrolobjectivedemandssuperiortrackingperformance,wechosetodesigntheadaptive controllerusingtheAMCapproach,whichcombinespowerfultoolsfrombothrobustmultivariablecontrol and adaptive control. The AMC approach, unlike many other MMAC approaches, does not necessarily aim to select one controller, nor does it use a discontinuous selection strategy. Using a continuous control law is important in this application in order to avoid poor transients, control chattering, and exciting the unmodeledstructuraldynamics. TheglobaluncertainAHFVmodelisrepresentedbyasetofeightcandi- date models corresponding to the eight parameter subsets obtained by dividing the interval of each of the three main uncertain parameters into two subintervals (cf. Table 4.1). A candidate controller is developed foreachcandidatemodelbyutilizingmixed- μsynthesisinordertoaccountfornotonlytheuncertainties, but also the robust-performance objective. A Q-mixing strategy, which preserves stability on the param- eter overlap, implements the mixing strategy, completing the multicontroller design. The design of the robust adaptive supervisor comprises two steps. First, a robust adaptive law generates an estimate of each 107 unknown parameter. Second, a mixing law continuously tunes the multicontroller via the mixing signal β(t). A comparative study in the velocity and altitude tracking aptitude of the AMC and nonadaptive mixed- μ approaches is presented. In the AMC case, smaller magnitudes in tracking errors are observed. ThisstudydemonstratesthepotentialoftheAMCapproachforhigh-performancecontrolofsystemswith largedegreesofparametricuncertainties. Appendix The following key results are used in the stability and robustness analysis of AMC schemes. The results arewellknown,and,unlessstatedotherwise,theirproofscanbefoundin[30]andthereferencestherein. Theorem11. LetΩ⊂R 2n becompactandθ beanyconstantinΩ. Lettheparameterizeddetectiblepair (C(θ),A(θ))beLipschitzwithrespecttoθ∈ Ω,whereA(θ)∈R n×n andC(θ)∈R l×n . Thenthereexists aLipschitzfunctionL : Ω→R n×l ,suchthat 1. ifθ(t)∈ Ω for allt≥ 0 and ˙ θ ∈L 2 , then the equilibriumx e = 0 of ˙ x = A I (t)x is exponentially stable(e.s.) 2. ifθ(t)∈ Ωforallt≥ 0and ˙ θ∈S(μ 2 S ),thenthereexistsaconstantμ ∗ > 0suchthatifμ S ∈ [0,μ ∗ ) theequilibriumx e = 0of ˙ x =A I (t)xise.s. whereA I (t) 4 =A(θ(t))−L(θ(t))C(θ(t)) TheproofofTheorem11isacombinationofthewell-knownresultsof[12]andthelineartimevarying (LTV)stabilityresultsfoundin[30]. ThefollowingresultconcernstheLTVsystemgivenby ˙ x = A(t)x+B(t)u, x(0) =x 0 (4.91) y = C(t)x+D(t)u (4.92) 108 wherex(t)∈R n ,y(t)∈R r ,u(t)∈R m , and the elements of the matricesA(t)∈R n×n ,B(t)∈R n×m , C(t)∈R l×n ,andD(t)∈R l×m areboundedcontinuousfunctionsoftime. Lemma12. IftheLTVsystem(4.91),(4.92)ise.s. andu∈L 2e then 1. foranyδ∈ [0,δ 1 )where0<δ 1 < 2α 0 isarbitrary,wehave kx t k 2δ ≤ cλ 0 p (δ 1 −δ)(2α 0 −δ) ku t k 2δ + t wherec = sup t kBkand t isanexponentiallydecayingtozerotermbecausex 0 6= 0. 2. u∈L 2 ⇒x∈L 2 ∩L ∞ , ˙ x∈L 2 ,andlim t→∞ |x(t)| = 0 3. u∈S(μ S )⇒x∈S(μ S )∩L ∞ Lemma 13. Consider the LTI system given by y = H(s)u where H(s) is a strictly proper rational function ofs. IfH(s) is analytic inRe[s] ≥ −δ/2 for someδ ≥ 0 andu ∈ L 2e then we have|y(t)| ≤ kH(s)k 2δ ku t k 2δ . ThefollowingBellman-Gronwall(B-G)lemmaisusefulforestablishingboundedness. Lemma 14 (B-G Lemma) . Letc 1 ,c 2 be positive constants andg(t) be a piece-wise continuous function oft. Ifforallt≥t 0 ≥ 0,thefunctiony(t)satisfiestheinequality y(t)≤c 1 +c 2 Z t t0 e −δ(t−τ) g 2 (τ)y(τ)dτ thenforallt≥t 0 ≥ 0 y(t)≤c 1 e −δ(t−t0) e c2 R t t 0 g 2 (τ)dτ +c 1 δ R t t0 e −δ(t−s) e c2 R t s g 2 (τ)dτ ds. 109 ProofofTheorem10 The analysis presented is a generalization of the proof found in [35]. For analysis purposes, let us rewrite thez i equationsof(4.11)bysubstitutingin(4.3). Theresultcanbewrittencompactlyas Z = Θ ∗ φ+η a (4.93) where Z 4 = [z 1 ... z n ] T , Θ ∗ 4 = [p ∗ 1 ... p ∗ n ] T = [A P (p ∗ ) B P (p ∗ )] (4.94) η a 4 =F(s) B w d w +B Δ Δ(s)W Δ (s)C Δ [x T P d T w u T ] T +F(s)(sI−A P )d n =F(s)(B w d w +B Δ ΔW Δ C Δ2 )d w +F(s)B Δ ΔW Δ C Δ1 y+F(s)B Δ ΔW Δ C Δ3 u +(F(s)(sI−A P )−F(s)B Δ ΔW Δ C Δ1 )d n . (4.95) Theη a (t) term is a modeling error term and acts as an estimation disturbance. The(n+n u )×(n+n u ) matrixΘ(t)denotestheestimateofΘ ∗ ,whichisformedbysubstituting ˆ p(t)forp ∗ . Thepropertiesofthe adaptivelaw(4.10)arewellknown(cf. [29]),andwesummarizethekeyresults: ˆ p(t)∈ Ω, ∀t≥ 0 (4.96) ,m, ˙ ˆ p∈S(|η a | 2 /m 2 )∩L ∞ (4.97) wherewehavemadeuseoftheshorthandnotation = [ 1 ... n ] T . 110 Because ˆ p(t) is a bounded function of time, we can view the closed-loop interconnection of the plant (4.3), estimator filters (4.11), and multicontroller (4.18) as an LTV system. Let us write this system compactlyas ˙ x =A(ˆ p(t))x+B     d n d w     , x 4 = [x T P x T C x T E ] T (4.98) u =C(ˆ p(t))x, u 4 =Z−Θφ (4.99) wherewerecallthatthestatesx C andx E correspondtothestatesofstabilizableanddetectablerealizations of (4.18) and (4.11), respectively; u = m 2 is the unnormalized estimation error vector; the matrices A(ˆ p),B,andC(ˆ p)aredefinedobviously;and ˆ pistunedbytheadaptivelaw(4.10). Thedependencieson p ∗ havenotbeenwrittenoutinordertosimplifynotation. The closed-loop system written in the form of (4.98)-(4.99) is suitable for the application of Morse’s tunability analysis [44] approach. Consider the arbitrary initializationsx(0) =x 0 and ˆ p(0) =p 0 ∈ Ω. It hasbeenestablishin[56]thatthereexistsauniqueglobalsolution[x T (t) ˆ p T (t) n d (t)] T ,∀t∈ [0,∞). Step1: Establishthatforallfixed ˆ p∈ Ω,{C(ˆ p),A(ˆ p)}isadetectablepair. To check detectability we show that if the output u and the exogenous inputsd n andd w are zero for allt the closed-loop statevector xconvergesto zero. Tothis end, wefix u ,d n ,d w ≡ 0 for this step only. After detectability is shown, we analyze the closed-loop stability with the standard assumptions that d w andd n arebounded,andwedonotassume u ≡ 0. Itfollowsfrom(4.10)thatthereisnoadaptation,i.e., ˆ p≡p 0 and,equivalently,Θ≡ Θ 0 = [A P0 B P0 ], whereA P0 ∈R n×n andB P0 ∈R n×nu aretheinitialestimatesofA P andB P ,respectively,givenby ˆ p(0). Notethattheclosed-loopsystemisLTIbecausethereisnoadaptation. Because u ≡ 0,wehavethatZ = Θ 0 φfrom(4.99). Substituting(4.11)yields F(s)˙ x P =F(s)(A P0 x P +B P0 u) (4.100) 111 wherewehaveusedd n ≡ 0tosubstitutex P fory. SinceF(s)isastable,minimum-phasefilter, ˙ x P =A P0 x P +B P0 u. (4.101) ThemulticontrolleristhefixedLTIsystem ˙ x C = A C (β(p 0 ))x C +B C y u = C C (β(p 0 ))x C . (4.102) Bycombining(4.101)and(4.102),weobtain(ford w = 0d n = 0)     ˙ x P ˙ x C     =     A P (p 0 ) B P (p 0 )C C (β(p 0 )) B C A C (β(p 0 ))         x PA x C     . (4.103) It then follows from the design of the multicontroller and mixer thatx P ,x C → 0, and in turnx E → 0 as t→∞. Therefore,sincex→∞ast→∞if u ,d w ,d n ≡ 0,thepair(C(p 0 ),A(p 0 ))isdetectable. Step 2: Establish that along the solutions of (4.10), (4.12), and (4.98) there exists a functionL : Ω→ R ¯ n×¯ n suchthatA I (t) 4 =A(ˆ p(t))−L(ˆ p(t))C(ˆ p(t))ise.s. ApplyingLemma13to(4.95),togetherwith|d n (t)|≤d 0 and|d w (t)|≤d 1 ,yields |η a (t)|≤ Δ 0 +Δ 1 +Δ 2 ky t k 2δ0 +Δ 3 ku t k 2δ0 , ∀t≥ 0 (4.104) where Δ 0 4 =kF (sI−A P ) −1 −FB Δ ΔW Δ C Δ1 k ∞−gn d 0 , Δ 1 4 =kF (B w +B Δ ΔW Δ C Δ2 )k ∞−gn d 1 Δ 2 4 =kFB Δ ΔW Δ C Δ1 k 2δ0 , Δ 3 4 =kFB Δ ΔW Δ C Δ3 k 2δ0 (4.105) 112 Sincem 2 = 1+ku t k 2 2δ0 +ky t k 2 2δ0 andm≥ 0itfollowsthat |η a (t)| 2 m 2 ≤μ 2 S 4 =c(Δ 2 0 +Δ 2 1 +Δ 2 2 +Δ 2 3 ) (4.106) forsomeconstantc> 0. Therefore,wehave ,m, ˙ ˆ p∈S(μ 2 S ). (4.107) The matrixA(ˆ p) depends on ˆ p asA C (β(ˆ p)) andC C (β(ˆ p)) depend on ˆ p. A C (β(ˆ p)) andC C (β(ˆ p)) are linear inβ, andβ is Lipschitz in ˆ p. Therefore,A is Lipschitz in ˆ p. C is affine in ˆ p, and, therefore, also Lipschitz. Furthermore, because the adaptive law guarantees that ˆ p(t) ∈ Ω and ˙ ˆ p ∈ S(μ 2 S ), it follows from the detectability result of Step 1 and result 2) of Theorem 11 that there exists a continuous function L : Ω→R ¯ n×¯ n suchthatthatA I (t) 4 =A(ˆ p(t))−L(ˆ p(t))C(ˆ p(t))ise.s.,i.e.,thetransitionmatrixΦ(t,τ) ofA I (t) satisfieskΦ(t,τ)k ≤ λ 0 e −α0(t−τ) for some positive constantsλ 0 ,α 0 andt ≥ τ ≥ 0, provided that μ 2 S <μ ∗ (4.108) for someμ ∗ > 0. Assume the filterF(s) is chosen so thatc(Δ 2 0 +Δ 2 1 ) is sufficiently small, sayc(Δ 2 0 + Δ 2 1 )<μ ∗ /2sothatforc(Δ 2 2 +Δ 2 3 )<μ ∗ /2,condition(4.108)isalwayssatisfied 4 . NotethatifΔ,d 0 ,d 1 = 0, the adaptive law guarantees that ˙ ˆ p ∈ L 2 , and from result 2) of Theorem 12, we have thatA I (t) is e.s. Alsoobservethatineithercase,kLk∈L ∞ sinceLiscontinuousandΩiscompact. Step3: Establishboundednessandconvergenceofx. Letδ∈ [0,δ 1 ),whereδ 1 < min{2α 0 ,δ 0 },andc> 0denotesanyfiniteconstant. 4 Boundednessoftheclosed-loopsignalscanbeprovenindependentofthesizeof c(Δ 2 0 +Δ 2 1 )byusingtheanalysisapproachof [30, Section 9.9.1], which involves a complicated, lengthy contradiction argument. The analysis presented here, however, has been chosenforsimplicity. 113 AddingandsubtractingL(ˆ p(t))e u totheleft-handsideof(4.98)yields ˙ x =A(ˆ p(t))x+B     d n d w     +L(ˆ p(t)) u −L(ˆ p(t)) u A I (t)x+B     d n d w     +L(ˆ p(t)) u (4.109) wherewehaveused(4.99)toobtainthesecondequality. InStep2,weestablishede.s. ofthehomogeneous partof(4.109). Weestablishthatm∈L ∞ : ByResult1ofLemma12andthee.s. propertyofA I ,wehavethat kx t k 2δ ≤ck( u ) t k 2δ +c. (4.110) ApplyingtheL 2δ normtou c =C C (t)x C ,wherekC C (t)kisbounded(becauseC C isLipschitzin ˆ pandΩ iscompact),yields ku t k 2δ ≤ck(x C ) t k 2δ ≤ck( u ) t k 2δ +c (4.111) where the third inequality is obtained by first recognizing thatx C is a subvector ofx and then applying inequality(4.110). Considerthefictitiousnormalizationsignal m 2 f 4 = 1+ku t k 2 2δ +ky t k 2 2δ . (4.112) Note that becauseδ < δ 0 , it follows from the definitions ofm,m f thatm ≤ m f . Substituting (4.111), and u =m 2 into(4.112)yields m 2 f ≤ck(m 2 ) t k 2 2δ +c≤ck(mm f ) t k 2 2δ +c (4.113) 114 wherethesecondinequalityisobtainedbyusingm≤m f . Fromthedefinitionofk(·) t k 2δ itfollowsthat m 2 f ≤c Z t 0 e −δ(t−τ) ((τ)m(τ)) 2 m 2 f (τ)dτ +c. (4.114) ApplyingtheB-GLemmato(4.114)with g(τ) =(τ)m(τ)yields m 2 f ≤ce −δt e c R t 0 g 2 (τ)dτ +cδ Z t 0 e −δ(t−s) e c R t s g 2 (τ)dτ ds. (4.115) Let us assume 5 that F(s) is chosen such that c(Δ 2 0 + Δ 2 1 ) ≤ δ/2. Because m ∈ S(μ 2 S ) implies c R t s ((τ)m(τ)) 2 dτ ≤ cμ 2 S (t−s), it follows that for c(Δ 2 2 + Δ 2 3 ) ≤ δ/2, we have m f ∈ L ∞ . Since m≤m f ,wehavethatm∈L ∞ ,andtogetherwithm∈L 2 ∩L ∞ impliesthat u =m 2 ∈L 2 ∩L ∞ . We now turn our attention to the injected system (4.109). First, we consider part (1). If we have that Δ,d n ,d w = 0, the termL(t) u can be viewed as an input into the e.s. linear system ˙ x = A I (t)x + ¯ u. BecausekL(t)k∈L ∞ and u ∈L 2 ∩L ∞ , the input ¯ u =Lm 2 belongs toL 2 ∩L ∞ . SinceA I (t) is e.s. andL u ∈L 2 ∩L ∞ ,itfollowsfromresult2)ofLemma12and(4.109)thatx∈L 2 ∩L ∞ , ˙ x∈L 2 ∩L ∞ , andx → 0 ast → ∞. From the convergence ofx, and consequentlyL u , it follows from (4.109) that ˙ x→ 0ast→ 0. ForResult2,wehavethatΔ,d 0 ,d 1 6= 0. Thedynamicsofxaregovernedby ˙ x =A I (t)x+ ¯ u+B     d n d w     (4.116) where d n ,d w are bounded and belong to S(μ 2 S ), and we have shown that ¯ u = L u ∈ S(μ 2 S )∩L ∞ . Therefore,x∈S(μ 2 S )∩L ∞ ,and,inturn, ˙ x∈L ∞ . Because|x P (t)|≤|x(t)|,wehavex P ∈S(μ 2 S ),i.e., Z t+T t |x P (τ)| 2 dτ ≤c 0 (Δ 2 2 +Δ 2 3 +d 2 0 +d 2 1 )T +c 1 . (4.117) 5 Onceagain(cf. footnote4),thisassumptioncanberelaxedbyusingtheanalysisapproachof[30,Section9.9.1]. 115 116 Chapter5 ConcludingRemarksandSuggestionsforFutureWork In this dissertation we focus on the control of systems whose parametric uncertainty represents the major obstacle in achieving acceptable performance objectives. AMC is proposed as an approach that combines robust control and adaptive control techniques. The stability and robustness analysis of Chapter 2 jus- tifies theoretically the AMC approach, while the results of Chapters 3 and 4 illustrate the high-level of performancethatcanbeachieved. WebelievethestrengthsoftheAMCarethefollowing: • In contrast to conventional adaptive control, powerful LTI tools are readily incorporated into the design. • TherearetheoreticallyguaranteedstabilityandrobustnesspropertiesfortheAMCscheme. • In contrast to logic-based switching, the level of controller tuning is of the order of the estimation error. This avoids unnecessary switching that may occur when models are difficult identify, and allowsthesupervisortotunethecontrollerparametersinresponsetoasmallbutpersistentestimation error. • AMCperformswellforawideclassofdisturbancesandnoises. There are a number of worthwhile extensions to the current version of AMC. In Chapter 4 we de- signedanAMCschemeforaspecialclassMIMOsystems,buttherehasnotbeenanattempttoextendthis 117 approach to a more general class of MIMO systems. The major obstacle is determining plant parameter- izations suitable for estimation. Another future direction is AMC for nonlinear systems. We believe that utilizingthenonlinearversionoftheconceptdetectabilitywillbecrucialfornonlinearAMC. Other possible extensions and future work will involve reducing the complexity of the controller to meet real-time computational demands. To reduce the computational costs, the designer should limit the number of candidate controllers. One way to do this is to limit, if possible, the number unknown parame- ters, which is also good practice for conventional adaptive control. Another option is to use a windowing technique whereby only the neighboring candidate controllers (with respect to the current estimate) are activated. Thechoiceofcontrollerinterpolationmethodalsoaffectsthecomputationalrequirementsofthe AMC scheme. Certain controller interpolation methods are more computational efficient than others. For example, gain scheduling interpolation methods result in a low-order multicontroller, with an order equal to the maximum candidate controller order. The impact on the performance of these schemes should be investigated. Future work can focus on improving the performance of AMC for off-nominal conditions. Possible extensions may involve integrating switching-based logic or conventional adaptive control into the AMC scheme. One of the motivations for controller mixing was to avoid certain switching behaviors. But switchingcanbeeffectiveforadaptingtoabrupt,largeparameterchanges. Acombinedswitching-mixing schemewouldbeabletoadaptrapidlytolargechangesintheestimationmodel,andthenmixthecandidate controllers in response to smaller estimation errors. Alternatively, conventional adaptive control can be usedtoovercometheassumptionthatΩisbounded. Wenowconsidertheseextensioninmoredetail. 118 5.1 AdaptiveMixingControlWithSwitching To motivate and illustrate the integration of switching logic into AMC, let us revisit the benchmark mass- spring-dashpot example of Chapter 3 but now with the unknown parameter θ ∗ given by the discontinuous waveform θ ∗ (t) =        θ 4 , 0≤t≤ 25 θ 1 , t≥ 25. (5.1) Theconstantsθ i (i = 1,2,3,4)denotethemodelcenter,i.e., θ 1 = 1.385, θ 2 = 0.83 θ 3 = 0.52, θ 4 = 0.325. (5.2) Thespringconstantvariation(5.1)modelsanabruptchangeintheplantdynamicsfromModel4toModel 1att = 25. Onecouldusethistimevariationtomodelseveremechanicaldamage,forexample. WesimulatetheAMCschemeofChapter3infeedbackwiththetime-varyingplant. Fig.5.1showsthe simulation results for this case. A large transient occurs soon after the parameter jump att = 25 because thesupervisormustmixincontrollersK 3 andthenK 2 beforeinsertingthecorrectcontrollerK 1 . While the investigations of Chapter 3 demonstrate that the AMC scheme can achieve excellent per- formance when the plant is either time invariant or slowly time varying, the preceding simulation result clearly illustrates that it may lack the ability to respond with sufficient speed to large parameter changes. Switching, on the other hand, is particularly adept at responding to rapid, large parameter changes but canresultindegradedperformancewhensubtleadaptationisrequired,aswasdemonstratedinChapter3. Intuitively,whatisneededisadaptationthatiscommensuratewiththeparametervariation,withrespectto speedandmagnitude. We propose a modification to the AMC scheme so that mixing behavior occurs over a long-time scale but is capable of switching over a short-time scale. Fig. 5.2 depicts the modification, where the robust 119 parameter estimator can be reset to a new estimate within Ω. Resetting allows the supervisor to switch to anewcontrollerinresponsetolargeparameterchanges. Theresetlogic,calledreset-and-hold ,isdepicted schematicallyinFig.5.3. The reset-and-hold logic is a variant of the hysteresis switching logic commonly used in supervisory adaptive control. The signals μ i (i = 1,2,3,4) play the role of the monitoring signals of supervisory adaptivecontrolandaredefinedasa2δ S -normoftheestimationerrorsgivenbymodel i,i.e., μ i (t) 4 =k(z(t)−θ i φ(t)) t k 2 2δ S = Z t 0 e −δ S (t−τ) (z(t)−θ i φ(t)) 2 dτ, δ S > 0. (5.3) Themonitoringsignalsconveyhowclosetheestimationmodelsaretothetruemodel. Recallthatθ i takes onthefixedvalueofthecenterofthei-thmodelandis not tunedbytheadaptivelaw. Thedesignvariable δ S > 0isusedtotakeintoaccountpastdata. Foreachp∈I−I θ(t) ,thereset-and-holdlogicteststheresettingcondition (1+h)μ p < min q∈I θ (t) μ q (5.4) where the design variableh > 0 is the hysteresis constant and ensures infinitely fast switching does not occur;I 4 ={1,2,3,4} is the candidate model index set; andI θ(t) 4 ={i∈I :θ(t)∈ Ω i } is the index set of active models. Note thatI andI θ(t) are defined as in Chapter 2. If condition (5.4) holds, the estimate θ(t) is reset toθ q . Moreover, the adaptive gain Γ is set to the zero matrix, turning off mixing. Mixing is turned on (by settingΓ to its initial valueΓ 0 ) only after a hold period oft h seconds has elapsed since the lastreset. The reset-and-hold logic differs from conventional hysteresis switching logic in two ways. First, as specified in condition (5.4), the supervisor will not reset the estimate to θ i if Model i is active. This restriction ensures the supervisor remains in mixing mode for “local” changes in the model for which the mixingapproachispreferred,andonlyswitchesinresponseto“global”changes. 120 Second, when a switch does occur, the supervisor initiates the hold period lasting t h > 0 seconds during which its mixing capabilities are turned off. This additional logic has been introduced because if the supervisor detects that a switch is to be performed, then the plant may have changed so drastically that past data no longer accurately reflect the current plant dynamics. Thus, if the adaptive law relies on past data, estimation may be unreliable. As an example of an estimator that uses past data, consider the adaptivelawofChapter3,whichisbasedonminimizingthecostfunction J(θ) = 1 2 Z t 0 e β(t−τ) 2 (t,τ)m 2 (τ)dτ. (5.5) The integral form of the above cost function takes advantage of past data to mitigate the effects of esti- mation disturbances. Least-squares based adaptive laws also use past data to generate online estimates. For this class of adaptive laws, the hold period oft h provides the adaptive law time to discount unreliable data that could adversely affect estimation. Note that during a hold period, the supervisor can adapt via switching,i.e.,estimateresetting. LetusexaminetheperformanceoftheAMCwithresetandholdincontextofthebenchmarkexample subjecttotheparametervariation(5.1). Forthisdemonstration,thedesignvariableswerechosenwiththe valuesofChapter3and δ S = 0.04, h = 0.001, t h = 5. (5.6) The results are shown in Fig 5.4(a), where we have also plotted the results produced by the RMMAC and supervisory adaptive controllers of Chapter 3 for comparison. Observe that reset-and-hold logic has improved the performance of the AMC scheme considerably by endowing AMC with the capability to switch, while also preserving the mixing behavior during nominal conditions. The supervisory adaptive controlleralsoyieldsacceptableperformance,butstillthereistheriskoftheundesirableswitchingbehav- iorsdescribedanddemonstratedinChapter3. FortheRMMACscheme, aftertheparameterchangethere 121 isasmalltransient,butlargerthaneitheroftheotherschemes. ThetransientoccursbecausetheRMMAC modelassumptionsareviolatedforabriefperiodoftime,buttheRMMACschemedoesregainacceptable performanceeventually. 5.2 UnpredictableParameterChanges ThroughoutithasbeenassumedthattheplantparametersbelongtothecompactsetΩ. Thus,afinitesetof candidate controllers were designed off line such that for anyθ ∗ ∈ Ω there exists a stabilizing controller. But the assumptionθ ∗ ∈ Ω may not always hold. Severe failures or a lack of reliable a priori knowledge of Ω, for example, may result in a violation of this assumption. It is, therefore, desirable to modify the AMCschemetocopewiththissituation. Because AMC utilizes an online parameter estimator, a conventional adaptive controller, in a straight forward fashion, can be included in the candidate controller set. Fig. 5.5 presents a simulation result of the modified AMC in closed-loop with the benchmark MSD system. In this example, we have included an adaptive pole placement controller (APPC) in the candidate controller set that is mixed into feedback whenθ(t)isontheboundaryoroutsideofΩ. Theplantparameterofthisexampleisgivenby θ ∗ (t) =        1+ 3 50 t, t≥ 0 4, otherwise. (5.7) For comparison, a mixed- μ controller is simulated side-by-side with the AMC. The mixed- μ controller was design in [16] to provide robust performance for allθ ∗ ∈ Ω. Forθ ∗ = 4, the mixed- μ controller is destabilizing. AMC , on the other hand, mixes the APPC into feedback and maintains stability. It should be noted that any multiple model adaptive control approach that uses the finite candidate controller set {K i } 4 i=1 willnotbeabletomaintainstability. 122 0 5 10 15 20 25 30 35 40 45 50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time, sec Unknown spring constant (a) AbruptparameterjumpfromcenterofModel4tocenterofModel1. 0 5 10 15 20 25 30 35 40 45 50 -150 -100 -50 0 50 100 150 Time, sec Plant output (b) PlantoutputyieldedbyAMC. 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 β 1 (t) 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 β 2 (t) 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 β 3 (t) 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 Time, sec β 4 (t) (c) Mixingsignal. Figure5.1: SimulationresultsforAMCwithanabruptparameterjump. 123 G(s; θ*) K 1 K N … Multicontroller E Robust Supervisor Mixer Robust Estimator Mixing Strategy M disturbance and noise u u y Reset and Hold ( ) t θ Figure5.2: ModifiedAMCschemewithestimateresetting. () () () : 1min t t pqI q pI I h θ θ μ μ ∈ ∃∈ − +≤ ( ) t θ Initialize ( ) 0 p f t tt θ θ = Γ= = f h tt t ≤+ 0 Γ= Γ yes no yes no Reset and Hold Resume Mixing Figure5.3: Thereset-and-holdalgorithm. 124 0 5 10 15 20 25 30 35 40 45 50 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Time, sec Plant output AMC w/ reset and hold RMMAC Supervisory switching (a) Plant outputs yielded by AMC with reset and hold, RMMAC, and supervisory switching control. 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 β 1 (t) 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 β 2 (t) 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 β 3 (t) 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 Time, sec β 4 (t) (b) Controlweights. 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Abstract (if available)

Abstract Despite the remarkable theoretical accomplishments and successful applications of adaptive control, the field is not sufficiently mature to solve challenging control problems requiring strict performance and safety guarantees. Towards addressing these issues, a novel deterministic multiple-model adaptive control approach called adaptive mixing control is proposed.

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Asset Metadata

Creator Kuipers, Matthew (author)

Core Title Multiple model adaptive control with mixing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 08/05/2009

Defense Date 04/01/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag adaptive control,control theory,intelligent control,multiple model adaptive control,OAI-PMH Harvest,robust adaptive control,robust control

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ioannou, Petros A. (committee chair), Flashner, Henryk (committee member), Safonov, Michael G. (committee member)

Creator Email kuipers@usc.edu,mck76@cornell.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2478

Unique identifier UC194253

Identifier etd-Kuipers-3000 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-173607 (legacy record id),usctheses-m2478 (legacy record id)

Legacy Identifier etd-Kuipers-3000.pdf

Dmrecord 173607

Document Type Dissertation

Rights Kuipers, Matthew

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu