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A model reference adaptive control scheme for linear delay systems

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A model reference adaptive control scheme for linear delay systems

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zed) Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A MODEL REFERENCE ADAPTIVE CONTROL SCHEME FOR LINEAR DELAY SYSTEMS by Michael Busson A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (Applied Mathematics) May 1997 Copyright 1997 Michael Busson Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI N u m b e r : 1 3 8 4 8 8 6 UMI Microform 1384886 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY O F S O U T H E R N CALIFORNIA T H E G RAD U A TE S C H O O L U N IV E R SIT Y PA R K LO S A N G E L E S. CALI F O R N IA 9 0 0 0 7 This thesis, written by 'B a S - S o * ________________ under the direction of hjJ>.— Thesis Committee, and approved by all its members, has been pre~ sented to and accepted by the Dean of The Graduate School, in partial fulfillment of the requirements for the degree of Jlaster__of_ Sc ie n ce January 29, 1997 THESIS COMMITTEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements First of all I would like to thank my advisor Professor Gary Rosen who assisted my studies through out my stay at USC. While he gave me full freedom to work on the thesis at my own pace, he also guided me through the problems that I faced and gave me his full support. He carefully read the manuscript and made many comments which have led to significant improvements. I’m also thankful to Professor Cunming Wang and Professor Wlodek Proskurowski for serving as members on my thesis committee and reviewing and grading the thesis. Lastly, I would like to thank all my Indian friends at USC with whom I had some great times and who made my stay in California memorable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents A cknow ledgem ents ii List O f Figures iv A b stract v 1 In tro d u ctio n 1 2 T he M odel R eference A daptive C ontrol (M RA C) Problem For D elay System s 3 2.1 Tracking Error C onvergence........................................................................ 6 2.2 Parameter Convergence................................................................................. 13 3 E xam ples and N um erical R esults 17 3.1 Introduction..................................................................................................... 17 3.2 A First Order Delay E q u a tio n .................................................................... 17 3.3 A Second Order Delay E qu atio n ................................................................. 19 3.4 A Second Order Delay Equation with Distributed Delay .................... 24 3.5 A Nonlinear Delay E q u a tio n ........................................................................ 30 4 C onclusion 35 A ppendix A Source Code for Example 3.4 .............................................................................. 36 References 46 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List Of Figures 3.1 The tracking error, |e(t)| = |u(i) — v(£)|, of a first order delay equation. 19 3.2 The convergence of the parameters, q0(t) and qri(t), of a first order delay equation........................................................................................................20 3.3 The tracking error, |e(f)| = \y(t) — u>(f)|, of a second order delay equation...................................................................................................................23 3.4 The convergence of the parameters, qo(t), qi(t) and q 2(t), of a second order delay equation.............................................................................................23 3.5 The tracking error, |e(t)| = \y(t) — u>(f)|, of a second order delay equation with distributed d e la y ...................................................................... 28 3.6 The convergence of the parameters qi(t) and g2(0i °f a second order delay equation with distributed delay.............................................................29 3.7 The convergence of the parameter qs(t), of a second order delay equa tion with distributed delay.............................................................................. 29 3.8 The tracking error, |e(f)| = |u(f) — u(f)|, of a nonlinear delay equation. 33 3.9 The convergence of the parameter qo{t), of a nonlinear delay equation. 34 3.10 The convergence of the parameter qi{t), of a nonlinear delay equation. 34 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract A model reference adaptive control scheme for linear delay systems is considered. The tracking error is shown to go to zero asymptotically under certain conditions. With the assumption of persistence of excitation, the parameter error is shown to go to zero as well. Examples, involving first and second order delay differential equations with and without distributed delay are discussed and numerical simulation results are presented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction A control problem of interest involves the forcing of a plant to follow, or track, the trajectory of a given system or model. It is frequently true that the characteristics of the plant to be controlled depend on a set of parameters. In real processes, it is often the case that the designer does not know the true value of these parameters. In this case (see, for example, [ 6]) the resulting problem is one of adaptive control. The objective then is to design the controller and an estimator so that the closed-loop system is intelligent enough to adjust its characteristics in a changing environment and so that it is operating in an optimum way according to some desired level of performance. To construct such a system we must provide it with a feedback controller. The aim of the control is force the error between plant output and desired ouput to be small. If the desired output is changing over time, the problem is also referred to as a tracking problem. In this thesis the desired output is specified by a reference model. Model reference adaptive control (MRAC) for delay systems is then an adaptive control problem, where the output or state of the unknown plant has to track the output or state of the specified reference model with the plant and the reference model governed by delay differential equations, and while the unknown parameters are identified in real time. In an ordinaxy differential equation, an unknown function x and its derivatives x are all evaluated at the same instant t, while in a delay differential equation the derivative x with the highest order is evaluated at t, lower order derivatives of x and x itself are evaluated at time t and/or at some earlier times. Delay differential equations are also known as hereditary systems and functional differential equations (FDE). For specifying an initial value problem we do not only need a constant at 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. some initial value, but we also have to specify an initial value function, or history, on an interval of length r, where r is the maximum delay in the equation. We are then looking for a continuous extension of the initial value function that satisfies the delay equation. The initial value function in general does not satisfy the delay equation. In this thesis we present the MRAC problem for delay systems. The plant is gov erned by a lineax delay differential equation, whose system parameters are unknown and whose input is to be determined by a designer. The reference model is also represented by a linear delay equation with reference input g. We then determine the feedback control law, such that the state of the plant asymptotically tracks the state of the given reference model. We obtain the closed loop system that consists of the plant, the reference model and the estimator for the parameters. Under cer tain conditions on the reference model we establish tracking error convergence. At the same time the parameters are estimated and updated. With the assumption of persistence of excitation we establish parameter convergence. We apply the theory to several examples in which the plants are governed by linear delay equations of first or second order with and without distributed delay. We also give an example involving the control of a nonlinear plant. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 The M odel Reference Adaptive Control (M RAC) Problem For Delay Systems We consider a hereditary system with m different delays r,-, 0 < i < m. All delays are between — r and 0, — r < — r m < ... < — rq < — r0 = 0. The state of the plant to be controlled is given by u which maps t — > u(t) from [— r, oo) into R nxn. For every i = 0 ,1, . . . , m, the matrix Ai(q) is an element of R nXn and Ai(q) depends on a parameter q from the parameter space Q. We assume that {Q,< >q, | • |q} is a Hilbert space. For each q € Q, A(q)(0) is an n x n m atrix, that depends on a paramter q, but also on 0 G [— r, 0]. We assume that for each 6 R " the maps q — > • Ai(q)tp and q — > A(q)(0)i(> from Q into R" are linear for all 6 G [— r, 0] and 0 < i < m. We are interested in adaptively controlling the plant which is governed by the linear delay differential equation T P fO “ (0 = E Ai{q)u{t - n) + / A{q){0)u{t + 6)d6 + f(t), t > 0, (2.1) 1 = 0 J~r with initial conditions u(0) = //, u0 = tp. (2.2) The parameter q G Q is assumed to be unknown and is to be identified in real time as the plant is controlled. The function < p maps [— r, 0] into R n and defines u(t) for all t G [— r, 0]. We let L2 = L2(—r, 0; R n) be the Hilbert space of all functions x : [— r, 0] — > ■ R n that are square integrable, and denote by < >£,2 the usual inner product and by 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. | • | t h e usual norm on L2. For u € Z - 2( - r ,7 ’;R n), T > 0, we define ut € L2 by ut(9) = u(t + 9) for all 9 € [— r, 0] and t > 0. We introduce for every q € Q the linear operator L(q) that maps f> — ► L(q)4> from C([— r,0 ];R n) into R " and is given by ™ ro £ (« )* = £ A ( « W - r i ) + / A ( q ) (0 ) m M , ^ € C ( [ - r ,0 ] ;R " ) . (2.3) i=0 J~r We can then write the plant (2.1), (2.2) as u(f) = L{q)ut + /(<), t > 0, (2.4) u(0) = 7 7 , uQ = ip. (2.5) The function / is called the control input. The model reference adaptive control problem (MRAC) consists of finding an / such that the unknown plant u tracks the state, v, of a given reference model, v(t) = Y^Civ{t — r-) + [ C{9)v{t + 9)d9+g(t), t > 0, (2.6) i=0 J~r with initial conditions u(0)=771, 1 7 0 = ^ , (2.7) where C, 6 R nXn for i = 0 ,..., k, —r < r\ < r ^— 1 < ... < = 0, C(9) {9 € [— r, 0]) is a n x n matrix with C square integrableon [— r , 0], g is a function that maps [0, 00) into R n, 7 7 1 € R n and p l 6 L2. The delays r] may be different from the delays of the plant. We again simplify notation by introducing the linear operator Lq that maps c f> — Lq4 > from C{[—r ,0]; R ”) into R n. The operator is defined by ^ = E C , ^ ( - r l ) + [° C(9)<f>(9)d9, ^ C ( ( - r , 0 ] ; R n). (2.8) i=o J~r Now the reference model (2.6), (2.7) can be written as v(t) = L0vt +g{t), t > 0, (2.9) u(0) = 7 /1, v0 = ipl . (2.10) 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We note that technically the operators L and L q are only defined on the dense subspace of L2(— r, 0 ;R n), C([— r, 0];R n). However, since in general we deal with the solutions, u and v to the delay differential systems (2.4), (2.5) and (2.9), (2.10), the right hand sides of (2.4) and (2.9) can be thought of as being under an integral sign and consequently, in this case, it is well defined to talk about L and L q applied to functions in L2( —r,0; R n). If the plant q is known and L q is uniformly exponentially stable, choosing / to be f(t) = L0ut - L{q)ut + g{t), t > 0, (2.11) will suffice to achieve the control aim. The nonadaptive closed loop system is then u(t) = Lout +g(t), t> 0, (2. 12) u( 0) = 7 7 , u0 = tp. (2.13) Then if e(<) = u ( t) - v ( t) , t>0, (2.14) we have e(t) = L0eu t > 0, (2.15) e(0) = 7 7 — r/1, e0 = i f - i p l. (2.16) The uniform exponential stability of Lq then yields lim^oo |e(f)|Rn = 0. But since q is unknown, we define the control input / as f(t) = L0ut -L(q(t))ut + g(t), t> 0, (2.17) where for each t > 0 the parameter q(t) denotes an updated estimate for q. The parameter q will be adaptively identified via the estimator q in real time. In order to state the adaptive law for q it is helpful to introduce a new operator. We define for every 0 G C{[—r, 0];Rn) the linear operator that maps q — > ■ B(cf>)q from Q into R n and is defined by B(<f))q = L{q)< t> , q e Q . (2.18) 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Because of the assumption that A,(q) and A(q)(9) axe linear in q, for every < f) £ C ([— r, 0]; R n), B(cp) is linear. Let (B(<p))* denote the Hilbert space adjoint of B (< j> ), that maps from R n into Q. We can define the adaptive update law for q as follows: q(t) = (B{ut))m (u(t) t > 0, (2.19) 9(0) = ?o, (2.20) where q0 is an initial guess for q. By combining the plant, (2.4), (2.5), the control input, (2.17), the reference model, (2.9), (2.10), and the adaptive law for q, (2.19), (2.20), the closed loop system is then given by u (0 = L0ut - L(q{t) - q ) u t + g{t), t> 0 (2.21) v(t) = L0vt +g{t), t > 0, (2.22) 9(0 = {B(ut))‘{u{t) - u(t)), t > 0, (2.23) u(0) = T }, uQ = v = > , (2.24) v(0) = q l, v0 = < p l , (2.25) I I o ' (2.26) 2.1 Tracking Error Convergence We first want to show that the state of the plant, u, tracks the state of the reference model, v. For that we define the error, e(t) = u(t) — v(t), t > 0, (2.27) et = ut — vt, t > 0, (2.28) r{t) = q{t)-q , t > 0. (2.29) The function r is called the parameter error. We have to show that |e(t)|R.» — $ ■ 0 as t oo. It is straight forward to show that the error, e and r obey the system e(i) = L0et - L(r(t))(et + vt), t > 0, (2.30) 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v{t) = L0vt + g{t), t > 0, r(t) = (B(et + vt))m e(t), t > 0, e(0) — T ] — r}1, eo = tp — ip1, u(0) = r)1, vQ = p \ r (0) = qo~q. (2.31) (2.32) (2.33) (2.34) (2.35) Let H = R n x Z < 2(— r, 0 ;R n) be the Hilbert space with inner product < (»7l, )» (^2, q>2) > H = < > R n + < <^17^2 >£27 i = 1, 2. (2.36) On the Hilbert space H we define the linear operator A q, that maps (7 7 , p) — y Ao(rj,p) from V (A q) C H into H , where V(Ao) = {(i],ip) e H : p absolutely continuous with < p € £ 2, and = V ’(O)} and Ao{<p(0),<p) = {L0p,p) for (<^(0),y?) € V(Aq). We assume that there exists a constant a0 > 0 such that < Ao{p{Q),p),(<p{0),p) >H < -aol^CO)!2, (^ (0 ),^) 6 V{Aq). (2.37) That is, we assume that the reference model is uniformly exponentially stable. For each t > 0, we define e(t) = (e(t),et), where e satisfies (2.30) and (2.33). T heorem 1 We define E : [0,oo) -> R + as £(<) = 5 {!«(()&+ |r(i)l|}, t> 0. (•2.38) Then the function E is non-increasing and (2.39) Proof. D3E{s) = = < Dse(s),e{s) >H + < D3r(s),r{s) > q = < ( e (s ), e3), ( e (s ), e3) >H + < D3r(s), r(s) >Q 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = < (L0es - L(r(s))(e3 + v3),e3),{e(s),e3) >H + < (B(e„ + u,))*e(s), r(s) >Q = < (L0e3, es), (e(s), e3) >H - < (L(r(s))(e3 + v3), 0), (e(s), es) >H + < (B(es + us))*e(s),r(s) >Q = < A )(e(5), es), (e(s), e3) >H - < L(r($))(es + v3), e(s) > Rn - < 0, e3 >Ll + < e(s), B(es + t> s)r(s) > Rn < - a 0|e (s)|^ „ - < L{r{s)){e3 + us),e(s) > Rn + < e(s), L{r(s))(es + us) > Rn = - a 0|e(s)j^„ Consequently E is nonincreasing. By integrating both sides from 0 to t we obtain E(t) — E(0) = f DsE(s)ds < — q 0 f |e(s)|Rnds, t > 0, (2.40) Jo Jo and therefore we have the result E(t) + a0 t \e{t)\2 Kndt < £ ( 0), t> 0. (2.41) Jo © T h e o rem 2 If g € £ 2(0, oo;R n), then e,v and u 6 L2(0, oo;R n). P roof. By Theorem 1 it follows that T \e{t)\l_ndt < — E{0), t > 0. (2.42) Jo C lq Because /J |e(i)|Rndi is monotone increasing and bounded, we know that lim^oo /0 e |e(f )|Rndf exists and roo f t 1 / \e{t)\Rndt = lim / \e{t)\Rndt < — E(0). (2.43) Jo t-*co Jo O c q Therefore e € Z /2(0, oo; R 7 1 ). By our assumption (2.37) on A o , we have that < A )(u(s),us),(u(s),v,) >h< - a o b (s )|Rn, s > 0, 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for some a0 > 0. It follows that | z ) * l ( u (s),V 3)|tf = < £>5( u ( s ) , u s ) , ( u ( s ) , u s ) > h = < {L qv 3 + 5(s), *>,,), ( u ( s ) , u a) > h = < {Lov„ V3), (v(s), v3) > H + < (g(s), 0 ), (u (s ), V3) > H = < . 4 o M s ) , v,), (v(s), V,) > H + < g(s), v{s) > Rn + < 0, Va > I* < - Q o|» ( « ) | r » + I« ( s ) | r » |0 ( * ) | r " = - « o | u ( 3 ) | a n + — | a 0t;(a)|R n|flr(a)|R n do < - a 0|u (s)|R n + | | qto|w( « ) | r » + = - ^ o|u ( « ) | r » + 5 “ |flf(3)|Rn. Integrating from 0 to t we find that IMO.wOljy ~ l(y(0),uo)ltf = / ^,|(u(5),w s)|^(/s Jo ^ — I \g(s)\R"ds - a0 [ |u(s)|R„ds, t > 0, Oo Jo Jo and thus K * ) I r » + \Vt\ h + a oJQ Iu (5 )Ir." ^ 5 = \{v{t),vt)\2 H + a0 [ |u(s)&„ds JO < \(v(Q ),V o)\2 H + ^ - f |flf(3)|^„</3 cto Jo ^ \vl \kn + I / I L + ~ M i ( o ,o o ; R n) = : M . (2 .4 4 ) Obviously /0 ‘ |u(s)|R„ds is monotone increasing and bounded by Therefore limt_,.o/o exists and is less than or equal to It follows immediately that v € £ 2(0, 00; R n). We know that e and v ( E £ 2(0, 00; R") and u(t) = e{t) + v{t), for all t > 0. Consequently u € Z /2(0,o o ;R n). 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We use the results of the previous theorems to prove error convergence. We as sume that there exist positive constants ao,.. -, am and a function a 6 £ 2(— r,0; R +) such that K C ^ I r ." < a«-klQMR" and |A,(q)(0)7 7 |Rn < a(0)|g|g|7 7 |Rn, i = q€Q, T j €. R n. T h eo rem 3 For (e, u ,r) the solution to the initial value problem (2.30)- have |e ( t ) |R n —^ 0, as t — ^ oo. P roof. By Theorem 1 we have that for all* > 0 | e ( 0 l a » le *li2 KOIq . ► < K 0Ir» + M L + K 0IJ H e (0 ll/ + lr(0lQ = 2E(t)<2E(Q) = e0- When taking the square root on both sides we obtain K O I r " KUa KOIq . < ft, t > 0. From (2.44) it follows that M 0 I r » N In where M is defined as in (2.44). For all t2 > ti > 0, we have that e ( ^ ) | R n - | e ( f ! ) | | n = I f D3\e{s)\l_nds M il I f t2 = / 2 < D3e(s),e{s) >B.n ds \Jt i < \/M, t > 0, (2.45) (2.35) we (2.46) (2.47) (2.48) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f t 2 < 2 / I < L0es - £(r(s)){e, + us}, e(s) > Rn \ds J t i < 2 [h |(L0ea)Te(s)|rfs + 2 fh |(L (r(s)){e, + v .})Te(s)|<fa J t i J t i < 2 f \LQ es\nn\e{s)\Knds+ 2 [ |£ (r(s)){e, + u,}|R n|e(s)|Rn(/s J t i J t i = 2 j f £ c '*e(5 - r .-) + /_° c (0)e(s + 0)de 1 = 0 h \e{s)\Rnds R" 53 ^«(r(5)){e(5 - r.) 4- u(s - r,)} +21 Jti + 1 ° . 4 ( r ( s ) ) ( 9 ) { e ( s + 0) + v(s + B)}de i 2 / ' ! ( e I I C . I I + / : I I C ( « ) | | | e ( s + » ) | r - < m ) & * |e (s)|R»cfe ft2 / ^ + 2 £ ailr(5)|g|e(s - r.) + u(s - r,-)|R» ■ '‘i Vi=o + J a(0)|r(s)|Q|e(s + 6) + t>(s + 0)|Rn£/0j f0 ds < ^ +y y mf d e y y ° ^ s + e ^ d e y ds r^ 2 + 2fo / 5 3 a<So{|e(s - r,-)|R» + |«(s - r,-)|R»} •/ t » i= 0 + J a(#)£o{le(s + ^)|r" + lv(s + 0)\Kn}dO ds < ' X o f E W C i U o + W C W ^ -r,0;Rn *n) | e s |l,2 <fe J t t i=0 + 2^0 / 5 3 a,'(£° + V^M) J t l 1=0 + { J \ ( $ ) 2 dO}i ( y y le f s + S )!^ } 1 + {jT M *+ « ) & .< /« ] — / £ l|C »ll£o + l|C llL o(-r.0:R nx,» ^ 0 d s J t i i=0 / t2 J7 1 ^ _________ __ 5 3 a'(^° + vM ) + |a|L2(-r,0;R)(|es|L2 + 1 1=0 < 2e o 2 (E l|C ',|| + ||C ||l2(_ r,0;Rnxn)^ (^2 — ^ l) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m + 2 f o ( 0 ‘ + la l^ ( - r ,0 ; R ) ) ( f o + n/ M ) ( * 2 ~ * l) i= 0 < K(t2 — *i), for some K > 0. Suppose |e(£)|Rn 0. Then we know that |e(<)|R_„ -ft 0. This implies that there exists an e > 0 such that for all T > 0 there exists a t > T with |e(f)|n.„ > £ ■ Define S = and let {£ ,}^0 be such that |e(t,-)|^_n > e and U+l —t{ > 6 for i > 0. It then follows that L em m a 4 Let h > 0 and p £ N. Let x be a function that maps from [0, oo) -* R n with |x(<)|r,™ 0 as t -» oo and x £ Lp(t,t + h; R n) for all t > 0. Then we have |e(£)lkn — k(*.-)lk-| — ^ ( t — tf) < KS ^ 2 ’ ^ ^ [U,U + < £ ] > * > 0, and therefore that l®(^)Ir** — Ir"I — — 9 ) ^ £ [tiiU + £], i > 0. This implies that c( 0 I r » = le(0lR« ~ |e(<i)la» + |c(*.-)Ir « > - | | e ( < ) l a « - |e (t,-)lR » | + N M I r " > — — + e = — , t £ [t,-, t{ -f- £], i > 0. Integrating from 0 to 00, Theorem 1 implies that This is a contradiction and consequently |e(f)|Rn — » • 0, as t — > ■ 00. O as t -> 00. (2.49) 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof. We have to show that for every e > 0, there exists a T > 0 such that f)+ h |:r(s)|pds < e for all t > T. For all > 0, there exists a To > 0 such that |z(0lRn < -fa for all t > To. Then we obtain the following estimate s:+h |*(»)|*de =< = h f = eP < e, t > T — T0, £ < 1. © With these results, we can establish the following Corollary. Corollary 5 For (e, v,r) the solution of the initial value problem (2.30)- (2.35) we have t 0, as t oo (2.50) Proof. By Theorem 3 we know, that |e(s)|Rn — > • 0. We apply Lemma 4 with h = r and p = 2 and obtain the desired result. © Corollary 6 The function |e(f)|# — > 0, as t -4 oo. Proof. By definition of e we have Urn \e{t)\2 H = Um{|e(<)|R» + |et|£ j = Urn |c ( < )|r » + /jm \et\\2 = 0, and therefore |e(t)|// — * ■ 0, as t — > oo. © 2.2 Parameter Convergence We want to show next that, under certain conditions on the reference model, pa rameter convergence is achieved. That is that l i m t_+oo lr (0lQ = 0- For this we make the following definition: Definition 1 The reference model, (L o ,g ,v o ), is said to be persistently exciting, if there exist positive constants To, 8q, Tq and So such that for each q € Q and t > To, there exists a t € [t,t + To] for which rt+So J L(q)u3 ds >£o\q\Q, (2.51) R" 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where u is the state of the plant in the closed loop system. L em m a 7 Let h > 0 and x 6 £ 2(0, oo;R n). Then we have rt+ h J |:r(s)|Rn(/.s — y 0, as t oo. (2.52) Proof. Suppose |x(s)|Rnds -f± 0, as t — » • 0. Then we know there exists an e > 0 such that for all T > 0 there exists a t > T with rt+h J | x (5) | Rn ds > e. Let {f,-}£o be such that / t* ,+ A |.T(s)|Rnrfs > e and ft+i — U > h for i > 0. It then follows that rt,+ h ( rti+ h ) ? f rti+ h ) 2 0 < £ < j |x(s)|Rnds < j y |x(s)|R„ds> j j f 1 ds > = oo. U li+h o 1 1 1 |x(s)|Rnds> /iz, i> 0. and therefore 0 < — < / k(s)|nnds, i > 0. h Jt, This implies that |zU2(o,oo;R") = / |z(s )Ir n d s > f ^ f \x(s)\tln d s > jr )^ - = J0 t= 0Jt' :=0 n which is contradiction to x € £ 2(0, 00; R n). 0 T heorem 8 / / g € £ 2(0, 00; R") and the reference model is persistently exciting, then K O Iq '+ O , as t — » • 00. (2.53) P roof. We first show that |r(t)|g converges as t — > 00. By Theorem 1 we know that E is monotone decreasing and by definition of E, E(t) > 0 for t > 0. This 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. means, that E has a lower bound and therefore that lim*-**, E(t) exists. Theorem 6 showed, that limt_ * o o |e(t)|j/ exists. We can now write Um |r(i)|J = Um{2E(t) - \e(t)\2 H} = Um 2E{t) - Urn \e{t)\2 H = 2 Um E(t) t— yoo t-+ oo < — ► 0 0 t-yoo and therefore limbec |r (£) |q exists. Next, we show that |r(i)|Q — ► 0. Suppose |r(£) |< g -fa 0. Then there exists {i,}yi0 with t0 > To, tj+i — tj > t0 for j > 0 and tj -¥ oo as j — y oo such that lr (^)l<3 > 7 , j > 0, for some 7 > 0. If the reference model is persistently exciting, it follows that for each j > 0 and some tj 6 [tj, tj + ro] we have rij+ So r(tj) r t j + o 0 /. ^(r(ij))us ds R n < f 1 Ai{r(tj))u(s - r.) + I A(r(tj))(0)u(s - 0) J h S J ~ r r t j + S a m fO ^ a «'lr ( ^ ' ) lQ l u ( s _ r «')lRn + / a ( 0 ) l r ( * i ) l o K s ~ 0 ) I r ndG d s J h £ 0 J ~ r ( rt]+So JIL rt]+ & o fO \ - & y ji ai\u(s - n)|R"^s + J a(0)|u(s - 9)\RndO dsj f rh + S 0 rtj+So \ — I } Q» /. I^("® ^’» ) |R n ^^ d" I |o|/^2(_r,0;R) I n s l ^ j d s J \t'=0 * -> Jt3 ) ( rtj—rt+S0 rfj+<S0 \ = fo 2^a« ’ /- |u(s)|R nds + |a|ia (_r , 0;R ) I {Usli^ds . \t = 0 J t ] J We know by Theorem 2 that u G £ 2(0, 00; R n), and by applying Lemma 7 to u we obtain that rij-ri+So / |u(s)|Rn — > ■ 0, as tj — » • 00. (2.54) J t j - r , 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Suppose that -fr 0, as s -> • oo. Then there exist e2 > 0 and {s/}^0 such that s0 > r, s/+1 — s/ > r and |u „ |^ > e2, for I > 0. V V e have then M L (0,oo;R ") = / lU(s)|R n<fc > £ f ‘ |« ( 0 ) |r » < 0 JO /=0 a,_r - E M L > f > 2 = oo. 1=0 1=0 This is a contradiction because by Theorem 2 u € T2(0,00; R n) and therefore we know that \u3\l? -> 0 for s -> 00. Let k(s) = |u4|^ , then A:(s) — > 0 as s -4 0. We apply Lemma 4 and obtain that / i+ (5o r t+ S o \u3\i^ds = j k(s) ds 0, as t — t 00. (2.55) The estimates (2.54), (2.55) imply that / .m ,' f t n + S o - r i 7£o < fo 2^ a .- / |u(s)|R n < /s rin+S0 \ + |a|£,2(_r,0;R ) J. |Usl^dsj ->-0, as n -> 00. This is a contradiction because -y£0 > 0 . © 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Examples and Num erical Results 3.1 Introduction V V e will present several examples of MRAC for delay systems. They include a first or der delay system and a second order delay system with and without distributed delay term. All computations have been done on a SPARCstation 4 in the programming language C. For numerical integration, solving differential equations and matrix ma nipulation we used functions, that are provided by the IMSL library. We used Gear’s method for stiff systems for solving the delay equations because the problem shows equilibrium behaviour, which implies that it might be stiff. For identifying a func tion 7 6 L2{ — 1, 0;R ), as we need to do in example 3.4, we approximate 7 by using standard linear B-splines, on the interval [— 1,0] and approximate L2( —1,0; R) by L™ (-1,0;R) = spanfe1 ?}. 3.2 A First Order Delay Equation We consider the MRAC control of the one dimensional first order delay equation ii(t) = q0u(t) + qiu{t - 1) + /(f), t > 0, (3.1) together with the initial conditions u(0) = 2, u{0) = sin(0) + 2cos(3 6), 0 6 [-1 ,0 ). (3.2) 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We take the reference model to be given by v{t) = a 0u(*) +g{t), t> 0, (3.3) u(0) = 0, (3.4) where g(t) = cos(f) — 6sin(3f) + 2sin(t) + 4cos(3t), t > 0, (3.5) a 0 = -2 . (3.6) In this example, H = R x L2(—1,0; R), Q = R 2, n = l, r = 1, m = l and k = 0, which means there is no delay in the reference model. We assume that q = (go,qi) € Q is unknown and is to be indentified as the system is adaptively controlled. We define the inner product, < •, • >q on the Hilbert space Q by < 9,P >Q= loqoPo +7i9iPi, q,P€Q, (3.7) where 71, 72 € R + are positive weights. These serve as adaptive gains in the closed loop system. Their value can be chosen so as to tune the system and speed conver gence. In a sense, they are related to the level of persistence of excitation. By (2.17) we have as control input f(t) = aQ u(t) - q0(t)u{t) - qi(t)u(t - I) + g(t), t> 0, (3.8) and the closed loop system becomes then u(t) = a0u{t) - (qo(t) - q0)u(t) - (^(f) - qi)u(t - 1) + g(t), t > 0, (3.9) v(t) = a0v(t) +g{t), t > 0, (3.10) < 7 o (* ) = 7o«(0(u ( 0 - u(0)> < > 0, (3.11) qi{t) = 7 iu ( < - l) ( u ( i) - u ( i) ) , t> 0 , (3-12) u(0) = 2, u(t) = sin(f) + 2cos(3t), £ € [— 1,0) (3.13) n(0) = 2, (3.14) <7o(0) = 0, gi(0) = l , (3.15) 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where q(0) = (0, 1) axe the initial guesses for q. We chose 70 = 5 and -yi = 5. The true value for q, (q0, qi) = (1,1.5), was used. We simulated the closed loop system over the time interval [0,20]. We have plotted in Figure 3.1 the tracking error, |e(t)| = |u(f) — u(f)|, for t E [0,20]. The convergence of the parameters qo(t) and qi(t) as t — ¥ 00 can be seen in Figure 3.2. 0.4 0.35 0.3 0.25 ^ 0.2 0.15 0.1 0.05 10 t 12 14 16 18 20 Figure 3.1: The tracking error, |e(f)| = |u(t) — u(<)|, of a first order delay equation. 3.3 A Second Order Delay Equation In this example we consider the control of the second order delay equation y(t) + qoi/(t) + qiy{t - I) + qiy{t) = t > o, (3.16) with initial conditions y(0) = 2, y(9) = sin(0) + 2cos(30), 0 € [ - l,O ) , (3.17) 2 /(0) = 1, y(0) = cos(0)-6sin(30), 0 6 [-1 ,0 ). (3.18) 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.8 1 .6 1.4 1.2 qO(t) 0.8 0.6 0.4 0.2 lO t 12 14 16 18 20 Figure 3.2: The convergence of the parameters, qQ (t) and qi(t), of a first order delay equation. This is a damped linear harmonic oscillator with delayed damping The reference model, a second order differential equation without delay, is given by w{t) + a0w{t) +0Q w(t) = g-z{t), t > 0, (3.19) u;(0) = 4.0, tw(0) = 4.5, (3.20) where g2{t) = 2sin(£) + 2cos(£) — 12sin(3<) + 2cos(3f), t > 0, (3.21) q 0 = 2, 0o = 3. (3.22) We transform the linear second order delay equation into a system of two linear first order delay equations by defining «(0 = , t > - U and f ( t ) = i 0 ) , t > 0. (3.23) V 2/(0 J V h ( t ) J 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We also transform the reference model, described by a linear second order differential equation, into a system of two linear first order differential equations with “ d ^ ■ ( ® (t) 1 ■ <3-24) Consequently we have MO = 2/(0 = MO, < > 0 , (3.25) M O = 2/(0 = -<7oMO - tfiM* - 1) - f c M O + /2(0, i > 0, (3-26) ^ ( 0 = ^ ( 0 = M O , < > 0, (3.27) v2{t) = w{t) = -or0u2(0 ~/?oM O -^2(0, * > 0- (3-28) In this example, H = R 2 x La(— 1,0; R 2), Q = R 3, n = 2, r = l, m = l and k = 0. Again, we assume that 9 = (9o,9i,? 2) £ (? is unknown and is to be indentified as the system is adaptively controlled. We define the inner product, < •,• > q on the Hilbert space Q by 2 < q,p >Q=^~tiqiPi, q,peQ, (3.29) 1= 0 where 70, 71,72 € R + are positive weights, or adaptive gains. We can write the equations (3.23)-(3.28) in matrix form, then the notation is consistent with (2.1), (2.2), (2.6) and (2.7), and we have for the plant «(0 = ( W ) + ( ° °- (3'3°) v -<h -9o / V 0 -q i J \ f 2{t) J = Ao(<7M O + M <7)M -1) + /(0 , t > 0, (3.31) and for the reference model we have = ( 1 1 | M ) + ( ! , ) = c ov(t) +g{t), t> 0. (3.32) V -0o ~ “ o J \ M O / From (2.17) we have as control input / ( 0 = CQ u{t) - A0{q(t))u(t) + g{t), t > 0 (3.33) 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and therefore ii(t) = CQ u(t) — Ao{q(t) — q)u(t) — Ai(q(t) — q)u(t — 1) + g(t) (3.34) 0 1 \ v / 0 0 \ u(t) + I _ _ u(t) -0 0 -O C Q j \ 92(f) - q2 q0(t) - q0 ) + (o (3 '3 5 ) v(t) = ( ° 1 I v(t) + ( ° ) , t > 0, (3.36) V -00 -<*0 / V 92{t) / with initial conditions u(0) = | 2 ) , u(i) = ( -^ )+ 2 c o s (3 6 » ) 1 1 ; \ cos(0) — 6sin(30) 1 "(0) = ^ ) • (3.38) The adaptive law for q is given by < 7 o (0 = -lou2(t){u2{t) - v2{t)), t > 0, (3.39) qi{t) = - li u 2( t - l)(u 2 (0 - v 2(t)), t > 0, (3.40) <72(0 = ~ l 2Ui{t){u2{t) - v2(t)), t > 0, (3.41) with the initial guess < 7 o (0 ) = 3, 9i(0) = 3, 92(0) = 3. (3.42) We chose 70 = 4, 71 = 2 and 72 = 16. The true value of 9, 9 = ( < 70, 91, 92) = (1,1.5,2.5), is used. We simulated the closed loop system over the time interval [0,80]. We have plotted the tracking error in Figure 3.3, |e(t)[ = |i/(f) — u>(f)|, for f 6 [0,80]. The convergence of the parameters 90(f), 91(f) and q2(t) as t — » • 00 can be seen in Figure 3.4. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.5 0.45 0.4 0.35 0.3 0.2 0.15 0.1 0.05 10 20 30 50 60 40 70 80 t Figure 3.3: The tracking error, |e(t)| = |y(f) — u;(f)|, of a second order delay equation. 15 10 qO(t) -5 20 50 60 70 O 10 30 40 80 t Figure 3.4: The convergence of the parameters, qo(t), qi(t) and q2[t), of a second order delay equation. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 A Second Order Delay Equation with D istributed Delay In this example we consider the control of the second order delay equation y(0 + %y{t) + qiy(t - l) + / ° t 93(0) y(t + 0)d9 + q2y(t) = f 2{t), t > 0, (3.43) where a distributed delay occurs under the integral sign, with initial conditions y(0) = 2, y{9) = sin(0) + 2cos(3$), 9 € [-1,0), (3-44) 2/(0) = 1, y(9) = cos(9) — 6sin(30), 0 € [ -l,O ). (3.45) The reference model, a second order differential equation without delay, is given by w(t) + Q0w(t) + (3Q w(t) = g2{t), t> 0, (3.46) to(0) = 2.5, ti>(0) = 1.5, (3.47) where g2 is defined as in (3.21), a0 = 2 and (30 = 3. We transform these linear second order delay/differential equations into a system of two linear first order de lay/differential equations by defining u and / as in (3.23) and v and g as in (3.24). Consequently we have «i (t)=y(t) = ti2(<), (3.48) u2{t) = y{t) = -q0u2{t) - q iu 2( t - 1) - q 2ui(t) - J te{0)u2{t + 9)d6 + f 2{t), t > 0, (3.49) vi(t) = w(t) = v2(t), t > 0, (3.50) v2(t) = ui{t) = - a Q v2(t) - PoVi(t) +g2{t), t> 0. (3.51) In this example, H — R 2 x L2(—1,0; R 2), Q = R 3 x L2(—1,0; R), n = 2, r = 1, m = 1 and k = 0. Again, we assume that ( < 7 o5< ? ii< ? 2, 93) € Q is unknown and is to 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be indentified as the system is adaptively controlled. We define the inner product, < •, • > q on the Hilbert space Q by 2 -o <liP >Q~ WiPi + / 7 ( % 3(0)P3(0) de, q , p e Q , (3.52) i = 0 where 70, 71,72 € R + are positive weights and 7 € Lco(R.+) is a positive weighting function. We can write the equations (3.23), (3.24), (3.48)— (3.51) in m atrix form, then the notation is consistent with (2.1), (2.2), (2.6) and (2.7), and we have for the plant i i { t ) = | \ ) t t ( 0 + ( | J ° | “ ( * ~ 1 ) -<?2 -qo J \ 0 — < ? i + f° I ° ° I u(t + e) dO, i > 0 + ( ° I (3.53) 1 ■ 1 0 - m I ~ v /*(<) £( = 4 0(g)u(f) + Ai{q)u{t - 1) + A(q)u(t + 9){0) dO + f{t), t> 0, (3.54) while for the reference model we have "(*) = ( 1 ) u(f)+( , ) = C0v(t) +g{t), t> 0. (3.55) \ -A) -oo / V ) By (2.17) the control input is given by /(f) = C0u(t) - A0(q{t))u(t) - A t(q (t))u (t- 1) + J A(q(t))(9)u(t + 0) dO + g(t), t > 0, (3.56) and therefore ii{t) = C0u{t) - A0(q(t) - q)u{t) - Ai{q(t) - q)u{t - 1) - j ° A(q(t) - q)(9)u(t + 0)d0 + g(t) (3.57) = ( ° 1 ) u(f) + ( ° ° ) u(t) \ - fa — a 0 / I q2(t) - q2 q0{t) - qQ I 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with initial conditions ,A , ( 2 \ I sin(0) + 2cos(30) \ 8 (0)“ ( i ) ' “ ( , ) “ ( « ( * ) - . J * , ) ’ 9 € H ' 0)’ (360) " <0) = ( 1*5 ) ' (3'61) The adaptive law for q is given by < 7 o(0 = ~lou2{t){u2{t) - v2(t)), t > 0, (3.62) ?i(0 = -7i«a(< ~ 1)(«2(0 -«2(0)» (3-63) = ~72Ui(0(u 2(0 - «a(<))» 1 ^ (3.64) J ^(0)D tqz{t,9)pz{0) dO = J ^pz(9)u2(t + 9)(u2( t ) - v 2(t)) d9, (3.65) t > 0, P3 € L2( — 1,0;R ) with initial guess qo(0) = 1, qi(0) = 1.5, fc(Q) = 2.5, (3.66) < fe M = l, 9 G [— 1,0]. (3.67) As was noted above, we approximate L2(—1,0; R ) by (— 1,0; R) = span-fy?^}, j = 0 ,..., N, where are linear B-splines. Therefore, (3.65) only has to hold for ipf G (— 1,0; R), j = 0 ,..., IV , and we replace (3.65) by j\(9)DMt,9)<p?(9) d9 = J % ? (9 )u 2(t + 9)(u2( t ) - v 2(t))d9, (3.68) f > 0, j = Q,...,N 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We approximate qz(t, •) for every t > 0 by an element BN(t, ■ ) 6 1,0; R ), that is defined as follows N = « € [ - ! , 0], i > 0, (3.69) x'=0 where B ^ € Ci(0, oo; R ), i = 0 ,..., N. For convenience we define the vectors BN{t),BN(t) 6 R ^ 1, t > 0 by B "(f) = (B " (f),B ''(i),...,B # (< )) , ( > o, B K(t) = ( 4 w(i), BP(t) ,. . . , Btt(t))T , t > 0. (3.70) (3.71) We can then substitute BN for q3 in (3.68) and obtain J + B)(ui(t) - v2(i)) 00 = f ^ ( 0 ) D ,B ( t,6 ) ^ ( e ) i6 = 1° 7(0) £ B . ^ V f W v f («) di J -■ SS = £ f° 7 (0 V f W v f W de 1= 0 7-1 /?, 7(0)vi5(«vf («) 00 y and therefore 1 f-i <${0)u2(t + 0)(u2(<) - v3(<)) d0 ^ V /-I + 0)(«2(*) - u 2(0) d O ) = (BN(t))TP, i > 0, (3.72) where P = ( /_ “ 7(0)»’?'(0)¥'j'(0) 00) " € R(A r+ 1 » x(ATH-l) (3.73) 2 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Because of the symmetry of P we obtain ( /°r vffW ua(t + 0)(u2(t) - v2(t)) dO \ (B N( t ) f = p ~ l : ^ /-I <dv(0 W * + ^)(u2(i) - u2(f)) dO ) t > 0. (3.74) We then solve (3.74) and obtain an approximation for 93 on [— 1,0]. We set N = 8 and chose for 70 = 0, 71 = 2 and 72 = 1, so that in this example we are only attempting to identify q\, q2 and 93. We defined the weighting function 7(0) = 2 — 5 0,0 €. [— 1,0]. The true value 9, q = (90, 91, 92, 93) = (1,1.5,2.5,93) with 93(0) = r i f l 'e*, ^ ^ [— 1,0], is used. We simulated the closed loop system over the time interval [0,180]. We have plotted the tracking error, |e(f)| = |t/(f) — tt>(£)|, for t S [0,180] in Figure 3.5. We can see the convergence of the parameters 91(f) and 92(f) in Figure 3.6. The convergence of the parameter 93(f), that occurs at about f = 170, is shown in Figure 3.7. o.s 0.4 ^ 0.3 0.2 — 1 — 0.1 - 180 Figure 3.5: The tracking error, |e(f)| = |y(f) — tu(f)|, of a second order delay equation with distributed delay. 28 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3.5 2.5 1.5 0.5 20 40 60 80 100 120 160 140 180 time t Figure 3.6: The convergence of the parameters qi(t) and of a second order delay equation with distributed delay. 2.5 q3(o> q3(180) dotted line d ash ed line solid line q3bar 1.5 0.5 L - 1 -0.9 - 0.8 -0.7 - 0.6 -0.5 -0.4 -0.3 - 0.2 - 0 .1 Figure 3.7: The convergence of the parameter 93(f), of a second order delay equation with distributed delay. 29 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3.5 A Nonlinear Delay Equation We note that strictly speaking, this example is not covered by the theory in Chapter 2. However, the theory can be extended, so that the desired convergence for the following problem is guaranteed. We consider the control of the nonlinear delay equation u(t) = q0u{t) + qi(u{t))u(t - 1) + /(<), t > 0, (3.75) with initial conditions u(0) = 2, u(6) =sin(0) + 2cos(3 0), 0 6 [-1,0]. (3.76) The reference model, a lineax differential equation without delay, is given by i)(t) = a0v(t) + g(t), t> 0, (3.77) and u(0) = 4, (3.78) where g{t) = 2 sin(f) + cos(f) — 6 sin(3i) -t- 4 cos(3f), t > 0, (3.79) and Q 0 = -2 . (3.80) In this example, H = R, Q = R x L2(— oo,oo;R ), n = 1, r = 1 and k = 0. Again, we assum that q = (qo,qi) € Q is unknown and is to be identified as the system is adaptively controlled. We define the inner product, < •, • > q on the Hilbert space Q by < q ,P > Q = loQoPo + f 7(% i(0)Pi(0) dO, p ,q € Q , (3.81) J — oo where 70 £ R + is a positive weight and 7 6 £<»( — 00, 00; R +) is a positive weighting function. We choose as control input f{t) = a0u {t)-q 0{t)u(t)-ql{t,u(t))u(t-l)+g(t), t > 0, (3.82) 30 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. and the closed loop system becomes then u{t) = a0u(t) - ( < 7 o(*) - qo)u(t) - {gj(t,u(t)) - qi{u{t))} u{t - 1) + g(t),{3.83) v(t) = a0v(t) + g(t), t > 0, (3.84) with initial conditions u(0) = 2, u(0) = sin(0) + 2cos(3 8), d £ [— 1,0] y(0) = 4. The adaptive law for q is given by with initial guess (3.85) (3.86) <7o(0 = 7ou(*)(«(0 - u(<))> t > 0, (3.87) i-<x l{G)Dtqi{t,0)pl{e) d.0 = pl{u{t))u{t-l){u{t)-v{t)), (3.88) t > 0, pi € L2{— oo, o o ; R ), (3.89) <7o(0) = 2, <?i(O,0)=O, Q e ( - 00, 00). (3.90) Similar to the example before, we approximate Li{—oo, oo; R ) by (— oo, oo; R) span{ip^’M}, j = 0,..., N, where <Pj (0 ) p - j + 1 6 e [ V - l ) % J $ ) , l < j < N - l -%o + j + i ee[j%,(j + i ) % ) , o < j < N - \ 0 otherwise (3.91) and M € R + is some positive number. We assume that qi is symmetric with respect to the y-axis and that qi(t) — * ■ 0 as t — > ■ oo. Therefore, (3.88) only has to hold for G Li?{—oo, oo; R ), j = 0 ,..., N — 1, and we replace (3.88) by / : ‘y{9)Dtql(t,0)'p?'M(0) d6 = ^ " ( u ( 0 M « - l ) ( « ( 0 -« (< )), (3-92) t> 0, j = 0 ,..., N - 1 31 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. We approximate qi(t, •) for every t > 0 by an element B N(t,-) € L^i—00, 00; R), that is defined as follows N-l (3.93) 1= 0 where Bf1 £ C i(0, 00; R), i = 0, . . . , N — 1. For convenience we define the vectors B"(t),BN{t) € R " , t > 0 by B'v(() = (B0 A '(i),B f'((),...,e jy _ 1(()) , < > 0, BN(t) = (s* '(i). B f'to . • • •. B «-.(*))T . ( > 0. We can then substitute BN for qi in (3.92) and obtain f j M(u(t))u(t - l)(u(i) - »(<)) = r 7WD<BK(t,6)v”M(e) d0 J — 0 0 = r 7 ( 9 ) E B?(t)v “- M(e )v fM(i> ) ds ,= 0 = E B "(0 r 7 ( 9 ) v ? U '(9VfW ( 9 ) de i=0 • /-°° (3.94) (3.95) = (£ "(* ))r and therefore / ^ • " ( « ( 0 )ti(«- ! ) ( « ( < ) -® ( 0 ) N V - 1)(«(0 - u( 0 ) ) = (BN{t))TP, t> 0, (3.96) where € R yvxjv (3.97) 32 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. and because of the symmetry of P we obtain - l \ - !)(«(<) - u(0) J t > 0. (3.98) We solve (3.98) and can then approximate qx on (— 00, 00). We set N = 8, M = 12 and chose 70 = 3. We defined the weighting function 7(0) = 0 otherwise _ , e-(8 /l.S)2 . The true value q, q = (qo,qi) = (l,<7 i) with qi(9) = 4-— ^ — , 9 € (— 00, 00), is used. We simulated the closed loop system over the time interval [0,190]. We have plotted the tracking error, |e(t)| = |u(£) — u(i)|, for t € [0,190] in Figure 3.8. The convergence of the parameter qi(t) occurs at about t = 180. We can see the convergence of the parameter q0(t) in Figure 3.9 and qi(t) in Figure 3.10. 0.45 0.35 ^ 0.25 0.15 0.05 40 60 80 100 120 140 160 180 Figure 3.8: The tracking error, |e(<)| = |u(£) — «(£)!> of a nonlinear delay equation. 33 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2.5 1.5 o cr 0.5 -0.5 - 1 40 20 60 80 100 120 140 160 180 t Figure 3.9: The convergence of the parameter qo(t), of a nonlinear delay equation. 2.5 qi(0) q1(190) qlbar dotted line dashed line solid line 1.5 0.5 - 0 .5 Figure 3.10: The convergence of the parameter qi{t), of a nonlinear delay equation. 34 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Chapter 4 Conclusion We presented a model reference adaptive control scheme for delay systems, where both the plant and the reference model axe linear delay equations. A feedback control law for the plant was defined. We obtained the closed loop system. We established tracking error convergence under certain conditions on the reference model dynamics. The unknown parameters were estimated at the same tim e and were updated. With the persistence of excitation assumption, we showed th at the parameter error goes to zero. We illustrated the theory on several examples, including a linear first and second order delay differential equation, and a nonlinear delay differential equation. A problem for further study involves the persistence of excitation assumption. In general it is very difficult to verify this assumption. Is it possible to find a more readily verifiable condition which insures parameter convergence? Also, we restricted g, the input of the reference model, to be an element of Z /2(0, oo; R"). Could we choose g 6 Lp(0, oo;R n) with 1 < p < oo, so as to obtain a more general result? Finally, the well posedness of the closed loop system was not investigated. Therefore, a careful analysis of the question of existence, uniqueness and continuous dependence of the closed loop system would be another interesting topic for further study.. Our last example, in which we considered a nonlinear plant, is not covered by the theory in this thesis. The extension of the theory to the nonlinear case should be possible and is also a possible extension of the effort described here. 35 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. A ppendix A Source Code for Example 3.4 /* * D lin2Int.c lin ear, second order delay d iffe re n tia l equations with * distribu ted delay * / /* * the follow in g function of the IM S L library are used: ivpag_, sset_ , * crgrb_, lftq s _ , lfsqs_, qdag_ */ /* constants */ #define G A M M A O 0.0 #define G A M M A 1 2.0 #define G A M M A 2 1.0 #define G A M M A 4 7.0 #define G A M M A 5 2.0 #define A 0B A R 1.0 #define A 1B A R 1.5 #define B0BA R 2.5 #define A L P H A 0 2.0 #define BET A 0 3.0 / * in it ia l values */ #define A0INITIAL 1.5 #define A1INITIAL 2.0 #define B0INITIAL 2.0 #define AINITIAL 1.0 #define E T A 2.0 #define XSI 1.0 #define V00 2.5 #define V01 1.5 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. #define NSIZE 1000 #define Ninterval 8 #define Na 8 #define SE C O N D S 190.0 #define F IL E N A M E "Dlin2Intb" #define M M A X 200 #define D 0T SPER SE C 0N D 10 #define A F R E Q U E N Z 20 /* tech n ical constants (necessary for IM SL) */ #def ine N T (NSIZE* (MMAX+D+l) #define C 0FA C T0R S 1 #define M X P A R M 50 #define N E Q 7 #define N X 2 #define N SY S (NEQ+Na+1) #include <stdlib.h> #include <stdio.h> #include <math.h> #include <string.h> #include "phi.c" #include "imsl.h" flo a t xx[NT][NX]; /* d iscretizied values for x l ( t ) , x2(t) */ flo a t sp lin e [Ninterval] [Na + 1] [2] , aBartheta [Ninterval] [2] ; int K = 0, I , J ; int NaPlusl = Na + 1, cofactors = C0FACT0RS, ldbandfac = C 0FA C T0R S + 1; flo a t W [Na + 1] , t t t , bandfac[Na + l] [C0FACT0RS + l] ; flo a t gam m aO = G A M M A O , gammal = G A M M A 1 , gamma2 = G A M M A 2 , gamma4 = G A M M A 4 , gamma5 = G A M M A 5 ; flo a t phiO (float theta) { /* y (t) on [-1,0] */ return (sin (th eta) + 2.0 * cos(3.0 * th e ta )); > R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. flo a t psiO (float theta) { /* y '(t) on [-1 ,0 ] */ return (cos(th eta) - 6.0 * sin (3.0 * th e ta )); } flo a t aBar(float theta) { /* -1.0 <= th eta <= 0.0 */ return (1.5 / (1.0 - exp(-l.O )) * exp(theta)); } flo a t gammaf(float theta) ■ C /* -1.0 <= th eta <= 0.0 * / flo a t gam; theta = theta + 1; gam = (1.0 - theta) * gam m a4 + theta * gamma5; return (gam); > flo a t sumAspl ine (char nameD , int nr, flo a t A[]) i FILE *poly; in t j , k, step s = 200; flo a t theta, d eltath eta = 1.0 / steps, At, err, sumErr; char filename [30] , number [10]; strcpy(filenam e, name); sprintf(number, ’"/,03i", nr); strcat(filenam e, number); poly = fopen(filenam e, "wt"); theta = -1.0; sumErr = 0.0; for (j = 0; j <= steps; j++) { R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. At = 0.0; for (k = 0; k <= Na; k++) At = At + A[k] * phi(theta + 1 .0, k, Na); err = fabs(At - aBar(theta)); sumErr = sumErr + err * err; fp rin tf (poly, '/.f, */.f, '/,f\n", th eta, At, aBar(theta), err); theta = theta + deltatheta; > fc lo se (p o ly ); return (sumErr); > flo a t x(in t index, flo a t theta) / * index means xl or x2 * / int k, ind exleft; flo a t h, w eigh tleft, weightright; k = theta * NSIZE; i f (k > theta * NSIZE) k—; indexleft = k + NSIZE; weightright = fabs(theta * NSIZE - k ); w eightleft = 1.0 - weightright; h = (w eightleft * xx[ind exleft] [index - 1] + weightright * xx[indexleft + l] [index - 1]); return (h ); > void dummy(int *n, flo a t * t, flo a t yD , flo a t yprimeQ) >; flo a t f(flo a t t) flo a t hh; hh = B E T A 0 * (sin (t) + 2.0 * cos(3.0 * t) ) 39 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. + A L P H A O * (co s(t) - 6.0 * sin (3 .0 * t)) - sin (t) - 18.0 * cos(3.0 * t) ; return (hh); > flo a t h (float *xpointer) { /* -1.0 <= th eta <= 0.0 */ flo a t y, x = *xpointer; y = 1.0 * gammaf(x) * phi(x + 1.0, I, Na) * phi(x + 1.0, J, Na); return (y ) ; > extern void rhsEquation(int *nsyspointer, flo a t *T, flo a t Y d , flo a t YprimeQ) /* procedure represents system of d iffe re n tia l equations */ flo a t t = *T, x2tminusl, tt; int nsys = *nsyspointer; x2tminusl = x(2, t - 1.0); / * System of equations */ / * Y[0] = x l, Y[l]=x2, Y[2] =vl, Y[3]=v2, Y[4]=a0, Y[5]=al, Y[6]=b0 * / Yprime[0] = Y[l] ; YprimeCl] = (-B0BAR - B E T A 0 + Y[6]) * Y[0] + (-A0BAR - A L P H A O + Y[4]) * Y[1] + (-A1BAR + Y[5]) * x2tminusl + f ( t ) ; Yprime[2] = Y[3] ; Yprime [3] = -BETA0 * Y[2] - A L P H A O * Y[3] + f ( t ) ; Yprime[4] = -gam m aO * Y[l] * C Y Cl] - Y[3]); Yprime [5] = -gammal * x2tminusl * (Y[l] - Y[3]); Yprime [6] = -gamma2 * Y[0] * (Y[l] - Y[3]); / * co efficien ts Aj for sum(Aj*spline(theta)) * / flo a t th e ta l, theta2, x2thetal, x2theta2, c, h, aNal, aNa2; 40 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. int i /* 0<=i<=Na * / , ] / * l<=j<=Ninterval */ , ido; h = 1.0 / Ninterval; for ( i = 0; i <= Na; i++) Yprime[NEQ + i] = 0.0; c = sq rt(3.0) / 6.0; for (j = 1; j <= Ninterval; j++) { /* consider j-th interval of [-1,0] and [ t - l ,t ] */ th eta l = (j -0 .5 -c ) * h; theta2 = C j - 0. 5+c) * h; x2thetal = x(2, t - 1.0 + th eta l); x2theta2 = x(2, t - 1.0 + th eta2); aNal =0.0; aNa2 = 0.0; for ( i = 0; i <= Na; i++) { aNal = aNal + Y [N E Q + i] * spline [j - 1] [i] [0] ; aNa2 = aNa2 + Y [N E Q + i] * spline [j - 1] [i] [1] ; Yprime [N E Q + i] = Yprime [N E Q + i ] + h / 2 . 0 * (Y[l] - Y[3]) * (x2thetal * sp lin e [j - 1] [i] [0] + x2theta2 * spline [j - 1] [i] [1] ) ; > /* for */ Yprime [1] = Yprime [1] + ((aBartheta[j - 1] [0] - aNal) * x2th etal + (aBartheta[j - 1] [1] - aNa2) * x2theta2) * h / 2.0; > / * for * / } { flo a t z[Na + 1] ; int i ; Ifsqs_(&NaPlusl, bandfac, &ldbandfac, fccofactors, & (Yprime[NEQ]) , z ) ; for ( i = 0; i <= Na; i++) Yprime [i + N E Q ] = z [i] ; > } /* procedure rhsEquation * / void main(int argc, char *argvD) { int nsys = NSYS, m xparm = M X P A R M ; flo a t param [M X P A R M ] ; flo a t z[NSYS], T, tend, tolera, Y[NSYS], step size = 1.0 / NSIZE; int i /* 0<=i<=Na * / , j / * l<=j<=Ninterval */ , ido; 41 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. char f ilen am e[100]; in t m ; /* how many seconds to run */ m = SE C O N D S; strcpy(filenam e, FIL E N A M E ); /* I n itia liz a tio n */ for ( i = 0; i < NSIZE; i++) { T = ( i - NSIZE) * stepsize; x x [ i] [0] = phiO(T); x x [ i] [1] = psiO(T); > T = 0.0; Y[0] = E T A ; Y[l] = XSI; Y[2] = V00; Y[3] = V01; Y[4] = A O INITIAL; Y[5] = A1INITIAL; Y[6] = BOINITIAL; for (j = 0; j < N X ; j++) { xx [NSIZE] [j] = Y[j] ; } / * I n itia liz e A[j,N] and Q */ { flo a t qband[Na + 1] [C 0FA C T0R S + 1] , q[Na + 1] [Na + 1] ; flo a t a, b, errabs, errest, e r r e l, resu lt; int nlc = 0, nuc = 1 /* # of lower and upper codiagonals * / , co2 = C0FA CT0R S + 1; int iru le = 6; for ( i = 0; i <= Na; i++) { for (j = 0 ; j <= Na; j++) q [i] [j] =0.0; Y [N E Q + i] = AINITIAL; > a = -1.0; b = 0.0; / * Interval bounds * / R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. errabs = 0.0; errel = 0.001; for ( i = 0; i <= Na; i++) { /* diagonal of Q * / / * Integrate function without sin g u la rities * / I » i; J = i; /* printfC"('/.i,*/.i) ",I,J); */ qdag_(h, Aa, Ab, Aerrabs, Aerrel, Airule, A result, A errest); q [i] Ci] = r e su lt; } for ( i = 0; i < Na; i++) { /* f ir s t upper codiagonal of * Q * / /* Integrate function without sin g u la rities * / I = i; J = i + 1; qdag_(h, &a, &b, Aerrabs, Aerrel, Airule, A result, A errest); q[I] [J] = r e su lt; /* printfC "C /.i,7.i) M , J ) ; */ q[J] [I] = r e s u lt ; > / * transform q in bandmatrix * / crgrb_(ANaPlusl, q, ANaPlusl, Anlc, Anuc, qband, Aldbandfac); / * cholesky-factorization for qband * / Iftqs_(ANaPlusl, qband, Aldbandfac, Acofactors, bandfac, Aldbandfac); } /* evaluate aBair and phi in thetal and theta2 and store it * / { flo a t th e ta l, theta2, c, h; c = sq rt(3.0) / 6.0; h = 1.0 / Ninterval; for (j = 1; j <= Ninterval; j++) { thetal = (j - 0.5 - c) * h; theta2 = (j - 0.5 + c) * h; aBarthetaCj - 1] [0] = aLBar(thetal - 1.0); aBartheta[j - 1] [1] = aLBax(theta2 - 1.0); for (i = 0; i <= Na; i++) { sp lin e[j - 1] [ i ] [0] = p h i(th etal, i , Na) ; sp lin e[j - 1] [ i ] [ l ] = phi(theta2, i , Na) ; > 43 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. > > /* technical things (necessary for IN T E G R A T O R ) * / { flo a t nu ll = 0.0; in t index = 1; sset_(&mxparm, &null, param, Aindex); > tolera = 0.002; param[0] = stepsize; param[3] = 100000.0; param[9] = 1.0; param[11] = 2.0; / * gear’s method */ param[12] = 2.0; / * divided difference * / ido = 1; printf ("Outputformat in '/.s: t, Ie(t) I, y(t), w(t) , a0(t), al(t), b 0(t)\n " , filename); printf ("Outputformat in '/.s20: t , q(t), q b ar(t), lr(t)|\n", filename); /* Iteration for d iff’ equation solver * / { in t dots = NSIZE / D 0TSPER SEC 0N D , index; FILE *acoeff, *out; flo a t norm; char AA[20] ; strcpy(AA, fileneime); strcat(AA, ".AA"); acoeff = fopen(AA, "wt") ; out = fopen(filenam e, "wt"); i f (out == N U L L ) { printf ("Could not open '/.s. \nExit\n", filenam e); e x it(l) ; > for (index = 1; index < m * NSIZE; index++) { tend = index * step size; ivpag_(&ido, &nsys, rhsEquation, dum m y, z, & T , fttend, fetolera, param, Y); for (i = 0; i < N X ; i++) { 44 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. xx [NSIZE + index] [i] = Y[i] ; > i f (index '/, dots == 0) { /* * t , e r r o r (t), y(t), w(t), a0(t), al(t), * b0(t) */ f printf (out, "%f , '/.f, '/.f, '/.f, */.f, '/,f, '/. f \n", T, fabs(Y[0] - Y[2] ) , Y[0] , Y[2] , Y[4] , Y[5] , Y[6] ) ; fp rin tf (a co eff, "*/tf ", T ); for (i = 0; i <= Na; i++) fp rin tf (a co eff, ",'/.f ", Y [N E Q + i]); fp r in tf(a c o e ff, "\n"); > i f (index '/. (NSIZE * A F R E Q U E N Z ) == 0) norm = sumAspline(filename, T, &Y[NEQ]); > sumAspline(filename, T, &Y[NEQ]); fc lo se (o u t); fc lo se (a c o e ff); > /* IM S L * / ido = 3; ivpag_(&ido, &nsys, rhsEquation, dum m y, z, & T , fttend, fttolera, param, Y ); > R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Reference List [1] H. T. Banks, J. A. Bums, and E. M. Cliff. Parameter estimation and identifica tion. SIAM J. Control, 19(6):791— 828, 1981. [2] H. T. Banks and T. Tsukazan. An abstract framework for approximate solutions to optimal control problems governed by hereditary systems. In International Conference on Differential Equations, H. A. Antosiewicz, editor. Academic Press, Inc., 1974. [3] M. Bohm, M.A. Demetriou, S. Reich, and I.G. Rosen. A model reference adaptive control scheme for nonlinear infinite dimensional systems. SIAM J. Control, to appear. [4] R. D. Driver. Ordinary and Delay Differential Equation, volume 20 of Applied Mathematical Sciences. Springer-Verlag, 1977. [5] E. Kreyszig. Introductory Functional Analysis With Applications. John Wiley k Sons, 1978. [6] K. S. Narendra and A. M. Annaswamy. Stable Adaptive Systems. Prentice Hall Information and System Sciences Series. Prentice-Hall, 1989. 46 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

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Creator Busson, Michael (author)

Core Title A model reference adaptive control scheme for linear delay systems

School Graduate School

Degree Master of Science

Degree Program Applied Mathematics

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

Tag engineering, industrial,Mathematics,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Rosen, Gary (committee chair), Proskurowski, Wlodek (committee member), Wang, Chunming (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-12299

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

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