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Linear quadratic control, estimation, and tracking for random abstract parabolic systems with application to transdermal alcohol biosensing
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Linear quadratic control, estimation, and tracking for random abstract parabolic systems with application to transdermal alcohol biosensing
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Linear Quadratic Control, Estimation, and Tracking for Random Abstract Parabolic Systems with Application to Transdermal Alcohol Biosensing by Mengsha Yao A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Applied Mathematics) August 2021 Copyright 2021 Mengsha Yao Dedication To my parents. ii Acknowledgements First and foremost, I would like to express my sincere gratitude to my advisor Prof. Gary Rosen. I appreciate all his eorts, enthusiasm, and patience towards my research over the last ve years. I admire his immense knowledge. No matter how small questions I ask, he oers inspired response, which helps me a lot on understanding my questions. He has a wide range of knowledge and has the ability to bring me happiness, from advanced mathematics to basic language knowledge, from American culture to interesting life stories; every conversation with him is a wonderful journey. He knew that I love playing piano and personally led me to a piano room in another building on campus. Words cannot fully express how thankful I am to have such a good advisor. I could not have nished this dicult journey without his constant encouragement. It is he who taught me how to be a good researcher, scientist, and professor. In addition, I would like to thank the rest of my committee members, Prof. Susan Luczak, and Prof. Chunming Wang for their valuable comments on my project. Many thanks for their support and kind counsel. Also I would like to thank Prof. Susan Luczak for her advice on our poster and her contribution to the data we used in this research. I am also very grateful to the whole mathematical department for any and every role they have played in my doctoral education. Every talented professor, sta member and friend I meet here contribute a lot to my colorful study life and in uence me to become a better person. In their company, the time I spent at University of Southern California has been an extremely memorable experience, and it is also the huge treasure for my whole life. iii Last but not least, I would like to thank my parents: Ludi Yao and Li Lin, for encouraging me to pursue my dreams, for supporting me in life, and showering me with endless love. Thanks for all your encouragement! iv Table of Contents Dedication ii Acknowledgements iii List Of Tables vii List Of Figures viii Abstract x Chapter 1: Introduction 1 1.1 The Alcohol Biosensor Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 A Brief Survey of the Relevant Literature . . . . . . . . . . . . . . . . . . . . 7 1.3 Outline of the Remainder of the Thesis . . . . . . . . . . . . . . . . . . . . . 10 Chapter 2: Preliminaries 12 Chapter 3: A Model for the Metabolism and Transdermal Alcohol Transport 18 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Linearization about an Equilibrium Solution . . . . . . . . . . . . . . . . . . 21 Chapter 4: The Population Model in State Space Form 23 4.1 Abstract Parabolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Random Parameters as Auxiliary Spatial Variables . . . . . . . . . . . . . . 30 4.3 Approximation and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 32 Chapter 5: Linear Quadratic Gaussian Control Problems 35 5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 LQG Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.3 Approximation and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.4 Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 6: Linear Quadratic Gaussian Tracking Problems 51 6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.2 Approximation and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.3 Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 v Chapter 7: Numerical Results 58 7.1 The Numerical Convergence of the Approximation . . . . . . . . . . . . . . . 58 7.2 The Values of the Performance Index by Dierent Approximating Optimal Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.3 Simulations of an Intravenously-Infused Alcohol Study with Transdermal Sens- ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.4 Tracking BAC in Intravenous Alcohol Studies . . . . . . . . . . . . . . . . . 67 7.4.1 A Linearization-Based Approach . . . . . . . . . . . . . . . . . . . . 68 7.4.2 A Nonlinear Observer and Feedback Controller . . . . . . . . . . . . 71 7.5 Deconvolving BAC or BrAC from TAC . . . . . . . . . . . . . . . . . . . . . 73 Chapter 8: Discussion, Concluding Remarks, and Future Research 76 Reference List 79 vi List Of Tables 7.1 L 2 norm of the dierence between the approximating optimal functional con- trol gains and the innite-dimensional optimal functional control gains. . . . 59 7.2 L 2 norm of the dierence between the approximating optimal functional ob- server gains and the innite-dimensional optimal functional observer gains. . 60 7.3 Optimal functional control gains ^ f 1 and ^ f 2 for various values of q 1 and the expected values of these gains whenq Beta(;) with = 3 and = 2. . 61 7.4 Value of the performance index from simulations using dierent controllers/- compensators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.5 Value of the performance index when q 1 =q 1;k Beta(3; 2) using dierent controllers/compensators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 vii List Of Figures 1.1 (Left) Giner, Inc. (Newton, Massachusetts) WristTAS TM 7 and (Right) https://www.scramsystems.com/ SCRAM Systems (Alcohol Monitoring Systems, Littleton Colorado) Transder- mal Continuous Alcohol Monitoring Devices. . . . . . . . . . . . . . . . . . . 3 7.1 Optimal functional control gains for n =m = 4; 8; 16; 32. . . . . . . . . . . . 59 7.2 Optimal functional observer gains for n =m = 4; 8; 16; 32. . . . . . . . . . . 60 7.3 Optimal functional control gains ^ f 3 for various values of q 1 and the expected values of these gains whenq 1 Beta(;) with = 3 and = 2. . . . . . . 61 7.4 First simulation example: BAC curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compensator but designed with incorrect value for, 3) no control is applied (i.e. open loop), and 4) simulated innite-dimensional compensator designed with correct or true value of . . . 65 7.5 First simulation example: Control (intravenous infusion rate) trajectory curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compensator but designed with incorrect value for, and 3) simulated innite- dimensional compensator designed with correct or true value of . 4) control is steady. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.6 First simulation example: Control (intravenous infusion rate) increment to steady state control trajectory curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compensator but designed with incorrect value for , and 3) simulated innite-dimensional compensator de- signed with correct or true value of . . . . . . . . . . . . . . . . . . . . . . . 67 7.7 Second simulation example: BAC curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compensator but designed with incorrect values for and b, 3) no control is applied (i.e. open loop), and 4) simulated innite-dimensional compensator designed with correct or true values of and b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 viii 7.8 Second simulation example: Control (intravenous infusion rate) trajectory curves when 1) compensator designed under uncertainty, 2) simulated innite- dimensional compensator but designed with incorrect values for andb, and 3) simulated innite-dimensional compensator designed with correct or true values of and b. 4) control is steady. . . . . . . . . . . . . . . . . . . . . . 69 7.9 Second simulation example: Control (intravenous infusion rate) increment to steady state control trajectory curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compensator but designed with incorrect values for and b, and 3) simulated innite-dimensional compen- sator designed with correct or true values of and b. . . . . . . . . . . . . . 70 7.10 Tracked BAC and tracking control by the linear approach . . . . . . . . . . . 71 7.11 Tracked BAC and tracking control by the nonlinear approach . . . . . . . . . 72 7.12 Tracked TAC and Estimated BrAC . . . . . . . . . . . . . . . . . . . . . . . 75 ix Abstract This research is motivated by the following two applications involving biosensor-measured transdermal alcohol concentration (TAC). A TAC biosensor measures the ethanol content in perspiration. The rst application is the control of intravenously-infused alcohol studies based on a population model for the study participant or subject and TAC sensing, while the second application is estimating blood or breath alcohol concentration (respectively, BAC or BrAC) from TAC. A dynamical model for the underlying control system is established. It takes the form of a semi-linear, parabolic PDE/ODE hybrid system describing the transport of ethanol from the blood through the skin, its excretion within perspiration, and nally its measurement on the surface of the skin by an electro-chemical biosensor. Since the parameters of this dynamical model can vary with the individual wearing the sensor, the particular sensor being worn, and environmental factors such as ambient temperature and humidity, we allow the model parameters to be random with either known or estimated distribution. A state space formulation of the model set in an appropriately constructed Gelfand triple of Bochner spaces is derived wherein the random parameters are treated as additional spatial variables. The resulting population model takes the form of an abstract parabolic hybrid system involving coupled partial and ordinary dierential equations with random parameters. For intravenously-infused alcohol studies, there are two aims. One is clamping: to have the subject's BAC either reach a specied level of intoxication and then remain at that level for an extended period of time. To solve this problem, a discrete time linear quadratic (LQ) regulator is coupled with a linear quadratic Gaussian (LQG) compensator for the linearized x random abstract parabolic systems obtained from the semi-linear population model. A nite- dimensional Galerkin-based approximation and convergence theory for the closed-loop linear quadratic control and estimation of abstract parabolic systems with random parameters is developed. The ecacy of our approach and the performance of our control design are demonstrated with numerical studies. A realistic simulation of an intravenously-infused alcohol clamping study with transdermal sensing and our compensator is included. The second aim of intravenously-infused alcohol studies is tracking: to force the blood al- cohol concentration follow a pre-specied target trajectory of intoxication level by controlled infusion. This problem as well as the problem of deconvolving BAC from TAC is reformu- lated as a discrete time LQG tracking problem based on the same population model used in the clamping studies. Similarly, the convergence of nite-dimensional approximations is es- tablished, simulation studies for both a linear and nonlinear compensator are presented, and numerical results for the regularized deconvolution of BAC or BrAC from TAC are presented and discussed. xi Chapter 1 Introduction 1.1 The Alcohol Biosensor Problem This research is motivated by two important applications of transdermal alcohol biosen- sors which allow for the measurement of alcohol excreted from the body through the skin via perspiration. The rst is the control of an intravenously-infused alcohol clamping study whose aim is to have the subject or participant either 1) (clamping) reach a specied level of intoxication and to then remain at the level for an extended period of time, or alterna- tively, 2) (tracking) to follow a pre-specied target time varying trajectory of intoxication, controlled in real time by intravenous ethanol infusion. The second application takes the form of an estimation problem. It involves the deconvolution of blood or breath alcohol concentration (respectively, BAC or BrAC) from biosensor-measured transdermal alcohol concentration (TAC). The crucial aspect of this research in the context of both applications is related to being able to obtain an estimate for BAC or BrAC from TAC. Before transdermal alcohol biosensors had been fully developed, there were, and still are, direct methods for obtaining BAC/BrAC data. Directly measuring alcohol concentration in the blood by administering a blood test is generally accepted as the most reliable method. However this method has the obvious drawback of being invasive, expensive, and essentially impossible to do in real time. For these reasons it is unacceptable for use as a sensor in a closed-loop feedback control law. On the other hand, BrAC measurements are much easier 1 to obtain from deep breaths exhaled into a relatively portable instrument known as a breath analyzer. The breath analyzer produces an estimate of BrAC based on empirical models from elementary chemistry, a crude model for the interaction between blood and breath in the lungs, and an analog electro-chemical sensor that produces a current from ethanol molecules detected in the breath sample. If one accepts BrAC as a reasonable surrogate for BAC, it is possible, at least in a relatively crude manner, to use a breath analyzer as the sensor in clamping studies. This is typically done using one of two protocols. They can be described as follows. 1) Following a pre-calculated oral dosing to bring the BrAC up to the desired level, which typically takes 40 minutes, the researcher would have the study participant periodically breathe into a breath analyzer and based on the reading, manually adjust the intravenous ow of ethanol in the hope of maintaining BrAC at the desired clamped constant level. 2) A "quick clamping" method has been developed in which a sophisticated pharma- cokinetic model with participant-dependent parameters is used to calculate an appropriate intravenous dose that will rapidly bring the participant's BrAC up to the target level (typi- cally taking 20 minutes) and then maintain it at that level for an extended period of time. Unfortunately however, obtaining breath measurements also presents a number of chal- lenges that make it also unacceptable as a sensor in a real time controller. It requires the active participation of the study subject which can interfere with the experimental design. Moreover, deep breath samples can be dicult to produce, even for an expert, and the cycle time is really not fast enough for a feedback control. Recent and ongoing advances in biosensor technology, on the other hand, have now made it possible to passively, nearly continuously, and in real time, measure the amount of alcohol excreted in vapor through the skin ([31], [40], [41], [42], [43]). This measurement is known as transdermal alcohol concentration or TAC. The sensor hardware used is also basically electro-chemical in nature; it takes the form of what is essentially a miniaturized fuel cell. It performs a redox reaction that yields four electrons, and thus an electrical current for 2 Figure 1.1: (Left) Giner, Inc. (Newton, Massachusetts) WristTAS TM 7 and (Right) https://www.scramsystems.com/ SCRAM Systems (Alcohol Monitoring Systems, Littleton Colorado) Transdermal Continuous Alcohol Monitoring Devices. each molecule of ethanol. The sensor is packaged to resemble a digital watch, a wearable personal tness monitor, or an ankle bracelet (see two existing devices in Fig. 1.1). These wearable biosensors have now made it possible for researchers and clinicians who study and treat people suering from alcohol use disorder to collect data passively, unobtrusively, and nearly continuously in the eld and in naturalistic settings. One of the foci of the research reported on in this thesis is the investigation of whether or not these devices can be used as the real-time sensor in a fully automated alcohol intox- ication clamping or tracking study. However, a major obstacle to alcohol researchers and clinicians in attempting to use these TAC devices to quantitatively measure alcohol levels in the human body is that the TAC measurement they provide does not consistently map onto the BAC, which is the well-understood measure of alcohol concentration in the body. It seems that TAC sensors exhibit signicant sensitivity with respect to variations in the physi- ological characteristics of the individual being tested, ambient environmental conditions, and variability in sensor hardware. Indeed, variations in skin thickness, porosity, and tortuosity, blood ow, level of physical activity, perspiration/sweating, and ambient temperature and humidity can all aect the levels of TAC. This is in contrast to the breath analyzer in that BrAC yields a more robust estimate of BAC. Indeed, as we noted earlier, the breath analyzer also uses a redox reaction to produce a current from molecules of ethanol in the breath sam- ple, but the conversion of this current to an estimate of BAC is based on Henry's Law ([26]). 3 Henry's law involves a single parameter known as the partition coecient, which is reason- ably robust with respect to the individual who is being tested and ambient environmental conditions. Hardware variance can eectively be dealt with via calibration. Consequently, before TAC biosensors can be used in the lab, clinic or in the eld, and in particular, as a real-time senor as part of a controller for alcohol clamping and tracking studies, a reliable and consistent method for converting TAC into either BAC or BrAC must be developed. In this dissertation, we are proposing to use novel transdermal alcohol biosensor technology together with sophisticated mathematical models and strategies to realize this conversion and to then use this technology to design feedback control laws that allow for the complete automation of alcohol clamping and tracking studies. In addition, as a by-product of our investigation of the development of these controllers, we also came upon an ecient and eective method for estimating or deconvolving BAC from TAC in the eld that has the potential to make wearable transdermal alcohol sensors more practical and useful to clinicians, researchers, and to the ordinary consumers. Although the two aims (clamping/tracking studies and deconvolution of BAC from TAC) and their solutions are closely related, somewhat dierent advanced mathematical strategies are involved. The rst aim in controlling intravenously-infused alcohol clamping studies involves nding the optimal infused alcohol ow rate with the aid of a TAC sensor so that a desired level of BAC can be reached and maintained for an extended period of time at the least cost. To solve this control problem, we employ a discrete time linear quadratic (LQ) regulator coupled with a linear quadratic Gaussian (LQG) observer. The regulator together with the observer is known as an LQG compensator. The control problem is formulated in an innite- dimensional state space determined by the underlying model or state equation that takes the form of a random abstract parabolic system. Then by taking advantage of relatively recent advances (over the last few decades) in linear quadratic control and estimation theory for innite-dimensional systems, and a new way to deal with population models in the form of state and output equations involving random parameters, we are able to 1) completely 4 automate a clamping or tracking study, 2) eliminate the need to calibrate the underlying model to a particular individual, device, and environmental conditions, and 3) no longer require the active involvement of the participant. Our population model for the transport of ethanol from the blood through the dermal (with an active blood supply) and epidermal (without an active blood supply) layers of the skin, its excretion via perspiration, and its eventual measurement by the TAC biosensor on the surface of the skin takes the form of an abstract, semi-linear, parabolic partial/or- dinary dierential equation (PDE/ODE) hybrid system. The model includes a Michaelis- Menten nonlinear reaction term which captures the alcohol dehydrogenase enzyme-catalyzed metabolism of ethanol that takes place in the liver. Control is eected by linearizing the semi- linear hybrid PDE/ODE model about a nominal equilibrium solution or operating regime and then by designing an LQG compensator to make adjustments to the intravenous ethanol ow rate so as to optimally maintain the desired clamped BAC. The second aim of our investigation involves intravenously-infused alcohol tracking stud- ies wherein the TAC sensor is used to eect closed-loop feedback control the ethanol infusion ow rate so as to have the study participant's BAC track a pre-specied target trajec- tory. This would represent a signicant advance over the open-loop or manually titrated intravenously-infused alcohol clamping (i.e. involving a constant target trajectory) studies that are the current state of the art ([32], [34]). Here the model is the same but we reformu- late it as a tracking control problem to nd the optimal tracking control (estimated optimal infused alcohol ow rate) such that the BAC tracks a given target BAC signal. In addition, we were also able to reformulate the problem of estimating or deconvolving BAC from the biosensor measured TAC problem as a tracking control problem. In this case, the blood alcohol level or BAC is the input rather than a state variable and consequently there is no need to model the metabolism of ethanol in the liver. Thus the Michaelis- Menten nonlinear reaction term in the dynamical model is no longer required. The estimation problem is reformulated as a tracking problem by considering the model to be the plant and 5 letting the target reference signal be the TAC. Then the optimal tracking control is the estimated BAC or BrAC. We are also able to introduce regularization to guard against undesirable or non-physical artifacts such as excessive oscillations which result from the inherent ill-posedness of the inversion process. To solve both tracking problems, a discrete time LQG tracking compensator in the innite-dimensional state space with state equation taking the form of a random abstract parabolic system is designed. The motivation for allowing the parameters in our reaction diusion model to be random is that we have found that although our basic model captures the underlying physics or physiology quite well, the parameters that appear in the model in the form of coecients can vary with the individual wearing the sensor, the particular sensor being worn and envi- ronmental factors such as ambient temperature and humidity. It is not clear how one would construct a rst principles model for this variation and therefore we deal with it by allowing the model parameters to be random with their distribution assumed to be either known a-priori or estimated. Moreover, even though our present eort is focused on two rather spe- cialized problems for a particular plant modeled as an ODE/PDE hybrid reaction-diusion system, the approach we develop here is readily applicable to any plant whose dynamics are described by a random abstract parabolic system. Once the model for the clamping, tracking and estimation problems is formulated as a random abstract parabolic system, we can apply the results from linear semigroup theory to this system to facilitate the application of established LQG optimization, approximation and convergence theories. In summary, the central focus of this thesis is the development of a real-time closed- loop feedback control strategy based on a population model for the transdermal transport of alcohol and a transdermal alcohol biosensor using LQG control theory in innite-dimensional Hilbert space for the purpose of solving both the regulator ([45]) and the tracking problems ([46]). In particular, our approach is based on reformulating our population model plant as a random abstract parabolic system in a Gelfand triple setting. We then rely on recent ideas detailed in [19] and [36] which allow us to reformulate the model weakly in state 6 space form in a Gelfand triple of Bochner spaces in such a way that the random parameters now are eectively treated like additional or auxiliary spatial variables. We note though that the resulting weak form does not involve derivatives with respect to these auxiliary variables. This is the novelty in our approach. We are then able to design LQG control and tracking compensators using what can now be considered to be a standard LQG regulator or tracking problem in Hilbert space. Finally, we develop nite-dimensional approximations via a Galerkin approach and convergence theories for both the discrete time LQG control and LQG tracking problems coupled with an LQG observer for this abstract parabolic system with random parameters originating from the transdermal alcohol biosensor problem. 1.2 A Brief Survey of the Relevant Literature The modern theory of optimal control has been largely developed over the course of the past century to the point where it is now recognized as an extremely powerful tool with applications in the physical, biological, and social sciences and engineering. It deals with the problem determining and analyzing an input to, or control law for, a discrete or continuous time dynamical system that will cause the system to perform optimally over a nite or innite time horizon as determined by an appropriately dened performance index or cost functional. Of primary interest to us in this thesis is a particular class of problems known as the linear quadratic regulator (LQR) problem. The LQR problem is characterized by a performance index that is quadratic in both the state and the control with the underlying model or state equation or constraint being linear in both the state and the control. The appeal of the LQR formulation is that it admits an optimal control law in what is essentially closed form as a linear state feedback of the state variable. If the state of the linear system cannot be measured completely either due to the basic structure of the model or possibly because the system is perturbed by additive (white Gaussian) noise, it is possible to design an output feedback control law through the use of an observer, a dynamical system which serves 7 as a lter and yields an estimate of the state. In practice, when the system is corrupted by Gaussian white noise, the observer typically takes the form of a Kalman lter. The combined feedback control law together with the observer is known as a linear quadratic Gaussian (LQG) compensator. A common extension of the LQR theory solves the LQR tracking problem in which the state or the output in the form of a linear function of the state is forced to track a given reference signal at minimal cost. The LQ tracking problem is central to the work we report on here. The theory of LQ control largely originated by Kalman ([20] [21], [22], and [23]) with countless extensions and enhancements being added to the literature ever since. Of particular interest to us in this thesis is the LQ control theory for systems set in innite-dimensional Hilbert space. This aspect of LQ control theory along with its nite- dimensional approximation and associated convergence theory dates back to the mid late 1970's. The approximation theory for the continuous time linear quadratic regulator (LQR) problem in Hilbert space was developed in [15] and specically for abstract parabolic systems in [4]. For discrete time LQR problems in Hilbert space, LQR approximation results can be found in [18]. The nite-dimensional approximation and convergence theory for the continuous time LQG compensator (i.e. the combined LQ regulator and observer) in Hilbert space was developed in [16] and corresponding discrete time LQG compensator in [17]. More recent results on LQ control in Hilbert space can be found in [27] and [29]. An extensive bibliography and survey of the literature on the LQ control theory for innite-dimensional systems can be found in [1] and [28]. Here we investigate the application of these results into abstract parabolic systems with random parameters by exploiting some more recent results on systems of this type developed in [19] and [36]. In these treatments, the underlying parabolic systems are considered in weak form in appropriately constructed Bochner spaces wherein the random parameters are eectively treated as additional spatial variables. In this way, their LQ control and estimation can be formulated in Hilbert space and their nite-dimensional approximation 8 can be facilitated via a Galerkin approach. Some standard LQG tracking approaches in nite-dimensional space and in innite-dimensional Hilbert space can be found in [5], [25], and [47]. The early development of real-time closed-loop feedback for human subject laboratory studies involving the intravenous infusion of alcohol based on transdermal sensing can be found in [32] and [34]. Recently, the research group of applied mathematicians and psychol- ogists from University of Southern California developed a number of successful approaches based on both rst principles mathematical models for the diusion of ethanol from the blood, through the skin, and measurement by the TAC biosensor, including data-driven machine learning techniques such as physics informed articial neural networks and hidden Markov models. In their earlier eorts to convert TAC into BAC or BrAC ([6], [12], [35]), they dealt with these issues by rst calibrating the underlying transport models using si- multaneous BrAC and TAC data collected in the laboratory from a single drinking episode conducted under controlled conditions. In more recent treatments, they have been able to eliminate this need to calibrate the models to individual subjects and environmental condi- tions by quantifying the uncertainty through the use of a population model ([37], [38], [39]). Rather than train the model parameters with data from a single drinking episode, they al- low the model parameters to be random and t the distributions to BrAC/TAC data from numerous drinking episodes from an entire cohort of individuals. In this way, an optimal infused alcohol ow rate or an estimate of BrAC from TAC signals together with an error band can be obtained without requiring that the underlying models rst be calibrated to each individual subject and current environmental conditions. In addition, several other research groups have developed standard regression models ([9], [10], [11]) and have investigated machine learning techniques such as random forests ([14]) to convert TAC into either BAC or BrAC. 9 1.3 Outline of the Remainder of the Thesis An outline of the remainder of this thesis is as follows. In Chapter 2, we include some preliminary results; we state any key denitions and theorems that will be used in the re- mainder of the thesis. In Chapter 3, we derive our PDE/ODE mathematical model which can describe both intravenously-infused alcohol studies and the BAC deconvolution prob- lem. We also derive its linearization about an equilibrium solution to the alcohol biosensor problem. In Chapter 4, we reformulate the dynamical model as an abstract parabolic system and discuss its properties, and in particular, its solution in terms of linear semigroup theory. We then show how when the system parameters are assumed to be random, this abstract parabolic system can be reformulated as an equivalent deterministic system in a Gelfand triple of appropriately chosen Bochner spaces wherein the random parameters are treated as additional spatial variables albeit without any derivatives. We also discuss nite-dimensional approximation and convergence results for this random abstract parabolic system. In Chap- ter 5, we formulate a standard discrete time linear quadratic Gaussian control and estimation problem associated with our abstract parabolic system derived in Chapter 4. We show how the basic LQG compensator theory in Hilbert space can then be applied to our problem. Our nite-dimensional approximation and convergence results along with the corresponding matrix representations for the operators appearing in the model equations are also presented and discussed in this chapter. In Chapter 6, the nite-dimensional approximation, con- vergence, and matrix representation results for the discrete time linear quadratic Gaussian tracking problem are given and discussed. Numerical experiments for both the LQG control and LQG tracking problems are given in Chapter 7. Results from our numerical studies are included to demonstrate the ecacy and performance of our LQG compensator approach. A realistic simulation of an intravenously-infused alcohol clamping study with transdermal sensing and our compensator is included. Then for the tracking problem, we have included a simulation of an intravenously-infused alcohol study with a non-steady state target blood al- cohol concentration and where the output feedback compensator is based on observations of 10 transdermal alcohol concentration from the biosensor. Finally we present numerical results for the regularized deconvolution of blood alcohol concentration from transdermal alcohol concentration. A nal chapter contains some discussion of our numerical ndings, a few concluding remarks, and discussion of possible avenues for future research. 11 Chapter 2 Preliminaries In this chapter, we summarize some of the basic denitions and theorems we will use in the following chapters, so that we can refer to this chapter when we need to introduce these ideas and results in what is to follow. The theorems and their proofs in this chapter can be found in [13, 33, 44]. Denition 1. (Gelfand Triple) Let H be a Hilbert space and let V be a dense subspace of H such that V can be endowed with a topological vector space structure for which the inclusion map i is continuous, or equivalently, bounded. Identifying H with its dual space H via the Riesz map, the adjoint to i is the map i :H =H !V : It follows that H is densely and continuously embedded in V and the duality pairing betweenV andV , denoted by<;> V ;V is then compatible with the inner product onH, in the sense that V ;V = H ; whenever u2H =H V and v2V H. The triple V ,!H ,!V is then referred to as a Gelfand triple. 12 Denition 2. (Sesquilinear Form)A sesquilinear form on a vector space H over a eldF is a map a(;) :HH!F which is conjugate linear in the rst argument, and linear in the second, i.e. a(v 1 ;cw 1 ) =ca(v 1 ;w 1 ); a(v 1 ;w 1 +w 2 ) =a(v 1 ;w 1 ) +a(v 1 ;w 2 ) a(cv 1 ;w 1 >= ca(v 1 ;w 1 ); a(v 1 +v 2 ;w 1 ) =a(v 1 ;w 1 ) +a(v 2 ;w 1 ) for all c2F and v 1 ;v 2 ;w 1 ;w 2 2H. Note: a bilinear form is a special case of the sesquilinear form with linearity on both arguments. Dene the inner product on H byh;i H and the norms on H and V byjj H ,kk V , respectively. Dene a sesquilinear form a (;) : VV ! C which satises the following properties: Assumption 1. (Boundedness) There exists a constant 0 > 0 such that for each'; 2V , we have a('; ) 0 k'k V k k V : Assumption 2. (Coercivity) There exist constants 0 2 R and 0 > 0 such that for each '2V , we have a(';') + 0 j'j 2 H 0 k'k 2 V : It is assumed that all of the constants, 0 , 0 , and 0 do not depend on q, for q2Q. For eachq2Q, we require that besides satisfying Assumptions 1 and 2, the sesquilinear form a (q;;) :VV!C also satises: Assumption 3. (Continuity) Forq;e q2Q, we have for all'; 2V ,ja(q;'; )a(~ q;'; )j d Q (q; ~ q)k'kk k, where d Q (;) denotes any p-metric on R p . 13 Denition 3. (Semigroup) A family of bounded linear operatorsfT (t) :t 0g on a real or complex Banach space X is called a semigroup on X if (i)T (0) =I, (ii)T (t +s) =T (t)T (s) for every t;s 0. Denition 4. (Innitesimal Generator of Semigroup) LetfT (t) :t 0g be a semigoup on X, a linear operator A dened by Ax = lim t#0 T (t)xx t with its domain of denition Dom(A) :=fx2X : lim t#0 T (t)xx t existsg is called the innitesimal generator of the semigroupfT (t);t 0g. Denition 5. (C 0 -Semigroup) A semigroupfT (t) : t 0g is called a strongly continuous or C 0 -semigroup on X if lim t#0 jjT (t)xxjj X = 0; forx2X Theorem 1. For each C 0 -semigroup T (t), there exists real numbers M 1 and !2R such thatkT (t)kMe !t for t 0. It is written thatA2G(M;!) when a linear operatorA is the innitesimal generator of a C 0 -semigroup satisfying such an exponential growth. Denition 6. (Resolvent Set and Operator) LetX be a normed space and letA :dom(A)! X be a linear operator withdom(A)X. The resolvent set(A) ofA is the set of all2C for which the range of the linear transformation IA : dom(A)! X is dense and and IA has a continuous inverse on this set. For any 2 (A), the resolvent operator is denoted by R (A) = (IA) 1 . 14 Theorem 2. Let A2G(M;!) be the innitesimal generator for T (t). Then, R (A)x = Z 1 0 e t T (t)xdt for any with Re>! and all x2X. The Hille-Yosida theorem provides necessary and sucient conditions for an operator to be the innitesimal generator of a strongly continuous semigroup. Theorem 3. (Hille-Yosida) For M 1, !2R, we have A2G(M;!) if and only if 1. A is closed and densely dened, i.e. dom(A) =X. 2. For real >!, we have 2(A) and R (A) satises kR (A) n k M (!) n ; n = 1; 2; : Denition 7. (Exponentially Stable) A semigroupfT (t) :t 0g is exponentially stable if there exist real constant M 1 and !> 0 such thatkT (t)kMe !t for t 0. Theorem 4. For eachC 0 -semigroupT (t), if for somep, 1p<1, we have R 1 0 kT (t)xk p dt< 1, for every x2X, then there exists real constants M 1 and ! > 0 such thatkT (t)k Me !t for t 0. Denition 8. (Analytic Semigroup) LetfT (t);t 0g be a C 0 -semigroup on X with in- nitesimal generator A.fT (t);t 0g is said to be an analytic semigroup if (i)for some 0<<=2, the bounded linear operator T (t) :X!X can be extended to t2 , =f0g[ft2C :jarg(t)j<g and the usual semigroup conditions hold for s;t2 :T (0) =I;T (t +s) =T (t)T (s), and for each x2X, T (t)x is continuous in t; (ii)For allt2 nf0g,T (t) is analytic in t in the sense of the uniform operator topology. 15 Theorem 5. (Necessary and Sucient Condition for the Generation of an Analytic Semi- group) IfA +I is of type (M;!) for some real , where 0 ! < =2, then for each >,f : Re> g(A), and in this half plane,k(AI) 1 kC(1 +j j) 1 , for someC > 0. Conversely if A is a densely dened closed operator satisfying the above bound, then there exists a with < and !2 [0;=2] such thatA +I is of type (M;!). If A +I is of type (M;!) where !2 [0;=2] and M 1, then A is the generator of an analytic semigroup. Theorem 6. (Trotter-Kato) Let H and H N be Hilbert spaces such that H N H. Let P N : H! H N be an orthogonal projection of H onto H N . Assume P N x! x as N!1 for allx2H. LetA N ;A be innitesimal generators ofC 0 semigroupsT N (t);T (t) onH N ;H respectively satisfying (i)There exists M;! such thatjT N (t)jMe !t for each N, (ii)There exists D dense in H such that for some , (IA)D is dense in H and A N P N x!Ax for all x2D. Then T N (t)x!T (t)x uniformly in t on compact intervals [0;T ] for each x2H. Theorem 7. (Hilbert Projection Theorem) LetH be a Hilbert space andM a closed subspace of H, there exists a unique point y2M for whichkxyk is minimized over M. Denition 9. (Initial Value Problem) Let X be a Banach space and A is an innitesimal generator of C 0 semigroups in X. Consider the initial value problem du(t)=dt =Au(t) +f(t); 0tT; u(0) =u 0 ; (2.1) whereu 0 2Dom(A) andf2C([0;T ];X). A functionu is called a solution of this initial value problem if, besides satisfying (2.1), it has the properties u2 C 1 ([0;T ];X), u(t)2 Dom(A) for each t2 [0;T ] and Au2C([0;T ];X). 16 Theorem 8. LetfT (t)g denote the semigroup generated by A. Then u(t) =T (t)u 0 + Z t 0 T (ts)f(s)ds (2.2) is a solution of the initial value problem (2.1). If a solution to the initial value problem (2.1) exists, it is represented by (2.2). Therefore, the solution is unique. 17 Chapter 3 A Model for the Metabolism and Transdermal Alcohol Transport We consider a dynamical model which describes the process of alcohol metabolism and the transdermal transport of ethanol from the blood through the epidermal layer of the skin, excreted within perspiration, and nally measured on the surface of the skin by an alcohol biosensor. This dynamical model is reasonable to explain the alcohol transport process both with the metabolism involved in liver and only from the dermal layer of skins to the biosensor wearing on the surface of skins, by endowing state and input variables with dierent meanings. The dynamical model takes the form of a hybrid, semi linear, one dimensional, PDE/ODE reaction diusion equation coupled via Dirichlet boundary conditions to two well-mixed compartments, one representing the concentration of ethanol in either the dermal layer with an active blood supply or the blood (depends on dierent situations, i.e. infused intravenous studies or deconvolution of BAC from TAC), and the other the concentration of ethanol in the transdermal alcohol biosensor. The in ow to the two compartments is proportional to the ux at the boundary of the epidermal layer of the skin. Aside from the relatively small amount of ethanol excreted from the body through urine, tears, breast milk, sweat and perspiration, the primary mechanism by which ethanol is pro- cessed out of the body is via a reaction that takes place in the liver and which is catalyzed by 18 a group of enzymes known as alcohol dehydrogenase (ADH). The enzyme catalyzed reaction in the blood compartment (liver) is modeled by a Michaelis-Menten term which typically exhibits rst-order kinetics at low concentrations and zero-order kinetics at higher concen- trations with enzyme saturation. Once saturation occurs, alcohol begins to accumulate in the blood, it crosses the blood-brain barrier which results in intoxication. In the transdermal alcohol biosensor, the ethanol is consumed in an oxidation-reduction reaction wherein each molecule of ethanol produces four electrons. The resulting current is measured with the measurement being bench calibrated with a source of ethanol vapor with known concentration. In addition, since the values of the parameters which appear in the model for an individual subject will in all likelihood be unknown and un-measurable, we will consider the parameters to be random with distribution that has previously been t to a cohort from an appropriately stratied population (see, for example, [2], [37], and [39]). Consequently, the resulting control problem is one in which the process is to be regulated for an individual based on a population model. We note that we use the term individual here to mean a member of a population whose characteristics include not only physiological considerations (e.g. skin porosity, tortuosity, thickness, blood pressure, etc.) but also environmental conditions (e.g. ambient temperature, humidity, etc.) and variations in sensor hardware (e.g. manufacturer, calibration, age, etc.). 3.1 Mathematical Model The underlying dynamical system as described in the previous paragraph takes the fol- lowing form: 19 @~ x @t (t;) = @ 2 ~ x @ 2 (t;); t> 0; 2 (0; 1); d ~ w dt (t) = @~ x @ (t; 0) ~ w(t) +! 1 (t); t> 0; d~ v dt (t) = @~ x @ (t; 1) K~ v(t) M + ~ v(t) ~ v(t) +b~ u(t) +! 2 (t); t> 0; (3.1) with respectively, boundary conditions, controlled variable, observation, and initial condi- tions: ~ x(t; 0) = ~ w(t); ~ x(t; 1) = ~ v(t); t> 0; ~ z(t) =~ v(t) (or ~ w(t)); ~ y(t) = ~ w(t) +(t); t> 0; ~ x (0;) =' 0 (); 2 (0; 1); ~ w(0) = 0 ; ~ v(0) = 0 ; (3.2) where the parameters appearing in the model equations (3.1)-(3.2), ,, ,,M,K,, and b are all assumed to be positive, and the initial conditions ' 0 , 0 , and 0 are all assumed to be nonnegative. In the above system, ~ x(t;) is the concentration of ethanol at time t 0 and depth 2 [0; 1] in the epidermal layer of the skin. Without loss of generality, we have normalized the thickness of the epidermal layer to be one. ~ w(t) is the concentration of ethanol in the transdermal alcohol biosensor vapor collection chamber at time t 0. The Michaelis-Menten nonlinear reaction term involving the parametersK andM in the equation for ~ v in the system (3.1) models the liver's alcohol dehydrogenase enzyme-catalyzed metabolism of ethanol into aldehydes and ketones which are then either further processed, used or excreted from the body through urine or breath. Note that when ~ v<<M, this term yields rst-order kinetics (i.e. exponential decay) and when M << ~ v, we have zero-order kinetics. The term~ v(t) represents all the other losses of ethanol from the blood; in general ~ v(t)<< K~ v(t) M+~ v(t) . Based on dierent purposes, this model can be used to describe either intravenously- infused alcohol clamping studies or deconvolution of BAC from TAC. In clamping studies, ~ v(t) is the concentration of ethanol in the blood at time t 0. ~ u(t) is the infused alcohol ow rate. The state variable to be controlled or regulated is ~ v. In deconvolution, since the 20 process described by this model dose not involve the metabolic reaction in the liver, there is no Michaelis-Menten nonlinear reaction term (i.e., setK = 0). And ~ v(t) is the concentration of ethanol in the dermal layer at time t 0. ~ u(t) is BAC at time t 0. The state variable to be tracked is ~ w. In addition, ! 1 , ! 2 , and denote uncorrelated, zero-mean, stationary, Gaussian white noise processes with variances 2 1 , 2 2 , and 2 ( is positive), respectively. 3.2 Linearization about an Equilibrium Solution As what we introduce in Chapter 1, no matter for either the intravenously-infused alcohol studies or deconvolution, we use LQG compensator (or LQG tracking) which is a linear control method. So one of the strategies to deal with the nonlinear system (3.1)-(3.2) (i.e. with the Michaelis-Menten nonlinear term) is based upon a linearization about a nominal operating regime. In this case, ~ v represents BAC. If this nominal BAC level is set to be ~ v(t) = ~ v 0 , then an equilibrium solution to the system (3.1), (3.2) is given by ~ x(t;) = ~ x 0 () = ~ v 0 + + ~ v 0 + ; ~ w(t) = ~ w 0 = ~ v 0 + ; ~ v(t) = ~ v 0 ; ~ u(t) = ~ u 0 = ~ v 0 b( +) + K~ v 0 b(M + ~ v 0 ) +~ v 0 : 21 Then to synthesize the optimal controller, we linearize about ~ x 0 , ~ w 0 , ~ v 0 and ~ u 0 , by writing ~ x = ~ x 0 +x, ~ w = ~ w 0 +w, ~ v = ~ v 0 +v, and ~ u = ~ u 0 +u and obtain the linearized system for x, w, v, and u given by: @x @t (t;) =q 1 @ 2 x @ 2 (t;); t> 0; 2 (0; 1); dw dt (t) =q 3 @x @ (t; 0)q 4 w(t) +! 1 (t); t> 0 dv dt (t) =q 5 @x @ (t; 1)q 6 v(t) +q 2 u(t) +! 2 (t); t> 0; (3.3) with boundary conditions, controlled variable, observation and initial value: x(t; 0) =w(t); x(t; 1) =v(t); z(t) =v(t) (or w(t)); y(t) =w(t) +(t); x(0;) =' 0 () ~ x 0 () =x 0 (); (3.4) for t > 0, respectively, where in the equations (3.3), (3.4), the parameters q 1 = , q 2 = b, q 3 =,q 4 = ,q 5 =, andq 6 = KM (M+~ v 0 ) 2 + are all positive. The state of the system is given by the triple (w;v;x). We only know the distribution of the parametersq = (q 1 ;q 2 ;q 3 ;q 4 ;q 5 ;q 6 ) in the subject cohort of interest, rather than the precise values of the parameters themselves. That is we design our optimal controller based on a population model rather than on one for the actual plant being controlled. Note that if we set K = 0 to make (3.1)-(3.2) describe a deconvoluting BAC from TAC, then the linearized model (3.3)-(3.4) with q 6 = is in fact the plant (3.1)-(3.2). So the linearized model (3.3)-(3.4) will be the main system we are going to discuss in the rest of this thesis. 22 Chapter 4 The Population Model in State Space Form Before we dene our population model, we rst recast the linearized hybrid ODE/PDE initial-boundary value problem (3.3)-(3.4) as an abstract parabolic system in a Gelfand triple of Hilbert spaces which can be treated as the general form of the special case of the alcohol biosensor problem. We therefore obtain abstract parabolic systems originated from the original alcohol biosensor problem with deterministic parameters. Then we consider the abstract parabolic systems with random parameters by appropriately constructing Bochner spaces wherein the random parameters are eectively treated as additional spatial variables. Finally, we consider the nite-dimensional approximation via a Galerkin approach. 4.1 Abstract Parabolic Systems LetQ be a compact subset of the positive orthant ofR 6 , letH =R 2 L 2 (0; 1) be endowed with the standard inner product and norm, and for any q2Q, let H q =R 2 L 2 (0; 1) with the inner product byh;i q (;;'); ( ; ; ') q = q 1 q 3 + q 1 q 5 + Z 1 0 '() '()d; ; ;; 2R; '; '2L 2 (0; 1) and normj(;;')j q =j (;;'); (;;') q j 1=2 . It is clear that for any ^ '; ^ 2H xed, the map q7! D ^ '; ^ E q is continuous from Q into R. 23 Let V be the Hilbert space dened by V =f(;;')2H :'2H 1 (0; 1); ='(0); ='(1)g; with inner product byh;i V ('(0);'(1);'); ( '(0); '(1); ') V ='(0) '(0) +'(1) '(1) + ' 0 ; ' 0 L 2 (0;1) ; and normk('(0);'(1);')k V =j ('(0);'(1);'); ('(0);'(1);') V j 1=2 , whereh;i L 2 (0;1) de- notes the standard inner product on L 2 (0; 1). Standard arguments yield the dense and continuous embeddings V ,! H q ,! V and that the embeddings are uniformly bounded with respect to q for q2Q. Next, we rewrite the system (3.3) as a vector version. That is 0 B B B B @ dw dt (t) dv dt (t) @x @t (t;) 1 C C C C A = 0 B B B B @ q 3 @x @ (t; 0)q 4 w(t) q 5 @x @ (t; 1)q 6 v(t) q 1 @ 2 x @ 2 (t;) 1 C C C C A + 0 B B B B @ 0 q 2 0 1 C C C C A u(t) + 0 B B B B @ ! 1 (t) ! 2 (t) 0 1 C C C C A ; and for any (;;') T ; ( ; ; ') T 2V , by the inner product dened on H q , compute * 0 B B B B @ q 3 @' @ =0 q 4 q 5 @' @ =1 q 6 q 1 @ 2 ' @ 2 1 C C C C A ; 0 B B B B @ ' 1 C C C C A + V ;V = Z 1 0 q 1 @ 2 ' @ 2 'd + q 1 q 5 q 5 @' @ =1 q 6 ! + q 1 q 3 q 3 @' @ =0 q 4 ! =q 1 '() @' @ () 1 0 Z 1 0 @' @ @ ' @ d ! + q 1 q 5 q 5 @' @ =1 '(1)q 6 '(1) '(1) ! + q 1 q 3 q 3 @' @ =0 '(0)q 4 '(0) '(0) ! (4.1) 24 = q 1 q 4 q 3 '(0) '(0) q 1 q 6 q 5 '(1) '(1)q 1 Z 1 0 ' 0 () ' 0 ()d: Based on the computations, dene the bilinear form a(q;;): VV!R by a(q; ('(0);'(1);'); ( '(0); '(1); ')) = q 1 q 4 q 3 '(0) '(0) + q 1 q 6 q 5 '(1) '(1) +q 1 Z 1 0 ' 0 () ' 0 ()d: (4.2) Now we show that the bilinear form a(q;;) satises the Assumption 1, 2, and 3. For any ('(0);'(1);'); ( '(0); '(1); '))2V , by Cauchy-Schwarz inequality, ja(q; ('(0);'(1);'); ( '(0); '(1); '))j q 1 q 4 q 3 j'(0) '(0)j + q 1 q 6 q 5 j'(1) '(1)j +q 1 j Z 1 0 ' 0 () ' 0 ()dj M 1 j'(0) '(0)j +M 2 j'(1) '(1)j +M 3 Z 1 0 j' 0 ()j 2 d !1 2 Z 1 0 j ' 0 ()j 2 d !1 2 ; where M 1 , M 2 , M 3 exist and are positive constant upper bound of q 1 q 4 =q 3 , q 1 q 6 =q 5 , q 1 , respectively, and independent ofq i fori = 1; ; 6 sinceQ is a compact set. In addition, by the application of Morrey's inequality with n = 1,p = 2, and = 1 2 , there existsC 1 ; C 2 > 0 such that j'(0) '(0)jC 1 Z 1 0 j'()j 2 +j' 0 ()j 2 d ! 1=2 Z 1 0 j '()j 2 +j ' 0 ()j 2 d ! 1=2 j'(1) '(1)jC 2 Z 1 0 j'()j 2 +j' 0 ()j 2 d ! 1=2 Z 1 0 j '()j 2 +j ' 0 ()j 2 d ! 1=2 : 25 We consider a upper bound of R 1 0 j'()j 2 d. By Fundamental Theorem of Calculus, Z 1 0 j'()j 2 d = Z 1 0 '(0) + Z 0 ' 0 (t)dt 2 d 2 Z 1 0 ' 2 (0) + Z 0 ' 0 (t) 1dt 2 d (CauchySchwarz)2' 2 (0) + 2 Z 1 0 Z 0 ' 0 (t) 2 dt Z 0 1 2 dtd 2' 2 (0) + 2 Z 1 0 Z 0 ' 0 (t) 2 dtd 2' 2 (0) + 2 Z 1 0 ' 0 () 2 d: Therefore, j'(0) '(0)jC 1 2' 2 (0) + 3 Z 1 0 j' 0 ()j 2 d ! 1=2 2 ' 2 (0) + 3 Z 1 0 j ' 0 ()j 2 d ! 1=2 C 0 1 k('(0);'(1);')k V k( '(0); '(1); ')k V ; where C 0 1 is a constant and independent of q i for i = 1; ; 6. Similarly, there exists a constant C 0 2 independent of q i for i = 1; ; 6 such that j'(1) '(1)jC 2 2' 2 (0) + 3 Z 1 0 j' 0 ()j 2 d ! 1=2 2 ' 2 (0) + 3 Z 1 0 j ' 0 ()j 2 d ! 1=2 C 0 2 k('(0);'(1);')k V k( '(0); '(1); ')k V : Therefore, there exist constants C 1 , C 2 independent of q i for i = 1; ; 6 such that ja(q; ('(0);'(1);'); ( '(0); '(1); '))j C 1 k ('(0);'(1);')k V k ( '(0); '(1); ')k V + C 2 k ('(0);'(1);')k V k ( '(0); '(1); ')k V +M 3 k ('(0);'(1);')k V k ( '(0); '(1); ')k V = 0 k ('(0);'(1);')k V k ( '(0); '(1); ')k V ; 26 where 0 exist and is a positive constant which is independent of q i for i = 1; ; 6. This shows the boundedness of the bilinear form a(q;;). Next, for 0 = 0, a(q; ('(0);'(1);'); ('(0);'(1);')) + 0 j ('(0);'(1);')j 2 q = q 1 q 4 q 3 '(0) '(0) + q 1 q 6 q 5 '(1) '(1) +q 1 Z 1 0 ' 0 () ' 0 ()d 0 k ('(0);'(1);')k 2 V ; (4.3) where a positive constant 0 exists since Q is a compact set. This shows the coercivity of the bilinear form a(q;;) with 0 = 0. Dene d Q to be a proper Euclidean distance on R. Then for any q, q 0 2Q, ja(q; ('(0);'(1);'); ( '(0); '(1); '))a(q 0 ; ('(0);'(1);'); ( '(0); '(1); '))j =j q 1 q 4 q 3 q 0 1 q 0 4 q 0 3 '(0) '(0) + q 1 q 6 q 5 q 0 1 q 0 6 q 0 5 '(0) '(1) + Z 1 0 q 1 q 0 1 ' 0 () ' 0 ()dj K 1 d Q (q;q 0 )k ('(0);'(1);')k V k ( '(0); '(1); ')k V +K 2 d Q (q;q 0 )k ('(0);'(1);')k V k ( '(0); '(1); ')k V +K 3 d Q (q;q 0 ) Z 1 0 ' 0 () 2 d !1 2 Z 1 0 ' 0 () 2 d !1 2 K 0 d Q (q;q 0 )k ('(0);'(1);')k V k ( '(0); '(1); ')k V ; where K 0 , K 1 , K 2 , K 3 exist and are positive constants which are independent of q i for i = 1; ; 6 sinceQ is a compact set. In addition, the second inequality is true by Morrey's Inequality withn = 1,p = 2, and = 1 2 (for details, see the proof of boundedness ofa(q;;) above). This shows the continuity of the bilinear form a(q;;) with respect to q2Q. Therefore, we can dene A(q): Dom(A(q))H!H by: hA(q) ^ '; ^ i V ;V =a(q; ^ '; ^ ); 27 for any ^ '2Dom A(q) , ^ 2V , where Dom A(q) = ^ ' = ('(0);'(1);')2V :'2H 2 (0; 1) is independent ofq2Q. Equation (4.1) shows that for any ^ ' = ('(0);'(1);')2Dom(A(q)), we have A(q) ^ ' =A(q)('(0);'(1);') = q 3 ' 0 (0)q 4 '(0);q 5 ' 0 (1)q 6 '(1);q 1 ' 00 ; and that the operator A(q) is densely dened on H q and self-adjoint. Consequently A(q) is the innitesimal generator of a self-adjoint, analytic semigroup of bounded linear operators, T (q;t) :t 0 , on H q . (see, for example, [3] and [44]). Note that the analytic semigroupfT (q;t) :t 0g is also uniformly exponentially stable by the coercivity of a(q;;) with 0 = 0. In fact, when 0 = 0, the resolvent operator (IA(q)) 1 exists as a bounded operator onH orV (see [3]), so the spectrum ofA(q), say (A(q)) (1; 0), i.e. supf :2(A(q))g< 0. Therefore by Theorem 4.3 from Section 4.3 in [33] and the compactness of Q, the analytic semigroupfT (q;t) : t 0g is uniformly exponentially stable on H q with respect to q2Q. By dierent purposes, we dene the controlled variable operatorD2L(H q ;R) by either D(;;') = p orD(;;') = p , for some weight > 0. The observed state variable is w and therefore the observation or output operatorC2L(H q ;R) is given byC(;;') =. For q2 Q, the input operator B(q)2 L(R;H q ) is given by B(q)u = (0;q 2 u; 0), and the random noise in uence operator B 1 2L(R 2 ;H q ) by B 1 ! = (! 1 ;! 2 ; 0) for ! = (! 1 ;! 2 ) T . 28 Formally, the state variable is x = (w;v;x)2 H. According to the operators dened above, (3.3)-(3.4) can be rewritten in state space form as the evolution system _ x(t) =A(q)x(t) +B(q)u(t) +B 1 !(t); y(t) =Cx(t) +(t); z(t) =Dx(t); t> 0: x(0) = x 0 2H: (4.4) From linear semigroup theory (see [44]), the mild solution of (4.4) is unique and given by: x(t) =T (q;t)x 0 + Z t 0 T (q;ts) B(q)u(s) +B 1 (q)w(s) ds; t 0: (4.5) Let denote the length of the sampling interval, and consider zero-order hold inputs of the form u(t) = u k , for t2 [k; (k + 1)), k = 0; 1; 2; . If we then dene x k = x(k), y k = y(k), z k = z(k), k = 0; 1; 2; and according to (4.5), dene ^ A(q) = T (q;)2 L(H q ;H q ), ^ B(q) = R 0 T (q;s)B(q)ds2L(R;H q ), and ^ B 1 (q) = R 0 T (q;s)B 1 ds2L(R 2 ;H q ). Recall that sincea(q;;) is coercive with 0 = 0,A(q) is invertible for anyq2Q. So ^ B(q) = A(q) 1 ( ^ A(q)I)B(q), ^ B 1 (q) =A(q) 1 ( ^ A(q)I)B 1 . In addition, dene ^ C =C2L(H q ;R), and ^ D =D2L(H q ;R), where I denotes the identity operator on H q . In addition, for the convenience in future descriptions, we dene ^ G = ^ ^ D ^ D inL(H q ;H q ), for some nonnegative weight ^ . Moreover, we will also require an operator ^ Q inL(H q ;H q ) given by ^ Q = ^ D ^ D and a positive constant ^ r> 0. Therefore, we obtain the discrete-time dynamical system with parameter q2 Q corre- sponding to (4.4) and (4.5) given by x k+1 = ^ A(q)x k + ^ B(q)u k + ^ B 1 (q)!(k); y k = ^ Cx k +(k); z k = ^ Dx k ; x 0 =x 0 : (4.6) 29 4.2 Random Parameters as Auxiliary Spatial Variables As was noted in Chapter 1, the precise value of the parameter vector q2 Q will not be known. Consequently, we will design our control and tracking compensator (combined controller and observer) based on a population model. We formulate this model using an idea discussed in [19] and [36] wherein we allow the parameterq =q to be a random vector with supportQ and distribution described by the either known or t (see, for example, [37], [38] or [39]) push-forward probability measure (i.e. q ) with the assumption that all functions involvingq are-measurable. We note that in this approach, the state equation is formulated in weak form wherein the parametersq are treated as additional space variables; that is the spatial dimension of the state equation eectively now becomes 1 + dimq = 7 with the spatial variable given by (;q 1 ;q 2 ;:::;q 6 ). LetV be the Bochner spaceV =L 2 (Q;V ) and letV be its dual. LetH =L 2 (Q;H q ), and identify the Hilbert spaceH with its dual to obtain the Gelfand tripleV ,!H ,!V . Dene the average bilinear forma(;) onVV by: a('; ) =E a q;'(q); (q) = Z Q a q;'(q); (q) d(q); for '; 2V. Now we check boundedness and coercivity of a(;). By using the boundedness and coercivity of a(q;;), and the Cauchy-Schwartz inequality, we have for any '; 2V, ja('; )j Z Q ja(q;'(q); (q))jd(q) 0 Z Q k'(q)k V k (q)k V d(q) 0 Z Q k'(q)k 2 V d(q) ! 1=2 Z Q k (q)k 2 V d(q) ! 1=2 = 0 k'k V k k V ; 30 and a(';') = Z Q a(q;'(q);'(q))d(q) 0 Z Q k'(q)k 2 V d(q) = 0 k'k 2 V ; which show thata(;) is also bounded and coercive with 0 = 0 onVV. Therefore, we can use this bilinear form to dene a self-adjoint operatorA2L(V;V ) by hA'; i V ;V =a('; ); '; 2V: Similarly, the operator A can be restricted to Dom(A) =f'2 V : A'2 Hg as the innitesimal generator of an exponentially stable, analytic semigroupfT(t) : t 0g of bounded, self-adjoint, operators onH. Dene the operatorsB2L(R;H),B 1 2L(R 2 ;H), andC;D2L(H;R) byBu = E fB(q)gu, u2 R,B 1 ! = E fB 1 g! = B 1 !, !2 R 2 ,C ^ ' = E fC ^ 'g, andD ^ ' = E fD ^ 'g, ^ '2H, respectively. Then as in the deterministic case (4.4), we can write a population model corresponding to (3.3)-(3.4) in state space form as _ x(t) =Ax(t) +Bu(t) +B 1 !(t); y(t) =Cx(t) +(t); z(t) =Dx(t); x(0) = x 0 : (4.7) It is shown in [19] and [36] that the solutions to (4.4) and (4.7) agree almost surely or a:e q2Q. From linear semigroup theory (see [44]), the mild solution of (4.7) is unique and given by: x(t) =T(t)x 0 + Z t 0 T(ts) Bu(s) +B 1 w(s) ds; t 0: (4.8) For the same length of the sampling interval dened in Section 4.1, consider zero-order hold inputs of the formu(t) =u k , for t2 [k; (k + 1)), k = 0; 1; 2; . If we then dene 31 x k =x(k),y k =y(k),z k =z(k), k = 0; 1; 2; and according to (4.8), dene ^ A = T()2 L(H;H), ^ B = R 0 T(s)Bds2 L(R;H), and ^ B 1 = R 0 T(s)B 1 ds2 L(R 2 ;H). Recall that sincea(;) is coercive with 0 = 0,A is invertible. So ^ B =A 1 ( ^ AI)B, ^ B 1 = A 1 ( ^ AI)B 1 . In addition, dene ^ C =C2L(H;R), and ^ D =D2L(H;R), where I denotes the identity operator onH. Therefore, we obtain the discrete-time dynamical system corresponding to (4.7) and (4.8) given by x k+1 = ^ Ax k + ^ Bu k + ^ B 1 !(k); y k = ^ Cx k +(k); z k = ^ Dx k ; x 0 =x 0 : (4.9) For the nite-time horizon problem which will be discussed later, if a terminal penalty is to be included, dene an operator ^ G2L(H;H) which would most likely be chosen to be ^ G = ^ ^ D ^ D, for some nonnegative weight . Moreover, we will also require an operator ^ Q in L(H;H) given by ^ Q = ^ D ^ D and a positive constant ^ r> 0. 4.3 Approximation and Convergence Let N represent the multi-index N = (n;m 1 ;m 2 ;::::;m 6 ) and we write N!1 which means n!1 and m i !1, i = 1; 2;:::; 6. We assume that the random parameter q i has support [a i ;b i ], i = 1; 2;:::; 6, all assumed to be bounded. Let Q be the compact subset of R 6 given by Q = 6 i=1 [a i ;b i ]. For i = 1; 2;:::; 6, partition [a i ;b i ] into m i equal subintervals, and let m i j denote the characteristic function of the j-th subinterval, j = 1; 2;:::;m i . For n = 1; 2;:::, letf' n j g n j=0 denote the standard linear B-splines on [0; 1] with respect to the uniform meshf0; 1 n ; 2 n ;:::; 1g and set ^ ' n j = (' n j (0);' n j (1);' n j )2V . LetJ denote a multi-index of the form J = (j 0 ;j 1 ;:::;j 6 ) where j 0 2f0; 1; 2;:::;ng and j i 2f1; 2;:::;m i g, i = 1; 2;:::; 6. Then set N J = ^ ' n j 0 6 i=1 m i j i , let V N = span J f N J g, and let P N : H! V N denote the orthogonal projection ofH ontoV N . Standard arguments from the theory of splines and 32 piecewise constant approximation inL 2 can be used to argue thatP N converges strongly to the identity in bothH andV. We consider a Galerkin approximation ofA (see [4] for details). We dene the operators A N onV N by essentially restricting the forma to the subspaceV N V N . To be more specic, for ' N ; N 2V N , we have hA N ' N ; N i V N ;V N =a(' N ; N ) = Z Q a(q;' N (q); N (q))d(q): Obviously, sinceA N is a linear operator on a nite-dimensional space, it is the innites- imal generator of a uniformly continuous semigroup T N (t) = e A N t for t 0. So we can dene ^ A N 2L(V N ;V N ) by ^ A N =T N () =e A N . We use a variational corollary of Trotter-Kato theorem (see Theorem 2.1, 2.2 and 2.3 in [3]) to obtain the convergence of the approximating semigroupsT N (t) for t 0 on any compact sub intervals. Thereby, we obtain the convergence of the operatorsA N . Theorem 9. Given the bilinear form a(q;;) by equation (4.2) (actually we just need the bilinear form a(q;;) satises the Assumption 1, 2, and 3), then for each x 2 H, T N (t)P N x!T(t)x in theV norm (so also in theH norm) for t> 0, uniformly in t on compact sub intervals. Remark 1. The Trotter-Kato theorem (see [3]) requires the following assumption: For each x2 V, there exists x N 2 V N such thatk xx N k V ! 0. However, we note that this assumption is actually equivalent to the strong convergence of orthogonal projectionP N of V ontoV N , i.e. for anyx2V,P N x!x asN!1 inVnorm. Indeed, for anyx2V satisfying the assumption in [3], we havekP N xxk V k xx N k V ! 0. On the other hand, for any x2V satisfyingP N x!x as N!1 inVnorm, take x N =P N x2V N , then byP N !I strongly inV, we havekxx N k V =kxP N xk V ! 0. 33 From the denition ^ A =T() and ^ A N =T N (), we immediately get that ^ A N P N ! ^ A strongly inV (so also inH) as N!1. Note that sinceT(t) andT N (t) are self adjoint inH, we have ( ^ A N ) P N ! ^ A strongly inH as N!1. We then set ^ B N = (A N ) 1 ( ^ A N I N )P N B2 L(R;V N ). Similarly we set ^ B N 1 = (A N ) 1 ( ^ A N I N )P N B 1 2L(R 2 ;V N ), and then set ^ C N = ^ CP N 2L(V N ;R), ^ D N = ^ DP N 2 L(V N ;R). By previous denitions, we set ^ G N = P N ^ GP N = ^ ( ^ D N ) ^ D N 2 L(V N ;V N ), ^ Q N = P N ^ QP N = ( ^ D N ) ^ D N 2 L(V N ;V N ). Standard arguments from functional analysis (specically, linear semigroup theory) (see, for example [17] and [18]) can then be used to argue the requisite convergence of them inH. Therefore, we obtain an approximating discrete time dynamical system corresponding to (4.7) and (4.8) given by x N k+1 = ^ A N x N k + ^ B N u k + ^ B N 1 !(k); y N k = ^ C N x N k +(k); z N k = ^ D N x N k ; x N 0 =P N x 0 =P N x 0 : (4.10) 34 Chapter 5 Linear Quadratic Gaussian Control Problems In the intravenously-infused alcohol clamping studying, one of the main purposes is to obtain the optimal infused alcohol ow rate based on the data from the TAC biosensor so that a desired level of BAC can be reached and maintained for an extended period of time at the least cost. To solve this problem, we develop a discrete time linear quadratic (LQ) control theory coupled with a linear quadratic Gaussian (LQG) compensator for the random abstract parabolic system (4.9). A nite-dimensional approximation and convergence theory for the closed-loop LQ control and estimation of the same abstract parabolic system with random parameters is also developed. Since in discrete or sampled time, all of the relevant operators are bounded (in partic- ular, the state transition operator), the discrete or sampled time LQ theory in (innite- dimensional) Hilbert space is entirely analogous to the nite-dimensional case with the ex- ception of the fact that the Riccati matrix equations, the expressions for the gains, and the matrices themselves are simply replaced by their operator counterparts. This is also true for the LQ tracking theory which will be discussed in the next chapter. 5.1 Problem Formulation We formulate a standard LQG control in the Bochner space for the random abstract parabolic system obtained in Chapter 4. To formulate this standard LQG control problem, 35 we rst need to consider the quadratic performance index in H q subject to the dynamical system (4.6) ^ J (u) =E k 1 1 X k=k 0 h ^ Qx k ; x k i Hq + ^ ru 2 k +h ^ Gx k 1 ; x k 1 i Hq ; (5.1) where ^ Q, ^ G, and ^ r are given in Section 4.1, and k 1 can be either nite or innite (in the latter case, ^ = 0). Now consider the time invariant nite horizon discrete time LQG control problem inH subject to the dynamical system (4.9): (P 1) Choose an input u2l 2 (k 0 ;k 1 1;R) for which the quadratic performance index ^ J(u) = k 1 1 X k=k 0 h ^ Qx k ;x k i H + ^ ru 2 k +h ^ Gx k 1 ;x k 1 i H (5.2) is minimized. Here ^ Q, ^ G are given in Section 4.2. ^ r is given in Section 4.1. Correspondingly, the time invariant innite horizon discrete time LQG control problem inH subject to the dynamical system (4.9) is given by (P 2) Choose an input u2l 2 (k 0 ;1;R) for which the quadratic performance index ^ J(u) = 1 X k=k 0 h ^ Qx k ;x k i H + ^ ru 2 k (5.3) is minimized. Note that in light of our denitions, the quadratic performance index given in (5.2) or (5.3) is the same as the one given in (5.1). If there is no noise in the dynamical system (4.9), (P 1) and (P 2) are linear quadratic regulator (LQR) problems and the solutions can be obtained by general discrete time LQR theory (see [18]). Without the noise terms in (4.9) (then there is no expected value in 36 the corresponding quadratic performance index), for every initial value x k 0 , the optimal closed-loop solution to (P 1) is unique and generated by the linear control law u k = ^ F k x k =h ^ f k ; x k i H ; k =k 0 ; k 0 + 1;; k 1 1; (5.4) where ^ F k = n ^ r + ^ B ^ k+1 ^ B o 1 ^ B ^ k+1 ^ A; (5.5) ^ f k = ^ F k are the corresponding functional gains obtained by Riesz Representation Theorem. The operators ^ k ,k =k 0 ; k 0 +1;; k 1 1, are the unique self-adjoint positive semi-denite operators satisfying the following Riccati dierence equation ^ k = ^ A ^ k+1 ^ k+1 ^ B ^ r + ^ B ^ k+1 ^ B 1 ^ B ^ k+1 ^ A + ^ Q; (5.6) k =k 0 ; k 0 + 1;; k 1 1, with ^ k 1 = ^ G. Moreover, it follows that ^ J( u) =h ^ k 0 x k 0 ;x k 0 i H , and that the optimal trajectoryf x k g k 1 k=k 0 is given by x k+1 = ^ A ^ B ^ F k x k , x k 0 =x k 0 . In terms of the innite horizon problem (P 2), we need to rstly consider the existence and uniqueness of its solution. An operator ^ 2 L(H;H) is a solution to the algebraic Riccati equation (ARE) associated with (P 2) if ^ = ^ A ^ ^ ^ B ^ r + ^ B ^ ^ B 1 ^ B ^ ^ A + ^ Q: (5.7) A control sequenceu2l 2 (k 0 ;1;R) is dened to be an admissible control for the initial conditionx k 0 if ^ J (u) <1. The existence of a positive semi-denite self-adjoint solution to the ARE is equivalent to the existence of an admissible control for any initial condition x k 0 . As in the case of nite-dimensional systems, the existence of an admissible control is equivalent to saying that the system is stabilizable (See Theorem 2.3 of [18]). 37 Since the solution to ARE (5.7) can only in uence the optimal control law for (P 2) rather than the estimator (we will see this fact in the next section), to obtain the exis- tence of the solution to ARE (5.7), we only need to consider the corresponding LQR prob- lem associated with (P 2) (i.e. there is no noise terms in the dynamical system (4.9) and no expected value in the quadratic performance index). Let us consider the zero control, say u(t) 0 in the corresponding LQR problem associated with (P 2), then the dynam- ical system (4.9) becomes x k+1 = ^ Ax k , x k 0 = x k 0 ; the quadratic performance index be- comes ^ J(0) = P 1 k=k 0 h ^ Qx k ;x k i H , plug the solutionx k =T k ()x k 0 =T(k)x k 0 into the quadratic performance index and estimatej ^ J(0)j. By the exponential stability of T(t), we have for any x k 0 2 H, P 1 k=0 j T k ()x k 0 j 2 = P 1 k=0 j T(k)x k 0 j 2 <1 (this is the discrete version of Lemma 2 in [7]). Then by the positive semi-denite property of ^ Q, we havejJ(0)j P 1 k=k 0 jhQ 1 2 T k ()x k 0 ;Q 1 2 T k ()x k 0 i H j P 1 k=k 0 jQ 1 2 j 2 H jT k ()x k 0 j 2 H P 1 k=0 jQ 1 2 j 2 H jT k ()x k 0 j 2 H <1, for any initial condition x k 0 . Therefore, u(t) 0 is an admissible control. It follows that there exists a positive semi-denite self-adjoint solution to the ARE (5.7) and thereby the solution to corresponding LQR problem associated with (P 2) exists. In addition, ^ J(u) is a strictly convex function, and R is a subspace which is convex, so if an optimal control exists, it is unique. Then without the noise terms in (4.9), for every initial valuex k 0 , the optimal closed-loop solution to (P 2) is unique and generated by the linear control law u k = ^ F x k =h ^ f; x k i H ; k =k 0 ; k 0 + 1;; (5.8) where ^ F = n ^ r + ^ B ^ ^ B o 1 ^ B ^ ^ A; (5.9) ^ f = ^ F is the corresponding functional gain obtained by Riesz Representation Theo- rem, ^ J( u) =h ^ x k 0 ;x k 0 i H , and that the optimal trajectoryf x k g 1 k=k 0 is given by x k+1 = ^ A ^ B ^ F x k , x k 0 =x k 0 . 38 5.2 LQG Compensator However, since the dynamical system (4.9) is corrupted by Gaussian noise, and in ad- dition, it depends on the random vector q, the state cannot be measured completely. We therefore design what is known as an LQG compensator which combines the LQR theory with a Kalman lter-based state estimator or observer (see, for example, [16], [24]). In the observer or estimator, the state covariance operator and output covariance matrix are given by ~ Q = ^ B 1 ^ B 1 2L(H;H) where = diag( 2 1 ; 2 2 )2 R 22 , and ~ R = 2 > 0, respectively. The observer or state estimator for nite-time horizon problem (P 1) takes the form ~ x k+1 = ^ A ~ x k + ^ Bu k + ~ L k (y k ^ C ~ x k ); ~ x k 0 = ~ ' 0 ; (5.10) where ~ ' 0 2H is arbitrary. The operators which are known as Kalman gains ~ L k 2L(R;H) for k 0 are given by ~ L k = ^ A ~ S k ^ C n 2 + ^ C ~ S k ^ C o 1 ; with positive semi-denite, self-adjoint operators ~ S k be given by recurrence Riccati dieren- tial equation ~ S k+1 = ^ A ~ S k ~ S k ^ C 2 + ^ C ~ S k ^ C 1 ^ C ~ S k ^ A + ~ Q; ~ S k 0 = 0: (5.11) Based on separation principle (see Theorem 5.3 in [24]), the LQG compensator for (P 1) is given by u k = ^ F k ~ x k =h ^ f k ; ~ x k i H ; k =k 0 ;k 0 + 1;k 0 + 2;::: where ~ x k is generated by (5.10), and the feedback operators ^ F k and functional control gains ^ f k are given by (5.5) and (5.4), respectively. Note that since ~ L k 2L(R;H), it follows that in fact ~ L k = ~ l k 2H. The elements ~ l k inH are the optimal functional observer gains. 39 The observer or state estimator ~ x k for (P 2) takes the form ~ x k+1 = ^ A ~ x k + ^ Bu k + ~ L(y k ^ C ~ x k ); ~ x k 0 = ~ ' 0 ; (5.12) where ~ ' 0 2H is arbitrary, the Kalman gain ~ L2L(R;H) is given by ~ L = ^ A ~ S ^ C n 2 + ^ C ~ S ^ C o 1 ; with positive semi-denite, self-adjoint operator S be given by algebraic Riccati equation (ARE) ~ S = ^ A ~ S ~ S ^ C 2 + ^ C ~ S ^ C 1 ^ C ~ S ^ A + ~ Q: (5.13) Note that by comparison with ARE (5.7), (5.13) is also related to a same innite horizon LQR problem involved in the dynamical system (4.9) mentioned in Section 5.1 by only re- placing ^ A, ^ B, ^ Q, ^ r, by ^ A , ^ C , ~ Q, 2 . Consider the exponentially stable, analytic semigroup of bounded linear operatorsT(t) onH, sinceH is re exive, the adjoint semigroupT (t) is also an exponentially stable, analytic semigroup on H whose innitesimal generator is A (see Corollary 10.6 in [33]). Therefore by the same reason mentioned in Section 5.1, the solution to ARE (5.13) exists. Similarly, based on separation principle, the LQG compensator for (P 2) is given by u k = ^ F ~ x k =h ^ f; ~ x k i H ; k =k 0 ;k 0 + 1;k 0 + 2;::: where ~ x k is generated by (5.12), and the feedback operators ^ F and functional control gains ^ f are given by (5.9) and (5.8), respectively. Similarly, note that since ~ L2 L(R;H), it follows that in fact ~ L = ~ l2H. The elements ~ l inH are the optimal functional observer gains. 40 5.3 Approximation and Convergence We are going to estimate the LQG control problems on both nite and innite horizons listed in Section 5.1-5.2 in the sequence of nite-dimensional spacesV N constructed in Sec- tion 4.3. Consider the sequence of nite-dimensional approximating LQG control problems on the nite time horizon subject to the approximating dynamical system(4.10): P 1 N For every N, choose an input u N 2l 2 (k 0 ;k 1 1;R) for which the approximating quadratic performance index ^ J N (u N ) = k 1 1 X k=k 0 h ^ Q N x N k ;x N k i H + ^ r(u N k ) 2 +h ^ G N x N k 1 ;x N k 1 i H is minimized. Correspondingly, the sequence of nite-dimensional approximating LQG control problems on the innite time horizon subject to the dynamical system (4.10) is given by P 2 N For every N, choose an input u N 2 l 2 (k 0 ;1;R) for which the approximating quadratic performance index ^ J N (u N ) = 1 X k=k 0 h ^ Q N x N k ;x N k i H + ^ r(u N k ) 2 is minimized. Set ~ Q N = ^ B N 1 ( ^ B N 1 ) 2L(V N ;V N ). Then the approximating observer or state esti- mator for approximating nite time horizon problem (P 1 N ) takes the form ~ x N k+1 = ^ A N ~ x N k + ^ B N u N k + ~ L N k (y N k ^ C N ~ x N k ); ~ x N k 0 =P N ~ ' 0 ; (5.14) where the approximating Kalman gains ~ L N k 2L(R;V N ) are given by ~ L N k = ^ A N ~ S N k ( ^ C N ) n 2 + ^ C N ~ S N k ( ^ C N ) o 1 ; 41 with positive semi-denite, self-adjoint operators ~ S N k be given by the approximating recur- rence Riccati dierential equation ~ S N k+1 = ^ A N ~ S N k ~ S N k ( ^ C N ) 2 + ^ C N ~ S N k ( ^ C N ) 1 ^ C N ~ S N k ( ^ A N ) + ~ Q N ; ~ S N k 0 = 0: (5.15) As in the innite-dimensional case, the solution to (P 1 N ) is unique and is given by u N k = ^ F N k ~ x N k =h ^ f N k ; ~ x N k i H ; k =k 0 ; k 0 + 1; where ^ F N k = n ^ r + ( ^ B N ) ^ N k ^ B N o 1 ( ^ B N ) ^ N k ^ A N ; and ^ N k is the unique positive semi-denite, symmetric solution to the approximating Riccati dierence equation, ^ N k = ( ^ A N ) ^ N k+1 ^ N k+1 ^ B N ^ r + ( ^ B N ) ^ N k+1 ^ B N 1 ( ^ B N ) ^ N k+1 ^ A N + ^ Q N ; (5.16) k =k 0 ; k 0 +1;; k 1 1, with ^ N k 1 = ^ G N ; the approximating state estimator ~ x N k is generated by (5.14). The approximating observer or state estimator for approximating innite time horizon problem (P 2 N ) takes the form ~ x N k+1 = ^ A N ~ x N k + ^ B N u N k + ~ L N (y N k ^ C N ~ x N k ); ~ x N k 0 =P N ~ ' 0 ; (5.17) where the approximating Kalman gains ~ L N 2L(R;V N ) are given by ~ L N = ^ A N ~ S N ( ^ C N ) n 2 + ^ C N ~ S N ( ^ C N ) o 1 ; 42 with positive semi-denite, self-adjoint operators ~ S N be given by the approximating ARE ~ S N = ^ A N ~ S N ~ S N ( ^ C N ) 2 + ^ C N ~ S N ( ^ C N ) 1 ^ C N ~ S N ( ^ A N ) + ~ Q N : (5.18) SinceT N (t) and (T N ) (t) are uniformly continuous semigroups, which is also exponential stability, analytic onH, by the same reasons as those in Section 5.2, the solution ~ S N to the approximating ARE (5.18) exists. The solution to (P 2 N ) is unique and is given by u N k = ^ F N ~ x N k =h ^ f N ; ~ x N k i H ; k =k 0 ; k 0 + 1; where ^ F N = n ^ r + ( ^ B N ) ^ N ^ B N o 1 ( ^ B N ) ^ N ^ A N ; and ^ N is the unique positive semi-denite, symmetric solution to the approximating ARE, ^ N = ( ^ A N ) ^ N ^ N ^ B N ^ r + ( ^ B N ) ^ N ^ B N 1 ( ^ B N ) ^ N ^ A N + ^ Q N ; (5.19) the solution ^ N to this approximating ARE (5.19) exists by the exponential stability of T N (t); the approximating state estimator ~ x N k is generated by (5.17). Next, let us obtain the convergence results for approximating problems (P 1 N ) and (P 2 N ). By the approximating operators dened in Section 4.3, we obtain that ^ A N , ( ^ A N ) , ^ B N , ( ^ B N ) , ^ B N 1 , ( ^ B N 1 ) , ^ C N , ( ^ C N ) , ^ D N , ( ^ D N ) , ^ Q N , ~ Q N strongly converge correspondingly to their innite-dimensional counterparts ^ A, ( ^ A) , ^ B, ( ^ B) , ^ B 1 , ( ^ B 1 ) , ^ C, ( ^ C) , ^ D, ( ^ D) , ^ Q, ~ Q. Therefore, by Theorem 3.3 in [18], we immediately have the following convergence results of (P 1 N ): 43 Theorem 10. The discrete time LQG control problem (P 1) and its approximating problem (P 1 N ) are dened in Section 5.1 and 5.3, respectively. Then (P 1 N ) is convergent to (P 1) in the following way: (i) lim N!1 j u N uj l 2 = 0; (ii) lim N!1 j ^ N k P N x ^ k xj H = 0; (iii) lim N!1 j ~ S N k P N x ~ S k xj H = 0; (iv) lim N!1 j ^ J N ( u N ) ^ J( u)j = 0; (v) lim N!1 k ^ F N k P N ^ F k k L(H;R) = 0; 8x2V, andk =k 0 ; ;k 1 1, where thel 2 norm is dened byhx;yi l 2 = P k 1 k=k 0 hx k ;y k i X , for any x;y2l 2 (k 0 ;k 1 ;X) and any Hilbert space X. Then we consider the convergence of approximation problem (P 2 N ) on an innite hori- zon k = k 0 ;k 0 + 1; . We need to consider the uniform boundedness of the solution to approximation AREs (5.18) and (5.19). Lemma 11.f ^ N g is a uniformly bounded sequence of bounded linear operators in terms of N. Proof. For anyx2H,T N ()P N !T() inH-norm, i.e. lim N!1 kT N ()P N xT()xk H = 0; so there exists a constant C, such that for any N, kT N ()P N xk H kT N ()P N xT()xk H +kT()xk H <C; (5.20) i.e. sup N kT N ()P N xk H <1: (5.21) 44 By uniform boundedness principle, we have sup N kT N ()k H <1: (5.22) Therefore, we can nd a constant ~ M <1 independent ofN, such thatkT N ()k H ~ M, i.e. ^ A N is uniformly bounded in terms of N withk ^ A N k H ~ M. Since ^ B N ! ^ B, ^ Q N P N ! ^ Q, similarly to (5.20)- (5.22), we get the uniform boundedness of ^ B N and ^ Q N . Since the approximation problem is discussed in the nite-dimensional spaceV N , mod- ifying the equation (2.6) in [8] to our case, for each N, we obtain ^ N ( ^ A N + ^ B N ~ F N ) ^ N ( ^ A N + ^ B N ~ F N ) + ^ Q N + ^ r( ~ F N ) ~ F N ; (5.23) where ~ F N is chosen such thatk ^ A N + ^ B N ~ F N k 1. This can be achieved since we have the existence of the positive-denite self-adjoint solution of approximation ARE (5.19) and therefore, for each N, (P 2 N ) exists a stable solution. Furthermore, we consider the uniform boundedness of ~ F N . Sincejk ^ A N kk ^ B N ~ F N kj< 1, andk ^ A N k < ~ M, we havek ^ B N ~ F N k < 1 + ~ M. Thereforek( ~ F N ) ( ^ B N ) k < 1 + ~ M. i.e. for anyx2V,k( ~ F N ) ( ^ B N ) xkk( ~ F N ) ( ^ B N ) kkxk< (1 + ~ M)kxk. Assume that ~ F N is not uniformly bounded, then for any large enough constant C > 0, there exists N, such thatk ~ F N k > C. So the norm of its adjointk( ~ F N ) k > C. By the denition of operator norm, for this C and N, we can pick up a ~ x2V, such that k( ~ F N ) k = sup x2V k( ~ F N ) xk kxk k( ~ F N ) ( ^ B N ) ~ xk k( ^ B N ) ~ xk >C: (5.24) 45 Since R is a nite-dimensional space, ( ^ B N ) converges strongly to ( ^ B N ) , we have k ^ B ~ xk< 2k( ^ B N ) ~ xk. So (5.24) implies thatk( ~ F N ) ( ^ B N ) ~ xk>Ck( ^ B N ) ~ xk> 1 2 Ck ^ B ~ xk. Therefore, 1 + ~ M >k( ~ F N ) ( ^ B N ) k = sup ~ x2V k( ~ F N ) ( ^ B N ) ~ xk k ~ xk > C 2 k ^ B ~ xk k ~ xk ; i.e.k ^ B ~ xk< 2(1+ ~ M) C k ~ xk. By the denition of operator norm, we know thatk ^ B k< 2(1+ ~ M) C , soC has to be less than 2(1+ ~ M) k ^ B k , which is contradiction to arbitrary large C. Therefore, ~ F N is uniformly bounded. Now we assume that there exists constants C Q , C F , such that for any N,k ^ Q N k C Q , k ~ F N kC F . (5.23) tells us that (1k ^ A N + ^ B N ~ F N k 2 )k ^ N kC Q + ^ rC 2 F ; (5.25) sok ^ N kc(C Q + ^ rC 2 F ), where c is a constant which is greater than 1 and independent of N. We proved thatf ^ N g is uniformly bounded in N. Similarly, we can also provef ~ S N g is uniformly bounded in terms of N. By Theorem 3.10 in [18], the solutions to the approximating AREs related to generating approximating feedback gains (5.19) and generating approximating Kalman gains (5.18) in (P 2 N ), ^ N and ~ S N , respectively, converge strongly inH to the solutions to the AREs (5.7), ^ and (5.13), ~ S, respectively. And applying Theorem 3.9 in [18], since the control space is nite, we have Theorem 12. The discrete time LQG control problem (P 2) and its approximating problem (P 2 N ) are dened in Section 5.1 and 5.3, respectively. Then (P 2 N ) is convergent to (P 2) in the following way: (i) lim N!1 j u N uj l 2 = 0; (ii) lim N!1 k ^ F N P N ^ Fk L(H;R) = 0; (iii) lim N!1 j ^ J N ( u N ) ^ J( u)j = 0 46 where the l 2 norm here is dened by hx;yi l 2 = 1 X k=k 0 hx k ;y k i X ; for any x;y2l 2 (k 0 ;1;X), any Hilbert space X. Finally, we note that both the innite-dimensional and approximating nite-dimensional optimal compensator on both the nite time and innite time horizon can be represented (and in the nite-dimensional case, implemented) in terms of functional gains with the approximating nite-dimensional functional gains converging to their innite-dimensional counterparts. For example, in the nite-dimensional, innite time horizon case, for k = k 0 ;k 0 + 1;:::, the approximating compensator is then given by: u N k = ^ F N ~ x N k =h ^ f N ; ~ x N k i H = Z Q q 1 q 3 ^ f N 1 (q) ~ x N 1;k (q) + q 1 q 5 ^ f N 2 (q) ~ x N 2;k (q) + Z 1 0 ^ f N 3 (;q) ~ x N 3;k (;q)d d(q); where in the above expression, we have used the following notational convention ^ f N = ( ^ f N 1 ; ^ f N 2 ; ^ f N 3 )2 V N and ~ x N k = ( ~ x N 1;k ; ~ x N 2;k ; ~ x N 3;k )2 V N given by (5.17). The other three cases are analogous. 5.4 Matrix Representation Note that the equations listed in Section 5.3 are all operator equations, albeit nite- dimensional ones. In order to actually carry out computations (i.e. by using standard ARE solvers), these equations must be converted to equivalent matrix equations. Since the basis we have chosen for V N is not orthonormal, some care must be exercised in making this conversion so as to obtain a standard symmetric matrix ARE. Now consider the matrix representation of the approximating Riccati equations. For the operators ^ A N , ^ B N , ^ B N 1 , ^ C N , ^ D N , we can obtain the matrix representations of them with 47 respect to the basisf N J g J dened in Section 4.3 by the standard method and denote them by ^ A N , ^ B N , ^ B N 1 , ^ C N , ^ D N . Dene the N by N (N = (n + 1) 6 i=1 m i ) transfer matrix M N whose the element at the position (i;j) of matrix is given by [M N ] ij =h N i ; N j i H . Then the matrix representations of ( ^ A N ) , ( ^ B N ) , ( ^ B N 1 ) , ( ^ C N ) , ( ^ D N ) are ( ^ A N ) = (M N ) 1 ( ^ A N ) T M N ( ^ B N ) = ( ^ B N ) T M N ; ( ^ B N 1 ) = ( ^ B N 1 ) T M N ; ( ^ C N ) = (M N ) 1 ( ^ C N ) T ; ( ^ D N ) = (M N ) 1 ( ^ D N ) T : By the expressions of ^ Q N , ^ G N , ~ Q N , the matrix representations of them are ^ Q N = (M N ) 1 ( ^ D N ) T ^ D N ; ^ G N = ^ (M N ) 1 ( ^ D N ) T ^ D N ; ~ Q N = ^ B N 1 ( ^ B N 1 ) T M N : Dene matrices ~ ^ Q N =M N ^ Q N = ( ^ D N ) T ^ D N and ~ G N =M N ^ G N = ^ ( ^ D N ) T ^ D N . Then the matrix representation of approximating Riccati dierence equation (5.16) is given by ~ N k = ( ^ A N ) T ~ N k+1 ~ N k+1 ^ B N (^ r + ( ^ B N ) T ~ N k+1 ^ B N ) 1 ( ^ B N ) T ~ N k+1 ^ A N + ~ ^ Q N ; ~ N k 1 = ~ G N : This is a standard symmetric matrix Riccati dierence equation on ~ N k , which isM N times the matrix representation of ^ N k . 48 Dene matrix ~ ~ Q N = ~ Q N (M N ) 1 = ^ B N 1 ( ^ B N 1 ) T . Similarly, the matrix representation of the approximating Riccati dierence equation (5.15) is given by ~ S N k+1 = ^ A N ~ S N k ~ S N k ( ^ C N ) T 2 + ^ C N ~ S N k ( ^ C N ) T 1 ^ C N ~ S N k ( ^ A N ) T + ~ ~ Q N ; ~ S N k 0 = 0; where ~ S N k is the matrix representation of ~ S N k times (M N ) 1 . So for the matrix representation of the solution to (P 1 N ), we have u N k =[ ^ F N k ]~ x N k ; k =k 0 ;; k 1 1; [ ^ F N k ] = n ^ r + ( ^ B N ) T ~ N k+1 ^ B N o 1 ( ^ B N ) T ~ N k+1 ^ A N ; where ~ x N k is the coecient vector of ~ x N k generated by ~ x N k+1 = ^ A N ~ x N k + ^ B N u N k + ~ L N k (y N k ^ C N ~ x N k ); ~ x N k 0 = [P N ~ ' 0 ]; (5.26) where [P N ~ ' 0 ] is the coecient vector ofP N ~ ' 0 and the matrix approximating Kalman gains ~ L N k are given by ~ L N k = ^ A N ~ S N k ( ^ C N ) T n 2 + ^ C N ~ S N k ( ^ C N ) T o 1 : In terms of the matrix representation of approximating ARE (5.19), it is given by ~ N = ( ^ A N ) T ~ N ~ N ^ B N (^ r + ( ^ B N ) T ~ N ^ B N ) 1 ( ^ B N ) T ~ N ^ A N + ~ ^ Q N : This is a standard symmetric matrix ARE on ~ N , which is M N times the matrix represen- tation of ^ N . Similarly, the matrix representation of the approximating ARE (5.18) is given by ~ S N = ^ A N ~ S N ~ S N ( ^ C N ) T 2 + ^ C N ~ S N ( ^ C N ) T 1 ^ C N ~ S N ( ^ A N ) T + ~ ~ Q N ; 49 where ~ S N is the matrix representation of ~ S N times (M N ) 1 . So for the matrix representation of the solution to (P 2 N ), we have u N k =[ ^ F N ]~ x N k ; k =k 0 ; k 0 + 1; [ ^ F N ] = n ^ r + ( ^ B N ) T ~ N ^ B N o 1 ( ^ B N ) T ~ N ^ A N ; where ~ x N k is the coecient vector of ~ x N k generated by ~ x N k+1 = ^ A N ~ x N k + ^ B N u N k + ~ L N (y N k ^ C N ~ x N k ); ~ x N k 0 = [P N ~ ' 0 ]; (5.27) where [P N ~ ' 0 ] is the coecient vector ofP N ~ ' 0 and the matrix approximating Kalman gain ~ L N is given by ~ L N = ^ A N ~ S N ( ^ C N ) T n 2 + ^ C N ~ S N ( ^ C N ) T o 1 : 50 Chapter 6 Linear Quadratic Gaussian Tracking Problems In this chapter, we develop a discrete time LQG tracking compensator for the closed loop output feedback control for intravenously-infused alcohol studies with non-steady state target blood alcohol concentration and deconvoluting blood or breath alcohol concentration from biosensor measured transdermal alcohol concentration. We force both models to track a given TAC signal. A nite-dimensional approximation and convergence theory for the closed- loop LQ tracking and estimation of the abstract parabolic system with random parameters (4.9) is also developed. 6.1 Problem Formulation We formulate a standard LQG tracking problem for the random abstract parabolic system obtained in Chapter 4. We assume thatf k g is the given reference signal to be tracked. Associated with our alcohol biosensor problem, this reference signal refers to TAC signal. To formulate this standard LQG tracking problem, we rst need to consider the quadratic performance index in H q subject to the dynamical system (4.6) ^ J (u) =E k 1 1 X k=k 0 j ^ Dx k k j 2 + ^ ru 2 k + ^ j ^ Dx k 1 k 1 j 2 ; (6.1) where k 1 can be either nite or innite (in the latter case, ^ = 0). 51 Now consider the time invariant discrete time LQG tracking problem on nite horizon subject to the dynamical system (4:9): (P 1) Choose an input u2l 2 (k 0 ;k 1 1;R) for which the quadratic performance index ^ J (u) = k 1 1 X k=k 0 j ^ Dx k k j 2 + ^ ru 2 k + ^ j ^ Dx k 1 k 1 j 2 ; (6.2) is minimized. Correspondingly, the time invariant discrete time LQG tracking problem on innite hori- zon subject to the dynamical system (4:9) is given by (P 2) Choose an input u2l 2 (k 0 ;1;R) for which the quadratic performance index ^ J (u) = 1 X k=k 0 j ^ Dx k k j 2 + ^ ru 2 k ; (6.3) is minimized. Note that in light of our denitions, the quadratic performance index given in (6.2) or (6.3) is the same as the one given in (6.1). The closed-loop solutions to the discrete-time LQ tracking problem on the nite-horizon (P 1) are given in [5], [25], and [47]. These results are all consistent with each other; here we mainly cite the results from [5]. Similarly, since we are not able to observe the full state of the system (4.9), our feedback tracking controller will require an observer or state estimator. Here we use the same com- pensator discussed in Section 5.2 and integrate the results with separation principle. That is, the optimal feedback tracking control laws for (P 1) and (P 2) are obtained by applying the corresponding deterministic tracking laws to the Kalman estimators ~ x k and ~ x generated by the recurrence relation (5.10) and (5.12), respectively. 52 For every initial valuex 0 and reference trajectory signalf k g, the optimal input for the problem (P 1) is unique and generated by the following control law u k = ^ F k ~ x k + k = ( ^ B K k+1 ^ B + ^ r) 1 ^ B (K k+1 ^ A ~ x k + k+1 ); where ~ x k is generated by (5.10), and ^ F k = ( ^ B K k+1 ^ B + ^ r) 1 ^ B K k+1 ^ A2L(H;R) and k = ( ^ B K k+1 ^ B + ^ r) 1 ^ B k+1 2 R withK k , k computed backward from k 1 to k 0 in the following way: K k 1 = ^ G; K k = ^ A [K k+1 K k+1 ^ B( ^ B K k+1 ^ B + ^ r) 1 ^ B K k+1 ] ^ A + ^ Q; k 1 = ^ ^ D k 1 ; k = ^ A (I + ^ r 1 K k+1 ^ B ^ B ) 1 k+1 + ^ D k : In terms of the innite horizon tracking problem (P 2), we assume that the reference signal converges to a xed reference signalfg. By Theorem 2:4 in [25], ^ A open loop stable, implies that the solution to the algebraic Riccati equation (ARE) K = ^ A [KK ^ B( ^ B K ^ B + ^ r) 1 ^ B K] ^ A + ^ Q exists, is unique, self-adjoint, and positive semi-denite, and the ecacy of the tracking controller u k = ^ F ~ x k + = ( ^ B K ^ B + ^ r) 1 ^ B (K ^ A ~ x k +); is guaranteed, where ~ x k is generated by (5.12), and ^ F = ( ^ B K ^ B+ ^ r) 1 ^ B K ^ A2L(H;R) and = ( ^ B K ^ B + ^ r) 1 ^ B 2R with = ^ A (I + ^ r 1 K ^ B ^ B ) 1 + ^ D : 53 6.2 Approximation and Convergence We are going to estimate the LQG tracking problems on both nite and innite horizons listed in Section 6.1 in the sequence of nite-dimensional spacesV N constructed in Section 4.3. Consider the sequence of nite-dimensional approximating LQG tracking problems on the nite time horizon subject to the approximating dynamical system (4.10) P 1 N For every N, choose an input u N 2l 2 (k 0 ;k 1 1;R) for which the approximating quadratic performance index ^ J N u N = k 1 1 X k=k 0 j ^ D N x N k k j 2 + ^ r(u N k ) 2 + ^ j ^ D N x N k 1 k 1 j 2 ; is minimized. Correspondingly, the sequence of nite-dimensional approximating LQG tracking prob- lems on the innite time horizon subject to the dynamical system (4.10) is given by P 2 N For every N, choose an input u N 2 l 2 (k 0 ;1;R) for which the approximating quadratic performance index ^ J N u N = 1 X k=k 0 j ^ D N x N k k j 2 + ^ r(u N k ) 2 ; is minimized. The solution to (P 1 N ) is given in terms ofK N k and N k on [k 0 ;k 1 ] respectively by K N k = ( ^ A N ) [K N k+1 K N k+1 ^ B N (( ^ B N ) K N k+1 ^ B N + ^ r) 1 ( ^ B N ) K N k+1 ] ^ A N + ^ Q N ; N k = ( ^ A N ) [I + ^ r 1 K N k+1 ^ B N ( ^ B N ) ] 1 N k+1 + ( ^ D N ) k : withK N k 1 = ^ G N and N k 1 = ^ ( ^ D N ) k 1 . The optimal tracking control, u N , for problem (P 1 N ) is u N k = ^ F N k ~ x N k + N k ; 54 where the approximating observer ~ x N k is generated by (5.14), and ^ F N k = [( ^ B N ) K N k+1 ^ B N + ^ r] 1 ( ^ B N ) K N k+1 ^ A N ; N k = [( ^ B N ) K N k+1 ^ B N + ^ r] 1 ( ^ B N ) N k+1 : It was shown in [18], thatK N k P N converges strongly to K k , k2 [k 0 ;k 1 ], andF N k P N converges in the uniform operator topology to F k , k2 [k 0 ;k 1 ]. In addition, by (2:47) in [25], [I + ^ r 1 K N k+1 ^ B N ( ^ B N ) ] 1 in the denition of N k can be transformed into an expression involving [^ r + ( ^ B N ) K N k+1 ^ B N ] 1 , and therefore that N k ! k as N ! 1. Combined with separation principle, the approximating closed loop control, u N converges to u from Problem (P 1) in l 2 , where the l 2 inner product and corresponding norm are dened by hx;yi l 2 = P k 1 k=k 0 hx k ;y k i H for anyx andy inl 2 (k 0 ;k 1 ;H). The approximating cost functions ^ J N ( u N ) also converge to ^ J( u) as N!1. In steady state, by Theorems 3.9 and 3.10 in [18], a positive semi-denite self-adjoint solution,K, to the ARE K N =( ^ A N ) fK N K N ^ B N [( ^ B N ) K N ^ B N + ^ r] 1 ( ^ B N ) K N g ^ A N + ^ Q N ; exists, the optimal control u N for approximating tracking problem (P 2 N ) is given by u N k = ^ F N ~ x N k + N = [( ^ B N ) K N ^ B N + ^ r] 1 ( ^ B N ) (K N ^ A N ~ x N k + N ); where the approximating observer ~ x N k is generated by (5.17), ^ F N = [( ^ B N ) K N ^ B N + ^ r] 1 ( ^ B N ) K N ^ A N , N = [( ^ B N ) K N ^ B N + ^ r] 1 ( ^ B N ) N , with N =( ^ A N ) (I + ^ r 1 K N ^ B N ( ^ B N ) ) 1 N + ( ^ D N ) ; 55 and, analogous to the nite time horizon case, convergence of K N , F N , u N , and ^ J N to their innite-dimensional counterparts, as N!1, can be established (see [18] and [45]). Finally, we note that both the innite-dimensional and approximating nite-dimensional optimal compensator on both the nite time and innite time horizon can be represented (and in the nite-dimensional case, implemented) in terms of functional gains with the approximating nite-dimensional functional gains converging to their innite-dimensional counterparts. For example, in the nite-dimensional, nite time horizon case, fork =k 0 ;k 0 + 1;:::, the optimal compensator is given by u N k = ^ F N k ~ x N k + N k =h ^ f N k ; ~ x N k i H + N k = Z Q q 1 q 3 ^ f N 1;k (q) ~ x N 1;k (q) + q 1 q 5 ^ f N 2;k (q) ~ x N 2;k (q) + Z 1 0 ^ f N 3;k (;q) ~ x N 3;k (;q)d d(q) + N k ; where ^ f N k = ( ^ f N 1;k ; ^ f N 2;k ; ^ f N 3;k )2V N H with ^ f N k =( ^ f N 3;k (0;); ^ f N 3;k (1;); ^ f N 3;k (;)) = ( ^ F N k ) =( ^ A N ) K N k+1 ^ B N [( ^ B N ) K N k+1 ^ B N + ^ r] 1 2V N : The other three cases are analogous (see [45]). 6.3 Matrix Representation For the same reasons as what we mentioned in Section 5.4, we need to consider the matrix representation of the solutions for both (P 1 N ) and (P 2 N ). Using the same notations in Section 5.4, the matrix representations ofK N k and N k are given by ~ K N k = ( ^ A N ) T [ ~ K N k+1 ~ K N k+1 ^ B N (( ^ B N ) T ~ K N k+1 ^ B N + ^ r) 1 ( ^ B N ) T ~ K N k+1 ] ^ A N + ~ ^ Q N ; ~ N k = ( ^ A N ) T [I + ^ r 1 ~ K N k+1 ^ B N ( ^ B N ) T ] 1 ~ N k+1 + ( ^ D N ) T k ; 56 with ~ K N k 1 = ^ ( ^ D N ) T ^ D N and ~ k 1 = ^ ( ^ D N ) T k 1 . Here the matrix ~ K N k isM N times the matrix representation ofK N k , the vector ~ N k is M N times the coecient vector of N k . So for the matrix representation of the solution to (P 1 N ), we have u N k = [( ^ B N ) T ~ K N k+1 ^ B N + ^ r] 1 ( ^ B N ) T ( ~ K N k+1 ^ A N ~ x N k + ~ N k ); where the vector ~ x N k is generated by (5.26). In terms of the steady state,K N and N are given by ~ K N = ( ^ A N ) T [ ~ K N ~ K N ^ B N (( ^ B N ) T ~ K N ^ B N + ^ r) 1 ( ^ B N ) T ~ K N ] ^ A N + ~ ^ Q N ; ~ N = ( ^ A N ) T [I + ^ r 1 ~ K N ^ B N ( ^ B N ) T ] 1 ~ N + ( ^ D N ) T ; where the matrix ~ K N is M N times the matrix representation ofK N , the vector ~ N is M N times the coecient vector of N . So for the matrix representation of the solution to (P 2 N ), we have u N k = [( ^ B N ) T ~ K N ^ B N + ^ r] 1 ( ^ B N ) T ( ~ K N ^ A N ~ x N k + ~ N ); where the vector ~ x N k is generated by (5.27). 57 Chapter 7 Numerical Results To demonstrate the ecacy and the level of performance of our approach to the LQG control and tracking problems talked about in previous chapters, we here design several numerical experiments based on intravenously-infused alcohol clamping studies and decon- voluting BAC from TAC. In all the examples, we assume the underlying population model given in that the distribution of the random parameters is either known a-priori or it has been t using. 7.1 The Numerical Convergence of the Approximation We choose examples that demonstrate the ecacy of our approach to LQG control with regard to the convergence of the nite-dimensional approximations. We consider a system of the general form of the one given in (3.3) and (3.4). In particular, we letq 1 = 0:2,q 2 = 0:5,q 3 = 0:5,q 4 = 0:5,q 5 = 0:5,q 6 = 0:5, 1 = 0:05, 2 = 0:05, = 0:05, k 0 = 0, k 1 =1, = 1, = 0, and ^ r = 0:1. We assume further that we do not actually know the precise value of q 1 , but rather only that it is random withq 1 Beta(;) with = 3 and = 2. We take the sampling interval to be = 0:1 and the discretization level of 2 [0; 1] and q 1 2 [0; 1] to be given by the multi-index N = (n;m). We note that all the results we present here are for an innite rather than nite time horizon design since 58 Figure 7.1: Optimal functional control gains for n =m = 4; 8; 16; 32. it is much easier to display steady state (functional) gains than it is when they are time dependent. Figures 7.1 and 7.2 show the functional control and observer gains for (from lower to upper) n =m = 4; 8; 16; and 32. The plots are o-set so they can be distinguished from one another. Tables 7.1 and 7.2 contain theL 2 norm of the dierence between the approximating gains and the innite-dimensional (computed withn =m = 32) control and observer gains. m =n 4 8 12 16 20 24 28 Norm (10 4 ) 18.00 10.00 5.18 2.61 1.25 0.54 0.17 Table 7.1: L 2 norm of the dierence between the approximating optimal functional control gains and the innite-dimensional optimal functional control gains. 59 Figure 7.2: Optimal functional observer gains for n =m = 4; 8; 16; 32. m =n 4 8 12 16 20 24 28 Norm (10 5 ) 12.62 7.39 4.81 3.21 2.09 1.24 0.56 Table 7.2: L 2 norm of the dierence between the approximating optimal functional observer gains and the innite-dimensional optimal functional observer gains. In Table 7.3, we show the optimal functional control gains ^ f 1 and ^ f 2 and in Figure 7.3 we plot the optimal functional control gains ^ f 3 for the full state feedback controller when q 1 = :1j, j = 1; 2;:::; 9; 10 all computed with n = 32. In these tables and gures, we have also tabulated and plotted the expected value of the optimal functional control gains, E [ ^ f] computed using our approach withn =m = 16 andq 1 Beta(;) with = 3 and = 2. In addition, since our scheme yields the approximating optimal control (and observer) gains as a function ofq 1 , we can readily compute 90% credible intervals and bands for the optimal control gains computed with our method. In Figure 7.3 the shaded region is the 90% credible 60 Figure 7.3: Optimal functional control gains ^ f 3 for various values of q 1 and the expected values of these gains whenq 1 Beta(;) with = 3 and = 2. band centered at the mean for the optimal functional control gains ^ f 3 computed using our method. q 1 0:1 0:3 0:5 0:7 0.9 E [q 1 ] ^ f 1 0.2630 0.2128 0.1740 0.1462 0.1258 0.1603 ^ f 2 3.9239 1.8792 1.2699 0.9654 0.7807 1.2339 Table 7.3: Optimal functional control gains ^ f 1 and ^ f 2 for various values of q 1 and the expected values of these gains whenq Beta(;) with = 3 and = 2. 61 7.2 The Values of the Performance Index by Dierent Approximating Optimal Compensators We used our nite time horizon theory to design the compensator and show how well our controller, observer, and compensator designs perform in practice by comparing the values for dierent performance index ^ J(u). Table 7.4 shows the values of the performance index, ^ J(u), when the system (3.3) was simulated with dierent approximating optimal controllers/compensators. We set x(0;) = 1:0, 0 1, w(0) = 1:0, v(0) = 1:0 and computed the approximating controllers with eithern = 32 orn =m = 32. The plant parameter values wereq = [0:2; 0:5; 0:5; 0:5:0:5; 0:5] T , the nal time was T = 10:0, and the length of the sampling interval was = 0:1. The standard deviations of the noise processes were taken to be 1 = 2 = = 0:05 and the control penalty weight was ^ r = 0:1. We set the seed in Matlab's random number generator to be equal to one in all of the simulations. We simulated the linearized plant using our spline model with n = 64 and for our scheme we assumed that q 1 =q 1 was random with q 1 Beta(3; 2). In Table 7.4, Controller/Compensator 1 was no control (i.e. u k = 0, k = 0; 1; 2;:::; 99), Controller/Compensator 2 was the optimal innite-dimensional (n = 64) full state feedback controller computed with the plant's value for q 1 to be q 1 = 0:2, Controller/Compensator 3 was the optimal nite-dimensional (n = 32) output feedback compensator computed with the plant's value for q 1 to be the plant value of q 1 , q 1 = 0:2, Controller/Compensator 4 was the optimal nite-dimensional (n = 32) output feedback compensator but computed with the incorrect value for q 1 , q 1 = 0:8, Controller/Compensator 5 was the optimal nite- dimensional (n = 32) output feedback compensator but computed with q 1 = E[q 1 ] = 0:6, and Controller/Compensator 6 was the optimal nite-dimensional (n = 32;m = 32) output feedback compensator computed using the approach we developed in this paper. 62 Finally, in Table 7.5 we show results of simulating controller/compensator 1, 2, and 3 along with compensator 6, the one developed here, for the case whereq 1 =q 1;k Beta(;) with = 3 and = 2, k = 0; 1; 2;:::; 100, where k is time, q 1 in the plant changed every time, so this will in uence the state x and therefore in uence the compensator. Con/Comp 1 2 3 4 5 6 ^ J(u)) 21.99 10.88 10.89 11.92 11.70 11.57 Table 7.4: Value of the performance index from simulations using dierent controllers/com- pensators. Con/Comp 1 2 3 6 ^ J(u) 9.07 5.08 7.78 5.60 Table 7.5: Value of the performance index when q 1 = q 1;k Beta(3; 2) using dierent controllers/compensators. 7.3 Simulations of an Intravenously-Infused Alcohol Study with Transdermal Sensing We present two simulation results for an intravenously-infused alcohol clamping exper- iment with transdermal sensing based on model (3.1)-(3.2): one example in which all the parameters are assumed known except for the diusivity =q 1 of the epidermal layer, which is assumed to be random with a beta distribution. Then in the second example, we assume that both the diusivity of the epidermal layer =q 1 and the control actuator gain b =q 2 are random. We assume that they are jointly independent with marginal distributions that are both Beta distributions, but with dierent, but known, parameters. In both examples, we set the target clamped value for the BAC to be ~ v 0 = 60:0 mg/dl or 0:06 percent alcohol. For the plant (3.1)-(3.2), we set the epidermal layer diusivity parameter = 0:48, the epidermal layer/TAC biosensor interface parameter = 0:56, the TAC biosensor collection reservoir clearance parameter = 0:05, the dermal layer 63 (blood)/epidermal layer interface parameter = 0:26, and the other losses of ethanol from the blood parameter = 0. These values were chosen to be consistent with our prior experi- ence tting our models to human subject data (see, for example,[6, 12, 35]). The Michaelis- Menten parametersK andM were taken to beK = 25 mg/(dl hr) andM = 10:5 mg/dl, respectively, which are considered to be approximately typical for a 70 kg male (see, for example, [30]). The control ~ u is the ow rate of the intravenous drip in units of ml/hr. Thus, to compute the plant control actuator gain, we assumed a 6% V/V ethanol intravenous drip, took the density of ethanol to be 790 mg/ml, and that the average human has 420 dl of body water. It follows that b =:06 790=420 =:1129 mg/(ml dl) and therefore that b~ u has the units of mg/(dl hr) (i.e. mg of ethanol per dl of body water per hour). We assume that the standard deviations for all the noise processes is 1 = 2 = = 5 mg/dl. The specied values for the plant parameters yield the equilibrium solution: ~ x(t;) = ~ x 0 () = ~ v 0 + + ~ v 0 + = 4:9180 + 55:082, 0 1, ~ w(t) = ~ w 0 = ~ v 0 + = 55:0820, ~ v(t) = ~ v 0 = 60:0, and ~ u(t) = ~ u 0 = ~ v 0 b( +) + K~ v 0 b(M+~ v 0 ) +~ v 0 = 199:7811. The parameter values in the linearized model, (3.3), (3.4), are q 1 = 0:48, q 2 = 0:1129, q 3 = 0:56, q 4 = 0:05, q 5 = 0:26, and q 6 = 0:053. We assumed that at time zero, the target clamped BAC of 60 mg/dl is perturbed by v(0) =10:0 mg/dl, so ~ v(0) = 0 = 50:0 mg/dl or 0.05 percent alcohol. The other initial conditions are ~ x(0;) = ' 0 () = ~ x 0 () = 4:9180 + 55:082, 0 1, and ~ w(0) = 0 = ~ w 0 = 55:0820, x(0;) = 0, 0 1, and w(0) = 0. In the nite time horizon regulator performance index (5.2), we set the penalty parameters to be = 100:0, ^ r = 0:001, and = 100:0. We set the sampling time at = 0:05 hours and ran the simulation for T = 6:0 hours (i.e. k 0 = 0 and k 1 = 121). In all of our simulations, we simulated the plant using a linear spline based Galerkin approximation for deterministic abstract parabolic system in the spirit of the one described in Section 4.3 with N = 64 and with n = 64 to simulate the fact that the plant is innite-dimensional. 64 Figure 7.4: First simulation example: BAC curves when 1) compensator designed under un- certainty, 2) simulated innite-dimensional compensator but designed with incorrect value for , 3) no control is applied (i.e. open loop), and 4) simulated innite-dimensional com- pensator designed with correct or true value of . In our rst simulation example, we assumed that only was not known exactly, but rather that =q 1 Beta(3,2) and used the approach, we developed here to design a nite- dimensional approximating compensator. In doing this, we set n = 8, and m 1 = 4. In this way, our regulator and observer were each (8 + 1) 4 = 36 dimensional. In our second simulation, we assumed that both and b were not known exactly but rather that = q 1 Beta(3,2), b = q 2 Beta(2,10), and that they are independent. We then used the approach we developed here to design a nite-dimensional approximating compensator. In doing this, we set n = 8, and m i = 4, i = 1; 2. In this way, our regulator and observer were each (8 + 1) 4 4 = 144 dimensional. 65 Figure 7.5: First simulation example: Control (intravenous infusion rate) trajectory curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compen- sator but designed with incorrect value for, and 3) simulated innite-dimensional compen- sator designed with correct or true value of . 4) control is steady. In Fig. 7.4 (Fig. 7.7), we plot 1) the simulated closed loop BAC curve that results when our compensator is designed under uncertainty in = q 1 (and b = q 2 ). For comparison we have also plotted on the same set of axes the simulated closed loop BAC trajectories that result when 2) an innite-dimensional (n = 64) compensator is designed but the incorrect value(s) for the parameter(s) (and b), q A 1 = 0:7 (and q A 2 = 0:05) is (are) used, 3) no control is used, i.e the open loop system, and 4) the optimal innite-dimensional (n = 64) compensator, the one designed using the actual plant or true value(s) of (andb),q 1 = 0:48 (and q 2 = 0:1129). In Fig. 7.5 (Fig. 7.8) and Fig. 7.6 (Fig. 7.9), we plot the corresponding optimal control trajectories, ~ u = ~ u 0 +u and the corresponding optimal control trajectory increments, u , 66 Figure 7.6: First simulation example: Control (intravenous infusion rate) increment to steady state control trajectory curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compensator but designed with incorrect value for , and 3) simulated innite-dimensional compensator designed with correct or true value of . respectively. Note that both of these gures are plotted on a logarithmic scale to accom- modate the signicant dierences in magnitude between the various curves depicted in the plots. 7.4 Tracking BAC in Intravenous Alcohol Studies In this section, we consider the LQG tracking problem involving the simulation of an intravenously-infused alcohol study where the objective is to have the subject's BAC track a pre-specied target BAC trajectory. For this simulation, we designed two dierent tracking compensators. The rst one involves a linear feedback law and a linear observer based on the linearized model (3.3)-(3.4) for the plant (3.1)-(3.2), while the second one is based on a 67 Figure 7.7: Second simulation example: BAC curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compensator but designed with incorrect val- ues for andb, 3) no control is applied (i.e. open loop), and 4) simulated innite-dimensional compensator designed with correct or true values of and b. design involving a nonlinear feedback tracking controller and a nonlinear observer. Similar to the examples from control problems in the previous sections, we assume that a subset of the model parameters are only known up to their distribution in a cohort of the population, and in all the following examples, plant and compensator parameter values were set based on our earlier eorts involving this data and models of this general form (see, for example, [6, 35, 37, 39]). 7.4.1 A Linearization-Based Approach In this example, the tracking controller is designed based on having the v-component of the linearized model, (3.3), (3.4), track k ~ v 0 , wheref k g is the target BAC signal (so in this 68 Figure 7.8: Second simulation example: Control (intravenous infusion rate) trajectory curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compen- sator but designed with incorrect values for and b, and 3) simulated innite-dimensional compensator designed with correct or true values of and b. 4) control is steady. caseD2L(H q ;R) isD(;;') = p ). Iffu k g denotes the resulting tracking compensator for the linearized system, thenf~ u = ~ u 0 +u k g is the corresponding tracking controller for the nonlinear plant. For the plant, we set = 0:48, = 0:56, = 0:05, = 0:26, = 0, b = 0:1129, ^ r = 0:015, ^ = 100, = 100, 1 = 2:75, 2 = 3:75, and = 2:75. The Michaelis- Menten parameters were taken to beK = 25:0,M = 10:5 (see [30]). The sampling time was taken to be = :01, and we simulated the experiment for almost 15 hours. We linearized about a nominal BAC of ~ v 0 = 60 mg/dl. This results in the equilibrium solution ~ w 0 = 55:08, ~ u 0 = 199:78, ~ x 0 () = ~ a 0 + ~ b 0 with ~ a 0 = 4:92 and ~ b 0 = 55:08. These values result in the following parameters for the linearized system: q 1 = 0:48, q 2 = 0:113, q 3 = 0:56, q 4 = 0:05, 69 Figure 7.9: Second simulation example: Control (intravenous infusion rate) increment to steady state control trajectory curves when 1) compensator designed under uncertainty, 2) simulated innite-dimensional compensator but designed with incorrect values for and b, and 3) simulated innite-dimensional compensator designed with correct or true values of and b. q 5 = 0:26 andq 6 = 0:053. The plant was simulated using the same Galerkin scheme we used to design our compensator but with n = 64 (to simulate innite dimensionality). We designed a nite time horizon (controller and observer) tracking compensator using our scheme. We assumed that the values for q 1 and q 2 in the linearized system ( and b in the plant) were not known but that their distributions were. We assumed thatq = (q 1 ;q 2 ) with q 1 Beta(5; 2) and q 2 Beta(1:8; 14:0) and that they were independent. For our nite-dimensional approximation, we set N = 8 and m 1 = m 2 = 4 (larger values for N, m 1 and m 2 had little if any eect on performance). In Fig. 7.10, we have plotted the results of our simulation. We have plotted both the controlled BAC and the control (the intravenous infusion rate of a 6% by volume solution of ethanol). For comparison, along 70 Figure 7.10: Tracked BAC and tracking control by the linear approach with the compensator designed using our scheme, we have plotted a compensator designed with n = 64 using the true plant parameters q 1 = 0:48 and q 2 = 0:1129, and a compensator designed using incorrect values for the plant parameters q 1 = 0:714 and q 2 = 0:143. On the same set of axes we have also plotted the target BAC signal,f k g. 7.4.2 A Nonlinear Observer and Feedback Controller Since the nonlinearity in the plant (3.1)-(3.2) is of the form ^ B K~ v b(M+~ v) , if the value of b, K, and M are known, a nonlinear feedback tracking compensator can be designed that eectively cancels the nonlinearity. Indeed, we subtracted this nonlinear feedback from a linear compensator that forces the linear part of the plant (i.e withK set to zero in (3.1),(3.2) 71 Figure 7.11: Tracked BAC and tracking control by the nonlinear approach or (3.3),(3.4)) to track the given reference signalf k g. We assumedb,K,M andq 2 are known and designed a compensator of the form ~ u k = ^ F k ~ x k + k + K ^ D ~ x k q 2 ( p M + ^ D ~ x k ) ; wheref ^ F k g andf k g are designed to have the system (3.3),(3.4) with q 6 = trackf k g. We paired this controller with the nonlinear observer given by ~ x k+1 = ^ A ~ x k + ^ Bu k + ~ L k (y k ^ C ~ x k ) ^ B K ^ D ~ x k q 2 ( p M + ^ D ~ x k ) : All parameter values were the same as in Section 7.4.1 except we set q 6 = , ^ r = 0:08, and 1 = 2 = = 12:0, assumed the value of b = q 2 = 0:1129 is known, and once again we 72 assumed thatq 1 Beta(5; 2). We obtained the tracking and the controller plotted in Fig 7.11. 7.5 Deconvolving BAC or BrAC from TAC In this section, we illustrate how our LQG tracking scheme can be used to deconvolve an estimate of BAC from the biosensor observed TAC. We also show how to introduce regularization into the process to mitigate the inherent ill-posedness of the inverse problem being solved. As what we have introduced in Chapter 3, this is a linear problem since the compartment at = 1 is the dermal layer of the skin and the input, ~ u, is now BAC. Therefore we do not include the metabolic processes in the liver in the model, i.e., there is no Michaelis-Menten nonlinear term in the plant (3.1)-(3.2). The model output is once again TAC. Moreover, note that in this case, the linearized model (3.3)-(3.4) with q 6 = is in fact the plant (3.1)-(3.2) with K = 0 since the objective is to determine an input (i.e. BAC) signal that causes the model to track a given TAC signal that causes the model to track a given TAC signal (so, in this case, D2L(H q ;R) is D(;;') = p and k is the given TAC signal to be tracked). In addition, since this is in general an ill-posed inverse problem, it is desirable to include regularization in the form of a penalty on both the magnitude and rate of change of BAC. To do this, we rewrite the abstract parabolic system (4.9) as x k+1 = ^ Ax k + ^ Bu k + 04u k + ^ B 1 !(k); u k+1 =0x k +u k +4u k y k = ^ Cx k +(k); z k = ^ Dx k ; 73 by introducing4u k =u k+1 u k . Take the input to the system to now be4u k whereu k is BAC. Set x k = [x k ;u k ] T , therefore, the rewritten system is x k+1 = ~ Ax k + ~ B4u k + ~ B 1 !(k); y k = ~ Cx k +(k); z k = ~ Dx k ; where ~ A = 0 B @ ^ A; ^ B 0; 1 1 C A ; ~ B = 0 B @ 0 1 1 C A ; ~ B 1 = 0 B @ 1 0 1 C A ; ~ C = ^ C; 0 ; ~ D = ^ D; 0 : In this way,u k becomes a state variable, the state now being given by the vector x k , and we adjoin the additional state equation4u k = u k , with initial conditionu 0 = 0, u k now being the control or input. So the quadratic performance index on for example, the nite time horizon, is given by ^ J (4u k ) = k 1 1 X k=k 0 j ~ Dx k k j 2 + ^ r4u 2 k + ^ j ~ Dx k 1 k 1 j 2 : It is now possible to penalize both u k (by making it a component of the output) and 4u k =u k in the quadratic cost functional,u k as part of the state and4u k as the control. Once the optimal u k is determined, the estimated BAC, u k is obtained from u k = P k1 j=0 u j . Since the model is the plant, full state feedback is possible, but instead we use the optimally data-corrected estimated state from the TAC residual-driven Kalman observer. We set the sampling rate at = 1=60, we letq 1 Beta(3; 2),q 2 Beta(2; 12),q 3 = = 3:986, q 4 = = 0:050, q 5 = = 0:086, q 6 = = 0:850, ^ = 100, = 500, the penalty weight onu k (now part of the ^ D operator) to be 50 and on4u k =u k to be 10 4 , 1 = 2 = = 2:0, n = 8, and m 1 = m 2 = 4. For data, we used TAC collected by one of the co-authors (S. E. L.) using a WrisTAS 7 TM transdermal alcohol biosensor manufactured by Giner, Inc. of 74 Figure 7.12: Tracked TAC and Estimated BrAC Waltham, MA. Simultaneous BrAC data was also collected, so we could evaluate the ecacy of our approach. The target TAC signal to be tracked along with the resulting simulated TAC and the estimated and true BrAC are plotted in Fig. 7.12. 75 Chapter 8 Discussion, Concluding Remarks, and Future Research Beyond the rigorous mathematical theory upon which they are based, our numerical studies (see Section 7.1) have demonstrated near optimal performance and convergence of nite-dimensional approximations of LQG compensators designed using our general approach for a plant of the form (4.9). We also demonstrated that our nite-dimensional LQG com- pensators perform well in both the case where the plant system parameters are xed but unknown (with known distribution) (Table 7.4), and where they take on dierent values in each sampling interval (Table 7.5). We do note, however, that the important question re- mains open as to whether the closed loop system consisting of the innite-dimensional plant with either unknown or random parameters together with our nite-dimensional compen- sator could keep stable. In our simulations of the intravenously-infused alcohol clamping study based on the ob- jective which is to have the subject either reach a specied level of intoxication and to then remain at the level for an extended period of time, our approximating LQG compensator performed nearly as well as the optimal innite-dimensional compensator designed using the actual plant parameters (See Section 7.3). Based on the rst example in Section 7.3, we observed that the performance of the compensator is not that sensitive to imprecise knowl- edge of the diusivity of the epidermal layer (see Fig. 7.5), but in the second example, we saw that imprecise knowledge of both the diusivity of the epidermal layer and the actua- tor gain seriously degrades performance. In this second case, our population model-based 76 nite-dimensional compensator design, which uses only knowledge of the joint distribution of these parameters, performed close to the optimal (i.e. close to the performance of the innite-dimensional compensator designed using the plant values of these parameters; see Fig. 7.8). In our simulations of the intravenously-infused alcohol tracking study based on the ob- jective which is to have the subject's BAC track a pre-specied target BAC trajectory, the simulated BAC obtained using our approach tracks reasonably well on the target BAC, and the infused alcohol ow rate (control) from our approach nearly overlaps the control com- puted when the plant parameters are known exactly (see Fig. 7.10). Fig. 7.11 demonstrates that applying our approach in which we deal with the system with random parameters in the Bochner spaces to even a nonlinear controller and observer yields a BAC signal that closely tracks the target BAC. By Fig. 7.12, we observe that our tracking scheme also yields an ecient and stable algorithm that can be used to estimate (or deconvolve) BAC or BrAC from a TAC signal in real time since the estimated BrAC and simulated TAC appear to closely track the actual BrAC and target TAC, respectively. Finally, in our computations, we were able to use the steady state Riccati operators,K and ~ S, and control and observer gains, ^ F and ~ L, which can all be pre-computed o-line. This is despite the fact that the TAC reference signal to be tracked is time dependent. Then estimating BAC or BrAC from TAC can conceivably be carried out in real time directly on a mobile device (e.g. an i-Phone, Fitbit or Apple Watch) since now only relatively simple calculations are required at each time to obtain ~ u k : the backward integration of k and the forward integration of ~ x k as the solutions of non- homogeneous linear recurrence relations with forcing terms depending on the observed TAC signal being tracked. This is a signicant advancement towards providing reliable estimated BrAC based on TAC in real time or near real time. As for future research, the next step would be to attempt to use our control schemes in an IRB approved human subjects trial of a clamping and/or tracking experiment. 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Abstract (if available)
Abstract
This research is motivated by the following two applications involving biosensor-measured transdermal alcohol concentration (TAC). A TAC biosensor measures the ethanol content in perspiration. The first application is the control of intravenously-infused alcohol studies based on a population model for the study participant or subject and TAC sensing, while the second application is estimating blood or breath alcohol concentration (respectively, BAC or BrAC) from TAC. A dynamical model for the underlying control system is established. It takes the form of a semi-linear, parabolic PDE/ODE hybrid system describing the transport of ethanol from the blood through the skin, its excretion within perspiration, and finally its measurement on the surface of the skin by an electro-chemical biosensor. Since the parameters of this dynamical model can vary with the individual wearing the sensor, the particular sensor being worn, and environmental factors such as ambient temperature and humidity, we allow the model parameters to be random with either known or estimated distribution. A state space formulation of the model set in an appropriately constructed Gelfand triple of Bochner spaces is derived wherein the random parameters are treated as additional spatial variables. The resulting population model takes the form of an abstract parabolic hybrid system involving coupled partial and ordinary differential equations with random parameters. For intravenously-infused alcohol studies, there are two aims. One is clamping: to have the subject’s BAC either reach a specified level of intoxication and then remain at that level for an extended period of time. To solve this problem, a discrete time linear quadratic (LQ) regulator is coupled with a linear quadratic Gaussian (LQG) compensator for the linearized random abstract parabolic systems obtained from the semi-linear population model. A finite-dimensional Galerkin-based approximation and convergence theory for the closed-loop linear quadratic control and estimation of abstract parabolic systems with random parameters is developed. The efficacy of our approach and the performance of our control design are demonstrated with numerical studies. A realistic simulation of an intravenously-infused alcohol clamping study with transdermal sensing and our compensator is included. The second aim of intravenously-infused alcohol studies is tracking: to force the blood alcohol concentration follow a pre-specified target trajectory of intoxication level by controlled infusion. This problem as well as the problem of deconvolving BAC from TAC is reformulated as a discrete time LQG tracking problem based on the same population model used in the clamping studies. Similarly, the convergence of finite-dimensional approximations is established, simulation studies for both a linear and nonlinear compensator are presented, and numerical results for the regularized deconvolution of BAC or BrAC from TAC are presented and discussed.
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Creator
Yao, Mengsha
(author)
Core Title
Linear quadratic control, estimation, and tracking for random abstract parabolic systems with application to transdermal alcohol biosensing
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Applied Mathematics
Degree Conferral Date
2021-08
Publication Date
07/13/2021
Defense Date
06/17/2021
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Los Angeles
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University of Southern California
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abstract parabolic system,discrete-time,finite-dimensional approximation and convergence,intravenously-infused alcohol,linear quadratic compensator,linear quadratic control,OAI-PMH Harvest,semigroups of operators,transdermal alcohol biosensor
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Rosen, Gary (
committee chair
), Luczak, Susan (
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), Wang, Chunming (
committee member
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mengshay@icloud.com,mengshay@usc.edu
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https://doi.org/10.25549/usctheses-oUC15491151
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Tags
abstract parabolic system
discrete-time
finite-dimensional approximation and convergence
intravenously-infused alcohol
linear quadratic compensator
linear quadratic control
semigroups of operators
transdermal alcohol biosensor