For linear time-invariant descriptor models, this dissertation studies the extended strictly positive real (ESPR) design of continuous-time systems and the strictly positive real (SPR) analysis and design of discrete-time systems, respectively, all in the LMI framework. For a continuous-time system, by the LMI-based ESPR Lemma, a controller is designed such that the closed-loop system has its transfer matrix being ESPR while admissibility of the compensated descriptor system is guaranteed. Three forms of synthesis are considered, i.e. the static state feedback synthesis, estimated state feedback synthesis, and the dynamic output feedback synthesis. Moreover, design criterion of a dynamic output feedback controller in the state-space model is also addressed. For a discrete-time system, an LMI-based SPR characterization is developed. After giving the definition of SPR, the Cayley transformation is used to establish formulas bridging the admissible realizations for SPR and strictly bounded real (SBR) transfer matrices. Based on them, an LMI-based necessary and sufficient condition for a descriptor system to be, simultaneously, admissible and SPR is derived. When the descriptor variables are transformed into the SVD coordinate, it is shown that such a condition will have solution in the block diagonal form. Based on this result, the problem of static state feedback design to make transfer matrix of the closed-loop systems SPR is tackled. The problems of robust ESPR and SPR analysis and design when the considered systems have norm-bounded unstructured uncertainty are also addressed. Similarly, LMI-based conditions to guarantee robust admissibility with transfer matrices being ESPR for continuous systems or being SPR for discrete systems are proposed. Based on them, for continuous systems, a static state feedback controller and a dynamic output feedback controller are designed to make the entire family of uncertain closed-loop systems robustly admissible with transfer matrices being ESPR. While for discrete systems, only static state feedback controller is designed to achieve the robust admissibility and robust SPR property. Finally, based on ESPR lemma (or SPR lemma), we propose a new LMI-based robust admissibility analysis for a class of LTI continuous-time (or discrete-time) descriptor systems with convex polytopic uncertainties appearing on all the system matrices. Moreover, the development of state feedback controllers stemmed from these analysis results is also investigated. It is shown that the provided method has the capability to tackle the problem of computing a required feedback gain matrix for systems with either constant or polytopically dependent derivative (or advanced) state matrix in a unified way. Besides, the application of SPR property to absolute stability problem involving an LTI discrete-time descriptor system and a memoryless time-varying nonlinearity is also addressed. Since all conditions are expressed in LMIs, the obtained results are numerically tractable. It is illustrated by several numerical examples.

Based on the Lyapunov stability theorem, two different variable structure controllers are proposed in this thesis for two different classes of multi-variable descriptor systems subject to matched nonlinear perturbations. The integral variable structure controller is proposed first for solving the stabilization problems, and model reference variable structure controller is the second for solving the state tracking problems. Both proposed control schemes can guarantee the trajectories of the controlled systems to lie in the sliding surface from initial time, so that the properties of regularity, impulse free, and stability can be obtained. Two numerical examples are given for demonstrating the feasibility of the proposed control schemes.

In this thesis, an LMI-based pole-clustering characterization for descriptor systems is investigated. A necessary and sufficient condition for checking simultaneously the regularity, impulse immunity, and finite eigenvalues locating in the generalized LMI regions is derived. Since uncertainty exists inevitably in control systems, we propose two sufficient conditions to guarantee the robust pole clustering in the generalized LMI regions for uncertain descriptor systems with two types of uncertainties, i.e. the norm bounded uncertainty and the convex polytopic uncertainty. The LMI-based state feedback controller design methods are developed as well. Finally, the validity and the feasibility of our theoretical results are verified by the numerical simulation results of several examples.

The paper investigates problems of designing controllers to linear time-invariant continuous descriptor systems subject to norm-bounded structured uncertainty so that the closed-loop systems are admissible or D-admissible with their transfer matrices having H-infinite norm bounded by a prescribed value. The constant state feedback and the dynamic output feedback designs are addressed. In both design methods, sufficient LMI conditions are derived to guarantee achievement of the desired specifications, such as robust H-infinite norm and pole-clustering constraints. Finally, two numerical examples are shown for the illustration.

We study linear descriptor systems with rectangular variable coefficient matrices. Using local and global equivalence transformations we introduce normal and condensed forms and get sets of characteristic quantities. These quantities allow us to decide whether a linear descriptor system with variable coefficients is regularizable by derivative and/or proportional state feedback or not. Regularizable by feedback means for us that their exist a feedback which makes the closed loop system uniquely solvable for every consistent initial vector.

By the name HALFSPACE SYSTEMS, this dissertation refers to systems whose dynamics are modeled by linear constraints of the form Exk+1 <= Fxk + Buk (where E, F 2 andlt;mn, B 2 andlt;mp). This dissertation explores the concepts of BOUNDEDNESS, STABILITY, IRREDUNDANCY, and MAINTAINABILITY (which is the same as REACHABILITY OF A TARGET TUBE) that are related to the control of halfspace systems. Given that a halfspace system is bounded, and that a given static target tube is reachable for this system, this dissertation presents algorithms to MAINTAIN the system in this target tube. A DIFFERENCE INCLUSION has the form xk+1 = Axk + Buuk, where xk, xk+1 2 andlt;n, uk 2 andlt;p, A 2 andlt;nn, Bu 2 andlt;np, Ai 2 andlt;nn, Bj 2 andlt;np, and A and Bu belong to the convex hulls of (A1,A2, . . . ,Aq) and (B1, B2, . . . , Br) respectively. This dissertation investigates the possibility that halfspace systems have equivalent difference inclusion representation for the case of uk = 0. An affirmitive result in this direction may make it possible to apply to halfspace systems the control theory that exists for difference inclusions.

Dans cette thèse, la conception d’observateurs pour les systèmes singuliers linéaires incertains et leurs applications au contrôle et au diagnostic. En effet, nous avons développé des méthodes de reconstruction d’état et d’estimation de défauts est étudié. Les systèmes algèbro-différentiels ou systèmes singuliers peuvent être considérés comme une généralisation des systèmes dynamiques. Ils constituent un puissant outil de modélisation dans la mesure où ils peuvent décrire des processus régis à la fois par des équations différentielles (dynamiques) et des équations algébriques (statiques). La nouvelle structure d’observateurs utilisée dans cette thèse est nommée l’Observateur Dynamique Généralisé (ODG), elle est plus générale que celle d’Observateurs Proportionnels (OP) et d’Observateurs Proportionnels Intégrals (OPI). Cette structure présente une estimation d’état alternative qui peut être considérée comme plus générale que les OP et les OPI, ceux-ci pouvant être considérés comme des cas particuliers de cette structure. L’approche proposée repose sur la paramétrisation des solutions des équations de Sylvester pour éliminer le biais entre l’erreur d’observation et la paire (entrée état). La thèse est organisée comme suit : Dans l’introduction générale, nous présentons la problématique et les objectifs de la thèse ainsi que les principales contributions. Dans le premier chapitre, nous présentons la classe des systèmes singuliers considérée. Nous faisons des rappels sur l’analyse de stabilité et l’utilisation des outils numériques LMI avec lesquels nous vérifions l’existence de conditions de stabilité. Ensuite, nous présentons les méthodes de reconstruction d’état des systèmes singuliers linéaires à savoir l’ODG, l’OP et l’OPI. Dans le deuxième chapitre, nous présentons en détail la procédure de synthèse d’ODG pour les systèmes singuliers continus avec et sans perturbations. Ensuite, nous faisons une extension aux systèmes singuliers en temps discret avec et sans perturbations. Dans le chapitre 3, nous donnons les conditions d’existence et de stabilité robuste de l’ODG pour les systèmes singuliers à paramètres incertains, où l’incertitude est bornée. Dans le chapitre 4, nous présentons une méthode de synthèse de commande stabilisante par retour d’état basée observateur pour une classe de systèmes singuliers linéaires avec et sans perturbations. Le chapitre 5, est consacré au diagnostic. L’étude que nous avons menée est traitée en deux étapes : La première étape est consacrée à la détection et l’isolation des défauts en utilisant un ODG. Cet observateur génère des résidus qui sont en mesure de représenter seulement la présence d’un défaut, de sorte que nous pouvons localiser des défauts multiples. Enfin, la deuxième étape est consacrée à l’estimation des défauts en utilisant un ODG avec une structure modifiée. Ces approches sont développées pour les systèmes singuliers et pour les systèmes singuliers incertains avec ou sans perturbations. Nous terminerons ce mémoire de thèse par une conclusion générale et quelques perspectives. / In this thesis the observer design for uncertain linear descriptor systems and their applications to control and fault diagnosis is studied. Descriptor systems can be considered as a generalization of dynamical systems. This class of systems include algebraic and differential equations. The observer used in this work has a new structure more general than those presented in the literature. The observer structure proposed has additional degrees of freedom, which provides it robustness in face to variations not considered in the model. The new observer structure used in this thesis, named as generalized dynamic observer (GDO), is designed for different classes of descriptor systems. The asymptotic stability of the observer is proved by Lyapunov analysis through a set of linear matrix inequalities (LMIs). In all cases, the LMI obtained from the Lyapunov analysis is treated by the elimination lemma. The use of the elimination lemma is essential in the development of the stability analysis of the observers, since it allows to obtain the GDO structure. Proportional observers (PO) and proportional-integral observers (PIO) can be considered as particular cases of our observer. The thesis is organized as follows: In the general introduction, the problem formulation is presented, the objectives of the thesis are pointed out, the scope of the investigation and the main contributions are also presented. Chapter 1 introduces descriptor systems as the class of systems considered in this work and presents a review of the state of the art focused on the observers design for these systems. Also we introduce the GDO as an observer with structure more general than that of the PO and the PIO. Chapter 2 develops the GDO for descriptor systems with or without disturbances. Extension of these approaches for discrete-time descriptor systems with or without disturbances are also presented. In Chapter 3, the robust approach of the GDO is treated for parametric uncertain descriptor systems, where the uncertainty is bounded, and for linear parameter varying (LPV) descriptor systems, where the parameters vary inside a polytope. Chapter 4 presents the GDO application to observer-based control with the objective to stabilize descriptor systems that normally are unstable. An extension of this approach to disturbed descriptor systems is also developed. Chapter 5 presents the GDO application to fault diagnosis, which is divided in two parts. The first one is to detect and isolate faults by using a GDO that provides residuals that are able to represent only the presence of one fault, so that we can isolate multiple faults. And the second part is to estimate the faults by using a GDO with a modified structure. These approaches are developed for descriptor systems and for uncertain descriptor systems. The last part is dedicated to general conclusions and some perspectives.

<p>I denna avhandling studeras optimal återkopplad styrning av olinjära deskriptorsystem. Ett deskriptorsystem är en matematisk beskrivning som kan innehålla både differentialekvationer och algebraiska ekvationer. En av anledningarna till intresset för denna klass av system är att objekt-orienterade modelleringsverktyg ger systembeskrivningar på denna form. Här kommer det att antas att det, åtminstone lokalt, är möjligt att eliminera de algebraiska ekvationerna och få ett system på tillståndsform. Teoretiskt är detta inte så inskränkande för genom att använda någon indexreduktionsmetod kan ganska generella deskriptor\-system skrivas om så att de uppfyller detta antagande.</p><p>För system på tillståndsform kan Hamilton-Jacobi-Bellman-ekvationen användas för att bestämma den optimala återkopplingen. Ett liknande resultat finns för deskriptor\-system där istället en Hamilton-Jacobi-Bellman-liknande ekvation ska lösas. Denna ekvation innehåller dock en extra term för att hantera de algebraiska ekvationerna. Eftersom antagandena i denna avhandling gör det möjligt att skriva om deskriptorsystemet som ett tillståndssystem, undersöks hur denna extra term måste väljas för att båda ekvationerna ska få samma lösning.</p><p>Ett problem med att beräkna den optimala återkopplingen med hjälp av Hamilton-Jacobi-Bellman-ekvationen är att det leder till att en olinjär partiell differentialekvation ska lösas. Generellt har denna ekvation ingen explicit lösning. Ett lättare problem är att beräkna en lokal optimal återkoppling. För analytiska system på tillståndsform löstes detta problem på 1960-talet och den optimala lösningen beskrivs av serieutvecklingar. I denna avhandling generaliseras detta resultat så att även deskriptor-system kan hanteras. Metoden illustreras med ett exempel som beskriver en faslåsande krets.</p><p>I många situationer vill man veta om ett område är möjligt att nå genom att styra på något sätt. För linjära tidsinvarianta system fås denna information från styrbarhetgramianen. För olinjära system används istället styrbarhetsfunktionen. Tre olika metoder för att beräkna styrbarhetsfunktionen har härletts i denna avhandling. De framtagna metoderna är också applicerade på några exempel för att visa beräkningsstegen.</p><p>Dessutom har observerbarhetsfunktionen studerats. Observerbarhetsfunktionen visar hur mycket utsignalenergi ett visst initial tillstånd svarar mot. Ett par olika metoder för att beräkna observerbarhetsfunktionen för deskriptorsystem tagits fram. För att beskriva en av metoderna, studeras ett litet exempel bestående av en elektrisk krets.</p> / <p>In this thesis, optimal feedback control for nonlinear descriptor systems is studied. A descriptor system is a mathematical description that can include both differential and algebraic equations. One of the reasons for the interest in this class of systems is that several modern object-oriented modeling tools yield system descriptions in this form. Here, it is assumed that it is possible to rewrite the descriptor system as a state-space system, at least locally. In theory, this assumption is not very restrictive because index reduction techniques can be used to rewrite rather general descriptor systems to satisfy this assumption.</p><p>The Hamilton-Jacobi-Bellman equation can be used to calculate the optimal feedback control for systems in state-space form. For descriptor systems, a similar result exists where a Hamilton-Jacobi-Bellman-like equation is solved. This equation includes an extra term in order to incorporate the algebraic equations. Since the assumptions made here make it possible to rewrite the descriptor system in state-space form, it is investigated how the extra term must be chosen in order to obtain the same solution from the different equations.</p><p>A problem when computing the optimal feedback law using the Hamilton-Jacobi-Bellman equation is that it involves solving a nonlinear partial differential equation. Often, this equation cannot be solved explicitly. An easier problem is to compute a locally optimal feedback law. This problem was solved in the 1960's for analytical systems in state-space form and the optimal solution is described using power series. In this thesis, this result is extended to also incorporate descriptor systems and it is applied to a phase-locked loop circuit.</p><p>In many situations, it is interesting to know if a certain region is reachable using some control signal. For linear time-invariant state-space systems, this information is given by the controllability gramian. For nonlinear state-space systems, the controllabilty function is used instead. Three methods for calculating the controllability function for descriptor systems are derived in this thesis. These methods are also applied to some examples in order to illustrate the computational steps.</p><p>Furthermore, the observability function is studied. This function reflects the amount of output energy a certain initial state corresponds to. Two methods for calculating the observability function for descriptor systems are derived. To describe one of the methods, a small example consisting of an electrical circuit is studied.</p> / Report code: LiU-TEK-LIC-2006:8

I denna avhandling studeras optimal återkopplad styrning av olinjära deskriptorsystem. Ett deskriptorsystem är en matematisk beskrivning som kan innehålla både differentialekvationer och algebraiska ekvationer. En av anledningarna till intresset för denna klass av system är att objekt-orienterade modelleringsverktyg ger systembeskrivningar på denna form. Här kommer det att antas att det, åtminstone lokalt, är möjligt att eliminera de algebraiska ekvationerna och få ett system på tillståndsform. Teoretiskt är detta inte så inskränkande för genom att använda någon indexreduktionsmetod kan ganska generella deskriptor\-system skrivas om så att de uppfyller detta antagande. För system på tillståndsform kan Hamilton-Jacobi-Bellman-ekvationen användas för att bestämma den optimala återkopplingen. Ett liknande resultat finns för deskriptor\-system där istället en Hamilton-Jacobi-Bellman-liknande ekvation ska lösas. Denna ekvation innehåller dock en extra term för att hantera de algebraiska ekvationerna. Eftersom antagandena i denna avhandling gör det möjligt att skriva om deskriptorsystemet som ett tillståndssystem, undersöks hur denna extra term måste väljas för att båda ekvationerna ska få samma lösning. Ett problem med att beräkna den optimala återkopplingen med hjälp av Hamilton-Jacobi-Bellman-ekvationen är att det leder till att en olinjär partiell differentialekvation ska lösas. Generellt har denna ekvation ingen explicit lösning. Ett lättare problem är att beräkna en lokal optimal återkoppling. För analytiska system på tillståndsform löstes detta problem på 1960-talet och den optimala lösningen beskrivs av serieutvecklingar. I denna avhandling generaliseras detta resultat så att även deskriptor-system kan hanteras. Metoden illustreras med ett exempel som beskriver en faslåsande krets. I många situationer vill man veta om ett område är möjligt att nå genom att styra på något sätt. För linjära tidsinvarianta system fås denna information från styrbarhetgramianen. För olinjära system används istället styrbarhetsfunktionen. Tre olika metoder för att beräkna styrbarhetsfunktionen har härletts i denna avhandling. De framtagna metoderna är också applicerade på några exempel för att visa beräkningsstegen. Dessutom har observerbarhetsfunktionen studerats. Observerbarhetsfunktionen visar hur mycket utsignalenergi ett visst initial tillstånd svarar mot. Ett par olika metoder för att beräkna observerbarhetsfunktionen för deskriptorsystem tagits fram. För att beskriva en av metoderna, studeras ett litet exempel bestående av en elektrisk krets. / In this thesis, optimal feedback control for nonlinear descriptor systems is studied. A descriptor system is a mathematical description that can include both differential and algebraic equations. One of the reasons for the interest in this class of systems is that several modern object-oriented modeling tools yield system descriptions in this form. Here, it is assumed that it is possible to rewrite the descriptor system as a state-space system, at least locally. In theory, this assumption is not very restrictive because index reduction techniques can be used to rewrite rather general descriptor systems to satisfy this assumption. The Hamilton-Jacobi-Bellman equation can be used to calculate the optimal feedback control for systems in state-space form. For descriptor systems, a similar result exists where a Hamilton-Jacobi-Bellman-like equation is solved. This equation includes an extra term in order to incorporate the algebraic equations. Since the assumptions made here make it possible to rewrite the descriptor system in state-space form, it is investigated how the extra term must be chosen in order to obtain the same solution from the different equations. A problem when computing the optimal feedback law using the Hamilton-Jacobi-Bellman equation is that it involves solving a nonlinear partial differential equation. Often, this equation cannot be solved explicitly. An easier problem is to compute a locally optimal feedback law. This problem was solved in the 1960's for analytical systems in state-space form and the optimal solution is described using power series. In this thesis, this result is extended to also incorporate descriptor systems and it is applied to a phase-locked loop circuit. In many situations, it is interesting to know if a certain region is reachable using some control signal. For linear time-invariant state-space systems, this information is given by the controllability gramian. For nonlinear state-space systems, the controllabilty function is used instead. Three methods for calculating the controllability function for descriptor systems are derived in this thesis. These methods are also applied to some examples in order to illustrate the computational steps. Furthermore, the observability function is studied. This function reflects the amount of output energy a certain initial state corresponds to. Two methods for calculating the observability function for descriptor systems are derived. To describe one of the methods, a small example consisting of an electrical circuit is studied. / Report code: LiU-TEK-LIC-2006:8

Differential-algebraic equations (DAEs) form the natural way in which models of physical systems are delivered from an object-oriented modeling tool like Modelica. Differential-algebraic equations are also known as descriptor systems, singular systems, and implicit systems. If some constant parameters in such models are unknown, one might need to estimate them from measured data from the modeled system. This is a form of system identification called gray box identification. It may also be of interest to estimate the value of time-varying variables in the model. This is often referred to as state estimation. The objective of this work is to examine how gray box identification and estimation of time-varying variables can be performed for models described by differential-algebraic equations. If a model has external stimuli that are not measured or uncertain measurements, it is often appropriate to model this as stochastic processes. This is called noise modeling. Noise modeling is an important part of system identification and state estimation, so we examine how well-posedness of noise models for differential-algebraic equations can be characterized. For well-posed models, we then discuss how particle filters can be implemented for estimation of time-varying variables. We also discuss how constant parameters can be estimated. When estimating time-varying variables, it is of interest to examine if the problem is observable, that is, if it has a unique solution. The corresponding property when estimating constant parameters is identifiability. In this thesis, we discuss how observability and identifiability can be determined for DAEs. We propose three approaches, where one can be seen as an extension of standard methods for state-space systems based on rank tests. For linear DAEs, a more detailed analysis is performed. We use some well-known canonical forms to examine well-posedness of noise models and to implement estimation of time-varying variables and constant parameters. This includes formulation of Kalman filters for linear DAE models. To be able to implement the suggested methods, we show how the canonical forms can be computed using numerical software from the linear algebra package LAPACK.