Convex Duality Approach to Robust Stabilization of Uncertain Plants

In this thesis we are study the problem of designing the controllers that are robust with respect to the parametric uncertainty. In Part I "The Rank-One Problem" we consider the class of systems with restriction that the structure of uncertainty is limited to a vector. In Chapter " Canonical Parametrization of the Dual Problem in Robust Optimization: Non-Rational Case" we extend the class of allowed systems. The main result is the canonical parametrization of all destabilizing uncertainties in the dual problem. The corresponding result in the rational case was previously stated in terms of unstable zero-pole cancellations. For non-rational systems the situation with common zeros is more complicated. The nominal factors can contain a singular component and cannot be treated by unstable cancellations. We have shown that in the general case the common zeros of the plant factors are naturally replaced by a scalar function with the positive winding number. To illustrate the duality principle, the result is applied to a system with delay. By dual parametrization we can easily calculate the optimal uncertainty bound and the optimal controller. Since the optimal controller is not robustly stabilizing in the strong sense,as it is only a limit of suboptimal robustly stabilizing controllers,we have to regularize the limiting controller. In Chapter "Regularization of the Limiting Optimal Controller in Robust Stabilization" we present a method of obtaining the suboptimal controller of lower order that provides the stability margin as close to the optimal one as we wish. The method is illustrated with some scalar examples. In Chapter "Robust Control via Linear Programming" we propose the numerical algorithm for the optimal robust control synthesis. The algorithm proposed is a sequence of the standard linear programming problems of growing dimensions which approximate the initial problem. In the special case, when the uncertainty parameter is real-valued, it is shown that the initial problem can be considered as finite-dimensional in the space of variables. In Part II "Convex Duality: Matrix Case" we generalize the results to the system with matrix uncertainties. We obtain a canonical factorization of a plant with unstructured uncertainty in terms of an unitary matrix function with finite winding number and an outer matrix function. We introduce a metric in the space of factorization and discuss connection with nu-gap metric

Regular Implementability and Stabilization Using Controllers With Pre-Specified Input/Output Partition

This paper deals with the problems of regular imple- mentability and stabilization of a given plant in the context of finite- dimensional linear differential system behaviors. In particular we solve the problems of regular implementability and stabilization using controllers in which a pre-specified subset of the plant con- trol variables is free. We will also extend the results to the situation in which the set of plant control variables is partitioned into two complementary subsets. Variables from one subset should become controller inputs, while variables from the other should become controller outputs. In other words, we consider the problems of regular implementability and stabilization using controllers with a priori given input/output structure

Locally symmetric submanifolds lift to spectral manifolds

In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of the locally symmetric manifold

On the computation of π\pi-flat outputs for differential-delay systems

We introduce a new definition of $\pi$-flatness for linear differential delay systems with time-varying coefficients. We characterize $\pi$- and $\pi$-0-flat outputs and provide an algorithm to efficiently compute such outputs. We present an academic example of motion planning to discuss the pertinence of the approach.Comment: Minor corrections to fit with the journal versio

Implementation of behavioral systems

In this chapter, we study control by interconnection of a given linear differential system (the plant behavior) with a suitable controller. The problem formulations and their solutions are completely representation free, and specified only in terms of the system dynamics. A controller is a system that constrains the plant behavior through a certain set of variables. In this context, there are two main situations to be considered: either all the system variables are available for control, i.e., are control variables (full control) or only some of the variables are control variables (partial control). For systems evolving over a time domain (1D) the problems of implementability by partial (regular) interconnection are well understood. In this chapter, we study why similar results are not valid in themultidimensional (nD) case. Finally, we study two important classes of controllers, namely, canonical controllers and regular controllers

Control of 2D behaviors by partial interconnection

Abstract-In this paper we study the stability of two dimensional (2D) behaviors with two types of variables: the variables that we are interested to control (the to-be-controlled variables) and the variables on which we are allowed to enforce restrictions (the control variables). We derive conditions for the stabilization of the to-be-controlled variables by regular partial interconnection, i.e., by imposing non-redundant additional restrictions to the control variables

Regulation and robust stabilization: a behavioral approach

In this thesis we consider a number of control synthesis problems within the behavioral approach to systems and control. In particular, we consider the problem of regulation, the H! control problem, and the robust stabilization problem. We also study the problems of regular implementability and stabilization with constraints on the input/output structure of the admissible controllers. The systems in this thesis are assumed to be open dynamical systems governed by linear constant coefficient ordinary differential equations. The behavior of such system is the set of all solutions to the differential equations. Given a plant with its to-be-controlled variable and interconnection variable, control of the plant is nothing but restricting the behavior of the to-be-controlled plant variable to a desired subbehavior. This restriction is brought about by interconnecting the plant with a controller (that we design) through the plant interconnection variable. In the interconnected system the plant interconnection variable has to obey the laws of both the plant and the controller. The interconnected system is also called the controlled system, in which the controller is an embedded subsystem. The interconnection of the plant and the controller is said to be regular if the laws governing the interconnection variable are independent from the laws governing the plant. We call a specification regularly implementable if there exists a controller acting on the plant interconnection variable, such that, in the interconnected system, the behavior of the to-becontrolled variable coincides with the specification and the interconnection is regular. Within the framework of regular interconnection we solve the control problems listed in the first paragraph. Solvability conditions for these problems are independent of the particular representations of the plant and the desired behavior.

On differential-algebraic control systems

In der vorliegenden Dissertation werden differential-algebraische Gleichungen (differential-algebraic equations, DAEs) der Form \ddt E x = Ax + f betrachtet, wobei $E$ und $A$ beliebige Matrizen sind. Falls $E$ nichtverschwindende Einträge hat, dann kommen in der Gleichung Ableitungen der entsprechenden Komponenten von $x$ vor. Falls $E$ eine Nullzeile hat, dann kommen in der entsprechenden Gleichung keine Ableitungen vor und sie ist rein algebraisch. Daher werden Gleichungen vom Typ \ddt E x = Ax + f differential-algebraische Gleichungen genannt. Ein Ziel dieser Dissertation ist es, eine strukturelle Zerlegung einer DAE in vier Teile herzuleiten: einen ODE-Anteil, einen nilpotenten Anteil, einen unterbestimmten Anteil und einen überbestimmten Anteil. Jeder Anteil beschreibt ein anderes Lösungsverhalten in Hinblick auf Existenz und Eindeutigkeit von Lösungen für eine vorgegebene Inhomogenität $f$ und Konsistenzbedingungen an $f$. Die Zerlegung, namentlich die quasi-Kronecker Form (QKF), verallgemeinert die wohlbekannte Kronecker-Normalform und behebt einige ihrer Nachteile. Die QKF wird ausgenutzt, um verschiedene Konzepte der Kontrollierbarkeit und Stabilisierbarkeit für DAEs mit~$f=Bu$ zu studieren. Hier bezeichnet $u$ den Eingang des differential-algebraischen Systems. Es werden Zerlegungen unter System- und Feedback-Äquivalenz, sowie die Folgen einer Behavioral-Steuerung $K_x x + K_u u = 0$ für die Stabilisierung des Systems untersucht. Falls für das DAE-System zusätzlich eine Ausgangs-Gleichung $y=Cx$ gegeben ist, dann lässt sich das Konzept der Nulldynamik wie folgt definieren: die Nulldynamik ist, grob gesagt, die Dynamik, die am Ausgang nicht sichtbar ist, d.h. die Menge aller Lösungs-Trajektorien $(x,u,y)$ mit $y=0$. Für rechts-invertierbare Systeme mit autonomer Nulldynamik wird eine Zerlegung hergeleitet, welche die Nulldynamik entkoppelt. Diese versetzt uns in die Lage, eine Behavior-Steuerung zu entwickeln, die das System stabilisiert, vorausgesetzt die Nulldynamik selbst ist stabil. Wir betrachten auch zwei Regelungs-Strategien, die von den Eigenschaften der oben genannten System-Klasse profitieren: Hochverstärkungs- und Funnel-Regelung. Ein System \ddt E x = Ax + Bu, $y=Cx$, hat die Hochverstärkungseigenschaft, wenn es durch die Anwendung der proportionalen Ausgangsrückführung $u=-ky$, mit $k>0$ hinreichend groß, stabilisiert werden kann. Wir beweisen, dass rechts-invertierbare Systeme mit asymptotisch stabiler Nulldynamik, die eine bestimmte Relativgrad-Annahme erfüllen, die Hochverstärkungseigenschaft haben. Während der Hochverstärkungs-Regler recht einfach ist, ist es jedoch a priori nicht bekannt, wie groß die Verstärkungskonstante $k$ gewählt werden muss. Dieses Problem wird durch den Funnel-Regler gelöst: durch die adaptive Justierung der Verstärkung über eine zeitabhängige Funktion $k(\cdot)$ und die Ausnutzung der Hochverstärkungseigenschaft wird erreicht, dass große Werte $k(t)$ nur dann angenommen werden, wenn sie nötig sind. Eine weitere wesentliche Eigenschaft ist, dass der Funnel-Regler das transiente Verhalten des Fehlers $e=y-y_{\rm ref}$ der Bahnverfolgung, wobei $y_{\rm ref}$ die Referenztrajektorie ist, beachtet. Für einen vordefinierten Performanz-Trichter (funnel) $\psi$ wird erreicht, dass $\|e(t)\|<\psi(t)$. Schließlich wird der Funnel-Regler auf die Klasse von MNA-Modellen von passiven elektrischen Schaltkreisen mit asymptotisch stabilen invarianten Nullstellen angewendet. Dies erfordert die Einschränkung der Menge der zulässigen Referenztrajektorien auf solche die, in gewisser Weise, die Kirchhoffschen Gesetze punktweise erfüllen.In this dissertation we study differential-algebraic equations (DAEs) of the form Ex'=Ax+f. One aim of the thesis is to derive the quasi-Kronecker form (QKF), which decomposes the DAE into four parts: the ODE part, nilpotent part, underdetermined part and overdetermined part. Each part describes a different solution behavior. The QKF is exploited to study the different controllability and stabilizability concepts for DAEs with f=Bu, where u is the input of the system. Feedback decompositions, behavioral control and stabilization are investigated. For DAE systems with output equation y=Cx, we may define the concept of zero dynamics, which are those dynamics that are not visible at the output. For right-invertible systems with autonomous zero dynamics a decomposition is derived, which decouples the zero dynamics of the system and allows for high-gain and funnel control. It is shown, that the funnel controller achieves tracking of a reference trajectory by the output signal with prescribed transient behavior. Finally, the funnel controller is applied to the class of MNA models of passive electrical circuits with asymptotically stable invariant zeros

Applications of the Quillen-Suslin theorem to multidimensional systems theory

The purpose of this paper is to give four new applications of the Quillen-Suslin theorem to mathematical systems theory. Using a constructive version of the Quillen-Suslin theorem, also known as Serre's conjecture, we show how to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems. We prove that a flat multidimensional linear system is algebraically equivalent to the controllable 1-D dimensional linear systems obtained by setting all but one functional operator to zero in the polynomial matrix defining the system. In particular, we show that a flat ordinary differential time-delay linear system is algebraically equivalent to the corresponding ordinary differential system without delay, i.e., the controllable ordinary differential linear system obtained by setting all the delay amplitudes to zero. We also give a constructive proof of a generalization of Serre's conjecture known as Lin-Bose's conjecture. Moreover, we show how to constructively compute (weakly) left-/right-/doubly coprime factorizations of rational transfer matrices over a commutative polynomial ring. The Quillen-Suslin theorem also plays a central part in the so-called decomposition problem of linear functional systems studied in the literature of symbolic computation. In particular, we show how the basis computation of certain free modules, coming from projectors of the endomorphism ring of the module associated with the system, allows us to obtain unimodular matrices which transform the system matrix into an equivalent block-triangular or a block-diagonal form. Finally, we demonstrate the package QuillenSuslin which, to our knowledge, contains the first implementation of the Quillen-Suslin theorem in a computer algebra system as well as the different algorithms developed in the paper