PrefaceThis book originates from several editions of lecture notes that were used as teaching material for the course 'Control Theory for Linear Systems', given within the framework of the national Dutch graduate school of systems and control, in the period from 1987 to 1999. The aim of this course is to provide an extensive treatment of the theory of feedback control design for linear, finite-dimensional, time-invariant state space systems with inputs and outputs.One of the important themes of control is the design of controllers that, while achieving an internally stable closed system, make the influence of certain exogenous disturbance inputs on given to-be-controlled output variables as small as possible. Indeed, in the appropriate sense this theme is covered by the classical linear quadratic regulator problem and the linear quadratic Gaussian problem, as well as, more recently, by the H 2 and H ∞ control problems. Most of the research efforts on the linear quadratic regulator problem and the linear quadratic Gaussian problem took place in the period up to 1975, whereas in particular H ∞ control has been the important issue in the most recent period, starting around 1985.In, roughly, the intermediate period, from 1970 to 1985, much attention was attracted by control design problems that require to make the influence of the exogenous disturbances on the to-be-controlled outputs equal to zero. The static state feedback versions of these control design problems, often called disturbance decoupling, or disturbance localization, problems were treated in the classical textbook 'Linear Multivariable Control: A Geometric Approach', by W.M. Wonham. Around 1980, a complete theory on the disturbance decoupling problem by dynamic measurement feedback became available. A central role in this theory is played by the geometric (i.e., linear algebraic) properties of the coefficient matrices appearing in the system equations. In particular, the notions of (A, B)-invariant subspace and (C, A)-invariant subspace play an important role. These notions, and their generalizations, also turned out to be central in understanding and classifying the 'fine structure' of the system under consideration. For example, important dynamic properties such as system invertibility, strong observability, strong detectability, the minimum phase property, output stabilizability, etc., can be characterized in terms of these geometric concepts. The notions of (A, B)-invariance and (C, A)-invariance also turned out to be instrumental in other synthesis problems, like observer design, problems of tracking and regulation, etc. In this book, we will treat both the 'pre-1975' approach represented by the linear quadratic regulator problem and the H 2 control problem, as well as the 'post-1985' approach represented by the H ∞ control problem and its applications to robust control. However, we feel that a textbook dedicated to control theory for linear state space systems should also contain the central issues of the 'geometric approach', namely a treatment...

The use of persistently exciting data has recently been popularized in the context of data-driven analysis and control. Such data have been used to assess system theoretic properties and to construct control laws, without using a system model. Persistency of excitation is a strong condition that also allows unique identification of the underlying dynamical system from the data within a given model class. In this paper, we develop a new framework in order to work with data that are not necessarily persistently exciting. Within this framework, we investigate necessary and sufficient conditions on the informativity of data for several data-driven analysis and control problems. For certain analysis and design problems, our results reveal that persistency of excitation is not necessary. In fact, in these cases data-driven analysis/control is possible while the combination of (unique) system identification and model-based control is not. For certain other control problems, our results justify the use of persistently exciting data as data-driven control is possible only with data that are informative for system identification.

Abstract. This paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems. In many applications, we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables. The obvious cases where such functionals are important are in Lyapunov theory and in LQ and H∞ optimal control. With some exceptions, these theories have almost invariably concentrated on first order models and state representations. In this paper, we develop a theory for linear time-invariant differential systems and quadratic functionals. We argue that in the context of systems described by one-variable polynomial matrices, the appropriate tool to express quadratic functionals of the system variables are two-variable polynomial matrices. The main achievement of this paper is a description of the interaction of one-and two-variable polynomial matrices for the analysis of functionals and for the application of higher order Lyapunov functionals.Key words. quadratic differential forms, linear systems, polynomial matrices, two-variable polynomial matrices, Lyapunov theory, positivity, spectral factorization, dissipativeness, storage functions AMS subject classifications. 93A10, 93A30, 93D05, 93D20, 93D30, 93C05, 93C45 PII. S03630129963030621. Introduction. In the theory of models for dynamical systems, it has been customary to consider both external input/output as well as state space models. Also, there is a well developed theory for passing from one type of model to another. Thus, there are efficient algorithms for passing from a convolution, to a transfer function, to a state model, and back. Even for stochastic and nonlinear systems, there are methods for associating a first order state representation to a high order model.However, in addition to understanding the interaction between system variables, we need in many applications to understand also the behavior of certain functionals of these variables. The obvious cases where such functionals are crucial are in Lyapunov theory, in the theory of dissipative systems, and in optimal control. In these contexts it is remarkable to observe that the theory of dynamics has almost invariably concentrated on first order models and state representations. Thus, in studying system stability using Lyapunov methods, we are constrained to consider state representations, and optimal control problems invariably assume that the cost is an integral of a function of the state and the input. The question thus occurs of whether it is possible to develop an external theory-for example, Lyapunov theory-for systems and functionals so that analysis of stability and passivity, for instance, could proceed on the basis of a first principles model instead of first having to find a state representation. In this paper, we consider models that are not in state form (even though some proofs use state representations). Our models are externally specified yet they are not completely general first principles mod...

This paper deals with robust synchronization of uncertain multi-agent networks. Given a network with for each of the agents identical nominal linear dynamics, we allow uncertainty in the form of additive perturbations of the transfer matrices of the nominal dynamics. The perturbations are assumed to be stable and bounded in -norm by some a priori given desired tolerance. We derive state space formulas for observer based dynamic protocols that achieve synchronization for all perturbations bounded by this desired tolerance. It is shown that a protocol achieves robust synchronization if and only if each controller from a related finite set of feedback controllers robustly stabilizes a given, single linear system. Our protocols are expressed in terms of real symmetric solutions of certain algebraic Riccati equations and inequalities, and also involve weighting factors that depend on the eigenvalues of the graph Laplacian. For undirected network graphs we show that within the class of such dynamic protocols, a guaranteed achievable tolerance can be obtained that is proportional to the quotient of the second smallest and the largest eigenvalue of the Laplacian. We also extend our results to additive nonlinear perturbations with -gain bounded by a given tolerance. Index Terms-Laplacian matrix.Manuscript

Abstract-The problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative. Our formulation and solution of this problem is completely representation-free. The system dynamics are specified by a differential behavior, and the performance is specified through a quadratic differential form. We view control as interconnection: a controller constrains a distinguished set of system variables, the control variables. The resulting behavior of the to-be-controlled variables is called the controlled behavior. The constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior, obtained by setting the control variables equal to zero, and the plant behavior, obtained by leaving the control variables unconstrained. The main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications. The essential requirement is a coupling condition, an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior.Index Terms-Behaviors, controller implementability, coupling condition, dissipative systems, hidden behavior, quadratic differential forms, storage functions.