Jan C. Willems

Dissipative dynamical systems part I: General theory

The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper.

Least squares stationary optimal control and the algebraic Riccati equation

The optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated. Both the case in which the terminal state is free and that in which the terminal state is constrained to be zero are treated. The integrand of the performance criterion is allowed to be fully quadratic in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem. Frequency-domain and time-domain conditions for the existence of solutions are derived. The algebraic Riccati equation is then examined, and a complete classification of all its solutions is presented. It is finally shown how the optimal control problems introduced in the beginning of the paper may be solved analytically via the algebraic Riccati equation.

Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems

Conditions under which a nonlinear system can be rendered passive via smooth state feedback are derived. It is shown that, as in the case of linear systems, this is possible if and only if the system in question has relative degree one and is weakly minimum phase. It is proven that weakly minimum phase nonlinear systems with relative degree one can be globally asymptotically stabilized by smooth state feedback, provided that suitable controllability-like rank conditions are satisfied. This result incorporates and extends a number of stabilization schemes recently proposed for global asymptotic stabilization of certain classes of nonlinear systems. >

Paradigms and puzzles in the theory of dynamical systems

A self-contained exposition is given of an approach to mathematical models, in particular, to the theory of dynamical systems. The basic ingredients form a triptych, with the behavior of a system in the center, and behavioral equations with latent variables as side panels. The author discusses a variety of representation and parametrization problems, in particular, questions related to input/output and state models. The proposed concept of a dynamical system leads to a new view of the notions of controllability and observability, and of the interconnection of systems, in particular, to what constitutes a feedback control law. The final issue addressed is that of system identification. It is argued that exact system identification leads to the question of computing the most powerful unfalsified model. >

Dissipative dynamical systems Part II: Linear systems with quadratic supply rates

This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.

From time series to linear system-part I. Finite dimensional linear time invariant systems

Abstract In the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed. In this context the notions of linearity, time invariance, and finite dimensionality will be introduced. It will be shown that a given family of time series may be represented by a system of (AR) equations: Riw(t + l) + Rl − 1w(t + l − 1) + … + R0w(t) = 0, or, equivalently, by a finite dimensional linear time invariant system: x(t + 1) = Ax(t) + Bu(t); y(t) = Cx(t) + Du(t); w = (u, y), if and only if this family is linear, shift invariant and complete (or, as is equivalent, closed in the topology of pointwise convergence). This yields a very high level and elegant set of axioms which characterize these familiar objects. It is emphasized, however, that no a priori choice is made as to which components of w are inputs and which are outputs. Such a separation always exists in any specific linear time invariant model. Starting from these definitions, the structural indices of such systems are introduced and it is shown how an (AR) representation of a system having a given behaviour can be constructed. These results will be used in a modelling context in Part II of the paper.

The Riccati equation

1 Count Riccati and the Early Days of the Riccati Equation.- 2 Solutions of the Continuous and Discrete Time Algebraic Riccati Equations: A Review.- 3 Algebraic Riccati Equation: Hermitian and Definite Solutions.- 4 A Geometric View of the Matrix Riccati Equation.- 5 The Geometry of the Matrix Riccati Equation and Associated Eigenvalue Methods.- 6 The Periodic Riccati Equation.- 7 Invariant Subspace Methods for the Numerical Solution of Riccati Equations.- 8 The Dissipation Inequality and the Algebraic Riccati Equation.- 9 The Infinite Horizon and the Receding Horizon LQ-Problems with Partial Stabilization Constraints.- 10 Riccati Difference and Differential Equations: Convergence, Monotonicity and Stability.- 11 Generalized Riccati Equation in Dynamic Games.

Dissipative Dynamical Systems

Dissipative systems provide a strong link between physics, system theory, and control engineering. Dissipativity is first explained in the classical setting of input/state/output systems. In the context of linear systems with quadratic supply rates, the construction of a storage leads to a linear matrix inequality (LMI). It is in this context that LMIs first emerged in the field. Next, we phrase dissipativity in the setting of behavioral systems, and present the construction of two canonical storages, the available storage and the required supply. This leads to a new notion of dissipativity, purely in terms of boundedness of the free supply that can be extracted from a system. The storage is then introduced as a latent variable associated with the supply rate as the manifest variable. The equivalence of dissipativity with the existence of a non-negative storage is proven. Finally, we deal with supply rates that are given as quadratic differential forms and state several results that relate the existence of a (non-negative) storage to the two-variable polynomial matrix that defines the quadratic differential form. In the ECC presentation, we mainly discuss distributed dissipative systems described by constant coefficient linear PDEs. In this setting, the construction of storage functions leads to Hilberts 17-th problem on the representation of non-negative polynomials as a sum of squares.

Models for Dynamics

The purpose of this paper is to give a tutorial exposition of what we consider to be the basic mathematical concepts in the theory of dynamical systems.

On interconnections, control, and feedback

The purpose of this paper is to study interconnections and control of dynamical systems in a behavioral context. We start with an extensive physical example which serves to illustrate that the familiar input-output feedback loop structure is not as universal as we have been taught to believe. This leads to a formulation of control problems in terms of interconnections. Subsequently, we study interconnections of linear time-invariant systems from this vantage point. Let us mention two of the results obtained. The first one states that any polynomial can be achieved as the characteristic polynomial of the interconnection with a given plant, provided the plant is not autonomous. The second result states that any subsystem of a controllable system can be implemented by means of a singular feedback control law. These results yield pole placement and stabilization of controllable plants as a special case. These ideas are finally applied to the stabilization of a nonlinear system around an operating point.