Solution of polynomial Lyapunov and Sylvester equations

A two-variable polynomial approach to solve the one-variable polynomial Lyapunov and Sylvester equations is proposed. Lifting the problem from the one-variable to the two-variable context gives rise to associated lifted equations which live on finite-dimensional vector spaces. This allows for the design of an iterative solution method which is inspired by the method of Faddeev for the computation of matrix resolvents. The resulting algorithms are especially suitable for applications requiring symbolic or exact computation

A two-variable approach to solve the polynomial Lyapunov equation

A two-variable polynomial approach to solve the one-variable polynomial Lyapunov equation is proposed. Lifting the problem from the one-variable to the two-variable context allows to use Faddeev-type recursions in order to solve the polynomial Lyapunov equation in an iterative fashion. The method is especially suitable for applications requiring exact or symbolic computation

On the linear quadratic data-driven control

The classical approach for solving control problems is model based: first a model representation is derived from given data of the plant and then a control law is synthesized using the model and the control specifications. We present an alternative approach that circumvents the explicit identification of a model representation. The considered control problem is finite horizon linear quadratic tracking. The results are derived assuming exact data and the optimal trajectory is constructed off-line

Recursive exact H-infinity identification from impulse-response measurements

We study the H∞-partial realization problem from a behavioral point of view; we give necessary and sufficient conditions for solvability, and a characterization of all solutions. Instrumental in such analysis is the notion of time- and space-symmetrization of the data, which allows to transform the realization problem with metric- and stability constraints into an unconstrained behavioral modeling one

Autonomous linear lossless systems

We define a lossless autonomous system as one having a quadratic differential form associated with it called an energy function, which is positive and which is conserved. We define an oscillatory system as one which has all its trajectories bounded on the entire time axis. In this paper, we show that an autonomous system is lossless if and only if it is oscillatory. Next we discuss a few properties of energy functions of autonomous lossless systems and a suitable way of splitting a given energy function into its kinetic and potential energy components

The behavioral approach to systems and modeling

An introduction to behavioral system theory, and a brief review of the content of the Special Issue are given

Conserved- and zero-mean quadratic quantities in oscillatory systems

We study quadratic functionals of the variables of a linear oscillatory system and their derivatives. We show that such functionals are partitioned in conserved quantities and in trivially- and intrinsic zero-mean quantities. We also state an equipartition of energy principle for oscillatory systems

Hamiltonian and Variational Linear Distributed Systems

We use the formalism of bilinear- and quadratic differential forms in order to study Hamiltonian and variational linear distributed systems. It was shown in [1] that a system described by ordinary linear constant-coefficient differential equations is Hamiltonian if and only if it is variational. In this paper we extend this result to systems described by linear, constant-coefficient partial differential equations. It is shown that any variational system is Hamiltonian, and that any scalar Hamiltonian system is contained (in general, properly) in a particular variational system

Linear Hamiltonian behaviors and bilinear differential forms

We study linear Hamiltonian systems using bilinear and quadratic differential forms. Such a representation-free approach allows us to use the same concepts and techniques to deal with systems isolated from their environment and with systems subject to external influences and allows us to study systems described by higher-order differential equations, thus dispensing with the usual point of view in classical mechanics of considering first- and second-order differential equations only

Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation

We study the existence of positive and negative semidefinite solutions of algebraic Riccati equations (ARE) corresponding to linear quadratic problems with an indefinite cost functional. The problem to formulate reasonable necessary and sufficient conditions for the existence of such solutions is a long-standing open problem. A central role is played by certain two-variable polynomial matrices associated with the ARE. Our main result characterizes all unmixed solutions of the ARE in terms of the Pick matrices associated with these two-variable polynomial matrices. As a corollary of this result we obtain that the signatures of the extremal solutions of the ARE are determined by the signatures of particular Pick matrices