Hendrikus Trentelman

On Quadratic Differential Forms

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

Every storage function is a state function

H_{infty}

The dissipation inequality and the algebraic Riccati equation

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.

On the regular implementability of nD systems

It is shown that for linear dynamical systems with quadratic supply rates, a storage function can always be written as a quadratic function of the state of an associated linear dynamical system. This dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate.

On flat systems behaviors and observable image representations

Undoubtedly one of the most important concepts in linear systems and control, both from a theoretical as well as from a practical point of view, is the algebraic Riccati equation. Since its introduction in control theory by Kaiman [16] the beginning of the sixties, the algebraic Riccati equation has known an impressive range of applications, such as linear quadratic optimal control, stability theory, stochastic filtering and stochastic control, stochastic realization theory, synthesis of linear passive networks, differential games and, most recently, H ∞ optimal control and robust stabilization. The purpose of the present paper is to give an expository survey of the main concepts, results and applications related to the algebraic Riccati equation.

Guaranteed roll-off in a class of high-gain feedback design problems

Abstract In this short paper we study the problem of finding necessary and sufficient conditions for regular implementability by partial interconnection for nD system behaviors. In [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744.] such conditions were obtained in the context of 1D systems. In the present paper we show that the conditions obtained in [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744.] are no longer valid in general in the nD context. We also show that under additional assumptions, the conditions still remain relevant. We also reinvestigate the conditions for regular implementability by partial interconnection in terms of the canonical controller that were obtained in [P. Rocha, Canonical controllers and regular implementation of nD behaviors, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.]. Using the geometry of the underlying modules we generalize a result on regular implementability from the 1D to the nD case. Finally, we study how, in the 1D context, the conditions from [M.N. Belur, H.L. Trentelman, Stabilization, pole placement and regular implementability, IEEE Trans. Automat. Control 47(5) (2002) 735–744; P. Rocha, Canonical controllers and regular implementation of nD behaviors, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.] are connected.

Dissipativity preserving model reduction by retention of trajectories of minimal dissipation

In this short paper we propose a definition of flatness for systems not necessarily given in input/state/output representation. A flat system is a system for which there exists a mapping such that the manifest system behavior is equal to the image of this mapping, and such that the latent variable appearing in this image representation can be written as a function of the manifest variable and its derivatives up to some order. For linear differential systems, flatness is equivalent to controllability. We will generalize the main theorem of Levine and Nguyen (Systems Control Lett. 48 (2003) 69) to general linear differential systems.

Almost disturbance decoupling with bounded peaking

A number of synthesis problems associated with (almost) disturbance decoupling by state or measurement feedback is considered. Starting from a mathematical definition of the notion of high-frequency roll-off, known results on the solvability of these problems are generalized to the situation in which we require their solvability together with a certain guaranteed roll-off between disturbance and control. The conditions are formulated in terms of the solvability and approximate solvability of certain matrix equations in rational functions.

Bounded real and positive real balanced truncation using Σ-normalised coprime factors

We present a method for model reduction based on ideas from the behavioral theory of dissipative systems, in which the reduced order model is required to reproduce a subset of the set of trajectories of minimal dissipation of the original system. The passivity-preserving model reduction method of Antoulas (Syst Control Lett 54:361–374, 2005) and Sorensen (Syst Control Lett 54:347–360, 2005) is shown to be a particular case of this more general class of model reduction procedures.

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

In this paper we study a generalization of the almost disturbance decoupling problem by state feedback. Apart from approximate decoupling from the external disturbances to a first to-be-controlled output, we require a second output to be uniformly bounded with respect to the accuracy of decoupling. We also study the situation in which additionally we require the closed loop system to be internally stable. These problems are studied using the geometric approach to linear systems. We introduce some new almost controlled invariant subspaces and study their geometric structure. Necessary and sufficient conditions for the solvability of the above problems are then formulated in terms of these almost controlled invariant subspaces.