Philip Losse

Controllability and Observability of Second Order Descriptor Systems

We analyze controllability and observability conditions for second order descriptor systems and show how the classical conditions for first order systems can be generalized to this case. We show that performing a classical transformation to first order form may destroy some controllability and observability properties. As an example, we demonstrate that the loss of impulse controllability in constrained multibody systems is due to the representation as a first order system. To avoid this problem, we will derive a canonical form and new first order formulations that do not destroy the controllability and observability properties.

The Modified Optimal

The

\mathcal{H}_\infty

\mathcal{H}_\infty

Control Problem for Descriptor Systems

control problem is studied for linear constant coefficient descriptor systems. Necessary and sufficient optimality conditions are derived for systems of arbitrary index. These conditions are formulated in terms of deflating subspaces of even matrix pencils containing only the parameters of the original system. It is shown that this approach leads to a more numerically robust and efficient method in computing the optimal value

Numerical Linear Algebra Methods for Linear Differential-Algebraic Equations

\gamma

H ∞ -control for descriptor systems - A structured matrix pencils approach

in contrast to other methods such as the widely used Riccati- and linear matrix inequality (LMI)-based approaches. The results are illustrated by a numerical example.

THE MODIFIED OPTIMAL H∞ CONTROL PROBLEM FOR DESCRIPTOR SYSTEMS

A survey of methods from numerical linear algebra for linear constant coefficient differential-algebraic equations (DAEs) and descriptor control systems is presented. We discuss numerical methods to check the solvability properties of DAEs as well as index reduction and regularization techniques. For descriptor systems we discuss controllability and observability properties and how these can be checked numerically. These methods are based on staircase forms and derivative arrays, obtained by real orthogonal transformations that are discussed in detail. Then we use the reformulated problems in several control applications for differential-algebraic equations ranging from regular and singular linear-quadratic optimal and robust control to dissipativity checking. We discuss these applications and give a systematic overview of the theory and the numerical solution methods. In particular, we show that all these applications can be treated with a common approach that is based on the computation of eigenvalues and deflating subspaces of even matrix pencils. The unified approach allows us to generalize and improve several techniques that are currently in use in systems and control.

ℋ∞ Control for Descriptor Systems – a Pencil Based Approach

The H∞ control problem is studied for linear constant coefficient descriptor systems. Necessary and sufficient optimality conditions are derived in terms of deflating subspaces of even matrix pencils for problems of arbitrary index. It is shown that this approach leads to a more robust method in computing the optimal value γin contrast to other methods such as the widely used Riccati or LMI based approaches. The results are illustrated by a numerical example.