Timo Reis

Balanced truncation model reduction of second-order systems

In this paper we consider structure-preserving model reduction of second-order systems using a balanced truncation approach. Several sets of singular values are introduced for such systems, which lead to different concepts of balancing and different second-order balanced truncation methods. A comparison of these methods with other second-order balanced truncation techniques is presented. We also show that, in general, none of the existing structure-preserving balanced truncation methods for second-order systems preserves stability in the reduced models. Numerical examples are given that demonstrate the properties of the new methods.

Controllability of Linear Differential-Algebraic Systems—A Survey

Different concepts related to controllability of differential-algebraic equations are described. The class of systems considered consists of linear differential-algebraic equations with constant coefficients. Regularity, which is, loosely speaking, a concept related to existence and uniqueness of solutions for any inhomogeneity, is not required in this article. The concepts of impulse controllability, controllability at infinity, behavioral controllability, and strong and complete controllability are described and defined in the time domain. Equivalent criteria that generalize the Hautus test are presented and proved.

Lyapunov Balancing for Passivity-Preserving Model Reduction of RC Circuits

We apply a Lyapunov-based balanced truncation model reduction method to differential-algebraic equations arising in modeling of RC circuits. This method is based on diagonalizing the solution of one projected Lyapunov equation. It is shown that this method preserves passivity and delivers an error bound. By making use of the special structure of circuit equations, we can reduce the numerical effort for balanced truncation drastically.

Brief paper: Balanced truncation model reduction for systems with inhomogeneous initial conditions

We present a rigorous approach to extend balanced truncation model reduction (BTMR) to systems with inhomogeneous initial conditions, we provide an estimate for the error between the input-output maps of the original and of the reduced initial value system, and we illustrate numerically the superiority of our approach over the naive application of BTMR. When BTMR is applied to linear time invariant systems with inhomogeneous initial conditions, it is crucial that the initial data are well represented by the subspaces generated by BTMR. This requirement is often ignored or it is avoided by making the restrictive assumption that the initial data are zero. To ensure that the initial data are well represented by the BTMR subspaces, we add auxiliary inputs determined by the initial data.

Zero dynamics and funnel control of linear differential-algebraic systems

We study the class of linear differential-algebraic m-input m-output systems which have a transfer function with proper inverse. A sufficient condition for the transfer function to have proper inverse is that the system has ‘strict and non-positive relative degree’. We present two main results: first, a so-called ‘zero dynamics form’ is derived; this form is—within the class of system equivalence—a simple form of the DAE; it is a counterpart to the well-known Byrnes–Isidori form for ODE systems with strictly proper transfer function. The ‘zero dynamics form’ is exploited to characterize structural properties such as asymptotically stable zero dynamics, minimum phase, and high-gain stabilizability. The zero dynamics are characterized by (A, E, B)-invariant subspaces. Secondly, it is shown that the ‘funnel controller’ (that is a static non-linear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.

A survey on model reduction of coupled systems

In this paper we give an overview of model order reduction techniques for coupled systems. We consider linear time-invariant control systems that are coupled through input-output relations and discuss model reduction of such systems using moment matching approximation and balanced truncation. Structure-preserving approaches to model order reduction of coupled systems are also presented. Numerical examples are given.

Stability analysis and model order reduction of coupled systems

In this paper we discuss the stability and model order reduction of coupled linear time-invariant descriptor systems. Sufficient conditions for the asymptotic stability of a closed-loop system are given. We present a model reduction approach for coupled systems based on reducing the order of the subsystems and coupling the reduced-order models through the same interconnection relations as for the original system. Such an approach allows us to obtain error bounds for the reduced-order closed-loop system in terms of the errors in the reduced-order subsystems. Model reduction of coupled systems with unstable or improper subsystems is also considered. Numerical examples are given.

Positivity Preserving Balanced Truncation for Descriptor Systems

We propose a model reduction method for positive systems that ensures the positivity of the reduced-order model. In the standard as well as in the descriptor case, for continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Positivity and stability are preserved, and an error bound in the

FINITE-RANK ADI ITERATION FOR OPERATOR LYAPUNOV EQUATIONS ∗

\mathcal{H}_\infty

Zero dynamics and funnel control for linear electrical circuits

-norm is provided.