Tobias Damm

Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems

We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems, the relation of Gramians and energy functionals for bilinear systems is less obvious. We discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic Gramians satisfying generalized Lyapunov equations. In any of the considered cases, the definition of algebraic Gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems. We illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems.

Krylov subspace methods for model order reduction of bilinear control systems

We discuss the use of Krylov subspace methods with regard to the problem of model order reduction. The focus lies on bilinear control systems, a special class of nonlinear systems, which are closely related to linear systems. While most existent approaches are based on series expansions around zero, we will extend the underlying ideas to a more general context and show that there exist several ways to reduce bilinear systems. Besides, we will briefly address the problem of stability preserving model reduction and further explain the benefit of using two-sided projection methods. By means of some numerical examples, we will illustrate the performance of the presented reduction methods.

Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations

We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In this paper, we describe a direct and an iterative method based on this idea. The direct method is applicable if the generalized Lyapunov operator is a low-rank perturbation of a standard Lyapunov operator; it is related to the Sherman–Morrison–Woodbury formula. The iterative method requires a stability assumption; it uses convergent regular splittings, an alternate direction implicit iteration as preconditioner, and Krylov subspace methods. Copyright

Technical communique: The Lambert W function and the spectrum of some multidimensional time-delay systems

In this note we find an explicit expression for the eigenvalues of a retarded time-delay system with one delay, for the special case that the system matrices are simultaneously triangularizable, which includes the case where they commute. Using matrix function definitions we define a matrix version of the Lambert W function, from which we form the expression. We prove by counter-example that some expressions in other publications on Lambert W for time-delay systems do not always hold.

Newton's method for a rational matrix equation occurring in stochastic control

We study a general class of rational matrix equations, which contains the continuous (CARE) and discrete (DARE) algebraic Riccati equations as special cases. Equations of this type were encountered in [SIAM J. Control and Optimization 36 (1998) 1504–1538; Stochastics and Stochastics Reports, 65 (1999) 255–297], where H∞-type problems of disturbance attenuation for stochastic linear systems were studied. We develop a unifying framework for the analysis of these equations based on the theory of (resolvent) positive operators and show that they can be solved by Newtons method starting at an arbitrary stabilizing matrix.

AN EXPONENTIAL TURNPIKE THEOREM FOR DISSIPATIVE DISCRETE TIME OPTIMAL CONTROL PROBLEMS

We investigate the exponential turnpike property for finite horizon undiscounted discrete time optimal control problems without any terminal constraints. Considering a class of strictly dissipative systems, we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property. Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration.

Brief paper: On detectability of stochastic systems

We discuss notions of detectability for stochastic linear control systems of Ito type. A natural concept of detectability requires a non-zero output, if the state process is unstable. We show that this property can equivalently be characterized by a generalized version of the Hautus-test. The proof is based on spectral theory for positive operators.

State-feedback H∞-type control of linear systems with time-varying parameter uncertainty

This paper is concerned with stabilisation and disturbance attenuation problems for linear systems with uncertain parameters. We consider the case, where the uncertainties are possibly unbounded but have a certain normal distribution and the case, where the uncertainties are arbitrary but bounded. Criteria for the solvability of the disturbance attenuation problem are presented in terms of generalized Riccati-type matrix inequalities and equations. The main part of the paper is devoted to the analysis and solution of these equations. A numerical example is given to illustrate our results.

Newton's method for concave operators with resolvent positive derivatives in ordered Banach spaces

Abstract We prove a non-local convergence result for Newton’s method applied to a class of nonlinear equations in ordered real Banach spaces. The key tools in our approach are special notions of concavity and the spectral theory of resolvent positive operators.

Model reduction of time-delay systems using position balancing and delay Lyapunov equations

Balanced truncation is a standard and very natural approach to approximate dynamical systems. We present a version of balanced truncation for model order reduction of linear time-delay systems. The procedure is based on a coordinate transformation of the position and preserves the delay structure of the system. We therefore call it (structure-preserving) position balancing. To every position, we associate quantities representing energies for the controllability and observability of the position. We show that these energies can be expressed explicitly in terms of the solutions to corresponding delay Lyapunov equations. Apart from characterizing the energies, we show that one block of the (operator) controllability and observability Gramians in the operator formulation of the time-delay system can also be characterized with the delay Lyapunov equation. The delay Lyapunov equation undergoes a contragredient transformation when we apply the position coordinate transformation and we propose to truncate it in a classical fashion, such that positions which are only weakly connected to the input and the output in the sense of the energy concepts are removed.