Partial realization for singular systems in standard form

AbstractThe partial realization problem under consideration consists in finding, for a given sequence s=(sk)0N−1 of blocks, matrices (A,E,B,C) of appropriate size such that si=CEN−1−iAiB and the identity matrix is a linear combination of A and E. We discuss the question whether there is always a realization of this form for which the state space dimension is equal to the maximal rank of the underlying Hankel matrices. We show that this question has an affirmative answer if the block size is less than or equal to 2 and some other cases but not in general. The paper strengthens results obtained by Manthey et al. [cf. W. Manthey, U. Helmke, D. Hinrichsen, in: U. Helmke et al. (Eds.), Operators, Systems, and Linear Algebra, Teubner, Stuttgart, 1997, pp. 138–156]. The main tools are the results of the authors obtained in connection with Vandermonde factorization of block Hankel matrices. Finally, an interpretation of the problem in periodic discrete-time systems is given

On computing minimal realizations of periodic descriptor systems

Abstract: We propose computationally efficient and numerically reliable algorithms to compute minimal realizations of periodic descriptor systems. The main computational tool employed for the structural analysis of periodic descriptor systems (i.e., reachability and observability) is the orthogonal reduction of periodic matrix pairs to Kronecker-like forms. Specializations of a general reduction algortithm are employed for particular type of systems. One of the proposed minimal realization transformations for which the backward numerical stability can be proved

Computation of Kalman Decompositions of Periodic Systems

We consider the numerically reliable computation of reachability and observability Kalman decompositions of a periodic system with time-varying dimensions. These decompositons generalize the controllability/observability Kalman decompositions for standard state space systems and have immediate applications in the structural analysis of periodic systems. We propose a structure exploiting numerical algorithm to compute the periodic controllability form by employing exclusively orthogonal similarity transformations. The new algorithm is computationally efficient and strongly backward stable, thus fulfils all requirements for a satisfactory algorithm for periodic systems

Balanced truncation model reduction of periodic systems

The balanced truncation approach to model reduction is considered for linear discrete-time periodic systems with time-varying dimensions. Stability of the reduced model is proved and a guaranteed additive bound is derived for the approximation error. These results represent generalizations of the corresponding ones for standard discrete-time systems. Two numerically reliable methods to compute reduced order models using the balanced truncation approach are considered. The square-root method and the potentially more accurate balancing-free square-root method belong to the family of methods with guaranteed enhanced computational accuracy. The key numerical computation in both methods is the determination of the Cholesky factors of the periodic Gramian matrices by solving nonnegative periodic Lyapunov equations with time-varying dimensions directly for the Cholesky factors of the solutions

A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $\Omega=(0,1)$, $\Omega=\mathbb{T}^1$ and dim($\Omega$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations

Stabilization of systems with asynchronous sensors and controllers

We study the stabilization of networked control systems with asynchronous sensors and controllers. Offsets between the sensor and controller clocks are unknown and modeled as parametric uncertainty. First we consider multi-input linear systems and provide a sufficient condition for the existence of linear time-invariant controllers that are capable of stabilizing the closed-loop system for every clock offset in a given range of admissible values. For first-order systems, we next obtain the maximum length of the offset range for which the system can be stabilized by a single controller. Finally, this bound is compared with the offset bounds that would be allowed if we restricted our attention to static output feedback controllers.Comment: 32 pages, 6 figures. This paper was partially presented at the 2015 American Control Conference, July 1-3, 2015, the US