Minimal-norm static feedbacks using dissipative Hamiltonian matrices

Abstract

In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose an algorithm to tackle this problem. We illustrate the effectiveness of our algorithm compared to state-of-the-art methods for the SOF problem on numerous numerical examples from the COMPLeIB library. IntroductionConsider a continuous linear-time invariant (LTI) system in the forṁwhere, for all t ∈ R, x(t) ∈ R n is the state space, u(t) ∈ R m is the control input, A ∈ R n,n , B ∈ R n,m , and C ∈ R p,n . The notion of stabilizing the system pair (A, B) using feedback controllers is a fundamental one, and is referred to as the static-state feedback (SSF) problem. It requires to find K ∈ R m,n such that A − BK is stable, that is, all eigenvalues of the matrix A − BK are in the left half of the complex plane and those on the imaginary axis are semisimple; see for example [4,2]. The first goal of this paper is to solve the SSF problem; this can be divided into two parts: 1) Feasibility. Check the existence of a feedback matrix K such that A − BK is stable. IIT Delhi.2) Optimization. If the problem is feasible, minimize the norm of the feedback matrix, that is,where · is a given norm such as the ℓ 2 norm or the Frobenius norm.The second goal is to consider the analogous problem for system triplets (A, B, C), referred to as the static-output feedback (SOF) problem. The SOF problem requires to find K ∈ R m,p such that A − BKC is stable; see [27] for a survey on the SOF problem. This decision problem is believed to be NP-hard as no polynomial-time algorithm is known. Moreover, if extra constraints are imposed on the entries of the static controller, then this decision problem is NP-hard [19,6]. As a consequence, the minimal-norm SOF problem, for which the norm of K is minimized, is hard as well. For more discussion on the hardness of this problem, we refer to the recent paper by Peretz [21] and the references therein. The solution of the SOF problem is important for systems which models structural dynamics, and naturally needs a static feedback that can be built into the structure [25,31,32,22,21]. It was shown that optimal SOFs may achieve similar performance as optimal dynamic feedbacks. Contribution and outline of the paperRecently, in [11], a parametrization of the set of all stable matrices was obtained in terms of dissipative Hamiltonian (DH) systems. DH systems are special cases of port-Hamiltonian systems, which recently have received a lot attention in energy based modeling; see for example [13,23,24], and also [12,5,18] for ...

Solving matrix nearness problems via Hamiltonian systems, matrix factorization, and optimization

“…In Section 3.4, we briefly discuss other approaches to tackle the nearest stable matrix problem. In Section 3.5, we show how the ideas from Sections 3.2 and 3.3 can be used to solve the static-state and static-output feedback problems, which is the result from [47].…”

“…where • is a given norm such as the 2 norm, • 2 , or the Frobenius norm, • F . In [47], we used the DH form to solve this problem. In view of Theorem 3.1, the following theorem is relatively straightforward.…”

“…The main goal of these lecture notes is to survey a series of recent works [44,43,45,41,42,25,47] that aim at solving several nearness problems for a given system. As we will see, these problems can be written as distance problems of matrices or matrix pencils.…”

On Parametrizations of State Feedbacks and Static Output Feedbacks and Their Applications

“…In [30], a parametrization of all the SFs and SOFs of Linear-Time-Invariant (LTI) continuous-time systems is achieved by using a characterization of all the (marginally) stable matrices as dissipative Hamiltonian matrices, leading to a highly performance sequential semi-definite programming algorithm for the minimal-gain SOF problem. The proposed method there can be applied also to LTI discrete-time systems by adding semi-definite conditions for placing the closed-loop eigenvalues in the unit disk.…”