Optimal Robustness of Port-Hamiltonian Systems

Mehrmann, Volker; Dooren, Paul M. Van

doi:10.1137/19m1259092

Cited by 26 publications

(23 citation statements)

References 32 publications

(36 reference statements)

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“…Recently, 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 robustness analysis. A matrix A ∈ R n,n is called a DH matrix if A = (J − R)Q for some J, R, Q ∈ R n,n such that J T = −J, R is positive semidefinite and Q is positive definite.…”

Section: Contribution and Outline Of The Papermentioning

confidence: 99%

Minimal-norm static feedbacks using dissipative Hamiltonian matrices

Gillis

Sharma

2021

Linear Algebra and its Applications

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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 ...

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Section: Contribution and Outline Of The Papermentioning

confidence: 99%

Minimal-norm static feedbacks using dissipative Hamiltonian matrices

Gillis

Sharma

2021

Linear Algebra and its Applications

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“…Passive systems and their relationships with positive-real transfer functions are well studied, starting with the works [ 13 , 18 , 21 – 24 ] and the topic has recently received a revival in the work on port-Hamiltonian (pH) systems , [ 15 , 19 , 20 ]. For a summary of the relationships see [ 2 , 21 ], where also the characterization of passivity via the solution set of an associated linear matrix inequality (LMI) is highlighted.…”

Section: Introductionmentioning

confidence: 99%

Computation of the Analytic Center of the Solution Set of the Linear Matrix Inequality Arising in Continuous- and Discrete-Time Passivity Analysis

et al. 2020

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In this paper formulas are derived for the analytic center of the solution set of linear matrix inequalities (LMIs) defining passive transfer functions. The algebraic Riccati equations that are usually associated with such systems are related to boundary points of the convex set defined by the solution set of the LMI. It is shown that the analytic center is described by closely related matrix equations, and their properties are analyzed for continuous-and discrete-time systems. Numerical methods are derived to solve these equations via steepest descent and Newton methods. It is also shown that the analytic center has nice robustness properties when it is used to represent passive systems. The results are illustrated by numerical examples.

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“…For continuous-time systems, [9] presents an algorithm that constructs a sequence of successive stable iterates to compute a nearby stable approximation X. A reformulation of (1.1) is studied by using linear dissipative Hamiltonian systems in [7]. The authors of [4] use the projected gradient descent method to solve such a problem.…”

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confidence: 99%