Finding the Nearest Positive-Real System

Abstract

The notion of positive realness for linear time-invariant (LTI) dynamical systems, equivalent to passivity, is one of the oldest in system and control theory. In this paper, we consider the problem of finding the nearest positive-real (PR) system to a non PR system: given an LTI control system defined by Eẋ = Ax+Bu and y = Cx+Du, minimize the Frobenius norm of (∆ E , ∆ A , ∆ B , ∆ C , ∆ D ) such that (E + ∆ E , A + ∆ A , B + ∆ B , C + ∆ C , D + ∆ D ) is a PR system. We first show that a system is extended strictly PR if and only if it can be written as a strict port-Hamiltonian system. This allows us to reformulate the nearest PR system problem into an optimization problem with a simple convex feasible set. We then use a fast gradient method to obtain a nearby PR system to a given non PR system, and illustrate the behavior of our algorithm on several examples. This is, to the best of our knowledge, the first algorithm that computes a nearby PR system to a given non PR system that (i) is not based on the spectral properties of related Hamiltonian matrices or pencils, (ii) allows to perturb all matrices (E, A, B, C, D) describing the system, and (iii) does not make any assumption on the original given system. generate energy: the system (1.1) is called passive if there exists a nonnegative scalar valued function V, called the storage function, such that V(0) = 0 and the dissipation inequalityholds for all admissible u, t 0 , and t 1 ≥ t 0 ; see for example [2,27]. The energy is defined via the inner product of the input and output vectors u(t) and y(t) of the system hence these vectors need to be of the same length.Given (E, A, B, C, D), the goal of this paper is to findWe will refer to this problem as the nearest PR system problem. We will also consider the case of nearest standard PR systems when E = I n imposing ∆ E = 0. Since passivity and positive realness are equivalent for LTI systems, the distance to positive realness has direct applications in passive model approximations. For example, when a real-world problem is approximated by a model (1.1), the passivity of the physical system may not be preserved, that is, the approximation process (for example, finite element or finite difference models, linearization, or model order reduction) makes the passive system nonpassive. The nonpassive system has to be approximated by a nearby passive system by perturbing E, A, B, C, and D.Several algorithms tackle this problem using the spectral properties of the related Hamiltonian/skew-Hamiltonian matrices or pencils for the input systems that are asymptotically stable, controllable, observable and almost passive; see [19,37,50,48] and the references therein. The algorithms in [19] and [37] impose additional assumptions on the input system, namely E = I n and D + D T nonsingular, and are restricted to perturbations of the matrix C only. These algorithms are based on the displacement of the imaginary eigenvalues of the related Hamiltonian matrix. The methods in [50] and [48] can deal with general systems (i.e., when E is ...

Robust port-Hamiltonian representations of passive systems

“…This freedom, which results from particular choices of solutions to the KYP-LMI as well as subsequent state space transformations, may be used to make the representation more robust to perturbations. In many ways the pH representation seems to be the most robust representation [17,18] and it also has many other advantages: it encodes the geometric and algebraic properties directly in the properties of the coefficients [25]; it allows easy ways for structure preserving model reduction [12,21]; it easily extends to descriptor systems [2,24]; and it greatly simplifies optimization methods for computing stability and passivity radii [8,9,10,20].…”

“…A natural measure for optimality is a large passivity radius ρ M , which is the smallest perturbation (in an appropriate norm) to the coefficients of a model M that makes the system non-passive. Computational methods to determine ρ M were introduced in [20], while the converse question, what is the nearest passive system to a non-passive system has recently been discussed in [8,10]. Once we have determined a solution X ∈ X > to the LMI (4), we can determine the representation (12) as in Section 2.3 and the system is automatically passive (but not necessarily strictly passive).…”

Minimal-norm static feedbacks using dissipative Hamiltonian matrices

“…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.…”

“…A matrix A is stable if and only if it is a DH matrix. This parametrization has been used to solve several nearness problems for LTI systems [17,10,12,9]. In this paper, we provide a complete characterization for the static-state stabilizing feedbacks of a given pair (A, B) in terms of DH matrices.…”

Structured eigenvalue/eigenvector backward errors of matrix pencils arising in optimal control

“…It can be interpreted as the Frobenius norm of the smallest perturbation that makes (λ, x) being an eigenpair of the perturbed pencil. Minimizing this expression over all (λ, x) ∈ (iR) × (C 2n+m ) we obtain the distance of L(z) to the next pencil having eigenvalues on the imaginary axis and thus, the passivity radius of L(z), see [13,22]. If the even structure of the pencil is taken into account, then a structured eigenpair backward error with respect to structure-preserving perturbations can be defined as η even (L, λ, x) = inf [∆ M ∆ N ] F ∆ M ∈ Herm(2n + m), ∆ N ∈ SHerm(2n + m)…”