Stability robustness of perturbed structured systems (original) (raw)

On structured realizability and stabilizability of linear systems

2013 American Control Conference, 2013

We study the notion of structured realizability for linear systems defined over graphs. A stabilizable and detectable realization is structured if the state-space matrices inherit the sparsity pattern of the adjacency matrix of the associated graph. In this paper, we demonstrate that not every structured transfer matrix has a structured realization and we reveal the practical meaning of this fact. We also uncover a close connection between the structured realizability of a plant and whether the plant can be stabilized by a structured controller. In particular, we show that a structured stabilizing controller can only exist when the plant admits a structured realization. Finally, we give a parameterization of all structured stabilizing controllers and show that they always have structured realizations.

On Structured Perturbations for Two Classes of Linear Infinite-Dimensional Systems

This paper considers two classes of infinite-dimensional systems described by an abstract differential equation 2(0 = (A + BAC)x(t), x(O) = xo, on a Hilbert space, where A, B, C are linear, possibly unbounded operators and A is an unknown, linear, bounded perturbation. The two classes of systems are defined in terms of properties imposed on the triple {A, B, C}. It is proved that for every A the perturbed system {A + EAF, B, C1 inherits all the properties of the unperturbed system {A, B, C} if {A, E, F} and {A, B, C} are in the same class.

Robust stability of patterned linear systems

Electronic Journal of Differential Equations

For a Hurwitz stable matrix A∈ℝ n×n , we calculate the real structured radius of stability for A with a perturbation P=BΔ(t)C, where A,B,C,Δ(t) form a patterned quadruple of matrices; i.e., they are polynomials of a common matrix of simple structure M∈ℝ n×n .

Generic properties and control of linear structured systems: a survey

Automatica, 2003

In this survey paper, we consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A; B; C and D, is either a ÿxed zero or a free parameter. The location of the ÿxed zeros in these matrices constitutes the structure of the system. Indeed a lot of man-made physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly, a lot of classical properties of linear systems can be studied in terms of genericity. Moreover, these generic properties can, in general, be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system-speciÿc properties like controllability, the ÿnite and inÿnite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input-output decoupling, and so on. In this paper, we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool. We emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only. Moreover, the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well-known polynomially bounded combinatorial techniques for larger systems. ?

On the Design of Structured Stabilizers for LTI Systems

IEEE Control Systems Letters, 2019

Designing a static state-feedback controller subject to structural constraint achieving asymptotic stability is a relevant problem with many applications, including network decentralized control, coordinated control, and sparse feedback design. Leveraging on the Projection Lemma, this work presents a new solution to a class of state-feedback control problems, in which the controller is constrained to belong to a given linear space. We show through extensive discussion and numerical examples that our approach leads to several advantages with respect to existing methods: first, it is computationally efficient; second, it is less conservative than previous methods, since it relaxes the requirement of restricting the Lyapunov matrix to a block-diagonal form.

Robust structured control design via LMI optimization

IFAC Proceedings Volumes (IFAC-PapersOnline), 2011

This paper presents a new procedure for discrete-time robust structured control design. Parameter-dependent nonconvex conditions for stabilizable and induced L 2-norm performance controllers are solved by an iterative linear matrix inequalities (LMI) optimization. A wide class of controller structures including decentralized of any order, fixed-order dynamic output feedback, static output feedback can be designed robust to polytopic uncertainties. Stability is proven by a parameter-dependent Lyapunov function. Numerical examples on robust stability margins shows that the proposed procedure can obtain less conservative results than traditional stability criteria.

Linear Perturbation Theory for Structured Matrix Pencils Arising in Control Theory

SIAM Journal on Matrix Analysis and Applications, 2006

We investigate the effect of linear perturbations on several structured matrix pencils arising in control theory. These include skew-symmetric/symmetric pencils arising in the computation of optimal H∞ control and linear quadratic control for continuous and discrete time systems.

Eigenvalue characterization of some structured matrix pencils under linear perturbation

˜The œelectronic journal of linear algebra, 2024

We study the effect of linear perturbations on three families of matrix pencils. The matrix pairs of the first two families are Hermitian/skew-Hermitian with special 3 × 3 block cases appeared in continuous-time control, and the matrix pairs of the third family are special 3 × 3 non-Hermitian block matrices appeared in discrete-time control. For the first family of matrix pencils and more general cases of the second family of matrix pencils, based on the properties of the involved matrices, we obtain some upper or lower bounds on the set of eigenvalues of linearly perturbed matrix pencils which are on the imaginary axis. Studying a special 3 × 3 block matrix pencil, which is associated with continuous-time control, leads us to some linear perturbation that do not preserve (properly) the structure of the matrices. This, in turn, leads to a numerical technique for finding the nearest Hermitian/skew-Hermitian matrix pencil which can satisfy conditions such that, for some nonzero real perturbation parameter, some or all of its eigenvalues lie on the imaginary axis. We also study the linearly perturbed matrix pencils, associated with discrete-time control, using an one-to-one equivalence between the matrix pencil of continuous-time problem and the matrix pencil of discrete-time problem.

Systems with structured uncertainty: relations between quadratic and robust stability

IEEE Transactions on Automatic Control, 1993

The purpose of this note is to investigate the relation between the notions of robust stability and quadratic stability for uncertain systems with structured uncertainty due to both real and complex parameter variations. We present examples which demonstrate that for systems containing at least two uncertain blocks, the notions of robust stability for complex parameter variations and quadratic stability for real parameter variations are not equivalent; in fact neither implies the other. A byproduct of these examples is that, for this class of systems, quadratic stability for real perturbations need not imply quadratic stability for complex perturbations. This is in stark contrast with the situation in the case of unstructured uncertainty, for which it is known that quadratic stability for either real or complex perturbations is equivalent to robust stability for complex perturbations, and thus equivalent to a small gain condition on the transfer matrix that the perturbation experiences.

Robust stability of linear systems described by higher-order dynamic equations

IEEE Transactions on Automatic Control, 2000

In this note we study the stability radius of higher order di erential and difference systems with respect to various classes of complex a ne perturbations of the coe cient matrices. Di erent perturbation norms are considered. The aim is to derive robustness criteria which are expressed directly in terms of the original data. Previous results on robust stability of Hurwitz and Schur polynomials 13] are extended to monic matrix polynomials. For disturbances acting via a uniform input matrix, computable formulae are obtained whereas for perturbations with multiple input matrices structured singular values are involved.