Stability radii of infinite-dimensional systems subjected to unbounded stochastic perturbations

In this paper, we use the framework of stability radii to study the robust stability of linear deterministic systems on real Hilbert spaces which are subjected to unbounded stochastic perturbations. First, we establish an existence and uniqueness theorem of the solution of the abstract equation describing the system. Then we characterize the stability radius in terms of a Lyapunov equation or equivalently in terms of the norm of an input-output operator.     

Stability radii of higher order positive difference systems

In this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples.     

Stability of switching infinite-dimensional systems

In this note, we generalize the results from Narendra and Balakrishnan (IEEE Trans. Automatic Control 39 (1994) 2469) to the infinite-dimensional system theoretic setting. The paper gives results on the stability of a switching system of the form , i∈{1,2}, when the infinitesimal generators A₁ and A₂ commute. In addition, the existence of a common quadratic Lyapunov function is demonstrated.     

New characterization of positive realness and control of a class of uncertain polytopic discrete-time systems

This paper deals with the problems of positive real analysis and control synthesis for a class of discrete-time polytopic systems with uncertainties. The systems under consideration are modelled in a polytopic form with linear fractional uncertainties in its vertices. A new linear matrix inequality (LMI) characterization of positive realness for this class of systems is given. It enables one to check the positive realness by using parameter-dependent Lyapunov function. This new characterization exhibits a kind of decoupling between the Lyapunov matrix and the system matrices, which is subsequently exploited for control design. Based on the new result, sufficient conditions with reduced conservatism are obtained. A numerical example is also included to demonstrate the applicability of the proposed results.     

-output feedback of infinite-dimensional systems via approximation

As in the finite-dimensional case, a state-space based controller for the infinite-dimensional disturbance-attenuation problem may be calculated by solving two Riccati equations. These operator Riccati equations can rarely be solved exactly. We approximate the original infinite-dimensional system by a sequence of finite-dimensional systems. The solutions to the corresponding finite-dimensional Riccati equations are shown to converge to the solution of the infinite-dimensional Riccati equations. Furthermore, the corresponding finite-dimensional controllers yield performance arbitrarily close to that obtained with the infinite-dimensional controller.     

Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S^−1/2B^*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H_∞ for positive real transfer functions of the form D+S^−1/2B^*(author−A)⁻¹,B.     

Stability of positive linear switched systems on ordered Banach spaces

We provide sufficient criteria for the stability of positive linear switched systems on ordered Banach spaces. The switched systems can be generated by finitely many bounded operators in infinite-dimensional spaces with a general class of order-inducing cones. In the discrete-time case, we assume an appropriate interior point of the cone, whereas in the continuous-time case an appropriate interior point of the dual cone is sufficient for stability. This is an extension of the concept of linear Lyapunov functions for positive systems to the setting of infinite-dimensional partially ordered spaces. We illustrate our results with examples.     

Real stability radii of linear time-invariant time-delay systems

In this paper, the real stability radii for perturbed linear differential equations of the retarded and neutral type are investigated. We present the corresponding readily computable formulae with respect to an arbitrary stability region in the complex plane. The results are further extended to the case of linear discrete time-delay systems.     

Monomial reachability and zero controllability of discrete-time positive switched systems

In this paper, monomial reachability and zero controllability properties of discrete-time positive switched systems are investigated. Necessary and sufficient conditions for these properties to hold, together with some interesting examples and some testing algorithms, are provided.     

Stability radii of positive linear systems under fractional perturbations

In this paper we study stability radii of positive linear discrete‐time systems under fractional perturbations. It is shown that real and complex stability radii coincide and can be computed by a simple formula. From the obtained results, we apply to derive estimates and computable formulae for the stability radii of positive linear delay systems. Finally, a simple example is given to illustrate the obtained results. Copyright © 2008 John Wiley & Sons, Ltd.     

Open-loop stabilizability of infinite-dimensional systems

In this paper we study open-loop stabilizability, a general notion of stabilizability for linear differential equations =Ax+Bu in an infinite-dimensional state space. This notion is sufficiently general to be implied by exact controllability, by optimizability, and by various general definitions of closedloop stabilizability. Here,A is the generator of a strongly continuous semigroup, and we make very few a priori restrictions on the class of controlsu. Our results hinge upon the control operatorB being smoothly left-invertible, which is a very mild restriction when the input space is finite-dimensional. Since open-loop stabilizability is a weak concept, lack of open-loop stability is quites strong. A focus of this paper is to give necessary conditions for open-loop stabilizability, thus identifying classes of systems which are not open-loops stabilizable. First we give useful frequency domain conditions that are equivalent to our definitions of open-loop stabilizability, and lead to a version of the Hautus test for open-loop stabilizability. When the input space is finite-dimensional, we give necessary conditions for open-loop stabilizability which involve spectral properties ofA. We show that these results are not true if the conditions onB are weakened. We obtain analogous results for discrete-time systems. We show that, for a class of systems without spectrum determined growth, optimizability is impossible. Finally, we show that a system is open-loop stabilizable with a class of controlu if and only if the system with the sameA but a more boundedB is open-loop stabilizable with a larger class of controls. This work was partially supported by NSF Grant DMS-9623392.     

Practical output regulation for bounded linear infinite-dimensional state space systems

We develop the mathematical foundations of practical state space output regulation for bounded infinite-dimensional linear systems. By practical output regulation we mean asymptotic tracking of references and rejection of disturbances with a given accuracy. Our main results are general upper bounds for the norms perturbations to the parameters of the exosystem, the plant and a controller which achieves exact output regulation. These bounds depend explicitly on the desired tracking accuracy ε>0. In this paper, all perturbations are assumed to be bounded, additive and linear. Our results apply for both feedforward and error feedback controllers, and for arbitrary bounded uniformly continuous reference/disturbance signals.     

Stability conditions for a class of time-varying discrete-time systems

This paper deals with the stability conditions for a class of dynamical discrete-time systems, called ‘P-invariants’ and ‘not P-invariants’. Stability conditions are given in terms of polyhedral convex cones and are obtained by using some extensions on M-matrices.     

Stability radii of differential algebraic equations with structured perturbations

This paper deals with a formula for computing stability radii of a differential algebraic equation of the form AX^′(t)−BX(t)=0, where A,B are constant matrices. A computable formula for the complex stability radius is given and a key difference between the ordinary differential equation (ODEs for short) and the differential algebraic equation (DAEs for short) is pointed out. A special case where the real stability radius and the complex one are equal is considered.     

On delay-dependent stability conditions for time-varying systems

In this paper, some recent results on additional dynamics for transformed time-delay systems are extended to the case of time-varying systems. Special equations which describe these dynamics are derived. Additional restrictions on stability and robust stability imposed by the transformations are obtained.     

Remarks on -input bounded-state stability of linear controllable infinite-dimensional systems

It is shown that every eP-input bounded-state stable linear (infinite-dimensional) system x_k+1=A_kx_k+B_ku_k is uniformly power equistable, if it is uniformly equicontrollable.     

On the positive invariance of polyhedral sets for discrete-time systems

In this note, necessary and sufficient conditions for a polyhedral set to be a positively invariant set of a linear discrete-time system are established.     

Feedback invariance of SISO infinite-dimensional systems

We consider a linear single-input single-output system on a Hilbert space X, with infinitesimal generator A, bounded control element b, and bounded observation element c. We address the problem of finding the largest feedback invariant subspace of X that is in the space c ^⊥ perpendicular to c. If b is not in c ^⊥, we show this subspace is c ^⊥. If b is in c ^⊥, a number of situations may occur, depending on the relationship between b and c.