Stability radii of infinite-dimensional positive systems

We show that for infinite-dimensional discrete-time positive systems the complex and real stability radii coincide. Furthermore, we provide a simple formula for the complex stability radius of positive systems by the associated transfer function. We illustrate our results with an example dealing with a simple type of differential-difference equations.The author would like to thank the Deutsche Forschungsgemeinschaft for its support during this work.     

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.     

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.     

On stability of a class of positive linear functional difference equations

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae. Pham Huu Anh Ngoc and Toshiki Naito are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049.     

Stability radii of positive discrete-time systems under affine parameter perturbations

In this paper we study stability radii of positive linear discrete-time systems under affine parameter perturbations. It is shown that real and complex stability radii of positive systems coincide for arbitrary perturbation structures, in particular, for blockdiagonal disturbances as considered in μ-analysis. Estimates and computable formulae are derived for these stability radii. The results are derived for arbitrary perturbation norms induced by monotonic vector norms (e.g. p-norms, 1⩽p⩽∞). © 1998 John Wiley & Sons, Ltd.     

Strong stability radii of positive linear time‐delay systems

In this paper, we study robustness of the strong delay‐independent stability of linear time‐delay systems under multi‐perturbation and affine perturbation of coefficient matrices via the concept of strong delay‐independent stability radius (shortly, strong stability radius). We prove that for class of positive time‐delay systems, complex and real strong stability radii of positive linear time‐delay systems under multi‐perturbations (or affine perturbations) coincide and they are computed via simple formulae. Apart from that, we derive solution of a global optimization problem associated with the problem of computing of the strong stability radii of a positive linear time‐delay system. An example is given to illustrate the obtained results. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust stability of a special class of polynomial matrices with control applications

The robust stability problem of uncertain continuous-time systems described by higher-order dynamic equations is considered in this paper. Previous results on robust stability of Metzlerian matrices are extended to matrix polynomials, with the coefficient matrices having exactly the same Metzlerian structure. After defining the structured uncertainty for this class of polynomial matrices, we provide an explicit expression for the real stability radius and derive simplified formulae for several special cases. We also report on alternative approaches for investigating robust Hurwitz stability and strong stability of polynomial matrices. Several illustrative examples throughout the paper support the theoretical development. Moreover, an application example is included to demonstrate uncertainty modeling and robust stability analysis used in control design.     

A note on the stability of fractional order systems

In this paper, a new approach is suggested to investigate stability in a family of fractional order linear time invariant systems with order between 1 and 2. The proposed method relies on finding a linear ordinary system that possesses the same stability property as the fractional order system. In this way, instead of performing the stability analysis on the fractional order systems, the analysis is converted into the domain of ordinary systems which is well established and well understood. As a useful consequence, we have extended two general tests for robust stability check of ordinary systems to fractional order systems.     

Stability and robust stability of positive Volterra systems

We study positive linear Volterra integro‐differential systems with infinitely many delays. Positivity is characterized in terms of the system entries. A generalized version of the Perron–Frobenius theorem is shown; this may be interesting in its own right but is exploited here for stability results: explicit spectral criteria for L¹‐stability and exponential asymptotic stability. Also, the concept of stability radii, determining the maximal robustness with respect to additive perturbations to L¹‐stable system, is introduced and it is shown that the complex, real and positive stability radii coincide and can be computed by an explicit formula. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability of a general class of difference systems

In this paper, we shall investigate several notions of stability of a general ordinary difference system using the prolific Second Method of Lyapunov.     

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.     

Stability radii of infinite dimensional systems with stochastic uncertainty and their optimization

In this paper we consider infinite dimensional systems which are subjected to stochastic structured multiperturbations. We first characterize the stability radii of these systems in terms of a Lyapunov equation and the corresponding Lyapunov inequalities. Then we investigate the problem of maximizing the stability radius by linear state feedback. We show that the supremal achievable stability radius can be determined via the resolution of a parametrized Riccati equation. Illustrative examples are included. Copyright © 2006 John Wiley & Sons, Ltd.     

On the degree of polynomial parameter-dependent Lyapunov functions for robust stability of single parameter-dependent LTI systems: A counter-example to Barmish's conjecture

In this paper, we consider the robust Hurwitz stability analysis problems of a single parameter-dependent matrix A(θ)?A₀+θA₁ over θ∈[-1,1], where A₀,A₁∈R^n×n with A₀ being Hurwitz stable. In particular, we are interested in the degree N of the polynomial parameter-dependent Lyapunov matrix (PPDLM) of the form that ensures the robust Hurwitz stability of A(θ) via . On the degree of PPDLMs, Barmish conjectured in early 90s that if there exists such P(θ), then there always exists a first-degree PPDLM P(θ)=P₀+θP₁ that meets the desired conditions, regardless of the size or rank of A₀ and A₁. The goal of this paper is to falsify this conjecture. More precisely, we will show a pair of the matrices A₀,A₁∈R^3×3 with A₀+θA₁ being Hurwitz stable for all θ∈[-1,1] and prove rigorously that the desired first-degree PPDLM does not exist for this particular pair. The proof is based on the recently developed techniques to deal with parametrized LMIs in an exact fashion and related duality arguments. From this counter-example, we can conclude that the conjecture posed by Barmish is not valid when n?3 in general.     

Stability robustness of linear normal distributed parameter systems

This paper considers the stability robustness analysis problem for linear distributed parameter systems containing known perturbation operators multiplied by uncertain parameters. The nominal system operators are assumed to be normal, but allowed to be unbounded. The perturbation operators are confined to some relative bounded set, but may be unbounded also. By using the Lyapunov stability criterion, simple bounds on uncertain parameters are derived to ensure the stability of the perturbed systems. Examples are provided to illustrate the usage of the theoretical results.     

Stability and robust stability of positive linear Volterra difference equations

We first introduce a class of positive linear Volterra difference equations. Then, we offer explicit criteria for uniform asymptotic stability of positive equations. Furthermore, we get a new Perron–Frobenius theorem for positive linear Volterra difference equations. Finally, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to these perturbations are given. Copyright © 2008 John Wiley & Sons, Ltd.     

挠性系统的鲁棒控制设计

提出一种新的鲁棒设计思想：将挠性模态部分的Nyquist图线安排到右半平面。由于综合运用了频域分析、极点配置、正实性引理、线性矩阵不等式等概念和算法，使得这种鲁棒控制设计得以实现，且简单直观。通过两个算例说明了该设计方法的有效性。     

Stability margin of uncertain control systems

Achieving stability is the essential issue in the control system design. In this paper, four approaches that can be used to calculate the stability margin of the interval plant family are summarized and compared. The μ approach gives the bounds of the stability margin, and good estimation can be obtained with the numerical method. The eigenvalue approach yields accurate value, and the MATLAB＇s function robuststab sometimes provides wrong results. Since the eigenvalue approach is both accurate and computationally efficient, it is recommended for the calculation of the stability margin, while utilization of the function robuststab should be avoided due to the unreliable results it gives.     

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.     

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.