ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws

A family of time-varying hyperbolic systems of balance laws is considered. The partial differential equations of this family can be stabilized by selecting suitable boundary conditions. For the stabilized systems, the classical technique of construction of Lyapunov functions provides a function which is a weak Lyapunov function in some cases, but is not in others. We transform this function through a strictification approach to obtain a time-varying strict Lyapunov function. It allows us to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of input-to-state stability (ISS) type. Two examples illustrate the results.     

Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters

This paper is concerned with the problem of robustly stochastically exponential stability and stabilization for a class of distributed parameter systems described by uncertain linear first-order hyperbolic partial differential equations (FOHPDEs) with Markov jumping parameters, for which the manipulated input is distributed in space. Based on an integral-type stochastic Lyapunov functional (ISLF), the sufficient condition of robustly stochastically exponential stability with a given decay rate is first derived in terms of spatial differential linear matrix inequalities (SDLMIs). Then, an SDLMI approach to the design of robust stabilizing controllers via state feedback is developed from the resulting stability condition. Furthermore, using the finite difference method and the standard linear matrix inequality (LMI) optimization techniques, recursive LMI algorithms for solving the SDLMIs in the analysis and synthesis are provided. Finally, a simulation example is given to demonstrate the effectiveness of the developed design method.     

Lyapunov stability tests for linear time-delay systems

An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proofs are presented in details for the case of systems with multiple delays. The state of the art, ongoing research and potential extensions to other classes of delay systems are discussed.     

Robust exponential stability of time-varying linear systems

This paper investigates the robustness of time-varying linear systems under a large class of complex time-varying perturbations. Previous results⁸ which were restricted to bounded linear perturbations of output feedback type are generalized to unbounded and nonlinear perturbations of multi-output feedback type. We establish a lower bound for the stability radius of these systems and show how it may be possible to improve the bound using time-varying scalar transformations of the state, input and output variables. The results are applied to derive Gershgorin type stability criteria for time-varying linear systems.     

On exponential stability of linear networked control systems

This paper addresses exponential stability of linear networked control systems. More specifically, the paper considers a continuous‐time linear plant in feedback with a linear sampled‐data controller with an unknown time varying sampling rate, the possibility of data packet dropout, and an uncertain time varying delay. The main contribution of this paper is the derivation of new sufficient stability conditions for linear networked control systems taking into account all of these factors. The stability conditions are based on a modified Lyapunov–Krasovskii functional. The stability results are also applied to the case where limited information on the delay bounds is available. The case of linear sampled‐data systems is studied as a corollary of the networked control case. Furthermore, the paper also formulates the problem of finding a lower bound on the maximum network‐induced delay that preserves exponential stability as a convex optimization program in terms of linear matrix inequalities. This problem can be solved efficiently from both practical and theoretical points of view. Finally, as a comparison, we show that the stability conditions proposed in this paper compare favorably with the ones available in the open literature for different benchmark problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Uniform exponential stability of linear time-varying systems: revisited

Some classical results known in the adaptive control literature are often used as analysis tools for nonlinear systems by evaluating the nonlinear differential equations along trajectories. While this technique is widely used, as we remark through examples, one must take special care in the consideration of the initial conditions in order to conclude uniform convergence. One way of taking care explicitly of the initial conditions is to study parameterized linear time-varying systems. This paper re-establishes known results for linear time-varying systems via new techniques while stressing the importance of imposing that the formulated sufficient and necessary conditions must hold uniformly in the parameter. Our proofs are based on modern tools which can be interpreted as an “integral” version of Lyapunov theorems; rather than on the concept of uniform complete observability which is most common in the literature.     

Piecewise Lyapunov functions for robust stability of linear time-varying systems

In this paper, we investigate the use of two-term piecewise quadratic Lyapunov functions for robust stability of linear time-varying systems. By using the so-called S-procedure and a special variable reduction method, we provide numerically efficient conditions for the robust asymptotic stability of the linear time-varying systems involving the convex combinations of two matrices. An example is included to demonstrate the usefulness of our results.     

Analysis of practical stability and Lyapunov stability of linear time-varying neutral delay systems

Practical stability guarantees trajectories of a dynamical system being bounded within a prespecified region during a specified time interval and is of great interest in many applications. For a class of linear time-varying systems described by delay differential equations of neutral type, concepts of practical stability involving both variations of the state and its derivative are introduced in terms of given estimate sets. Sufficient conditions of practical stability are established on the basis of the mixed differential difference comparison principle presented in this paper, in terms of coefficients of the systems and the given allowable trajectory bounds. For the case of time-invariant estimate sets, these conditions are also independent of delay. These results are then applied to the Lyapunov stability and positively invariant tubes of the corresponding homogeneous systems. Specifically, an algebraic condition of globally exponentially asymptotical stability is derived, which is related to the well known property of the M-matrix and is independent of delay. Finally, an illustrative example is given.     

Diagonally invariant exponential stability and stabilizability of switching linear systems

The paper proposes analysis and design techniques for switching linear systems (whose commutations occur in an arbitrary manner from the internal dynamics point of view, being determined by exogenous agents). We define and characterize (by “if and only if” conditions) two properties, namely (i) diagonally invariant exponential stability and (ii) diagonally invariant exponential stabilizability. Both properties rely on the existence of contractive invariant sets described by Hölder p  -norms, 1≤p≤∞$1\le p\le \infty$, and imply the standard concepts of “exponential stability” and “exponential stabilizability”, respectively (whereas the counter-parts are, in general, not true). We prove that properties (i), (ii) are equivalent to a set of inequalities written for the matrix measure (associated with the p-norm) applied to the matrices of the open-loop system (property (i)), and, respectively, to the matrices of the closed-loop system (property (ii)). We also develop computational instruments for testing the properties (i), (ii) in the cases of the usual p  -norms with p∈{1,2,∞}$p\in \{\mathrm{1,2},\infty \}$. These instruments represent computable necessary and sufficient conditions for the existence of the properties (i), (ii), and whenever the property (ii) exists, a suitable state-feedback matrix is provided. Two numerical examples are presented in order to illustrate the exploration of properties (i), (ii), as well as the use of software resources available on a powerful environment (such as MATLAB).     

Preservation of exponential stability for linear non-autonomous functional differential systems

We consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation. As particular cases, the system includes models with concentrated delays and systems of integrodifferential equations. Our method is based on Bohl–Perron type theorems.     

Analysis and stabilization for networked linear hyperbolic systems of rationally dependent conservation laws

In this paper, for a networked linear hyperbolic partial differential equations (PDEs) system of conservation laws, the propagation periods of which are rationally dependent, with coupled boundary conditions, we propose a novel approach to analyze its controllability and observability. In addition, we propose a design method of a stabilizing controller, where a boundary-input with boundary-valued feedback is considered. First, we characterize the control properties, such as controllability of such a PDE system, in terms of the corresponding ones of a finite-dimensional discrete-time system defined on the boundaries of the PDE system, which is derived by fully exploiting the method of characteristics. Since the obtained discrete-time system is low-dimensional, its analysis is relatively easier. Next, we propose a design method of a stabilizing controller based on this discrete-time system. Finally, numerical simulations are presented to show that the proposed method is effective.     

Asymptotical stability of 2-D linear discrete stochastic systems

The main goal of the present paper is to find computable stability criteria for two-dimensional stochastic systems based on Kronecker product and nonnegative matrices theory. First, 2-D discrete stochastic system model is established by extending system matrices of the well-known Fornasini–Marchesini?s second model into stochastic matrices. The elements of these stochastic matrices are second-order, weakly stationary white-noise sequences. Second, a necessary and sufficient condition for 2-D stochastic systems is presented, this is the first time that has been proposed. Third, computable mean-square asymptotic stability criteria are derived via Kronecker product and the nonnegative matrix theory. The criteria are only sufficient conditions. Finally, illustrative examples are provided.     

On exponential stability for linear discrete-time systems in Banach spaces

In this paper we investigate four concepts of exponential stability for difference equations in Banach spaces. Characterizations of these concepts are given. They can be considered as variants for the discrete-time case of the classical results due to Barbashin [6] and Datko [5]. An illustrative example clarifies the relations between these concepts.     

Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems

This paper proposes new sufficient conditions for the exponential stability and stabilization.of linear uncertain polytopic time-delay systems. The conditions for exponential stability are expressed in terms of Kharitonov-type linear matrix inequalities （LMIs） and we develop control design methods based on LMIs for solving stabilization problem. Our method consists of a combination of the LMI approach and the use of parameter-dependent Lyapunov functionals, which allows to compute simultaneously the two bounds that characterize the exponetial stability rate of the solution. Numerical examples illustrating the conditions are given.     

A necessary condition for quantitative exponential stability of linear state-space systems

For an arbitrary n×n constant matrix A the two following facts are well known:

• (1/n)Re(traceA)−max_j=1,…,nRe λ_j(A)0;

• If U is a unitary matrix, one can always find a skew-Hermitian matrix A so that U=e^A.

In this note we present the extension of these two facts to the context of linear time-varying dynamical systemsAs a by-product, this result suggests that, the notion of “slowly varying state-space systems”, commonly used in literature, is mathematically not natural to the problem of exponential stability.     

Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems

This paper proposes new sufficient conditions for the exponential stability and stabilization of linear uncertain polytopic time-delay systems. The conditions for exponential stability are expressed in terms of Kharitonov-type linear matrix inequalities (LMIs) and we develop control design methods based on LMIs for solving stabilization problem. Our method consists of a combination of the LMI approach and the use of parameter-dependent Lyapunov functionals, which allows to compute simultaneously the two bounds that characterize the exponetial stability rate of the solution. Numerical examples illustrating the conditions are given.     

Decomposition of 2-D linear systems into 1-D systems in thefrequency domain

A method is presented for the decomposition of the frequency domain of 2-D linear systems into two equivalent 1-D systems having dynamics in different directions and connected by a feedback system. It is shown that under some assumptions the decomposition problem can be reduced to finding a realizable solution to the matrix polynomial equation X(z₁)P(z₂ )+Q(z₁)Y(z₂ )=D(z₁, z₂). A procedure for finding a realizable solution X(z₁ ), Y(z₂) to the equation is given     

An analysis of the exponential stability of linear stochastic neutral delay systems

This paper is concerned with the analysis of the mean square exponential stability and the almost sure exponential stability of linear stochastic neutral delay systems. A general stability result on the mean square and almost sure exponential stability of such systems is established. Based on this stability result, the delay partitioning technique is adopted to obtain a delay‐dependent stability condition in terms of linear matrix inequalities (LMIs). In obtaining these LMIs, some basic rules of the Ito calculus are also utilized to introduce slack matrices so as to further reduce conservatism. Some numerical examples borrowed from the literature are used to show that, as the number of the partitioning intervals increases, the allowable delay determined by the proposed LMI condition approaches h_max, the maximal allowable delay for the stability of the considered system, indicating the effectiveness of the proposed stability analysis. Copyright © 2013 John Wiley & Sons, Ltd.