Stability analysis of linear time-varying systems: Improving conditions by adding more information about parameter variation

In this paper a new Lyapunov function is proposed for stability analysis of linear time-varying systems. This new function carries more information regarding parameter variation leading to less conservative conditions. Using Finsler’s lemma and a suitable form to describe the high-order time-derivatives of the parameters, finite sets of LMIs are obtained which are progressively less conservative as a pair of parameters grow. Previous results can be seen as a special case and numerical examples are carried out for the sake of illustration.     

Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties

In this paper, we consider discrete time systems with polytopic time varying uncertainty. We look for a class of parameter dependent Lyapunov functions which are quadratic on the system state and depend in a polytopic way on the uncertain parameter. We show that extending the new discrete time stability condition proposed by de Oliveira et al. (Systems Control Lett. 36 (1999) 135.) to the case of time varying uncertainty leads to a necessary and sufficient condition for the computation of such a Lyapunov function. This allows to check asymptotic stability of the system under study. The obtained linear matrix inequalities condition can also be used to cope with the control synthesis problem.     

Robust static output feedback control for linear discrete-time systems with time-varying uncertainties

This paper is concerned with the problem of designing robust static output feedback controllers for linear discrete-time systems with time-varying polytopic uncertainties. Sufficient conditions for robust static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities, and the results are extended to H₂ and H_∞ static output feedback controller designs. Numerical examples are given to illustrate the effectiveness of the proposed design methods.     

Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators

This paper deals with the stabilization of a class of nonlinear discrete-time systems under control saturations including time-varying parameter dependency. The control law studied consists of the gain-scheduled feedback of the measured output and of the nonlinearity present in the dynamics of the controlled system. Furthermore, the saturations are taken into account by modeling the nonlinear saturated system through a dead-zone nonlinearity satisfying a modified sector condition. Thus, as for precisely known systems, linear matrix inequality (LMI) stabilization conditions are proposed for such a generic system. These conditions can be cast into convex programming problems for design purposes. An illustrative example stresses the efficiency of the main result.     

Robust stabilisation of discrete-time time-varying linear systems with Markovian switching and nonlinear parametric uncertainties

In this paper, the problem of the robustness of the stability of a discrete-time linear stochastic system is addressed. The nominal plant is described by a discrete-time time-varying linear system subject to random jumping according with a non-homogeneous Markov chain with a finite number of states. The class of admissible uncertainties consists of multiplicative white noise type perturbations with unknown intensity. It is assumed that the intensity of white noise type perturbations is modelled by unknown nonlinear functions subject to linear growth conditions. The class of admissible controls consists of stabilising state feedback control laws. We show that the best robustness performance is achieved by the stability provided by a state feedback design based on the stabilising solution of a suitable discrete-time Riccati-type equation.     

On the conservatism of the sum-of-squares method for analysis of time-delayed systems

In this paper, we present a converse Lyapunov result which shows that using polynomial Lyapunov functionals to prove the stability of linear time-delay systems is not conservative. This result motivates the sum-of-squares approach to stability analysis of linear time-delay systems. This main result is based on an extension of the Weierstrass approximation theorem to show that linear varieties of polynomials can be used to approximate linear varieties of the space of continuous functions.     

Robust H-infinity filtering for time-delay systems with missing measurements： a parameter-dependent approach

Robust H-infinity filtering for a class of uncertain discrete-time linear systems with time delays and missing measurements is studied in this paper. The uncertain parameters are supposed to reside in a convex polytope and the missing measurements are described by a binary switching sequence satisfying a Bernoulli distribution. Our attention is focused on the analysis and design of robust H-infinity filters such that, for all admissible parameter uncertainties and all possible missing measurements, the filtering error system is exponentially mean-square stable with a prescribed H-infinity disturbance attenuation level. A parameter-dependent approach is proposed to derive a less conservative result. Sufficient conditions are established for the existence of the desired filter in terms of certain linear matrix inequalities （LMIs）. When these LMIs are feasible, an explicit expression of the desired filter is also provided. Finally, a numerical example is presented to illustrate the effectiveness and applicability of the proposed method.     

Nonlinear control design for linear differential inclusions via convex hull of quadratics

This paper presents a nonlinear control design method for robust stabilization and robust performance of linear differential inclusions (LDIs). A recently introduced non-quadratic Lyapunov function, the convex hull of quadratics, will be used for the construction of nonlinear state feedback laws. Design objectives include stabilization with maximal convergence rate, disturbance rejection with minimal reachable set and least L₂ gain. Conditions for stabilization and performances are derived in terms of bilinear matrix inequalities (BMIs), which cover the existing linear matrix inequality (LMI) conditions as special cases. Numerical examples demonstrate the advantages of using nonlinear feedback control over linear feedback control for LDIs. It is also observed through numerical computation that nonlinear control strategies help to reduce control effort substantially.     

A less conservative LMI condition for the robust stability of discrete-time uncertain systems

In this paper, a less conservative condition for the robust stability of uncertain discrete-time linear systems is proposed. The uncertain parameters, assumed to be time-invariant, are supposed to belong to convex bounded domains (polytope type uncertainty). The stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain. A simple and numerically efficient feasibility test provides a set of Lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain. Examples illustrate the results.     

Interval-model-based global optimization framework for robust stability and performance of PID controllers

PID controller structure is regarded as a standard in the control-engineering community and is supported by a vast range of automation hardware. Therefore, PID controllers are widely used in industrial practice. However, the problem of tuning the controller parameters has to be tackled by the control engineer and this is often not dealt with in an optimal way, resulting in poor control performance and even compromised safety. The paper proposes a framework, which involves using an interval model for describing the uncertain or variable dynamics of the process. The framework employs a particle swarm optimization algorithm for obtaining the best performing PID controller with regard to several possible criteria, but at the same time taking into account the complementary sensitivity function constraints, which ensure robustness within the bounds of the uncertain parameters’ intervals. Hence, the presented approach enables a simple, computationally tractable and efficient constrained optimization solution for tuning the parameters of the controller, while considering the eventual gain, pole, zero and time-delay uncertainties defined using an interval model of the controlled process. The results provide good control performance while assuring stability within the prescribed uncertainty constraints. Furthermore, the controller performance is adequate only if the relative system perturbations are considered, as proposed in the paper. The proposed approach has been tested on various examples. The results suggest that it is a useful framework for obtaining adequate controller parameters, which ensure robust stability and favorable control performance of the closed-loop, even when considerable process uncertainties are expected.     

多区域时滞电力系统稳定判据的优化

在大规模互联电力系统进行全网动态分析与控制的过程中,通信时滞的存在易导致动力系统产生控制设备失效、系统性能恶化及失稳等现象.而现有时滞系统稳定判据的研究多基于Lyapunov分析方法,研究过程中需采用不等式放缩及构造Lyapunov泛函,使得稳定判据具有较高保守性,增加了系统的控制成本.针对这一问题,本文以多区域时滞电力系统为研究对象,通过引入Wirtinger不等式优化稳定判据推导过程和构造新的Lyapunov泛函两种方式,降低系统稳定判据的保守性.最后,利用典型的二阶时滞系统和两区域负荷频率控制系统,验证了本文算法的正确性和有效性.     

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.     

Explicit construction of quadratic lyapunov functions for the small gain,positivity, circle,and popov theorems and their application to robust stability. part II: Discrete-time theory

In a companion paper (‘Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. Part I: Continuous-time theory’), Lyapunov functions were constructed in a unified framework to prove sufficiency in the small gain, positivity, circle, and Popov theorems. In this Part II, analogous results are developed for the discrete-time case. As in the continuous-time case, each result is based upon a suitable Riccati-like matrix equation that is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Multivariable versions of the discrete-time circle and Popov criteria are obtained as extensions of known results. Each result is specialized to the linear uncertainty case and connections with robust stability for state-space systems is explored.     

Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions

In this article, the fuzzy Lyapunov function approach is considered for stabilising continuous-time Takagi-Sugeno fuzzy systems. Previous linear matrix inequality (LMI) stability conditions are relaxed by exploring further the properties of the time derivatives of premise membership functions and by introducing slack LMI variables into the problem formulation. The relaxation conditions given can also be used with a class of fuzzy Lyapunov functions which also depends on the membership function first-order time-derivative. The stability results are thus extended to systems with large number of rules under membership function order relations and used to design parallel-distributed compensation (PDC) fuzzy controllers which are also solved in terms of LMIs. Numerical examples illustrate the efficiency of the new stabilising conditions presented.     

A formal framework for connective stability of highly decentralized cooperative negotiations

Multiagent cooperative negotiation is a promising technique for modeling and controlling complex systems. Effective and flexible cooperative negotiations are especially useful for open complex systems characterized by high decentralization (which implies a low amount of exchanged information) and by dynamic connection and disconnection of agents. Applications include ad hoc network management, vehicle formation, and physiological model combination. To obtain an effective control action, the stability of the negotiation, namely the guarantee that an agreement will be eventually reached, is of paramount importance. However, the techniques usually employed for assessing the stability of a negotiation can be hardly applied in open scenarios. In this paper, whose nature is mainly theoretical, we make a first attempt towards engineering stable cooperative negotiations proposing a framework for their analysis and design. Specifically, we present a formal protocol for cooperative negotiations between a number of agents and we propose a criterion for negotiation stability based on the concept of connective stability. This is a form of stability that accounts for the effects of structural changes on the composition of a system and that appears very suitable for multiagent cooperative negotiations. To show its possible uses, we apply our framework for connective stability to some negotiations taken from literature.     

Stability of a vector integer quadratic programming problem with respect to vector criterion and constraints

The paper analyzes five types of stability against perturbations of initial data in criterion functions and constraints of vector integer quadratic optimization problems. Necessary and sufficient conditions are proved for all the types of stability. The relationship among stability with respect to changes of vector criterion coefficients, stability with respect to changes of initial data in constraints, and stability with respect to vector criterion and constraints is established. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 63–72, September–October 2006.     

Delay‐dependent LMI condition for global asymptotic stability of discrete‐time uncertain state‐delayed systems using quantization/overflow nonlinearities

A delay‐dependent criterion for the global asymptotic stability of a class of uncertain discrete‐time state‐delayed systems using various combinations of quantization and overflow nonlinearities is presented. The proposed criterion is in the form of a linear matrix inequality and, hence, computationally tractable. A numerical example highlighting the usefulness of the proposed criterion is given. Copyright © 2010 John Wiley & Sons, Ltd.     

Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

This paper presents new extended linear matrix inequality (LMI) characterizations for the synthesis of robust and state feedback controllers for continuous‐time linear time‐invariant systems with polytopic uncertainty. Based on a suitable change of variables and the Elimination Lemma, the proposed robust control design techniques are stated as extended LMI conditions parameterized in terms of 2 scalar parameters. One parameter is shown to belong to a bounded domain, thus limiting the scalar search domain. For the other parameter, a bounded subset is provided from numerical experiments. The benefits of the methodology are illustrated through numerical simulations performed on an uncertain model borrowed from the literature. It is shown that the proposed LMI relaxations can provide less conservative results with fewer scalar searches than some existing methods in the literature.     

Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.     

A novel methodology to solve the CQLF problem for a finite set of stable second-order systems

A new methodology to provide conclusive information about the existence/non-existence of a common quadratic Lyapunov function (CQLF) for a finite set of stable second-order systems is presented. Despite the high complexity of the CQLF problem, even in the case of N second-order systems, the results presented in this paper have a very simple and intuitive theoretical support, including topics such as classical intersection of convex sets and properties of convex linear combinations. Illustrative examples to show the performance of the proposed methodology are provided.