Lyapunov-Based Robustness Analysis Techniques for Clearance

Stability and state-feedback stabilization of linear systems with uncertain coefficients and uncertain time-varying delays are considered. The system under consideration may be unstable without delay, but it becomes asymptotically stable for positive values of the delay. A new descriptor discretized Lyapunov-Krasovskii functional (LKF) method is introduced, which combines the application of the complete LKF and the discretization method of K. Gu with the descriptor model transformation. For the first time, the new method allows to apply the discretized LKF method to synthesis problems. Moreover, the analysis of systems with polytopic time-invariant uncertainties is less restrictive by the new discretized method. Sufficient conditions for robust stability and stabilization of uncertain neutral type systems are derived in terms of linear matrix inequalities (LMIs) via input-output approach to stability. Numerical examples illustrate the efficiency of the new method.

This technical note addresses robust stability of uncertain systems with rational dependence on unknown time-varying parameters constrained in a polytope. First, the technical note proves that a sufficient linear matrix inequality (LMI) condition that we previously proposed, based on homogeneous polynomial Lyapunov functions (HPLFs) and on the introduction of an extended version of Polya's theorem, is also necessary. Second, the technical note proposes a new sufficient and necessary LMI condition by exploiting properties of the simplex and sum-of-squares (SOS) parameter-dependent polynomials. Lastly, the technical note investigates relationships among these conditions and conditions based on the linear fractional representation (LFR). It is worth remarking that sufficient and necessary LMI conditions for this problem have not been proposed yet in the literature.

This paper presents a design scheme of a minimal order observer-based guaranteed cost controller for uncertain neutral systems, in which some state variables can not be measured. The uncertainties are assumed to be norm-bounded. The initial state is assumed unknown but their mean and covariance are assumed known. A sufficient condition for robust stability analysis and robust stabilization are derived via linear matrix inequalities (LMIs). To show the advantage of the proposed method, a numerical example is given.

: This AASERT program supported research in robust simulation, hierarchical uncertainty representation, and novel methods for robustness analysis of uncertain systems. In the context of this program, robust simulation means simulating simultaneously sets of initial conditions and disturbance or noise signals. Thus sets of state space must be propagated by the dynamics of the model. Initial investigations have focused on piecewise linear discrete time systems, which map polyhedra to polyhedra at each time step. Linear programming can be used to refine the resulting bounds. This is important if the potentially exponential growth in set descriptions is to be overcome. Hierarchical uncertainty modeling is a new framework to include explicit representation of uncertainty in component modeling. The focus has been on LFTs and implicit (DAE) representations. A variety of examples including parasitics and non linearities illustrate the key ideas. Finally, this report describes new bounds on a spherical mu problem that allows for correlations between uncertainties in an LFT framework. Interestingly, this setting provides quite elegant bounds and simplified computation.

A state-space characterization of robustness analysis and synthesis for nonlinear uncertain systems is proposed. The robustness of a class of nonlinear systems subject to L/sub 2/-bounded structured uncertainties is characterized in terms of a nonlinear matrix inequality (NLMI), which yields a convex feasibility problem. As in the linear case, scalings are used to find a Lyapunov or storage function that give sufficient conditions for robust stability and performances. Sufficient conditions for the solvability of robustness synthesis problems are represented in terms of NLMIs as well. With the proposed NLMI characterizations, it is shown that the computation needed for robustness analysis and synthesis is not more difficult than that for checking Lyapunov stability; the numerical solutions for robustness problems are approximated by the use of finite element methods or finite difference schemes, and the computations are reduced to solving linear matrix inequalities. Unfortunately, while the development in this paper parallels the corresponding linear theory, the resulting computational consequences are, of course, not as favourable.

In this note, a new result that extends the scaled small gain theorem is presented. As is well known, the scaled small gain theorem gives necessary and sufficient conditions for robust stability of a nominal linear time-invariant system in feedback with structured operators of norm less than or equal to unity. We propose alternative linear matrix inequality conditions that give necessary and sufficient conditions for robust stability against the class of structured unitary operators (invertible operators of exactly unit norm). It is shown that this result, besides being a less conservative version of the scaled small gain theorem, has connections to several recent results on the control of spatially interconnected systems and serves to unify and quantify the conservatism of those results.

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0<α<2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0<α<1 and 1<α<2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.

The robust stabilization problem of uncertain discrete system with delay and saturating actuators is investigated. A polytopic representation is used to describe the saturation behavior. The sufficient condition of robust stabilization is obtained by using parameter dependent Lyapunov function. Furthermore, the attractive domain is estimated. Finally, numerical example shows the effectiveness of the results.

The robust stability analysis for discrete large-scale uncertain systems with multiple time delays is addressed in this paper. We establish a method for selecting properly a positive definite matrix Q to derive a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach method with this upper bound, several sufficient conditions are presented to guarantee the robust stability of the overall systems. Comparisons between the proposed results with a previous one are also given.

This article concerns the finite-time robust guaranteed cost control problem for a class of linear continuous-time singular systems with norm-bounded uncertainties. In this study, the problem is to design a state feedback controller such that the closed-loop system is finite-time stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. By constructing an appropriate Lyapunov function, a sufficient condition for the finite-time robust stability of the system based on linear matrix inequality (LMI) is established. Furthermore, the sufficient condition for the existence of the guaranteed cost controller is formulated in terms of LMIs, which can make the closed-loop uncertain singular system finite-time robust stable. Finally, two numerical examples are given for illustration of the proposed theoretical results.

We deal with the robust stabilizability of the class of uncertain linear time-delay systems with Markovian jumping parameters (ULTDSMJP). Under the assumption of complete access to the continuous state and the boundedness of the system's uncertainties, an independent delay sufficient conditions which guarantee the robustness stability of this class of systems are established.

This paper describes the synthesis of robust decentralized controllers for uncertain large-scale discrete-time systems with time delays in the subsystem interconnections. Based on the Lyapunov method, a sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI). The solutions of the LMIs can be obtained easily using efficient convex optimization techniques. A numerical example is given to illustrate the proposed method.

On the robustness of uncertain time-delay systems with structured uncertainties

In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered. By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties. In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh −. Some analytical methods are employed to investigate these stability conditions. Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay. Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper.

Eigenstructure assignment-based robust stability conditions for uncertain systems with multiple time-varying delays