Deconvolution filtering for stochastic systems via homogeneous polynomial Lyapunov functions

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.

Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach

This work researches the problem of searching for multiple homogeneous polynomial Lyapunov functions (HPLFs) for heterogeneous switched linear systems. First, a nonconvex optimization condition is constructed to study the stability property of heterogeneous switched systems, where each Lyapunov function candidate reduces dimension to their corresponding matrix eigenvalue. Based on the stability analysis condition, a controller-dependently multiple HPLFs condition is introduced to determine controllers and explores locally minimum mode-dependent average dwell time (LMMDADT). Additionally, the existing properties condition and solvable properties condition of controllers are given in the form of HPLFs. At last, a practical example and a contrast example are both presented to show feasibility of the proposed results.

LMI RELAXATIONS FOR HOMOGENEOUS POLYNOMIAL SOLUTIONS OF PARAMETER-DEPENDENT LMIS

본 논문에서는 변수 불확실성과 필터이득 섭동을 가지는 시스템에 대한 견실비약성 H∞ 칼만형필터 설계기법을 제안한다. 필터가 존재할 충분조건과 견실비약성 H∞ 필터 설계기법을 선형행렬부등식 (LMI: Linear Matrix Inequality 접근법으로 제안하고 시스템과 필터의 불확실성을 매개변수화 선형행렬부등식(PLMI: Parameterized Linear Matrix Inequality)으로 구조화된 불확실성의 형태로 표현한 후 Lyapunov 함수를 통해 시스템의 불확실성과 더불어 필터이득섭동을 고려한 칼만형 H∞ 필터가 존재할 충분조건과 필터설계기법을 PLMI 형태로 보인다. PLMI는 무한개의 LMI의 형태로 나타나므로 완화기법(relaxation technique)을 적용하여 유한개의 LMI의 형태로 변환한 후 견실하고 최적화된 필터이득과 필터섭동범위를 계산하고, 예제와 모의실험을 통해 제시된 필터의 타당성을 검증한다.

This paper addresses the admissibility analysis of uncertain time-varying continuous-time descriptor systems. The results proposed in our paper extend the stability analysis approach based on homogeneous polynomial Lyapunov functions (HPLFs) previously developed for uncertain standard statespace systems to the robust admissibility analysis of uncertain descriptor ones. A new power transformation of the state vector with respect to its dynamic and algebraic parts has been proposed leading to new LMI admissibility analysis conditions. These conditions are less restrictive than the ones based on quadratic Lyapunov functions as sustained by our numerical examples.

This paper investigates the stabilization problem by means of a static output feedback control for positive systems with/without interval uncertainties. The designed closed‐loop system is ensured to be positive and stable with a prescribed decay rate. Necessary and sufficient conditions for the existence of such controllers are established in terms of linear matrix inequalities together with a matrix equality constraint. Furthermore, the desired static output feedback controllers can be derived directly via the cone complementarity linearization techniques. Finally, an illustrative example is presented to show the validity of results obtained in this paper. Copyright © 2010 John Wiley & Sons, Ltd.

For continuous-time linear time-invariant (LTI) systems with polytopic uncertainties, we develop a robust sampled-data state-feedback control design scheme in terms of linear matrix inequalities (LMIs). Truncated power series expansions are used to approximate a discretized model of the original continuous-time system. The system matrices obtained by using the power series approximations are then expressed as homogeneous polynomial parameter-dependent (HPPD) matrices of finite degrees, and conditions for designing the controller are formulated as a HPPD matrix inequality, which can be solved by means of a recent LMI relaxation technique to test the positivity of HPPD matrices with variables in the simplex. To take care of the errors induced by the remainder terms of the truncated power series, the terms are considered as norm bounded uncertainties and then incorporated into the proposed LMI conditions. Finally, examples are used to illustrate the approach.

This paper presents new stability conditions satisfying decay rate for both continuous and discrete fuzzy control systems. Linear matrix inequality (LMI) based designs for decay rate are also considered using the concept of parallel distributed compensation (PDC). To design fuzzy control systems, nonlinear systems are represented by Takagi-Sugeno fuzzy models. The PDC is employed to design fuzzy controllers from the Takagi-Sugeno fuzzy models. The stability analysis discussed here is reduced to a problem of finding a common Lyapunov function for a set of LMIs. Convex optimization techniques involving LMIs are utilized to find a common Lyapunov function and stable feedback gains satisfying decay rate. A simple example demonstrates the utility of the LMI-based designs proposed in this paper.

The paper investigates the filter design problem based on linear matrix inequality (LMI) formulation. Filter design approaches to achieve bounded robust H 8 norm, robust H 2 norm, and robust l 2 – l 8 gain are developed. It is shown that these designs can all be cast in terms of LMIs. Moreover, the LMIs for robust performance assessment and filter design bear similar form with respect to these different measures. The fixed‐order filter design problem is also tackled in terms of LMIs. An example is provided to illustrate the filter design procedure.

Synchronization analysis of coupled connected neural networks with mixed time delays

SUMMARYThis paper investigates robust stability of time‐varying uncertain genetic regulatory networks (GRNs). In particular, the considered model includes, as special cases, SUM and PROD regulatory functions typically considered in the literature. It is supposed that the coefficients of the GRN are affine linear functions of an uncertain vector constrained in a polytope, and that the activation functions are uncertain into sector‐type regions. As the first problem, we consider to establish whether the GRN is robustly globally stable for all admissible uncertainties. It is shown that this problem can be addressed by solving a linear matrix inequality (LMI) feasibility test built by exploiting homogeneous polynomial Lyapunov functions. As the second problem, we consider to determine the slowest speed with which the concentrations of mRNAs and proteins reach their equilibrium values. It is shown that a guaranteed underestimate of such a speed can be provided by solving a generalized eigenvalue problem built from the proposed stability condition. Some numerical examples illustrate the proposed approaches. It is worth remarking that this paper proposes for the first time in the literature the use of nonquadratic Lyapunov functions for studying robust stability of uncertain GRNs, whereas existing works have addressed the problem only via quadratic Lyapunov functions (either common or parameter‐dependent), which are known to be conservative for time‐varying uncertainty. Copyright © 2011 John Wiley & Sons, Ltd.

The design of robust controllers is considered here for a class of two-dimensional (2-D) discrete switched systems described by the Roesser model with polytopic uncertainties. Attention focuses on the design of a switched state feedback controller, which guarantees the robust asymptotic stability and a prescribed performance for the closed-loop system. By using multiple parameter-dependent Lyapunov functionals, and introducing some switched free-weighting matrices, a new sufficient condition for the robust performance analysis of uncertain 2-D discrete switched systems is developed. Furthermore, the design of switched state feedback controller is proposed in terms of linear matrix inequalities (LMIs). Illustrative examples are given to illustrate the effectiveness of the developed theoretical results.

This paper studies the design problem of robust H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> filter for neutral stochastic uncertain delay systems. Aiming at neutral stochastic delay systems with polytopic uncertainty, we deduce sufficient conditions of the existence of H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> filter, and express them into a form of the linear matrix inequality (LMI), then utilize schur complement lemma transformed them into LMIs which can be solved. At last we utilize standard digital software to get filter parameters. The feasibility of the methods of design filter for neutral stochastic uncertain delay systems is demonstrated by numerical examples.

Because of intracellular and extracellular noise perturbations and environment fluctuations, gene regulation is an intrinsically noisy process. In this paper, we present a hybrid genetic regulatory network (GRN) model which is based on the Markov chain. The GRNs are composed of N modes and the network switches from one mode to another according to a Markov chain with known transition probability. Time‐delays here are mode‐dependent. Based on the Lyapunov stability theory and the linear matrix inequality (LMI) technique, sufficient conditions are given to ensure the stochastic stability of the GRNs with polytopic uncertainties and disturbance attenuation. All the conditions are presented in terms of LMIs which are easily verified via the LMI toolbox. Examples are provided to illustrate the effectiveness of the theoretical results.Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society