Strict Linear Matrix Inequalities Research Articles

Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances

In this paper, a sliding mode control (SMC) of uncertain discrete singular systems with external disturbances and time-varying delays is under consideration. By use of the free weighting matrices and the Lyapunov–Krasovskii functional, a delay-dependent sufficient condition is given in strict linear matrix inequality (LMI) format to guarantee the sliding mode dynamics to be admissible (regular, causal and stable). Furthermore, a proposed SMC law and an adaptive SMC law are synthesized to make sure that the trajectories of system can be driven to a region near equilibrium point in finite time. Finally, a numerical example is designed to display the effectiveness of the control scheme. All these results are expected to propose a new approach for the research on SMC of discrete time-delay singular systems.

New admissibility conditions for singular linear continuous-time fractional-order systems

This paper deals with the admissibility problem of singular fractional-order continuous time systems. It is based on new admissibility conditions of singular fractional-order systems expressed in a set of strict Linear Matrix Inequalities (LMI s). Then, a static output feedback controller is designed for the closed-loop system to be admissible. Numerical examples are given to illustrate the proposed methods.

Non-fragile quantized H∞ output feedback control for nonlinear systems with quantized inputs and outputs

This paper is concerned with non-fragile H∞ control problems for a class of continuous-time nonlinear systems with unknown nonlinearity and quantized inputs and outputs. The construction of both static output feedback (SOF) and observer-based output feedback (OBOF) control laws in the presence of additive interval-bounded controller coefficient variations can be divided into two parts, linear and nonlinear parts. The linear part plays a role in achieving the H∞ performance, while the nonlinear part is used to reduce the quantization effect. However, it should be pointed out that the effect of input and output quantization can be eliminated fully for SOF case by requiring knowledge of all signs of the states, but only the effect of input quantization can be eliminated for OBOF case. It is worth mentioning that three novel alternative methods with strict linear matrix inequality (LMI) conditions are proposed to design both SOF and OBOF controllers. In particular, these three new methods do not introduce any other auxiliary constraints as many existing results do where a matrix equality constraint between system matrix and Lyapunov matrix is often inserted. Finally, the effectiveness and advantages of the proposed control methods are demonstrated by a numerical example.

Stochastic admissibility and stabilization of singular Markovian jump systems with multiple time-varying delays

This paper is concerned with stochastic admissibility and state feedback stabilization for a class of singular Markovian jump systems with multiple time-varying delays. The singular matrix E with both modedependent and mode-independent is considered in the system. Firstly, based on Lyapunov functional method and free-weighting matrix method, sufficient condition is presented in the form of linear matrix inequalities (LMIs) to guarantee the considered system to be stochastically admissible. Secondly, by state feedback controller, sufficient condition is derived in terms of strict LMIs to ensure the closed-loop system to be stochastically stabilizable. Finally, numerical examples are provided to illustrate the effectiveness of the proposed approaches.

Robust and non-fragile static output feedback control for continuous-time semi-Markovian jump systems

This paper investigates the problem of robust and non-fragile [Formula: see text] static output feedback (SOF) controller design for a class of continuous-time semi-Markovian jump linear systems (S-MJLSs) with norm-bounded uncertainties. The systems under consideration are subject to a semi-Markov stochastic process with time-varying transition rates (TRs), which relax the traditional assumption on Markovian jump linear systems (MJLSs) that all of the TRs are constant. Based on a semi-Markovian Lyapunov function together with the property of cumulative distribution function, a sufficient condition for robust [Formula: see text] performance analysis is first derived. Then via some matrix inequality linearization procedures combined with sojourn-time decomposition idea, two approaches, namely, the convex linearization approach and iterative approach, are developed to solve the robust and non-fragile SOF controller synthesis problem. It is shown that the controller gains can be obtained by solving a set of strict linear matrix inequalities (LMIs) or a sequential minimization problem subject to LMI constraints. Finally, illustrative examples are provided to illustrate the effectiveness of the proposed approaches.

H∞ State Estimation for Stochastic Markovian Jumping Neural Network with Time-Varying Delay and Leakage Delay

The H∞ state estimation problem for a class of stochastic neural networks with Markovian jumping parameters and leakage delay is investigated in this paper. By employing a suitable Lyapunov functional and inequality technic, the sufficient conditions for exponential stability as well as prescribed H∞ norm level of the state estimation error system are proposed and verified, and all obtained results are expressed in terms of strict linear matrix inequalities (LMIs). Examples and simulations are presented to show the effectiveness of the proposed methods, at the same time, the effect of leakage delay on stability of neural networks system and on the attenuation level of state estimator are discussed.

Globally optimal distributed cooperative control for general linear multi-agent systems

This paper aims to design distributed consensus protocols which satisfy two design requirements for identical general linear multi-agent systems on fixed, undirected graphs: meeting the global optimality and guaranteeing a prescribed convergence speed. By using inverse optimal approaches, the optimal partial stabilization is developed and the globally optimal distributed consensus problem for leader following and leaderless problems are solved. To obtain prescribed convergence speed of the multi-agent system, novel globally optimal distributed consensus design procedures are proposed. First, combining with the regional pole assignment, the optimal control can be found by solving a strict linear matrix inequality (LMI) problem. It turns out that the increasing number of the agent nodes will not increase the computational complexity. Then, a modified linear quadratic regulator (MLQR) design method is developed which leads to a model free design procedure by employing the adaptive dynamic programming (ADP) technique. Finally, a numerical example is given to illustrate the effectiveness of the proposed procedures.

Delay-dependent stability and stabilization criteria for T-S fuzzy singular systems with interval time-varying delay by improved delay partitioning approach.

This paper deals with the stability analysis and fuzzy stabilizing controller design for a class of Takagi–Sugeno fuzzy singular systems with interval time-varying delay and linear fractional uncertainties. By decomposing the delay interval into two unequal subintervals and seeking a appropriate ρ, a new Lyapunov–Krasovskii functional is constructed to develop the improved delay-dependent stability criteria, which ensures the considered system to be regular, impulse-free and stable. Furthermore, the desired fuzzy controller gains are also presented by solving a set of strict linear matrix inequalities. Compared with some existing results, the obtained ones give the result with less conservatism. Finally, some examples are given to show the improvement and the effectiveness of the proposed method.

Modified Sliding Mode Control for Mismatched Uncertain Systems with Unknown Disturbances

In this paper, a new sliding mode control law is developed for a class of mismatched uncertain systems with more general exogenous disturbances. First, we derive a new existence condition of linear sliding surface in terms of strict linear matrix inequalities such that the reduce-order sliding mode dynamics is is asymptotically stable. Second, we propose an adaptive sliding mode control law such that the system states reach the sliding surface in finite time and stay on its thereafter. Final, a numerical example is used to demonstrate the efficacy of the proposed method.

SpecificD-Admissibility and Design Issues for Uncertain Descriptor Systems with Parametric Uncertainty in the Derivative Matrix

Stability analysis issues and controller synthesis for descriptor systems with parametric uncertainty in the derivative matrix are discussed in this paper. The proposed descriptor system can extend the system’s modeling extent of physical and engineering systems from the traditional state-space model. First, based on the extendedD-stability definitions for the descriptor model, necessary and sufficient admissibility andD-admissibility conditions for the unforced nominal descriptor system are derived and formulated by compact forms with strict linear matrix inequality (LMI) manner. In contrast, existing results need to involve nonstrict LMIs, which cannot be evaluated by current LMI solvers and need some extra treatments. Deducing from the obtained distinct results, the roust admissibility andD-admissibility of the descriptor system with uncertainties in both the derivative matrix and the system’s matrices thus can be coped. Furthermore, by involving a proportional and derivative state feedback (PDSF) control law, we further address the controller design for the resulting closed-loop systems. Since all the proposed criteria are explicitly expressed in terms of the strict LMIs, we can use applicable LMI solvers for evaluating the feasible solutions. Finally, the efficiency and practicability of the proposed approach are demonstrated by two illustrative examples.

Intelligent digital redesign for non‐linear systems: observer‐based sampled‐data fuzzy control approach

In this study, an intelligent digital redesign (IDR) technique is proposed for an observer-based sampled-data fuzzy controller of non-linear systems. By using a Takagi–Sugeno fuzzy model, the pre-designed analog and sampled-data fuzzy controllers are supposed, and these discretised closed-loop systems are obtained, respectively. Based on the IDR problem, the authors guarantee both stability and state-matching conditions. Unlike the previous techniques, the proposed IDR not only improves the state-matching performance using the state-matching error cost function, but is also derived in the strict linear matrix inequality format. In a numerical example, the effectiveness of the proposed technique and the results of the improved performance are shown.

Robust H∞ deconvolution filtering for uncertain singular Markovian jump systems with time‐varying delays

SummaryThe problem of H∞ deconvolution filter design for a class of singular Markovian jump systems with time‐varying delays and parameter uncertainties is considered in this paper. By constructing a more comprehensive stochastic Lyapunov‐Krasovskii functional, novel delay‐dependent conditions are established to guarantee the filtering error system is not only stochastically admissible, but also satisfies a prescribed H∞‐norm level for all admissible uncertainties. The desired filter parameters can be obtained by solving a set of strict linear matrix inequalities. Two examples and an electrical RLC circuit example are employed to verify the effectiveness and usefulness of the proposed methods in the paper. Copyright © 2015 John Wiley & Sons, Ltd.

Passivity based sliding mode control of uncertain singular Markovian jump systems with time-varying delay and nonlinear perturbations

In this paper, a sliding mode control (SMC) with passivity of uncertain singular Markovian jump systems (SMJS) with time-varying delay and nonlinear perturbations is investigated. Firstly, a new integral-type switching surface is designed. Secondly, by introducing free matrix method and Lyapunov functionals, a delay-dependent sufficient condition in terms of strict linear matrix inequalities (LMI) is derived for the resulting sliding mode dynamics such that the closed-loop system is stochastically admissible and robustly passive for all admissible uncertainties. Thirdly, a SMC law is synthesized to drive the system trajectories onto the predefined switching surface in a finite time. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed method.

A novel descriptor redundancy approach for non-quadratic robust H∞ control of T-S fuzzy nonlinear singularly perturbed systems

A new design procedure for robust H∞ control for a class of T-S fuzzy singularly perturbed systems based on fuzzy Lyapunov functions and non-PDC control scheme is proposed in this paper. Using the proposed controller, the closed-loop TS fuzzy singularly perturbed system becomes asymptotically stable in the absence of disturbances and satisfies the H∞-norm condition in the presence of disturbances all positive values of the singular perturbation parameter within the given desired bound. The main drawback of a fuzzy Lyapunov-based approach is that one needs to find some upper or lower bounds for the derivatives of the grades of the membership functions, which is hard to find in practice. By using natural properties of a T-S model, a novel procedure is proposed to tackle this problem. Using the descriptor redundancy approach, the design conditions are derived in the form of some strict linear matrix inequalities (LMIs). Moreover, to cover a more general problem, it is assumed that there exist some uncertainties in the local system matrices of the T-S fuzzy singularly perturbed system. Two examples are presented to illustrate that the proposed design conditions are feasible for larger bounds on the singular perturbation parameter compared to those of previous techniques.

New results on stability of singular stochastic Markov jump systems with state-dependent noise

This paper aims to develop the stability theory for singular stochastic Markov jump systems with state-dependent noise, including both continuous- and discrete-time cases. The sufficient conditions for the existence and uniqueness of a solution to the system equation are provided. Some new and fundamental concepts such as non-impulsiveness and mean square admissibility are introduced, which are different from those of other existing works. By making use of the $\mathcal{H}$-representation technique and the pseudo inverse $E^+$ of a singular matrix $E$, sufficient conditions ensuring the system to be mean square admissible are established in terms of strict linear matrix inequalities, which can be regarded as extensions of the corresponding results of deterministic singular systems and normal stochastic systems. Practical examples are given to demonstrate the effectiveness of the proposed approaches.

$$H_{\infty }$$ H ∞ Estimation for Markovian Jump Neural Networks With Quantization, Transmission Delay and Packet Dropout

This paper is concerned with the problem of robust $$H_{\infty }$$Hź estimation for a class of Markovian jump neural networks with norm-bounded parameter uncertainties, time-varying delay and limited communication capacity including signal transmission delay, measurement quantization and data packet dropout, which occur simultaneously in the same networked control system framework. Our objective is to design mode-dependent $$H_{\infty }$$Hź filter in the network environment in order to ensure the filtering error system is not only exponentially mean-square stable, but also satisfies a prescribed $$H_{\infty }$$Hź norm level for the limited communication capacity and for all admissible uncertainties. We use the delay system method to deal with signal transmission delay and data packet dropout, utilize the sector bound approach to handle measurement quantization. We also construct a comprehensive stochastic Lyapunov---Krasovskii functional to show the inherent mode-dependent state delays in the neural networks itself and the network-induced signal transmission delay. The free-weighting matrix technique and a set of strict linear matrix inequalities are employed to derive novel conditions for the desired mode-dependent $$H_{\infty }$$Hź filter. A numerical example is provided to verify the effectiveness and usefulness of the obtained main results.

Reliable $$H_{\infty }$$ H ∞ Stabilization of Fuzzy Systems with Random Delay Via Nonlinear Retarded Control

In this paper, the robust reliable $$H_\infty $$H? control problem has been investigated for a class of nonlinear discrete-time TS fuzzy systems with random delay. In particular, the proposed fuzzy system consists of local nonlinear models with set of fuzzy rules, but the conventional TS fuzzy systems has local linear models. Our attention is focused on the design of a feedback reliable nonlinear retarded control law to ensure the robust asymptotic stability for nonlinear discrete-time TS fuzzy system with admissible uncertainties as well as actuator failure cases and random delay. In particular, by using an input delay approach, the random delay with stochastic parameters in the system matrices is introduced in the system model. Based on the Lyapunov approach, firstly, a sufficient condition for asymptotic stability is proposed for TS fuzzy systems in the presence of actuator failures. Then, a robust reliable $$H_\infty $$H? control is designed for the uncertain TS fuzzy system by solving a strict linear matrix inequalities using the available numerical software. Finally, a numerical example based on real-time ball and beam system is provided to validate the effectiveness of the proposed design technique.

New admissibility analysis for discrete singular systems with time-varying delay

In this paper, the issue of admissibility for discrete singular systems with time-varying delay is addressed. The main contribution of this paper is to reduce the conservatism of the considered system by adopting an improved reciprocally convex combination approach to bound the forward difference of the double summation term, and utilizing the reciprocally convex combination approach to bound the forward difference of the triple summation term in the Lyapunov function. Without employing decomposition and equivalent transformation of the considered system, a strict delay-dependent linear matrix inequality (LMI) criterion is built to guarantee the considered system to be regular, causal and stable. A numerical example is given to illustrate the effectiveness of this proposed method.

State-space [formula omitted] controller design for descriptor systems

This paper proposes a new linear matrix inequality (LMI) method to design state-space H∞ controllers for linear time-invariant descriptor systems. Unlike preceding studies, where descriptor-type controllers are first computed and then numerically transformed to state-space controllers, the proposed method carries out the transformation analytically in the parameter domain. We derive a necessary and sufficient LMI condition for the existence of a state-space controller with the same dynamic order of the descriptor system to be controlled, which makes the closed-loop system regular, impulse-free, stable, and guarantees the H∞ norm bound imposed on the closed-loop transfer function. Furthermore, we present parameterization of all such state-space controllers by variables satisfying the LMI condition and an arbitrary nonsingular matrix. The LMIs utilized in this paper are strict ones, that is, those containing no equality, while LMIs with equality constraints have been extensively used in the analysis and design for descriptor systems. The strict LMIs play key roles in deriving the results of this paper.

A Robust $H_{\infty}$ Non-PDC Design Scheme for Singularly Perturbed T–S Fuzzy Systems With Immeasurable State Variables

This paper addresses an observer-based robust $H_{\infty}$ fuzzy controller design method for singularly perturbed nonlinear systems with external disturbances, which are represented by uncertain Takagi–Sugeno (T–S) fuzzy models. For a practical output feedback design, the premise variables of the fuzzy controller and the fuzzy observer are considered unknown in general. A fuzzy Lyapunov function is utilized for the synthesis of the nonparallel-distributed-compensation-based controller. The closed-loop singularly perturbed T–S fuzzy system is asymptotically robustly stable in the absence of disturbances and satisfies an $H_{\infty}$ -norm condition in the presence of disturbances for all positive values of the singular perturbation parameter within the given desired bound. Using the descriptor redundancy approach, some strict linear matrix inequality (LMI) conditions are derived for the $H_{\infty}$ robustness of the closed-loop system when uncertainties are simultaneously considered in all state-space matrices of each fuzzy subsystem. The main disadvantage of a fuzzy Lyapunov function-based design is that the bounds on the derivatives of the membership functions must be known a priori . Inspired by a recent work that provides the bounds of the membership derivatives for a given arbitrary compact set on the state space, a new method is proposed that guarantees that the closed-loop observer-based uncertain singularly perturbed T–S fuzzy system remains robustly stable in the compact set. Using Finsler's lemma, the resulting design conditions are converted to LMIs. Two numerical examples are provided to show the effectiveness of the proposed method.