Input–output approach to stability and -gain analysis of systems with time-varying delays

Stability and L₂ (l₂)-gain of linear (continuous-time and discrete-time) systems with uncertain bounded time-varying delays are analyzed under the assumption that the nominal delay values are not equal to zero. The delay derivatives (in the continuous-time) are not assumed to be less than q<1. An input–output approach is applied by introducing a new input–output model, which leads to effective frequency domain and time domain criteria. The new method significantly improves the existing results for delays with derivatives not greater than 1, which were treated in the past as fast-varying delays (without any constraints on the delay derivatives). New bounded real lemmas (BRLs) are derived for systems with state and objective vector delays and norm-bounded uncertainties. Numerical examples illustrate the efficiency of the new method.     

Stability, robust stabilization and control of singular-impulsive systems via switching control

In this paper, stability, robust stabilization and H_∞ control of singular-impulsive systems are studied. Some new fundamental properties are derived for switched singular systems subject to impulse effects. Applying the Lyapunov function theory, several sufficient conditions are established for exponential stability, robust stabilization and H_∞ control of the corresponding singular-impulsive closed-loop systems. Some numerical examples are given to demonstrate the effectiveness of the proposed control and stabilization methods.     

estimation for discrete-time piecewise homogeneous Markov jump linear systems

This paper concerns the problem of H_∞ estimation for a class of Markov jump linear systems (MJLS) with time-varying transition probabilities (TPs) in discrete-time domain. The time-varying character of TPs is considered to be finite piecewise homogeneous and the variations in the finite set are considered to be of two types: arbitrary variation and stochastic variation, respectively. The latter means that the variation is subject to a higher-level transition probability matrix. The mode-dependent and variation-dependent H_∞ filter is designed such that the resulting closed-loop systems are stochastically stable and have a guaranteed H_∞ filtering error performance index. Using the idea in the recent studies of partially unknown TPs for the traditional MJLS with homogeneous TPs, a generalized framework covering the two kinds of variations is proposed. A numerical example is presented to illustrate the effectiveness and potential of the developed theoretical results.     

Distributed robust consensus control in directed networks of agents with time-delay

This paper investigates consensus problems for directed networks of agents with external disturbances and model uncertainty on fixed and switching topologies. Both networks with and without time-delay are taken into consideration. In doing the analysis, we first perform a model transformation and turn the original system into a reduced-order system. Based on this reduced-order system, we then present conditions under which all agents reach consensus with the desired H_∞ performance. Finally, simulation results are provided to demonstrate the effectiveness of our theoretical results.     

Suboptimal robust synthesis for MIMO plant under coprime factor perturbations

Suboptimal robust synthesis for MIMO nominal system under coprime factor perturbations is considered in classical and non-classical statements. In the classical statement, weights of perturbations and upper bound on magnitude bounded exogenous disturbance are assumed to be known to controller designer. Suboptimal synthesis within ε tolerance is reduced to the solution of log₂(1/ε) standard mixed sensitivity problems of ℓ₁ optimization. In the non-classical statement, the upper bounds on perturbations and exogenous disturbance are to be estimated from measurement data and suboptimal synthesis is reduced to the solution of 1/ε mixed sensitivity problems.     

On delay-derivative-dependent stability of systems with fast-varying delays

Stability of linear systems with uncertain bounded time-varying delays is studied under the assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the L₂-norm of a certain integral operator. There exists a bound on this operator norm in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving a tight operator bound which is an increasing and continuous function of the delay derivative upper bound d?1. For d→∞ the new bound corresponds to the second case and improves the existing bound. As a result, for the first time, delay-derivative-dependent frequency domain and time domain stability criteria are derived for systems with the delay derivative greater than 1.     

Limitations of nonlinear discrete periodic control for disturbance attenuation and robust stabilization

This paper presents a performance analysis of nonlinear periodically time-varying discrete controllers acting upon a linear time-invariant discrete plant. Time-invariant controllers are distinguished from strictly periodically time-varying controllers. For a given nonlinear periodic controller, a time-invariant controller is constructed. Necessary and sufficient conditions are given under which the time-invariant controller gives strictly better control performance than the time-invariant controller from which it was obtained, for the attenuation of l_p exogenous disturbances and the robust stabilization of l_p unstructured perturbations, for all p[1,∞].     

On standard control problems for systems with infinitely many unstable poles

In this paper, H^∞ control for a class of linear time invariant systems with infinitely many unstable poles is studied. An example of such a plant is a high gain system with delayed feedback. We formulate the problem via a generalized plant which consists of a rational transfer matrix and the inverse of a scalar (possibly irrational) inner function. It is shown that the problem can be decomposed into a finite-dimensional H^∞ control problem and an additional rank condition.     

Robust nonblocking supervisory control of discrete-event systems under partial observation

In this paper, we examine the problem of robust nonblocking supervisory control. In the problem considered here, the exact model of the plant is not known but is assumed to be among a finite set of possible models. For each plant model a legal marked behavior is assumed given. We extend previous results for the case of control with full observation to the case of control under partial observation where only a subset of events are observable. Furthermore, we remove the limitations of previous results on ensuring the nonblocking property of the plant under supervision. We characterize the entire set of solutions of the robust control problem and obtain a set of necessary and sufficient conditions for the existence of a solution for the problem. As an illustrative example, we use our results on robust control to solve a fault recovery problem.     

control with limited communication and message losses

We propose an H^∞ approach to a remote control problem where the communication is constrained due to the use of a shared channel. The controller employs a periodic time sequencing scheme for message transmissions from multiple sensors and to multiple actuators of the system. It further takes into account the information on the random message losses that occur in the channel. An exact characterization for controller synthesis is obtained and is stated in terms of linear matrix inequalities. Furthermore, an analysis on the loss probabilities of the messages to accomplish stabilization is carried out. The results are illustrated through a numerical example.     

Observer-based networked control for continuous-time systems with random sensor delays

This paper is concerned with the networked control system design for continuous-time systems with random measurement, where the measurement channel is assumed to subject to random sensor delay. A design scheme for the observer-based output feedback controller is proposed to render the closed-loop networked system exponentially mean-square stable with H_∞ performance requirement. The technique employed is based on appropriate delay systems approach combined with a matrix variable decoupling technique. The design method is fulfilled through solving linear matrix inequalities. A numerical example is used to verify the effectiveness and the merits of the present results.     

control for fast sampling discrete-time singularly perturbed systems

This paper is concerned with the H_∞ control problem via state feedback for fast sampling discrete-time singularly perturbed systems. A new H_∞ controller design method is given in terms of solutions to linear matrix inequalities (LMIs), which eliminates the regularity restrictions attached to the Riccati-based solution. A method for evaluating the upper bound of singular perturbation parameter with meeting a prescribed H_∞ performance bound requirement is also given. Furthermore, the results are extended to robust controller design for fast sampling discrete-time singularly perturbed systems with polytopic uncertainties. Numerical examples are given to illustrate the validity of the proposed methods.     

Noncausal linear periodically time-varying scaling for robust stability analysis of discrete-time systems: Frequency-dependent scaling induced by static separators

This article is concerned with robust stability analysis of discrete-time systems and introduces a novel and powerful technique that we call noncausal linear periodically time-varying (LPTV) scaling. Based on the discrete-time lifting together with the conventional but general scaling approach, we are led to the notion of noncausal LPTV scaling for LPTV systems, and its effectiveness is demonstrated with a numerical example. To separate the effect of noncausal and LPTV characteristics of noncausal LPTV scaling to see which is a more important source leading to the effectiveness, we then consider the case of LTI systems as a special case. Then, we show that even static noncausal LPTV scaling has an ability of inducing frequency-dependent scaling when viewed in the context of the conventional LTI scaling, while causal LPTV scaling fails to do so. It is further discussed that the effectiveness of noncausal characteristics leading to the frequency-domain interpretation can be exploited even for LPTV systems by considering the νN-lifted transfer matrices of N-periodic systems.     

Robust control of uncertain differential linear repetitive processes

For two-dimensional (2-D) systems, information propagates in two independent directions. 2-D systems are known to have both system-theoretical and applications interest, and the so-called linear repetitive processes (LRPs) are a distinct class of 2-D discrete linear systems. This paper is concerned with the problem of L₂–L_∞ (energy to peak) control for uncertain differential LRPs, where the parameter uncertainties are assumed to be norm-bounded. For an unstable LRP, our attention is focused on the design of an L₂–L_∞ static state feedback controller and an L₂–L_∞ dynamic output feedback controller, both of which guarantee the corresponding closed-loop LRPs to be stable along the pass and have a prescribed L₂–L_∞ performance. Sufficient conditions for the existence of such L₂–L_∞ controllers are proposed in terms of linear matrix inequalities (LMIs). The desired L₂–L_∞ dynamic output feedback controller can be found by solving a convex optimization problem. A numerical example is provided to demonstrate the effectiveness of the proposed controller design procedures.     

Quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems

This paper presents a novel quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems via H^∞ approach. In the proposed algorithm, a self-constructing neural fuzzy network (SCNFN) is developed with both structure and parameter learning phases, so that the number of fuzzy rules and network parameters can be adaptively determined. Based on the SCNFN, an uncertainty observer is first introduced to watch compound system uncertainties. Subsequently, an optimal NFN-based controller is designed to overcome the effects of unstructured uncertainty and approximation error by integrating the NFN identifier, linear optimal control and H^∞ approach as a whole. The adaptive tuning laws of network parameters are derived in the sense of quadratic stability technique and Lyapunov synthesis approach to ensure the network convergence and H^∞ synchronization performance. The merits of the proposed control scheme are not only that the conservative estimation of NFN approximation error bound is avoided but also that a suitable-sized neural structure is found to sufficiently approximate the system uncertainties. Simulation results are provided to verify the effectiveness and robustness of the proposed control method.     

Exponential stability of singular systems with multiple time-varying delays

This paper deals with the class of continuous-time singular linear systems with multiple time-varying delays in a range. The global exponential stability problem of this class of systems is addressed. Delay-range-dependent sufficient conditions such that the system is regular, impulse-free and α-stable are developed in the linear matrix inequality (LMI) setting. Moreover, an estimate of the convergence rate of such stable systems is presented. A numerical example is employed to show the usefulness of the proposed results.     

Observers for a class of Lipschitz systems with extension to performance analysis

In this paper, observer design for a class of Lipschitz nonlinear dynamical systems is investigated. One of the main contributions lies in the use of the differential mean value theorem (DMVT) which allows transforming the nonlinear error dynamics into a linear parameter varying (LPV) system. This has the advantage of introducing a general Lipschitz-like condition on the Jacobian matrix for differentiable systems. To ensure asymptotic convergence, in both continuous and discrete time systems, such sufficient conditions expressed in terms of linear matrix inequalities (LMIs) are established. An extension to H_∞ filtering design is obtained also for systems with nonlinear outputs. A comparison with respect to the observer method of Gauthier et al. [A simple observer for nonlinear systems. Applications to bioreactors, IEEE Trans. Automat. Control 37(6) (1992) 875–880] is presented to show that the proposed approach avoids high gain for a class of triangular globally Lipschitz systems. In the last section, academic examples are given to show the performances and some limits of the proposed approach. The last example is introduced with the goal to illustrate good performances on robustness to measurement errors by avoiding high gain.     

Reduced-order filtering for singular systems

This paper solves the problem of reduced-order H_∞ filtering for singular systems. The purpose is to design linear filters with a specified order lower than the given system such that the filtering error dynamic system is regular, impulse-free (or causal), stable, and satisfies a prescribed H_∞ performance level. One major contribution of the present work is that necessary and sufficient conditions for the solvability of this problem are obtained for both continuous and discrete singular systems. These conditions are characterized in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint. Moreover, an explicit parametrization of all desired reduced-order filters is presented when these inequalities are feasible. In particular, when a static or zeroth-order H_∞ filter is desired, it is shown that the H_∞ filtering problem reduces to a convex LMI problem. All these results are expressed in terms of the original system matrices without decomposition, which makes the design procedure simple and directly. Last but not least, the results have generalized previous works on H_∞ filtering for state-space systems. An illustrative example is given to demonstrate the effectiveness of the proposed approach.     

filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities

In this paper, we deal with the robust H_∞ filtering problem for a class of uncertain nonlinear time-delay stochastic systems. The system under consideration contains parameter uncertainties, Itô-type stochastic disturbances, time-varying delays, as well as sector-bounded nonlinearities. We aim at designing a full-order filter such that, for all admissible uncertainties, nonlinearities and time delays, the dynamics of the filtering error is guaranteed to be robustly asymptotically stable in the mean square, while achieving the prescribed H_∞ disturbance rejection attenuation level. By using the Lyapunov stability theory and Itô’s differential rule, sufficient conditions are first established to ensure the existence of the desired filters, which are expressed in the form of a linear matrix inequality (LMI). Then, the explicit expression of the desired filter gains is also characterized. Finally, a numerical example is exploited to show the usefulness of the results derived.