On robustness of predictor feedback control of linear systems with input delays

This paper is concerned with the robustness of the predictor feedback control of linear systems with input delays. By applying certain equivalent transformations on the characteristic equation associated with the closed-loop system, we first transform the robustness problem of a predictor feedback control system into the stability problem of a neutral time-delay system containing an integral operator in the derivative. The range of the allowable input delay for this neutral time-delay system can be computed by exploring its delay dependent stability conditions. In particular, delay dependent stability conditions for the neutral time-delay system are established by partitioning the delay into segments. The conservatism of this method can be reduced when the number of segments in the partition is increased. Numerical examples are worked out to illustrate the effectiveness of the proposed method.     

On robust stability of LTI fractional-order delay systems of retarded and neutral type

This paper deals with the analysis of robust BIBO-stability of LTI fractional order delay systems in the presence of real parametric uncertainties. Two large classes of these systems, namely retarded and neutral types, are considered. Two theorems are given to check the robust BIBO-stability of these two families of fractional order systems. One of these theorems provides necessary and sufficient conditions for the case of retarded type and another one presents only sufficient conditions for the case of neutral type. Furthermore, upper and lower bounds (cutoff frequencies) are provided for drawing the value sets. To illustrate the results, two numerical examples are presented.     

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.     

Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays

This paper focuses on the problem of delay-dependent robust stochastic stability analysis and controller synthesis for Markovian jump systems with state and input delays. It is assumed that the delays are constant and unknown, but their upper bounds are known. By constructing a new Lyapunov-Krasovskii functional and introducing some appropriate slack matrices, new delay-dependent stochastic stability and stabilization conditions are proposed by means of linear matrix inequalities (LMIs). An important feature of the results proposed here is that all the robust stability and stabilization conditions are dependent on the upper bound of the delays. Memoryless state feedback controllers are designed such that the closed-loop system is robustly stochastically stable. Some numerical examples are provided to illustrate the effectiveness of the proposed method.     

Exponential stability of uncertain T-S fuzzy switched systems with time delay

This paper discusses the delay-dependent exponential stability of a class of uncertain T-S fuzzy switched systems with time delay. The method is based on Lyapunov stability theorem and free weighting matrices approach. Two illustrative examples are given to demonstrate the effectiveness of the proposed method.     

Gain-scheduled control via filtered scheduling parameters

Gain-scheduled control via LPV system models enjoys LMI-based synthesis methods and in particular parameter-dependent Lyapunov matrices have been employed to successfully reduce conservatism. Those controllers derived via parameter-dependent Lyapunov matrices, however, end up with depending on derivatives of scheduling parameters. Though this can be avoided by approximating derivatives or restricting Lyapunov matrices to be partly constant, the former loses guarantee of performance and stability and the latter can cause conservatism. This paper proposes a synthesis method of gain-scheduled controllers that depend on filtered scheduling parameters, instead of derivatives, with a concrete guarantee of a performance level. Moreover, it is shown that the performance level of conventional derivative-dependent gain-scheduled controllers is recovered with arbitrarily small errors.     

Improved LMI representations for delay-independent and delay-dependent stability conditions

Some new linear matrix inequality (LMI) representations for delay-independent and delay-dependent stability conditions are obtained by introducing additional matrices and eliminating the product coupling of the system matrices and the Lya-punov matrices. The results improve conservativeness of the given conditions for the analysis and the design of tune-delay systems with polytopic-type uncertainty.     

Robust stability analysis and stabilisation of uncertain neutral singular systems

This paper studies the problem of robust stability and stabilisation of uncertain neutral singular systems and develops a new stability criterion of the differential operator, ? by the final value theorem for Laplace transform. By utilising Lyapunov functional and free-weight matrix method, we employ the linear matrix inequality technique and the stability condition to derive some new criteria which ensure the robustly asymptotical stability of the studied system. State feedback controllers with disturbance are designed to guarantee the robustly asymptotical stability of the uncertain system. Numerical examples are provided to validate the effectiveness of the obtained results.     

On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale’s form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale’s form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.     

Robust finite-time stabilisation of uncertain linear systems

This article deals with the problem of finite-time stability and stabilisation of uncertain linear systems. For linear time-varying systems subject to norm-bounded uncertainties some conditions for finite-time stability are provided. These conditions are expressed in terms of differential linear matrix inequalities. Then the problem of controller design is tackled, both for the state feedback and for the output feedback case; in both cases the controller can be found solving a suitable set of LMIs. A typical engineering case-study is included, to illustrate the applicability of the devised conditions.     

Stability overlay for adaptive control laws

This paper proposes an architecture referred to as Stability Overlay (SO) for adaptive control of a class of nonlinear time-varying plants. The SO can be implemented in parallel with a wide range of “performance-based” adaptive control laws, i.e., adaptive control laws that seek to improve closed-loop performance, but may be susceptible to instability in the presence of unaccounted model uncertainty. In this architecture, the performance-based adaptive control law designates candidate controllers based on performance considerations, while the SO supervises this selection based upon online robust stability considerations. A particular selection of a performance-based adaptive control law is not specified. Rather, this selection can be from a wide range of adaptive control schemes. This paper provides stability proofs for the SO architecture and presents a simulation that illustrates the applicability of the proposed method.     

A delay-partitioning approach to the stability analysis of discrete-time systems

This paper revisits the problem of stability analysis for linear discrete-time systems with time-varying delay in the state. By utilizing the delay partitioning idea, new stability criteria are proposed in terms of linear matrix inequalities (LMIs). These conditions are developed based on a novel Lyapunov functional. In addition to delay dependence, the obtained conditions are also dependent on the partitioning size. We have also established that the conservatism of the conditions is a non-increasing function of the number of partitions. Numerical examples are given to illustrate the effectiveness and advantage of the proposed methods.     

Discrete-time stochastic control systems: A continuous Lyapunov function implies robustness to strictly causal perturbations

Discrete-time stochastic systems employing possibly discontinuous state-feedback control laws are addressed. Allowing discontinuous feedbacks is fundamental for stochastic systems regulated, for instance, by optimization-based control laws. We introduce generalized random solutions for discontinuous stochastic systems to guarantee the existence of solutions and to generate enough solutions to get an accurate picture of robustness with respect to strictly causal perturbations. Under basic regularity conditions, the existence of a continuous stochastic Lyapunov function is sufficient to establish that asymptotic stability in probability for the closed-loop system is robust to sufficiently small, state-dependent, strictly causal, worst-case perturbations. Robustness of a weaker stochastic stability property called recurrence is also shown in a global sense in the case of state-dependent perturbations, and in a semiglobal practical sense in the case of persistent perturbations. An example shows that a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case disturbances that are not strictly causal. Our positive results are also illustrated by examples.