Uri Shaked

An improved stabilization method for linear time-delay systems

In this paper, we combine a new approach for linear time delay systems based on a descriptor representation with a recent result on bounding of cross products of vectors. A delay-dependent criterion for determining the stability of systems with time-varying delays is obtained. This criterion is used to derive an efficient stabilizing state-feedback design method for systems with parameter uncertainty, of either the polytopic or the norm-bounded types.

Delay-dependent stability and H ∞ control: Constant and time-varying delays

Three main model transformations were used in the past for delay-dependent stability. Recently a new (descriptor) model transformation has been introduced. In the present paper, we compare methods under different transformations and show the advantages of the descriptor one. We obtain new delay-dependent stability conditions for systems with time-varying delays in terms of linear matrix inequalities. We also refine recent results on delay-dependent H ∞ control and extend them to the case of time-varying delays. Numerical examples illustrate the effectiveness of our method.

Improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty

Simple modifications to the representation of the bounded real lemma (BRL) in the linear continuous time-invariant case are introduced. These modifications reduce the overdesign that occurs in the analysis and the design of systems with polytopic-type uncertainties. A different Lyapunov function accompanies each of the vertices of the uncertainty polytope, thus eliminating the need for quadratic stability or stabilizability. The advantages of these new representations are demonstrated by way of two examples.

H∞-control of linear state-delay descriptor systems: an LMI approach

Abstract For continuous-time, linear descriptor system with state-delay a H ∞ -control problem is solved. Sufficient conditions for delay-dependent/delay-independent stability and L 2 -gain analysis are obtained in terms of linear matrix inequalities (LMIs). A bounded real lemma and state-feedback solutions are derived for systems which may contain polytopic parameter uncertainties. The filtering problem is also solved and an output-feedback controller is then found by solving two LMIs. The first LMI is associated with a proportional-derivative state-feedback control. The second LMI is derived in two different forms, the first one corresponds to the adjoint of the system that describes the estimation error and the other stems from the original system. These two forms lead to different results. Numerical examples are given which illustrate the effectiveness of the new theory.

Robust H/sub /spl infin// control of distributed delay systems with application to combustion control

The article is concerned with the H/sub /spl infin// analysis and synthesis of linear distributed delay systems. An efficient stability and L/sub 2/-gain criterion is established. It is based on a recent approach to the analysis and design of linear time delay systems which represents the system in an equivalent descriptor form. The obtained criterion is used to derive an efficient state-feedback control design which stabilizes the distributed delay system and achieves a guaranteed disturbance attenuation level in spite of a polytopic uncertainty in the system parameters. The new method is applied to the robust stabilization and control of combustion in rocket motor chambers.

Robust H/sub /spl infin// filtering of linear systems with time-varying delay

A robust delay-dependent H/sub /spl infin// filtering design is proposed for linear continuous systems with parameter uncertainty and time-varying delay. The resulting filter is of the general linear observer type and it guarantees that the induced L/sub 2/-norm of the system, relating the exogenous signals to the estimation error, is less than a prescribed level for all possible parameters that reside in a given polytope. Our design is based on the application of the descriptor model transformation and Parks inequality for the bounding of cross terms and is expected to be the least conservative as compared to existing design methods. A numerical example indeed demonstrates this advantage of the new filtering scheme.

Stability and guaranteed cost control of uncertain discrete delay systems

Robust stability and the guaranteed cost control problem are considered for discrete-time systems with time-varying delays from given intervals. A new construction of Lyapunov–Krasovskii functionals (LKFs), which has been recently introduced in the continuous-time case, is applied. To a nominal LKF, which is appropriate to the system with nominal delays, terms are added that correspond to the system with the perturbed delays and that vanish when the delay perturbations approach zero. The nominal LKF is chosen in the form of the descriptor type and is applied either to the original or to the augmented system. The delay-independent result is derived via the Razumikhin approach. Guaranteed cost state-feedback control is designed. The advantage of the new tests is demonstrated via illustrative examples.

Robust discrete-time minimum-variance filtering

The bounded-variance filtered estimation of the state of an uncertain, linear, discrete-time system, with an unknown norm-bounded parameter matrix, is considered. An upper bound on the variance of the estimation error is found for all admissible systems, and estimators are derived that minimize the latter bound. We treat the finite-horizon, time-varying case and the infinite-time case, where the nominal system model is time invariant. In the special stationary case, where it is known that the uncertain system is time invariant, we provide a robust filter for all uncertainties that still keep the system asymptotically stable.

Robust minimum variance filtering

This paper deals with the robust minimum variance filtering problem for linear systems subject to norm-bounded parameter uncertainty in both the state and the output matrices of the state-space model. The problem addressed is the design of linear filters having an error variance with a guaranteed upper bound for any allowed uncertainty. Two methods for designing robust filters are investigated. The first one deals with constant parameter uncertainty and focuses on the design of steady-state filters that yield an upper bound to the worst-case asymptotic error variance. This bound depends on an upper bound for the power spectrum density of a signal at a specific point in the system, and it can be made tighter if a tight bound on the latter power spectrum can be obtained. The second method allows for time-varying parameter uncertainty and for general time-varying systems and is more systematic. We develop filters with an optimized upper bound for the error variance for both finite and infinite horizon filtering problems.

Robust Stability and Stabilization of Linear Switched Systems With Dwell Time

Sufficient conditions are given for the stability of linear switched systems with dwell time and with polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, which is non-increasing at the switching instants is assigned to each subsystem. During the dwell time, this function varies piecewise linearly in time after switching occurs. It becomes time invariant afterwards. This function leads to asymptotic stability conditions for the nominal set of subsystems that can be readily extended to the case where these subsystems suffer from polytopic type parameter uncertainties. The method proposed is then applied to stabilization via state-feedback both for the nominal and the uncertain cases.