Emilia Fridman

Technical Communique: Robust sampled-data stabilization of linear systems: an input delay approach

A new approach to robust sampled-data control is introduced. The system is modelled as a continuous-time one, where the control input has a piecewise-continuous delay. Sufficient linear matrix inequalities (LMIs) conditions for sampled-data state-feedback stabilization of such systems are derived via descriptor approach to time-delay systems. The only restriction on the sampling is that the distance between the sequel sampling times is not greater than some prechosen h>0 for which the LMIs are feasible. For h->0 the conditions coincide with the necessary and sufficient conditions for continuous-time state-feedback stabilization. Our approach is applied to two problems: to sampled-data stabilization of systems with polytopic type uncertainities and to regional stabilization by sampled-data saturated state-feedback.

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.

New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems ☆

Abstract A new (descriptor) model transformation and a corresponding Lyapunov–Krasovskii functional are introduced for stability analysis of systems with delays. Delay-dependent/delay-independent stability criteria are derived for linear retarded and neutral type systems with discrete and distributed delays. Conditions are given in terms of linear matrix inequalities and for the first time refer to neutral systems with discrete and distributed delays. The proposed criteria are less conservative than other existing criteria (for retarded type systems and neutral systems with discrete delays) since they are based on an equivalent model transformation and since they require bounds for fewer terms. Examples are given that illustrate advantages of our approach.

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.

Brief paper: A refined input delay approach to sampled-data control

This paper considers sampled-data control of linear systems under uncertain sampling with the known upper bound on the sampling intervals. Recently a discontinuous Lyapunov function method was introduced by using impulsive system representation of the sampled-data systems (Naghshtabrizi, Hespanha, & Teel, 2008). The latter method improved the existing results, based on the input delay approach via time-independent Lyapunov functionals. The present paper introduces novel time-dependent Lyapunov functionals in the framework of the input delay approach, which essentially improve the existing results. These Lyapunov functionals do not grow after the sampling times. For the first time, for systems with time-varying delays, the introduced Lyapunov functionals can guarantee the stability under the sampling which may be greater than the analytical upper bound on the constant delay that preserves the stability. We show also that the term of the Lyapunov function, which was introduced in the above mentioned reference for the analysis of systems with constant sampling, is applicable to systems with variable sampling.

Stability of linear descriptor systems with delay: a Lyapunov-based approach

The Lyapunov second method is developed for linear coupled systems of delay differential and functional equations. By conventional approaches such equations may be reduced to the neutral systems and the known results for the latter may be exploited. In the present paper we introduce a new approach by constructing a Lyapunov–Krasovskii functional that corresponds directly to the descriptor form of the system. Moreover, by representing a neutral system in the descriptor form we obtain new stability criteria for neutral systems which are less conservative than the existing results. Sufficient conditions for delay-dependent/delay-independent stability and for robustness of stability with respect to small delays are given in terms of linear matrix inequalities. Illustrative examples show the effectiveness of the method.

Brief paper: Wirtinger's inequality and Lyapunov-based sampled-data stabilization

Discontinuous Lyapunov functionals appeared to be very efficient for sampled-data systems (Fridman, 2010; Naghshtabrizi, Hespanha, & Teel, 2008). In the present paper, new discontinuous Lyapunov functionals are introduced for sampled-data control in the presence of a constant input delay. The construction of these functionals is based on the vector extension of Wirtingers inequality. These functionals lead to simplified and efficient stability conditions in terms of Linear Matrix Inequalities (LMIs). The new stability analysis is applied to sampled-data state-feedback stabilization and to a novel sampled-data static output-feedback problem, where the delayed measurements are used for stabilization.

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.