Delay-dependent Dynamic Feedback H_∞ Control For Uncertain Singular Time-delay Systems

Singular time-delay systems, which are also referred to as generalized differential-difference equations or generalized functional differential equations, are emerged from electrical machinery, input-output model, econometric model, environmental pollution, spaceship attitude, to name a few. Since time delays are frequently encountered in engineering and singular systems can describe practical physical processes more comprehensively than regular ones, increasing attention has been devoted to the study of singular time-delay systems [2]-[7] . On the other hand, the problem of analysis and synthesis for singular time-delay systems is much more complicated than that of regular ones due to the fact that singular time-delay systems have some properties that need not be considered in regular ones, such as non-regularity and solutions with impulses, and so on. A sufficient condition for stability of nominal singular systems with single constant time-delay in terms of linear matrix inequalities (LMIs) was given in [2] and [3] independently. Based on this condition, and by using the idea of generalized quadratic stability and generalized quadratic stabilization, the robust stabilization problem via state feedback controller [2] , robust guaranteed cost control problem [3] and robust H_∞ control problem [4] were discussed, respectively. In [5], a reliable H_∞ controller for singular systems with multiple time delays and time-varying uncertainties was designed, which guaranteed that, for all admissible uncertainties and actuator failures occurring in the prescribed subset of faulty actuators, the closed-loop is exponentially stable and satisfy H∞, norm bound constraint. Above results are all delay-independent, so they are quite conservative, especially when the delay is comparatively small. Recently, in [6], by decomposing the systems into slow and fast subsystems, the authors proposed a delay-dependent stability condition and a state feedback stabilization controller design method for singular time-delay systems in terms of LMIs. However, the conditions obtained are only applicable to systems without uncertainty.In this paper, the problem of delay-dependent robust stabilization for two classes ofsingular time-delay systems with norm-bounded uncertainties is investigated, consider system:All the coefficient matrices except the matrix E include uncertainties. We consider the case of single constant time-delay, the value of which is not required to be precisely known.Based on the delay-dependent stability criterion for nominal singular time-delay systems, and by using the idea of generalized quadratic stabilization, dynamic feedback case is discussed.The sufficient conditions for the existence of the robust stabilization H feedback controllers are given, which ensure that the resulting closed-loop systems are regular, impulse free and asymptotically stable for all admissible uncertainties,while the closed-loop transfer function from the disturbance to the unknown output satisfies a prescribed H -norm bound constraint. And we can calculate the cofficients of the controller by LMIs and Cone Complementary.