Robust Exponential Stabilization of Uncertain Impulsive Bilinear Time-Delay Systems with Saturating Actuators

This paper investigates the problem of robust exponential stabilization for uncertain impulsive bilinear time-delay systems with saturating actuators. By using the Lyapunov function and Razumikhin-type techniques, two classes of impulsive systems are considered: the systems with unstable discrete-time dynamics and the ones with stable discrete-time dynamics. Sufficient conditions for robust stabilization are obtained in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the theoretical results.


Introduction
In practical control systems, impulsive dynamical systems are very important and have attracted considerable interest in science and engineering during the past decades.Two classical monographs are Lakshmikantham et al. [1] and Bainov and Simeonov [2].In general, as reported in [1], impulsive systems provide a natural framework for mathematical modeling of many real-world evolutionary processes where the states undergo abrupt changes at certain instants or at variable instants, or autonomous systems with impulsive effects.Stability properties of impulsive systems have been extensively studied in the literatures.We refer to Bainov and Simeonov [2], Li, Soh and Xu [3,4], Li, Wen and Soh [5], Yang [6], Chen and Zheng [7], Barreiro and Baños [8], and the references therein.The impulsive control method based on stability theory of impulsive dynamical systems has been widely used, see [7], Xu and Teo [9], and Liu, Eberhard and Teo [10].
Saturating nonlinear actuators is one of the common nonlinearities and exists in many practical systems.If a control system is designed to be stable without considering the saturation, the stability of the closed-loop-controlled system cannot be guaranteed, see Ma and Zhang [11].Two important papers on saturation control are Full [12] and Sontag and Sussmann [13].However, as we have known, no papers have considered the saturation control in impulsive systems.
This work is inspired by Wu and Wei [7], in which the authors considered the problems of robust stability and stabilization for uncertain impulsive time-delay systems.Unfortunately, they need all the impulsive time sequences to satisfy some strict conditions.That is, the length of the intervals between two adjacent time instants must have upper bound or lower bound.But in practical systems, it is always impossible or difficult to obtain it.In this paper, robust stabilization for uncertain impulsive time-delay systems with saturating actuators is considered.With saturation control, new conditions based on Lapunov-Razumikhin function and LMIs are established which can easily be used for systems with any time sequences.

Problem Formulation and Preliminaries
Throughout this paper, if not explicitly given, matrices are assumed to have compatible dimensions.For symmetric matrices A and B, the notation A ≥ B (A > B, A ≤ B, A < B) means A−B is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix.λ max (•) and λ min (•) represent the maximum and minimum eigenvalues of the corresponding matrix, respectively.• denotes Euclidean norm for vectors or the spectral norm of matrices.PC([−τ, 0], R n ) denotes the set of piecewise right continuous function φ : [−τ, 0] → R n with the norm defined by φ τ = sup −τ≤θ≤0 φ(θ) , where τ is a known positive constant.
In the following, we will divide two cases to consider the robust stabilization of system (1).We denote by N min (β) the class of impulsive time sequences that satisfy inf k {t k − t k−1 } ≥ β and denote by N max (β) the class of impulsive time sequences that satisfy sup k {t k − t k−1 } ≤ β.
We need the following lemmas.

Robust Stabilization
In this section, we consider the robust stabilization of system (1).
) and u(t) = 0 is said to be exponentially stable over a given class Definition 4. System ( 1) is said to be robustly exponentially stabilizable over a given class N with respect to static state feedback if there exists a memoryless linear state feedback law, such that the corresponding closed-loop system is robustly exponentially stable for all uncertainties satisfying (2).
Remark 7. By Theorem 6, in the case μ < 1, different from Chen and Zheng [7], we can stabilize impulsive systems with any time sequences by saturation control.Remark 8.In Chen and Zheng [7], the authors need continuous control input u c (t) and impulsive control input u d (t).But in this paper, we only need continuous control input u(t).
For μ = 1, we have the following result.Theorem 9. Assume that there exist matrices X > 0, Y and positive scalars α, d, ε 1 , ε 2 , and ε 3 such that (8) and (11) hold, then for any bounded time-delay τ(t) ≤ τ, system (1) can be robustly stabilized by the control law (5) with K = Y X −1 for any impulsive time sequence {t k }, where Proof.Choose a sufficiently small scalar h > 0, and set similar to the proof of Theorem 6, we can obtain the result of Theorem 9.The proof is complete.
For μ > 1, we have the following result.
Remark 11.On the conditions in Theorem 10, with the prescribed scalar β > 0, we can stabilize not only the impulsive systems over time sequence N min (β), but some ones over other time sequences.

Numerical Examples
In this section, two illustrative examples will be presented to show the effectiveness of the results obtained.

Conclusion
This paper studied a class of uncertain impulsive time-delay systems.Based on Lyapunov function and Razumikhintype techniques, we obtain conditions which can guarantee the above systems robustly exponentially stabiliable for any impulsive time sequences and improves some related results.