In this paper, we study the non-fragile controllability of stochastic systems with both state and control input delays. The main purpose is to design a state feedback controller under admissible uncertainties so that the closed-loop system is not only mean-square asymptotically stable, but also satisfies the given performance index.
By using the linear matrix inequality (LMI) method, the sufficient conditions for the solvability of the non-fragile controllability problem of the delay stochastic system are given, and the parametric expressions of the non-fragile controller are given by their feasible solutions. Finally, a numerical example is provided to illustrate the application of this method.
Because uncertainties and time delays are ubiquitous, and their existence is the main cause of system instability and serious damage to system performance, many experts and scholars have studied various systems and reached feasible conclusions [1-3]. Reference [1] discusses a class of switched singular systems consisting of any finite number of singular subsystems. In reference [2], the problem of fuzzy guaranteed cost control is studied by using T-S model. However, these results rarely consider the imprecise implementation of the controller. In practice, due to the inadequate accuracy of the instrument and the rounding error of the calculation, the parameters of the controller change, which leads to the performance degradation or stability destruction of the closed-loop system. Therefore, uncertainties, i.e. non-fragility, must be considered in the design of controllers.
In recent years, non-fragile control has achieved a series of results [4,5]. In reference [4], a parallel distributed compensation method based on state feedback is used to study the non-fragile problem of a class of T-S fuzzy model describing nonlinear time-delay systems. In addition, due to the important application of stochastic systems, many research methods and theories have been gradually extended from deterministic systems to stochastic systems [6-7]. As far as the literature is concerned, the problem of non-fragile control has not been discussed in stochastic systems with delays in both state and control inputs. Combining with the additive uncertainties appearing in the controller, this paper discusses the above systems and gives sufficient conditions for the solvability of the problem in the form of linear matrix inequality (LMI).
Finally, a numerical example is given to illustrate the effectiveness of the proposed method. In the above formulas, the state variable is the control input, the external interference input and the control output. Sum is a real constant, a continuous real number initial vector function on it. Definition 1: Given scalar and time-delay stochastic systems (1)~3) are non-fragile controllable. It means to design controllers to make closed-loop systems asymptotically stable in mean square and satisfy the uncertainties of any non-zero and all admissible controllers under zero initial conditions.
Among them, the corresponding satisfiable controllers (5)~7) are non-fragile controllers. And given by formulas (9) and (10) of [8], respectively, the delay stochastic systems (1)~3) are considered to be non-fragile controllable. Obviously, from the above formula (8), we can deduce the reference [8] (8). Therefore, the closed-loop system (16)() is mean-square asymptotically stable. The combination formula (24) and the sum formula (29) clearly show that the formula (16) is valid.
The proof is complete.
In this paper, a stochastic system with delays in state and input is discussed, and the problem of non-fragile control of the system is studied. Non-fragile control considers that when uncertainties exist in the designed controller, thermostatic element it can still ensure that the system is not only mean-square asymptotically stable, but also satisfies the given index. The research results can be expressed in the form of LMI, which can be easily solved directly by the LMI toolbox in matlab.