一类时滞不确定系统的最优滑模设计

Optimal Sliding Mode Design for a Class of Uncertain Systems with Time-delay

针对一类含状态时滞的线性不确定系统,研究具有二次型性能指标的最优滑动模态的设计问题.基于状态方程的标准型,将最优滑模设计问题转化成线性时滞系统的最优控制问题.针对由最优控制的必要条件导出的一族既含时滞项又含超前项的线性两点边值问题,采用时滞线性系统最优控制灵敏度法,将该问题转化为递推求解一族不含时滞项和超前项变量的线性两点边值问题.通过有限次递推,得到最优滑模的近似解.提出并证明了最优滑动运动渐近稳定的充分条件.仿真示例验证了该方法的有效性.

The problem of designing optimal sliding manifolds with a quadratic performance index is presented for a class of linear uncertain systems with state time-delay. First, based on the regular form of the state equation, the problem of designing optimal sliding manifolds is transformed into an optimal control problem for linear time-delay systems. Then according to the necessary conditions of optimal control, the two-point boundary value （TPBV） problems with both time- delay and time-advance terms are induced. ＂A sensitivity approach getting the optimal control law for linear time-delay system is adopted to convert the original TPBV problems into that of solving a series of TPBV iterative formulas without time-delay and time-advance variable terms. By the finite-step recursion, an approximate solution for the optimal sliding mode is obtained. Finally, a sufficient asymptotic stability condition of the optimal sliding motion is derived and proved. Simulation results validate the efficiency of the proposed method.