基于LMI的鲁棒单调收敛迭代学习控制设计
Robust Monotonic Convergent Iterative Learning Control Design: an LMI-based Method
摘要：
研究了不确定线性系统的鲁棒单调收敛迭代学习控制（ILC）问题，给出了基于误差信号$l_{2}$范数的迭代学习控制算法。首先，证明了迭代学习控制系统的鲁棒单调收敛性可以等价地由矩阵多项式在某个集合上的正定性来建立。然后，给出了一个以平方和（SOS）形式表示的正定性的充分必要条件，该条件与线性矩阵不等式（LMI）的可行性是一致的。在此条件下，通过求解一组凸优化问题，得到了最大化收敛速度的最优迭代学习控制算法。此外，学习函数的阶数可以任意选择，使得设计者可以灵活地决定学习算法的复杂度。
英文摘要：
This work investigates robust monotonic convergent iterative learning control (ILC) for uncertain linear systems in both time and frequency domains, and the ILC algorithm optimizing the convergence speed in terms of $l_{2}$ norm of error signals is derived. Firstly, it is shown that the robust monotonic convergence of the ILC system can be established equivalently by the positive definiteness of a matrix polynomial over some set. Then, a necessary and sufficient condition in the form of sum of squares (SOS) for the positive definiteness is proposed, which is amendable to the feasibility of linear matrix inequalities (LMIs). Based on such a condition, the optimal ILC algorithm that maximizes the convergence speed is obtained by solving a set of convex optimization problems. Moreover, the order of the learning function can be chosen arbitrarily so that the designers have the flexibility to decide the complexity of the learning algorithm.
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