Abstract
AbstractThis 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 l2 norm of error signals is derived. First, 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 for the positive definiteness is proposed, which is amendable to the feasibility of linear matrix inequalities. 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.
Highlights
Iterative leaning control (ILC) is a useful control strategy to improve tracking performance over repetitive trials ([1], [2], [3])
In some stable ILC systems the error can grow very large before converging to the desired output trajectory, which is undesirable in most practical applications [5], [6]
[13] studies the problem of robust convergent ILC by solving μ-synthesis problem; [14] presents sufficient conditions for robust monotonic convergence analysis based on μ analysis; [15], [16] provide H∞-based design method to synthesize high-order ILCs
Summary
This work investigates robust monotonic convergent ILC with learning function of flexible order for uncertain linear systems in both time and frequency domains, and the optimal ILC algorithm which optimizes the convergence speed in terms of l2 norm of error signals is derived. We provide a necessary and sufficient condition for the positive definiteness of the matrix polynomial over the constrained set, which is equivalent to the feasibility of linear matrix inequalities (LMIs) Based on such a condition, the optimal ILC algorithm that maximizes the robust monotonic convergence speed can be obtained by solving a set of convex optimization problems. Given a matrix polynomial P (x), the notation deg(P ) denotes the maximum of the degrees of the entries of. F (s) = (b(s) ⊗ I)T M (b(s) ⊗ I) , L(α) : Rω(r,2d,n) → Rnσ(r,d)×nσ(r,d) is a linear parametrization of the linear set L = {L = LT : (b(s) ⊗ I)T L (b(s) ⊗ I) = 0}, and α It follows that F (s) is a SOS matrix polynomial if and only if there exists α satisfying the LMI.
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