Full‐complexity polytopic robust control invariant sets for uncertain linear discrete‐time systems

Abstract

This paper presents an algorithm for the computation of full-complexity polytopic robust control invariant (RCI) sets, and the corresponding linear statefeedback control law. The proposed scheme can be applied for linear discretetime systems subject to additive disturbances and structured norm-bounded or polytopic uncertainties. Output, initial condition, and performance constraints are considered. Arbitrary complexity of the invariant polytope is allowed to enable less conservative inner/outer approximations to the RCI sets whereas the RCI set is assumed to be symmetric around the origin. The nonlinearities associated with the computation of such an RCI set structure are overcome through the application of Farkas' theorem and a corollary of the elimination lemma to obtain an initial polytopic RCI set, which is guaranteed to exist under certain conditions. A Newton-like update, which is recursively feasible, is then proposed to yield desirable large/small volume RCI sets. KEYWORDS full-complexity polytope, LMIs, optimization, robust control invariant sets, uncertain systems Int J Robust Nonlinear Control. 2019;29:3587-3605.wileyonlinelibrary.com/journal/rnc

Computation of robust control invariant sets with predefined complexity for uncertain systems

“…We allow the sets , and to be non‐symmetric, unlike References 10,11,17,18 where some or all the sets are assumed to be symmetric. The affine parameter dependence of the system, as considered in for example, References 11,18, is just a special case that corresponds to D p = 0 in (8). Moreover, the algorithm proposed in this paper does not impose any structural restriction on the matrix .…”

“…Depending upon the choice of the initial set, the result of such a recursive method is the arbitrarily close outer/inner approximation of the maximal RCI set 8,9 . Although effective, the geometric approach does not guarantee finite time termination of the procedure, and also the obtained set may have a very high representational complexity 2,4,8‐11 . For polytopic systems, the computational complexity grows exponentially with each additional vertex and system dimension 1 …”

“…As a result, References 11,15‐18 proposed different approaches for the computation of restricted complexity or low complexity polytopic RCI sets. Algorithms developed by References 15‐17 are applicable to polytopic linear systems, whereas those in References 11,18 can be applied to affinely parameter‐dependent systems. It is important to mention here that all these algorithms assume the invariance inducing controller to be based on linear state feedback, which is computed by the algorithm while optimizing the volume of the set.…”

“…By then using this condition along with the full block S‐procedure, 25 a sufficient linear matrix inequality (LMI) feasibility condition for invariance is derived. Also, an approach to directly maximize the volume of the RCI sets is presented, which is less conservative than using an indirect approach used in References 10,11. In contrast to References 10,11,15,17,18,22, we directly compute the RCI sets without imposing any structural restrictions on the control input so that possibly larger RCI sets can be obtained.…”

“…Also, an approach to directly maximize the volume of the RCI sets is presented, which is less conservative than using an indirect approach used in References 10,11. In contrast to References 10,11,15,17,18,22, we directly compute the RCI sets without imposing any structural restrictions on the control input so that possibly larger RCI sets can be obtained. The algorithm generates RCI sets of monotonically increasing volume at each iteration until convergence.…”