Low-Complexity Polytopic Invariant Sets for Linear Systems Subject to Norm-Bounded Uncertainty

Abstract

Abstract-We propose a novel algorithm to compute lowcomplexity polytopic Robust Control Invariant (RCI) sets, along with the corresponding state-feedback gain, for linear discretetime systems subject to norm-bounded uncertainty, additive disturbances and state/input constraints. Using a slack variable approach, we propose new results to transform the original nonlinear problem into a convex/LMI problem whilst introducing only minor conservatism in the formulation. Through numerical examples, we illustrate that the proposed algorithm can yield improved maximal/minimal volume RCI set approximations in comparison with the schemes given in the literature.

Synthesis of separable controlled invariant sets for modular local control design

“…While it is easy ex post to evaluate the effect of ignoring the positive term by looking at its magnitude, it is more subtle how much conservativeness that is introduced by restricting Θ j . Once an initial solution is obtained, it can be iteratively updated as in [19] by using the values of LMI variables Φ 0 j and H 0 x from a previous iteration. The idea is to use a different matrix inverse identity than the one employed in the proof Lemma 3.…”

“…x . Thus after the same re-definitions as above forD j x ,D j d andΦ j , we get the LMI (19). We move on to finding an expression that ensures (17).…”

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

“…Note that for m = n, (P, b) reduces to a low-complexity polytope. 20,25 The requirements on an RCI set 30 include invariance and output constraint satisfaction. For system (1), set (P, b) and given 0 <̄∈ R n , these can be written as…”

“…Furthermore, the fact that matrix P is considered to be nonsquare prevents the direct application of the linearization procedure presented in the work of Tahir and Jaimoukha. 25 In this section, we propose a linearization algorithm, extending the basic ideas proposed in the work of Liu and Jaimoukha 26 that involves the computation of an initial solution. An update algorithm is then presented in the next section.…”

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

“…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.…”