A Spectral Characterization of Controllability for Linear Discrete-Time Systems with Conic Constraints

Abstract

In this paper, we generalize all previously known results on the controllability of discrete-time linear systems with conic input and/or state constraints. In addition, we single out two cases of the problem which did not appear in the literature before. We provide a characterization for the first one of these new cases. For the second newly introduced case, we show that it is rare and pathological. Moreover, we show that the classical results cannot be extended to this last pathological case. These results altogether lead to an almost complete spectral characterization of controllability for discrete-time linear systems with conic input and/or state constraints. Introduction.Ever since introduced by Kalman, controllability has become one of the most fundamental system-theoretical notions. Well-known algebraic characterizations of Kalman [13] and Hautus [10] for the controllability of (unconstrained) linear systems are among the classical results of systems and control theory.In the presence of input constraints, controllability has been studied in [4, 28] for continuous-time and in [7] for discrete-time systems when the input constraint set is the positive orthant. In the more general context of convex input constraints, controllability and related notions have been studied in [6,17,19] for discrete-time and [18] for continuous-time systems. A unified treatment of both the continuousand discrete-time cases can be found in [31] for the case when the input constraint set is a compact set containing the origin in its interior. Based on the foundations laid by these papers, variants of controllability such as instantaneous controllability and small-time controllability as well as the structure of reachable sets for continuous-time linear systems with input constraints have been investigated in [2,3,5,8,9,14,32]. All these papers provide elegant characterizations for controllability as well as the related notions of reachability and null controllability for input constraints case.Although controllability has been extensively studied and well understood with input constraints, it is overlooked to a great extent in the presence of state constraints. To the best of the authors' knowledge, controllability was investigated in [11,12,15,16] under state constraints either for very particular constraint sets or with restrictive assumptions. As such, the controllability problem for linear systems subject to state constraints remains still unsolved in general. In a way, this is somewhat surprising as constrained control problems have been of great interest to the control community due to the importance of state constraints in practical control problems.In this paper, we focus on discrete-time systems that are subject to conic input/state constraints. Apart from being interesting on their own, conic constraints

Structured distance to non‐surjectivity of convex processes and its applications to robust controllability under structured perturbations

“…where ℱ is a strict convex process. Indeed, it is well-known (see, e.g [28,31] that a result similar to (31) holds for (46): Let ℱ ∈ S(ℝ n , ℝ n ), then (46) is controllable ⟺ ℱ is reproducing and Im(ℱ − λI) = ℝ n ,…”

“…In the case K = ℝ, a number of equivalent characterisations for controllability of inclusions (30) have been obtained in [1,5] (see also [28] for the case of discrete-time systems x k + 1 ∈ ℱ(x k ), k = 0, 1, …), one of which can be stated as follows (see Aubin et al [1]): Let ℱ ∈ S(ℝ n , ℝ n ), then (30) is controllable ⟺ ℱ is reproducing and Im(ℱ − λI) = ℝ n ,…”

On Reachability and Null-Controllability of Nonstrict Convex Processes

“…We call L − and L + , respectively, the minimal and maximal linear processes associated with H. If H is not clear from context, we write L − (H) and L + (H) in order to avoid confusion. Example 2: Let H be of the form (6). It can be shown that if the set {u | Bu = 0, Du ∈ Y} is a subspace, then…”

“…This means that cl(H(K)) ⊆ K since K is closed. Hence, H(K) ⊆ K. In other words, K is strongly H invariant due to Lemma 3. There is a link between weakly H invariant cones and eigenvalues of the dual of H, given in [6,Thm. 3.2].…”

“…Proof: To prove sufficiency, we assume that F (H) ⊆ R(H). By [6,Lemma 4.3] we know that if dom H + R − = R n , then F (H) + R − = R n . By our assumption, we therefore know that R n = F (H) + R − ⊆ R(H) + R − = R(H).…”