On a Relaxation of Time-Varying Actuator Placement

Abstract

We consider the time-varying actuator placement in continuous time, where the goal is to maximize the trace of the controllability Grammian. A natural relaxation of the problem is to allow the {0, 1} variable indicating whether an actuator is used at a given time to take on values in the closed interval [0, 1]. We show that all optimal solutions of both the original and the relaxed problems can be given via an explicit formula, and that, as long as the input matrix has no zero columns, the solutions sets of the original and relaxed problem coincide. arXiv:1912.09454v2 [eess.SY] 25 Dec 2019Proof of Proposition 3.5. That the second term under the integral integrates to δ f * (b)µ(S ) is clear. We only sketch the rest of the proof, as it is almost identical to the second part of Proposition 3.4. Indeed, we proceed exactly as in Proposition 3.4 until the penultimate line, where the inequality f * (t) > f * (b) is now replaced with t ≤ b l or t < b l .

Controllability maximization of large-scale systems using projected gradient method

“…If tr(C T (B)) is sufficiently large, there exists a direction that can be specified by an eigenvector of C T (B) such that the state x(t) of system (1) can move to the direction with a low input energy. Furthermore, it is well known that tr(C −1 T (B)) quickly leads to intractable problems [15]. Consequently, in the existing studies [10], [14], [15], tr(C T (B)) was adopted as a controllability index.…”

“…Furthermore, it is well known that tr(C −1 T (B)) quickly leads to intractable problems [15]. Consequently, in the existing studies [10], [14], [15], tr(C T (B)) was adopted as a controllability index. Note that unlike tr(C −1 T (B)), tr(C T (B)) can be defined even if system (1) is not controllable.…”

“…Specifically, BV(t) was considered as the matrix B, where B ∈ R n×m and V(t) ∈ {0, 1} m×m denote a fixed constant matrix and time-varying diagonal matrix, respectively, and an optimization problem to determine the diagonal entries of V(t) was examined. This optimization problem was then completely resolved, as reported in [15]. However, the methods to predetermine M candidate column vectors of B in [9]- [11] and the matrix B in [14], [15] for a large-scale networked dynamical system remain unclear.…”

“…This optimization problem was then completely resolved, as reported in [15]. However, the methods to predetermine M candidate column vectors of B in [9]- [11] and the matrix B in [14], [15] for a large-scale networked dynamical system remain unclear.…”

“…To avoid this limitation, in this letter, we consider the controllability maximization problems from a different perspective than those considered in [9]- [11], [14], [15]. In particular, we formulate two optimal decision problems of B in system (1) without using candidates of column vectors considered in [9]- [11] and a fixed B considered in [14], [15]. The first problem is a sparsity constraint optimization problem with a box constraint that specifies the possible values of each element in B.…”