The robust minimal controllability problem

Abstract

In this paper, we address two minimal controllability problems, where the goal is to determine a minimal subset of state variables in a linear time-invariant system to be actuated to ensure controllability under additional constraints. First, we study the problem of characterizing the sparsest input matrices that assure controllability when the autonomous dynamics' matrix is simple. Secondly, we build upon these results to describe the solutions to the robust minimal controllability problem, where the goal is to determine the sparsest input matrix ensuring controllability when specified number of inputs fail. Both problems are NP-hard, but under the assumption that the dynamics' matrix is simple, we show that it is possible to reduce these two problems to set multi-covering problems. Consequently, these problems share the same computational complexity, i.e., they are NP-complete, but polynomial algorithms to approximate the solutions of a set multi-covering problem can be leveraged to obtain close-to-optimal solutions to either of the minimal controllability problems.

Minimum cost constrained input-output and control configuration co-design problem: A structural systems approach

“…In addition, in [9], [7], [8] the submodularity properties of functions of the controllability Grammian are explored, and design algorithms are proposed that achieve feasible placement with certain guarantees on the optimality gap. The I/O selection problem considered in the present paper differs from the aforementioned problems in the following two aspects: first, the selection of the inputs is restricted to belong to a specific given set of possible inputs, i.e., we study constrained input placement, and, hence, differing from [4], [5], [6] in which unconstrained input placement is studied. Secondly, it contrasts with [7], [8], [9], [10] in the sense that we do not aim to ensure performance in terms of a function of the controllability Grammian, but we aim to minimize the overall actuation cost, measured in terms of manufacturing/installation/preference costs.…”

“…In [4], the minimal controllability problem (MCP), i.e., the problem of determining the sparsest input matrix that ensures controllability of a given the system dynamics matrix, was shown to be NP-hard, and some greedy algorithms provided. Exact solutions to MCP are explored in [5], and in [6], using graph theoretical constructions, the minimal controllability problem is shown to be polynomially solvable for almost all numerical realizations of the dynamic matrix, satisfying a predefined pattern of zeros/nonzeros. Alternatively, in [7], [8], [9], [10] the configuration of actuators is sought to ensure certain performance criteria; more precisely, [7], [8], [10] focus on optimizing properties of the controllability Grammian, whereas [9] studies leader selection problems, in which leaders are viewed as inputs to the system, and the selection criteria aims to speed up convergence.…”

On the Computation of the Distance-based Lower Bound on Strong Structural Controllability in Networks

“…Further studies in this direction include enumerating and counting strong structurally controllable graphs for a given set of network parameters (leaders and nodes) [22], [23], leader selection to achieve desired structural controllability (e.g., [24], [4], [25], [5], [7], [6], [26]), and network topology design for a desired control performance (e.g., [27], [28], [29], [30], [31]).…”

Strongly Uncontrollable Network Topologies

“…Roughly speaking, the existence of an equitable partition of the nodes of a network induce an invariant subspace for the uncontrolled dynamics and thus if the control nodes are chosen to preserve the invariant subspace then uncontrollability ensues. A closely related line of research is the so-called minimal controllability problem which is concerned with the scenario of controlling a large-scale multiagent system with the fewest possible number of control nodes [14], [15], [16], [17], [18]. Although it has been shown that solving minimal controllability problems is computationally intractable for generic systems (unless P = N P ), heuristic algorithms are known that produce approximate solutions [14], [16], [17], [18].…”

“…A closely related line of research is the so-called minimal controllability problem which is concerned with the scenario of controlling a large-scale multiagent system with the fewest possible number of control nodes [14], [15], [16], [17], [18]. Although it has been shown that solving minimal controllability problems is computationally intractable for generic systems (unless P = N P ), heuristic algorithms are known that produce approximate solutions [14], [16], [17], [18]. On the other hand, in the case of structured systems, the minimal controllability problem can be solved in polynomial time [19], [15].…”