Matroid-Constrained Approximately Supermodular Optimization for Near-Optimal Actuator Scheduling Luiz F. O. Chamon, Alexandre Amice, and Alejandro Ribeiro Abstract— This work considers the problem of scheduling instants one-by-one by selecting the feasible match that most actuators to minimize the Linear Quadratic Regulator (LQR) reduces the objective. What allows us to construct solutions objective. In general, this problem is NP-hard and its solution in this way is the fact that the constraint on the number can therefore only be approximated even for moderately large systems. Although convex relaxations have been used to obtain of actuators can be mapped into a combinatorial structure these approximations, they do not come with performance called a matroid, a family of sets that can be built element guarantees. Another common approach is to use greedy search. by element [20, Ch. 39]. Without a theoretical guarantee, Still, classical guarantees do not hold for the scheduling however, it is unclear how well the actuators selected by this problem because the LQR cost function is neither submodular heuristic perform due to the intractability of the problem. nor supermodular. Though surrogate supermodular figures of merit, such as the log det of the controllability Gramian, are Still, near-optimal guarantees exist for greedy solutions often used as a workaround, the resulting problem is not in certain circumstances. For generic matroid-constrained equivalent to the original LQR one. This work shows that no minimization problems, such as actuator scheduling (see Sec- change to the original problem is needed to obtain performance tion II), a common guarantee relies on the supermodularity of guarantees. Specifically, it proves that the LQR cost function is the objective, i.e., on a form of diminishing return displayed approximately supermodular and provides new near-optimality certificates for the greedy minimization of these functions over by set functions such as the log determinant or the rank a generic matroid. These certificates are shown to approach the of the controllability Gramian. Greedily minimizing these classical 1=2 guarantee of supermodular functions in relevant supermodular functions over an arbitrary matroid yields 1=2- application scenarios. optimal solutions [21]. Other methods based on continuous extensions of these functions can be used to improve this I. INTRODUCTION factor to 1 − 1=e [22]. Nevertheless, many cost functions of Many problems in control theory and application are interest are not supermodular. In particular, this is the case combinatorial in nature, such as selecting sensors for state of the LQR objective for actuator scheduling [9], [11]. estimation, actuators for regulation, or allocating tasks to In this work, we address this issue using the concept autonomous agents [1]–[4]. In these problems, we seek to of approximate supermodularity [4], [23]–[25]. We start choose from a discrete set of possible elements (sensors, by casting the actuator scheduling problem as a matroid- tasks, or actuators) so as to optimize some objective, e.g., the constrained optimization problem and showing how it can be state estimation mean-square estimation (MSE), the number solved greedily (Section II). Since the resulting constraints of agents required or the time to complete all tasks, or some are more general than bounding set cardinality, we derive a control performance measure. What makes these problems novel near-optimality certificate for the greedy minimization difficult is the fact that only a limited number of elements can of approximately supermodular set functions over a generic be chosen due to cost, power, or communication constraints. matroid (Section III). We then prove that the LQR cost This leads to discrete optimization problems that are NP- function is approximately supermodular by bounding how hard in general and whose solutions can therefore only be much it violates the diminishing returns property, giving approximated even for moderately large systems [5]–[11]. explicit guarantees on the greedy actuator scheduling solu- To make the discussion more concrete, consider the prob- tion (Section IV). We argue that these guarantees improve lem known as actuator scheduling in which we seek to in scenarios of practical interest and illustrate their typical select a given number of actuators per time instant so as to values in simulations (Section V). minimize some control cost. One approach to approximate Notation: Lowercase and uppercase boldface letters rep- the optimal solution of this problem is using a convex resent vectors (x) and matrices (X) respectively, while relaxation as in [12]–[15], typically including a sparsity pro- calligraphic letters denote sets (A). We write jAj for the moting regularization. Though practical, these methods do cardinality of A and X 0 (X 0) to say X is positive not come with approximation guarantees. Another common semi-definite (definite) matrix. Hence, X Y , bT Xb ≤ avenue is to build the schedule using discrete optimization T n b Y b for all b 2 R . We write λmax and λmin for the methods, such as tree pruning [16] or greedy search [17]– maximum and minimum eigenvalue of a matrix, respectively, [19]. The latter solution is attractive in practice due to its low and use R+ to denote the non-negative real numbers. complexity and iterative nature: actuators are matched to time II. PROBLEM FORMULATION Electrical and Systems Engineering, University of Pennsylvania. fluizf, amice, [email protected]. This work Consider a discrete-time, time invariant dynamical system is supported by the ARL DCIST CRA W911NF-17-2-0181. and denote its set of available inputs/actuators at time k by Vk. Suppose that this system is actuated only through Algorithm 1 Greedy search for minimization over a matroid n a subset Sk ⊆ Vk of its inputs, so that its states xk 2 R Given an objective function f and a matroid (E; I), evolve according to initialize G0 ;, Z E, and t 0 while Z 6= ; X xk+1 = Axk + biui;k, (1) g argminu2Z f(Gt [ fug) i2Sk Z Z n fgg if G [ fgg 2 I then G G [ fgg and t t + 1 where b 2 n is the input vector and u is the control t t+1 t i R i;k end action of the i-th input and A is the state transition matrix. Let V = V0 [V1 [···[VN−1 be the set of all actuators available throughout the time window k = 0; 1;:::;N − 1 Proof. This result follows directly from the classical dy- and call V ⊇ S = S0 [S1 [···[SN−1 a schedule, as it denotes the set of active inputs at each time instant. Assume namic programming argument for the LQR by taking into ac- count only the active actuators in S (e.g., see [26]). For ease without loss of generality that Vj \Vk = ; for all j 6= k. If an input v is available at different instants, it can be represented of reference, we provide a derivation in the appendix. j k by different elements, e.g., v 2 Vj and v 2 Vk. Although (PI) is NP-hard in general [5]–[11], its feasible Actuator scheduling seeks to choose which actuators are sets have a matroidal structure. Matroids are algebraic struc- used at each time step so as to minimize a control cost subject tures that generalize the notions of linear independence in a to a budget constraint. Formally, given a limit sk on the vector space. Formally, number of active inputs at iteration k, we seek a feasible schedule that minimizes the value of the LQR problem, i.e., Definition 1. A matroid M = (E; I) consists of a finite set of elements E and a family I ⊆ 2E of subsets of E, called minimize J(S) , V (S) − V (;) independent sets, that satisfy: S⊆V (PI) 1) ; 2 I; subject to jS \ Vkj ≤ sk, k = 0;:::;N − 1 2) if A ⊆ B and B 2 I, then A 2 I; with 3) if A; B 2 I and jAj < jBj, then there exists e 2 B n A such that A [ feg 2 I. " T The most common example of matroid is the uniform V (S) = min E xN QN xN U(S) matroid, for which the independent sets are defined as Iu = N−1 !# fA ⊆ E j jAj ≤ kg for some k ≥ 1. It is ready that Iu X T X 2 obeys the properties in Definition 1. These matroids are + xk Qkxk + ri;kui;k , (2) k=0 i2S\Vk found in cardinality constrained problems, such as actuator or sensor selection [4], [7]–[10], [27]. The feasible sets N−1 where U(S) = fui;k j i 2 S \ Vkgk=0 is the set of control of problem (PI), on the other hand, form a more general actions, Qk 0 are the state weights, and ri;k > 0, for structure known as a partition matroid: given a partition fEkg all i and k, are the input weights. The expectation in (2) of the ground set E, i.e., E = [kEk and Ej \Ek = ; for j 6= k, is taken with respect to the initial state x0 ∼ N (x¯0; Π0) and parameters ak, the independent sets of a partition matroid assumed to be a Gaussian random variable with mean x¯0 and are given by Ip = fA ⊆ E j jA \ Ekj ≤ akg. covariance Π0 0. Observe that the constant V (;) in the Notice that properties 1 and 2 from Definition 1 imply objective of (PI) does not affect the solution of the optimiza- that the independent sets of matroids can be constructed it- tion problem. It is used so that J(;) = 0, which simplifies the eratively, element-by-element, which suggests that a solution presentation of our near-optimal certificates (see Section III). of (PI) could be obtained by using the greedy procedure in Note from (2) that the objective of (PI) itself involves Algorithm 1 with E = V, f(S) = J(S), and I = fA ⊆ V j a minimization.