A perspective-based convex relaxation for switched-affine optimal control

We consider the switched-affine optimal control problem, i.e., the problem of selecting a sequence of affine dynamics from a finite set in order to minimize a sum of convex functions of the system state. We develop a new reduction of this problem to a mixed-integer convex program (MICP), based on perspective functions. Relaxing the integer constraints of this MICP results in a convex optimization problem, whose optimal value is a lower bound on the original problem value. We show that this bound is at least as tight as a similar bound obtained from another well-known MICP reduction (via conversion to a mixed logical dynamical system); our numerical study indicates it is often substantially tighter. Using simple integer-rounding techniques, we can also use our formulation to obtain an upper bound (and corresponding sequence of control inputs). In our numerical study, this bound was typically within a few percent of the optimal value, making it attractive as a stand-alone heuristic, or as a subroutine in a global algorithm such as branch and bound. We conclude with some extensions of our formulation to problems with switching costs and piecewise affine dynamics. 1 Switched-affine control A switched-affine system has the form xt+1 = A xt + b ut , t = 0, 1, . . . , where xt ∈ R n is the state at time t, ut ∈ {1, . . . , K} is the control input at time t, and A, . . . , A and b, . . . , b are given matrices and vectors. At each time period, the control input selects from a given finite set of affine dynamics. We assume, without loss of generality, that (A, b) 6= (A, b) for i 6= j. Switched-affine systems arise in various engineering applications, for example as models of switched-mode power supplies and power conversion circuits. Mechanical Engineering Department, Stanford University. [email protected] Electrical Engineering Department, Stanford University. [email protected] 1 The switched-affine control problem is minimize ∑T t=0 gt(xt) subject to xt+1 = A xt + b ut ut ∈ {1, . . . , K}, (1) where the constraints must hold for t = 0, . . . , T − 1. The problem variables are the system states x0, . . . , xT ∈ R n and the control inputs u0, . . . , uT−1. The problem parameters are the dynamics (A, b) for i = 1, . . . , K and the stage cost functions g0, . . . , gT . We assume the stage cost functions gt : R n → R∪{∞} are convex and extended valued, which allows us to represent convex state constraints in the stage cost function. We define the state constraint set as Xt = {x | gt(x) < ∞}, so the objective is infinite unless xt ∈ Xt holds for t = 1, . . . , T . We can use g0 to encode a given initial condition, so that X0 = {xinit}, for some xinit ∈ R . The switched-affine control problem (1) is NP-hard in general, and can be solved globally only at great computational cost (in the worst-case). However, by reformulating it as a mixed-integer convex program (MICP), lower bounds on the optimal value can be obtained by relaxing the integer constraints, and upper bounds can be obtained by applying an integerrounding heuristic to the relaxed solution. These bounds can be used as the basis for a global solver (using, e.g., branch and bound), or alternatively, the rounding procedure can be used as a heuristic to produce a good, if not optimal, sequence of control inputs. The success of both methods (i.e., the run-time of a global solution algorithm, or the quality of the heuristic control input sequence) depends crucially on the MICP reformulation (and the tightness of the bounds it produces). In this paper, we give a new MICP formulation than achieves better bounds than those obtained from another popular reformulation technique. Although we focus on the specific problem given in (1), we give some extensions of our approach to some related problems in §6. 1.1 Previous work Many approaches exist for optimal control of switched systems; a summary can be found in [Sag09]. Here we mention some particularly relevant techniques. Reformulation as a mixed logical dynamical system. Switched-affine systems are a special case of hybrid systems, i.e., systems involving continuous and logical dynamics. A standard approach to solve (1), proposed by Bemporad, Torrisi, and Morari [TB04], is to first convert the switched-affine system into an equivalent mixed logical dynamical (MLD) system, which expresses the system using a combination of linear and binary constraints on the original variables and some auxiliary variables (see [BM99] for details on MLD systems). Minimizing a sum of convex functions of the system states can therefore be expressed as an MICP. We will call this the MLD approach to solving (1), and will briefly describe it in §2.