Design of Affine Controllers via Convex Optimization

Abstract

Abstract-We consider a discrete-time time-varying linear dynamical system, perturbed by process noise, with linear noise corrupted measurements, over a finite horizon. We address the problem of designing a general affine causal controller, in which the control input is an affine function of all previous measurements, in order to minimize a convex objective, in either a stochastic or worst-case setting. This controller design problem is not convex in its natural form, but can be transformed to an equivalent convex optimization problem by a nonlinear change of variables, which allows us to efficiently solve the problem. Our method is related to the classical -design procedure for time-invariant, infinite-horizon linear controller design, and the more recent purified output control method. We illustrate the method with applications to supply chain optimization and dynamic portfolio optimization, and show the method can be combined with model predictive control techniques when perfect state information is available.Index Terms-Affine controller, dynamical system, dynamic linear programming (DLP), linear exponential quadratic Gaussian (LEQG), linear quadratic Gaussian (LQG), model predictive control (MPC), proportional-integral-derivative (PID).

Optimality of Affine Policies in Multistage Robust Optimization

“…It has been originally advocated in the context of stochastic programming (see Charnes et al [15], Garstka and Wets [20], and references therein), where such policies are known as decision rules. More recently, the idea has received renewed interest in robust optimization (Ben-Tal et al [7]), and has been extended to linear systems theory (Ben-Tal et al [4,5]), with notable contributions from researchers in robust model predictive control and receding horizon control (see Löfberg [27], Bemporad et al [1], Kerrigan and Maciejowski [25], Skaf and Boyd [32], and references therein). In all the papers, which usually deal with the more general case of multidimensional linear systems, the authors typically restrict attention, for purposes of tractability, to the class of disturbance-affine policies, and show how the corresponding policy parameters can be found by solving specific types of optimization problems, which vary from linear and quadratic programs (Ben-Tal et al [4], Kerrigan and Maciejowski [24,25]) to conic and semidefinite (Löfberg [27], Ben-Tal et al [4]), or even multiparametric, linear, or quadratic programs (Bemporad et al [1]).…”

An efficient method to estimate the suboptimality of affine controllers

“…The second part of Proposition 3.1 provides a method for optimizing over affine output feedback policies u = Uy by re-parameterizing the problem into one of optimizing over a different but equivalent class of parametric policies, for which u and x become affine functions of the parameters [4,15,24]. The resulting optimization problem can then be solved using Proposition 3.2.…”

“…An attractive feature of such affine parameterizations is that they can be shown to be equivalent (in the state feedback case) to parameterizations of control policies as affine functions of prior states [16,24], or (in the output feedback case) as affine functions of prior measurements [15,4]. The idea underpinning these equivalence results is akin to that of the well-known Youla parameterization (or Q-parameterization) in linear systems [28], and relies on a similar nonlinear transformation to produce a convex set of constraint admissible policies over which one can optimize.…”

Nonlinear Q-Design for Convex Stochastic Control

“…We solve the design problem using the sampling method, with (training) samples, and verify the results by simulation on another (validation) set of 10000 samples. We compare the performance of the nonlinear controller found using our method to the optimal linear controller, designed using the same training set (see [53]), and the certainty-equivalent model predictive control (CE-MPC). We also show the results obtained with a prescient controller, i.e., a controller that is not causal (which, of course, gives us a lower bound on achievable performance).…”