Fast Non-Parametric Learning to Accelerate Mixed-Integer Programming for Online Hybrid Model Predictive Control

Today's fast linear algebra and numerical optimization tools have pushed the frontier of model predictive control (MPC) forward, to the efficient control of highly nonlinear and hybrid systems. The field of hybrid MPC has demonstrated that exact optimal control law can be computed, e.g., by mixed-integer programming (MIP) under piecewise-affine (PWA) system models. Despite the elegant theory, online solving hybrid MPC is still out of reach for many applications. We aim to speed up MIP by combining geometric insights from hybrid MPC, a simple-yet-effective learning algorithm, and MIP warm start techniques. Following a line of work in approximate explicit MPC, the proposed learning-control algorithm, LNMS, gains computational advantage over MIP at little cost and is straightforward for practitioners to implement

Using ADMM for Hybrid System MPC

Model Predictive control (MPC) has been studied extensively because of its ability to handle constraints and its great properties in terms of stability and performance [Mayne et al., 2000]. We have in this thesis focused on MPC of Hybrid Systems, i.e. systems with both continuous and discrete dynamics. More specifically, we look at problems that can be cast as Mixed Integer Quadratic Programming (MIQP) problems which we are solving using a Branch and Bound technique. The problem is in this way reduced to solving a large number of constrained quadratic problems. However, the use in real time systems puts a requirement on the speed and efficiency of the optimization methods used. Because of its low computational cost, there have recently been a rising interest in the Alternating Direction Method of Multiplies (ADMM) for solving constrained optimization problems. We are in this thesis looking at how the different properties of ADMM can be used and improved for these problems, as well as how the Branch and Bound solver can be tailored to accompany ADMM. We have two main contributions to ADMM that mitigate some of the downsides with the often ill-conditioned problems that arise from Hybrid Systems. Firstly, a technique for greatly improving the conditioning of the problems, and secondly, a method to perform fast line search within the solver. We show that these methods are very efficient and can be used to solve problems that are otherwise hard or impossible to precondition properly

Model predictive control techniques for hybrid systems

This paper describes the main issues encountered when applying model predictive control to hybrid processes. Hybrid model predictive control (HMPC) is a research field non-fully developed with many open challenges. The paper describes some of the techniques proposed by the research community to overcome the main problems encountered. Issues related to the stability and the solution of the optimization problem are also discussed. The paper ends by describing the results of a benchmark exercise in which several HMPC schemes were applied to a solar air conditioning plant.Ministerio de Eduación y Ciencia DPI2007-66718-C04-01Ministerio de Eduación y Ciencia DPI2008-0581

A Finite-Time Cutting Plane Algorithm for Distributed Mixed Integer Linear Programming

Many problems of interest for cyber-physical network systems can be formulated as Mixed Integer Linear Programs in which the constraints are distributed among the agents. In this paper we propose a distributed algorithm to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. In the proposed algorithm, at each communication round, agents solve locally a small LP, generate suitable cutting planes, namely intersection cuts and cost-based cuts, and communicate a fixed number of active constraints, i.e., a candidate optimal basis. We prove that, if the cost is integer, the algorithm converges to the lexicographically minimal optimal solution in a finite number of communication rounds. Finally, through numerical computations, we analyze the algorithm convergence as a function of the network size.Comment: 6 pages, 3 figure

Overall Complexity Certification of a Standard Branch and Bound Method for Mixed-Integer Quadratic Programming

This paper presents a method to certify the computational complexity of a standard Branch and Bound method for solving Mixed-Integer Quadratic Programming (MIQP) problems defined as instances of a multi-parametric MIQP. Beyond previous work, not only the size of the binary search tree is considered, but also the exact complexity of solving the relaxations in the nodes by using recent result from exact complexity certification of active-set QP methods. With the algorithm proposed in this paper, a total worst-case number of QP iterations to be performed in order to solve the MIQP problem can be determined as a function of the parameter in the problem. An important application of the proposed method is Model Predictive Control for hybrid systems, that can be formulated as an MIQP that has to be solved in real-time. The usefulness of the proposed method is successfully illustrated in numerical examples.Comment: Paper accepted for presentation at, and publication in the proceedings of, the 2022 American Control Conferenc

Tailored Presolve Techniques in Branch-and-Bound Method for Fast Mixed-Integer Optimal Control Applications

Mixed-integer model predictive control (MI-MPC) can be a powerful tool for modeling hybrid control systems. In case of a linear-quadratic objective in combination with linear or piecewise-linear system dynamics and inequality constraints, MI-MPC needs to solve a mixed-integer quadratic program (MIQP) at each sampling time step. This paper presents a collection of block-sparse presolve techniques to efficiently remove decision variables, and to remove or tighten inequality constraints, tailored to mixed-integer optimal control problems (MIOCP). In addition, we describe a novel heuristic approach based on an iterative presolve algorithm to compute a feasible but possibly suboptimal MIQP solution. We present benchmarking results for a C code implementation of the proposed BB-ASIPM solver, including a branch-and-bound (B&B) method with the proposed tailored presolve techniques and an active-set based interior point method (ASIPM), compared against multiple state-of-the-art MIQP solvers on a case study of motion planning with obstacle avoidance constraints. Finally, we demonstrate the computational performance of the BB-ASIPM solver on the dSPACE Scalexio real-time embedded hardware using a second case study of stabilization for an underactuated cart-pole with soft contacts.Comment: 27 pages, 7 figures, 2 tables, submitted to journal of Optimal Control Applications and Method