Forward Stochastic Reachability Analysis for Uncontrolled Linear Systems using Fourier Transforms

We propose a scalable method for forward stochastic reachability analysis for uncontrolled linear systems with affine disturbance. Our method uses Fourier transforms to efficiently compute the forward stochastic reach probability measure (density) and the forward stochastic reach set. This method is applicable to systems with bounded or unbounded disturbance sets. We also examine the convexity properties of the forward stochastic reach set and its probability density. Motivated by the problem of a robot attempting to capture a stochastically moving, non-adversarial target, we demonstrate our method on two simple examples. Where traditional approaches provide approximations, our method provides exact analytical expressions for the densities and probability of capture.Comment: V3: HSCC 2017 (camera-ready copy), DOI updated, minor changes | V2: Review comments included | V1: 10 pages, 12 figure

Analysis of the explicit model predictive control for semi-active suspension

Explicit model predictive control (MPC) enhances application of MPC to areas where the fast online computation of the control signal is crucial, such as in aircraft or vehicle control. Explicit MPC controllers consist of several affine feedback gains, each of them valid over a polyhedral region of the state space. In this paper the optimal control of the quarter car semi-active suspension is studied. After a detailed theoretical introduction to the modeling, clipped LQ control and explicit MPC, the article demonstrates that there may exist regions where constrained MPC/explicit MPC has no feasible solution. To overcome this problem the use of soft constraints and combined clipped LQ/MPC methods are suggested. The paper also shows that the clipped optimal LQ solution equals to the MPC with horizon N=1 for the whole union of explicit MPC regions. We study the explicit MPC of the semi-active suspension with actual discrete time observer connected to the explicit MPC in order to increase its practical applicabili ty. The controller requires only measurement of the suspension deflection. Performance of the derived controller is evaluated through simulations where shock tests and white noise velocity disturbances are applied to a real quarter car vertical model. Comparing MPC and the clipped LQ approach, no essential improvement was detected in the control behavior

A Generalized Stopping Criterion for Real-Time MPC with Guaranteed Stability

Most of the real-time implementations of the stabilizing optimal control actions suffer from the necessity to provide high computational effort. This paper presents a cutting-edge approach for real-time evaluation of linear-quadratic model predictive control (MPC) that employs a novel generalized stopping criterion, achieving asymptotic stability in the presence of input constraints. The proposed method evaluates a fixed number of iterations independent of the initial condition, eliminating the necessity for computationally expensive methods. We demonstrate the effectiveness of the introduced technique by its implementation of two widely-used first-order optimization methods: the projected gradient descent method (PGDM) and the alternating directions method of multipliers (ADMM). The numerical simulation confirmed a significantly reduced number of iterations, resulting in suboptimality rates of less than 2\,\%, while the effort reductions exceeded 80\,\%. These results nominate the proposed criterion for an efficient real-time implementation method of MPC controllers

Minkowski addition of convex polytopes

This note summarizes recent results from computational geometry which determine complexity of computing Minkowski sum of k convex polytopes in R d, which are represented either in terms of facets or in terms of vertices. In particular, it is pointed out for which cases there exists an algorithm which runs in polynomial time. The note is based on papers of Gritzmann and Sturmfels [6] and Komei Fukuda [3]. An algorithm which aims at reducing the complexity of obtaining minimal representation of polytopes given by a set of inequalities is presented as well

Stabilizing Polynomial Approximation of Explicit MPC

A given explicit piecewise affine representation of an MPC feedback law is approximated by a single polynomial, computed using linear programming. This polynomial state feedback control law guarantees closed-loop stability and constraint satisfaction. The polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage.Funding Agencies|Scientific Grant Agency of the Slovak Republic|1/0079/091/0095/11|Slovak Research and Development Agency|W-0029-07LPP-0092-07|</p

An optimal path planning problem for heterogeneous multi-vehicle systems

A path planning problem for a heterogeneous vehicle is considered. Such a vehicle consists of two parts which have the ability to move individually, but one of them has a shorter range and is therefore required to keep in a close distance to the main vehicle. The objective is to devise an optimal path of minimal length under the condition that at least one part of the heterogeneous system visits all desired waypoints exactly once. Two versions of the problem are considered. One assumes that the order in which the waypoints are visited is known a priori. In such a case we show that the optimal path can be found by solving a mixed-integer second-order cone problem. The second version assumes that the order in which the waypoints are visited is not known a priori, but can be optimized so as to shorten the length of the path. Two approaches to solve this problem are presented and evaluated with respect to computational complexity

Stabilizing Polynomial Approximation of Explicit MPC

A given explicit piecewise affine representation of an MPC feedback law is approximated by a single polynomial, computed using linear programming. This polynomial state feedback control law guarantees closed-loop stability and constraint satisfaction. The polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage.Funding Agencies|Scientific Grant Agency of the Slovak Republic|1/0079/091/0095/11|Slovak Research and Development Agency|W-0029-07LPP-0092-07|</p