Proving Properties on PWA Systems Using Copositive and Semidefinite Programming

Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorithmic solution approaches for copositive programs.

International audience

The Lovasz theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovasz theta number toward the chromatic number of G, which is shown to be equal to the fractional chromatic number of G. Solving copositive programs is NP-hard. This motivates the study of tractable approximations of the copositive cone. We investigate the Parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs. We provide some computational results indicating that the Lovasz theta number can be strengthened significantly toward the fractional chromatic number of G on some Hamming graphs.

This paper presents a novel observer-based controller design method for discrete-time piecewise affine (PWA) systems. The basic idea is as follows: at first, a piecewise linear (without affine terms) state feedback controller and a PWA observer are designed separately, and then it is proved that the output feedback controller constructed by the resulting observer and state feedback controller gains can guarantee the stability of the closed-loop system. During the controller design, the piecewise-quadratic Lyapunov function technique is used. Moreover, the region information is taken into account to treat the affine terms, so the controller gains can be obtained by solving a set of linear matrix inequalities, which are numerically feasible with commercially available software. Three simulation examples are given finally to verify the proposed theoretical results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

We consider the problem of controlling a discrete-time piecewise affine (PWA) system from a specification given as a Linear Temporal Logic (LTL) formula over linear predicates in its state variables. We present a computational framework for finding initial states and feedback control strategies guaranteeing the satisfaction of such a specification by all the trajectories of the closed loop system. Our solution is based on abstracting the system to a finite transition system and on controlling the abstraction from an LTL specification.

In this paper we present an approach for designing of reference tracking controllers for constrained, discrete-time piecewise affine systems. The approach follows the idea of reference governor techniques where the desired set-point is filtered by a system called the “reference governor”. Based on the system current state, set-point, and prescribed constraints, the reference governor computes a new set-point for a low-level controller so that the state and input constraints are satisfied and convergence to the original set-point is guaranteed.

This paper addresses the hybrid control design of discrete-time piecewise-affine systems subject to actuator saturation under event-triggering mechanism. Conditions to ensure regional stability are proposed, and the region of asymptotic stability is determined for piecewise-affine systems by modelling actuator saturation as a piecewise-affine model for each subsystem. Event-triggered feedback control gains can be obtained for single-input piecewise-affine systems by solving a linear matrix inequality issue based on the event-triggering mechanism. In particular, an optimisation method for determining a larger domain of attraction is provided, using ellipsoidal approximations to estimate the domain of attraction. The effectiveness of the proposed results is demonstrated through simulation examples.

In this paper we consider discrete-time piecewise affine systems with boolean inputs, outputs and states and show that they can be represented in a logic canonical form where the logic variables influence the switching between different submodels but not the continuous-valued dynamics. We exploit this representation for studying Lagrange stability and developing performance analysis procedures based on linear matrix inequalities. Moreover, by using arguments from dissipativity theory for non-linear systems, we generalize our approach to solve the H ∞ analysis problem.

This paper presents robust stabilizability analysis and stabilizing state-feedback controller design for discrete-time piecewise affine systems in the presence of disturbance inputs. The piecewise affine plant leads to an increasing sequence of robust symbolic models by extending existing symbolic models to the case where bounded disturbance inputs, as well as control inputs, affect the state transfer. A numerical example illustrates the developed analysis and design approaches.

Non-synchronized Observer-based Control of Discrete-time Piecewise Affine Systems: an LMI Approach

In this paper we investigate the stability of discrete-time PWA systems in closed-loop with quadratic cost based model predictive controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability. We prove that Lyapunov stability can be achieved for the closed-loop system even though the considered Lyapunov function and the system dynamics may he discontinuous. The stabilization conditions are derived using a terminal cost and constraint set method. An S-procedure technique is employed to reduce conservativeness of the stabilization conditions and a linear matrix inequalities set-up is developed in order to calculate the terminal cost. A new algorithm for computing piecewise polyhedral positively invariant sets for PWA systems is also presented. In this manner, the on-line optimization problem associated with MPC leads to a mixed integer quadratic programming problem, which can be solved by standard optimization tools.

This paper is an attempt to exploit the opportunity of Semi Definite Programming (SDP), which is an area of convex and conic optimization. Indeed, Numerous NP-hard problems can be solveby using this approach. Hence, we intend to investigate the strength of SDP to model and provide tight relaxations of combinatorial and quadratic problems in order to present a new polynomial time algorithm for solving this robust model. This algorithm was firstly use to solve the nonlinear programs, the reason for which we search to extend it to the SDP programs. Actually, this algorithm designs the combination of two penalization methods. The first one is a primal-dual interior point (PDIM) method while the second one is a primal-dual exterior point (PDEM) method. Unlike the first method, which converges globally, the second one, also called the primal dual nonlinear rescaling method, has local super linear/quadratic convergence. Therefore, it seems appropriate to use a mixed algorithm based on the interior-exterior point method (IEPM). In fact, this resolution starts from the interior method, and at a certain level of execution, it proceeds to exterior method. Hence, a convergence evaluation function is use to know the level of permutation. Through evaluation, it has been approve that our approach is use to solve some instances of max-cut problem. This problem is a central graph theory model that occurs in many real problems and it is one of many NP-hard problems, which has attracted many researchers over the years. Then, we have used a semi definite programming solver SDPA (Semi Definite Programming Algorithm) that is modify to include the exterior point method subroutine. From the computational performance, we conclude that as the problem size increases, interior-exterior point algorithm gets relatively faster. The numerical results obtained are promising.

This paper dealt with the stabilization problem of a discrete-time piecewise affine (PWA) systems with time-delay. An H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> state feedback controller design is proposed, and the PWA system was described as ellipsoid which characterized by a set of vector inequalities..thereby the constraint of linear matrix inequalities (LMIs) was released. In terms of LMIs, the time-delay PWA system was stabilized in Lyapunov theory, and the minimum problem of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> performance index can be cast as a convex optimization problem with LMIs constraints. The simulation results verified the effectiveness of the proposed method.

In this paper, we introduce a rank-one Semi-Definite Programming (SDP) solution method for mobile source localization in sensor networks. The position and velocity of mobile source are jointly estimated using Time Delay (TD) measurements. To obtain the position and velocity of mobile source, a Relaxed Semi-Definite Programming (RSDP) algorithm is firstly designed by dropping the rank-one constraint. However, dropping the rank-one constraint leads to produce a suboptimal solution. To improve the performance, we further put forward the Penalty Function Semi-Definite Programming (PF-SDP) method to obtain the rank-one solution of the estimation problem by introducing the penalty terms. By adaptively choosing the penalty coefficient, an Adaptive Penalty Function Semi-Definite Programming (APF-SDP) algorithm is also proposed to avoid the excessive penalty. We also conduct experiments in both a simulated environment and a real system to demonstrate the effectiveness of the proposed methods. The results have demonstrated that the proposed APF-SDP outperforms the PF-SDP in terms of the position and velocity estimation whether the noise level is large or not.

The quadratic shortest path problem (QSPP) has a lot of applications in real-life problems. This thesis consists of four essays about the QSPP concerning its theory and computation, where linear and semidefinite programming were used as a major tool for solving and understanding the problem. In the second chapter, we investigate the computational complexity of the problem as well as the polynomial-time solvable special cases. In the third and fourth chapters, we show that the linearization problem of the quadratic shortest path problem on directed acyclic graphs can be solved efficiently. By using the linearization problem of binary quadratic problems as a unifying tool, the dominance between the Generalized Gilmore-Lawler bound and the first level RLT bound is also established. We also propose a new class of relaxation for binary quadratic optimization problems based on its linearization problem. In the last chapter, we derive several semidefinite programming relaxations with increasing complexity for the quadratic shortest path problem. We implement the alternating direction method of multipliers to solve our strongest semidefinite programming relaxation, and the obtained bound is currently the best one solved in the literature.