Didier Henrion

GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi

GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality relaxations of the (generally nonconvex) global optimization problem of minimizing a multivariable polynomial function subject to polynomial inequality, equality, or integer constraints. It generates a series of lower bounds monotonically converging to the global optimum without any problem splitting. Global optimality is detected and isolated optimal solutions are extracted automatically. Numerical experiments show that for most of the small-scale problems described in the literature, the global optimum is reached at low computational cost.

GloptiPoly 3: moments, optimization and semidefinite programming

We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming.

Positive polynomials and robust stabilization with fixed-order controllers

Recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial. An application to the design of fixed-order controllers robustly stabilizing a linear system subject to polytopic uncertainty is then proposed, based on linear matrix inequality optimization. The key ingredient in the design procedure resides in the choice of the central polynomial. Several numerical examples illustrate the relevance of the approach.

Convergent relaxations of polynomial matrix inequalities and static output feedback

Using a moment interpretation of recent results on sum-of-squares decompositions of nonnegative polynomial matrices, we propose a hierarchy of convex linear matrix inequality (LMI) relaxations to solve nonconvex polynomial matrix inequality (PMI) optimization problems, including bilinear matrix inequality (BMI) problems. This hierarchy of LMI relaxations generates a monotone sequence of lower bounds that converges to the global optimum. Results from the theory of moments are used to detect whether the global optimum is reached at a given LMI relaxation, and if so, to extract global minimizers that satisfy the PMI. The approach is successfully applied to PMIs arising from static output feedback design problems.

HIFOO - A MATLAB PACKAGE FOR FIXED-ORDER CONTROLLER DESIGN AND H∞ OPTIMIZATION

Abstract H∞ controller design for linear systems is a difficult, nonconvex and typically nonsmooth (nondifferentiable) optimization problem when the order of the controller is fixed to be less than that of the open-loop plant, a typical requirement in e.g. embedded aerospace control systems. In this paper we describe a new MATLAB package called HIFOO, aimed at solving fixed-order stabilization and local optimization problems. It depends on a new hybrid algorithm for nonsmooth, nonconvex optimization based on several techniques, namely quasi-Newton updating, bundling and gradient sampling. The user may request HIFOO to optimize one of several objectives, including H∞ norm, which requires either the Control System Toolbox for MATLAB or, for much better performance, the linorm function in the SLICOT package. No other external package is required, but the quadratic programming code quadprog from either MOSEK or the Optimization Toolbox for MATLAB is recommended. Numerical experiments on benchmark problem instances from the COMPleib database indicate that HIFOO could be an efficient and reliable computer-aided control system design (CACSD) tool, with a potential for realistic industrial applications.

Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations

We consider the class of nonlinear optimal control problems (OCPs) with polynomial data, i.e., the differential equation, state and control constraints, and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI- (linear matrix inequality)-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.

Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation

Topological separation is investigated in the case of an uncertain time-invariant matrix interconnected with an implicit linear transformation. A quadratic separator independent of the uncertainty is shown to prove losslessly the closed-loop well-posedness. Several applications for LTI descriptor system analysis are then given. First, some known results for stability and pole location of descriptor systems are demonstrated in a new way. Second, contributions to robust stability analysis and time-delay systems stability analysis are exposed. These prove to be new even when compared to results for usual LTI systems (not in descriptor form). All results are formulated as linear matrix inequalities (LMIs).

Multiobjective robust control with HIFOO 2.0

Abstract Multiobjective control design is known to be a difficult problem both in theory and practice. Our approach is to search for locally optimal solutions of a nonsmooth optimization problem that is built to incorporate minimization objectives and constraints for multiple plants. We report on the success of this approach using our public-domain matlab toolbox hifoo 2.0, comparing our results with benchmarks in the literature.

Convex Computation of the Region of Attraction of Polynomial Control Systems

We address the long-standing problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving a convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description.

Solving polynomial static output feedback problems with PENBMI

An algebraic formulation is proposed for the static output feedback (SOF) problem: the Hermite stability criterion is applied on the closed-loop characteristic polynomial, resulting in a non-convex bilinear matrix inequality (BMI) optimization problem for SIMO or MISO systems. As a result, the BMI problem is formulated directly in the controller parameters, without additional Lyapunov variables. The publicly available solver PENBMI 2.0 interfaced with YALMIP 3.0 is then applied to solve benchmark examples. Implementation and numerical aspects are widely discussed.