Dominikus Noll

Robust Control via Sequential Semidefinite Programming

This paper discusses nonlinear optimization techniques in robust control synthesis, with special emphasis on design problems which may be cast as minimizing a linear objective function under linear matrix inequality (LMI) constraints in tandem with nonlinear matrix equality constraints. The latter type of constraints renders the design numerically and algorithmically difficult. We solve the optimization problem via sequential semidefinite programming (SSDP), a technique which expands on sequential quadratic programming (SQP) known in nonlinear optimization. Global and fast local convergence properties of SSDP are similar to those of SQP, and SSDP is conveniently implemented with available semidefinite programming (SDP) solvers. Using two test examples, we compare SSDP to the augmented Lagrangian method, another classical scheme in nonlinear optimization, and to an approach using concave optimization.

An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory

We present a new approach to a class of non-convex LMI-constrained problems in robust control theory. The problems we consider may be recast as the minimization of a linear objective subject to linear matrix inequality (LMI) constraints in tandem with non-convex constraints related to rank deficiency conditions. We solve these problems using an extension of the augmented Lagrangian technique. The Lagrangian function combines a multiplier term and a penalty term governing the non-convex constraints. The LMI constraints, due to their special structure, are retained explicitly and not included in the Lagrangian. Global and fast local convergence of our approach is then obtained either by an LMI-constrained Newton type method including line search or by a trust-region strategy. The method is conveniently implemented with available semi-definite programming (SDP) interior-point solvers. We compare its performance to the wellknown D - K iteration scheme in robust control. Two test problems are investigated and demonstrate the power and efficiency of our approach.

Mixed

We present a new approach to mixed

H_2/H_\infty

H_2/H_\infty

Control via Nonsmooth Optimization

output feedback control synthesis. Our method uses nonsmooth mathematical programming techniques to compute locally optimal

Controller Design via Nonsmooth Multidirectional Search

H_2/H_\infty

Dynamic SPECT imaging using a single camera rotation (dSPECT)

-controllers, which may have a predefined structure. We prove global convergence of our method and present tests to validate it numerically.

Partially Augmented Lagrangian Method for Matrix Inequality Constraints

We propose an algorithm which combines multidirectional search (MDS) with nonsmooth optimization techniques to solve difficult problems in automatic control. Applications include static and fixed-order output feedback controller design, simultaneous stabilization,

A Spectral Quadratic-SDP Method with Applications to Fixed-Order H2 and H∞ Synthesis

H_2/H_\infty

Mixed H 2 /H ∞ control via nonsmooth optimization

-synthesis, and much else. We show how to combine direct search techniques with nonsmooth descent steps in order to obtain convergence certificates in the presence of nonsmoothness. Our technique is efficient when small and medium size controllers for plants with large state dimension are sought. Our numerical testing includes several benchmark examples. For instance, our algorithm needs 0.41\,s to compute a static output feedback stabilizing controller for the Boeing 767 flutter benchmark problem [E. E. J. Davison, IFAC Technical Committee Reports, Pergamon Press, Oxford, 1990], a system with 55 states. The first static controller without performance specifications for this system was obtained in [J. Burke, A. Lewis, and M. Overton, SIAM J. Optim., 15 (2003), pp. 751-779].