Fixed Low-Order Controller Design and H ∞ Optimization for Large-Scale Dynamical Systems

Large-scale linear time-invariant dynamical systems with inputs and outputs present major challenges for controller design. Model-order reduction has become popular in recent years, but controllers designed for reduced-order models may result in unstable closed-loop plants when applied to the larger-scale system. We investigate the practicality of fixed low-order controller design applied directly to large-scale continuous-time sparse systems. We assume that it is practical to compute the eigenvalues with largest real part of such systems using Matlab’s eigs, which requires only matrix-vector products, but that it is not possible to compute the norm using Matlab’s getPeakGain or SLICüt’s slinorm, which use the Boyd-Balakrishnan- Bruinsma-Steinbuch algorithm, requiring both Hamiltonian eigenvalue decompositions and singular value decompositions. Instead, we employ a recently developed efficient algorithm called Hybrid-Expansion-Contraction (HEC), which while not guaranteed to correctly compute the norm, finds, under certain assumptions, at least a local maximizer of the associated transfer function. Our controller design code uses nonsmooth optimization techniques first to attempt to stabilize the closed-loop system and then to minimize its norm proxy as computed by HEC. It is implemented in a new experimental Matlab code hifüüS, based on the public-domain hifoo toolbox first presented in ROCOND 2006, and will be made available for public use after further investigation and development.