Robust Control

This chapter considers stability and performance of closed-loop systems in the presence of unstructured and structured model uncertainties categorised into additive and multiplicative ones. It is common to pull them out of an uncertain plant model which results in a standard extended control configuration with a nominal plant model that is part of a controller feedback loop and a feedback block with a signal block collecting all weighted uncertainties assumed to be stable and norm-bounded. The design of a closed-loop system that is robustly stable and meets performance requirements with regard to a set of bounded uncertainties boils down to find a controller \(\mathbf {K}(s)\) so that the \(\mathrm {H}_{\infty }\) -norm of the transfer function matrix relating the input and output of the uncertainty feedback is less than one (small gain theorem). For structured uncertainties, Doyle introduced the structured singular value (SSV) \(\mu \) . Since its direct computation is, in general, difficult, it is usually approximated by an upper bound. The idea of the so-called D-K iteration used to synthesise a robust controller is outlined. Performance requirements can be achieved by choosing performance weights on the sensitivity and the complementary sensitivity function. Typical requirements such as reference tracking and noise attenuation or disturbance rejection can be met by finding a controller that minimises the \(\mathrm {H}_{\infty }\) -norm of the transfer function matrix \({\mathbf {G}}_{wz}(s)\) relating exogenous inputs \(\boldsymbol {w}\) to output control errors \(\boldsymbol {z}\) . Robust performance can be captured in an \(\mathrm {H}_{\infty }\) -framework by adding a fictitious uncertainty block accounting for \(\mathrm {H}_{\infty }\) -performance specifications to the block of model uncertainties. GNU Octave provides support for the design of \(\mathrm {H}_{\infty }\) -controllers but not for a \(\mu \) -synthesis.