Worst‐case stability and performance with mixed parametric and dynamic uncertainties

Abstract

Summary This work deals with computing the worst‐case stability and the worst‐case H∞ performance of linear time‐invariant systems subject to mixed real‐parametric and complex‐dynamic uncertainties in a compact parameter set. Our novel algorithmic approach is tailored to the properties of the nonsmooth worst‐case functions associated with stability and performance, and this leads to a fast and reliable optimization method, which finds good lower bounds of μ. We justify our approach theoretically by proving a local convergence certificate. Because computing μ is known to be NP‐hard, our technique should be used in tandem with a classical μ upper bound to assess global optimality. Extensive testing indicates that the technique is practically attractive. Copyright © 2016 John Wiley & Sons, Ltd.

Optimizing the Kreiss Constant

“…Remark 7. Program ( 16) is a special case of a much wider class of problems with parametric uncertainty discussed in [2,7], where a successful approach alternating between two non-smooth programs is used. The inner max-max program with controller K fixed is characterized by a light form of non-smoothness and can be addressed by a first-order non-smooth trust-region technique whose convergence certificates have been established in [7].…”

“…Program ( 16) is a special case of a much wider class of problems with parametric uncertainty discussed in [2,7], where a successful approach alternating between two non-smooth programs is used. The inner max-max program with controller K fixed is characterized by a light form of non-smoothness and can be addressed by a first-order non-smooth trust-region technique whose convergence certificates have been established in [7]. The outer min-max program with δ fixed corresponds to a more severe form of nonsmoothness and should be handled by dedicated bundle or bundle trust-region techniques [7,5,6].…”

Cutting plane oracles for non-smooth trust-regions

“…The first non-convex bundle method with convergence certificate under inexact subgradients and function values is [51], where downshifted tangents are used. The model-based trust-region bundle method has been used for parametric and mixed robust control, one of the most challenging problems in feedback control design [13,10,2,1]. We have also used the trust region bundle method within a branch-and-bound approach to global optimization problems like computing the distance to instability of a controlled system, see [56,55].…”

Adaptive tracking for MIMO nonlinear systems with Unknown fast time‐varying parameters

“…Adaptive control of linear time‐invariant systems with unknown constant parameters has been well explored, eg, in the works of Ioannou and Apkarian and Noll . However, it is not straightforward to generalize such adaptive control techniques to general MIMO nonlinear systems.…”