No Excuses to Avoid Observing Unstable Nonlinear Systems: An LMI-Based Discontinuous Solution

A novel discontinuous observer design for nonlinear systems, whether stable or not, is presented in this article; it removes former limitations on matching and minimum-phase conditions, Lipschitz bounds, and linearity of outputs. Based on a pair of diffeomorphisms an observer error system can be written whose nonlinear nominal model depends entirely on the error, input, and output signals, thus allowing error-dependent domains of attraction where unstable signals are estimated. Nonlinear nominal models constitute the key for the proposal power and flexibility; they are handled via convex embedding of nonlinearities and the direct Lyapunov method. The discontinuous nature of the observer makes it insensitive to matched signals; the influence of unmatched terms is attenuated via $\mathscr {H}_\infty$ techniques. All design conditions are in the form of linear matrix inequalities. The examples illustrate the advantages over the existing methodologies even for unstable nonminimum-phase systems that do not hold the standard matching or Lipschitz conditions.