Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise

Abstract

Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which are independently proven to be stable, results in overall stable system dynamics. However, even for linear systems, this property does not necessarily hold if polytopic parameter uncertainty and stochastic noise influence the system’s state and output equations. In this case, the control and observer design needs to be performed simultaneously to guarantee stabilization. However, the loss of the validity of the separation principle leads to nonlinear matrix inequalities instead of LMIs. For those nonlinear inequalities, the current paper proposes an iterative LMI solution procedure. If this algorithm produces a feasible solution, the resulting controller and observer gains ensure robust stability of the closed-loop control system for all possible parameter values. In addition, the proposed optimization criterion leads to a minimization of the sensitivity to stochastic noise so that the actual state trajectories converge as closely as possible to the desired operating point. The efficiency of the proposed solution approach is demonstrated by stabilizing the Zeeman catastrophe machine along the unstable branch of its bifurcation diagram. Additionally, an observer-based tracking control task is embedded into an iterative learning-type control framework.

Global Versus Local Lyapunov Approach Used in Disturbance Observer-Based Wind Turbine Control

“…The corresponding separation principle is also valid for observer-based TS controllers (Yoneyama et al (1998), Ma et al (1998)), but does not necessarily hold for parameter uncertainties or stochastic noise. Therefore, design procedures are investigated, which account for these restrictions (Zemouche et al, 2016), (Rauh et al, 2021). As an overall observer and controller design often results in conservative controller and control objective performances, respectively, pragmatic design syntheses are intended for real world application, rather than a guaranteed overall stability of the observer-based controller.…”

Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures

“…In future work, the approach will not only be used for a stability analysis of dynamic systems but also to optimize controllers so that the domains of attraction of stable operating points become as large as possible. Moreover, it is reasonable to consider not only set-valued uncertainty representations, but also links to techniques which simultaneously allow for robustifying control procedures in the presence of stochastic noise [11,28,32]. Finally, applications to continuous-time processes, in combination with the new invariance test sketched in Sec.…”

“…• development of iterative LMI techniques for synthesizing robust output and state feedback controllers for systems which are simultaneously subject to polytopic parameter uncertainty and/or stochastic noise [11,28,32,41];…”

Virtual Actuator Design for Uncertain Metzler Systems

“…The possible question to ask is whether (Eq. 65) happens to constitute the classical separation principle invalidity, if the single actuator faults are projected onto the uncertainty of input system matrix B f , see, e.g., the accompanying tasks in Do et al (2021); Rauh et al (2021).…”