Abstract
This paper deals with the design of linear observer-based state feedback controllers with constant gains for a class of nonlinear discrete-time systems in the form of a quasi-linear representation in presence of stochastic noise. For taking into account nonlinearities in the design of linear observer-based state feedback controllers, a polytopic modeling approach is investigated. An optimization problem is formulated to reduce the sensitivity of the controlled system towards stochastic input, state, and output noise with a predefined covariance. Due to the nonlinearities, the separation principle does not hold, thus, the controller and the observer have to be designed simultaneously. For this purpose, a Lyapunov-based method is used, which provides, in addition to the controller and observer gains, a stability proof for the nonlinear closed loop in a predefined polytopic domain. In general, this leads to nonlinear matrix inequalities. To solve these nonlinear matrix inequalities efficiently, we propose an approach based on linear matrix inequalities (LMIs) with a superposed iteration rule. When using this iterative LMI approach, a minimization task can be solved additionally, which desensitizes the closed loop to stochastic noise. The proposed method additionally enables the consideration of different linear closed loop structures by a unified Lyapunov-based framework. The efficiency of the proposed approach is demonstrated and compared with a classical LQG approach for a nonlinear overhead traveling crane.
Highlights
The research field of linear control theory is well investigated and facilitates generalized and efficient methods to design linear controllers for linear systems
We address the structured linear control (SLC) problem in which the dynamic controller is described as a structured state feedback approach in an augmented system model
This makes it possible to apply the method on a myriad of different types of closed loops, such as PID structures or observer-based feedback controllers and their combinations within a uniform approach, whereas other methods are solely applicable to one particular controller type
Summary
The research field of linear control theory is well investigated and facilitates generalized and efficient methods to design linear controllers for linear systems. LMI methods have been established for the robust controller design and used to prove asymptotic stability of the closed loop simultaneously. Robust Feedback Control with LMIs purpose, the nonlinear model can be transformed into a quasilinear form, whereby the bounded nonlinearities can be evaluated using interval arithmetic (Rauh and Romig, 2021; Rauh et al, 2021). For these systems, a convex LMI approach is just as applicable as for pure parameter uncertainty. By taking the uncertainties into account in the controller design, stability of the nonlinear closed loop can be guaranteed
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