Robust Feedback Control for Discrete-Time Systems Based on Iterative LMIs with Polytopic Uncertainty Representations Subject to Stochastic Noise

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.

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.…”

“…The number n v of the vertices to be considered in the polytopic uncertainty representation (38) can often be reduced by identifying physically motivated linear dependencies between individual entries of the matrix J (x) and by expressing them in terms of common interval parameters. For an example, where this has been done successfully, the reader is referred to [11].…”

“…• 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];…”

Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications

“…which accounts for a non-zero ellipsoid midpoint with x k , Φ, and p defined according to ( 26), (28), and (29). Inflate the outer ellipsoid bound described by the shape matrix (31) with…”

“…with α ∈ R as an optimization degree of freedom. The optimal choice of this parameter α is currently being investigated in a parallel research activity aiming at the use of an ellipsoidal calculus for analyzing the stability of discrete-time dynamic systems, as well as for extensions of the control procedure presented in [31].…”