Linear Matrix Inequalities for an Iterative Solution of Robust Output Feedback Control of Systems with Bounded and Stochastic Uncertainty

Abstract

Linear matrix inequalities (LMIs) have gained much importance in recent years for the design of robust controllers for linear dynamic systems, for the design of state observers, as well as for the optimization of both. Typical performance criteria that are considered in these cases are either H2 or H∞ measures. In addition to bounded parameter uncertainty, included in the LMI-based design by means of polytopic uncertainty representations, the recent work of the authors showed that state observers can be optimized with the help of LMIs so that their error dynamics become insensitive against stochastic noise. However, the joint optimization of the parameters of the output feedback controllers of a proportional-differentiating type with a simultaneous optimization of linear output filters for smoothening measurements and for their numeric differentiation has not yet been considered. This is challenging due to the fact that the joint consideration of both types of uncertainties, as well as the combined control and filter optimization lead to a problem that is constrained by nonlinear matrix inequalities. In the current paper, a novel iterative LMI-based procedure is presented for the solution of this optimization task. Finally, an illustrating example is presented to compare the new parameterization scheme for the output feedback controller—which was jointly optimized with a linear derivative estimator—with a heuristically tuned D-type control law of previous work that was implemented with the help of an optimized full-order state observer.

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

“…In the second case, the measured output does not necessarily coincide with the system output to be controlled. A linear filter-based output feedback (Case 3), in which the filter is designed in a model-free manner, is only mentioned for the sake of completeness and has been treated by the authors in a separate publication [21]. The reason for this separate investigation is the fact that the design criteria do not fully coincide with the requirements derived for the Cases 1 and 2 in this paper:…”

“…whereŷ f is a vector consisting of filtered system outputs and estimated output derivatives, cf. [21].…”

“…It should be noted that the different control structure of Case 3 leads to a similar, however not identical, LMI approach. Therefore, Case 3 is treated separately in [21]. Moreover, as already discussed in [21], Case 3 should typically only be used if a maximum of two time derivatives of measured output signals are estimated with the model-free filter.…”

“…Therefore, Case 3 is treated separately in [21]. Moreover, as already discussed in [21], Case 3 should typically only be used if a maximum of two time derivatives of measured output signals are estimated with the model-free filter. In all other cases, the observer-based concepts of Cases 1 and 2 are advantageous.…”

Verified Integration of Differential Equations with Discrete Delay

“…with a delay-free realization of the control input u(t) were presented in [26,27] as a benchmark scenario for the design of a robust output feedback controller in which the input u(t) was chosen to be proportional to the velocity x 2 (t). From an engineering viewpoint, this model describes the simplest linear representation of an actively controlled wheel suspension system with a first-order lag behavior of the actuator.…”