Robust Stability Criterion and Design of Optimal Robust Systems

Abstract

Based on the Kharitonov criterion, a robust stability limit is constructed. Methods for calculating this limit are developed. An algorithm for designing an optimal robust control system with a given value of this limit is presented. The effectiveness of the algorithm is demonstrated by numerical examples.

APPENDIX

The mathematical models of the constraints have the following form (see Example 2):

$$\Delta \chi 1(\delta ): = - 32768{{\delta }^{6}} - 57344{{\delta }^{5}} + 660992{{\delta }^{4}} - 660992{{\delta }^{2}} + 57344\delta + 32768,$$

$$\Delta \chi 2(\delta ): = 32768{{\delta }^{6}} + 8192{{\delta }^{5}} - 710144{{\delta }^{4}} - 1371136{{\delta }^{3}} - 710144{{\delta }^{2}} + 8192\delta + 32768,$$

$$\Delta \chi 3(\delta ): = - 32768{{\delta }^{6}} + 57344{{\delta }^{5}} + 660992{{\delta }^{4}} - 660992{{\delta }^{2}} - 57344\delta + 32768,$$

$$\Delta \chi 4(\delta ): = 32768{{\delta }^{6}} - 8192{{\delta }^{5}} - 710144{{\delta }^{4}} + 1371136{{\delta }^{3}} - 710144{{\delta }^{2}} - 8192\delta + 32768.$$

The performance criterion is I(δ) := (1 – δ). The initial condition is δ := 0. The program code in Maple 15 for solving this problem is presented below.

Given

Constraints: Δχ1(δ) ≥ 0, Δχ2(δ) ≥ 0, Δχ3(δ) ≥ 0, Δχ4(δ) ≥ 0, δ < 1.

a := Minimize (I, δ), a = 0.18613, Δχ1(a) = 2.13211 × 10⁴, Δχ2(a) = 0.01798, Δχ3(a) = 0.01233, Δχ4(a) = 1.46297 × 10⁴.