Design of Suboptimal Robust Controllers Based on a Priori and Experimental Data

Abstract

This paper develops a novel unified approach to designing suboptimal robust control laws for uncertain objects with different criteria based on a priori information and experimental data. The guaranteed estimates of the γ₀, generalized H₂, and H_∞ norms of a closed loop system and the corresponding suboptimal robust control laws are expressed in terms of solutions of linear matrix inequalities considering a priori knowledge and object modeling data. A numerical example demonstrates the improved quality of control systems when a priori and experimental data are used together.

APPENDIX

Proof of Lemma 3.1. We write the Lagrange function for this problem and express the optimal value of its dual function as

$$\mathop {\min }\limits_{{{P}_{0}}\, \geqslant \,0,{{\gamma }^{2}}\, \geqslant \,0} \mathop {\max }\limits_{{{K}_{x}}\, \geqslant \,0,{{K}_{w}}\, \geqslant \,0} \left[ {\operatorname{tr} {{C}_{\Theta }}{{K}_{x}}C_{\Theta }^{{\text{T}}} + \operatorname{tr} {{P}_{0}}({{A}_{\Delta }}{{K}_{x}}A_{\Delta }^{{\text{T}}} - {{K}_{x}} + B{{K}_{w}}{{B}^{{\text{T}}}}) + {{\gamma }^{2}}(1 - \operatorname{tr} {{K}_{w}})} \right]$$

$$ = \mathop {\min }\limits_{{{P}_{0}}\, \geqslant \,0,{{\gamma }^{2}}\, \geqslant \,0} \mathop {\max }\limits_{{{K}_{x}}\, \geqslant \,0,{{K}_{w}}\, \geqslant \,0} \left[ {{{\gamma }^{2}} + \operatorname{tr} {{K}_{x}}(A_{\Delta }^{{\text{T}}}{{P}_{0}}{{A}_{\Delta }} - {{P}_{0}} + C_{\Theta }^{{\text{T}}}{{C}_{\Theta }}) + \operatorname{tr} {{K}_{w}}({{B}^{{\text{T}}}}{{P}_{0}}B - {{\gamma }^{2}}I)} \right].$$

This value is finite under inequalities (3.4); then the maximum is reached at K_x = 0 and K_w = 0. In this case, the optimal value of the dual problem coincides with λ_max(B^TP₀B). Since the function is convex and there exists an interior point satisfying the constraint, the primal and dual problems have the same optimal value [18].