Parametrization of Optimal Anisotropic Controllers

Abstract

This paper provides a parametrization of optimal anisotropic controllers for linear discrete time invariant systems. The controllers to be designed are limited by causal dynamic output-feedback control laws. The obtained solution depends on several adjustable parameters that determine the specific type of controller, and is of the form of a system of the Riccati equations relating to a ${{\mathcal{H}}_{2}}$-optimal controller for a system formed by a series connection of the original system and the worst-case generating filter corresponding to the maximum value of the mean anisotropy of the external disturbance.

APPENDIX

Proof of Theorem 2. First, let us show that the conditions (i)–(vi) from the formulation of the Theorem 2 are equivalent to the following:

(a) ($\bar {A}$, ${{\bar {B}}_{u}}$) is stabilizable,

(b) (${{\bar {C}}_{y}}$, $\bar {A}$) is detectable,

(c) $\operatorname{im} ({{\bar {B}}_{Q}} - {{\bar {B}}_{u}}\bar {D}_{P}^{ + }\bar {R}) \subseteq \mathcal{W}({{\bar {F}}_{{{{P}_{u}}}}})$,

(d) $S({{\bar {F}}_{{{{Q}_{y}}}}}) \subseteq \ker ({{\bar {C}}_{P}} - \bar {R}\bar {D}_{Q}^{ + }{{\bar {C}}_{y}})$,

(e) $\mathcal{S}({{\bar {F}}_{{{{Q}_{y}}}}}) \subseteq \mathcal{W}({{\bar {F}}_{{{{P}_{u}}}}})$,

(f) $(\bar {A} - {{\bar {B}}_{u}}\bar {D}_{P}^{ + }\bar {R}\bar {D}_{Q}^{ + }{{\bar {C}}_{y}})\mathcal{S}({{\bar {F}}_{{{{Q}_{y}}}}}) \subseteq \mathcal{W}({{\bar {F}}_{{{{P}_{u}}}}})$,

where matrices ${{\bar {C}}_{P}}$, ${{\bar {D}}_{P}}$, ${{\bar {B}}_{Q}}$, ${{\bar {D}}_{Q}}$, and $\bar {R}$, as well as systems ${{\bar {F}}_{{{{P}_{u}}}}}$ and ${{\bar {F}}_{{{{Q}_{y}}}}}$ are set in accordance to the material presented in Subsection 2.3 in relation to the system (18). Note that the conditions (a)–(f) are a direct analogue of the conditions (i)–(vi) of Theorem 1 for system (18).

The equivalence of (i) ⇔ (a) and (ii) ⇔ ((b) is obvious due to the notation (19). For further proof, let us determine the relation of the sets $\mathcal{W}({{F}_{{{{P}_{u}}}}})$ and $\mathcal{S}({{F}_{{{{Q}_{y}}}}})$ from Theorem 1 to the sets $\mathcal{W}({{\bar {F}}_{{{{P}_{u}}}}})$ and $\mathcal{S}({{\bar {F}}_{{{{Q}_{y}}}}})$, respectively. Given the system $\bar {F}$, using Definitions 4 and 5, it can be verified that there exist matrices $\bar {\Pi }$ and $\bar {\Lambda }$ such that

$$\mathcal{W}({{\bar {F}}_{{{{P}_{u}}}}}) = \mathcal{W}({{F}_{{{{P}_{u}}}}})\,\, \times \,\,{{\mathbb{R}}^{{{{n}_{h}}}}},\quad \mathcal{S}({{\bar {F}}_{{{{Q}_{y}}}}}) = \mathcal{S}({{F}_{{{{Q}_{y}}}}})\,\, \times \,\,{{\{ 0\} }^{{{{n}_{h}}}}},$$

(A.1a)

$$(\bar {A} + {{\bar {B}}_{u}}\bar {\Pi })\mathcal{W}({{\bar {F}}_{{{{P}_{u}}}}}) \subseteq \mathcal{W}({{\bar {F}}_{{{{P}_{u}}}}}),\quad (\bar {A} + \bar {\Lambda }{{\bar {C}}_{y}})\mathcal{S}({{\bar {F}}_{{{{Q}_{y}}}}}) \subseteq \mathcal{S}({{\bar {F}}_{{{{Q}_{y}}}}}),$$

(A.1b)

$$\rho (\bar {A} + {{\bar {B}}_{u}}\bar {\Pi }) < 1,\quad \rho (\bar {A} + \bar {\Lambda }{{\bar {C}}_{y}}) < 1.$$

(A.1c)

After this, we can conclude that the equivalence of the conditions (iii) ⇔ (c) and (iv) ⇔ (d) holds due to the fact that

$$\ker ({{\bar {C}}_{P}} - \bar {R}\bar {D}_{Q}^{ + }{{\bar {C}}_{y}}) = \ker ({{C}_{P}} - ED_{Q}^{ + }{{C}_{y}})\,\, \times \,\,{{\mathbb{R}}^{{{{n}_{h}}}}},$$

(A.2a)

$$\operatorname{im} ({{\bar {B}}_{Q}} - {{\bar {B}}_{u}}\bar {D}_{P}^{ + }\bar {R}) = \operatorname{im} ({{B}_{Q}} - {{B}_{u}}D_{P}^{ + }R)\,\, \times \,\,{{\{ 0\} }^{{{{n}_{h}}}}}.$$

(A.2b)

Finally, by (A.1), the equivalence of the conditions (v) ⇔ (e) and (vi) ⇔ (e) is proved.

The structure of the controller (20) is determined by the content of Theorem 1.

Theorem 2 is proven.