On the Lagrange Duality of Stochastic and Deterministic Minimax Control and Filtering Problems

Abstract

As shown below, the linear operator norms in the deterministic and stochastic cases are optimal values of the Lagrange-dual problems. For linear time-varying systems on a finite horizon, the duality principle leads to stochastic interpretations of the generalized H₂ and H_∞ norms of the system. Stochastic minimax filtering and control problems with unknown covariance matrices of random factors are considered. Equations of generalized H_∞-suboptimal controllers, filters, and identifiers are derived to achieve a trade-off between the error variance at the end of the observation interval and the sum of the error variances on the entire interval.

APPENDIX

Proof of Theorem 4.1. We write the Lagrange function for problem S_∞ and find its dual function:

$$\mathop {\min }\limits_{\lambda \; \geqslant \;0,X(t)} \mathop {\max }\limits_{W(t)\; \geqslant \;0} \sum\limits_{t = {{t}_{0}}}^{{{t}_{f}} - 1} {\operatorname{tr} \left( {C(t)\;D(t)} \right)W(t){{{\left( {C(t)\;D(t)} \right)}}^{{\text{T}}}}} + \operatorname{tr} SMW({{t}_{f}}){{M}^{{\text{T}}}}$$

$$ - \;\lambda \left[ {\operatorname{tr} {{R}^{{ - 1}}}MW({{t}_{0}}){{M}^{{\text{T}}}} + \sum\limits_{t = {{t}_{0}}}^{{{t}_{f}} - 1} {\operatorname{tr} {{G}^{{ - 1}}}(t)HW(t){{H}^{{\text{T}}}} - 1} } \right]$$

$$ + \;\sum\limits_{t = {{t}_{0}}}^{{{t}_{f}} - 1} {\operatorname{tr} \left[ {\left( {A(t)\;B(t)} \right)W(t){{{\left( {A(t)\;B(t)} \right)}}^{{\text{T}}}} - MW(t + 1){{M}^{{\text{T}}}}} \right]X(t + 1)} $$

$$ = \mathop {\min }\limits_{\lambda \; \geqslant \;0,X(t)} \mathop {\max }\limits_{W(t)\; \geqslant \;0} \left\{ {\lambda + \sum\limits_{t = {{t}_{0}}}^{{{t}_{f}} - 1} {\operatorname{tr} W(t)\left[ {{{{\left( {C(t)\;D(t)} \right)}}^{{\text{T}}}}\left( {C(t)\;D(t)} \right) + {{{\left( {A(t)\;B(t)} \right)}}^{{\text{T}}}}X(t + 1)\left( {A(t)\;B(t)} \right)} \right.} } \right.$$

$$\left. {^{{^{{^{{^{{}}}}}}}}\left. { - \;{{M}^{{\text{T}}}}X(t)M - \lambda {{H}^{{\text{T}}}}{{G}^{{ - 1}}}(t)H} \right] + \operatorname{tr} W({{t}_{f}}){{M}^{{\text{T}}}}[S - X({{t}_{f}})]M} \right\},$$

where X(t₀) = λR^–1. The dual function is finite under the following inequalities:

$$\begin{gathered} {{\left( {C(t)\;D(t)} \right)}^{{\text{T}}}}\left( {C(t)\;D(t)} \right) + {{\left( {A(t)\;B(t)} \right)}^{{\text{T}}}}X(t + 1)\left( {A(t)\;B(t)} \right) - {{M}^{{\text{T}}}}X(t)M - \lambda {{H}^{{\text{T}}}}{{G}^{{ - 1}}}(t)H\;\leqslant \;0, \\ t = {{t}_{0}},\,\, \ldots ,\,\,{{t}_{f}} - 1,\quad S - X({{t}_{f}})\;\leqslant \;0. \\ \end{gathered} $$

(A.1)

(Otherwise, W(t) can be chosen so that the corresponding term will become infinite.) Thus, inequalities (A.1) must hold, but in this case, the minimum in the minimax problem is reached at W(t) = 0, t = t₀, …, t_f. As a result, we arrive at the dual problem: min λ subject to the constraints (A.1). With the introduced notations and the variable X(t) replaced by λX(t), these constraints are reduced to inequalities (4.7). Since the function is convex and there exists an interior point satisfying the constraints, the values of the primal and dual problems coincide.

We define the function V(t) = x^T(t)X(t)x(t), where X(t) satisfies inequalities (4.7). The increment of this function along the trajectories of system (4.1) satisfies the conditions

$$\begin{gathered} \Delta V(t) + {{\lambda }^{{ - 1}}}{{\left| {z(t)} \right|}^{2}} - {{{v}}^{{\text{T}}}}(t){{G}^{{ - 1}}}{v}(t)\;\leqslant \;0, \\ V({{t}_{0}}) = {{x}^{{\text{T}}}}({{t}_{0}}){{R}^{{ - 1}}}x({{t}_{0}}),\quad V({{t}_{f}})\; \geqslant \;{{\lambda }^{{ - 1}}}{{x}^{{\text{T}}}}({{t}_{f}})Sx({{t}_{f}}). \\ \end{gathered} $$

(A.2)

Hence,

$$\sum\limits_{t = {{t}_{0}}}^{{{t}_{f}} - 1} {{{{\left| {z(t)} \right|}}^{2}}} + {{x}^{{\text{T}}}}({{t}_{f}})Sx({{t}_{f}})\;\leqslant \;\lambda + \lambda \left[ {x({{t}_{0}}){{R}^{{ - 1}}}x({{t}_{0}}) + \sum\limits_{t = {{t}_{0}}}^{{{t}_{f}} - 1} {{{{v}}^{{\text{T}}}}(t){{G}^{{ - 1}}}(t){v}(t) - 1} } \right],$$

i.e., the minimum value λ making inequalities (4.7) solvable is the optimal value in problem D_∞ and coincides with the generalized H_∞ norm of system (4.1).

Proof of Theorem 5.1. Let us apply Theorem 4.1 to system (5.3): if inequalities (4.7) hold with the matrix A replaced by A – ΘC and the matrix B replaced by B – ΘC, then the generalized H_∞ norm of this system is smaller than λ. Using Schur’s complement lemma, we transform these inequalities to

$$\begin{gathered} Y(t + 1) - (A - \Theta C)Y(t){{(A - \Theta C)}^{{\text{T}}}} - (B - \Theta D)G{{(B - \Theta D)}^{{\text{T}}}} \\ - \;(A - \Theta C)Y(t)C_{z}^{{\text{T}}}{{\left( {\lambda I - {{C}_{z}}Y(t)C_{z}^{{\text{T}}}} \right)}^{{ - 1}}}{{C}_{z}}Y(t){{(A - \Theta C)}^{{\text{T}}}}\; \geqslant \;0 \\ \end{gathered} $$

provided that C_zY(t)$C_{z}^{{\text{T}}}$ < λI. Completing the square in Θ(t) on the left-hand side of the latter inequality yields

$$\begin{gathered} Y(t + 1) - A{{\left[ {{{Y}^{{ - 1}}}(t) + {{C}^{{\text{T}}}}G_{D}^{{ - 1}}C - {{\lambda }^{{ - 1}}}C_{z}^{{\text{T}}}{{C}_{z}}} \right]}^{{ - 1}}}{{A}^{{\text{T}}}} - {{G}_{B}} \\ - \;(\Theta - {{\Theta }_{\infty }})\left[ {{{Y}^{{ - 1}}}(t) + {{C}^{{\text{T}}}}G_{D}^{{ - 1}}C - {{\lambda }^{{ - 1}}}C_{z}^{{\text{T}}}{{C}_{z}}} \right]{{(\Theta - {{\Theta }_{\infty }})}^{{\text{T}}}}\; \geqslant \;0, \\ \end{gathered} $$

where Θ_∞ is given by (5.9) for P(t) = Y(t). (Here, we have involved the notations (5.5) and some manipulations.) Hence, if the filter parameters are given by (5.9), where the matrix P(t) satisfies Eq. (5.10), then $\gamma _{s}^{2}$ < λ.