Background results for robust minmax control of linear dynamical systems

TL;DR

This note compiles essential background results for robust minmax control of linear dynamical systems with a quadratic stage cost, consolidating corrected formulas into revised results such as Propositions 14.a and 20.a and generalizing key linear-algebra tools. It develops a comprehensive Lagrangian/minimax framework for constrained quadratic optimization, including trust-region type problems, and provides explicit conditions for existence, optimality, and duality across structured blocks and constraints. The work clarifies when strong duality holds and illustrates the distinction between robust control and worst-case feedforward control in constrained minmax settings, with a focus on tractable expressions involving pseudoinverses, Schur complements, and SVD-based decompositions. The practical impact lies in delivering precise, generalizable tools for deriving robust minmax controllers in linear-quadratic settings, including corrections and extensions that improve reliability of the foundational arguments and their numerical implementation.

Abstract

The purpose of this note is to summarize the arguments required to derive the results appearing in robust minmax control of linear dynamical systems using a quadratic stage cost. The main result required in robust minmax control is Proposition 20.a. Moreover, the solution to the trust-region problem given in Proposition 15 and Lemma 16 may be of more general interest. This revised (second) version provides the following corrections and extensions of the previous (first) version. 1. The optimal u and w formulas in the original Corollary 13, Proposition 14, Corollary 19, and Proposition 20 have been corrected in this revision. 2. Corollary 13 and Proposition 14 are combined in the revised Proposition 14.a. 3. Corollary 19 and Proposition 20 are combined in the revised Proposition 20.a. 4. The revised Proposition 12.a is a generalization of the previous Proposition 12. 5. Propositions 5.a, 5.b, and 5.c are new in this revision. 6. Figure 1 has been revised to illustrate the revised Proposition 14.a.

Figures (2)

• Figure 1: The optimal value function $L^0(\lambda)$ for $\min_u\max_wL$ and $\max_w\min_uL$ versus parameter $\lambda$. Top: $| \tilde{M}_{11} | = \left| M_{22} \right|$ showing cases 2(a) and 2(b). Bottom: $| \tilde{M}_{11} | < \left| M_{22} \right|$ showing cases 4(a) and 4(b).
• Figure 2: $L(w^0(\lambda), \lambda)$ versus $\lambda$ for the same $D$ but different $d$. Green lines: for $d \in R(D-\left| D \right|I)$, $L$ is bounded for all $\lambda\geq \left| D \right|$. Green dots: for $\left| (D-\left| D \right|I)^+d \right| \leq 1$, the optimum is on the boundary and $\lambda_P= \left| D \right|$. Red dots: for $\left| (D-\left| D \right|I)^+d \right| > 1$, the optimum is in the interior and $\lambda_P> \left| D \right|$. Blue line and dot: for $d \notin R(D-\left| D \right|I)$, $L$ is unbounded at $\lambda = \left| D \right|$, and the optimum is in the interior, and $\lambda_P > \left| D \right|$.

Theorems & Definitions (37)

• Proposition 1: Solving linear algebra problems.
• proof
• Definition 2: Convex function
• Proposition 3
• proof
• Proposition 4: Convex quadratic functions
• proof
• Proposition 5: Minimum of quadratic functions
• proof
• Proposition 2.a: Minimum of semidefinite quadratic functions subject to linear constraints