Passivity and complementarity

Abstract

This paper studies the interaction between the notions of passivity of systems theory and complementarity of mathematical programming in the context of complementarity systems. These systems consist of a dynamical system (given in the form of state space representation) and complementarity relations. We study existence, uniqueness, and nature of solutions for this system class under a passivity assumption on the dynamical part. A complete characterization of the initial states and the inputs for which a solution exists is given. These initial states are called consistent states. For the inconsistent states, we introduce a solution concept in the framework of distributions.

Appendices

Appendix 1: Quadratic programming

Theorem 11

Let $Q$ be a symmetric nonnegative definite matrix. Consider the following three quadratic programs

$$\begin{aligned}&\mathrm{minimize}\ \frac{1}{2}x^T Q x+b^Tx \ \mathrm{subject\ to}\ x\geqslant 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \! (\textit{QP}_1) \\&\mathrm{minimize}\ \frac{1}{2}x^T Q x+b^Tx \ \mathrm{subject\ to}\ Ax\geqslant c \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \! (\textit{QP}_2)\\&\mathrm{minimize}\ \frac{1}{2}x^T Q x-c^Tu\ \mathrm{subject\ to}\ A^Tu-Qx=b \text{ and } u\geqslant 0. \qquad \qquad \qquad \quad (\textit{QP}_3) \end{aligned}$$

The following statements hold.

• 1. Karush-Kuhn-Tucker conditions

  $$\begin{aligned}&x\geqslant 0\\&b+Qx\geqslant 0\\&x^T(b+Qx)=0 \end{aligned}$$

  are necessary and sufficient for the vector $x$ to be globally optimal solution of the quadratic program $(QP_1)$.

• 2. (Dorn’s duality theorem) [25, Theorem 8.2.4] If ${\bar{x}}$ solves $(QP_2)$ then there exists ${\bar{u}}$ such that $({\bar{x}},{\bar{u}})$ solves $(QP_3)$. Moreover, the two extrema are equal.

• 3. (Dorn’s converse duality theorem) [25, Theorem 8.2.6] If $({\bar{x}},{\bar{u}})$ solves $(QP_3)$ then there exists ${\hat{x}}$ with $Q{\hat{x}}=Q{\bar{x}}$ such that ${\hat{x}}$ solves $(QP_2)$.

Appendix 2: Geometric control theory

Consider the linear system

$$\begin{aligned} \dot{x}(t)&= Ax(t)+Bu(t)\\ y(t)&= Cx(t)+Du(t) \end{aligned}$$

where $x\in \mathbb{R }^n$, $u\in \mathbb{R }^m$, $y\in \mathbb{R }^p$, and all involved matrices are of appropriate dimensions. A subspace $\mathcal V \subseteq \mathbb{R }^n$ is said to be an output-nulling controlled-invariant subspace if there exists $L\in \mathbb{R }^{m\times n}$ such that

$$\begin{aligned} (A+BL)\mathcal V \subseteq \mathcal V \quad \text{ and } \quad (C+DL)\mathcal V =\{0\}. \end{aligned}$$

Since the set of all such subspaces are closed under subspace sum and intersection, there exists a unique subspace $\mathcal V ^*$ such that

$$\begin{aligned} \mathcal V \subseteq \mathcal V ^* \end{aligned}$$

holds whenever $\mathcal V$ is an output-nulling controlled invariant subspace. We call $\mathcal V ^*$ the largest output-nulling controlled invariant subspace and define

$$\begin{aligned} \mathcal{L }=\{L\mid (A+BL)\mathcal V ^*\subseteq \mathcal V ^* \text{ and } (C+DL)\mathcal V ^*=\{0\}\}. \end{aligned}$$

The following result (see e.g. [38, Theorem 7.11]) establishes a link between such subspaces and the so-called zero dynamics of linear systems.

Theorem 12

The pair $(u,x)$ satisfies the differential algebraic equations

$$\begin{aligned} \dot{x}(t)&= Ax(t)+Bu(t)\\ 0&= Cx(t)+Du(t) \end{aligned}$$

if and only if $x(0)\in \mathcal V ^*$ and the input $u$ has the form $u(t)=Lx(t)+v(t)$ where $L\in \mathcal{L }$, $v(t)\in \ker D\cap B^{-1}\mathcal V ^*$ for almost all $t$, and $B^{-1}\mathcal V ^*=\{\bar{v}\mid B\bar{v}\in \mathcal V ^*\}$.