A mixed-integer framework for analyzing neural network-based controllers for piecewise affine systems with bounded disturbances

TL;DR

This work presents a method for representing the closed-loop dynamics of piecewise affine (PWA) systems with bounded additive disturbances and neural network-based controllers as mixed-integer (MI) linear constraints and shows that such representations enable the computation of robustly positively invariant (RPI) sets for the specified system class by solving MI linear programs.

Abstract

We present a method for representing the closed-loop dynamics of piecewise affine (PWA) systems with bounded additive disturbances and neural network-based controllers as mixed-integer (MI) linear constraints. We show that such representations enable the computation of robustly positively invariant (RPI) sets for the specified system class by solving MI linear programs. These RPI sets can subsequently be used to certify stability and constraint satisfaction. Furthermore, the approach allows to handle non-linear systems based on suitable PWA approximations and corresponding error bounds, which can be interpreted as the bounded disturbances from above.

Figures (3)

• Figure 1: Illustration of the computation of the set $\overline{\mathcal{R}}_{\text{min}}=\overline{\mathcal{R}}^\mathcal{D}_{\overline{k}}(\mathcal{F}_{\text{max}})$ in blue starting from the set $\mathcal{F}_{\text{max}}$ in green. The black sets illustrate the shrinking sequence of sets $\overline{\mathcal{R}}^\mathcal{D}_{i}(\mathcal{F}_{\text{max}})$ with $i\in\{1,\dots\overline{k}-1\}$ during the computation of $\overline{\mathcal{R}}_{\text{min}}$. The computation terminates with the set $\overline{\mathcal{R}}^\mathcal{D}_{\overline{k}}(\mathcal{F}_{\text{max}})=\overline{\mathcal{R}}_{\text{min}}$ when \ref{['eq:TerminationFmin']} holds.
• Figure 2: The green and blue “tubes” illustrate in each case $100$ trajectories of the closed-loop system with a random additive disturbance $\left\lVert\boldsymbol{d}(k)\right\rVert_\infty\leq0.15$ in every time step. The black lines represent trajectories of the nominal system (i.e. $\boldsymbol{d}(k)=\boldsymbol{0}$). The small set around the origin is $\overline{\mathcal{R}}_{\text{min}}$ and the box $[-7.3,8.91]\times[-10,8.52]$ is the set $\mathcal{F}_{\text{max}}$. The thick grey lines indicate the regions of the PWA system.
• Figure 3: Nonlinear double integrator with an NN-based controller. The grey arrows represent the evolution of the closed-loop system within the set $\mathcal{F}_{\text{max}}$ (green set). The red line represents a trajectory $\boldsymbol{x}(0),\dots,\boldsymbol{x}(\overline{k})$ of the system \ref{['eq:NL_DI']} with \ref{['eq:UOfK']} starting at $\boldsymbol{x}(0)=(5.5 \ -1.5)^\top$. The small sets around the states $\boldsymbol{x}(i)$ are the reachable sets $\overline{\mathcal{R}}_i^\mathcal{D}( \{ \boldsymbol{x} \in \mathbb{R}^n \ | \ \boldsymbol{x} = (5.5 \ -1.5)^\top \} )$ with $i\in\{1,\dots,\overline{k}\}$ in which the disturbed PWA system (which is an approximation of \ref{['eq:NL_DI']}) and thus the nonlinear system is guaranteed to be in time step $i$. Thick grey lines represent the regions of the PWA system.

Theorems & Definitions (12)

• Definition 1: rakovic2005invariant
• Lemma 1
• Lemma 2
• Proposition 3
• proof
• Theorem 4
• proof
• Lemma 5
• Theorem 6
• proof