classes_PWA.qmd

---
title: "Piecewise affine (PWA) systems"
bibliography: 
    - ref_PWA.bib
    - ref_PWA_approx.bib
csl: ieee-control-systems.csl
format:
    html:
        html-math-method: katex
        code-fold: true
        code-summary: "Show the code"
crossref:
  fig-prefix: Fig. 
  eq-prefix: Eq.
#engine: julia
---

This is a subclass of switched systems where the functions on the right-hand side of the differential equations are affine functions of the state. For some (historical) reason these systems are also called *piecewise linear* (PWL).

We are going to reformulate such systems as switched systems with state-driven switching.

First, we consider the autonomous case, that is, systems without inputs: 
$$
\dot{\bm x} 
= 
\begin{cases}
\bm A_1 \bm x + \bm b_1, & \mathrm{if}\, \bm H_1 \bm x + \bm g_1 \leq 0,\\
\vdots\\
\bm A_m \bm x + \bm b_m, & \mathrm{if}\, \bm H_m \bm x + \bm g_m \leq 0.
\end{cases}
$$

The nonautonomous case of systems with inputs is then:
$$
\dot{\bm x} 
= 
\begin{cases}
\bm A_1 \bm x + \bm B_1 u + \bm c_1, & \mathrm{if}\, \bm H_1 \bm x + \bm g_1 \leq 0,\\
\vdots\\
\bm A_m \bm x + \bm B_m + \bm c_m, & \mathrm{if}\, \bm H_m \bm x + \bm g_m \leq 0.
\end{cases}
$$

::: {#exm-saturated-state-feedback}
## Linear system with saturated linear state feedback
In this example we consider a linear system with a saturated linear state feedback as in @fig-saturated-state-feedback.

![Linear system with a saturated linear state feedback](classes_figures/saturated_state_feedback.png){width=50% #fig-saturated-state-feedback}

The state equations for the close-loop system are
$$
\dot{\bm x} = \bm A\bm x + \bm b \,\mathrm{sat}(v), \quad v = \bm k^T \bm x,
$$
which can be reformulated as a piecewise affine system
$$
\dot{\bm x} = 
\begin{cases}
\bm A \bm x - \bm b, & \mathrm{if}\, \bm x \in \mathcal{X}_1,\\
(\bm A + \bm b \bm k^\top )\bm x, & \mathrm{if}\, \bm x \in \mathcal{X}_2,\\
\bm A \bm x + \bm b, & \mathrm{if}\, \bm x \in \mathcal{X}_3,\\
\end{cases}
$$
where the partitioning of the space of control inputs is shown in @fig-saturated-state-feedback-partitioning.

![Partitioning the control input space](classes_figures/saturated_state_feedback_partitioning.png){width=50% #fig-saturated-state-feedback-partitioning}

Expressed in the state space, the partitioning is
$$
\begin{aligned}
\mathcal{X}_1 &= \{\bm x \mid \bm H_1\bm x + g_1 \leq 0\},\\
\mathcal{X}_2 &= \{\bm x \mid \bm H_2\bm x + \bm g_2 \leq 0\},\\
\mathcal{X}_3 &= \{\bm x \mid \bm H_3\bm x + g_3 \leq 0\},
\end{aligned}
$$
where
$$
\begin{aligned}
\bm H_1 &= \bm k^\top, \quad g_1 = 1,\\
\bm H_2 &= \begin{bmatrix}-\bm k^\top\\\bm k^\top\end{bmatrix}, \quad \bm g_2 = \begin{bmatrix}-1\\-1\end{bmatrix},\\
\bm H_3 &= -\bm k^\top, \quad g_3 = 1.
\end{aligned}
$$
:::

## Approximation of nonlinear systems

While the example with the saturated linear state feedback can be modelled as a PWA system exactly, there are many practical cases, in which the system is not exactly PWA affine but we want to approximate it as such.

::: {#exm-nonlinear-approximation}
## Nonlinear system approximated by a PWA system
Consider the following nonlinear system
$$
\begin{bmatrix}
\dot x_1\\\dot x_2
\end{bmatrix}
=
\begin{bmatrix}
x_2\\
-x_2 |x_2| - x_1 (1+x_1^2)
\end{bmatrix}
$$

Our task is to approximate this system by a PWA system. Equivalently, we need to find a PWA approximation for the right-hand side function.

:::