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+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}
+
+
+
Example 1 (Linear system with saturated linear state feedback) In this example we consider a linear system with a saturated linear state feedback as in Fig 1.
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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 2.
+
+
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.
+
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Example 2 (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.
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