Robust Approximate Simulation for Hierarchical Control of Piecewise
Affine Systems under Bounded Disturbances

Consider a piecewise-affine (PWA) system as follows

$$\Sigma:\left\{\begin{array}[]{ll}\mathbf{\dot{x}}_{1}=\mathbf{A}_{i}\mathbf{x}_{1}+\mathbf{B}_{i}\mathbf{u}_{1}+\mathbf{c}_{i}&\\ \mathbf{y}_{1}=\mathbf{C}_{i}\mathbf{x}_{1}&\end{array}\right.$$

(1)

where the system state $\mathbf{x}_{1}\in\mathcal{X}_{1}^{i}:=\{\mathbf{x}_{1}\in\mathbb{R}^{n}|\ \mathbf{E}_{i}\mathbf{x}_{1}\geq\mathbf{f}_{i}\}$, and $\mathbf{E}_{i}\in\mathbb{R}^{b\times n}$, $\mathbf{f}_{i}\in\mathbb{R}^{b}$, $\mathbf{A}_{i}\in\mathbb{R}^{n\times n}$, $\mathbf{B}_{i}\in\mathbb{R}^{n\times p}$ and $\mathbf{C}_{i}\in\mathbb{R}^{k\times n}$ are given, for $i\in\mathcal{I}:=\{1,...,s\}$. The system output is $\mathbf{y}_{1}\in\mathbb{R}^{k}$. The term $\mathbf{c}_{i}\in\mathbb{R}^{n}$, which represents the lumped bounded external disturbances and piecewise linearization error, is assumed to be bounded such that $||\mathbf{c}_{i}||_{\infty}\leq\bar{c}_{i}$, where $\bar{c}_{i}$ is known. The cell bounding [13] is defined as $\mathbf{\bar{E}}_{i}=\begin{bmatrix}\mathbf{E}_{i}&-\mathbf{f}_{i}\end{bmatrix}$ for the partitions with some of the boundaries not crossing the origin (denoted as $\mathcal{I}_{1}$), with $\mathbf{\bar{E}}_{i}\begin{bmatrix}\mathbf{x}_{1}^{T}&1\end{bmatrix}^{T}\geq\mathbf{0}$ and it reduces to $\mathbf{E}_{i}$ for the partitions with all their boundaries crossing the origin (denoted as $\mathcal{I}_{0}$), with $\mathbf{E}_{i}\mathbf{x}_{1}\geq\mathbf{0}$. Besides, the continuity matrix [13] is defined as $\mathbf{\bar{J}}_{i}=\begin{bmatrix}\mathbf{J}_{i}&h_{i}\end{bmatrix}$ for $i\in\mathcal{I}_{1}$ (and $\mathbf{J}_{i}$ for $i\in\mathcal{I}_{0}$), with $\mathbf{\bar{J}}_{i_{1}}\begin{bmatrix}\mathbf{x}_{1}^{T}&1\end{bmatrix}^{T}=\mathbf{\bar{J}}_{i_{2}}\begin{bmatrix}\mathbf{x}_{1}^{T}&1\end{bmatrix}^{T}$ for $\mathcal{X}_{1}^{i_{1}}\cap\mathcal{X}_{1}^{i_{2}}$, $i_{1}$, $i_{2}\in\mathcal{I}$.

In this paper, we first consider the abstraction as a linear system of the following form

$$\Sigma^{\prime}:\left\{\begin{array}[]{ll}\mathbf{\dot{x}}_{2}=\mathbf{F}\mathbf{x}_{2}+\mathbf{G}\mathbf{u}_{2}&\\ \mathbf{y}_{2}=\mathbf{H}\mathbf{x}_{2}&\end{array}\right.$$

(2)

where $\mathbf{x}_{2}\in\mathbb{R}^{m}$ is the state of system (2), $\mathbf{y}_{2}\in\mathbb{R}^{k}$ is the system output. The matrices $\mathbf{F}\in\mathbb{R}^{m\times m}$, $\mathbf{G}\in\mathbb{R}^{m\times q}$ and $\mathbf{H}\in\mathbb{R}^{k\times m}$ are free to select. The abstraction (2) is typically simpler than each mode of the PWA system $\Sigma$ in terms of system dimension, i.e., $m\leq n$.

Then, if a single linear abstraction may not be completely adequate (typically if the concrete PWA system $\Sigma$ has many modes), the abstraction can be selected as a simpler PWA system of the form

$$\Sigma^{\prime\prime}:\left\{\begin{array}[]{ll}\mathbf{\dot{x}}_{2}=\mathbf{F}_{j}\mathbf{x}_{2}+\mathbf{G}_{j}\mathbf{u}_{2}&\\ \mathbf{y}_{2}=\mathbf{H}_{j}\mathbf{x}_{2}&\end{array}\right.$$

(3)

where $\mathbf{x}_{2}\in\mathcal{X}_{2}^{j}:=\{\mathbf{x}_{2}\in\mathbb{R}^{m}|\ \mathbf{E}_{aj}\mathbf{x}_{2}\geq\mathbf{f}_{aj}\}$, $\mathbf{E}_{aj}\in\mathbb{R}^{d\times m}$, $\mathbf{f}_{aj}\in\mathbb{R}^{d}$, $\mathbf{y}_{2}\in\mathbb{R}^{k}$, for $j\in\mathcal{I}_{a}:=\{1,...,r\}$ with $r\leq s$. The matrices $\mathbf{F}_{j}\in\mathbb{R}^{m\times m}$, $\mathbf{G}_{j}\in\mathbb{R}^{m\times q}$ and $\mathbf{H}_{j}\in\mathbb{R}^{k\times m}$ are free to select, for $j\in\mathcal{I}_{a}$. Let us denote the partitions with all their boundaries crossing the origin as $\mathcal{I}_{a0}$ and the partitions with some of the boundaries not crossing the origin as $\mathcal{I}_{a1}$. It is important to note that the abstraction $\Sigma^{\prime\prime}$ has less modes and much simpler dynamics than the concrete system $\Sigma$. The details of the partitions of the abstract PWA system will be studied in Section III.

Consider the abstraction in (2). Define $\mathbf{\tilde{x}}=\mathbf{x}_{1}-\mathbf{P}_{i}\mathbf{x}_{2}$, where $\mathbf{P}_{i}\in\mathbb{R}^{n\times m}$ is an injective map from the state space of $\mathbf{x}_{2}$ to that of $\mathbf{x}_{1}$ for $i\in\mathcal{I}$. Then,

	$\displaystyle\mathbf{\dot{\tilde{x}}}$	$\displaystyle=\mathbf{\dot{x}}_{1}-\mathbf{P}_{i}\mathbf{\dot{x}}_{2}$
		$\displaystyle=(\mathbf{A}_{i}\mathbf{x}_{1}+\mathbf{B}_{i}\mathbf{u}_{1}+\mathbf{c}_{i})-\mathbf{P}_{i}(\mathbf{F}\mathbf{x}_{2}+\mathbf{G}\mathbf{u}_{2})$
		$\displaystyle=\mathbf{A}_{i}\mathbf{\tilde{x}}+\mathbf{B}_{i}\mathbf{u}_{v}+\mathbf{A}_{i}\mathbf{P}_{i}\mathbf{x}_{2}+\mathbf{c}_{i}-\mathbf{P}_{i}(\mathbf{F}\mathbf{x}_{2}+\mathbf{G}\mathbf{u}_{2})$

where $\mathbf{u}_{v}:=\mathbf{u}_{1}(\mathbf{\tilde{x}},\mathbf{x}_{2},\mathbf{\bar{u}}_{2})$ is the control input of the concrete system, which is also known as the interface that we need to design. For the abstraction (2), an input transformation law is designed as

$$\mathbf{u}_{2}=\mathbf{L}\mathbf{x}_{2}+\mathbf{\bar{u}}_{2}$$

(4)

where the matrix $\mathbf{L}\in\mathbb{R}^{q\times m}$ is selected such that all the eigenvalues of the matrix $\mathbf{F}+\mathbf{G}\mathbf{L}$ of the transformed abstraction $\Sigma^{\prime}$ have negative real parts, and $\mathbf{\bar{u}}_{2}\in\mathbb{R}^{q}$ is the transformed input. Thus, we define our robust approximate simulation framework as the following joint system

$$\left\{\begin{array}[]{ll}\mathbf{\dot{\tilde{x}}}=\mathbf{A}_{i}\mathbf{\tilde{x}}+\mathbf{B}_{i}\mathbf{u}_{v}-[\mathbf{B}_{i}\mathbf{Q}_{i}+\mathbf{P}_{i}\mathbf{G}\mathbf{L}]\mathbf{x}_{2}-\mathbf{P}_{i}\mathbf{G}\mathbf{\bar{u}}_{2}+\mathbf{c}_{i}&\\ \mathbf{\dot{x}}_{2}=(\mathbf{F}+\mathbf{G}\mathbf{L})\mathbf{x}_{2}+\mathbf{G}\mathbf{\bar{u}}_{2}&\\ \mathbf{e}=\mathbf{y}_{1}-\mathbf{y}_{2}=\mathbf{C}_{i}\mathbf{\tilde{x}}&\end{array}\right.$$

(5)

for $i\in\mathcal{I}$, where $\mathbf{P}_{i}\in\mathbb{R}^{n\times m}$ and $\mathbf{Q}_{i}\in\mathbb{R}^{n\times m}$ satisfy the conditions:

$$\mathbf{H}=\mathbf{C}_{i}\mathbf{P}_{i},\ \mathbf{P}_{i}\mathbf{F}=\mathbf{A}_{i}\mathbf{P}_{i}+\mathbf{B}_{i}\mathbf{Q}_{i}$$

(6)

Define the state of the joint system as $\bm{\omega}=\begin{bmatrix}\mathbf{\tilde{x}}^{T}&\mathbf{x}_{2}^{T}\\ \end{bmatrix}^{T}$. Then, the joint partitions can be written as

$$\begin{split}\bm{\omega}\in\mathbf{\Omega}_{i}:&=\{\mathbf{E}_{i}(\mathbf{x}_{1}-\mathbf{P}_{i}\mathbf{x}_{2})+\mathbf{E}_{i}\mathbf{P}_{i}\mathbf{x}_{2}\geq\mathbf{f}_{i}\}\\ &=\{[\mathbf{E}_{i}\ \mathbf{E}_{i}\mathbf{P}_{i}]\bm{\omega}\geq\mathbf{f}_{i}\}\end{split}$$

(7)

Note that the joint partition (7) can be equivalently represented by:

$$\begin{split}\bm{\bar{\omega}}\in\mathbf{\bar{\Omega}}_{i}:&=\{\begin{bmatrix}\mathbf{E}^{\prime}_{i}&\mathbf{f}^{\prime}_{i}\end{bmatrix}\begin{bmatrix}\bm{\omega}\\ 1\end{bmatrix}\geq\mathbf{0}\}=\{\mathbf{\bar{E}}_{i}\bm{\bar{\omega}}\geq\mathbf{0}\}\end{split}$$

(8)

where $\mathbf{E}^{\prime}_{i}=\begin{bmatrix}\mathbf{E}_{i}&\mathbf{E}_{i}\mathbf{P}_{i}\end{bmatrix}$ and $\mathbf{f}^{\prime}_{i}=-\mathbf{f}_{i}$.

The joint system (5) corresponding to the PWA system abstraction (3) is similar to that of the linear system abstraction (2). In particular, it will result in a joint system of the form (5) for each tuple ($\mathbf{F}_{j}$, $\mathbf{G}_{j}$, $\mathbf{L}_{j}$), for $j\in\mathcal{I}_{a}$, and the conditions in (6) become

$$\mathbf{H}_{j}=\mathbf{C}_{i}\mathbf{P}_{i},\ \mathbf{P}_{i}\mathbf{F}_{j}=\mathbf{A}_{i}\mathbf{P}_{i}+\mathbf{B}_{i}\mathbf{Q}_{i}$$

(9)

for $i\in\mathcal{I}$. In other words, for each $i\in\mathcal{I}$ we need to determine matrices $\mathbf{P}_{i}$ and $\mathbf{Q}_{i}$ such that (9) holds true for a pair ($\mathbf{F}_{j}$, $\mathbf{H}_{j}$), $j\in\mathcal{I}_{a}$. The details of the joint partitions corresponding to this case will be discussed in Section III.

The objective of this paper is to design an interface $\mathbf{u}_{v}:=\mathbf{u}_{1}(\mathbf{\tilde{x}},\mathbf{x}_{2},\mathbf{\bar{u}}_{2})$ for the joint system (5) over the joint partitions (e.g.,(7)) to guarantee the boundedness of output error between $\mathbf{y}_{1}$ and $\mathbf{y}_{2}$, i.e., $||\mathbf{e}||=||\mathbf{y}_{1}-\mathbf{y}_{2}||\leq\delta$, where $\delta\in\mathbb{R}_{\geq 0}$ is some certain error bound. The basic control architecture proposed in this paper is shown in Figure 1.

In this part, we first design an interface for the joint system (5) for the linear abstraction in the form (2), and then a Lyapunov-like simulation function is presented to guarantee a formal error bound for the output tracking errors.

In this part, we consider the case where the abstraction is in form $(\ref{eq:PWA_abstraction})$, where we are free to choose the partitions as well as the system matrices on each partition. Compared to the previous part, the main difference in this part lies in the computation of the cell boundings of the joint system since the partitions of the abstract PWA system should be involved in the computation of the joint $\mathbf{\bar{E}}_{ij}$ as $\bm{\bar{\omega}}\in\mathbf{\bar{\Omega}}_{ij}$, where

$$\begin{split}\mathbf{\bar{\Omega}}_{ij}:&=\Big{\{}\begin{bmatrix}\mathbf{E}_{i}\\ \mathbf{E}_{cj}\end{bmatrix}(\mathbf{x}_{1}-\mathbf{P}_{i}\mathbf{x}_{2})+\begin{bmatrix}\mathbf{E}_{i}\mathbf{P}_{i}\\ \mathbf{E}_{cj}\mathbf{P}_{i}\end{bmatrix}\mathbf{x}_{2}\geq\begin{bmatrix}\mathbf{f}_{i}\\ \mathbf{f}_{cj}\end{bmatrix}\Big{\}}\\ &=\Big{\{}\begin{bmatrix}\mathbf{E}_{i}&\mathbf{E}_{i}\mathbf{P}_{i}\\ \mathbf{E}_{cj}&\mathbf{E}_{cj}\mathbf{P}_{i}\end{bmatrix}\bm{\omega}\geq\begin{bmatrix}\mathbf{f}_{i}\\ \mathbf{f}_{cj}\end{bmatrix}\Big{\}}\\ &=\{\mathbf{\bar{E}}_{ij}\bm{\bar{\omega}}\geq\mathbf{0}\}\end{split}$$

(20)

where $i\in\mathcal{I}$, $j\in\mathcal{I}_{a}$ and the matrices $\mathbf{E}_{cj}\in\mathbb{R}^{l\times n}$ and $\mathbf{f}_{cj}\in\mathbb{R}^{l}$ construct the desired partitions of $\mathbf{x}_{2}$ in the state space of $\mathbf{x}_{1}$ (i.e., $\mathbf{x}_{1}\in\mathcal{X}_{1}^{i}:=\{\mathbf{x}_{1}\in\mathbb{R}^{n}|\ \mathbf{E}_{i}\mathbf{x}_{1}\geq\mathbf{f}_{i}\}$, $i\in\mathcal{I}$). In particular, the desired partitions of $\mathbf{x}_{2}$ in the state space of $\mathbf{x}_{1}$ can be written in closed form as $\mathbf{E}_{cj}\mathbf{x}_{1}\geq\mathbf{f}_{cj}$, i.e., $\mathbf{E}_{cj}\mathbf{\tilde{x}}+\mathbf{E}_{cj}\mathbf{P}_{i}\mathbf{x}_{2}\geq\mathbf{f}_{cj}$, for $j\in\mathcal{I}_{a}$.

In fact, the partition of $\mathbf{x}_{2}$ can be viewed as $\mathbf{E}_{aj}\mathbf{x}_{2}\geq\mathbf{f}_{aj}$ as in (3), where $\mathbf{E}_{aj}=\mathbf{E}_{cj}\mathbf{P}_{i}$ and $\mathbf{f}_{aj}=\mathbf{f}_{cj}-\mathbf{E}_{cj}\mathbf{\tilde{x}}$. It is not surprising that the partition of $\mathbf{x}_{2}$ varies with $\mathbf{\tilde{x}}$ because we know that there must exist some linear transformation $\mathbf{\Pi}\in\mathbb{R}^{m\times n}$ between two different spaces, i.e. $\mathbf{x}_{2}=\mathbf{\Pi}\mathbf{x}_{1}$ from Lemma 1 of [16]. However, this relation cannot hold for our case where the disturbances exist in the state space of $\mathbf{x}_{1}$ or it may hold with very restrictive conditions (see proposition 4.1 of [17]). The relations $\mathbf{E}_{aj}=\mathbf{E}_{cj}\mathbf{P}_{i}$ and $\mathbf{f}_{aj}=\mathbf{f}_{cj}-\mathbf{E}_{cj}\mathbf{\tilde{x}}$ can be viewed as an online estimate to reshape the partitions of $\mathbf{x}_{2}$ in our case.