Enclosing the Sliding Surfaces of a Controlled Swing

In this paper, we consider a specific class of hybrid dynamical systems of the form

$$\mathcal{S}\left(\mathbb{A}\right):\begin{array}[]{c}\left\{\begin{array}[]{ccc}\dot{\mathbf{x}}=\mathbf{f}_{a}\left(\mathbf{x}\right)&&\text{if $\mathbf{x}\in\mathbb{A}$}\\ \dot{\mathbf{x}}=\mathbf{f}_{b}\left(\mathbf{x}\right)&&\text{if $\mathbf{x}\in\mathbb{B}=\overline{\mathbb{A}}$}\end{array}\right.\end{array}$$

(1)

where

• •

  $\mathbf{f}_{a},\mathbf{f}_{b}$ :$\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ are continuous and differentiable,

• •

  $\mathbb{A}$ is a closed subset of $\mathbb{R}^{n}$ that can be defined by inequalities linked by Boolean operators.

This definition is illustrated by the automaton of Figure 1, taking the conventions used for hybrid systems [3, 15]. The red arrows show transitions which may generate the sliding phenomena that are studied in this paper.

Given a set $\mathbb{X}\subset\mathbb{R}^{n}$, we denote by $\text{cl}(\mathbb{X})$ the smallest closed subset of $\mathbb{R}^{n}$ which contains $\mathbb{X}$. The intersection of closed sets and the finite union of closed sets if still a closed set. We define the closed complementary $\overline{\mathbb{A}}$ of the closed set $\mathbb{A}$ as

$$\overline{\mathbb{A}}=\text{cl}\{\mathbf{x}|\mathbf{x}\notin\mathbb{A}\}.$$

The boundary of the closed set $\mathbb{A}$ is denoted by $\partial\mathbb{A}$. The closed set $\mathbb{A}$ is topologically stable if $\partial\mathbb{A}=\partial\mathbb{\overline{A}}.$

For instance, the disk of $\mathbb{D}=\{\mathbf{x}\in\mathbb{R}^{2}|\|\mathbf{x}\|\leq 1\}$ is topologically stable with $\overline{\mathbb{D}}=\{\mathbf{x}\in\mathbb{R}^{2}|\|\mathbf{x}\|\geq 1\}$ and $\partial\mathbb{D}=\partial\mathbb{\overline{D}}=\{\mathbf{x}\in\mathbb{R}^{2}|\|\mathbf{x}\|=1\}$. But the circle $\mathbb{C}=\{\mathbf{x}\in\mathbb{R}^{2}|\|\mathbf{x}\|=1\}$ is not topologically stable. Indeed $\overline{\mathbb{C}}=\text{cl}\{\mathbf{x}\in\mathbb{R}^{2}|\|\mathbf{x}\|\neq 1\}=\mathbb{R}^{2}$ and $\partial\mathbb{C}=\{\mathbf{x}\in\mathbb{R}^{2}|\|\mathbf{x}\|=1\}\neq\partial\mathbb{\overline{C}}=\emptyset$.

In this paper, we will assume that

1. 1.

   closed sets $\mathbb{A}$ involved in the formulation (1) are topologically stable, i.e., they have the same boundary as their interior.

2. 2.

   the closed sets can be defined as a finite composition (with unions, intersections) of sets of the form

   $$\mathbb{A}=\left\{\mathbf{x}\in\mathbb{R}^{n}\,|\,c(\mathbf{x})\leq 0\right\}$$

   where $c$ is a smooth function.

We recall the notion of Lie derivative that will be used to define the sliding surfaces. Consider a function $c:\mathbb{R}^{n}\rightarrow\mathbb{R}$. The Lie derivative of $c$ with respect to the field $\mathbf{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ as

$$\mathcal{L}_{\mathbf{f}}^{c}\left(\mathbf{x}\right)=\frac{dc}{d\mathbf{x}}\left(\mathbf{x}\right)\cdot\mathbf{f}\left(\mathbf{x}\right).$$

(2)

We also define the Lie set as

$$\mathbb{L}_{\mathbf{f}}^{c}=\left\{\mathbf{x}|\mathcal{L}_{\mathbf{f}}^{c}\left(\mathbf{x}\right)\leq 0\right\}.$$

(3)

In our context, the field depends on $i\in\{a,b\}$ (see 1). We will write

$$\begin{array}[]{ccc}\mathcal{L}_{i}^{c}\left(\mathbf{x}\right)&=&\mathcal{L}_{\mathbf{f}_{i}}^{c}\left(\mathbf{x}\right)\\ \mathbb{L}_{i}^{c}&=&\mathbb{L}_{\mathbf{f}_{i}}^{c}\end{array}$$

(4)

The sliding surface $\mathbb{S}\left(\mathbb{A}\right)$ [13] for $\mathcal{S}\left(\mathbb{A}\right)$ (see Equation 1) is defined as the largest subset (with respect to the inclusion $\subset$) of the boundary $\partial\mathbb{A}$ of $\mathbb{A}$ such that the system can stay inside for a non-degenerate interval of time.

If $\mathbb{A}$ is defined by the inequality $c\left(\mathbf{x}\right)\leq 0$, then $\mathbb{B}=\overline{\mathbb{A}}$ is defined by $c\left(\mathbf{x}\right)\geq 0$ and the boundary by $c\left(\mathbf{x}\right)=0$ (see Subsection 2.2). The sliding surface is

$$\begin{array}[]{ccc}\mathbb{S}\left(\mathbb{A}\right)&=&\partial\mathbb{A}\cap\left\{\mathbf{x}\,|\,\mathcal{L}_{a}^{c}\left(\mathbf{x}\right)\geq 0\,\wedge\mathcal{L}_{b}^{c}\left(\mathbf{x}\right)\leq 0\right\}\\ &=&\partial\mathbb{A}\cap\overline{\mathbb{L}_{a}^{c}}\,\cap\mathbb{L}_{b}^{c}.\end{array}$$

(5)

Figure 2, taken from [17], illustrates the principle of this proposition in the case where $\mathbb{A}$ is described by one inequality $c\left(\mathbf{x}\right)\leq 0$. The boundary $\partial\mathbb{A}$ of $\mathbb{A}$ is composed of four parts :

$$\begin{array}[]{ccl}\partial\mathbb{A}\cap\overline{\mathbb{L}_{a}^{c}\left(q\right)}\,\cap\overline{\mathbb{L}_{b}^{c}\left(q\right)}&\rightarrow&\text{magenta}\\ \partial\mathbb{A}\cap\overline{\mathbb{L}_{a}^{c}\left(q\right)}\,\cap\mathbb{L}_{b}^{c}\left(q\right)&\rightarrow&\text{red}\\ \partial\mathbb{A}\cap\mathbb{L}_{a}^{c}\left(q\right)\,\cap\mathbb{L}_{b}^{c}\left(q\right)&\rightarrow&\text{yellow}\\ \partial\mathbb{A}\cap\mathbb{L}_{a}^{c}\left(q\right)\,\cap\overline{\mathbb{L}_{b}^{c}\left(q\right)}&\rightarrow&\text{black}\end{array}$$

One trajectory (dotted line) $\mathbf{x}(t)$ is also represented. Before the yellow arc, $c\left(\mathbf{x}\right)$ is positive and decreases. When it crosses the yellow arc, $c\left(\mathbf{x}\right)=0$ for some isolated time point $t_{1}$. Then $\mathbf{x}(t)$ remains inside $\mathbb{A}$ until it reaches the red arc. It slides in the red arc for some non-degenerate time interval. When $\mathbf{x}(t)$ reaches the magenta arc, it leaves $\mathbb{A}$.

We recall the following result that has been proved in [17].

Proposition 2.1 can be used to compute the sliding surface of a set $\mathbb{A}$ as soon as $\mathbb{A}$ can be defined by inequalities connected by Boolean operators such as and, or, not. The proposition is illustrated by Figure 4 in the case where $\mathbb{A}=\mathbb{A}_{1}\cup\left(\mathbb{A}_{2}\cap\mathbb{A}_{3}\right)$ and $\mathbb{A}_{i}=\left\{\mathbf{x}|c_{i}\left(\mathbf{x}\right)\leq 0\right\}$. The trajectory (green) slides twice, first on $\partial\mathbb{A}_{1}$, then it slides on $\partial\mathbb{A}_{2}$. The sliding surfaces are painted red.

In case of uncertainties, the system defined in (1) can be described as

$$\begin{array}[]{c}\mathcal{S}_{\mathbf{p}}:\begin{array}[]{c}\left\{\begin{array}[]{ccc}\dot{\mathbf{x}}=\mathbf{f}_{a}\left(\mathbf{x}\right)&&\text{if $\mathbf{x}\in\mathbb{A}$}\\ \dot{\mathbf{x}}=\mathbf{f}_{b}\left(\mathbf{x}\right)&&\text{if $\mathbf{x}\in\overline{\mathbb{A}}$}\end{array}\right.\end{array}\end{array}$$

but now, $\mathbb{A},\mathbf{f}_{a},\mathbf{f}_{b}$ are uncertain. For instance, they may depend on a parameter vector $\mathbf{p}\in[\mathbf{p}]$ which models the uncertainties. Recall that from Equation 5, the sliding surface satisfies

$$\begin{array}[]{ccl}\mathbb{S}&=&\partial\mathbb{A}\cap\overline{\mathbb{L}_{a}^{c}}\,\cap\mathbb{L}_{b}^{c}\end{array}$$

Assume that

$$\left\{\begin{array}[]{c}\mathbb{A}^{\subset}\subset\mathbb{A}\subset\mathbb{A}^{\supset}\\ \mathbb{L}_{a}^{c\,\subset}\subset\mathbb{L}_{a}^{c}\subset\mathbb{L}_{a}^{c\,\supset}\\ \mathbb{L}_{b}^{c\,\subset}\subset\mathbb{L}_{b}^{c}\subset\mathbb{L}_{b}^{c\,\supset}\end{array}\right.$$

The sliding surface satisfies

$$\underset{\mathbb{S}^{\subset}}{\underbrace{\partial\mathbb{A}^{\subset}\cap\overline{\mathbb{L}_{a}^{c\supset}}\,\cap\mathbb{L}_{b}^{c\subset}}}\subset\mathbb{S}\subset\underset{\mathbb{S}^{\supset}}{\underbrace{\partial\mathbb{A}^{\supset}\cap\overline{\mathbb{L}_{a}^{c\subset}}\,\cap\mathbb{L}_{b}^{c\supset}}}$$

and we can write

$$\mathbb{S}\in[\negthinspace[\mathbb{S^{\subset}},\mathbb{S}^{\supset}]\negthinspace]=\partial[\negthinspace[\mathbb{A^{\subset}},\mathbb{A}^{\supset}]\negthinspace]\cap\overline{[\negthinspace[\mathbb{L}_{a}^{c\,\subset},\mathbb{L}_{a}^{c\,\supset}]\negthinspace]}\,\cap[\negthinspace[\mathbb{L}_{b}^{c\,\subset},\mathbb{L}_{b}^{c\,\supset}]\negthinspace]$$

using the thick set formalism described in the following section. In our application, is is clear that $\mathbb{S}^{\subset}=\emptyset$, since the set to be enclosed is a surface that has no volume. The set $\mathbb{S}^{\supset}$ is an upper approximation of this surface as illustrated by Figure 5.

If an interval of $\mathbb{R}$ is an uncertain real number, a thick set is an uncertain subset of $\mathbb{R^{\textnormal{$n$}}}$. More precisely, a thick set is an interval of the powerset of $\mathbb{R}^{n}$ equipped with the inclusion $\subset$ as an order relation.

A thin set is a subset of $\mathbb{R^{\textnormal{$n$}}}$. It is qualified as thin because its boundary is thin.

Denote by $(\mathcal{P}(\mathbb{R}^{n}),\subset)$, the powerset of $\mathbb{R}^{n}$ equipped with the inclusion $\subset$ as an order relation. A thick set $[\negthinspace[\mathbb{X}]\negthinspace]$ of $\mathbb{R}^{n}$ is an interval of $(\mathcal{P}(\mathbb{R}^{n}),\subset)$. If $[\negthinspace[\mathbb{X}]\negthinspace]$ is a thick set of $\mathbb{R}^{n}$ , there exist two subsets of $\mathbb{R}^{n}$ , called the subset bound and the superset bound such that

$$[\negthinspace[\mathbb{X}]\negthinspace]=[\negthinspace[\mathbb{\mathbb{X}^{\subset}},\mathbb{X}^{\supset}]\negthinspace]=\{\mathbb{X}\in\mathcal{P}(\mathbb{R}^{n})|\mathbb{\mathbb{X}^{\subset}\subset\mathbb{X}\subset\mathbb{X}^{\supset}}\}.$$

(7)

The subset $\mathbb{X}^{\supset}\backslash\mathbb{X}^{\subset}$ is called the penumbra and plays an important role in the characterization of thick sets [10]. Thick sets can be used to represent uncertain sets (such as an uncertain map [12]) or soft constraints [7]. It can also can also be extended to fuzzy sets as shown in [14][6].

We now show how we can define operations for thick sets (as a union, intersection, difference, etc.). The main motivation is to be able to compute with thick sets.

Consider two thick sets $[\negthinspace[\mathbb{X}]\negthinspace]=[\negthinspace[\mathbb{\mathbb{X}^{\subset}},\mathbb{X}^{\supset}]\negthinspace]$ and $[\negthinspace[\mathbb{Y}]\negthinspace]=[\negthinspace[\mathbb{\mathbb{Y}^{\subset}},\mathbb{Y}^{\supset}]\negthinspace]$. We define the following operations [10] which corresponds to an interval extension in the sense of Moore [23] applied to thick sets:

$$\begin{array}[]{ccc}[\negthinspace[\mathbb{X}]\negthinspace]\cap[\negthinspace[\mathbb{Y}]\negthinspace]&=&[\negthinspace[\mathbb{\mathbb{X}^{\subset}}\cap\mathbb{Y}^{\subset},\mathbb{X}^{\supset}\cap\mathbb{Y}^{\supset}]\negthinspace]\\ {}[\negthinspace[\mathbb{X}]\negthinspace]\cup[\negthinspace[\mathbb{Y}]\negthinspace]&=&[\negthinspace[\mathbb{\mathbb{X}^{\subset}}\cup\mathbb{Y}^{\subset},\mathbb{X}^{\supset}\cup\mathbb{Y}^{\supset}]\negthinspace]\\ \overline{[\negthinspace[\mathbb{X}]\negthinspace]}&=&[\negthinspace[\overline{\mathbb{X}^{\supset}},\overline{\mathbb{X}^{\subset}}]\negthinspace]\\ \partial[\negthinspace[\mathbb{X}]\negthinspace]&=&[\negthinspace[\mathbb{X}]\negthinspace]\cap\overline{[\negthinspace[\mathbb{X}]\negthinspace]}\\ \mathbf{f}([\negthinspace[\mathbb{X}]\negthinspace])&=&[\negthinspace[\mathbf{f}(\mathbb{\mathbb{X}^{\subset}}),\mathbf{f}(\mathbb{X}^{\supset})]\negthinspace]\\ \mathbf{f}^{-1}([\negthinspace[\mathbb{X}]\negthinspace])&=&[\negthinspace[\mathbf{f}^{-1}(\mathbb{\mathbb{X}^{\subset}}),\mathbf{f}^{-1}(\mathbb{X}^{\supset})]\negthinspace]\end{array}$$

(8)

We could also have defined these operations as follows:

$$\,\begin{array}[]{ccc}[\negthinspace[\mathbb{X}]\negthinspace]\diamond[\negthinspace[\mathbb{Y}]\negthinspace]&=&[\negthinspace[\bigcap\left\{\mathbb{X}\diamond\mathbb{Y},\mathbb{X}\in[\negthinspace[\mathbb{X}]\negthinspace],\mathbb{Y}\in[\negthinspace[\mathbb{Y}]\negthinspace]\right\},\bigcup\left\{\mathbb{X}\diamond\mathbb{Y},\mathbb{X}\in[\negthinspace[\mathbb{X}]\negthinspace],\mathbb{Y}\in[\negthinspace[\mathbb{Y}]\negthinspace]\right\}]\negthinspace]\end{array}$$

where $\diamond$ is a binary operator on sets such as $\cup,\cap,\backslash,\dots$, and

$$\phi[\negthinspace[\mathbb{X}]\negthinspace]=[\negthinspace[\bigcap\left\{\phi(\mathbb{X}),\mathbb{X}\in[\negthinspace[\mathbb{X}]\negthinspace]\right\},\bigcup\left\{\phi(\mathbb{X}),\mathbb{X}\in[\negthinspace[\mathbb{X}]\negthinspace]\right\}]\negthinspace]$$

where $\phi$ is a unary operator on sets such as $\partial\mathbb{X},\overline{\mathbb{X}},\dots$

Some of these operations are illustrated by Figure 6. The first subfigure represents the thick set $[\negthinspace[\mathbb{X}]\negthinspace]=[\negthinspace[\mathbb{\mathbb{X}^{\subset}},\mathbb{X}^{\supset}]\negthinspace]$. The lower bound $\mathbb{\mathbb{X}^{\subset}}$ is painted red; the penumbra $\mathbb{X}^{\supset}\backslash\mathbb{X}^{\subset}$ is painted orange; and all $\mathbf{x}$ outside the upper bound $\mathbb{X}^{\supset}$ of $[\negthinspace[\mathbb{X}]\negthinspace]$ is in blue.

To compute with thick sets, we first need to generate elementary thick sets, called the atoms. They can be given by geometrical sets such as boxes, polygons or disks. They can also constructed from an expression of the form $f(\mathbf{x},\mathbf{p})\leq 0$, where $f:\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$ and $\mathbf{p}\in[\mathbf{p}]$ is the parameter vector. This can be done by the following operator

$$[\negthinspace[\sigma]\negthinspace](f,[\mathbf{p}])=[\negthinspace[\mathbb{\mathbb{X}^{\subset}},\mathbb{X}^{\supset}]\negthinspace]$$

with

$$\begin{array}[]{ccc}\mathbb{\mathbb{X}^{\subset}}&=&\{\mathbf{x}|\forall\mathbf{p}\in[\mathbf{p}]f(\mathbf{x},\mathbf{p})\leq 0\}\\ \mathbb{X}^{\supset}&=&\{\mathbf{x}|\exists\mathbf{p}\in[\mathbf{p}]f(\mathbf{x},\mathbf{p})\leq 0\}.\end{array}$$

Consider a pendulum depicted in Figure 7, left. Its state representation is

$$\begin{array}[]{lll}\left(\begin{array}[]{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array}\right)&=&\left(\begin{array}[]{c}x_{2}\\ -\sin x_{1}+p_{1}+u\end{array}\right)\end{array}$$

The pendulum represents a child swing, $u$ is a security break and $p_{1}$ is a perturbation which can be associated to the force generated by the child. In a nominal behavior, we have, $u=0$ but when the energy of the swing becomes too large, the controller generates a friction $u=-x_{2}$ to slow down the swing. More precisely, the following controller is:

$$\text{ if }c(\mathbf{x})<0\text{ then }u=0\text{ else $u=-x_{2}$,}$$

where $c(\mathbf{x})=x_{1}^{2}+x_{2}^{2}-1$ plays the role of the energy. The corresponding vector field is represented in Figure 7, right with two different trajectories (red) for $p_{1}=0$.

Assume that, when the controller alternates indefinitely between $u=0$ (break off) and $u=-x_{2}$ (break on), the controller may be damaged. We want to compute the sliding states which can be difficult to detect with simulations.

We have the two fields

$$\mathbf{f}_{a}=\left(\begin{array}[]{c}x_{2}\\ p_{1}-\sin x_{1}\end{array}\right),\mathbf{f}_{b}=\left(\begin{array}[]{c}x_{2}\\ p_{1}-\sin x_{1}-x_{2}\end{array}\right)$$

where $p_{1}\in[p_{1}]$ is an uncertain variable. We have

$$\begin{array}[]{ccl}\mathcal{L}_{a}^{c}\left(\mathbf{x}\right)&=&\frac{dc}{d\mathbf{x}}\left(\mathbf{x}\right)\cdot\mathbf{f}_{a}\left(\mathbf{x}\right)\\ &=&\left(\begin{array}[]{cc}2x_{1}&2x_{2}\end{array}\right)\cdot\left(\begin{array}[]{c}x_{2}\\ p_{1}-\sin x_{1}\end{array}\right)\\ &=&2x_{1}x_{2}+2x_{2}(p_{1}-\sin x_{1})\end{array}$$

(9)

and

$$\begin{array}[]{cll}\mathcal{L}_{b}^{c}\left(\mathbf{x}\right)&=&\frac{dc}{d\mathbf{x}}\left(\mathbf{x}\right)\cdot\mathbf{f}_{b}\left(\mathbf{x}\right)\\ &=&\left(\begin{array}[]{cc}2x_{1}&2x_{2}\end{array}\right)\cdot\left(\begin{array}[]{c}x_{2}\\ p_{1}-\sin x_{1}-x_{2}\end{array}\right)\\ &=&2x_{1}x_{2}+2x_{2}(p_{1}-\sin x_{1}-x_{2})\end{array}$$

To take into account the fact that $x_{1}$ and $x_{2}$ are given to the controller with a given bounded error $p_{2}\in[p_{2}],p_{3}\in[p_{3}]$, we take

$$\mathbb{A}=\left\{\mathbf{x}|\left(x_{1}+p_{2}\right)^{2}+\left(x_{2}+p_{3}\right)^{2}-1\leq 0\right\}.$$

To compute the paving associated to the expression (5), we need to build the atoms $[\negthinspace[\mathbb{A}]\negthinspace],[\negthinspace[\mathbb{L}_{a}^{c}]\negthinspace],[\negthinspace[\mathbb{L}_{b}^{c}]\negthinspace]$ as explained in Subsection 3.3.

If we choose $[p_{i}]=[-0.1,0.1],\forall i$, we get the approximation represented by Figure 8, Left. It is obtained by the following statements

$$\begin{array}[]{ccl}1&&[p_{1}]:=[p_{2}]:=[p_{3}]:=[-0.1,0.1]\\ 2&&[\negthinspace[\mathbb{L}_{a}^{c}]\negthinspace]:=[\negthinspace[\sigma]\negthinspace](2x_{1}x_{2}+2x_{2}(p_{1}-\sin x_{1}),[p_{1}])\\ 3&&[\negthinspace[\mathbb{L}_{b}^{c}]\negthinspace]:=[\negthinspace[\sigma]\negthinspace](2x_{1}x_{2}+2x_{2}(p_{1}-\sin x_{1}-x_{2}),[p_{1}])\\ 4&&[\negthinspace[\mathbb{A}]\negthinspace]:=[\negthinspace[\sigma]\negthinspace](\left(x_{1}+p_{2}\right)^{2}+\left(x_{2}+p_{3}\right)^{2}-1,[p_{2}]\times[p_{3}])\\ 5&&[\negthinspace[\mathbb{S}]\negthinspace]:=\partial[\negthinspace[\mathbb{A}]\negthinspace]\cap\overline{[\negthinspace[\mathbb{L}_{a}^{c}]\negthinspace]}\,\cap[\negthinspace[\mathbb{L}_{b}^{c}]\negthinspace]\end{array}$$

We note that $\mathbb{S}^{\subset}=\emptyset$, which is consistent with the fact that the surface to be approximated has no volume. This is the reason why the outer approximation $\mathbb{S}^{\supset}$ of the surface $\mathbb{S}$ corresponds to the penumbra of the thick set $[\negthinspace[\mathbb{S}]\negthinspace]$.