Finite controlled invariants for sampled switched systems

Abstract

We consider in this paper switched systems, a class of hybrid systems recently used with success in various domains such as automotive industry and power electronics. We propose a state-dependent control strategy which makes the trajectories of the analyzed system converge to finite cyclic sequences of points. Our method relies on a technique of decomposition of the state space into local regions where the control is uniform. We have implemented the procedure using zonotopes, and applied it successfully to several examples of the literature and industrial case studies in power electronics.

Appendices

Appendix 1: Sufficient condition of decomposition

Given a zone $R$, an invariant decomposition does not always exist. We give hereafter some geometrical conditions on the position of $R$ that guarantee the decomposability of $R$ when all the modes of the system are contractive, in the sense of Definition 1: For each $u\in U$, there exists $0<\beta _u<1$ such that, for all $x,y\in \mathbb {R}^n$:

$$\begin{aligned} \Vert \mathbf{x}(\tau ,x,u) - \mathbf{x}(\tau ,y,u) \Vert \le \beta _u \Vert x - y \Vert . \end{aligned}$$

For the sake of simplicity, we suppose that the switched system has only $|U|=2$ modes, and the state space dimension is $n=2$, but the reasoning extends to larger values of $|U|$ and $n$. We assume that matrix $A_u$ associated with mode $u$ ($u=1,2$) is invertible. Let $e_u = -A_u^{-1}B_u$ be the unique attractive equilibrium point associated with mode $u$ ($u=1,2$). Let us define the “pure” switching rule $\mathcal{S}_u$ ($u=1,2$) which applies repeatedly mode $u$ to any point $x \in \mathbb {R}^2$. Let $\mathcal{C}_1$ (resp. $\mathcal{C}_2$) be the $\tau$-sampled trajectory issued from $e_1$ (resp. $e_2$) under $\mathcal{S}_2$ (resp. $\mathcal{S}_1$) (i.e., $\mathcal{C}_1 = Post^{*}_{2}(e_1)$ and $\mathcal{C}_2 = Post^{*}_{1}(e_2)$). Since each mode is contractive, and $e_u$ is the unique equilibrium point associated with mode $u$, any trajectory under control $\mathcal{S}_u$ converges to the equilibrium point $e_u$ ($u=1,2$), whatever the starting point of $\mathbb {R}^2$. In particular, $\mathcal{C}_1$ ends to $e_2$ and $\mathcal{C}_2$ to $e_1$, as depicted in Fig. 11 for the boost converter (see Examples 1 and 2).

Fig. 11

Trajectories $\mathcal{C}_1$, and $\mathcal{C}_2$ and zone $R=[1.7,2]\times [1.1,1.2]$ for the DC–DC converter example

Theorem 4

Let $\varSigma$ be a sampled switched affine system as defined above. Suppose that the reference point $O$ is in $\mathcal{C}_1\cup \mathcal{C}_2$. If $R\subseteq \mathbb {R}^2$ is a box whose interior contains $O$, then there exists a positive integer $k$ such that $R$ is $k$-invariant.

Proof

Suppose $O\in \mathcal{C}_2$. (The case $O\in \mathcal{C}_1$ is symmetrical.) Consider a box $R$ of interior $\mathring{R}$ with $O\in \mathring{R}$. There exists $\Delta _O > 0$ such that $\mathcal{B}(O,\Delta _O) \subseteq R$. Since $O \in \mathcal{C}_2$, we have:

• (a) $e_2 \rightarrow ^{{\uppi }_1} O$, for some pattern ${\uppi }_1\in (1)^*$. Furthermore, for all $x\in R$, we have:

• (b) $x\rightarrow ^{{\uppi }_2} x_1$ for some $x_1\in \mathcal{B}(e_2,\Delta _O)$ and some pattern ${\uppi }_2\in (2)^*$, because $e_2$ is an attractive equilibrium point;

• (c) $x_1\rightarrow ^{{\uppi }_1} x_2$ for some $x_2\in \mathcal{B}(O,\Delta _O)$, because of (a) and because mode $1$ is contractive. This is depicted in Fig. 12. It follows from (b)-(c) that, for all $x \in R$: $x\rightarrow ^{{\uppi }_x} x_2$ for some $x_2\in \mathcal{B}(O,\Delta _O)$ and some pattern ${\uppi }_x\in (2^*1^*)$. Hence, we have: $\mathcal{B}(x_2,\Delta _x) \subseteq R$ for some $\Delta _x > 0$. Since, for any ${\uppi }_x$, $Post_{{\uppi }_x}$ is continuous, $Pre_{{\uppi }_x}(\mathcal{B}(x_2,\Delta _{x}))$ ⁵ is an open subset of $\mathbb {R}^2$ containing $x$. Since $R$ is a compact of $\mathbb {R}^2$, from the set $C=\{Pre_{{\uppi }_x}(\mathcal{B}(Post_{{\uppi }_x}(x),\Delta _{x}))\}_{x\in R}$, one can extract by Heine-Borel’s theorem, a subset $C'=\{Pre_{{\uppi }_{x_i}}(\mathcal{B}(Post_{{\uppi }_{x_i}}(x_i),\Delta _{x_i}))\}_{i\in I}$, for some finite set of indices $I$, such that $C'$ covers $R$ and $\mathcal{B}(x_i,\Delta _{x_i})$ is $R$-invariant via ${\uppi }_i$. This means that $C'\cap R$ is a $k$-invariant decomposition of $R$ of the form $\{(V_i,{\uppi }_i)\}_{i=1,\ldots ,m}$, where $m$ is the cardinal of $I$, $k$ the maximum length of ${\uppi }_1,\ldots ,{\uppi }_m$, and $V_i=\mathcal{B}(x_i,\Delta _{x_i})\cap R$ is such that $\bigcup _{i=1}^m V_i = R$ and $V_i$ is $R$-invariant via ${\uppi }_i$ ($1\le i\le m$). $\square$

The theorem gives an interesting locality condition on $O$ (location on one of the “pure” trajectories linking the equilibrium points), for ensuring the existence of $k$-invariant boxes $R$. This justifies a posteriori the existence of a decomposition for the zone $R$ of the DC–DC converter example 2 since it overlaps trajectory $\mathcal{C}_1$. Note also that $R$ can be arbitrarily small as far as it intersects $\mathcal{C}_1$ (or $\mathcal{C}_2$).

Fig. 12

Illustration of the proof

Appendix 2: Case studies

We now apply the decomposition procedure (enhanced for safety properties) for several examples. For each example, we will generate a $k$-invariant decomposition $\Delta$ of $R$ satisfying $Unf _{\Delta }(R)\subseteq S$, thus proving that $Unf _{\Delta }(R)$ is a controlled invariant, and that the system is safe.

Example 4

Let us come back to the boost DC–DC converter of example 1. For $R$, we now take the box $[1.75,1.95]\times [1.14,1.26]$,⁶ which corresponds to a medium value $1.85$ for $i_l$ with $\pm 0.1$ for variability, and medium value $1.20$ for $v_c$ with $\pm 0.06$ for variability. For the safety region, we take $S = [1.7,2.0]\times [1.10,1.30]$, which corresponds to an additional variability of $\pm 0.05$ for $i_l$ and $\pm 0.04$ for $v_c$.

The application of the algorithm Decomposition to $R$ and $S$, with $k=10$ and $d=4$ succeeds, yielding a $k$-invariant decomposition $\Delta$ of the form $\{(V_j,{\uppi }_j)\}_{j=1,\ldots ,16}$ of $R$ ⁷ satisfying $Unf _{\Delta }(R)\subseteq S$. The $k$-invariant decomposition $\Delta$ of $R$ is depicted in Fig. 13. The $\Delta$-unfolding of $R$ is depicted in Fig. 14. It is divided into regions of two different colors corresponding to the different control modes: blue (resp. red) indicates that mode 1 (resp. 2) should be applied. The safety zone $S = [1.7,2.0]\times [1.10,1.30]$ is delimited in green.

The controlled system has been implemented using Octave [17]. The unfolded $\Delta$-trajectory of the system starting at point $(1.75,1.26)$, is depicted in Fig. 15.

Fig. 13

Decomposition for boost converter for $R=[1.75,1.95]\times [1.14,1.26]$

Fig. 14

$\Delta$-unfolding for $R=[1.75,1.95[\times [1.14,1.26]$ where light red (resp. dark blue) indicates mode 1 (resp. 2), $S$ corresponds to plot area. (Color figure online)

Fig. 15

Unfolded $\Delta$-trajectory of the boost converter in plane $(i_l,v_c)$, starting at $(1.75,1.26)$ ($R$ in blue, $S$ corresponds to plot area). (Color figure online)

Other applications of the decomposition procedure are described in Sect. 5.

Example 5

(Helicopter motion [6]) The problem is to control a quadrotor helicopter to some position on top of a stationary ground vehicle, while satisfying constraint on the relative velocity. By controlling the pitch and roll angles, one can modify the speed and the position of the helicopter. Let $g$ be the gravitational constant, $x$ (resp. $y$) the position according to $x$-axis (resp. $y$-axis), ${\dot{x}}$ (resp. ${\dot{y}}$) the velocity according to $x$-axis (resp. $y$-axis), $\phi$ the pitch command, $\psi$ the roll command. The possible commands for the pitch and the roll are: $\phi ,\psi \in \{-10,0,10\}$. Since each mode corresponds to a pair $(\phi ,\psi )$, there are 9 modes. The dynamics are given by the equation:

$$\begin{aligned} {\dot{X}} = \begin{pmatrix} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{pmatrix} \cdot X + \begin{pmatrix} 0 \\ g.sin(-\phi ) \\ 0 \\ g.sin(\psi ) \end{pmatrix} \end{aligned}$$

where $X$ is $(x,{\dot{x}},y,\dot{y})^{T}$. The sampling period is $\tau = 0.1$. The variables $x$ and $y$ are decoupled in the equations and follow the same equations, so we focus on the control for $x$. We take $R=[-0.3,0.3]\times [-0.5,0.5]$ (i.e., $R = \{(x_1,x_2) \, |\, x_1 \in [-0.3,0.3], x_2 \in [-0.5,0.5]\}$). This corresponds to an equilibrium zone centered at the state $(0,0)$ of the ground vehicle, and a variability of $\pm 0.3$ for position and $\pm 0.5$ for velocity. We take $S = [-0.4,0.4]\times [-0.7,0.7]$ for the safety region, which corresponds to an additional variability of $\pm 0.1$ for position and velocity. As for $R$, we take a box guessed manually from the equilibrium zone appearing in the numerical experiments and simulations of [6]. We take the same safety zone $S$ as the one specified in [6].

The application of algorithm Decomposition to $R$ and $S$ with $k=6$ and $d=4$ succeeds, yielding a $k$-invariant decomposition $\Delta$ of $R$ of the form $\{(V_i,{\uppi }_i)\}_{i=1,\ldots ,10}$ ⁸ satisfying $Unf _{\Delta }(R)\subseteq S$. The decomposition $\Delta$ is represented in Fig. 16. The unfolding of $R$ is depicted in Fig. 17. The unfolding is divided into regions of three different colors corresponding to the different control modes: color red (resp. blue, green) indicates that mode $10$ (resp. $-10$, $0$) should be applied. The safety zone $S= [-0.4,0.4]\times [-0.6,0.6]$ in delimited in green.

Fig. 16

$k$-Invariant decomposition for helicopter motion

Fig. 17

$\Delta$-unfolding for helicopter motion where blue (resp. red, green) indicates mode 10 (resp. $-10$, $0$). (box $S$ delimited in green). (Color figure online)

The controlled system has been simulated using Octave. Figure 18 gives a simulation of the system represented in plane $(x, {\dot{x}})$, starting at point $(-0.3, 0.5)$.

Fig. 18

Unfolded $\Delta$-trajectory of helicopter motion in plane $(x,{\dot{x}})$ starting at $(-0.3,0.5)$ ($R$ in red, $S$ in green). (Color figure online)

Example 6

(Two-room building heating [8]) This is a simple heating model of a two-room building. Let $T=(T_1,T_2)^{T}$ be the state variable, where $T_i$ is the temperature of room $i$ ($i=1,2$). The dynamics are given by the equation: $\dot{T} = \begin{pmatrix} -\alpha _{21}-\alpha _{e1}-\alpha _{f}p &{} \alpha _{21}\\ \alpha _{12} &{} -\alpha _{12} - \alpha _{e2} \end{pmatrix} \times T + \begin{pmatrix} \alpha _{e1}T_e + \alpha _{f}T_fp\\ \alpha _{e2}T_e \end{pmatrix}$, where $p$ is a mode of value $0$ or $1$, and the heat transfer coefficients and external temperatures are given by the values: $\alpha _{12} = 5.10^{-2}$, $\alpha _{21} = 5.10^{-2}$, $\alpha _{e1} = 5.10^{-3}$, $\alpha _{e2} = 3.3.10^{-3}$, $\alpha _f = 8.3.10^{-3}$, $T_e = 10$, $T_f = 50$. The sampling period is $\tau = 5$. We take $R=[20.25,21.75]\times [20.25,21.75]$. This corresponds to an equilibrium zone centered at state $(21,21)$ with a variability of $\pm 0.75$. For the safety zone, we take $S = [20,22]\times [20,22]$, as in [8].

The application of the algorithm Decomposition to $R$ and $S$ with $k=4$ and $d=2$ succeeds, yielding a $k$-invariant decomposition⁹ $\Delta$ of the form $\{(V_j,{\uppi }_j)\}_{j=1,\ldots ,10}$ of $R$ satisfying $Unf _{\Delta }(R)\subseteq S$. The decomposition $\Delta$ is depicted in Fig. 19. The unfolding of $R$ is represented on Fig. 20. The unfolding is divided into regions of two colors corresponding to the different control modes: the red (resp. blue) color indicates that control 0 (resp. 1) should be applied. The safety zone $S$ in delimited green.

Fig. 19

$k$-Invariant decomposition for heating system

Fig. 20

$\Delta$-Unfolding for heating system where light red (resp. dark blue) indicates mode 0 (resp. 1). ($S$ corresponds to plot area). (Color figure online)

The controlled system has been simulated using Octave [17]. A simulation is depicted in Fig. 21 for starting temperature point $(20.25,21.75)$.

Example 7

(Multilevel converter) Using the decomposition procedure, we have synthesized a control for a case study originating from power electronics industry of high dimension ($n=7$), satisfying a safety constraint. The control has been implemented and successfully physically experimented with a prototype. See [7].

Fig. 21

Unfolded $\Delta$-trajectory of heating system in plane $(T_1,T_2)$, starting at $(20.25,21.75)$ ($R$ in blue, $S$ corresponds to plot area). (Color figure online)

Appendix 3: Control with disturbances

Zonotopes are useful for representing and manipulating efficiently convex polytopes. They allow us to extend easily the decomposition procedure in order to allow for small perturbations of the system dynamics (see, e.g., [12]). Using zonotopes, it is possible to introduce uncertainty or disturbance in our model. All the dynamics of the system are now of the form ${\dot{x}} = A_{u}x+B_u + \varepsilon$ where $\varepsilon$ represents disturbance under the form of a $n$-dimensional vector belonging to a given box $\varOmega$ of fixed size. We will use $\mathbf{x}(t,x,u,\varepsilon )$ to denote the point reached by the system at time $t$ under mode $u$ and disturbance $\varepsilon$ from the initial condition $x$. This defines a transition relation $\rightarrow ^{u,\varepsilon }_{\tau }$ given by:

$$\begin{aligned} x\rightarrow ^{u,\varepsilon }_{\tau } x'\quad \text{ iff } \mathbf {x}(\tau ,x,u,\varepsilon )=x' \quad \text{ for } x, x'\hbox { in }\mathbb {R}^n \ \hbox { and } \ \varepsilon \hbox { in }\varOmega \end{aligned}$$

We define $Post_u(X,\varOmega ) = \{x' \, | \, \exists \varepsilon \in \varOmega , \, x \rightarrow _{\tau }^{u,\varepsilon } x'\}$. This definition naturally extends to $Post_{{\uppi }}$ where ${\uppi }$ is a pattern. In order to handle disturbance, one replaces test $Post_{{\uppi }}(W) \subseteq R$ of algorithm $\hbox {Find}\_\hbox {Pattern}$ by $\square (Post_{{\uppi }}(W,\varOmega )) \subseteq R$, where ‘$\square$’ is the operator mapping a zonotope into the smallest enclosing box.

Example 8

(Boost converter) We consider the dynamics of the boost DC–DC converter (see Examples 1 and 2) in presence of disturbances belonging to $\varOmega = \{0\} \times \left[ \frac{-0.064}{x_l},\frac{0.064}{x_l}\right]$. These disturbances correspond to noise on the input voltage. They represent represent up to $8 \%$ of the value of the input voltage. With such disturbances, we have not found a control preserving the safety zone $S$ defined in example 2, and we have taken a larger (i.e., more tolerant) safety zone defined by $S' = [1.65,2.05] \times [1.10,1.30]$. The decomposition procedure then succeeds for $k=13$ and a $d=5$: it generates a $k$-invariant decomposition $\Delta '$ of $R$ satisfying $Unf _{\Delta '}(R)\subseteq S'$. The decomposition¹⁰ $\Delta '$ is depicted in Fig. 22. A simulation of the system starting at $(1.75,1.26)$ is depicted in Fig. 23.

Fig. 22

$k$-Invariant decomposition for boost converter with disturbances

Fig. 23

Unfolded $\Delta$-trajectory of the boost converter with disturbances in plane $(i_l,v_c)$, starting at $(1.75,1.26)$ ($R=$ in blue, $S$ corresponds to plot area). (Color figure online)

Example 9

(Helicopter motion) As done in [6], we will now solve the control problem with bounded disturbances to take into account a potential real-life environment. We use the same safe zone $S$ as in example 5 and add the following possible disturbances $\varepsilon \in [-0.02,0.02]\times [-0.1,0.1]$. The decomposition procedure succeeds, and generates a new $k$-invariant decomposition $\Delta '$ with $Unf _{\Delta '}(R)\subseteq S$. The decomposition $\Delta '$ is depicted in Fig. 24. A simulation of a run starting at point $(-0.3,0.5)$ is presented in Fig. 25.

Fig. 24

$k$-Invariant decomposition for helicopter motion with disturbances

Fig. 25

Unfolded $\Delta$-trajectory of helicopter motion with disturbances in plane $(x,\dot{x})$, starting at $(-0.3,0.5)$ ($R$ in blue, $S$ corresponds to plot area). (Color figure online)

Appendix 4: Example with $|R_{\Delta }^{\infty }| = \infty$

For this example, we use modes associated to repulsive homothetic transformation. There are $4$ modes such that, close to each corner of the global box $R = [-1,1] \times [-1,1]$, there exists a fixed point for one of the mode. We take for the dynamics of the modes: $A_1 = A_2 = A_3 = A_4 = \begin{pmatrix} 1.5 &{} 0 \\ 0 &{} 1.5 \end{pmatrix}$, $B_1 = \begin{pmatrix} 0.6 \\ 0.6 \end{pmatrix}$, $B_2 = \begin{pmatrix} -0.6 \\ 0.6 \end{pmatrix}$, $B_3 = \begin{pmatrix} -0.6 \\ -0.6 \end{pmatrix}$, and $B_4 = \begin{pmatrix} 0.6 \\ -0.6 \end{pmatrix}$. The $\Delta$-decomposition is presented in Fig. 26. We have $V_1 = [-1,0] \times [-1,0]$ associated to pattern ${\uppi }_1 = (1)$, $V_2 = [0,1] \times [-1,0]$ associated to pattern ${\uppi }_2 = (2)$, $V_3 = [0,1] \times [0,1]$ associated to pattern ${\uppi }_3 = (3)$, and $V_4 = [-1,0] \times [0,1]$ associated to pattern ${\uppi }_4 = (4)$. A $\Delta$-trajectory is depicted in Fig. 27. One can see that the system appears to be chaotic and will not converge to a limit cycle.

Fig. 26

Decomposition $\Delta$ of $R$ for the non contractive example

Fig. 27

$\Delta$-Trajectory for the non contractive example

Appendix 5: Visualizations of the iteration of $Post_{\Delta }$

In Fig. 28, we give the iterated images $Post_{\Delta }^k(R)$ of the boost converter benchmark for $k=0,10,20,40,80,100$. We see that for $k=100$, $Post_{\Delta }^k(R)$ is completely located inside $V_1$. The controlled system converges to a single limit point.

Fig. 28

Visualization of $Post_{\Delta }^k(R)$ of the boost DC/DC converter benchmark for $k=0,10,20,40,80,100$

Figure 29 shows the iterated images $Post_{\Delta }^k(R)$ of the two tanks benchmark for $k=0,5,10,15,20,25$. In the figures, it can be seens that starting with $k=10$, the set $Post_{\Delta }^k(R)$ does not intersect the borders of the decomposition $\Delta$. The system converges to a limit cycle (see also Fig. 10 on page 16).

Fig. 29

Visualization of $Post_{\Delta }^k$ of the two tanks benchmark for $k=0,5,10,15,20,25$