5-dimensional switching linear system | Theory of Hybrid Systems

Classification

# of variables	# of modes	# of jumps
5	5	5

Type	Continuous dynamics	Guards & Invariants	Resets
hybrid	linear polynomial	none	constant

Download

Flow*	switching_5.zip
SpaceEx	5_dim_linear_switch.zip

Model description

It is a piecewise linear system with different controlled continuous dynamics. The matrices $A, B, C, D$ describing the modes of the hybrid system are generated randomly by using the rss() Matlab function. The generated systems are then stabilized with an LQR controller to garantee a convergence to a stable attracting region. The transitions are determined heuristically by means of simulations.
The proposed benchmark consists of 5 modes und 5 transitions. The continuous dynamics in each mode $q_i, i=1,...,5$ is described by the following ODE:

$\dot{x} = A_i x+B_i u$

where $x \in \mathbb{R}^{5}$ is the state vector, $u$ is an input signal confined in compact bounded set $U$ and

$A_1=\scriptsize \begin{pmatrix} -0.8047 & 8.7420 & -2.4591 & -8.2714 & -1.8640\\ -8.6329 & -0.5860 & -2.1006 & 3.6035 & -1.8423\\ 2.4511 & 2.2394 & -0.7538 & -3.6934 & 2.4585\\ 8.3858 & -3.1739 & 3.7822 & -0.6249 & 1.8829\\ 1.8302 & 1.9869 & -2.4539 & -1.7726 & -0.7911\end{pmatrix}$ ,

$A_2 = \scriptsize \begin{pmatrix}-0.8316 & 8.7658 & -2.4744 & -8.2608 & -1.9033\\ -8.6329 & -0.5860 & -2.1006 & 3.6035 & -1.8423\\ 2.451 1 & 2.2394 & -0.7538 & -3.6934 & 2.4585\\ 8.3858 & -3.1739 & 3.7822 & -0.6249 & 1.8829\\ 1.5964 & 2.193 6 & -2.5872 & -1.6812 & -1.1324\end{pmatrix}$ ,

$A_3 =\scriptsize \begin{pmatrix}-0.9275 & 8.8628 & -2.5428 & -8.2329 & -2.0324\\ -8.6329 & -0.5860 & -2.1006 & 3.6035 & -1.8423\\ 2.4511 & 2.2394 & -0.7538 & -3.6934 & 2.4585\\ 8.3858 & -3.1739 & 3.7822 & -0.6249 & 1.8829\\ 0.7635 & 3.0357 & -3.1814 & -1.4388 & -2.2538\end{pmatrix}$ ,

$A_4=\scriptsize \begin{pmatrix}-1.0145& 8.9701& -2.6207& -8.2199& -2.1469\\ -8.6329& -0.5860& -2.1006& 3.6035& -1.8423\\ 2.4511& 2.2394& -0.7538& -3.6934& 2.4585\\ 8.3858& -3.1739& 3.7822& -0.6249& 1.8829\\ 0.0076& 3.9682& -3.8578& -1.3253& -3.2477\end{pmatrix}$ ,

$A_5=\scriptsize \begin{pmatrix}-1.4021& 10.1647& -3.3937& -8.5139& -2.9602\\ -8.6329& -0.5860& -2.1006& 3.6035& -1.8423\\ 2.4511& 2.2394& -0.7538& -3.6934& 2.4585\\ 8.3858& -3.1739& 3.7822& -0.6249& 1.8829\\ -3.3585& 14.3426& -10.5703& -3.8785& -10.3111\end{pmatrix}$ ,