Encoding sinusoidal functions in hybrid automata formalism

Herein, we illustrate how we can use linear ODEs to express sinusoids and how to describe them as hybrid automata. Note that our proposed scheme is suitable for sine waves (functions of time) and applies to formal verification tools, such as SpaceEx [4].

The differential equation

$\displaystyle\begin{aligned} \dot{x}&=y\\ \dot{y}&=-\omega^{2}(x-\mu t)\end{aligned}$

(1)

where $x$, $y$ are the state variables, $\omega$ and $\mu$ are parameters, and $t$ describes the time evolution, has the following analytical solution

	$\displaystyle y(t)$	$\displaystyle=A\sin(\omega t+\varphi)+\mu$		(2a)
	$\displaystyle x(t)$	$\displaystyle=-\frac{A}{\omega}\cos(\omega t+\varphi)+\mu t.$		(2b)

The initial conditions are $y(0)=A\sin(\varphi)+\mu$, and $x(0)=-\frac{A}{\omega}\cos(\varphi)$. If we ignore the initial phase, i.e. $\varphi=0$, the initial conditions become $y(0)=\mu$, $x(0)=-\frac{A}{\omega}$.

Hybrid automata are a common formalism for modeling hybrid systems and their semantics can be found at [3]. In this part, we show how to construct hybrid automata that respect the SX format [1]; a formalism that supports network of hybrid automata and is used by several reachability tools, such as SpaceEx.  
Base Component - Sine. For the sinusoid, we only need one base component that we call sin. We have one location, named loc1, with three variables $x$, $y$, and $t$. Using the equation 1, the flow is given by¹¹1Note that we opt for $omega*omega$ instead of $\omega^{2}$ to avoid any unexpected error caused by the SpacEx parser.

	$\displaystyle x^{\prime}$	$\displaystyle==y$
	$\displaystyle y^{\prime}$	$\displaystyle==-\omega*\omega*(x-\mu*t)$
	$\displaystyle t^{\prime}$	$\displaystyle==1$

The three variables $x$, $y$, $t$ are declared as controlled, while the parameters / constants $\omega$ and $\mu$ are uncontrolled. The variable $y$ is a global variable, i.e. it can be communicated/shared to other components, whereas $y$ and $t$ are local variables. The hybrid automaton requires no invariant, reset, guard or synchronization label. Detailed information about modeling hybrid automata in SpaceEx can be found at [2]. The complete base component is shown in Figure 1.

The initial conditions are selected according to Section 2 to match the desired behavior of the sinusoidal signal.

In this section, we show the applicability of this transformation via a numerical example as well as its usability within Simulink.