TTK4130: Modelling and Simulation
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% * Energy based systems and passivity
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# Energy-based methods and passivity> and passive systems are stable
## Energy-based methods
When studying stability properties of systems, we can use the fact that the energy in a system will decrease if the system is stable, e.g. if the kinetic energy of any mechanic system is
monotonically increasing, it is certainly unstable.
### The energy function
Let $V(\textbf{x},t) \geq 0$ be an energy function for the system
$$\dot{\textbf{x}} = \textbf{f}(\textbf{x},\textbf{u},t)$$
It may be the total energy of the system or some other property related to the energy. We then define the time derivative of the system to be
$$\dot{V} = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial \textbf{x}}\textbf{f}(\textbf{x},\textbf{u},t)$$
If $\dot{V}\leq 0$ the energy of the system is monotonically decreasing, which may be important concidering the stability properties of the system
For a second-order system $\ddot{x} = \textbf{f}(\textbf{x},\dot{\textbf{x}},t)$, the time derivative of the energy function $V(\textbf{x},\dot{\textbf{x}},t) \geq 0$ is
$$\dot{V} = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial \textbf{x}}\dot{\textbf{x}} +\frac{\partial V}{\partial \dot{\textbf{x}}}\textbf{f}(\textbf{x},\dot{\textbf{x}},t)$$
### Lyapunov methods
In this method, we are concidering a plant
$$\dot{\textbf{x}} = \textbf{f}(\textbf{x},\textbf{u})$$
we then select a suitable energy function $V(\textbf{x})$ called a **Lyapunov function candidate**, which is ** positive definite **, i.e. $V(\textbf{x}_0=0) = 0$, and $V(\textbf{x})>0,
\forall \textbf{x} \neq 0$. A typical Lyapunov function candidate is
$$V(\textbf{x}) = \textbf{x}^\top \textbf{P} \textbf{x}, \quad \textbf{P}=\textbf{P}^\top > 0$$
Given this function, we have that
$$\frac{\lambda_{min}(\textbf{P})}{2}\textbf{x}^\top\textbf{x}\leq V(\textbf{x}) \leq \frac{\lambda_{max}(\textbf{P})}{2}\textbf{x}^\top\textbf{x}$$
Where $ \lambda_{min}(\textbf{P}) > 0 $ is the smalles eigenvialue of $\textbf{P}$, and $ \lambda_{max}(\textbf{P}) > 0 $ is the largest eigenvialue of $\textbf{P}$.
## Passivity
When designing systems we often are dependent on dividing the system into smaller sub-sections and study them seperately. Energy-based methods is a practical way of examining the stability
properties of the whole system with respect to its sub-systems. Here is the general motivation:
> The interconnection of stable systems is not necessarily stable, however the interconnection of **passive** systems is passive