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 considering 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 $\textbf{P}$.

Positive real transfer functions

A system is only passive if, and only if the transfer function from input to output is positive real.

The function $H(s)$ is positive real if

1. $H(s)$ is analytic for all $Re[s] > 0$

2. $H(s)$ is real for all positive and real $s$

3. $Re[H(s)]\geq 0, \quad \forall Re[s] > 0$

If the function $H(s)$ is also rational, we can in addition affirm

The rational function $H(s)$ is positive real if and only if

1. All the poles of $H(s)$ have real parts less than or equal to zero

2. $Re[H(j\omega)] \geq 0, \quad \forall j\omega \neq \text{pole of } H(s)$

3. If $j\omega_0 $ is a pole of $H(s)$, then it is a simple pole and $$Res_{s = j\omega_0}[H(s)] = \lim\limits_{s\to j\omega_0}(s-j\omega_0)H(s) $$ is real and positive. If $H(s)$ is a pole at infitity, then it is a simple pole, and $$R_\infty = \lim\limits_{\omega\to \infty}\frac{H(j\omega)}{j\omega}$$ exsists and is real and positive

Bounded real transfer functions

The function B(s) is bounded real if

1. $B(s)$ is analytic for all $Re[s] > 0$

2. $|B(s)| \leq 1$ for all positive and real s

These properties can be usefull by defining

$$B(s) = \frac{H(s)-1}{H(s)+1}.$$

Then we have the property that $B(s)$ is bounded real if and only if H(s) is positive real

Storage function formulation

The scalar product

$$ \vec{u}\cdot \vec{v} = \sum_{i=3}^3 u_i v_i = \textbf{u}^\top \textbf{v} $$

The vector cross product

$$\textbf{u}^\times = \begin{pmatrix} 0 & -u_3 & u_2 \\ u_3 & 0 & -u_1 \\ -u_2 & u_1 & 0 \end{pmatrix} $$

$$\textbf{w} = \textbf{u}^\times\textbf{v} = \begin{pmatrix} u_2v_3 - u_3v_2 \\ u_3v_1 - u_1v_3 \\ u_1v_2 - u_2v_1 \end{pmatrix} $$

$$\vec{w} = \vec{u}\times \vec{v} \Leftrightarrow w_i = \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \varepsilon_{ijk}u_j v_k \Leftrightarrow \textbf{w} = \textbf{u}^\times\textbf{v} $$ $$ \varepsilon_{ijk} = \begin{cases} 1 \qquad \text{ when $i,j,k$ is a cyclic permutation} \\ -1 \quad ~ \text{ when $i,j,k$ is not a cyclic permutation} \\ 0 \qquad \text{ when $i=j$, $i=k$ or $k=j$} \end{cases}$$