# TTK4130: Modelling and Simulation

# Energy-based methods and passivity

When designing systems we often are dependent on dividing the system into smaller sub-sections and study them separately. 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

passivesystems is passive 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

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

If

For a second-order system

### Lyapunov methods

In this method, we are considering a plant

we then select a suitable energy function **Lyapunov function candidate**, which is ** positive definite **, i.e.

Given this function, we have that

Where

## Passivity

If a system can be described as a parallell or feedback interconnection of passive subsystems, then the total system will be passive, and it will not generate energy. This will under certain assumptions imply that the system is stable.

Concider a system with input

$u(t)$ and output$y(t)$ . Suppose that there is a constant$E_0 \geq 0$ so that for all control time histories of u and all$T\geq 0$ the integral of$u(t)y(t)$ satisfies$$ \int_0^T u(t)y(t) dt \geq -E_0 $$ then the system is said to be passive.

### 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

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

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

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

If the function **rational**, we can in addition affirm

The rational function

$H(s)$ is positive real if and only if

All the poles of

$H(s)$ have real parts less than or equal to zero

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

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

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

These properties can be usefull by defining

Then we have the property that

### Storage function formulation

# Rigid body dynamics

## Vectors

### The scalar product

### The vector cross product

## The rotation matrix

The coordinate transformation from frame b to frame a is given by

$$\textbf{v}^a = \textbf{R}_b^a\textbf{v}^b$$ where$$\textbf{R}_b^a = \{\vec{a}_i\cdot\vec{a}_j\}$$ is called therotation matrixfrom$a$ to$b$ . The elements$r_{ij} =\vec{a}_i\cdot\vec{a}_j\ $ of the rotation matrix$\textbf{R}_b^a$ are calleddirection cosinesThe rotation matrix is orthogonal and satisfies

$$\textbf{R}_a^b = (\textbf{R}_b^a)^\top = (\textbf{R}_b^a)^{-1}$$