L9

22/08/26 Sat 10:26am

System Modeling and Simulation

A system is a collection of components that interact to produce outputs.
EX:
Mechanical:

• car suspension
• inverted pendulum
• robotic arm

Electrical System:

• RLC circuit
• power converters
• motor drives

Thermal system:

• HVAC
• heat exchanger

Fluid System:

• hydraulics pneumatics
• pipe networks

Mixed Systems:

• Like Electric Vehicles which requires Mechanical, Electrical, Thermal.

Static and Dynamic Systems

A system who’s output is based on the current input of the system is known as a Static system

$$y=f(u)$$

Dynamic systems - output is dependent on past and current states

$$y=f(x,u)$$

Classification of Dynamic Systems

Classification	Types
Linearity	linear system / non-linear system
Time Dependency	time variant / time invariant
Signal Nature
I/P & O/P	SISO / MIMO

State Space Analysis

State and State Variables

The state of a system is the minimal set of variables that together with the inputs completely determine the future behaviour of the system.
The number of state variables in the system will give you the order of the system.

State Space Representation

$$\dot{x}(t)=Ax(t)+Bu(t)-\text{state eqn}$$ $$y(t)=Cx(t)+Du(t)-\text{o/p eqn}$$

where;
$x$ → State vector - n x 1
$u$ → input vector - m x 1
$y$ → o/p vector - p x 1
A → State matrix - n x n
B → Input matrix - n x m
C → Output matrix - p x n
D → feed through matrix - p x m

Ex: 1 Mass Spring Damper System

L10-2026-08-22-10.55.25.excalidraw

$$F=M\frac{{\partial }^{2}x}{\partial t^2}+B\frac{\partial x}{\partial t}+Kx$$ $$F=m\ddot{x}+B\dot{x}+kx$$ $$x_1=x\to 1$$ $$x_2=\dot{x_1}=\dot{x}-2$$

substitute 1 and 2 in the first equation

$$F=m{\dot{x}}_2+B{x}_2+K{x}_1$$ $$\dot{x_2}=\frac{F}{m}-\frac{B{x}_2}{m}-\frac{K{x}_1}{m}$$ $$\dot{x_1}=x_2$$ $$\left[\begin{array}{c}\dot{x_1}\\ \dot{x_2}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -\frac{K}{m}& -\frac{B}{m}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \frac{1}{m}\end{array}\right]F$$

$Y=x$

$$Y=Cx+Du$$ $$Y=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\end{array}\right]F$$

Question 2

$\ddot{\theta }+\frac{g}{l}\sin \theta =0$
as $\theta \to 0$
$\sin \theta \approx \theta$

$$\ddot{\theta }+\frac{g}{L}\theta =0$$

now this is a linear system

$$\theta =x_1$$ $$\dot{\theta }=\dot{x_1}=x_2$$ $$\dot{x_1}=x_2$$ $$\dot{x_2}+\frac{g}{L}{x}_1=0$$ $$\dot{x_2}=-\frac{g}{L}{x}_1$$ $$\left[\begin{array}{c}\dot{x_1}\\ \dot{x_2}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -\frac{g}{L}& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$$

Dc Motor Drive

Electrical:

$$L\frac{di}{dt}+Ri=V-K_e\omega$$

Mechanical:

$$J\dot{\omega }+b\omega =K_{t}i$$

can be written as

$$L\dot{x_2}+R{x}_2=V-K_e{x}_1$$ $$⟹\dot{x_2}=\frac{V-K_e{x}_1-R{x}_2}{L}$$ $$J\dot{x_1}+b{x}_1=K_t{x}_2$$ $$⟹\dot{x_1}=\frac{K_t{x}_2-b{x}_1}{J}$$ $$\boxed{\left[\begin{array}{c}\dot{x_1}\\ \dot{x_2}\end{array}\right]=\left[\begin{array}{cc}-\frac{b}{J}& \frac{K_t}{J}\\ -\frac{K_e}{L}& -\frac{R}{L}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \frac{1}{L}\end{array}\right]\left[\begin{array}{c}V\end{array}\right]}$$