Observe the output of a system for a given input from State-Space representation using Simulink

Suppose, we have a state-space representation of a system -

$\dot{\overrightarrow{x}} (t) = \begin{bmatrix} 0 & 9 \\ 1 & 0 \end{bmatrix} \overrightarrow{x} (t) + \begin{bmatrix} 1 \\ 0 \end{bmatrix} u(t)$

$y = \begin{bmatrix} 1 & -3 \end{bmatrix} \overrightarrow{x}$

By using state_space block of the simulink, we can see the behavior of this system. Here, we will see the output of the given system for a unit-step input u (t).

From the state space representation, in this system we have -

$A = \begin{bmatrix} 0 & 9 \\ 1 & 0 \end{bmatrix}$

$B = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

$C = \begin{bmatrix} 1 & -3 \end{bmatrix}$

D = 0

Initial Condition:

$\dot {x} (t) = \begin{bmatrix} 0 & 0 \end{bmatrix}$

So, update the state-space block according to the above -

After running the simulation, from the scope, we can see the output response of the given system.

Instead of using Simulink state-space block, we can draw the system and can see the output response from the scope.

Steps:

1) From the Simulink Library Browser > Simulink > Continuous, have as many integrator (1/s) as the no. of state variables

2) Put x_dot and x before and after each integrator

3) Insert the input at the leftmost. Here we are using a unit step input. From Simulink > Sources, insert one step block

4) Insert the output at the rightmost. Here, we will use a scope to see the output. From Simulink > Sinks, insert one scope block

5) Write the differential equations separately, and draw them using sum and gain blocks

$\dot {x_1} = 9 x_2 + u$

$\dot {x_2} = x_1$

6) Write the output equations and draw them using sum and gain blocks

$y = x_1 - 3 x_2$

From the scope we can see that we have the same output as the previous.