How could we rescale the step response of MIMO state space model system?.

I have a MIMO system and below its parameters

A=[0,1,0,0;0,-42.44,212.24,0;0,0,0,1;0,0,0,-42.44];

B=[0,0;106.12,-5;0,0;78.6,-42.44];

C=[1,0,0,0;0,1,0,0;0,0,1,0;0,0,0,1];

D=[0]; I found its step response, but I need to rescale its output, How could I rescale it?.

Answer 1

[Raj]

The system you have mentioned is fully controllable as the controllability matrix has full rank. By 'rescaling' I assume you want to control the system's step response. You can easily do that by using a feedback control. In this case a positive feedback control is going to make your system unstable. So you have to use a negative feedback controller. Something like this:

A=[0,1,0,0;0,-42.44,212.24,0;0,0,0,1;0,0,0,-42.44];
B=[0,0;106.12,-5;0,0;78.6,-42.44];
C=[1,0,0,0;0,1,0,0;0,0,1,0;0,0,0,1];
D=[0];
system=ss(A,B,C,D);
step(system) % Step response of original system
hold on
k=[10 10 10 10; 10 10 10 10]% This is a dummy gain matrix.Put your gain matrix here.
system1=ss(A-B*k,B,C,D) % Closed loop system with negative feedback
step(system1) % Step response of closed loop system

[HAITHAM AL SATAI]

I did all these steps sir, but I need to scale my response step to 1. This way doesn't give me the amplitude.

[Raj]

"I need to scale my response step to 1"- There are four outputs. Which one are you trying to control and to which input (out of 2) are you applying the step?

"This way doesn't give me the amplitude." - I am not sure what you mean here. The response you get from 'step' is the amplitude of output signals.

[HAITHAM AL SATAI]

You can see here the peak amplitude. I need to rescale until to be able to calculate the static error

[Raj]

Again, I am not sure what you are looking for. I am assuming when you say 'static error' ,you want to calculate steady state error. With my dummy gain matrix, I get the closed loop step response as this:

Now at a particular output, SSE is nothing but (Desired value-final value).

[HAITHAM AL SATAI]

Thank you so much sir for your cooperation with me.