Confusion calculating steady-state error in MATLAB

Question

I am trying to use MATLAB to calculate a steady-state error. I have found a weblink that gives code for finding the steady-state error, but when I use that code I get a value of the steady-error greater than 1. In my understanding, a steady-state error should be less than 1, because formula of steady-state error in case of step input = 1/(1+k)

The code:

sserror=abs(SP-y(end))

When I try to incorporate the above code into my code to get the steady-state error:

clc;clear all;close all
%defining parameters values
Ra=1
Kt=10
Kb=0.1
j=2
b=0.5
%defining numerator & denominator of system1 (1/js+b)
num=[1]
den=[j b]
% creating transfer function for system 1
sys1=tf(num,den)
%Applying block diagram reduction and simplification techniques
sys2=(-Kb)*Kt/Ra
%Creating a positive feedback system from above two systems sys1 and sys2
a=feedback(sys1,sys2,1)
%Introducing the effect of disturbance input 
OL=-1*a
%Calculating step response & extracting amplitude & time matrices
[y,t]=step(OL)
%Plotting  values of amplitude matrix extracted from  step resposne
plot(y)
%Calculating length(total number of elements) for timing matrix 
z=length(t)
%Finding value of last element of amplitude matrix
%That is also value of steady state speed
open_loop_steady_state_speed=y(z)
SP=1 % set point incase of step input
sserror=abs(SP-y(end))

I am trying to simulate a block diagram from a book, Modern Control Systems. I am also attaching a snapshot of that block diagram.

There are two block diagrams in the snapshot, one is for an open-loop control system and the other is for a closed-loop control system.

I am considering and trying to write code for an open-loop system.

Update: Note: Plot has been added in response to comment

Answer 1

a=feedback(sys1,sys2,1)

OL=-1*a

[y,t]=step(OL)

The output of the closed loop system is negated before it is supplied to the step function.

So even though the output of the original system a is going to 0.66 (steady state error 0.33), due to the negation of the output in OL, the calculation returns 1.66 > 1.

What is the reason / logic for doing OL = -1 * a ?

Note

There is one logical error also.

[y,t]=step(OL)         % SP is NOT the input given here!
SP=1                   % the set point has NO effect on y !
sserror=abs(SP-y(end)) % this calculation is hence invalid

Since the step function is called before setting the setpoint SP, the calculation SP - y(end) is invalid since SP was not the input actually supplied to the system.

The code in the link you provided in the question has the correct order of operations.

SP=5; %input value, if you put 1 then is the same as step(sys)
[y,t]=step(SP*sys); %get the response of the system to a step with amplitude SP
sserror=abs(SP-y(end)) %get the steady state error

Answer 2

I think you should be treating at the system like this, and analyze the system this way, taking \$T_d(s)\$ as your main input. With such a diagram, the definition for sys2 and sys1 changes, as you might notice. It becomes the formula below.

$$sys2=\frac{K_m(K_tK_b+K_a)}{R_a}$$ $$sys1=\frac{-1}{Js+b}$$

Rewriting the code according to this might yield a correct result.