Could you check this implementaion of Controller and observer design?

I'm trying to solve the system below, Could you check my work if it is correct or not?

close all;
clear all;
clc
%% 1) Set a requirement for step response, like settling time, overshoot 
%and static error of output.
A=[0,1,0,0;0,-42.44,212.24,0;0,0,0,1;0,0,0,-42.44];
B=[0;106.12;0;78.6];
C=[1,0,0,0;0,1,0,0;0,0,1,0;0,0,0,1];
D=0;
sys = ss(A, B, C, D)
t=0:0.1:10;
figure;
step(sys,t,'r')

%The parameters including overshoot, settling time and static error.
S = stepinfo(sys)

%% 2)Select the desired poles according to your requirements. If possible, 
%calculate the feedback matrix K to place the poles to the desired poles.

eig(sys)
% Now we check the stability
rank(ctrb(sys))
%The desired poles
P = [-1; -2; -3; -3.2850]
K = place(A,B,P)

%% 3) Simulate step response of the feedback system and calculate the parameters
%%including overshoot, settling time and static error. If the system does not satisfy the
%%requirement, go back to step 2).

Acl = A-B*K
syscl = ss(Acl,B,C,D)
figure;
step(syscl,'g')
%The parameters including overshoot, settling time.
S = stepinfo(syscl)
title('Step response of feedback control system')
grid on

%% 4) Design state observer of the system. The magnitude of observer poles must be 2~5
%times of system poles. Select different observer poles and compare the performance
%of observers.

sys=ss(A,B,C,D)
eig(sys)
rank(obsv(sys))
% Observer pole placement at  -10 and -9
% This observer will lead to a fast approximation of the states
L_T=place(A',C',[-10,-9,-8,-7])
L=L_T'
% Observer pole placement at  -1 and -2
% This observer will lead to a slower approximation of the states
% L_anastrofos=place(a',c',[-1,-2])
% L=L_anastrofos'
% State observer Feedback
sys=ss(A,B,C,D)
eig(sys)
rank(obsv(sys))
rank(ctrb(sys))
K=place(A,B,[-5 -6 -7 -8])

[Geoff Hayes]

Haitham - when you run the code/model, do you get unexpected results? Are there errors? Or are you just not sure if this has been implemented properly? Since it is homework, why not discuss with your professor or teaching assistant?

[HAITHAM AL SATAI]

@Geoff Hayes. I'm just not sure from my implmenation if it is correct or not. About my professor, she couldn't give me any answers.

Answer 1

[Raj]

Pretty good start!!

Now, please note that since you have mentioned that your doubt is in implementation, I have seen only your code.

I have not seen its exact correctness w.r.t the questionare that you have put.

1)" Set a requirement for step response, like settling time, overshoot and static error of output."- What you are doing is computing the step response of your system. Actually in addition to that, you have to set up some desired time domain step response critetia that your closed loop system (system+controller) should satisfy. Something like settling time should be 5 sec, overshoot should be less than 15% of peak value, settling error should be within 10% of final value etc.

2) rank(ctrb(sys)) will give you controllability of system not stability. eig(sys) will give you eigen values of system and if real part of eigen values are negative then system is stable or vice versa. Recommend giving a quick brush up on controllability and stability.

3) %The desired poles P = [-1; -2; -3; -3.2850] It will be good if you show how you arrived at these values.

4) Also in third step where you are simulating step response of the feedback system and calculating the parameters, you may have to go back and find different set of gains (k) if response is not meeting desired criteria set in s/n.1

Good Luck!!

[HAITHAM AL SATAI]

@Raj Thank you so much Mr. Raj for your feedback. Could you write me the first step as a code until to understand how to add it as a time please?

[Raj]

There is no code to be written specifically for that. What you have done is correct. Let me explain what i meant.

1) In your script, lines 13-15 you are finding out the step response of your open loop system. You get a set of time domain parameters there (like overshoot, settling time, steady state error etc).

2) Next you design a feedback controller based on desired pole position and get a closed loop system. Now in lines 30-32 you find the step response of your closed loop system and associated time domain parameters.

3) Now basically what you have to do is come up with a desired set of time response criteria for your closed loop system (Hint: You should have used this in computation of your desired poles) and see how much of what you got in s/n2 above (lines 30-32) meets this desired response. Tune your controller gain 'k' iteratively to match the time response of the closed loop system to the desired response.

Hope this helps!

[HAITHAM AL SATAI]

My desired poles I know they wrong. This is because When I got the eig(sys) to the sys I got these values :

0

-42.4400

0

-42.4400

and I don't know how to set the desired poles. @raj. Forgive me if I bother you.

[Raj]

See the example given here:

https://nptel.ac.in/courses/108103008/module5/lec2/1.html

It will give you an idea on how to compute desired pole position based on desired time domain parameters like overshoot, settling time etc.

[HAITHAM AL SATAI]

@Raj I got your idea Sir. But in my system when I apply this

%The parameters including overshoot, settling time and static error.

S = stepinfo(sys)

I got nothing :

S =

4×1 struct array with fields:

RiseTime

SettlingTime

SettlingMin

SettlingMax

Overshoot

Undershoot

Peak

PeakTime

How I extract these information ?

[Raj]

Your sys is a 4x1 state space model, so S = stepinfo(sys) should give you a 4×1 structure array S with required fields. See the section "Step-Response Characteristics of MIMO System" here:

https://in.mathworks.com/help/control/ref/stepinfo.html;jsessionid=d8cda1b3372359016839c379d2e3

[HAITHAM AL SATAI]

@Raj Thank you so much sir for your help