Question

Please use MATLAB, screenshot code and results

An automotive power train control system is described by the following matrix equations 1-12 -10 -57 [1] x= 1 0 0 +0 u To 100 y(t) = [ 35 – 5]x where u = -KX+r, and r is a unit step input. Use MATLAB/SIMULINK to plot different responses of the system output Y for the following feedback control gain matrix K: Casel: K = [1 44 67] Case2: K = [10 44 67] Case3: K = [44 1 1] Case4: K = [1 67 44]

Answer 1

Solution:

F12-10-5 u=-K.X+r. 1.12 i -0 o 5 o X + 10 - X- > (T-12 -10 -57 17 Or, XIi oo (o 10 -> x= Ax+ Br - 0 @here, gs is Unit step input. Y=135-5] cmY = [35-5] x + [o]ino Cami Y= ex+ Dr - Din are the state space equation.

Matlab Codes;:

For case1:

..............................

%Matlab code

%Clears the output screen
clc

%Case1: for K=[1 44 67]

%statement of K matrix
K = [1 44 67]

%statement of A matrix
A = [-12 -10 -5; 1 0 0; 0 1 0] - [1; 0; 0] * K

%statement of B matrix
B = [1; 0; 0]

%statement of C matrix
C = [3 5 -5]

%statement of D matrix
D = [0]

%statement of State space model, Let's name it G.
G = ss(A,B,C,D);

%Plot of Output for time t;
%where t starts from 0, increment in each step by 0.1, ends at 5
t= [0 : 0.01 : 5 ];

%plot of output
stepplot(G,t)
%Turning on Griding
grid on

Output:

K =

1 44 67

A =

-13 -54 -72
1 0 0
0 1 0

B =

1
0
0

C =

3 5 -5

D =

0

- O X Figure 1 File Edit View Insert Tools Desktop Window Help Ou ķ a .. . E Step Response 0.25 0.15 Amplitude 0.05 -0.05 0.5 1.5 2.5 3.5 4.5 Time (sec)

....................

For case2:

..............................

%Matlab code

%Clears the output screen
clc

%Case1: for K=[10 44 67]

%statement of K matrix
K = [10 44 67]

%statement of A matrix
A = [-12 -10 -5; 1 0 0; 0 1 0] - [1; 0; 0] * K

%statement of B matrix
B = [1; 0; 0]

%statement of C matrix
C = [3 5 -5]

%statement of D matrix
D = [0]

%statement of State space model, Let's name it G.
G = ss(A,B,C,D);

%Plot of Output for time t;
%where t starts from 0, increment in each step by 0.1, ends at 5
t= [0 : 0.01 : 5 ];

%plot of output
stepplot(G,t)
%Turning on Griding
grid on

Output:

K =

10 44 67

A =

-22 -54 -72
1 0 0
0 1 0

B =

1
0
0

C =

3 5 -5

D =

0

- O X Figure 1 File Edit View Insert Tools Desktop Window Help Ou ķ a .. . E Step Response 0.15 0.1 0.05 Amplitude -0.05 0.5 3.5 2.5 Time (sec)

....................

For case3:

..............................

%Matlab code

%Clears the output screen
clc

%Case1: for K=[44 1 1]

%statement of K matrix
K = [44 1 1]

%statement of A matrix
A = [-12 -10 -5; 1 0 0; 0 1 0] - [1; 0; 0] * K

%statement of B matrix
B = [1; 0; 0]

%statement of C matrix
C = [3 5 -5]

%statement of D matrix
D = [0]

%statement of State space model, Let's name it G.
G = ss(A,B,C,D);

%Plot of Output for time t;
%where t starts from 0, increment in each step by 0.1, ends at 5
t= [0 : 0.01 : 5 ];

%plot of output
stepplot(G,t)
%Turning on Griding
grid on

Output:

K =

44 1 1

A =

-56 -11 -6
1 0 0
0 1 0

B =

1
0
0

C =

3 5 -5

D =

0

- O X Figure 1 File Edit View Insert Tools Desktop Window Help Ou ķ a .. . E Step Response -0.1 Amplitude -0.5 0.5 3.5 2.5 Time (sec)

....................

For case4:

..............................

Output:

K =

1 67 44

A =

-13 -77 -49
1 0 0
0 1 0

B =

1
0
0

C =

3 5 -5

D =

0

- O X Figure 1 File Edit View Insert Tools Desktop Window Help Ou ķ a .. . E Step Response 0.25 0.15 -- Amplitude 0.05 -0.05 0.5 3.5 4 5 2.5 Time (sec)

.......................................

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