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Problem 7.22PP
For the system
design a state feedback controller that satisfies the following specifications: 
(a) Closed-loop poles have a damping coefficient ^ = 0.707.
(b) Step-response peak time is under 3.14 sec. 
Verify your design with Matlab.
Steo-bv-steo solution
Step-by-step solution
The system is.
‘■[i
[ i: ]
Here, 
F = 
And,
Step 1 of 5
G=|
Write the expression for peak time.
Substitute 3.14 s for in the equation to calculate the damping frequency. 
3.14s« —
* 3.14 s 
«1 rad/s
Step 2 of 5
Write the expression for damping frequency.
^4 ■
Substitute 1 rad/s for and 0.707 for ^ in the equation to calculate the natural frequency. 
-0.707'
___ 1 _
“ 0.707 
■ 1.414 lad/s
Write the general second order characteristic equation, 
s* + ® 0
Substitute0.707for ^ .a n d 1.414rad/s for 
»'+2(0.707)(1.414)i+( 1.414)’ =0 
s '+ 2 t + 2 = 0 (1)
Step 3 of 5
The linear state feedback expression is,
tt = -K x
Here.
K is the state feedback gain
K = [JC, AT,]
Calculate the value of the matrix, F -G K 
0 1
F -G K =
-6 -S
_ r ® '
■ [-6 -5
_r 0
Determine the matrix. ±
0 0 ]JC, K,i
- 5 - atJ
XS F -G K ] 
0
- 6 K, - 5 - j r J *
The closed-loop characteristic equation is,
s'+ ( 5 + a: ,)s + (6 + a: ,)= o ......(2)
Compare equations (1) and (2).
S + K^ = 2
K jm -3
And.
6+K, = 2 
K ,m -4
The state feedback gain is,
K = H -3]
The state feedback controller, u Is F H -3 1 4 
Write the MATLAB code to plot the step response.
» num=1;
» den=[1 2 2];
» sys=tf(num,den): 
» t=0:0.0001:6;
» y=step{sys,t);
» plot(t.y);
» grid;
» xiabel t 
» ylabel y(t)
Step 4 of 5
Step 5 of 5
The following is the unit step response.
t
From the unit step response the peak time Is,
/,= 3 . I4 s
The peak time obtained is at 3.14 s. Hence, it is verified using MATLAB.