Question

14. Consider the solar tracking servo with the following transfer function G,(s) = s(10s +1) G (5) U(s) X (s) X (s) Y(s) a. Draw the well labelled block diagram of a full state feedback digital control system with a closed loop observer and a reference. b. Design a full state digital feedback controller to place the system poles at R2--1+) by employing the feedback law from state space technique.

Answer 1

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. A block diagram of the system with an observer and a full state feedback is shown below:

D(S) Gp(s) Y_act LBS model R(S) = 0 Å ve(s) B X=Ax+ Bu U=-kx| te dback X $ = AŃ + Bu +L(y- cx) Observer

An observer is designed so that the states which are not available for measurement are estimated. The estimated states are then taken as input to the state feedback controller to enable a full state feedback law.

b. The state feedback law is designed via Matlab:

*********** Matlab Code *************

sys = tf(1,[10 1 0]);
[A,B,C,D] = tf2ss(1,[10 1 0]);

p1 = -1-1i;
p2 = -1+1i;
K = place(A,B,[p1 p2])
sys_cl = ss(A-B*K,B,C,D);

step(sys_cl);
grid;
xlabel('Time (sec)')
ylabel('Amplitude (m)')

********* End of Code *****

Output:

K =

    1.9000    2.0000

Step response after state feedback:

Step Response 0.06 0.05 0.04 Amplitude (m) 0.03 0.02 0.01 0 0 2. 3 4 5 6 7 Time (sec) (seconds)

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