Control Automatico Ll.docx

1)

2)

A=0 0 1

B= 0.3

1

G= -0.7

1.12

100

0

1

-1.17 1

010

-0.3

0.9

0.14

C=[1.9 1.3 1]

Qn=E(wwT)= 4 2 2

D=[0.53 −0.61]

1.5

H=[−1.2 −0.89]

Rn=E(vvT)=0.7

1

[ ] )  ( T T J(u)= 0.1x x+x2i+u 1002 u dt ∞0

To design an LQG servo controller for this system: 1. Create the state space system by typing the following in the MATLAB Command Window: 2. A = [0 1 0;0 0 1;1 0 0]; 3. B = [0.3 1;0 1;-0.3 0.9]; 4. G = [-0.7 1.12; -1.17 1; .14 1.5]; 5. C = [1.9 1.3 1];

6. D = [0.53 -0.61]; 7. H = [-1.2 -0.89]; sys = ss(A,[B G],C,[D H]); 8. Construct the optimal state-feedback gain using the given cost function by typing the following commands: 9. nx = 3;

%Number of states

10. ny = 1;

%Number of outputs

11. Q = blkdiag(0.1*eye(nx),eye(ny)); 12. R = [1 0;0 2]; 13. K = lqi(ss(A,B,C,D),Q,R); 14. Construct the Kalman state estimator using the given noise covariance data by typing the following commands: 15. Qn = [4 2;2 1]; 16. Rn = 0.7; 17. kest = kalman(sys,Qn,Rn); 18. Connect the Kalman state estimator and the optimal state-feedback gain to form the LQG servo controller by typing the following command: trksys = lqgtrack(kest,K) This command returns the following LQG servo controller: >> trksys = lqgtrack(kest,K)

a= x1_e

x2_e

x3_e

xi1

x1_e -2.373 -1.062 -1.649 0.772 x2_e -3.443 -2.876 -1.335 0.6351 x3_e -1.963 -2.483 -2.043 0.4049 xi1

0

0

0

0

b= r1

y1

x1_e

0 0.2849

x2_e

0 0.7727

x3_e

0 0.7058

xi1

1

-1

c= x1_e

x2_e

x3_e

xi1

u1 -0.5388 -0.4173 -0.2481 0.5578 u2 -1.492 -1.388 -1.131 0.5869

d= r1 y1 u1 0 0 u2 0 0