Pendulul invers

Pendulul invers

Cerinte:

1. Modelul discret pentru h = 0.01. Unde sunt polii?

>> A=[0 1 0 0; 0 -40 -20 0; 0 0 0 1; 0 25 17 0];

>> B=[ 0 7 0 -12]';

>> C=[1 0 0 0; 0 0 1 0];

>> D=zeros(2,1);

>> sys=ss(A,B,C,D);

>> H=tf(sys)

Transfer function from input to output

7 s^2 + 1.158e-014 s + 121

#1: ----- ----- --------- ----- -----

s^4 + 40 s^3 - 17 s^2 - 180 s

-12 s - 305

#2: ----- ----- ----- ----- -----

s^3 + 40 s^2 - 17 s - 180

>>

>> H_z=c2d(H,0.01,'zoh')

Transfer function from input to output

0.0003077 z^3 - 0.0003456 z^2 - 0.0002305 z + 0.0002694

#1: -------- ----- ------ ----- ----- ------------

z^4 - 3.672 z^3 + 5.014 z^2 - 3.012 z + 0.6703

-0.0005736 z^2 - 0.0001017 z + 0.0004239

#2: -------- ----- ------ -------

z^3 - 2.672 z^2 + 2.342 z - 0.6703

Sampling time: 0.01

2. Lege de reglare dupa stare à (K = ?) a.i. polii sunt

>> sys_z=c2d(sys,0.01,'zoh');

>> Ad=sys_z.a

Ad =

1.0000 0.0082 -0.0009 -0.0000

0 0.6703 -0.1649 -0.0009

0 0.0011 1.0008 0.0100

0 0.2061 0.1481 1.0008

>> Bd=sys_z.b

Bd =

0.0003

0.0577

>> K=place(Ad,Bd,[0.67 0.96 0.98+j*0.01 0.98-j*0.01])

K =

6.6171 -0.3514 -3.5253 -0.8726

3. Proiectarea unui estimator de stare cand p si theta sunt masurabile de forma:

Cat este L a.i. polii estimati sunt

>> Cd=sys_z.c

Cd =

1 0 0 0

0 0 1 0

>> L=place(Ad', Cd', [0.5 0.8 0.85+0.05*j 0.85-0.05*j]')'

L =

0.3083 -0.0708

0.0578 2.5542

4. Folosind starea estimata à comanda