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Ecole des JDMACS 2011- GT MOSAR Synth` ese de correcteurs : approche bas´ ee sur

la forme observateur/retour d’´ etat MATLAB Tutorial session

D. Alazard

Let us consider the simplified model of a launcher:

G

0

(s) = Y (s)

U(s) = 1 s

2

− 1 ,

and the following candidate controllers (positive feedback):

K

0

(s) = −23s − 32

s + 12 , K

1

(s) = − s

2

+ 27s + 26

s

2

+ 7s + 18 , K

2

(s) = − 1667s + 2753 s

2

+ 27s + 353 .

1 Observer-based realization

1.a) Give a state space realization of G

0

(s) with state vector x = [y y] ˙

T

, 1.b) Compute an observer-based realization of controller K

1

(s) (macro-function

cor2obr or cor2obra and obr2cor).

1.c) Plot the closed-loop response of plant and controller states to initial conditions: y(t = 0) = 1; ˙ y(t = 0) = −1.

1.d) Same things (1.b) and 1.c)) with controller K

2

(s).

2 Cross standard form

Let us consider K

0

(s). K

0

(s) was designed to assign the closed-loop dominant dynamics to −1 ± i.

2.a) Compute the matrix T

1×2

of the linear combination of plant states observed by controller state: x

K

= T x b (macro-function cor2tfg or cor2tfga),

1

(2)

2.b) Set-up the Cross Standard Problem associated with G

0

(s) ≡

"

A B C D

#

and K

0

(s) ≡

"

A

K

B

K

C

K

D

K

#

:

P

CSF

(s) :=

A T

#

B

K

− BD

K

B

−C

K

T − D

K

C D

K

DD

K

− D

K

I

m

− D

K

D

C I

p

− DD

K

D

2.c) Compute K c

(s) = arg min

K(s)

kF

l

(P

CSF

(s), K(s))k

for various right inverses T

#

of T and compare K c

(s) with K

0

(s).

2.d) The frequency-domain response of the controller must now fit the fol- lowing template:

10−1 100 101 102 103

−40

−30

−20

−10 0 10 20 30 40

Singular Values

Frequency (rad/sec)

Singular Values (dB)

Figure 1: Frequency-domain responses (magnitude) of K

0

(s) (solid line) and K c

(s) (dashed line) and template (grey patch).

Augment the standard problem P

CSF

(s) to take into account this frequency- domain specification (Figure 2) and compute such a controller K c

(s).

Check the closed-loop dynamics.

2.e) Now, the actuator dynamics A(s) =

s+55

"

−5 5

1 0

#

is taken into account. Compute stability margins of the controller K c

(s) on the plant G

0

(s)A(s).

2

(3)

u y w z

W (s) P CSF (s)

Figure 2: P

CSF

with frequency weight.

2.f ) Augment the previous standard problem with A(s) (Figure 3) and tune W (s) to find a controller K(s) fitting the template (Figure 1) and such c that the open loop transfer −G

0

(s)A(s) K c (s) has at least a 30 deg phase margin.

u y

w z

W (s)

P

CSF

(s) A(s)

Figure 3: P

CSF

with frequency weight and actuator dynamics.

2.g) Compute an observer-based realization of K(s) on the plant c G

0

(s)A(s).

2.h) Plot responses to initial conditons ˙ y(t = 0) = −1 and to a square reference signal using the controller K c (s) directly (Figure 4) or using its observer based realization (Figure 5).

y_dot 1 s

y 1 s moteur

1 s Signal

Generator

Scope K(s)

x’ = Ax+Bu y = Cx+Du

Gain 5

Figure 4: Closed-loop simulation with controller K c (s).

3

(4)

z_dot 1 s y_dot

1 s

y 1 s

pinv(T) K*u

moteur 1 s

TB K*u T*[1;0;−1]

K*u Signal Generator

Scope2

Scope1

Scope K.c

K*u

Gain 5

GGK*u

F K*u

Figure 5: Closed-loop simulation with observer based realization of controller K(s). c

4

Figure

Figure 1: Frequency-domain responses (magnitude) of K 0 (s) (solid line) and K c ∞ (s) (dashed line) and template (grey patch).
Figure 4: Closed-loop simulation with controller K c (s).
Figure 5: Closed-loop simulation with observer based realization of controller K(s).c

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