Adaptive Control Chapter 12: Indirect Adaptive Control 1 Adaptive - - PowerPoint PPT Presentation

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Chapter 12: Indirect Adaptive Control

Adaptive Control

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Chapter 12 Indirect Adaptive Control

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Pole placement It is a method that does not simplify the plant model zeros The pole placement allows to design a R-S-T controller for

• stable or unstable systems
• without restriction upon the degrees of A and B polynomials
• without restrictions upon the plant model zeros (stable or unstable)
• but A and B polynomials should not have common factors

(controllable/observable model for design)

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Structure ) ( ) ( ) (

1 1 1 − − − −

= q A q B q q H

d

A A

n n q

a q a q A

− − −

+ + + = ... 1 ) (

1 1 1

) ( ... ) (

1 * 1 2 2 1 1 1 − − − − − −

= + + + = q B q q b q b q b q B

B B

n n

Plant:

)

• 1

(q

• 1

)

• q
• d

B A PLANT R(q

• 1

)

• q
• d

B(q

• 1

) (q

• 1

) S P

r(t) y(t) +

• T( )

q

• 1

p(t) + +

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

1 1 1 1 1 1 1 1 1 1 − − − − − − − − − − − − −

= + = q P q B q T q q R q B q q S q A q B q T q q H

d d d BF

.... 1 ) ( ) ( ) ( ) ( ) (

2 2 1 1 1 1 1 1 1

+ + + = + =

− − − − − − − −

q p q p q R q B q q S q A q P

d

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

1 1 1 1 1 1 1 1 1 1 − − − − − − − − − − −

= + = q P q S q A q R q B q q S q A q S q A q S

d yp

Closed loop T.F. (r y) (reference tracking)

Defines the (desired )closed loop poles

Closed loop T.F. (p y) (disturbance rejection)

Output sensitivity function

Pole placement

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Choice of desired closed loop poles (polynomial P) ) ( ) ( ) (

1 1 1 − − −

= q P q P q P

F D Dominant poles Auxiliary poles

Specification in continuous time (tM, M)

2nd order (ω0, ζ)

discretization e

T ) (

1 −

q P

D 5 . 1 25 . ≤ ≤

e

T ω 1 7 . ≤ ≤ ζ

Choice of PD(q-1)(dominant poles)

• Auxiliary poles are introduced for robustness purposes
• They usually are selected to be faster than the dominant poles

Auxiliary poles

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Regulation( computation of R(q-1) and S(q-1)) ) ( ) ( ) ( ) ( ) (

1 1 1 1 1 − − − − − −

= + q P q R q B q q S q A

d

? ? ) ( deg

1 −

= q A nA ) ( deg

1 −

= q B nB A and B do not have common factors (Bezout) unique minimal solution for : 1 ) ( deg

1

− + + ≤ =

−

d n n q P n

B A P

1 ) ( deg

1

− + = =

−

d n q S n

B S

1 ) ( deg

1

− = =

− A R

n q R n ) ( * 1 ... 1 ) (

1 1 1 1 1 − − − − −

+ = + + = q S q q s q s q S

S S

n n

R R

n n q

r q r r q R

− − −

+ + = ... ) (

1 1 1

(*)

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Computation of R(q-1) and S(q-1) Equation (*) is written as: Mx = p

] ,..., , ,..., , 1 [

1

R S

n n T

r r s s x = ] ,..., , ,..., ,..., , 1 [

1

P

n i T

p p p p =

1 ... a1 1 . a2 1 a1 anA a2

. ...

anA ... ... b' 1 b' 2 b' 1 . b' 2 . . b' nB . . . . b' nB nA + nB + d

nB + d nA nA + nB + d

x = M-1p Use of WinReg or bezoutd.sci(.m) for solving (*)

b'i = 0 pour i = 0, 1 ...d ; b'i = bi-d pour i > d

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Structure of R(q-1) and S(q-1) R and S may include pre-specified fixed parts (ex: integrator) ) ( ) ( ' ) (

1 1 1 − − −

= q H q R q R

R

) ( ) ( ' ) (

1 1 1 − − −

= q H q S q S

S

HR, HS, - pre-specified polynomials

' '

' ... ' ' ) ( '

1 1 1

R R

n n q

r q r r q R

− − −

+ + =

' '

' ... ' 1 ) ( '

1 1 1

S S

n n q

s q s q S

− − −

+ + =

• The pre specified filters HR and HS will allow to impose certain properties of the

closed loop.

• They can influence performance and/or robustness

) ( ) ( ' ) ( ) ( ) ( ' ) ( ) (

1 1 1 1 1 1 1 − − − − − − − −

= + q P q R q H q B q q S q H q A

R d S

? ?

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Fixed parts (HR , HS). Examples Zero steady state error (Syp should be null at certain frequencies) Step disturbance : Sinusoidal disturbance :

s S

T q q H ω α α cos 2 ; 1

2 1

− = + + =

− −

1 1

1 ) (

− −

− = q q HS Signal blocking (Sup should be null at certain frequencies)

s R

T q q H ω β β cos 2 ; 1

2 1

− = + + =

− −

2 , 1 ; ) 1 (

1

= + = Sinusoidal signal: Blocking at 0.5fS:

−

n q H

n R ) ( ) ( ) ( ) ( ) (

1 1 1 1 1 − − − − −

′ = q P q S q H q A q S

S yp

) ( ) ( ) ( ) ( ) (

1 1 1 1 1 − − − − −

′ − = q P q R q H q A q S

R up

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Tracking (computation of T(q-1) )

• q
• 1

Bm A m Ideal case r (t) y* (t) desired trajectory for y (t) t y r

*

Tracking reference model (Hm)

2nd order (ω0, ζ)

discretization

s

T

) (

1 −

q Hm

5 . 1 25 . ≤ ≤

s

T ω

1 7 . ≤ ≤ ζ

) ( ) ( ) (

1 1 1 − − −

= q A q B q H

m m m

The ideal case can not be obtained (delay, plant zeros) Objective : to approach y*(t) ) ( ) ( ) ( ) (

1 1 ) 1 ( *

t r q A q B q t y

m m d − − + −

=

... ) (

1 1 1

+ + =

− −

q b b q B

m m m

... 1 ) (

2 2 1 1 1

+ + + =

− − −

q a q a q A

m m m

Specification in continuous time (tM, M)

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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) ( ) ( ) ( ) 1 (

1 1 *

t r q A q B d t y

m m − −

= + +

Build: Choice of T(q-1) :

• Imposing unit static gain between y* and y
• Compensation of regulation dynamics P(q-1)

⎩ ⎨ ⎧ = ≠ = ) 1 ( 1 ) 1 ( ) 1 ( / 1 B if B if B G T(q-1) = GP(q-1) Particular case : P = Am

⎪ ⎩ ⎪ ⎨ ⎧ = ≠ = =

−

) 1 ( 1 ) 1 ( ) 1 ( ) 1 ( ) (

1

B if B if B P G q T

F.T. r y: ) 1 ( ) ( ) ( ) ( ) (

1 * 1 1 ) 1 ( 1

B q B q A q B q q H

m m d BF − − − + − −

⋅ = Tracking (computation of T(q-1) )

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Pole placement. Tracking and regulation

+

• R

1 q

• d B

A S T A Bm

m

r(t) y (t+d+1) * u(t) y(t) q

• (d+1)

P(q -1 ) q

• (d+1)

B*(q )

• 1

B*(q )

• 1

B(1) q

• (d+1)

B m(q ) B*(q )

• 1
• 1

A m(q ) B(1)

• 1

) 1 ( * ) ( ) ( ) ( ) ( ) (

1 1 1

+ + = +

− − −

d t y q T t y q R t u q S

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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) ( ) ( ) ( ) 1 ( ) ( ) (

1 1 * 1 − − −

− + + = q S t y q R d t y q T t u ) 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) (

* 1 * 1 1 1

+ + = + + = +

− − − −

d t y q T d t y q GP t y q R t u q S ) ( 1 ) (

1 * 1 1 − − −

+ = q S q q S ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) (

1 1 * * 1

t y q R t u q S d t Gy q P t u

− − −

− − − + + = ) ( ) ( ) ( ) 1 (

1 1 *

t r q A q B d t y

m m − −

= + + ) ( 1 ) (

1 * 1 1 − − −

+ = q A q q A

m m

) ( ) ( ) ( ) ( ) 1 (

1 1 * *

t r q B d t y q A d t y

m m − −

+ + − = + + ... ) (

1 1 1

+ + =

− −

q b b q B

m m m

... 1 ) (

2 2 1 1 1

+ + + =

− − −

q a q a q A

m m m

Pole placement. Control law

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Indirect adaptive control Indirect adaptive control Step I : Estimation of the plant model ARX identification (Recursive Least Squares) At each sampling instant: ) ˆ , ˆ ( B A Step II: Computation of the controller Solving Bezout equation (for S’ and R’)

Remark: It is time consuming for large dimension of the plant model

P R H B q q S H A

R d S

= +

− −

' ˆ ) ( ' ˆ

1

⎪ ⎩ ⎪ ⎨ ⎧ = = ≠ = = = ) 1 ( ˆ 1 ˆ ) 1 ( ˆ ) 1 ( ˆ 1 ˆ ˆ B if G B if B G P G T ) ( ) ( ' ) (

1 1 1 − − −

= q H q R q R

R

) ( ) ( ' ) (

1 1 1 − − −

= q H q S q S

S

Compute:

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Supervision Estimation:

• Check if input is enough “persistently exciting”

(if not, do not take in account the estimations)

• Check if and are numerically “sound” (condition number)

(no close poles/zeros)

• If necessary, add external excitation (testing signal)

A ˆ B ˆ Control:

• Check if desired dominant closed loop poles are compatible

with estimated plant poles

• Check robustness margins

Additional problem:

• How to deal with neglected dynamics ?

(filtering of the data, robustification of PAA)

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Robust Control Design for Adaptive Control

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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The flexible transmission

Φ

m

axis motor d.c. motor Position transducer

axis position

Φ

ref load Controller u(t) y(t) A D C R-S-T controller D A C

Adaptive Control of a Flexible Transmission

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Adaptive Control of a Flexible Transmission Frequency characteristics for various load

Rem.: the main vibration mode varies by 100%

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Robust Control Design for Adaptive Control

parameter variations

(low frequency)

Adaptation Robust Design unstructured uncertainities

(high frequency)

Basic rule : The input sensitivity function (Sup) should be small in medium and high frequencies How to achieve this ? Pole Placement :

• Opening the loop in high frequencies (at 0.5fs)
• Placing auxiliary closed loop poles near the high frequency poles
• f the plant model

Generalized Predictive Control :

• Appropriate weighting filter on the control term in the criterion

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Robust Control Design for Adaptive Control

(Flexible Transmission)

a) Standard pole placement (1 pair dominant poles + h.f. aperiodic poles) b) Opening the loop at 0.5fs (HR = 1 + q-1) c) Auxiliary closed loop poles near high frequency plant poles

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Parameter Estimators for Adaptive Control Objective : to reduce the effect of the disturbances upon the quality of the estimation

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Classical Indirect Adaptive Control

PLANT disturb. u y

+ + +

q- dB A εCL q-dB A

• P.A.A.

y

Reference

Adjustable Controller

q -1

Filter Filter

Adaptation mechanism (design)

• Uses R.L.S. type estimator (equation error)
• Sensitive to output disturbances
• Requires « adaptation freezing » in the absence of persistent excitation
• The threshold for « adaptation freezing » is problem dependent

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Closed Loop Output Error Parameter Estimator for Adaptive Control

PLANT disturb. u y

+ + +

εCL q-dB A

• 1

S R û P.A.A. y

Reference

Adjustable Controller Adjustable Controller

q-1

Adaptation mechanism (design)

• Insensitive to output disturbances
• Remove the need for « adaptation freezing » in the absence of

persistent excitation

• CLOE requires stability of the closed loop
• Well suited for « adaptive control with multiple models »

q- dB A

+

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Adaptive Control – Effect of Disturbances Classical parameter estimator (filtered RLS) CLOE parameter estimator

Disturbances destabilize the adaptive system when using RLS parameter estimator (in the absence of a variable reference signal)