Adaptive Control Chapter 11: Direct Adaptive Control 1 Adaptive - - PowerPoint PPT Presentation

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

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Adaptive Control

Chapter 11: Direct Adaptive Control

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

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Chapter 11: Direct Adaptive Control

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

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Adaptive Control – A Basic Scheme

• Indirect adaptive control
• Direct adaptive control (the controller is directly estimated)

Adjustable Controller Plant

+

• SUPERVISION

Performance specifications Controller Design Plant Model Estimation Adaptation loop

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Outline

• Digital control systems
• Tracking and regulation with independent objectives

(known parameters)

• Adaptive tracking and regulation with independent objectives

(direct adaptive control)

• Pole placement (known parameters)
• Adaptive pole placement (indirect adaptive control)

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Actuator Sensor PLANT ADC DIGITAL COMPUTER CLOCK r(k) e(k) y(k) y(t) DAC + ZOH u(k) u(t) +

• Process

Digital Control System The control law is implemented on a digital computer ADC: analog to digital converter DAC: digital to analog converter ZOH: zero order hold

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• Sampling time depends on the

system bandwidth

• Efficient use of computer resources

DAC + ZOH PLANT ADC COMPUTER CLOCK DISCRETIZED PLANT

r(k) e(k) u(k) y(k) y(t) u(t) +

• Digital Control System

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

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Computer (controller)

D/A + ZOH

PLANT

A/D CLOCK

Discretized Plant

r(t) u(t) y(t)

The R-S-T Digital Controller

r(t)

m m

A B T S 1 A B q

d −

R

u(t) y(t) Controller Plant Model +

• )

1 ( ) (

1

− =

−

t y t y q

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

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Discrete time model – General form ) ( ) ( ) (

1 1

i d t u b i t y a t y

B A

n i i n i i

− − + − − =

∑ ∑

= =

d –delay (integer multiple of the sampling period)

) ( 1 ) ( 1

1 * 1 1 1 − = − − −

∑

+ = = + q A q q A q a

A

n i i i 1 1 2 1 1 *

... ) (

+ − − −

+ + + =

A A

n n q

a q a a q A ; ) ( ) (

1 * 1 1 1 − = − − −

∑

= = q B q q B q b

B

n i i i

;

1 1 2 1 1 *

... ) (

+ − − −

+ + + =

B B

n n q

b q b b q B

) ( ) ( ) ( ) (

1 1

t u q B q t y q A

d − − −

= ) ( ) ( ) ( ) (

1 1

t u q B d t y q A

− −

= + ) ( ) ( ) (

1

t u q H t y

−

= ) ( ) ( ) (

1 1 1 − − − −

= q A q B q q H

d

) ( ) ( ) (

1 1 1 − − − −

= z A z B q z H

z

• pulse transfer operator

(Predictive form)

;

1 1 − − → z

q

• transfer function

(*) (*) (*) (*)

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

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First order systems with delay T e G s H

s s

+ =

−

1 ) (

τ s s

T L L T d < < + = ; . τ

Fractional delay

Continuous time model

1 1 1 2 1 1 1 1 2 2 1 1 1

1 ) ( 1 ) ( ) (

− − − − − − − − −

+ + = + + = z a z b b z z a z b z b z z H

d d

Discrete time model

e a

T

T s

−

− =

1

) 1 (

1 T T L

s

e G b

−

− = ) 1 (

2

− =

− T L T T

e Ge b

s

Remark: For

) 1 ( 5 .

1 2 1 2

> − ⇒ > ⇒ > b b zero unstable b b T L

s

• - zero

x - pole z

1

• j

x

• b2

b1 b2 b1 >1 <1

• +j
• a 1

x

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Tracking and regulation with independent objectives It is a particular case of pole placement (the closed loop poles contain the plant zeros)) Allows to design a RST controller for:

• stable or unstable systems
• without restrictions upon the degrees of the polynomials A et B
• without restriction upon the integer delay d of the plant model
• discrete-time plant models with stable zeros!!!

It is a method which simplifies the plant zeros Allows exact achievement of imposed performances Remarks:

• Does not tolerate fractional delay > 0.5 TS (unstable zero)
• High sampling frequency generates unstable discrete time zeros !

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• 1
• 0.5

0.5 1

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 Zero Admissible Zone Real Axis Imag Axis f0/fs = 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

ζ = 0.1 ζ = 0.2

The model zeros should be stable and enough damped

Admissibility domain for the zeros of the discrete time model

Tracking and regulation with independent objectives

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(q )

• 1

S 1 R A B q d

− m m

A B T

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

P q

d ) 1 ( + − ) 1 ( + − d

q

m m d

A B q

) 1 ( + −

+

• )

( ) ( ) (

1 1 1 − − −

= q P q P q P

F D ) ( ) ( ) ( ) 1 (

1 1 *

t r q A q B d t y

m m − −

= + +

Reference signal (tracking): Tracking and regulation with independent objectives ) ( ) ( ) 1 ( ) ( ) ( ) (

1 * 1 1

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

− − −

− + + = Controller:

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T.F. of the closed loop without T:

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

1 1 * 1 * 1 1 1 1 1 * 1 1 1 1 * 1 1 − − − + − − + − − − + − − − − + − −

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

d d d d CL

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

1 1 * 1 1 * 1 1 1 − − − − + − − −

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

d

The following equation has to be solved : S should be in the form:

) ( ) ( ... ) (

1 1 * 1 1 1 − − − − −

′ = + + + = q S q B q s q s s q S

S S

n n

After simplification by B*,(*) becomes:

) ( ) ( ) ( ' ) (

1 1 1 1 1 − − + − − −

= + q P q R q q S q A

d

(*)

nP = deg P(q-1) = nA+d ; deg S'(q-1) = d ; deg R(q-1) = nA-1

Unique solution if:

1 1 1 1 1

... ) (

− − − − −

+ + =

A A

n n

q r q r r q R

d d q

s q s q S

− − −

+ + = ' ... ' 1 ) ( '

1 1 1

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

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(**) is written as: Mx = p

1 a1 1 a2 a1 : : 1 ad ad-1 ... a1 1 ad+1 ad a1 ad+2 ad+1 a2 . . ... anA . . . . 1 . . . 0 0 1 nA + d + 1

nA + d + 1 d + 1 nA ] ,..., , , ,..., , 1 [

1 1 1 −

′ ′ =

n d T

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

1 2 1 d n n n T

A A A

p p p p p p

+ +

=

Use of WinReg or predisol.sci(.m) for solving (**) x = M-1p Insertion of pre specified parts in R and S is possible Regulation ( computation of R(q-1) and S(q-1))

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Tracking (computation of T(q-1) ) Closed loop T.F.: r y

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

1 1 ) 1 ( 1 1 1 1 ) 1 ( 1 − − + − − − − − + − −

= = q P q A q q T q B q A q B q q H

m d m m m d BF

Desired T.F.

It results : T(q-1) = P(q-1) Controller equation: ) ( ) ( ) ( ) 1 ( ) ( ) (

1 1 * 1 − − −

− + + = q S t y q R d t y q P t u

[ ]

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

1 1 * * 1 1

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

− − −

− − − + + =

(s0 = b1)

) ( ) ( ) ( ) 1 (

1 1 *

t r q A q B d t y

m m − −

= + +

Reference signal (tracking) : ) ( ) ( ) 1 ( ) ( ) ( ) (

1 * 1 1

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

− − −

− + + =

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Tracking and regulation with independent objectives A time domain interpretation

Reformulation of the “design problem”: Find a controller which generate u(t) such that:

[ ]

) 1 ( * ) 1 ( ) 1 ( = + + − + + = + + d t y d t y P d t ε

(in case of correct tuning)

+

• R

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

P

• + P

) 1 ( + − d

q

) ( ≡ t ε

m m

A B A B q

d

*

) 1 ( + −

P q

d ) 1 ( + −

) 1 ( * ) ( ) (

1 ) 1 (

+ + =

− + −

d t y q P q t y

d

) 1 ( * ) ( ) (

) 1 ( 1

+ + =

+ − −

d t y q t y q P

d

) 1 ( * ) ( ) (

) 1 ( 1

= + + −

+ − −

d t y q t y q P

d

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

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Tracking and regulation with independent objectives Synthesis in the time domain – an example

*

) ( * ) (

1 1 − −

= q B q S ) ( * 1 ) ( ) ( * 1 ) ( ) ( * ) ( * ) ( ) ( ) ( ) (

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

+ = + = − = = + q A q q A q P q q P q A q P q R q P q R q q A

1 1 1 2 1 1

1 ) ( ); 1 ( ) ( ) ( ) 1 (

− −

+ = − + + − = + q p q P t u b t u b t y a t y

For d=0 (S’=1) [ ] [ ]

1 1 1 2 1 2 1 1 1

; ) ( ) 1 ( ) 1 ( * ) ( ) 1 ( * ) ( ) 1 ( ) ( ) ( ) 1 ( * ) ( ) 1 ( ) 1 ( * ) 1 ( ) 1 ( a p r b t y r t u b t Py t u t Py t y p t u b t u b t y a t Py t y p t y t y t y P t − = − − − + = = + − + − + + − = = + − + + = + − + = + ° ε

[ ] [ ]

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

2 1 2 1 1

t y t u t u t r b b t t y r t u b t u b t y q P

T T T

− = = = + − + = +

−

φ θ φ θ

+

• R

r(t) y* (t+1) u(t) y(t) S 1

P

• + P

1 −

q

) ( ≡ t ε

m m

A B A B q *

1 −

P q 1

−

Example: Controller satisfies:

Solve for u(t)

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Adaptive tracking and regulation with independent objectives Three techniques:

• Model reference adaptive control (direct)
• Plant model estimation + computation of the controller (indirect)
• Re-parametrized plant model estimation (direct)

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Model Reference Adaptive Control

[ ]

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

1

= + − + = +

− ∞ → ∞ →

t y t y q P t

t t

ε ) ( ˆ ) ( ) ( ˆ ) 1 ( ) ( ˆ ) 1 ( * ) (

1 2

t b t y t r t u t b t Py t u − − − + =

[ ]

[ ]

) ( ), 1 ( ), ( ) ( ; ) ( ˆ ), ( ˆ ), ( ˆ ) ( ˆ ) ( ) ( ˆ ) 1 ( * ) (

2 1 1

t y t u t u t t r t b t b t t t t y q P

T T T

− = = = +

−

φ θ φ θ ) ( ) 1 ( * ) ( ) 1 ( ) (

1 1

t t y q P t y q P

Tφ

θ = + = +

− −

[ ]

) ( ) ( ˆ ) 1 ( t t t

Tφ

θ θ ε − = +

[ ]

) ( ) 1 ( ˆ ) 1 ( t t t

Tφ

θ θ ε + − = +

Use P.A.A. However one should show in addition that is bounded (i.e. plant input and output are bounded)

) (t φ

Objective: Adjustable controller: But for the correct values of controller parameters one has: And therefore one has: Define the a posteriori adaptation error:

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Plant model estimation + computation of the controller (indirect) Step 1 : Plant model estimation

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

2 1 1

t t u b t u b t y a t y

T Pφ

θ = − + + − = +

[ ]

[ ]

) ( ), 1 ( ), ( ) ( ; ) ( ˆ ), ( ˆ ), ( ˆ ) ( ˆ ) ( ) ( ˆ ) 1 ( ) ( ˆ ) ( ) ( ˆ ) ( ) ( ˆ ) 1 ( ˆ

1 2 1 2 1 1

t y t u t u t t a t b t b t t t t u t b t u t b t y t a t y

T T P T P

− = − = = − + + − = + φ θ φ θ

[ ]

) ( ) ( ˆ ) 1 ( ˆ ) 1 ( ) 1 ( t t t y t y t

T P P

φ θ θ ε − = + − + = +

[ ]

) ( ) 1 ( ˆ ) 1 ( ˆ ) 1 ( ) 1 ( t t t y t y t

T P P

φ θ θ ε + − = + − + = +

Use PAA Step 2 : Computation of the controller

[ ]

[ ]

) ( ˆ ), ( ˆ ), ( ˆ ) ( ˆ ; ) ( ), 1 ( ), ( ) ( ) ( ) ( ˆ ) 1 ( * ) (

2 1 1

t r t b t b t t y t u t u t t t t y q P

T T T

= − = = +

−

θ φ φ θ ) ( ˆ t

P

θ

From

) ( ˆ ) ( ˆ

1 1

t a p t r − =

Plant model (unknown): Adjustable predictor: a priori prediction error: a posteriori prediction error: Compute at each instant t : Adjustable controller:

In the general case d > 0 one will have to solve equation (**)

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Re-parametrized plant model estimation (direct)

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

1 2 1 1 1 1 1 2 1 1

t t y p t u b t u b t y a p t y p t y p t u b t u b t y a t y

Tφ

θ + − = − + + − + − = ± − + + − = +

{

r

[ ]

[ ]

) ( ), 1 ( ), ( ) ( ; ) ( ), ( ˆ ), ( ˆ ) ( ˆ ) ( ) ( ˆ ) ( ) 1 ( ) ( ˆ ) ( ) ( ˆ ) ( ) ( ˆ ) ( ) 1 ( ˆ

2 1 1 2 1 1

t y t u t u t t r t b t b t t t t y p t u t b t u t b t y t r t y p t y

T T T

− = = + − = − + + + − = + φ θ φ θ

Plant model (unknown): Re-parametrized adjustable predictor:

[ ]

) ( ) ( ˆ ) 1 ( ˆ ) 1 ( ) 1 ( t t t y t y t

Tφ

θ θ ε − = + − + = +

[ ]

) ( ) 1 ( ˆ ) 1 ( ˆ ) 1 ( ) 1 ( t t t y t y t

Tφ

θ θ ε + − = + − + = +

a priori prediction error: a posteriori prediction error:

Use PAA One estimates directly the parameters of the controller

[ ]

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

1

= + − + = +

− ∞ → ∞ →

t y t y q P t

t t

ε

One has:

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Adaptive tracking and regulation with independent objectives

• Easy generalization for the case d > 0
• Elegant and simple solution for adaptation (direct)
• Unfortunately restricted use in practice because it requires that

the plant zeros remains always stable and well damped

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Direct Adaptive Control – Simulations results Tracking 1 ) (

1 = −

q P ) 4 . 1 ( ) (

1 1 − −

− = q q P The choice of the poles for the closed loop (regulation) has a great influence upon adaptation transient behavior!

Parameters change + adaptation

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Direct Adaptive Control – Simulations results Regulation 1 ) (

1 = −

q P ) 4 . 1 ( ) (

1 1 − −

− = q q P The choice of the poles for the closed loop (regulation) has a great influence upon adaption transient behavior!

Constant plant parameters Parameters change + adaptation