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

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Adaptive Control

Chapter 7: Digital Control Strategies

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

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Chapter 7: Digital Control Strategies

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

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r(t)

m m

A B T S 1 A B q

d −

R

u(t) y(t) Controller

Plant Model

+

• The R-S-T Digital Controller

Plant Model: ) ( ) ( * ) ( ) ( ) ( ) (

1 1 1 1 1 1 1 − − − − − − − − −

= = = q A q B q q A q B q q H q G

d d

A A

n n q

a q a q A

− − −

+ + + = ... 1 ) (

1 1 1

) ( * ... ) (

1 1 1 1 1 − − − − −

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

B B

n n

R-S-T Controller: ) ( ) ( ) 1 ( * ) ( ) ( ) (

1 1 1

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

− − −

− + + = Characteristic polynomial (closed loop poles): ) ( ) ( ) ( ) ( ) (

1 1 1 1 1 − − − − − −

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

d

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)

The digital PID can be designed using pole placement

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 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

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 1 − − − −

= q A q B q 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 si B si B G T(q-1) = GP(q-1) Particular case : P = Am

⎪ ⎩ ⎪ ⎨ ⎧ = ≠ = =

−

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

1

B si B si 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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Pole placement. Example

Plant : d=0 B(q-1) = 0.1 q-1 + 0.2 q-2 A(q-1) = 1 - 1.3 q-1 + 0.42 q-2 Bm(q-1) = 0.0927 + 0.0687 q-1 Tracking dynamics Am (q-1) = 1 - 1.2451q-1 + 0.4066 q-2 Ts = 1s , ω0 = 0.5 rad/s, ζ = 0.9 Regulation dynamics P (q-1) = 1 - 1.3741 q-1 + 0.4867 q-2 Ts = 1s , ω0 = 0.4 rad/s, ζ = 0.9 Pre-specifications : Integrator *** CONTROL LAW *** S (q-1) u(t) + R (q-1) y(t) = T (q-1) y*(t+d+1) y*(t+d+1) = [Bm(q-1)/Am(q-1)] r(t) Controller : R(q-1) = 3 - 3.94 q-1 + 1.3141 q-2 S(q-1) = 1 - 0.3742 q-1 - 0.6258 q-2 T(q-1) = 3.333 - 4.5806 q-1 + 1.6225 q-2 Gain margin : 2.703 Phase margin : 65.4 deg Modulus margin : 0.618 (- 4.19 dB) Delay margin: 2.1. s

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

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Pole placement. Example

10 20 30 40 50 60 70 80 90 100 0.2 0.4 0.6 0.8 1

Plant Output Time (t/Ts)

10 20 30 40 50 60 70 80 90 100 0.2 0.4 0.6 0.8 1 1.2

Control Signal Time (t/Ts)

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

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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 Does not tolerate fractional delay > 0.5 TS (unstable zero)

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

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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

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

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Tracking and regulation with independent objectives

+

• 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 − − −

= 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)

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

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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))

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

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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 – same as for pole placement Regulation ( computation of R(q-1) and S(q-1))

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

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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

+ + = +

− − −

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

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

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Tracking and regulation with independent objectives. Examples

Plant : d = 0 B(q-1) = 0.2 q-1 + 0.1 q-2 A(q-1) = 1 - 1.3 q-1 + 0.42 q-2 Bm (q-1) = 0.0927 + 0.0687 q-1 Tracking dynamics Am (q-1) = 1 - 1.2451q-1 + 0.4066 q-2 Ts = 1s , ω0 = 0.5 rad/s, ζ = 0.9 Regulation dynamics P (q-1) = 1 - 1.3741 q-1 + 0.4867 q-2 Ts = 1s , ω0 = 0.4 rad/s, ζ = 0.9 Pre-specifications : Integrator *** CONTROL LAW *** S (q-1) u(t) + R (q-1) y(t) = T (q-1) y*(t+d+1) y*(t+d+1) = [Bm (q-1)/Am (q-1)] . r(t) Controller : R(q-1) = 0.9258 - 1.2332 q-1 + 0.42 q-2 S(q-1) = 0.2 - 0.1 q-1 - 0.1 q-2 T(q-1) = P(q-1) Gain margin : 2.109 Phase margin : 65.3 deg Modulus margin : 0.526 (- 5.58 dB) Delay margin : 1.2

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

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10 20 30 40 50 60 70 80 90 100 0.2 0.4 0.6 0.8 1

Plant Output Time (t/Ts)

10 20 30 40 50 60 70 80 90 100

• 0.5

0.5 1 1.5

Control Signal Time (t/Ts)

Tracking and regulation with independent objectives. (d = 0)

Effect of low damped zeros The oscillations on the control input when there are low damped zeros can be reduced by introducing auxiliary poles

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

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Internal model control -Tracking and regulation It is a particular case of the pole placement The dominant poles are those of the plant model Allows to design a RST controller for:

• well damped stable systems
• without restrictions upon the degrees of the polynomial A and B
• without restrictions upon the delay of the discrete time model

The plant model should be stable and well damped ! Does not allow to accelerate the closed loop response Often used for the systems featuring a large delay

Remark: The name is misleading since it has nothing in common with the “internal model principle”

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

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

1 1 1 1 1 1 1 − − − − − − − −

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

F d

Dominant poles

( )

F P

n F

q q P

1 1

1 ) (

− −

+ = α ( typical choice)

(*) R should be in the form : R(q-1) = A(q-1).R’(q-1) ) ( ) ( ) ( ) (

1 1 1 1 − − − − −

= ′ + q P q R q B q q S

F d

After the cancellation of the common factor A(q-1),(*) becomes: ( )

) ( 1 ) (

1 1 1 − − −

′ − = q S q q S (typical choice)

Solution for: ) 1 ( ) 1 ( ) ( ) (

1 1

B P q A q R

F − −

= ) 1 ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) (

1 1 1 1 1

B P q B q q P q S q q S

F d F − − − − − −

− = ′ − = Regulation (computation of R(q-1) and S(q-1))

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

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) 1 ( / ) ( ) ( ) (

1 1 1

B q P q A q T

F − − −

=

Particular case : Am = APF (tracking dynamics = regulation dynamics)

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

1

B P A T q T

F

= =

−

(cancellation of the tracking reference model)

Tracking (computation of T(q-1) )

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

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Interpretation of the internal model control

Equivalent scheme

( )

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

1 1 1 1 1 1 1 1 1 − − − − − − − − −

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

F F R F

T0

+ +

• 1/S0

Plant

A B d q−

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

(t) y ˆ

y(t) R0 The plant model (prediction model) is an element of the control scheme Feedback on the Prediction error Rem.: For all the strategies one can show the presence of the plant model in the controller

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

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Internal model control of a system with large delay Plant: d = 7; A = 1 – 0.2q-1 ; B = q-1

The « delay margin » can be satisfied by introducing auxiliary poles

) 1 ( ) (

1 1 − −

+ = q q PF α

• 1< α < 0

α = −0.1; −0.3; −0.333 Good value

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

• 30
• 20
• 10

b) Syp Magnitude Frequency Responses Frequency (f/fs) Magnitude (dB)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

• 10
• 8
• 6
• 4
• 2

a) Syb Magnitude Frequency Responses Frequency (f/fs) Magnitude (dB)

α = -0.1 α = -0.333

Template for Modulus margin Template for Delay margin = Ts Template for Delay margin = Ts

α = -0.1 α = -0.333 α = -0.3

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

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1 1

1 ) (

− −

+ = q q H R

corresponds to the opening of the loop at 0.5fS

Internal model control of a system with large delay

See also: I.D. Landau (1995) : Robust digital control of systems with time delay (the Smith predictor revisited)

• Int. J. of Control, v.62,no.2 pp 325-347

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

• 30
• 20
• 10

b) Syp Magnitude Frequency Responses Frequency (f/fs) Magnitude (dB)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

• 10
• 8
• 6
• 4
• 2

a) Syb Magnitude Frequency Responses Frequency (f/fs) Magnitude (dB)

HR = 1, PF = 1 - 0.333q-1 HR = 1 + q-1, PF = 1 Template for Modulus margin Template for Delay margin = Ts Template for Delay margin = Ts HR = 1, PF = 1 - 0.333q-1 HR = 1 + q-1, PF = 1

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

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Minimum Variance Tracking and Regulation

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

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Disturbance Representation Deterministic disturbances Stochastic Disturbances Can not be described in a deterministic way, since they are not reproducible. Most of the stochastic disturbances can be described as: A white noise passed through a filter.

In a stochastic environment, the white noise play the role of the Dirac Pulse.

Can be described as a Dirac pulse passed through a filter

ramp step Dirac pulse

Disturbance model

1 s 1 s

2

Disturbance model

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

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Stochastic (random) Process

Example: record of a controlled variable in regulation (1 day)

• each evolution can be described by a different f(t)

(stochastic realization)

• for a fixed time (ex.: 10h) for each experiment (day) one gets a different measured

value (random variable)

• one can define a statistitics (mean value, variance) and probabilities of
• ccurrence of the various values
• if the stochastic process is ergodic the statistics over one experiment are significant
• if the stochastic process is gaussian the knowledge of the m.v. and variance allows

to give the probability of occurrence of a certain value (Gauss bell – App.A)

16h 10h 8h 1 day st 2 day nd 3 day rd

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

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Discrete-time Gaussian White Noise It is the fundamental generator signal {e(t)}: Sequence of independent equally distributed Gaussian random variables with rero mean and variance σ2 (0,σ ) standard deviation

{ }

) ( 1 lim ) ( . .

1

= = =

∑

= ∞ → N t N

t e N t e E V M

{ }

2 1 2 2

) ( 1 lim ) ( σ = = =

∑

= ∞ → N t N

t e N t e E var Independence : The knowledge of e(i) does not allow to predict an approximation for e(i+1), e(i+2)….

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

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Independence Test Autocorrelation (covariance) function:

{ }

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

1

i t e t e N i t e t e E i R

N t N

− = − =

∑

= ∞ →

Rem.: R(0) = var = σ2 Normalized autocorrelation (covariance) function : ) 1 ) ( ( ) ( ) ( ) ( = = RN R i R i RN Whiteness (independence test) :R(i)=RN(i) = 0 i= 1, 2, 3..-1, -2…

White Noise Spectral Density

1 RN i

• 1

2 3 4

• 1
• 2

Normalized Autocorrelations

0.5 ƒ

s

ƒ

• -
• R(0)

2 π

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

37

Moving Average Process – MA

e(t) 1+ c1 q-1 y(t)

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

1 1 1

t e q c t e c t e t y

−

+ = − + =

{ }

∑ ∑ ∑

= = =

= − + = = =

N t N t N t

t e N c t e N t y N t y E M V

1 1 1 1

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

{ }

2 2 1 1 2 2

) 1 ( ) ( 1 ) ( ) ( σ c t y N t y E R

N t y

+ = ∑ = =

=

{ }

2 2 1 1 2 1 1

) ( 1 ) 1 ( ) ( 1 ) 1 ( ) ( ) 1 ( σ c t e c N t y t y N t y t y E R

N t N t y

= = − = − =

∑ ∑

= =

.. ) 3 ( ) 2 ( = = =

y y

R R

+1 2

• 1
• 2
• c1

1+c1

2

(c >0)

1

RN

1 i

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

38

e(t) y(t)

• 1

C(q )

) ( ) ( ) ( ) ( ) (

1 1

t e q C i t e c t e t y

c

n i i − =

= − + =

∑

) ( * 1 1 ) (

1 1 1 1 − − = − −

+ = ∑ + = q C q q c q C

c

n i i i

) 1 ( 1 ) ( + − ≥ + ≥ =

C C

n i n i i R π σ π σ ω φ

ω ω ω

2 ) ( 2 ) ( ) ( ) (

2 2 2 j j j y

e C e C e C = =

−

Spectral density:

ω

φ φ

j e y

e z z z C z C z = =

−

; ) ( ) ( ) ( ) (

1

Relationship spectral density/transfer function:

e

φ Moving Average Process – MA

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

39

• 1

1+ a1 q

• 1

y(t) e(t)

• 1

A( q-1) y(t) e(t)

Auto-regressive Process – AR 1 1 ) ( ) ( ) 1 ( ) (

1 1 1 1

< + = + − − =

−

a q a t e t e t y a t y ) ( ) 1 ( ) (

1

t e t y a t y

A

n i i

+ − − = ∑

=

) ( * 1 1 ) (

1 1 1 1 − − = − −

+ = ∑ + = q A q q a q A

A

n i i i

) ( ) ( ) (

1

t e t y q A =

− Asymptotically stable

Spectral density: ) ( ) ( 1 ) ( 1 ) (

1

z z A z A z

e y

φ φ

−

=

ω

φ ω φ

j

e z y y

z

=

= ) ( ) (

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

40

Auto-regressive Moving Average Process - ARMA

C(q )

• 1

A(q )

• 1

y(t) e(t)

) ( ) ( ) ( ) (

1 1

t e i t e c i t y a t y

C A

n i i n i i

+ − + − − =

∑ ∑

= =

) ( ) ( ) ( ) (

1 1

t e q C t y q A

− −

=

Asymptotically stable

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

1 1

z z A z C z A z C z

e y

φ φ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

− −

Spectral density:

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

41

ARMAX Process (A.R.M.A. with « exogenous » input))

• q
• d

B(q

• 1

) A(q

• 1

) y(t)

• C(q
• 1

) A(q

• 1

)

+ +

u(t) e(t)

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

1 1 1 1

t e q A q C t u q A q B q t y

d − − − − −

+ =

Disturbance

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

1 1 1

t e q C t u q B q t y q A

d − − − −

+ = ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (

1 1 * 1 * 1 1 1

+ + − + − = + + + ∑ ∑ − + + − − + ∑ + − + − = +

− − − = = =

t e q C d t u q B t y q A t y t e i t e c i d t u b i t y a t y

B C A

n i n i i i n i i

Example: ; 1 ; 1 ; 1 = = = = d n n n

C B A

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

1 1 1

+ + + + − = + t e t e c t u b t y a t y Remark: in general nC = nA

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

42

Generality of the ARMAX Process

q

• d

B 1 A 1

• C2

A 2 y(t) e(t) u(t)

+ +

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

1 2 1 2 1 1 1 1

t e q A q C t u q A q B q t y

d − − − − −

+ = ) ( ) ( ) ( ) ( ) (

2 1 1 2 2 1 2 1

t e A C t u A B q t e A A A C t u A A A B q t y

d d

+ = + =

− − 1 2 2 1 2 1

; ; A C C A B B A A A = = =

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

43

Optimal Prediction = + ) / 1 ( ˆ t t y

Prediction of y(t+1) based on the measures of u and y available up to t

Prediction error:

) 1 ( ˆ ) 1 ( ) 1 ( + − + = + t y t y t ε

Objective: ),...) 1 ( ), ( ),..., 1 ( ), ( ( ) 1 ( ˆ ) / 1 ( ˆ − − = + = + t u t u t y t y f t y t t y such that :

[ ]

{ } min

) 1 ( ˆ ) 1 (

2 =

+ − + t y t y E

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

1 1 1

+ + + + − = + t e t e c t u b t y a t y ) 1 ( )] 1 ( ˆ ) ( ) ( ) ( [ ) 1 ( ˆ ) 1 ( ) 1 (

1 1 1

+ + + − + + − = + − + = + t e t y t e c t u b t y a t y t y t ε

[ ]

{ }

[ ]

{ } { }

{ }

)] ( ) ( ) ( )[ 1 ( 2 ) 1 ( ) 1 ( ˆ ) ( ) ( ) ( ) 1 ( ˆ ) 1 (

1 1 1 2 2 1 1 1 2

t e c t u b t y a t e E t e E t y t e c t u b t y a E t y t y E + + − + + + + + − + + − = + − + = 0

{

[ ]

{ }

) 1 ( ˆ ) ( ) ( ) (

2 1 1 1

= + − + + − t y t e c t u b t y a E Optimality condition: ) ( ) ( ) ( ) 1 ( ˆ

1 1 1

t e c t u b t y a t y

• pt

+ + − = + ) 1 ( ) 1 ( ˆ ) 1 ( ) 1 ( + = + − + = + t e t y t y t

• pt
• pt

ε Example : ) ( ) ( t e t = ε ) ( ) ( ) ( ) 1 ( ˆ

1 1 1

t c t u b t y a t y

• pt

ε + + − = +

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

44

Optimal prediction ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 (

1 1 * 1 *

+ + − + − = +

− − −

t e q C d t u q B t y q A t y ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ˆ

1 * 1 * 1 *

t e q C d t u q B t y q A t y

− − −

+ − + − = +

) 1 ( ) 1 ( ˆ ) 1 ( ) 1 ( + = + − + = + t e t y t y t

• pt

ε

) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ˆ

1 * 1 * 1 *

t q C d t u q B t y q A t y ε

− − −

+ − + − = +

ARMAX: Optimal predictor (theoretical): Prediction error: Optimal predictor (implementation):

One replaces the unknown white noise by the prediction error

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

45

Minimum Variance Tracking and Regulation

• random disturbances
• the discrete time plant model has stable zeros

Objective: minimization of the output variance (standard deviation)

[ ] [ ]

min ) ( ) ( 1 ) ( ) ( )) ( (

1 2 * 2 *

= ∑ − ≈ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − =

= N t

t y t y N t y t y E t u J

• A model for the disturbance has to be considered
• Plant + disturbance: ARMAX model

minimum variance control

y y

* *

t t y y

%

minimum value controlled

• utput

large variance reference reference small variance

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

46

y(t+1) = - a1 y(t) + b1 u(t) + b2 u(t-1)+ c1 e(t) + e(t+1) Plant + disturbance: Reference trajectory: y*(t+1) Criterion computation: E{[y(t+1) - y*(t+1)]2} = E{[-a1y(t) + b1u(t) + b2u(t-1) + c1e(t) - y*(t+1)]2} +E{e2(t+1)}+2E{e(t+1)[- a1y(t) + b1u(t) + b2u(t-1) + c1e(t) - y*(t+1)]}

{

= 0 Optimality condition: E{[-a1y(t) + b1u(t) + b2u(t-1) + c1e(t) - y*(t+1)]2} = 0 Control law (theoretical):

1 2 1 1 1 *

) ( ) ( ) 1 ( ) (

−

+ + − + = q b b t y a t e c t y t u y(t+1) - y*(t+1) = e(t+1) y(t) - y*(t) = e(t) ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) 1 ( ) (

1 1 * 1 1 2 1 1 1 * 1 1 − − − − −

− + = + − − + + = q S t y q R t y q T q b b t y a c t y q c t u Control law (implementation): Same control law as for « Tracking and regulation with independent objectives » by taking P(q-1) = C(q-1)

Minimum Variance Tracking and Regulation

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

47

Closed Loop Poles

Disturbance

T y *(t+d+1) 1 R

+

• PLANT

u(t) S

• q
• d

B A y(t)

• C

A

+ +

e(t)

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

1 1 * 1 − − −

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

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

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

d BF

A(q-1) = 1 + a1 q-1 ; B(q-1) = q-1 B*(q-1) ; B*(q-1) = b1 + b2 q-1 ; d = 0 T(q-1) = C(q-1) = 1 + c1 q-1; S(q-1) = B*(q-1) = b1 + b2 q-1 ; R(q-1) = r0 = c1 - a1

1 1 1 1 1 1 1 1 1 1

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

− − − − − − − − − −

= = + = q q C q q T q R q q A q q T q HBF Closed loop poles

The disturbance model (C(q-1)) defines the closed loop poles and therefore the regulation performance

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

48

Minimum Variance Tracking and Regulation – general case Same computations as for « tracking and regulation with independent

• bjectives » by taking P(q-1) = C(q-1) (see Chapter 3)

r(t)

DISTURBANCE

• Bm

Am T y *(t+d+1) 1 R

+

• PLANT
• q
• (d+1)

C(q-1) q

• (d+1)
• q
• (d+1)

u(t) S

• q
• d

B A y(t)

• C

A

+ +

e(t) Am(q-1) Bm(q-1)

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

49

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

1 1 * 1 − − −

− + + = q S t y q R d t y q T t u ) ( ' ) ( * ) ( ; ) ( ) (

1 1 1 1 1 − − − − −

= = q S q B q S q C q T ) ( ) ( ) ( * ) ( ' ) (

1 1 1 ) 1 ( 1 1 − − − + − − −

= + q C q R q B q q S q A

d Solving with predisol.sci(.m) or with WinReg (Adaptech)

) 1 ( ) ( ) 1 ( ) 1 (

1 *

+ + ′ = + + − + +

−

d t e q S d t y d t y

Prediction error :

MA of order d

{

[ ] [ ]

∑ = − − − ⋅ − =

= N t

i i t y i t y t y t y N i R

1 * *

,... 2 , 1 , ) ( ) ( ) ( ) ( 1 ) (

≈ ≥

1 ) ( + ≥ ≈ d i i RN 1 17 . 02 ) ( + ≥ ≤ d i N i RN Optimality test:

theoretical pratical

Minimum Variance Tracking and Regulation – General Case

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

50

Plant:

• d = 0
• B(q-1) = 0.2 q-1 + 0.1 q-2
• A(q-1) = 1 - 1.3 q-1 + 0.42 q-2

Tracking dynamics Ts = 1s, ω0 = 0.5 rad/s, ζ = 0.9

• Bm = +0.0927 +0.0687 q-1
• Am = 1 - 1.2451 q-1 + 0.4066 q-2

Disturbance polynomial C(q-1) = 1 -1.34 q-1 + 0.49 q-2 Pre-specifications: Integrator *** CONTROL LAW *** S(q-1) u(t) + R(q-1) y(t) = T(q-1) y*(t+d+1) y*(t+d+1) = [(Bmq-1)/Am(q-1)] . ref(t) Controller:

• R(q-1) = 0.96 - 1.23 q-1 + 0.42 q-2
• S(q-1) = 0.2 - 0.1 q-1 - 0.1 q-2
• T(q-1) = C(q-1)

Gain margin: 2.084 Phase margin: 61.8 deg Modulus margin: 0.520 (- 5.68 dB) Delay margin: 1.3 s

Minimum Variance Tracking and Regulation. Example

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

51

Poursuite et régulation à variance minimale.Exemple Attention: For robustness and actuator stress one may be obliged to add auxiliary poles (see book pg. 190)

50 100 150 200 250 300 350 400

• 0.2

0.2 0.4 0.6 0.8 1

Process Output Controller OFF Controller ON Amplitude

Measure Reference 50 100 150 200 250 300 350 400

• 1
• 0.5

0.5 1

Control Signal Amplitude Time (t/Ts)

Minimum Variance Tracking and Regulation. Example

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

52

Minimum Variance Tracking and Regulation The case of unstable zeros In this case minimum variance control can not be applied Solutions:

• Use of pole placement with a special choice of the closed loop poles
• Generalized minimum variance tracking and regulation

(modified criterion)

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

53

) ( ) ( ) (

1 1 1 * − − − + −

= q B q B q B ) ( '

1 − − q

B Reciprocical polynomial (stable) of ) (

1 − − q

B ) ( ) ( ) ( ) ( ) ( ) ( ' ) ( ) (

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

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

d

Use of pole placement

Unstable factor Closed loop poles

For details and examples, see book pg.192-195

(one reverses the order of the coefficients)

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

54

Generalized Minimum Variance Tracking and Regulation Criterion:

min ) ( ) ( ) ( ) 1 ( ) 1 (

2 1 1 *

= ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + − + +

− −

t u q C q Q d t y d t y E

1 1 1

1 ) 1 ( ) (

− − −

+ − = q q q Q α λ Particular case : α = 0

min ) 1 ( )] 1 ( ) ( [ ) ( ) 1 (

2 * 1

= ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + − − − + + +

−

d t y t u t u q C d t y E λ

Controllere: ) ( ) ( ) ( ) ( ) 1 ( ) ( ) (

1 1 1 * 1 − − − −

+ − + + = q Q q S t y q R d t y q C t u

Allows to stabilize the controller and the system (but not always!) Weighting the control variations

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

55

Generalized minimum variance tracking and regulation Design:

• One computes a minimum variance tracking/regulation controller

without taking in account the unstable nature of B. (Q(q-1)=0)

• One introduces Q(q-1) and search for λ > 0 wich stabilizes the

controller and the closed loop A solution does not always exists in particular when there are several unstable zeros

Q

u(t)

R T

+

• y*(t+d+1)

q

• d B(q
• 1)

A(q

• 1)Q(q
• 1) + B*(q
• 1) P(q
• 1)

r(t)

Bm A m 1 S +

y(t) PLANT

q

• d

B A C A

e(t) + +