Adaptive Control - GIPSA Lab

Adaptive Control – Landau,Lozano, M’Saad, Karimi1

Adaptive Control

Chapter 7: Digital Control Strategies


Chapter 7:Digital Control Strategies


r(t)

m

m

AB

TS1

ABq d−

R

u(t) y(t)

Controller

PlantModel

+

-

The R-S-T Digital Controller

Plant Model:)(

)(*)(

)()()(1

11

1

111

−

−−−

−

−−−− ===

qAqBq

qAqBqqHqG

dd

A

A

nn qaqaqA −−− +++= ...1)( 1

11 )(*...)( 111

11 −−−−− =++= qBqqbqbqB B

B

nn

R-S-T Controller: )()()1(*)()()( 111 tyqRdtyqTtuqS −−− −++=

Characteristic polynomial (closed loop poles):

)()()()()( 11111 −−−−−− += qRqBqqSqAqP d


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


Structure

)()()( 1

11

−

−−− =

qAqBqqH

d

AA

nn qaqaqA −−− +++= ...1)( 1

11 )(...)( 1*12

21

11 −−−−−− =+++= qBqqbqbqbqB B

B

nn

Plant:

) ------------1

(q- 1

)---------q

- dB

A

PLANT

R(q- 1

)

--------------------q

- dB(q

- 1)

(q- 1

)

S

P

r(t) y(t)+

-

T( ) q - 1

p(t)

++


)()()(

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

11

1111

111

−

−−−

−−−−−

−−−− =

+=

qPqBqTq

qRqBqqSqAqBqTqqH

d

d

d

BF

....1)()()()()( 22

11

11111 +++=+= −−−−−−−− qpqpqRqBqqSqAqP d

)()()(

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

1

11

1111

111

−

−−

−−−−−

−−− =

+=

qPqSqA

qRqBqqSqAqSqAqS

dyp

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


Choice of desired closed loop poles (polynomial P)

)()()( 111 −−− = qPqPqP FD

Dominant poles Auxiliary poles

Specificationin continuous time(tM, M)

2nd order (ω0, ζ)discretization

eT )( 1−qPD

5.125.0 0 ≤≤ eTω17.0 ≤≤ 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


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

)()()()()( 11111 −−−−−− =+ qPqRqBqqSqA d

? ?

)(deg 1−= qAnA )(deg 1−= qBnBA and B do not have

common factors

(Bezout)

unique minimal solution for :1)(deg 1 −++≤= − dnnqPn BAP

1)(deg 1 −+== − dnqSn BS 1)(deg 1 −== −AR nqRn

)(*1...1)( 1111

1 −−−−− +=++= qSqqsqsqS S

S

nn

R

R

nn qrqrrqR −−− ++= ...)( 1

101

(*)


Computation of R(q-1) and S(q-1)

Equation (*) is written as: Mx = p

],...,,,...,,1[ 01 RS nnT rrssx = ]0,...,0,,...,,...,,1[ 1 Pni

T pppp =

1 0 ... 0a1 1 .a2 0

1a1

anA a20 .

0 ... 0 anA

0 ... ... 0b' 1b' 2 b' 1. b' 2. .

b' nB .0 . . .0 0 0 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


Structure of R(q-1) and S(q-1)

R and S include pre-specified fixed parts (ex: integrator)

)()(')( 111 −−− = qHqRqR R )()(')( 111 −−− = qHqSqS S

HR, HS, - pre-specified polynomials

'

''...'')(' 1

101 R

R

nn qrqrrqR −−− ++= '

''...'1)(' 1

11 S

S

nn qsqsqS −−− ++=

•The pre specified filters HR and HS will allow to impose certain properties of the closed loop.•They can influence performance and/or robustness


Fixed parts (HR , HS). Examples

Zero steady state error (Syp should be null at certain frequencies)

Step disturbance : Sinusoidal disturbance : sS TqqH ωαα cos2;1 21 −=++= −−

11 1)( −− −= qqHS

Signal blocking (Sup should be null at certain frequencies)

sR TqqH ωββ cos2;1 21 −=++= −−

2,1;)1( 1 =+=Sinusoidal signal:Blocking at 0.5fS:

− nqH nR

)()()()(

)( 1

1111

−

−−−− ′

=qP

qSqHqAqS S

yp

)()()()()( 1

1111

−

−−−− ′

−=qP

qRqHqAqS Rup


Tracking (computation of T(q-1) )

------------q

- 1Bm

Am

Ideal case

r (t) y* (t)

desiredtrajectory for y (t)

t

y

r

*

Tracking referencemodel (Hm)

2nd order (ω0, ζ)discretization

sT )( 1−qHm

5.125.0 0 ≤≤ sTω17.0 ≤≤ ζ

)()(

)( 1

111

−

−−− =

qAqBq

qHm

mm

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

)()(

)()( 1

1)1(* tr

qAqBq

tym

md

−

−+−

=

...)( 110

1 ++= −− qbbqB mmm

...1)( 22

11

1 +++= −−− qaqaqA mmm

Specificationin continuous time(tM, M)


)()()(

)1( 1

1* tr

qAqB

dtym

m−

−

=++Build:

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

⎩⎨⎧

=≠

=0)1(1

0)1()1(/1Bsi

BsiBGT(q-1) = GP(q-1)

Particular case : P = Am⎪⎩

⎪⎨⎧

=

≠==−

0)1(1

0)1()1()1(

)( 1

Bsi

BsiBP

GqT

F.T. r y:)1(

)(

)()()(

1*

1

1)1(1

BqB

qAqBqqH

m

md

BF

−

−

−+−− ⋅=



Pole placement. Tracking and regulation

+

-

R

1 q-d

BAS

TA

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(*)()()()()( 111 ++=+ −−− dtyqTtyqRtuqS


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

1*1

−

−− −++=

qStyqRdtyqTtu

)1()()1()()()()()( *1*111 ++=++=+ −−−− dtyqTdtyqGPtyqRtuqS

)(1)( 1*11 −−− += qSqqS

)()()1()()1()()( 11**1 tyqRtuqSdtGyqPtu −−− −−−++=

)()()(

)1( 1

1* tr

qAqB

dtym

m−

−

=++

)(1)( 1*11 −−− += qAqqA mm

)()()()()1( 11** trqBdtyqAdty mm−− ++−=++

...)( 110

1 ++= −− qbbqB mmm ...1)( 22

11

1 +++= −−− qaqaqA mmm

Pole placement. Control law


Pole placement. Example

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

Bm(q-1) = 0.0927 + 0.0687 q-1Tracking dynamics

Am (q-1) = 1 - 1.2451q-1 + 0.4066 q-2Ts = 1s , ω0 = 0.5 rad/s, ζ = 0.9

Regulation dynamics P (q-1) = 1 - 1.3741 q-1 + 0.4867 q-2Ts = 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-2T(q-1) = 3.333 - 4.5806 q-1 + 1.6225 q-2

Gain margin : 2.703 Phase margin : 65.4 degModulus margin : 0.618 (- 4.19 dB) Delay margin: 2.1. s


Pole placement. Example

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Plant Output

Time (t/Ts)

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

Control Signal

Time (t/Ts)


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 zerosAllows exact achievement of imposed performances

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


-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Zero Admissible Zone

Real Axis

Imag

Axi

sf0/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




+

-

R

1 q-d

BAS

TA

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

)()()( 111 −−− = qPqPqP FD

)()()(

)1( 1

1* tr

qAqB

dtym

m−

−

=++Reference signal:(tracking)


T.F. of the closed loop without T:

)()()(

)()()()()()()( 11*

1*1

1

1

11*111

1*11

−−

−+−

−

+−

−−+−−−

−+−− ==

+=

qPqBqBq

qPq

qRqBqqSqAqBqqH

dd

d

d

CL

)()()()()()( 11*11*111 −−−−+−−− =+ qPqBqRqBqqSqA d

The following equation has to be solved :

S should be in the form: )()(...)( 11*110

1 −−−−− ′=+++= qSqBqsqssqS SS

nn

After simplification by B*,(*) becomes:)()()(')( 11111 −−+−−− =+ qPqRqqSqA d

(*)

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

11

101 ...)( −−

−−− ++= A

A

nn qrqrrqR d

d qsqsqS −−− ++= '...'1)(' 11

1

(**)

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


(**) is written as: Mx = p

1 0 a1 1 a2 a1 0: : 1ad ad-1 ... a1 1

ad+1 ad a1ad+2 ad+1 a2

. .0 0 ... 0 anA

0 . .

. .

1 . 0 .

. 00 0 1

nA + d + 1

nA + d + 1

d + 1 nA],...,,,,...,,1[ 1101 −′′= nd

T rrrssx ],...,,,...,,,1[ 121 dnnnT

AAApppppp ++=

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))



Closed loop T.F.: r y

)()()()(

)()(

)( 11

)1(11

1

1)1(1

−−

+−−−

−

−+−− ==

qPqAqqTqB

qAqBq

qHm

dm

m

md

BF

Desired T.F.

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

Controller equation:

)1()()()()()( *111 ++=+ −−− dtyqPtyqRtuqS

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

1*1

−

−− −++=

qStyqRdtyqPtu

[ ])()()1()()1()(1)( 11**1

1tyqRtuqSdtyqP

btu −−− −−−++= (s0 = b1)


Tracking and regulation with independent objectives. Examples

Plant : d = 0B(q-1) = 0.2 q-1 + 0.1 q-2

A(q-1) = 1 - 1.3 q-1 + 0.42 q-2Bm (q-1) = 0.0927 + 0.0687 q-1

Tracking dynamics Am (q-1) = 1 - 1.2451q-1 + 0.4066 q-2Ts = 1s , ω0 = 0.5 rad/s, ζ = 0.9

Regulation dynamics P (q-1) = 1 - 1.3741 q-1 + 0.4867 q-2Ts = 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-2T(q-1) = P(q-1)

Gain margin : 2.109 Phase margin : 65.3 degModulus margin : 0.526 (- 5.58 dB) Delay margin : 1.2


0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Plant Output

Time (t/Ts)

0 10 20 30 40 50 60 70 80 90 100-0.5

0

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 bereduced by introducing auxiliary poles


Internal model control -Tracking and regulationIt 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”


)()()()()()()( 1111111 −−−−−−−− ==+ qPqPqAqRqBqqSqA Fd

Dominant poles ( ) FPnF qqP 11 1)( −− += α

( typical choice)

(*)

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

)()()()( 1111 −−−−− =′+ qPqRqBqqS Fd

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

( ) )(1)( 111 −−− ′−= qSqqS (typical choice) Solution for:

)1()1()()( 11

BPqAqR F−− =

)1()1()()()()1()( 11111

BPqBqqPqSqqS Fd

F−−−−−− −=′−=

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


)1(/)()()( 111 BqPqAqT F−−− =

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

)1()1()1(

)1()( 1

BPA

TqT F==− (cancellation of the tracking reference model)



Interpretation of the internal model control

Equivalent scheme

( )

)()()1(

1)()1(

1)(

)()(

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

1111

0

11

0

111

0

−−−−

−−

−−−

==

=

==

qPqAB

qPB

qT

qPqS

qHforqABPqR

F

F

RF

T0

+

+

-

-1/S0 Plant

ABdq−

Model

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

(t)y

y(t)

R0

The plant model(prediction model)is an element of thecontrol scheme Feedback on the

Prediction error

Rem.: For all the strategies one can show the presence of the plant model in the controller


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()( 11 −− += qqPF α -1< α < 0

α = −0.1; −0.3; −0.333

Good value

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-30

-20

-10

0

b) Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

0 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

0a) Syb Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

α = -0.1α = -0.333Template for Modulus marginTemplate for Delay margin = Ts

Template for Delay margin = Tsα = -0.1α = -0.333α = -0.3


11 1)( −− += qqH 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-30

-20

-10

0

b) Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

0 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

0a) Syb Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

HR = 1, PF = 1 - 0.333q-1

HR = 1 + q-1, PF = 1

Template for Modulus marginTemplate for Delay margin = Ts

Template for Delay margin = TsHR = 1, PF = 1 - 0.333q-1

HR = 1 + q-1, PF = 1


Minimum Variance Tracking and Regulation


Disturbance RepresentationDeterministic disturbances

Stochastic DisturbancesCan 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 Diracpulse passed through a filter

rampstep Dirac pulse

Disturbance model

1

s1

s2

Disturbance model


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 measuredvalue (random variable)

• one can define a statistitics (mean value, variance) and probabilities of occurrence 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 allowsto give the probability of occurrence of a certain value (Gauss bell – App.A)

16h10h8h

1 day st

2 day nd

3 day rd


Discrete-time Gaussian White Noise

It is the fundamental generator signal{e(t)}: Sequence of independent equally distributed Gaussian randomvariables with rero mean and variance σ2 (0,σ ) standard deviation

{ } 0)(1lim)(..1

=== ∑=∞→

N

tNte

NteEVM

{ } 2

1

22 )(1lim)( σ=== ∑=

∞→

N

tN

teN

teEvar

Independence : The knowledge of e(i) does not allow to predictan approximation for e(i+1), e(i+2)….


Independence Test

Autocorrelation (covariance) function:

{ } )()(1lim)()()(1

iteteN

iteteEiRN

tN

−=−= ∑=

∞→

Rem.: R(0) = var = σ2

Normalized autocorrelation (covariance) function :

)1)0(()0()()( == RN

RiRiRN

Whiteness (independence test) :R(i)=RN(i) = 0 i= 1, 2, 3..-1, -2…White Noise

SpectralDensity

1

RN

i

o 1 2 3 4-1-2

NormalizedAutocorrelations

0.5 ƒ s

ƒ

- - - - - - - - - R(0)2 π


Moving Average Process – MA

e(t)1+ c

1q -1

y(t) )()1()1()()( 111 teqctectety −+=−+=

{ } ∑∑∑===

=−+===N

t

N

t

N

t

teN

cteN

tyN

tyEMV1

111

0)1(1)(1)(1)(..

{ } 221

1

22 )1()(1)()0( σctyN

tyERN

ty +=∑==

=

{ } 221

1

21

1

)(1)1()(1)1()()1( σctecN

tytyN

tytyERN

t

N

ty ==−=−= ∑∑

==

0..)3()2( === yy RR

0 +1 2-1-2

-----------c1

1+c12

(c >0)1RN1

i


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

1

teqCitectetycn

ii

−

=

=−+= ∑)(*11)( 11

1

1 −−

=

−− +=∑+= qCqqcqCcn

i

ii

)1(10)( +−≥+≥= CC niniiR

πσ

πσωφ ωωω

2)(

2)()()(

222jjj

y eCeCeC == −

Spectral density:

ωφφ jey ezzzCzCz == − ;)()()()( 1

Relationship spectral density/transfer function:

eφ

Moving Average Process – MA


------------------1

1+ a1 q- 1

y(t)e(t)

-------------1

A( q-1)

y(t)e(t)

Auto-regressive Process – AR

11

)()()1()( 111

1 <+

=+−−=−

aqa

tetetyaty

)()1()(1

tetyatyAn

ii +−−= ∑

=

)(*11)( 11

1

1 −−

=

−− +=∑+= qAqqaqAAn

i

ii

)()()( 1 tetyqA =−

Asymptotically stable

Spectral density:

)()(

1)(

1)( 1 zzAzA

z ey φφ−

=ωφωφ jezyy z

== )()(


Auto-regressive Moving Average Process - ARMA

C(q )-1

A(q )-1

y(t)e(t) )()()()(11

teitecityatyCA n

ii

n

ii +−+−−= ∑∑

==

)()()()( 11 teqCtyqA −− =

Asymptotically stable

)()()(

)()()( 1

1z

zAzC

zAzCz ey φφ ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−

Spectral density:


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

1te

qAqCtu

qAqBqty

d

−

−

−

−−

+=

Disturbance

)()()()()()( 111 teqCtuqBqtyqA d −−−− +=

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

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

11*1*1 11

++−+−=+

++∑ ∑ −++−−+∑ +−+−=+

−−−= ==

teqCdtuqBtyqAty

teitecidtubityatyB CA n

i

n

iii

n

ii

Example: 0;1;1;1 ==== dnnn CBA

)1()()()()1( 111 ++++−=+ tetectubtyaty

Remark: in general nC = nA


Generality of the ARMAX Process

q- d

B1

A1

------C2

A2

y(t)

e(t)

u(t) ++

)()()()(

)()()( 1

2

12

11

11 te

qAqCtu

qAqBqty

d

−

−

−

−−+=

)()()()()(21

12

21

21 teACtu

ABqte

AAACtu

AAABqty

dd+=+=

−−

122121 ;; ACCABBAAA ===


Optimal Prediction

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

Prediction error: )1(ˆ)1()1( +−+=+ tytytε

Objective: ),...)1(),(),...,1(),(()1(ˆ)/1(ˆ −−=+=+ tututytyftytty

such that : [ ]{ } min)1(ˆ)1( 2 =+−+ tytyE)1()()()()1( 111 ++++−=+ tetectubtyaty

)1()]1(ˆ)()()([)1(ˆ)1()1( 111 +++−++−=+−+=+ tetytectubtyatytytε

[ ]{ } [ ]{ } { }{ })]()()()[1(2

)1()1(ˆ)()()( )1(ˆ)1(

111

22111

2

tectubtyateEteEtytectubtyaEtytyE

++−+++++−++−=+−+

= 0

{

[ ]{ } 0)1(ˆ)()()( 2111 =+−++− tytectubtyaEOptimality condition:

)()()()1(ˆ 111 tectubtyaty opt ++−=+ )1()1(ˆ)1()1( +=+−+=+ tetytyt optoptε

Example :

)()( tet =ε )()()()1(ˆ 111 tctubtyaty opt ε++−=+


Optimal prediction

)1()()()()()()1( 11*1* ++−+−=+ −−− teqCdtuqBtyqAty

)()()()()()()1(ˆ 1*1*1* teqCdtuqBtyqAty −−− +−+−=+

)1()1(ˆ)1()1( +=+−+=+ tetytyt optε

)()()()()()()1(ˆ 1*1*1* tqCdtuqBtyqAty ε−−− +−+−=+

ARMAX:

Optimal predictor (theoretical):

Prediction error:

Optimal predictor (implementation):

One replaces the unknown white noise by the prediction error



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

Objective: minimization of the output variance (standard deviation)

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

2 *2 * =∑ −≈⎭⎬⎫

⎩⎨⎧ −=

=

N

ttyty

NtytyEtuJ

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

minimum variance control

y

y

**

t t

y

y

%

minimumvalue

controlled output

largevariance

reference

reference

smallvariance


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)]}

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

Control law (theoretical):

121

11* )()()1()(

−+

+−+=

qbbtyatectytu y(t+1) - y*(t+1) = e(t+1) y(t) - y*(t) = e(t)

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

1*1

121

11*1

1−

−−

−

− −+=

+−−++

=qS

tyqRtyqTqbb

tyactyqctu

Control law (implementation):

Same control law as for « Tracking and regulation with independent objectives »by taking P(q-1) = C(q-1)



Closed Loop Poles

Disturbance

T

y *(t+d+1)1

R

+

-PLANT

u(t)

S ---------q

- dB

A

y(t)

-----C

A

+

+

e(t)

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

1*1

−

−− −+=

qStyqRtyqTtu

)()()()()()()( 11*111

1*)1(11

−−−−−

−+−−−

+=

qRqBqqSqAqBqqTqH

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

11

11

111

111

)()(

)()()()( −

−

−−

−−−

−−− ==

+= q

qCqqT

qRqqAqqTqHBF

Closed loop poles

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


Minimum Variance Tracking and Regulation – general case

Same computations as for « tracking and regulation with independentobjectives » by taking P(q-1) = C(q-1) (see Chapter 3)

r(t)

DISTURBANCE

------Bm

AmT

y *(t+d+1)1

R

+

-PLANT

- - - - - - - - - - -----------q -(d+1)

C(q-1)

q- (d+1)

----------------------------q

- (d+1)

u(t)

S ---------q

- dB

A

y(t)

-----CA

+

+

e(t)

Am(q-1)

Bm(q-1)


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

1*1

−

−− −++=

qStyqRdtyqTtu

)(')(*)(;)()( 11111 −−−−− == qSqBqSqCqT

)()()(*)(')( 111)1(11 −−−+−−− =+ qCqRqBqqSqA d

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

)1()()1()1( 1* ++′=++−++ − dteqSdtydtyPrediction error : MA of order d

{

[ ] [ ]∑ =−−−⋅−==

N

tiityitytyty

NiR

1

** ,...2,1,0)()()()(1)(

≈≥

10)( +≥≈ diiRN 117.02)( +≥≤ diNiRN

Optimality test:

theoretical pratical

Minimum Variance Tracking and Regulation – General Case


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-2Disturbance polynomial C(q-1) = 1 -1.34 q-1 + 0.49 q-2Pre-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 degModulus margin: 0.520 (- 5.68 dB) Delay margin: 1.3 s

Minimum Variance Tracking and Regulation. Example


Poursuite et régulation à variance minimale.Exemple

Attention: For robustness and actuator stress one may be obligedto add auxiliary poles (see book pg. 190)

50 100 150 200 250 300 350 400

-0.2

0

0.2

0.4

0.6

0.8

1

Process Output

Controller OFF Controller ON

Am

plitu

de MeasureReference

50 100 150 200 250 300 350 400-1

-0.5

0

0.5

1

Control Signal

Am

plitu

de

Time (t/Ts)

Minimum Variance Tracking and Regulation. Example


Minimum Variance Tracking and RegulationThe 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)


)()()( 111* −−−+− = qBqBqB

)(' 1−− qB Reciprocical polynomial (stable) of )( 1−− qB

)()()()(

)()(')()(11*)1(11

1111

−−+−−−

−−−−+−

+=

=

qRqBqqSqA

qCqBqBqPd

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)


Generalized Minimum Variance Tracking and Regulation

Criterion:

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

2

1

1* =

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡+++−++ −

−tu

qCqQdtydtyE

1

11

1)1()( −

−−

+−

=qqqQ

αλ

Particular case : α = 0

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

)1(2

*1

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡++−−−+++

−dtytutu

qCdtyE λ

Controllere:)()(

)()()1()()( 11

1*1

−−

−−

+−++

=qQqS

tyqRdtyqCtu

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

Weighting the control variations


Generalized minimum variance tracking and regulation

Design:• One computes a minimum variance tracking/regulation controllerwithout 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 areseveral 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

Am

1

S +

y(t)

PLANT

q- d

B

A

CA

e(t)

+

+

Adaptive Control - GIPSA Lab

Documents

Transcript of Adaptive Control - GIPSA Lab