robust digital controller design methodsioandore.landau/adaptivecontrol... · i.d. landau, g. zito...

I.D. Landau, G. Zito - "Digital Control Systems" - Chapter 3 1

Chapter III

Robust Digital Controller Design Methods


Chapter 3. Robust Digital Controller Design Methods

3.1 Introduction3.2 Digital PID Controller

3.2.1 Structure of the Digital PID 1 Controller3.2.2 Design of the Digital PID 1 Controller3.2.3 Digital PID 1 Controller: Examples3.2.4 Digital PID 2 Controller3.2.5 Effect of Auxiliary Poles 3.2.6 Digital PID Controller: Conclusions

3.3 Pole Placement3.3.1 Structure3.3.2 Choice of the Closed Loop Poles (P(q-1))3.3.3 Regulation (Computation of R(q-1) and S(q-1))3.3.4 Tracking (Computation of T(q-1))3.3.5 Pole Placement: Examples

3.4 Tracking and Regulation with Independent Objectives3.4.1 Structure 3.4.2 Regulation (Computation of R(q-1) and S(q-1))3.4.3 Tracking (Computation of T(q-1))3.4.4 Tracking and Regulation with Independent Objectives:

Examples


Chapter 3. Robust Digital Controller Design Methods

3.5 Internal Model Control (Tracking and Regulation)3.5.1 Regulation3.5.2 Tracking3.5.3 An Interpretation of the Internal Model Control3.5.4 The Sensitivity Functions3.5.5 Partial Internal Model Control (Tracking and Regulation)3.5.6 Internal Model Control for Plant Models with Stable Zeros3.5.7 Example: Control of Systems with Time Delay

3.6 Pole Placement with Sensitivity Function Shaping3.6.1 Properties of the Output Sensitivity Function3.6.2 Properties of the Input Sensitivity Function3.6.3 Definition of the “Templates” for the Sensitivity Functions3.6.4 Shaping of the Sensitivity Functions3.6.5 Shaping of the Sensitivity Functions: Example 13.6.6 Shaping of the Sensitivity Functions: Example 2

3.7 Concluding Remarks3.8 Notes and References


Computer control (discrete-time controllers)Possibilities and advantages

• Large choice of strategies for controller design• Use of more complex algorithms but with

better performance than the PID• Techniques well suited for the control of:

- systems with delay (dead time)- systems characterized by high order dynamic models- systems with low damped vibration modes

• Easy combination of control design and system identification


Digital controllers – Design methods

• Digital PID controller• Pole placement (tracking and regulation)• Tracking and regulation with independent objectives• Internal model control (tracking and regulation)• Pole placement with sensitivity function shaping

Remarks:• All the controllers will have the RST structure

(two degrees of freedom controller)• The « memory » (number of parameters) depends upon the complexity of the model used for design

• All the design methods can be viewed as particular cases of the pole placement

• The design and tuning of the controllers require the knowledge of a discrete time model of the plant


Digital PID controller

• It results from the discretization of an analog PID controller

• The computation can be rigorously applied only to:- plants described by a model whose order is- plants with a time delay smaller than Ts

• The algorithm for the parameter calculation is a particular case of the pole placement

2≤n


Digital PID controller 1

Analog PID : K – proportional gain,Ti – integral actionTd – derivative actionTd/N - filtering on the

derivative action ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+++=

sNT

sTsT

KsHd

d

iPID

1

11)(

ss Tqs

Tqs 11

111;/)1(

−−

−→−→Discretization:

Digital PID controller 1:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

−+

+−

⋅+==−

−

−−

−−

1

1

11

11

11

)1(

111

)()()(

qNTT

T

qNTT

NT

qTT

KqSqRqH

sd

d

sd

s

i

sPID



)()()( 1

11

1 −

−− =

qSqRqHPID

22

110

1)( −−− ++= qrqrrqR 22

11

11

11 1)'1)(1()( −−−−− ++=+−= qsqsqsqqS

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛++= 12111 N

TT

sKri

s )1(12 NKsr +−=⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 10 1 Ns

TT

Kri

s

sd

d

NTTT

s+

−=′1

Remark:• The digital PID controller has 4 parameters (as the analog PID)• Common factor in the denominator: (integrator)• filtering action: factor in the denominator

)1( 1−− q)'1( 1

1−+ qs



y(t)

+

-

r(t) u(t)

R/S B/A

PLANT

r(t)

R

1/S B/A

PLANT

u(t) y(t)+

-

T=R

⇔

Structure RST with T = R

T.F. of the closed loop (r y)

)()()(

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

11

1111

111

−

−−

−−−−

−−− =

+=

qPqRqB

qRqBqSqAqRqBqHCL

)( 1−qP defines the closed loop polesThe controller introduces supplementary zeros (R)


Discrete-time model of the plant

Z.O.H.

H (z )-1

H (s)

sTGesH

s

+=

−

1)(

τ

20

20

20

2)(

sse

sHs

++=

−

ζωωω τ

)( sTor <τ

22

11

22

11

1

11

1)()()( −−

−−

−

−−

+++

==qaqa

qbqbqAqBqH

The discrete-time model is obtained:• directly by system identification (general case)• by discretization of the continuous-time model


Parameter computation of the digital PID controller 1

Performances specifications :

)()(

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

1

1111

111

−

−

−−−−

−−− =

+=

qPqB

qRqBqSqAqRqBqH M

CL (*)

)( 1−qBM cannot be imposed (as B is kept andThe controller introduces supplementary zeros)The characteristic polynomial (P) of the closed loop is specified :

22

11

1 ''1)( −−− ++== qpqpqP

Continuous-timespecification(tM, M)

2nd order (ω0, ζ)discretization

sT )( 1−qP5.125.0 0 ≤≤ sTω

17.0 ≤≤ ζ


Parameter computation of the digital PID controller 1

)(/)( 11 −− qAqB)( 1−qP

- Known (or identified) plant model :- Desired performances (CL poles):

)(;)( 11 −− qSqRTo be computed : From (*) – slide 11, one solves:

)()()()()( 11111 −−−−− += qRqBqSqAqP

? ?

)()()()(

))((

)1)(1)(1(

)()()()(1)(

1111

22

110

22

11

11

122

11

111122

11

1

−−−−

−−−−

−−−−

−−−−−−−

+′′=

++++

′+−++=

+=′+′+=

qRqBqSqA

qrqrrqbqb

qsqqaqa

qRqBqSqAqpqpqP

11

1 1)( −− ′+=′ qsqS)1()1)(()( 33

22

11

111 −−−−−− ′+′+′+=−=′ qaqaqaqqAqA

Tools : WinREG, bezoutd.sci(.m)


Choice of the polynomial P

- Fourth order Polynomial Equation.- P can also be chosen as a fourth order poly. (by adding aux. poles)

)1)(1)(1(

)1()(2

21

12

21

1

44

33

22

11

1

−−−−

−−−−−

++′+′+=

=++++=

qqqpqp

qpqpqpqpqP

ααj

05.0,5.0 21 −≤≤− ααx xx

x

1

Dominants poles2nd order (ω0 , ζ)

auxiliary poles

-1

The auxiliary poles improves the closed loop robustness


Equivalent analog PID controller parameters

21

21110

)1()2(

srsrsr

K′+

′+−−′=

210

1)1(rrr

sKTT si ++′+

⋅=

1

1

1 sTs

NT sd

′+′−

=3

1

21101

)1( sKrrsrs

TT sd ′+

+′−′⋅=

The continuous equivalent does not always exist!

0'1 1 ≤≤− sExistence condition: (Td/N > 0)

Digital PID controller always can be implemented even if :(no equivalent achievable performance with an analog PI)

1'0 1 ≤≤ s


Digital PID controller 1. Examples

sTGesH

s

+=

−

1)(

τPlant:

Discretized plant: B(q-1) = 0.1813 q-1 +0.2122 q-2A(q-1) = 1 -0.6065 q-1

Ts = 5s, G = 1, T = 10s, τ = 3Performances Ts = 5s , ω0 = 0.05 rad/s, ζ = 0.8*** CONTROL LAW *** S(q-1) . u(t) + R(q-1) . y(t) = T(q-1) . r(t)Controller : R(q-1) = 0.0621 + 0.0681 q-1

S(q-1) = (1 - q-1) . (1 - 0.0238 q-1)T(q-1) = R(q-1)

Gain Margin : 7.712 Phase Margin: 67.2 degModulus Margin : 0.751 (-2.49dB) Delay Margin : 45.4 sAnalog PID : k = -0.073, Ti = -2.735, Td = - 0.122, Td/N = 0.122


Performances: ω0 = 0.05 rad/s, ζ = 0.8

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

0.5

1

1.5

Control Signal

Time (t/Ts)

Closed Loop response slower than Open Loop response.The specified ω0 should be increased



Discretized plant : B(q-1) = 0.1813 q-1 + 0.2122 q-2A(q-1) = 1 - 0.6065 q-2

Ts = 5s, G = 1, T = 10s, τ = 3Performances Ts = 5s , ω0 = 0.15 rad/s, ζ = 0.8*** CONTROL LAW *** S(q-1) . u(t) + R(q-1) . y(t) = T(q-1) . r(t)Controller : R(q-1) = 1.6874 - 0.8924 q-1

S(q-1) = (1 - q-1) . (1 + 0.3122 q-1)T(q-1) = R(q-1)

Gain Margin : 3.681 Phase Margin : 58.4 degModulus Margin : 0.664 (- 3.56 dB) Delay Margin : 9.4 sAnalog PID : (no equivalent analog PID)

No equivalent analog PID as s’1> 0 (0.3122)



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

0.5

1

1.5

Control Signal

Time (t/Ts)

- Faster response- An overshoot appears because of the zeros introduced by R


The « good » digital PID controller (PID 2)

No supplementary zero is introduced

)()(

)1()1()( 1

11

−

−− ⋅=

qPqB

BPqHCLDesired T.F. for the closed loop: 1)1( =CLH

)()()]1(/)1([

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

1

1111

111

−

−

−−−−

−−− =

+=

qPqBBP

qRqBqSqAqBqTqHCL

)1()1(

)1()1()1()1()( 1 R

BRB

BPqT ===− R and S are unchanged

Only one coefficient instead of two coeffs.


Continuous time PID corresponding to digital PID 2

k d s

1 + - - - -T N

s

+

+ -

- PLANT

Kd

-

KT d s

K____Ti s

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

1/ ( )

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

1 + -----T

N s

d

The proportional and derivative actions only act on the measure

1

1

1 sTs

NT sd

′+′−

=)1)(2(

)1(

121

2111

srrrsrsTT sd ′++

−′+′⋅=

210

21 )2(rrr

rrTT si +++−

⋅=1

21

1)2(

srrK

′++−

=


Performances of the digital PID controller 2

Discretized plant : B(q-1) = 0.1813 q-1 +0.2122 q-2A(q-1) = 1 - 0.6065 q-1Ts = 5s, G = 1, T = 10s, τ = 3

Performances Ts = 5s , ω0 = 0.15 rad/s, ζ = 0.8*** CONTROL LAW *** S(q-1) u(t) + R(q-1) y(t) = T(q-1) r(t)Controller : R(q-1) = 1.6874 - 0.8924 q-1

S(q-1) = (1 - q-1) (1 + 0.3122 q-1)T(q-1) = 0.795

Gain Margin : 3.681 Phase Margin : 58.4 degModulus Margin : 0.664 ( - 3.56 dB) Delay Margin : 9.4 sAnalog PID : (no equivalent analog PID)

No equivalent analog PI as s’1> 0 (0.3122)

To be compared with PID 1, slide 17


Performances of the digital PID controller 2

ω0 = 0.15 rad/s, ζ = 0.8

Reduced overshoot (corresponding to ζ = 0.8). Same response for disturbance rejection

To be compared with slide 18

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

0.5

1

1.5

Control Signal

Time (t/Ts)


Auxiliary poles effects

The auxiliary poles reduce the input sensitivity function Sup at highfrequencies without degrading the closed loop performances

Better robustness and reduction of actuator stress


Digital PID controller : conclusions

• RST Canonical structure

• Equivalent analog PID if

• Used with 1st or 2nd order systems with delay < Ts

• For a delay the analog PID leads to closed loop responsesslower than open loop responses

• The digital PID controller gives better performances for systemswith delay (but there is no equivalent in continuous-time)

• The digital PID controller 2 leads to a step response with a smallerovershoot than PID 1

0'1 1 ≤≤− s

T25.0≥τ


Pole placement

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)

It is a method that does not simplify the plant model zeros

The digital PID can be designed using pole placement


Structure

) ------------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

−

−−− =

qAqBqqH

dPlant:

)(...)( 1*122

11

1 −−−−−− =+++= qBqqbqbqbqB BB

nn

AA

nn qaqaqA −−− +++= ...1)( 1

11


Pole placement

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

)()()(

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

11

1111

111

−

−−−

−−−−−

−−−− =

+=

qPqBqTq

qRqBqqSqAqBqTqqH

d

d

d

BF

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

11

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

Defines the (desired )closed loop poles

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

)()()(

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

1

11

1111

111

−

−−

−−−−−

−−− =

+=

qPqSqA

qRqBqqSqAqSqAqS

dyp

Output sensitivity function


Choice of desired closed loop poles (polynomial P)

)()()( 111 −−− = qPqPqP FD

Dominant poles Auxiliary poles

Choice of PD(q-1)(dominant poles)Specificationin continuous time(tM, M)


eT )( 1−qPD

5.125.0 0 ≤≤ eTω17.0 ≤≤ ζ

Auxiliary poles

• Auxiliary poles are introduced for robustness purposes• They usually are selected to be faster than the dominant 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 x = M-1p

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

Use of WinReg or bezoutd.sci(.m) for solving (*)


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

R and S include pre-specified fixed parts (ex: integrator))()(')( 111 −−− = qHqSqS S)()(')( 111 −−− = qHqRqR R

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)

)()()()(

)( 1

1111

−

−−−− ′

=qP

qSqHqAqS S

yp

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

11 1)( −− −= qqHS

Signal blocking (Sup should be null at certain frequencies)

)()()()(

)( 1

1111

−

−−−− ′

−=qP

qRqHqAqS R

up

sR TqqH ωββ cos2;1 21 −=++= −−

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

− nqH nR


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)

)()(

)( 1

111

−

−−− =

qAqBq

qHm

mm

...)( 110

1 ++= −− qbbqB mmm

...1)( 22

11

1 +++= −−− qaqaqA mmm

Specificationin continuous time(tM, M)


sT )( 1−qHm

5.125.0 0 ≤≤ sTω17.0 ≤≤ ζ

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

)()(

)()( 1

1)1(* tr

qAqBq

tym

md

−

−+−

=



)()()(

)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)

F.T. r y:)1(

)(

)()()(

1*

1

1)1(1

BqB

qAqBqqH

m

md

BF

−

−

−+−− ⋅=

⎪⎩

⎪⎨⎧

=

≠==−

0)1(1

0)1()1()1(

)( 1

Bsi

BsiBP

GqTParticular case : P = Am


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


Pole placement. Control law

)()()()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. 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))

It is a method which simplifies the plant zerosAllows exact achievement of imposed performances

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!

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



The model zeros should be stable and enough damped

-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

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)


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

T.F. of the closed loop without T:

)()()(

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

1*1

1

1

11*111

1*11

−−

−+−

−

+−

−−+−−−

−+−− ==

+=

qPqBqBq

qPq

qRqBqqSqAqBqqH

dd

d

d

CL

The following equation has to be solved :(*))()()()()()( 11*11*111 −−−−+−−− =+ qPqBqRqBqqSqA d

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

1 −−−−− ′=+++= qSqBqsqssqS SS

nn

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

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

11

110

1 ...)( −−−

−− ++= 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



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-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) = 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


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

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)

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 regulation

It is a particular case of the pole placementThe dominant poles are those of the plant modelDoes not allow to accelerate the closed loop response

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 !

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”


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

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

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

)()()()( 1111 −−−−− =′+ qPqRqBqqS Fd

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

)1()1()()( 11

BPqAqR F−− =

)1()1(

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

BP

qBqqPqSqqS FdF

−−−−−− −=′−=



)1(/)()()( 111 BqPqAqT F−−− =

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

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

BPATqT 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 thePrediction 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


Pole placement with sensitivity functions shaping

Performance specification for pole placement :• Desired dominant poles for the closed loop• The reference trajectory (tracking reference model)

Questions:• How to take into account the specifications in certain frequencyregions?

• How to guarantee the robustness of the controllers ?• How to take advantage from the degree of freedom forthe maximum number of poles which can be assigned ?

Answer:Shaping the sensitivity functions by:

- introducing auxiliary poles- introducing filters in the controllers


Sensitivity functions - reviewOutput sensitivity function:

)()()()()()()( 1111

111

−−−−−

−−−

+=

qRqBqqSqAqSqAqS dyp

Input sensitivity function:

)()()()()()()( 1111

111

−−−−−

−−−

+−=

qRqBqqSqAqRqAqS dup

Controller structure :

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

)()()()()()()( 1111111 −−−−−−−− ==+ qPqPqPqRqBqqSqA FDd

Pre specified parts (filters)Dominant and auxiliary filters:

Study of the properties of the sensitivity functions in the frequency domain: q=z=ejω


Properties of the output sensitivity function

P.1- The modulus of the output sensitivity function at a certainfrequency gives the amplification or attenuation factor of thedisturbance on the output

Syp(ω) > 1 amplificationSyp(ω) < 1(0 dB) attenuationSyp(ω) = 1 operation in open loop

( ) 1max

)(−

=∆ ωjSM ypModulus margin

P.2



P.3 – The open loop (KG) being stable one has the property:

∫ =−SS

f.πf/fj

yp df)(eS50

0

2 0log

The sum of the areas between the curve of Syp and the axis 0dB taken withtheir sign is null

Disturbance attenuation in a frequency region implies amplificationof the disturbances in other frequency regions!



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5

Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

ω = 0.4 rad/secω = 0.6 rad/secω = 1 rad/secTemplate for Modulus marginTemplate for Delay margin = Ts

Augmenting the attenuation or widening the attenuation zone

Higher amplification of disturbancesoutside the attenuation zone

Reduction of the robustness(reduction of the modulus margin)



P.4 – Cancellation of the disturbance effect at a certain frequency:

sjj

Sjjj ffeSeHeAeSeA /20)()()()()( ; πωωωωωω ==′= −−−−−

{Zeros of Syp Allows introduction of zeros at desired frequencies

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-40

-35

-30

-25

-20

-15

-10

-5

0

5


Frequency (f/fs)

Mag

nitu

de (d

B)

HS = 1 - q-1

HS = (1 + q-2)(1 - q-1)



P.5 - at frequencies where:)0(1)( dBjS yp =ω

sjj

Rjjj ffeReHeBeReB /20)()()()()( ** ; πωωωωωω ==′= −−−−−

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5


Frequency (f/fs)

Mag

nitu

de (d

B)

HR = 1HR = 1 + q-2

Allows introduction of zeros at desired frequencies


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5


Frequency (f/fs)

Mag

nitu

de (d

B)

PF = 1PF = (1 - 0.375q-1)2

Template for Modulus marginTemplate for Delay margin = Ts

P.6 – Asymptotically stable auxiliary poles (PF) lead (in general) to the reduction of in the attenuationband of 1/PF

)( ωjS yp

FPnF qpqP )1()( 11 −− ′+= 05.05.0 −≤′≤− p

DF PPP nnn −≤

In many applications, introduction of high frequency auxiliary polesis enough for assuring the required robustness margins




P.7 – Simultaneous introduction of a fixed part HSi and of a pairof auxiliary poles PFi having the form:

22

11

22

11

1

1

11

)(

)(−−

−−

−

−

++

++=

qqqq

qP

qH

i

i

F

S

ααββ

resulting from the dicretization of :

200

2

200

2

22

)(ωωζωωζ

++

++=

ssss

sFden

num1

1

112

−

−

+−

=zz

Ts

e

with:

introduces an attenuation at the normalized discretized frequency:

⎟⎠

⎞⎜⎝

⎛=

2arctan2 0 e

discTω

ω with the attenuation: ⎟⎟⎠

⎞⎜⎜⎝

⎛=

den

numtM

ζζ

log20 dennum ζζ <( )

and with negligible effect at f << fdisc and at f >> fdisc



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5


Frequency (f/fs)

Mag

nitu

de (d

B)

HS = 1, PF = 1HS = ( ω = 1.005, ζ = 0.21), PF = ( ω = 1.025, ζ = 0.34)

Effective computation with the function: filter22.sci (.m)


Properties of the input sensitivity function

P.1 – Cancellation of the disturbance effect on the input at a certain frequency (Sup = 0):

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-40

-30

-20

-10

0

10

20Sup Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

HR = 1HR = 1 + 0.5q-1

HR = 1 + q-1

sjj

Rj ffeReHeA /20)()()( ; πωωωω ==′ −−−

101)( 11 ≤<+= −− ββqqH R( active at 0.5fS)

Allows introduction of zeros at desired frequencies

Rem: The system operate in open loop at this frequency



P.2 – At frequencies where:

sjj

Sj ffeSeHeA /20)()()( ; πωωωω ==′ −−−

)()()( ω

ωω

j

jj

up eBeAeS

−

−− =0)( =ωjS yp

One has:

Inverse ofthe systemgain

Consequence : strong attenuation of the disturbances should bedone only in the frequency regions where the system gainis enough large ( in order to preserve robustness and avoidtoo much stress on the actuator)

Remember: gives the tolerance with respect to additive uncertainties on the model (high = weak robustness)

1)(

−ωjSup

)( ωjSup



P.3 – Simultaneous introduction of a fixed part HRi and of a pairof auxiliary poles PFi having the form:

resulting from the dicretization of :

22

11

22

11

1

1

11

)(

)(−−

−−

−

−

++

++=

qqqq

qP

qH

i

i

F

R

ααββ

200

2

200

2

22

)(ωωζωωζ

++

++=

ssss

sFden

num1

1

112

−

−

+−

=zz

Ts

s

with:

introduces an attenuation at the normalized discretized frequency:

⎟⎠

⎞⎜⎝

⎛=

2arctan2 0 e

discTω

ω with the attenuation: ⎟⎟⎠

⎞⎜⎜⎝

⎛=

den

numtM

ζζ

log20 dennum ζζ <( )

and with negligible effect at f << fdisc and at f >> fdisc


Templates for the output sensitivity functions Syp

Syp dB

0,5fe

Syp max= - M∆

0

Robustness

Performances

Syp dB

0,5fe

Syp max= - M∆

0

Attenuation zone

openingthe loop


Templates for the input sensitivity function Sup

S yp dB

0.5fs

Syp max= - M∆

0

attenuation regions

loop opening


Shaping the sensitivity functions

1. Choice of the dominants et auxiliary poles of the closed loop2. Choice of the fixed part of the controller (HS and HR )3. Simultaneous choice of the fixed parts and the auxiliary poles

Procedure:

Basic shaping : use 1 and 2Fine shaping: use 3

Tools for sensitivity shaping: WinReg (Adaptech) and ppmaster.m

There exist also tools for automatic sensitivity function shapingbased on convex optimization (Optreg from Adaptech)


Shaping the sensitivity functions - Example I

Plant: sTdqBqA e 1;2;3.0;7.01 11 ===−= −−

Specifications:• Integrator• Dominant poles: discretization of a cont. time 2nd order system : ω0 = 1 rad/s, ζ = 0.9

Controller A :Attenuation band: 0 up to 0.058 Hz but ∆M < -6 dB and ∆τ < Ts

Objective: same attenuation band but with ∆M > -6 dB and ∆τ > Ts- insertion of auxiliary poles: ( )214.01 −−= qPF

Controller B : good margins but reduction of the attenuation band-insertion of pole-aero filter HS/PF centered at ω0 = 0.4 rad/s (0.064 Hz)

Controller C : good attenuation band but Syp > 6 dB - larger (slower) auxiliary poles (0.4 0.44)

Controller D : Correct


Shaping the sensitivity functions - Example I

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5

10Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

ABCDTemplate for Modulus marginTemplate for Delay margin = Ts


Shaping the sensitivity functions - Example II

Plant (integrator): sTdqBqA s 1;2;5.0;1 11 ===−= −−

q-dBA

u(t) y(t)

Sinusoidal disturbance (0.25 Hz)

Low frequencies disturbances

+

+

+

Specifications:1. No attenuation of the sinusoidal disturbance at (0.25 Hz)2. Attenuation band at low frequencies : 0 à 0.03 Hz3. Disturbances amplification at 0.07 Hz: < 3dB 4. Modulus margin > -6 dB and Delay margin > T5. No integrator in the controller



1;1 2 =+= −SR HqH

Opening the loop at 0.25 Hz

- Fixed parts design :

-Dominant poles: discretization of a cont. time 2nd order system:ω0 = 0.628 rad/s, ζ = 0.9

Controller A : the specs. at 0.07 Hz are not fulfilled- insertion of a pole-zero filter HS/PF centered at ω0 = 0.44 rad/s

Controller B : Attenuation band smaller than that specified- dominant poles acceleration: ω0 = 0.9 rad/s

Controller C : Correct



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-15

-10

-5

0

5

10Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

ABCTemplate for Modulus marginTemplate for Delay margin = Ts


Some concluding remarks

• All the digital controllers has a three branches structure(R-S-T).• They have two degrees of freedom (tracking and regulation)• Controller design is done in two steps:

1) R et S (regulation) 2) T (tracking) • Controller complexity depends upon the plant model complexity• Pole placement is the basic control strategy• Tracking and regulation with independent objectives is applicable

to discrete time models with stable zeros• Internal model control is applicable only to stable and well

damped plants • Design of digital PID is a particular case of pole placement. Can

be used for the control of simple plants (order max. = 2)• All the digital controllers presented implement a predictive control

and they contain implicitly a predicition model of the plant

robust digital controller design methodsioandore.landau/adaptivecontrol... · i.d. landau, g. zito...

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