Robust discrete time control

Robust discrete time control I.D. Landau Emeritus Research Director at C.N.R.S Laboratoire d’Automatique de Grenoble, (INPG/CNRS), France

April 2004, Valencia I.D. Landau A course on robust discrete time control, part1

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Outline Part I Introduction, examples and basic concepts Part II Design of robust discrete time controllers Part III Special topics

I.D. Landau A course on robust discrete time control, part1

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Controller Design and Validation Performance specs. Robustness specs.

DESIGN METHOD

MODEL(S)

IDENTIFICATION

Ref. CONTROLLER

+ +

u

y

PLANT

1) Identification of the dynamic model 2) Performance and robustness specifications 3) Compatible robust controller design method 4) Controller implementation 5) Real-time controller validation (and on site re-tuning) 6) Controller maintenance (same as 5)

(5) and (6) require identification in closed-loop


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Part I Outline • The R-S-T digital controller • Robust control. What is it ? • Examples (4 applications) • Sensitivity functions • Stability of closed loop discrete time systems • Robustness margins • Robust stability • Small gain theorem/ Passivity theorem / Circle criterion • Description of uncertainties and robust stability • Templates for the sensitivity functions


4

The R-S-T Digital Controller CLOCK r(t)

Computer (controller)

u(t)

D/A + ZOH

y(t)

PLANT

A/D

Discretized Plant r(t)

Bm Am

T

+ -

Controller

1 S

u(t)

R

q −d B A

y(t)

Plant Model

q −1 y (t ) = y (t − 1) I.D. Landau A course on robust discrete time control, part1

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The R-S-T Digital Controller r(t)

Bm Am

+

T

-

1 S R

Controller

u(t) q − d B

y(t)

A Plant Model

−d −1 − d −1 −1 q B ( q ) q B * ( q ) −1 −1 G ( q ) = H ( q ) = = Plant Model: A(q −1 ) A( q −1 ) A(q −1 ) = 1 + a1q −1 + ... + an q −n B(q −1 ) = b1q −1 + ... + bn q −n = q −1 B * ( q −1 ) A

A

R-S-T Controller:

B

B

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

Characteristic polynomial (closed loop poles):

P(q −1 ) = A( q −1 ) S (q −1 ) + q −d B( q −1 ) R( q −1 ) I.D. Landau A course on robust discrete time control, part1

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Robust control Robustness of what ? with respect to what ? Robustness of an index of performance with respect to plant model uncertainties. Index of performance : • closed loop stability • time domain performance • frequency domain performance Plant model uncertainties: • low quality design model • uncertainties in a frequency range • variations of the dynamic characteristics of the plant I.D. Landau A course on robust discrete time control, part1

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Robust control Robust controller : assures the desired index of performance of the closed loop for a set of given uncertainties

Robust closed loop : the index of performance is guaranteed for a set of given uncertainties


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Robustness of a control system A control system is said to be robust for a set of given uncertainties upon the nominal plant model if it guarantees stability and performance for all plant models in this set.

•To characterize the robustness of a closed loop system a frequency domain analysis is needed • The sensitivity functions play a fundamental role in robustness studies • The study of closed loop stability in the frequency domain gives valuable information for characterizing robustness


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Robustness. Some questions •How to characterize “model uncertainty”? •How to deal with “model uncertainty” ? •How to characterize “controller robustness” ? Structured (parametric uncertainty)

Model uncertainty Unstructured (specified in the frequency domain)

We will consider the “uncertainties” described in the frequency domain (i.e. effect of parameter uncertainty should be converted in the frequency domain). Important concepts : • nominal model, uncertainty model, family of plant models • nominal stability, robust stability • nominal performance, robust performance I.D. Landau A course on robust discrete time control, part1

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Nominal model, uncertainty model, family of plant models Nominal model : the model used for design Uncertainty model : the description of the uncertainties in the frequency domain (magnitude, phase) Family of plant models (P): characterized by the nominal plant model and the uncertainty model The different true “plant models” belong to the family P


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Nominal and robust stability and performance Nominal stability : closed loop as. stable for the “nominal model” Robust stability: closed loop as. Stable for all the plant models belonging to the family (set) P Nominal performance: performance for the “nominal model” Robust performance: performance guaranteed for all plant models belonging to the family (set) P


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Robust stability and robust performance Robust stability: -1

-1

The closed loop poles remain inside the unit circle (discrete time case) for all the models belonging to the family P

1

r c

1

Robust performance: The closed loop poles remain inside the circle (c,r) for all the models belonging to the family P

• robust performance condition can be translated in a robust stability condition • this is not the only way for solving robust performance design I.D. Landau A course on robust discrete time control, part1

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Examples to illustrate robust control design • Continuous steel casting • Hot dip galvanizing • Flexible transmission • 360° flexible arm


14

Continuous steel casting

Level to be controlled

Shaking (to prevent adherence)


15

Continuos steel casting Plant (integrator):

A = 1 − q −1 ; B = 0.5q −1 ; d = 2 ; TS = 1s Low frequency disturbances

u(t)

-d

q B A

+

Sinusoidal disturbance (0.25Hz)

+ y(t)

+

Specifications: 1. No attenuation of the sinusoidal disturbance (0.25 Hz) 2. Attenuation band in low frequencies: 0 à 0.03 Hz 3. Disturbance amplification at 0.07 Hz: < 3dB 4. Modulus margin > -6 dB and Delay margin > TS 5. No integrator in the controller I.D. Landau A course on robust discrete time control, part1

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Hot dip galvanizing. Control of the deposited zinc Finished product

Steel strip

Measurement of deposited mass

Air knife Preheat oven Zinc bath

air

air

zinc steel strip I.D. Landau A course on robust discrete time control, part1

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Hot dip galvanizing. The control loops Desired deposited zinc mass per cm2

PRBS (± ∆P)

m*

RST controller

Measured deposited zinc mass per cm2

P0

+

Pressure controller

Pressurized Gaz supply

Coating process

+

Measurement delay

m ± ∆m

• important time delay with respect to process dynamics • time delay depends upon the steel strip speed • sampling frequency tied to the steel strip speed • constant integer delay in discrete time • parameter variations of the process as a function of the type of product I.D. Landau A course on robust discrete time control, part1

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Hot dip galvanizing. Model and specifications Plant model for a type of product :

q −7 (b q −1 ) 1 + a1 q −1 1

Model : TS = 12 sec; b1 = 0.3; a1 = −0.2(− 0.3) )

Specifications (performance) : •Modulus margin: ∆M ≥ 0.5 ∆τ ≥ 2TS •Delay margin: •Integrator


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Hot dip galvanizing. Performance


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Control of a Flexible Transmission

The flexible transmission

motor load

axis

Φm d.c. motor

axis position

Position transducer y(t)

Controller

D u(t) A C


R-S-T controller

A D C

Φref

21

Control of a Flexible Transmission Sampling frequency : 20 Hz

A( q −1 ) = 1 − 1.609555q −1 + 1.87644q −2 − 1.49879q −3 + 0.88574q −4 B( q −1 ) = 0.3053q −1 + 0.3943q −2 d =2

Model

Vibration modes:

ω1 = 11.949 rad / sec, ζ 1 = 0.042 ; ω 2 = 31.462 rad / sec, ζ 2 = 0.023 Specifications:

ω 0 = 11 .94 rad / sec, ζ = 0.9 Dominant poles: ω 0 = 11 .94 rad / sec, ζ = 0.8 Tracking:

Robustness margins: ∆M ≥ 0.5 ∆τ ≥ 2TS Zero steady state error (integrator) Constraints on Sup :

Sup

max

≤ 10 dB for f ≥ 0.35 f S = 7 Hz


22

360° Flexible Arm LIGHT SOURCE

RIGID FRAMES

MIRROR

ALUMINIUM

DETECTOR

COMPUTER TACH. POT.

LOCAL POSITION SERVO.

ENCODER


23

360° Flexible Arm

Frequency characteristics

Poles-Zeros Unstable zeros !

(Identified Model)


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360° Flexible Arm Sampling frequency : 20 Hz A ( q − 1 ) = 1 − 2 .1049 q − 1 + 1 .04851 q − 2 + 0 . 33836 q − 3 + 0 .46 q − 4

Model

− 1 .5142 q − 5 + 0 . 7987 q − 6 B ( q −1 ) = 0 .0064 q −1 + 0 . 0146 q − 2 − 0 .0697 q − 3 + 0 .044 q − 4 + 0 . 0382 q − 5 − 0 . 007 q − 6 d =0

Vibration modes: ω1 = 2.617rad / sec, ζ 1 = 0.018; ω2 = 14.402rad / sec, ζ 2 = 0.025; ω3 = 48.117rad / sec, ζ 3 = 0.038

Specifications: Tracking: Dominant poles:

ω 0 = 2.6173 rad / sec, ζ = 0.9 ω 0 = 2.6173 rad / sec, ζ = 0.8

Robustness margins: ∆M ≥ 0 .5 Zero steady state error (integrator) Constraints on Sup :

∆τ ≥ 2TS

Sup ≤ 15 dB for f < 4 Hz; Sup ≤ 0 dB for 4.5 ≤ f < 6.5 Hz; S up < 15 dB for 6.5 ≤ f < 8 Hz; S up < 10 dB for 8 ≤ f ≤ 10 Hz I.D. Landau A course on robust discrete time control, part1

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Robust Control. Basic concepts


26

Digital control in the presence of disturbances and noise (disturbance) p(t) r(t) +

T

1/S

-

v(t) + +

u(t)

B/A

y(t)

+

+

PLANT +

R

Output sensitivity function (p y) Input sensitivity function (p u)

+

b(t) (measurement noise)

−1

S yp ( z ) = −1

S up ( z ) =

Noise-output sensitivity function (b y)

−1

S yb ( z ) =

A( z −1 )S ( z −1 ) A( z −1 )S ( z −1 ) + B( z −1 )R( z −1 ) − A( z −1 )R (z −1 ) A( z −1 ) S ( z −1 ) + B( z −1 )R (z −1 ) − B( z −1 ) R( z −1 ) A( z −1 )S ( z −1 ) + B( z −1 ) R( z −1 )

B( z −1 )S ( z −1 ) Input disturbance-output sensitivity function S ( z −1 ) = yv (v y) A( z −1 )S ( z −1 ) + B( z −1 )R( z −1 ) All four sensitivity functions should be stable ! (see book pg.102 - 103) I.D. Landau A course on robust discrete time control, part1

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All four sensitivity functions must be stable !

z − d B ( z −1 ) −1 Plant model: A ( z ) is unstable −1 A( z ) Supose : R ( z −1 ) = A( z −1 ) (poles compensation by the controller zeros) −1

S yp ( z ) =

A( z −1 ) S ( z −1 ) −1

−1

−1

−1

=

S ( z −1 )

S ( z −1 ) + B ( z −1 ) A( z −1 ) −1 Sup ( z ) = − =− −1 −1 −1 −1 A( z ) S ( z ) + B ( z ) A( z ) S ( z −1 ) + B( z −1 ) B( z −1 ) A( z −1 ) B ( z −1 ) −1 S yb ( z ) = − =− −1 −1 −1 −1 A( z ) S ( z ) + B( z ) A( z ) S ( z −1 ) + B ( z −1 ) B( z −1 ) S ( z −1 ) B( z −1 ) S ( z −1 ) −1 S yv ( z ) = = −1 −1 −1 −1 A( z ) S ( z ) + B( z ) A( z ) A( z −1 )[ S ( z −1 ) + B ( z −1 )] A( z ) S ( z ) + B( z ) A( z ) A( z −1 ) A( z −1 )

Syv is unstable while Syp, Sup,and Syb may be stable if : S ( z −1 ) + B ( z −1 ) = 0 ⇒ z < 1 I.D. Landau A course on robust discrete time control, part1

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Complementary sensitivity function (disturbance) p(t) r(t) +

T

1/S

-

v(t) + +

u(t)

B/A

+

+

y(t)

PLANT +

R

+

b(t) (measurement noise)

For T = R one has: −1

S yr ( z ) =

B ( z −1 ) R ( z −1 ) −1

−1

−1

−1

A( z ) S ( z ) + B( z ) R( z )

= −S yb ( z −1 )

S yp ( z −1 ) − S yb ( z −1 ) = S yp ( z −1 ) + S yr ( z −1 ) = 1


29

Stability of closed loop discrete time systems The Nyquist is used like in continuous time (can be displayed with WinReg ou Nyquist_OL.sci(.m)) Im H

− jω − jω B ( e ) R ( e ) H OL (e − jω ) = A(e − jω ) S (e − jω )

Critical point

ω=π

-1

-1

S yp = 1 + H (e -jω ) BO

A( z −1 ) S ( z −1 ) + B( z −1 ) R( z −1 ) S yp ( z ) = 1 + H OL ( z ) = A( z −1 ) S ( z −1 ) −1

−1

−1

Re H

H (e -j ω ) BO

ω =0

Nyquist criterion (discrete time –O.L. is stable) The Nyquist plot of the open loop transfer fct. HOL(e-jω) traversed in the sense of growing frequencies (from 0 to 0.5fS) leaves the critical point[-1, j0] on the left I.D. Landau A course on robust discrete time control, part1

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Stability of closed loop discrete time systems Nyquist criterion (discrete time –O.L. is unstable) The Nyquist plot of the open loop transfer fct. HOL(e-jω) traversed in the sense of growing frequencies (from 0 et f S) leaves the critical point[-1, j0] on the left and the number of encirclements of the critical pointcounter clockwise should be equal to the number of unstable poles in open loop.

Remarks: Im H

ω = 2π

1 unstable pole in Open Loop Stable Closed Loop (a)

ω=π -1 ω=π

ω =0

-The controller poles may become unstable if high performances are required without using an appropriate design method

Re H Stable Open Loop Unstable Closed Loop(b)

-The Nyquist plot from 0.5fS to fS is the symmetric with respect to the real axis of the Nyquist plot from 0 to 0.5fS

ω =0


31

Marges de robustesse The minimal distance with respect to the critical point characterizes the robustness of the CL with respect to uncertainties on the plant model parameters( or their variations) Im H

1 ∆G

1

-1 ∆Μ Crossover frequency

∆Φ

Re H

-Gain margin ∆G -Phase margin ∆φ -Delay margin ∆τ -Modulus margin ∆M

|H OL |=1

ωCR


32

Robustness margins Gain margin ∆G =

1 H BO ( jω 180 )

pour

∠φ (ω180 ) = −180 o

Phase margin ∆φ = 180 0 − ∠φ (ω cr )

∆φ = min ∆ φ i

pour H BO ( jω cr ) = 1

If there several intersections with the unit circle

i

Delay margin ∆τ =

∆φ ω cr

Several intersections points:

∆τ = min i

∆φ i

ω cr i

Modulus margin ∆M = 1 + H BO ( jω ) min =

−1 S yp (

jω )

min

 =  S yp ( jω )  max 


−1

33

Robustness margins – typical values Gain margin : ∆G ≥ 2 (6 dB) [min : 1,6 (4 dB)] Phase margin : 30° ≤ ∆φ ≤ 60° Delay margin : fraction of system delay (10%) or of time response (10%) (often 1.TS) Modulus margin : ∆ M ≥ 0.5 (- 6 dB) [min : 0,4 (-8 dB)] A modulus margin ∆ M ≥ 0.5 implies ∆G ≥ 2 et ∆φ > 29° Attention ! The converse is not generally true The modulus margin defines also the tolerance with respect to nonlinearities I.D. Landau A course on robust discrete time control, part1

34

Robustness margins

Good gain and phase margin Bad modulus margin

Good gain and phase margin Bad delay margin


35

Modulus margin and sensitivity function

∆M = 1 + H OL ( z ) min = S yp ( z ) min = ( S yp ( z ) max ) = −1

−1

−1

−1

  A( z ) S ( z )   −1 −1 −1 −1  A( z ) S ( z ) + B( z ) R( z ) max  −1

−1

S yp (e − jω ) dB

max

−1

−1

pour z −1 = e − j 2πf

dB = ∆M −1dB = − ∆M dB

-1

S yp S yp

= - ∆M

max

0

ω -1

S yp

S yp

= - S yp

min

= ∆M

max


36

Robust stability To assure stability in the presence of uncertainties (or variations) on the dynamic chatacteristics of the plant model HOL – nominal F.T.; H’OL –Different from HOL (perturbed) Robust stability condition (sufficient cond.):

H OL′ ( z −1 ) − H OL ( z −1 ) < 1 + H OL ( z −1 ) = S yp−1 ( z −1 ) = A( z −1 ) S ( z −1 ) + B ( z −1 ) R( z −1 ) P( z −1 ) = ; z −1 = e − jω −1 −1 −1 −1 A( z ) S ( z ) A( z ) S ( z ) dB

Im H -1 1 + H OL ' H OL

(*)

-1

S yp S yp

Re H

= - ∆M

max

0

ω

HOL

-1

S yp

S yp

= - S yp

min

= ∆M

max

Size of the tolerated uncertainity on HOL at each frequency (radius) I.D. Landau A course on robust discrete time control, part1

37

Tolerance to plant additive uncertainty From previous slide :

B ′( z −1 ) R ( z −1 ) B ( z −1 ) R ( z −1 ) R ( z −1 ) − = ⋅ −1 −1 −1 −1 −1 ′ A ( z ) S ( z ) A( z ) S ( z ) S ( z )

H’OL

B′( z −1 ) B ( z −1 ) A( z −1 ) S ( z −1 ) + B ( z −1 ) R ( z −1 ) − < −1 −1 ′ A ( z ) A( z ) A( z −1 )S ( z −1 )

H’

HOL

H

B ′( z −1 ) B ( z −1 ) A( z −1 ) S ( z −1 ) + B ( z −1 ) R ( z −1 ) P ( z −1 ) − < = = S up−1 ( z −1 ) −1 −1 −1 −1 −1 −1 A′( z ) A( z ) A( z ) R ( z ) A( z ) R ( z )

H’

H

S

up

(*)

(**)

dB Limitation of the actuator stress

0 0.5f

e

S up -1 (Size of tolerated uncertainties) I.D. Landau A course on robust discrete time control, part1

38

Tolerance to plant normalized uncertainty (multiplicative uncertainty) From (**), previous slide:

B′( z −1 ) B ( z −1 ) − A′( z −1 ) A( z −1 ) A( z −1 ) S ( z −1 ) + B ( z −1 ) R ( z −1 ) P ( z −1 ) < = = S yb−1 ( z −1 ) −1 −1 −1 −1 −1 B( z ) B( z ) R( z ) B( z ) R( z ) A( z −1 )

The inverse of the modulus of the “complementary sensitivity function” gives at each frequency the tolerance with respect to “normalized (multiplicative) uncertainty” Relation between additive and multiplicative uncertainty:

H '−H H ' = H + ( H '−H ) = H (1 + ) H I.D. Landau A course on robust discrete time control, part1

39

Passivity (Hyperstability) Theorem u1

H1

y1

strictly positive real

-

H2 y2

passive

u2

H1: Strictly positive real transfer function (state x ) H2: linear or nonlinear, time invariant or time-varying

η 2 (0, t1 ) ≥ t∑= 0 y 2T (t )u2 (t ) ≥ −γ 2 ; γ 22 < ∞ ; ∇t1 ≥ 0 t1

2

Then :

lim x(t ) = 0 ; lim u1 (t ) = 0 ; lim y1 (t ) = 0 t →∞ t →∞ t →∞ I.D. Landau A course on robust discrete time control, part1

40

Strictly positive real transfer function (SPR) Frequency domain continuous jω

s

Im H

s =j ω

H(j ω) Re H

σ

discrete Im H

Im z

+j

z

jω

z =e 1

-1

Re H < 0 Re H > 0

j

-

Re H

Re z jω

z =e

jω

H(e )

Re H < 0 Re H > 0

- asimptotycally stable

- Re H (e ) > 0 for all e = 1, (0 < ω < π ) jω

jω

(discrete time case)

Input – output property (time domain)

η1 (0, t1 ) ≥ t∑=0 y1 (t )u1 (t ) ≥ −γ + κ u1 t1

T

2

2

1

2T

; γ 12 < ∞ ; κ > 0 ; ∇t1 ≥ 0


41

Stability criteria for nonlinear time/varying feedback system Im H (e-j ω )

y2 ku2

- k1

Re H (e -j ω )

-1

u2 H (e -j ω )

Im H (e -j ω )

critical circle

u2

β u α u

y2

1 -α

-1

Re H ( z ) > −1 / k ; ∀ z = 1

-1 β

H (e -j ω )


Re H (e -j ω )

Circle criterion 42

Stability in the presence of nonlinearities (tolerance of nonlinearities) Im H ( e -j ω )

critical circle

u

βu αu

H

BO

(z

-1

)

- 1 α

-

-1

-1 β

Re H ( e -j ω )

∆M

H ( e -j ω )

The modulus margin defines the tolerance with respect to nonlinear and/or time varying elements. The tolerance sector is defined by : Min gain: 1 /(1 + ∆M ) Max gain: 1 /(1 − ∆M ) I.D. Landau A course on robust discrete time control, part1

43

The circle criterion : a particular case KM=1 -1

1

Km=-1 H(z) should lie inside the unit circle

The modulus of the H(z) should be smaller than 1 at all frequencies i.e.:

H ( z) ∞ < 1

This is the “small gain theorem” I.D. Landau A course on robust discrete time control, part1

44

Small gain theorem u1

-

S1 ∞ < 1

S2 y2

y1

S1

S2

∞

≤1

u2

S1: linear time invariant (state x)

S1 ∞ < 1 S2:

S2

∞

≤1

Then:

lim x(t ) = 0 ; lim u1 (t ) = 0 ; lim y1 (t ) = 0 t →∞ t →∞ t →∞ It will be used to characterize “robust stability” I.D. Landau A course on robust discrete time control, part1

45

Relationship between passivity theorem and small gain theorem

H −1 S∞= ≤1 H +1 ∞ If S ∞ ≤ 1 ⇒ H = 1 + S is passive 1− S H −1

Robust discrete time control

Robust discrete time control

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