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Lecture 11 - Processes with Deadtime, Internal Model Control

· · · · · · Processes with deadtime Model-reference control Deadtime compensation: Dahlin controller IMC Youla parametrization of all stabilizing controllers Nonlinear IMC

­ Receding Horizon - MPC - Lecture 14

EE392m - Spring 2005 Gorinevsky

Control Engineering

11-1

Processes with Deadtime

· Examples: transport deadtime in paper, mining, oil · Deadtime = transportation time

EE392m - Spring 2005 Gorinevsky

Control Engineering

11-2

Processes with Deadtime

· Example: transport deadtime in food processing

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Control Engineering

11-3

Processes with Deadtime

· Example: resource allocation in computing

Computing Tasks

Difference Equation

Modeling Resource Queues Resource

Desired Performance

Feedback Control

EE392m - Spring 2005 Gorinevsky

Control Engineering

11-4

Control of process with deadtime

· PI control of a deadtime process yd

PLANT: P = z

-5

-

C

P

y

P = e - sTD P=z

-d

; PI CONTROLLER: k P = 0.3, k I = 0.2

continuous time discrete time

1 0.8 0.6 0.4 0.2 0

· Can we do better? PC ­ Make = z -d 1 + PC

­ Deadbeat controller

-d

0

5

10 15 20 DEADBEAT CONTROL

25

30

1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 30

z PC = 1 - z -d

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1 C= 1 - z -d

u ( t ) = u ( t - d ) + e( t )

11-5

Control Engineering

Model-reference Control

· Deadbeat control has bad robustness, especially w.r.t. deadtime · More general model-reference control approach

­ make the closed-loop transfer function as desired

P( z )C ( z ) = Q( z) 1 + P( z )C ( z ) 1 Q( z) C( z) = P( z ) 1 - Q ( z )

Q (z ) is the reference model for the closed loop C P y y

d

· Works if Q(z) includes a deadtime, at least as large as in P(z). Then C(z) comes out causal.

EE392m - Spring 2005 Gorinevsky Control Engineering 11-6

Causal Transfer Function

B ( z ) b0 z M + b1 z M -1 + ... + bN C (z) = = N A( z ) z + a1 z N -1 + ... + a N b0 z M - N + b1 z M - N -1 + ... + bN z - N = 1 + a1 z -1 + ... + a N z - N

· Causal implementation requires that N M

(1 + a44+24444 )u (t ) = (b 44444 24+ ...4444 )e(t ) z ... + a z z +bz +b z 14 4 3 1 4 4 3

1 -1 N -N 0 M -N 1 M - N -1 B(z) N -N A( z )

EE392m - Spring 2005 Gorinevsky

Control Engineering

11-7

Dahlin's Controller

· Eric Dahlin worked for IBM in San Jose (?) then for Measurex in Cupertino. 1 Q( z) C( z) = · Dahlin's controller, 1967 P( z ) 1 - Q ( z ) g (1 - b) -d · plant, generic first order response P( z ) = z -1 1 - bz with deadtime 1 - -d Q( z) = z · reference model: 1st order+deadtime 1 - z -1

1 - bz -1 1- C( z) = g (1 - b) 1 - z -1 - (1 - ) z -d

· Dahlin's controller

EE392m - Spring 2005 Gorinevsky

· Single tuning parameter: - tuned controller a.k.a. - tuned controller

Control Engineering

11-8

Dahlin's Controller

· Dahlin's controller is broadly used through paper industry in supervisory control loops - Honeywell-Measurex, 60%. · Direct use of the identified model parameters.

CLOSED-LOOP STEP RESPONSE WITH DAHLIN CONTROLLER

· Industrial tuning guidelines: Closed loop time constant = 1.5-2.5 deadtime.

1 0.8 0.6 0.4 0.2 0 0 10 20 30 40 Ta=2.5TD Ta=1.5TD Open-loop 50 60

CONTROL STEP RESPONSE 1.5 1 0.5 0

0

10

20

30

40

50

60

EE392m - Spring 2005 Gorinevsky

Control Engineering

11-9

Internal Model Control - IMC

General controller design approach; some use in process industry

e

P P0

e = r - ( y - P0u ) u = Qe

· continuous time s · discrete time z

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Q C= 1 - QP0

Reference model: T = QP0 Filter Q

Control Engineering

Internal model: P0

11-10

IMC and Youla parametrization

reference yd disturbance d

Q C= 1 - QP0 Q=

C

P

-

y output e error u control dy yd y d u

· Sensitivities

S = 1 - QP0 T = QP0 Su = Q

C · If Q is stable, then S, T, and the loop are stable 1 + CP0 · If the loop is stable, then Q is stable

· Choosing various stable Q parameterizes all stabilizing controllers. This is called Youla parameterization · Youla parameterization is valid for unstable systems as well

EE392m - Spring 2005 Gorinevsky Control Engineering 11-11

Q-loopshaping

· Systematic controller design: select Q to achieve the controller design tradeoffs · The approach used in modern advanced control design: H2/H, LMI, H loopshaping · Q-based loopshaping:

Loopshaping

-1

S = 1 - QP0

S << 1 Q (P0 )

· in band

· Recall system inversion

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Inversion

11-12

Control Engineering

Q-loopshaping

· Loopshaping

S = 1 - QP0 T = QP0

Q = P0 = F (P0 ) ,

-1

S << 1 Q (P0 )

-1

· in band · out of band

T << 1 QP0 << 1

· For a minimum phase plant

T = QP0 = F

F=

1

(1 + s )n

S = 1 - QP0 = 1 - F

· F is called IMC filter, F T, reference model for the output · Lambda-tuned IMC

EE392m - Spring 2005 Gorinevsky Control Engineering 11-13

IMC extensions

· Multivariable processes · Nonlinear process IMC · Multivariable predictive control - Lecture 14

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Control Engineering

11-14

Nonlinear process IMC

· Can be used for nonlinear processes

­ linear Q ­ nonlinear model N ­ linearized model L

e

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Control Engineering

11-15

Industrial applications of IMC

· Multivariable processes with complex dynamics · Demonstrated and implemented in process control by academics and research groups in very large corporations. · Not used commonly in process control (except Dahlin controller)

­ detailed analytical models are difficult to obtain ­ field support and maintenance

· process changes, need to change the model · actuators/sensors off · add-on equipment

EE392m - Spring 2005 Gorinevsky

Control Engineering

11-16

Dynamic inversion in flight control

& v = F ( x , v ) + G ( x , v )u & u = G -1 ( v des - F )

· Honeywell MACH · Dale Enns X-38 - Space Station Lifeboat Reference model: 1 & v = v des s

LCV v = MCV NCV

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Control Engineering

11-17

Dynamic inversion in flight control

· · · · NASA JSC study for X-38 Actuator allocation to get desired forces/moments Reference model (filter): vehicle handling and pilot `feel' Formal robust design/analysis (µ-analysis etc)

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Control Engineering

11-18

Summary

· Dahlin controller is used in practice

­ easy to understand and apply

· IMC is not really used much

­ maintenance and support issues ­ is used in form of MPC ­ Lecture 14

· Youla parameterization is used as a basis of modern advanced control design methods.

­ Industrial use is very limited.

· Dynamic inversion is used for high-performance control of air and space vehicles

­ this was presented for breadth, the basic concept is simple ­ need to know more of advanced control theory to apply in practice

EE392m - Spring 2005 Gorinevsky Control Engineering 11-19

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