Download - EECE 460 - Decoupling Control of MIMO Systems · university-logo EECE 460 Decoupling Control of MIMO Systems Guy A. Dumont Department of Electrical and Computer Engineering University

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EECE 460Decoupling Control of MIMO Systems

Guy A. Dumont

Department of Electrical and Computer EngineeringUniversity of British Columbia

January 2011

Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 1 / 28

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Contents

1 Introduction

2 Feedforward Action

3 Converting MIMO Problems to SISO Problems

4 Decoupling

5 Industrial Case Study

6 Summary

7 And in the end...


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Feedforward Action

Feedforward Action in Decentralized Control

Although it usually will not aide robust stability, the performance ofdecentralized controllers is often significantly enhanced by the judiciouschoice of feedforward action to reduce decoupling. Consider, for example, theoutput response at port #1, i.e.

Y1(s) = G11(s)U1(s)+m

∑i=2

G1i(s)Ui(s)

and, for simplicity, we consider only the effect of the j loop on the ith loop.We can then apply the feedforward ideas developed in Chapter 10 to obtainthe architecture shown on the next slide.


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Feedforward Action

Figure 21.6

Figure: Feedforward action in decentralized control.


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Feedforward Action

The feedforward gain Gjiff (s) should be chosen in such a way that the coupling

from the jth loop to the ith loop is compensated in a particular,problem-dependent frequency band [0 ωff ] – i.e.

Gjiff (jω)Gii (jω)+Gij (jω)≈ 0 ∀ω ∈ [0 ωff ]

This can also be written as

Gjiff (jω)≈− [Gii (jω)]−1 Gij (jω) ∀ω ∈ [0 ωff ]

from which we observe the necessity to build an inverse. Hence all of theissues associated with building inverses discussed in earlier chapters ariseagain.


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Feedforward Action

Example 21.6

Consider again the system

G0(s) =

2s2+3s+2

k12s+1

k21s2+2s+1

6s2+5s+6

with k12 =−1 and k21 = 0.5. We recall the results presented earlier for thiscase.


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Feedforward Action

We see that there is little coupling from the first to the second loop, butrelatively strong coupling from the second to the first loop. This suggests thatfeedforward from the second input to the first loop may be beneficial.

To illustrate, we choose Gjiff (s) to completely compensate the coupling at d.c.,

i.e. Gjiff (s) is chosen to be a constant Gji

ff (s) = α , satisfying

αG11(0) =−G12(0) =⇒= 1


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Feedforward Action

The resulting modified MIMO system can be seen to be modelled by

Y(s) = G0(s)[

U1(s)U2(s)

]= G0(s)

[1 10 1

][U′1(s)U2(s)

]= G′0(s)

[U′1(s)U2(s)

]where

G′0(s) =

[2

s2+3s+2−s

s2+3s+20.5

s2+2s+16.5s2+14.5s+9

(s2+2s+1)(s2+5s+6)

]


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Feedforward Action

The RGA is now ˜ = diag(1, 1) and when we redesign the decentralizedcontroller, we obtain the results presented below.

Figure: Performance of a MIMO decentralized control loop with interactionfeedforward.


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Converting MIMO Problems to SISO Problems


Many MIMO problems can be modified so that decentralized controlbecomes a more viable (or attractive) option. For example, one cansometimes use a precompensator to turn the resultant system into a morenearly diagonal transfer function.To illustrate, say the nominal plant transfer function is G0(s). If weintroduce a precompensator P(s), then the control loop appears as in thefigure below.

Figure: Feedback control with plant precompensation.


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The design of Cp(s) can then be based on the equivalent plant.

H(s) = G0(s)P(s)

Several comments are in order regarding this strategy:1 A first attempt at designing P(s) might be to approximate G0(s)−1 in

some way. For example, one might use the d.c. gain matrix G0(0)−1 as aprecompensator, assuming this exists.

2 If dynamic precompensators are used, then one needs to check that nounstable pole-zero cancellations are introduced between thecompensatory and the original plant.

3 Various measures of resultant interactions can be introduced. Forexample, the following terminology is frequently employed in thiscontext.


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Decoupling

Types of Decoupling

Dynamically decoupled: Here, every output depends on one and only oneinput. The transfer-function matrix H(s) is diagonal for all s. In this case, theproblem reduces to separate SISO control loops.

Band-decoupled and statically decoupled systems: When thetransfer-function matrix H(jω) is diagonal only in a finite frequency band, wesay that the system is decoupled in that band. In particular, we will say, whenH(0) is diagonal, that the system is statically decoupled.

Triangularly coupled systems: A system is triangularly coupled when theinputs and outputs can be ordered in such a way that the transfer-functionmatrix H(s) is either upper or lower triangular, for all s. The coupling is thenhierarchical.


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Decoupling

Example of Decoupling

Consider the decoupling control system below

Figure: 2×2 decoupling control scheme


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Decoupling


The control scheme consists of 4 controllers:Two feedback controllers Gc1 and Gc2

Two decouplers D12 and D21

The decoupler D21 is designed to cancel C21 arising from the processinteraction between M1 and C2. The cancellation will occur at the C2 summerif the decoupler output M21 satisfies

Gp21M11 +Gp22M21 = 0

with M21 = D21M11 this gives

(Gp21 +Gp22D21)M11 = 0−→ Gp21 +Gp22D21 = 0

which gives the ideal decoupler

D21(s) =−Gp21(s)Gp22(s)


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Decoupling


In an analogous manner, we can write

Gp12M22 +Gp11M12 = 0

(Gp12 +Gp11D12)M22 = 0

which gives the ideal decoupler

D12(s) =−Gp12(s)Gp11(s)


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Decoupling


Consider the system

G′0(s) =

[5e−5s

4s+12e−4s

8s+13e−3s

12s+16e−3s

10s+1

]


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Industrial Case Study

Strip Flatness Control

As an illustration of the use of simple precompensators to convert aMIMO problem into one in which SISO techniques can be employed, weconsider the problem of strip flatness control in rolling mills.Very similar issues arise in many other problems including paper makingand plastic extrusion.There are several control options to achieve improved flatness. Here, wewill focus on a particular aspect of the cooling spray option.

Figure: Typical flatness-control set-up for rolling mill.Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 17 / 28

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Figure: Internal roll heat flows.


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Based on the above discussion, a simplified model for this system (ignoringnonlinear heat-transfer effects, etc.) is shown in the block diagram on the nextslide, where U denotes a vector of spray valve positions and Y denotes theroll-thickness vector. (The lines indicate vectors rather than single signals.)

Figure: Simplified flatness-control feedback loop.


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The sprays affect the roll in a roughly exponential fashion as described by thematrix M:

M =

1 α α2 · · ·α 1

α2 . . ....

... 1 α

· · · α 1

The parameter α represents the level of interactivity in the system and isdetermined by the number of sprays present and how close together they are.


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An interesting thing about this simplified model is that the interaction iscaptured totally by the d.c. gain matrix M. This suggests that we could designan approximate precompensator by simply inverting this matrix. This leads to

M−1 =

11−α2

−α

1−α2 0 · · · 0−α

1−α21+α2

1−α2

...

0. . . 0

... 1+α2

1−α2−α

1−α2

0 · · · 0 −α

1−α21

1−α2


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In summary, we can (approximately) decouple the system simply bymultiplying the control vector by the appropriate vector. This set-up is shownin the block diagram below.

Figure: Flatness control with precompensation.


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The nominal decoupled system then becomes simply H(s) = diag 1(τs+1) . With

this new model, the controller can be designed by using SISO methods. Forexample, a set of simple PI controllers linking each shape meter with thecorresponding spray would seem to suffice. (We assume that the shape metersmeasure the shape of the rolls perfectly).

This idea is routinely used in this particular application and leads to excellentresults. (Of course, the practical problem has many other features that weleave aside so as not to distract from our key point here).

Actually, control problems almost identical to the above can be found in manyalternative industrial situations where there are longitudinal and traverseeffects. Examples are paper making and plastic extrusion.


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Impact of MIMO Controller


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Summary

Decentralized vs full MIMO control

In completely decentralized control, the MIMO system is approximatedas a set of independent SISO systems.

When applicable, the advantage of completely decentralized control is thatone can apply the simpler SISO theory.Applicability of this approximation depends on the neglected interactiondynamics, which can be viewed as modelling errors; robustness analysiscan be applied to determine their impact.Chances of success are increased by judiciously pairing inputs and outputs(for example, by using the Relative Gain Array, RGA) and by usingfeedforward.

Feedforward is often a very effective tool in MIMO problems.Interactions are then thought of as disturbances; this is an approximation,because the interactions involve feedback, whereas disturbance analysisactually presumes disturbances to be independent inputs.Some MIMO problems can be better treated as SISO problems if aprecompensator is first used. This leads to different types of decouplingcontrol


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Summary

There are several ways to quantify interactions in multivariable systemsincluding their structure and their strength.Interactions can have a completely general structure (every inputpotentially affects every output) or display particular patterns, such astriangular or dominant diagonal; they can also displayfrequency-dependent patterns, such as being statistically coupled orband-decoupled.The lower the strength of interaction, the more nearly a system behaveslike a set of independent systems that can be analyzed and controllerseparately.Weak coupling can be due to the nature of the interacting dynamics or toa separation in frequency range or time scale.The stronger the interaction, the more important it becomes to view themulti-input multi-output system and its interactions as a whole.Viewing the MIMO systems and its interactions as a whole requiresgeneralized synthesis and design techniques and insight. These are topicsof further control courses such as EECE 568.


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And in the end...


Download - EECE 460 - Decoupling Control of MIMO Systems · university-logo EECE 460 Decoupling Control of MIMO Systems Guy A. Dumont Department of Electrical and Computer Engineering University

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