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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...

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 2 / 28

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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.

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 3 / 28

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    Feedforward Action

    Figure 21.6

    Figure: Feedforward action in decentralized control.

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 4 / 28

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    Feedforward Action

    The feedforward gain Gjiff (s) should be chosen in such a way that the couplingfrom the jth loop to the ith loop is compensated in a particular,problem-dependent frequency band [0 ff ] i.e.

    Gjiff (j)Gii (j)+Gij (j) 0 [0 ff ]

    This can also be written as

    Gjiff (j) [Gii (j)]1 Gij (j) [0 ff ]

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

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 5 / 28

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    Feedforward Action

    Example 21.6

    Consider again the system

    G0(s) =

    2s2+3s+2 k12s+1k21

    s2+2s+16

    s2+5s+6

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

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 6 / 28

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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 G

    jiff (s) = , satisfying

    G11(0) =G12(0) == 1

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 7 / 28

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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

    ][U1(s)U2(s)

    ]= G0(s)

    [U1(s)U2(s)

    ]where

    G0(s) =

    [2

    s2+3s+2s

    s2+3s+20.5

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

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

    ]

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 8 / 28

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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.

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 9 / 28

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    Converting MIMO Problems to SISO Problems

    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.

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 10 / 28

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    Converting MIMO Problems to SISO Problems

    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.

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 11 / 28

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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(j) 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.

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 12 / 28

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    Decoupling

    Example of Decoupling

    Consider the decoupling control system below

    Figure: 22 decoupling control scheme

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 13 / 28

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    Decoupling

    Example of Decoupling

    The control scheme consists of 4 controllers:Two feedback controllers Gc1 and Gc2Two 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)

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 14 / 28

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    Decoupling

    Example of 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)

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 15 / 28

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    Decoupling

    Example of Decoupling

    Consider the system

    G0(s) =

    [5e5s4s+1

    2e4s8s+1

    3e3s12s+1

    6e3s10s+1

    ]

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 16 / 28

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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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    Industrial Case Study

    Figure: Internal roll heat flows.

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 18 / 28

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    Industrial Case Study

    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.

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 19 / 28

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    Industrial Case Study

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

    M =

    1 2 1

    2 . . ....

    ... 1 1

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

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 20 / 28

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    Industrial Case Study

    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

    M1 =

    112

    12 0 0

    12

    1+212

    ...

    0. . . 0

    ... 1+2

    12

    120 0 12

    112

    Guy A. Dumont (UBC EECE) EECE 460 - Decoupling Control January 2011 21 / 28