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    EECE 460 Decentralized Control of MIMO Systems

    Guy A. Dumont

    Department of Electrical and Computer Engineering University of British Columbia

    January 2012

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 1 / 32

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    Contents

    1 Introduction

    2 Decentralized Control Example

    3 Pairing of Inputs and Outputs Relative Gain Array Robustness Issues

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 2 / 32

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    Introduction

    Introduction

    In the case of SISO control, we found that one could use a wide variety of synthesis methods. Some of these carry may over directly to the MIMO case.

    However, there are several complexities that arise in MIMO situations. For this reason, it is often desirable to use synthesis procedures that are in some sense automated.

    Full MIMO control design is the topic of advanced control courses.

    In this course we investigate when, if ever, SISO techniques can be applied to MIMO problems directly. We will study

    Decentralized control as a mechanism for directly exploiting SISO methods in a MIMO setting Robustness issues associated with decentralized control.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 3 / 32

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    Decentralized Control

    Completely Decentralized Control

    Before proceeding to a fully interacting multivariable design, it is often useful to check on whether a completely decentralized design can achieve the desired performance objectives. When applicable, the advantage of a completely decentralized controller, compared to a full MIMO controller, is that it

    is often simpler to understand is often easier to maintain can be enhanced in a straightforward fashion in case of a plant upgrade.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 4 / 32

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    Decentralized Control

    Decentralized Control

    Exposure to practical control reveals that a substantial proportion of real-world systems utilize decentralized architectures.

    Thus, one is led to ask the question, is there ever a situation in which decentralized control will not yield a satisfactory solution?

    In Chapter 22 of the textbook several real-world examples that require MIMO thinking to get a satisfactory solution are presented.

    As a textbook example of where decentralized control can break down, consider the following MIMO example.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 5 / 32

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    Decentralized Control Example

    Example

    Consider a two-input, two-output plant having the transfer function

    G0(s) = [

    G011(s) G 0 12(s)

    G021(s) G 0 22(s)

    ]

    G011(s) = 2

    s2 +3s+2 G012(s) =

    k12 s+1

    G021(s) = k21

    s2 +2s+1 G022(s) =

    6 s2 +5s+6

    Assume that k12 and k21 depend on the operating point (a common situation, in practice).

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 6 / 32

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    Decentralized Control Example

    Example

    Operating point 1 (k12 = k21 = 0)

    Clearly, there is no interaction at this operating point. Thus, we can safely design two SISO controllers. To be specific, say we aim for the following complementary sensitivities:

    T01(s) = T02(s) = 9

    s2 +4s+9

    The corresponding controller transfer functions are C1(s) and C2(s), where

    C1(s) = 4.5 ( s2 +3s+2

    ) s(s+4)

    ; C2(s) = 1.5 ( s2 +5s+6

    ) s(s+4)

    The two independent loops perform as predicted by the choice of complementary sensitivities.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 7 / 32

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    Decentralized Control Example

    Example

    Operating point 2 (k12 = k21 = 0.1)

    We leave the controller as previously designed for operating point 1. We apply a unit step in the reference for output 1 at t = 1 and a unit step in the reference for output 2 at t = 10.

    The closed-loop response is shown on the next slide.

    The results would probably be considered very acceptable, even though the effects of coupling are now evident in the response.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 8 / 32

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    Decentralized Control Example

    Example

    Figure: Effects of weak interaction in control loops with SISO design.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 9 / 32

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    Decentralized Control Example

    Example

    Operating point 3 (k12 =−1,k21 = 0.5)

    With the same controllers and for the same test as used at operating point 2, we obtain the results on the next slide.

    We see that a change in the reference in one loop now affects the output in the other loop significantly.

    In a real application, those results would probably not be acceptable.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 10 / 32

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    Decentralized Control Example

    Example

    Figure: Effects of strong interaction in control loops with SISO design.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 11 / 32

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    Decentralized Control Example

    Example

    Operating point 4 (k12 =−2,k21 =−1)

    Now a simulation with the same reference signals indicates that the whole system becomes unstable.

    We see that the original SISO design has become unacceptable at this final operating point.

    The strong interactions need to be accounted for in the control design

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 12 / 32

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    Pairing of Inputs and Outputs

    Pairing of Inputs and Outputs

    If one is to use a decentralized architecture, then one needs to pair the inputs and outputs in the best possible way.

    In the case of an m×m plant transfer function, there are m! possible pairings.

    However, physical insight can often be used to suggest sensible pairings.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 13 / 32

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    Pairing of Inputs and Outputs Relative Gain Array

    RGA: A Simple Example

    One method that can be used to suggest pairings is a quantity known as the Relative Gain Array (RGA).

    Consider a simple 2×2 system

    y1 = K11u1 +K12u2 y2 = K21u1 +K22u2

    Assume we want to control y1 with u1.

    When the other loop is open, i.e. u2 = 0, we have y1 = K11u1

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 14 / 32

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    Pairing of Inputs and Outputs Relative Gain Array

    Now consider the situation when the other loop provides perfect control, i.e. y2 = 0. This happens when

    u2 =− K21u1 K22

    We then have

    y1 = [

    K11− K21 K22

    K12

    ] u1

    This means that closing the other loop has changed the gain between y1 and u1. The gain change is characterized by the ratio

    λ11 = 1

    1− K12K21K11K22

    Intuitively, for decentralized control, we prefer to pair variables uj and yi so that λij is close to 1, because this means that the gain from uj to yi is unaffected by closing the other loops.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 15 / 32

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    Pairing of Inputs and Outputs Relative Gain Array

    Relative Gain Array

    For a system with matrix transfer function G0(s), the RGA is defined as a matrix Λ with the ijth element

    λij = [G0(0)]ij [ G−10 (0)

    ] ji

    where [G0(0)]ij and [ G−10 (0)

    ] ji denote the ij

    th element of the plant d.c. gain and the jith element of the inverse of the d.c. gain matrix respectively.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 16 / 32

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    Pairing of Inputs and Outputs Relative Gain Array

    RGA

    Note that [G0(0)]ij corresponds to the d.c. gain from the ith input, ui, to the jth output, yj, while the rest of the inputs, u1 for 1 ∈ {1, 2, . . . , i−1, i+1, . . . , m} are kept constant. Also

    [ G−10

    ] ij is the reciprocal of the d.c. gain from the i

    th input, ui, to the jth output, yj, while the rest of the outputs, y1 for 1 ∈ {1, 2, . . . , j−1, j+1, . . . , m} are kept constant. Thus, the parameter λij provides an indication of how sensible it is to pair the ith input with the jth output.

    One usually aims to pick pairings such that the diagonal entries of Λ are large. One also tries to avoid pairings that result in negative diagonal entries in Λ.

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 17 / 32

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    Pairing of Inputs and Outputs Relative Gain Array

    RGA Example

    Consider again the system

    G0(s) =

     2s2+3s+2 k12s+1 k21

    s2+2s+1 6

    s2+5s+6

     The RGA is then

    Λ =

     11−k12k21 −k12k211−k12k21 −k12k21

    1−k12k21 1

    1−k12k21

    

    Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 18 / 32

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