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EECE 460Decentralized Control of MIMO Systems
Guy A. Dumont
Department of Electrical and Computer EngineeringUniversity 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 ControlExample
3 Pairing of Inputs and OutputsRelative Gain ArrayRobustness 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 varietyof synthesis methods. Some of these carry may over directly to theMIMO case.
However, there are several complexities that arise in MIMO situations.For this reason, it is often desirable to use synthesis procedures that arein 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 beapplied to MIMO problems directly.We will study
Decentralized control as a mechanism for directly exploiting SISOmethods in a MIMO settingRobustness 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 oftenuseful to check on whether a completely decentralized design canachieve 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 understandis often easier to maintaincan 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 ofreal-world systems utilize decentralized architectures.
Thus, one is led to ask the question, is there ever a situation in whichdecentralized control will not yield a satisfactory solution?
In Chapter 22 of the textbook several real-world examples that requireMIMO 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) G0
12(s)G0
21(s) G022(s)
]
G011(s) =
2s2 +3s+2
G012(s) =
k12
s+1
G021(s) =
k21
s2 +2s+1G0
22(s) =6
s2 +5s+6
Assume that k12 and k21 depend on the operating point (a commonsituation, 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 weaim 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 ofcomplementary 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. Weapply a unit step in the reference for output 1 at t = 1 and a unit step inthe 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 thoughthe 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 point2, we obtain the results on the next slide.
We see that a change in the reference in one loop now affects the outputin 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 thewhole system becomes unstable.
We see that the original SISO design has become unacceptable at thisfinal 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 theinputs and outputs in the best possible way.
In the case of an m×m plant transfer function, there are m! possiblepairings.
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 asthe 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
K22K12
]u1
This means that closing the other loop has changed the gain between y1and 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 isunaffected 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 asa matrix Λ with the ijth element
λij = [G0(0)]ij[G−1
0 (0)]
ji
where [G0(0)]ij and[G−1
0 (0)]
ji denote the ijth element of the plant d.c.gain and the jith element of the inverse of the d.c. gain matrixrespectively.
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, tothe jth output, yj, while the rest of the inputs, u1 for1 ∈ {1, 2, . . . , i−1, i+1, . . . , m} are kept constant.
Also[G−1
0
]ij is the reciprocal of the d.c. gain from the ith input, ui, to the
jth output, yj, while the rest of the outputs, y1 for1 ∈ {1, 2, . . . , j−1, j+1, . . . , m} are kept constant.
Thus, the parameter λij provides an indication of how sensible it is to pairthe ith input with the jth output.
One usually aims to pick pairings such that the diagonal entries of Λ arelarge. One also tries to avoid pairings that result in negative diagonalentries 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
k21s2+2s+1
6s2+5s+6
The RGA is then
Λ =
11−k12k21
−k12k211−k12k21
−k12k211−k12k21
11−k12k21
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 18 / 32
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Pairing of Inputs and Outputs Relative Gain Array
RGA Example
For 1 > k12 > 0, 1 > k21 > 0, the RGA suggests the pairing (u1, y1),(u2, y2).
We recall from our earlier study of this example that this pairing workedvery well for k12 = k21 = 0.1 and quite acceptably fork12 =−1, k21 = 0.5.
In the latter case, the RGA is
Λ =13
[2 11 2
]
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 19 / 32
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Pairing of Inputs and Outputs Relative Gain Array
RGA Example
However, for k12 =−2, k21 =−1 we found that the centralizedcontroller based on the pairing (u1, y1), (u2, y2) was actually unstable.
The corresponding RGA in this case is
Λ =
[−1 22 −1
]which indicates that we probably should have changed to the pairing(u1, y2), (u2, y1).
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 20 / 32
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Pairing of Inputs and Outputs Relative Gain Array
Frequency-Dependent RGA
The frequency-dependent RGA Λ(s) is defined as
λij(s) = [G0(s)]ij[G−1
0 (s)]
ji
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 21 / 32
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Pairing of Inputs and Outputs Relative Gain Array
Frequency-Dependent RGA
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 22 / 32
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Pairing of Inputs and Outputs Relative Gain Array
Quadruple-tank Apparatus
Consider the quadruple-tank apparatus shown below.
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 23 / 32
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Pairing of Inputs and Outputs Relative Gain Array
Quadruple-tank Apparatus
Recall that this system has an approximate transfer function,
G(s) =
3.71γ162s+1
3.7(1−γ2)(23s+1)(62s+1)
4.7(1−γ1)(30s+1)(90s+1)
4.7γ290s+1
The RGA for this system is
Λ =
[λ 1−λ
1−λ λ
]where λ =
γ1γ2
γ1γ2−1
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 24 / 32
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Pairing of Inputs and Outputs Relative Gain Array
Quadruple-tank Apparatus
For 1 < γ1 + γ2 < 2, we recall from a previous lecture that the system isminimum phase. If we take, for example, γ1 = 0.7 and γ2 = 0.6, then theRGA is
Λ =
[1.4 −0.4−0.4 1.4
]This suggests that we can pair (u1, y1) and (u2, y2).Because the system is of minimum phase, the design of a decentralizedcontroller is relatively easy in this case. For example, the followingdecentralized controller gives the results shown on the next slide.
C1(s) = 3(
1+1
10s
); C2(s) = 2.7
(1+
120s
)Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 25 / 32
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Pairing of Inputs and Outputs Relative Gain Array
Quadruple-tank Apparatus
Figure: Decentralized control of a minimum-phase four-tank system.
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 26 / 32
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Pairing of Inputs and Outputs Relative Gain Array
Quadruple-tank Apparatus
For 0 < γ1 + γ2 < 1, we recall from a previous lecture that the system isnonminimum phase. If we take, for example γ1 = 0.43 and γ2 = 0.34,then the system has a NMP zero at s = 0.0229, and the relative gain arraybecomes
Λ =
[−0.64 1.641.64 −0.64
]This suggests that (y1, y2) should be commuted for the purposes ofdecentralized control. This is physically reasonable, given the flowpatterns produced in this case. This leads to a new RGA of
Λ =
[1.64 −0.64−0.64 1.64
]
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 27 / 32
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Pairing of Inputs and Outputs Relative Gain Array
Quadruple-tank Apparatus
Note, however, that control will still be much harder than in theminimum-phase case. For example, the following decentralized controllersgive the results below.
C1(s) = 0.5(
1+1
30s
); C2(s) = 0.3
(1+
150s
)
Figure: Decentralized control of a nonminimum-phase four-tank system.Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 28 / 32
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Pairing of Inputs and Outputs Robustness Issues
Robustness Issues in Decentralized Control
One way to carry out a decentralized control design is to use a diagonalnominal model. The off-diagonal terms then represent under-modelling.Thus, we say we have a model G0(s), then the nominal model fordecentralized control could be chosen as
Gd0(s) = diag
{g0
11, . . . , g0mm(s)
}and the additive model error would be
G∈(s) = G0(s)−Gd0(s); G∆I(s) = G∈(s)
[Gd
0(s)]−1
A sufficient condition for robust stability is
σ (G∆1 (jω)T0 (jω))< 1 ∀ω ∈ R
where σ (G∆1 (jω)T0 (jω)) is the maximum singular value ofG∆1 (jω)T0 (jω).
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 29 / 32
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Pairing of Inputs and Outputs Robustness Issues
Example
Consider a MIMO system with
G(s) =
1
s+1 0.25 10s+1(s+1)(s+2)
0.25 10s+1(s+1)(s+2)
2s+2
; G0(s) =
1s+1 0
0 2s+2
We first observe that the RGA for the nominal model G0(s) is given by
Λ
[1.0159 −0.0159−0.0159 1.0159
]
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 30 / 32
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Pairing of Inputs and Outputs Robustness Issues
This value of the RGA might lead to the hypothesis that a correct pairingof inputs and outputs has been made and that the interaction is weak.
We thus proceed to do a decentralized design leading to a diagonalcontroller C(s) to achieve a complementary sensitivity T0(s), where
T0(s) =9
s2 +4s+9
[1 00 1
]; C(s) =
[ 9(s+1)s(s+4) 0
0 9(s+2)2s(s+4)
]
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 31 / 32
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Pairing of Inputs and Outputs Robustness Issues
However, this controller, when applied to control the full plant G(s),leads to closed-loop poles located at −6.00, −2.49± j4.69, 0.23± j1.36,and −0.50 – an unstable closed loop!
The lack of robustness in this example can be traced to the fact that therequired closed-loop bandwidth includes a frequency range where theoff-diagonal frequency response is significant.
More sophisticated techniques are required to adequately control thissystem.
Guy A. Dumont (UBC EECE) EECE 460 - Decentralized Control January 2012 32 / 32