Dr. Ali Karimpour Feb 2017
Lecture 9
Multivariable Control
Systems
Ali Karimpour
Associate Professor
Ferdowsi University of Mashhad
Lecture 9
References are appeared in the last slide.
Dr. Ali Karimpour Feb 2017
Lecture 9
2
Topics to be covered include:
• Decoupling
• Decoupling by State Feedback
• Diagonal controller (decentralized control)
• Decoupling
• Decoupling by Transfer Matrix (Nyquist-array methods)
Multivariable Control System Design(Decoupling, Diagonal controller and Nyquist-array method)
Dr. Ali Karimpour Feb 2017
Lecture 9
3
Introduction
DuCxy
BuAxx
DBAsICsG 1)()(
)()()()()(
)()()()()(
2221212
2121111
susgsusgsy
susgsusgsy
We see that every input controls more than one output and that every
output is controlled by more than one input.
Because of this phenomenon,which is called interaction, it is generally
very difficult to control a multivariable system.
Interaction
Dr. Ali Karimpour Feb 2017
Lecture 9
4
Definition 9-1
A multivariable system is said to be decoupled if its transfer-function matrix is diagonal
and nonsingular.
A conceptually simple approach to multivariable control is given by a two-steps
procedure in which
1. We first design a compensator to deal with the interactions in G(s) and
2. Then design a diagonal controller using methods similar to those for SISO systems.
)()()( sWsGsG ss
)()()( sKsWsK ss)(sKs
Decoupling
Decoupling
Dr. Ali Karimpour Feb 2017
Lecture 9
5
1. We first design a compensator to deal with the interactions in G(s) and
)()()( sWsGsG ss
Decoupling
Decoupling
2. Then design a diagonal controller using methods similar to those for SISO systems.
Dr. Ali Karimpour Feb 2017
Lecture 9
6
Decoupling by State Feedback
110
2
1
5.11
5.1
110
2
)(
ss
sssG
Example 9-1 Derive a decoupling Transfer matrix for following system.
11
75.05.71
75.05.71
)(
s
ss
s
sWs
.....0
0.....)()( sWsG s
Let K=diag(0.01+0.01/s, 0.01+0.01/s)
Dr. Ali Karimpour Feb 2017
Lecture 9
7
Decoupling
• Dynamic decoupling
• Steady-state decoupling
• Approximate decoupling at frequency ω0
s.frequencie allat diagonal is )(sGs
1. We first design a compensator to deal with the interactions in G(s) and
)()()( sWsGsG ss Decoupling
)()( choosecan we with exampleFor 1 sGsWIG ss
(s)l(s)GsKsWK(s)IslsK -
sss
1)()( have we)()(by Then
It usually refers to an inverse-based controller.
diagonal. is )0(sG
This may be obtained by selecting a constant pre compensator )0(1 GWs
.at possible as diagonal as is )(0
sGs
This is usually obtained by choosing a constant pre compensator1
0
GWs
)( ofion approximat real a is 00 jGG s for selection good a is frequency 0BW
Dr. Ali Karimpour Feb 2017
Lecture 9
8
Decoupling
The idea of using a decoupling controller is appealing, but there are several difficulties.
a. We cannot in general choose Gs freely. For example, Ws(s) must not cancel any
RHP-zeros and RHP poles in G(s)
b. As we might expect, decoupling may be very sensitive to modeling errors and
uncertainties.
c. The requirement of decoupling may not be desirable for disturbance rejection.
One popular design method, which essentially yields a decoupling controller, is the
internal model control (IMC) approach (Morari and Zafiriou).
Another common strategy, which avoids most of the problems just mentioned, is to
use partial (one-way) decoupling where Gs(s) is upper or lower triangular.
Dr. Ali Karimpour Feb 2017
Lecture 9
9
• Decoupling
• Decoupling by State Feedback
• Diagonal controller (decentralized control)
• Decoupling by Transfer Matrix (Nyquist-array methods)
Multivariable Control System Design(Decoupling, Diagonal controller and Nyquist-array method)
Dr. Ali Karimpour Feb 2017
Lecture 9
10
Decoupling by State Feedback
Decoupling of a control system in state space representation.
DuCxy
BuAxx
Let DBAsICsG 1)()( Suppose
diagonalsGsG 1)()( zeros RHP no is thereand 0|D| If
11111111 )()()( DBDCBDAsICDDBAsICsG
Otherwise (|D|=0):
)()()( tTrtKytu Static output feedback
Dynamic output feedback
)()()( tTrtKxtu Static state feedback
Dr. Ali Karimpour Feb 2017
Lecture 9
11
Cxy
BuAxx
Let
)()()( Suppose 1 trtFxEtu
Cxy
rBExFBEAx
11 )(have Then we
The transfer function matrix is 111 )()( BEFBEAsICsG
We shall derive in the following the condition on G(s) under which the system can be
decoupled by state feedback.
Decoupling by State Feedback
Decoupling of a control system in state space representation.
Dr. Ali Karimpour Feb 2017
Lecture 9
12
Theorem 9-1 A system represented by
with the transfer function matrix G(s) can be decoupled by state feedback of the form
Cxy
BuAxx
)()()( 1 trtFxEtu
if and only if the constant matrix E is nonsingular.
)(
0
0
lim
.
.
1
2
1
sG
s
s
E
E
E
Epd
d
s
p
Proof: See “Linear system theory and design” Chi-Tsong Chen
md
d
new
s
s
sG
0
0
)(
1
Furthermore the new system is in the form:
pd
p
d
d
AC
AC
AC
F
.
.
2
1
2
1
Decoupling by State Feedback
Dr. Ali Karimpour Feb 2017
Lecture 9
13
Example 9-1 Use state feedback to decouple the following system.
xyuxx
110
001
00
10
01
6116
100
010
Solution: Transfer function of the system is
65
6
65
6
6116
6
6116
116
)()(
22
2323
2
1
ss
s
ss
sss
s
sss
ss
BAsICsG
The differences in degree of the first row of G(s) are 1 and 2, hence d1=1 and
]01[6116
6
6116
116lim
2323
2
1
sss
s
sss
sssE
s
The differences in degree of the second row of G(s) are 2 and 1, hence d2=1 and
]10[65
6
65
6lim
222
ss
s
sssE
s
Decoupling by State Feedback
Dr. Ali Karimpour Feb 2017
Lecture 9
14
Now E is unitary matrix and clearly nonsingular so decoupling by state feedback is
possible and
10
01E
Solution (continue):
5116
010
2
1
2
1
d
d
AC
ACF
)()(
5116
010
10
01)()()( 1 trtxtrtFxEtu
The decoupled system is
xCxy
rxrBExFBEAx
110
001
00
10
01
6116
6116
000
)( 11
Exercise 9-1: Derive the corresponding decoupled transfer function matrix.
Decoupling by State Feedback
Dr. Ali Karimpour Feb 2017
Lecture 9
15
Decoupling by State Feedback
Example 9-2 Use state feedback to decouple the following system. The system is related
to a continuous chemical process with two reactor connected in series [Luyben ,1997]
2 0 1 0.0025 1 0,
2 4 0 0.0025 0 1x x u y x
1
1 0.00252 2
2 0.0025( 2)( 4) ( 2)
( ) ( ) ( ) s s
s s s
G s c sI A B D G s
So E is non-singular
1 0.0025lim [ ] [1 0.0025]1
2 2min(1,1) 11
min(2,1) 12 2 0.0025lim [ ] [0 0.0025]2
( 2)( 4) 2
1 0.00252 2
2 0.0025( 2)( 4) ( 2)
( )
E ss ssd
dE s
s s ss
s s
s s s
G s
1 0.0025( ) 2
0 0.0025E rank E
Dr. Ali Karimpour Feb 2017
Lecture 9
16
Decoupling by State Feedback
1
2
1
1 1
1
22
1 0 2 0
2 4[0 1]
d
d
C A C A AF
AC AC A
1 1( )x A BE F x BE r
y Cx
1 1
2 0 1 0.0025 1 0.0025 2 0 1 0.0025 1 0.0025( )
2 4 0 0.0025 0 0.0025 2 4 0 0.0025 0 0.0025
1 0
0 1
x x r
y x
0 0 1 0
0 0 0 1 0 0( ) , ( ) ( ) 0
0 01 0
0 1
x x r
G s rank A rank
y x
Dr. Ali Karimpour Feb 2017
Lecture 9
17
Decoupling by State Feedback
Now the MIMO system is decoupled and G(s) is diagonal, we can design two SISO
controller for the diagonalized G(s), using state feedback again we have :
1 1 1
1 10 01 0 1 0( ) ( )
0 1 1 0 1 10 0
s sG s C sI A BE F BE
s s
1 1 1( ) ( )G s C sI A BE F BE
Dr. Ali Karimpour Feb 2017
Lecture 9
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Decoupling by State Feedback
System model with state feedback decoupler
Dr. Ali Karimpour Feb 2017
Lecture 9
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Decoupling by State Feedback
Diagonalized system model with state feedback controller plus a gain to reduce Ess
Dr. Ali Karimpour Feb 2017
Lecture 9
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Decoupling by State Feedback
0 5 10 15 20 25 30-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Am
plitu
de (
mol*
A/ft3
)
Step responsse for closed loop system (Controller designed for decoupled system)
r1
y1
r2
y2
Applying the designed controller to the system result in the following response to specified
reference input r1,r2
Dr. Ali Karimpour Feb 2017
Lecture 9
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Property of Decoupling by State Feedback
1- All poles of decoupled are on origin.
3- No transmission zero in decoupled system.
4- Transmission zero of the system are deleted .
5- Unstable transmission zero is the main limitation of method.
2- Decoupled system is:
ndd
decouple ssdiagsG
...,,)( 1
Dr. Ali Karimpour Feb 2017
Lecture 9
22
• Decoupling
• Decoupling by State Feedback
• Diagonal controller (decentralized control)
• Decoupling by Transfer Matrix (Nyquist-array methods)
Multivariable Control System Design(Decoupling, Diagonal controller and Nyquist-array method)
Dr. Ali Karimpour Feb 2017
Lecture 9
23
Diagonal controller (decentralized control)
Another simple approach to multivariable controller design is to use a diagonal or
block diagonal controller K(s). This is often referred to as decentralized control.
Clearly, this works well if G(s) is close to diagonal, because then the plant to be
controlled is essentially a collection of independent sub plants, and each element in
K(s) may be designed independently.
However, if off diagonal elements in G(s) are large, then the performance with
decentralized diagonal control may be poor because no attempt is made to counteract
the interactions.
Dr. Ali Karimpour Feb 2017
Lecture 9
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The design of decentralized control systems involves some steps:
3- The design (tuning) of each controller ki(s)
Diagonal controller (decentralized control)
1- How many control loops is necessary?
loop-assignment problem or input-output pairing
2- The choice of pairings (control configuration selection)
Dr. Ali Karimpour Feb 2017
Lecture 9
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Input-Output Pairing
Definition of RGA (Relative Gain Array)
Physical Meaning of RGA: Let
TGGGRGAG )()(
ijijij hg /
2221212
2121111
ugugy
ugugy
jkuj
ijiij
k
u
y uyg
,0
or 0inputsother if and between relation
222121
2121111
0 ugug
ugugy
1
22
212 u
g
gu
ikyj
ijiij
k
u
y uyh
,0
or 0outputsother if and between relation
Relative gain?
1111
22
21
12111)( uhu
g
gggy
Dr. Ali Karimpour Feb 2017
Lecture 9
26
Input-Output Pairing
Let
1
1)( TGGG
2221212
2121111
ugugy
ugugy
λ=1 Open loop and closed loop gains are the same,
so interactions has no effect.
λ=0 g11=0 so u1 has no effect on y1.
0<λ Closing second loop, no sign change the gain between y1 and u1.
λ<0 Closing second loop leads to changing the sign of the gain between
y1 and u1.(Very Bad)
1_ To avoid instability caused by interactions in the crossover region one should
prefer pairings for which the RGA matrix in this frequency range is close to identity.
2_ To avoid instability caused by interactions at low frequencies one should avoid
pairings with negative steady state RGA elements.
In this section we provide two useful rules for pairing inputs and outputs.
Dr. Ali Karimpour Feb 2017
Lecture 9
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Input-Output Pairing
RGA property:
1- It is independent of input and output scaling.
2- Its rows and columns sum to 1.
3- The RGA is identity matrix if G is upper or lower triangular.
4- Plant with large RGA elements are ill conditioned.
5- Suppose G(s) has no zeros or poles at s=0. If λij() and λ(0) exist
and have different signs then one of the following must be true.
* G(s) has an RHP zeros. * Gij(s) has an RHP zeros.
* gij(s) has an RHP zeros.
6- If gijgij(1-1/λij) then the perturbed system is singular.
7- Changing two columns/rows of G leads to same changes to its RGA
Dr. Ali Karimpour Feb 2017
Lecture 9
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Diagonal controller (decentralized control)
Example 9-3 Select suitable pairing for the
following blending system. (ω is output
flow and x is the composition and defined as
percent of of ωA to total flow)
Solution:
BA
A
BA
ww
wx
www
B
A
w
w
w
x
w
xx
w
0
0
0
0111
RGA of the system is
00
00
1
1
xx
xx
If x0=0.1
1.09.0
9.01.0AB wxww &
If x0=0.9
9.01.0
1.09.0BA wxww &
Dr. Ali Karimpour Feb 2017
Lecture 9
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Diagonal controller (decentralized control)
Example 9-4 Select suitable pairing for the following plant
8.14.01.18
7.04.85.15
4.16.52.10
)0(G
Solution: RGA of the system is
98.107.09.0
43.037.094.0
41.145.196.0
)0(
Dr. Ali Karimpour Feb 2017
Lecture 9
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Diagonal controller (decentralized control)
Combination of SVD, C.N. and RGA can help in this matter.
1- How many control loops is necessary?
2- The choice of pairings (control configuration selection)
Dr. Ali Karimpour Feb 2017
Lecture 9
31
Diagonal controller (decentralized control)
3
2
1
3
2
1
3
2
1
)0(
020.095.090.0
008.095.052.0
006.09.048.0
u
u
u
G
u
u
u
y
y
y
Example 9-5 Determine the preferred multiloop control strategy for a process with
the following steady-state gain matrix, which has been scaled by dividing the process
variables by their maximum values.
Solution:
01.037.065.0
56.079.036.0
45.016.071.0
99.002.001.0
01.005.099.0
02.099.005.0
01.000
014.10
0062.1
01.083.056.0
68.041.060.0
73.038.057.0
)0(G
233211 && uyuyuy
132231 && uyuyuy
162..
NC
Dr. Ali Karimpour Feb 2017
Lecture 9
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Diagonal controller (decentralized control)
99.002.001.0
01.005.099.0
02.099.005.0
01.000
014.10
0062.1
01.083.056.0
68.041.060.0
73.038.057.0
)0(G 162..
NC
Determine three control loop is not suitable so:
55.0133..,
55.072..,
38184..,
,
1132
1131
1121
21
NCuu
NCuu
NCuu
andyy
46.1139..,
64.069..,
36.051.1..,
,
1132
1131
1121
31
NCuu
NCuu
NCuu
andyy
71.068..,
25.3338..,
37.045.1..,
,
1132
1131
1121
32
NCuu
NCuu
NCuu
andyy
233211 && uyuyuy 132231 && uyuyuy
1321 & uyuy
1322 & uyuy
Dr. Ali Karimpour Feb 2017
Lecture 9
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The RGA based techniques have many important advantages, such as very simple in
calculation as it only uses process steady-state gain matrix and scaling independent.
Moreover, using steady-state gain alone may result in incorrect interaction measures and
consequently loop pairing decisions, since no dynamic information of the process is
taken into consideration.
Many improved approaches, RGA-like, have been proposed and described in all
process control textbooks, for defining different measures of dynamic loop
interactions.
[1] D.Q. Mayne, “The design of linear multivariable systems,” Automatica, vol. 9, no. 2, pp.201–207, Mar. 1973.
Relative Omega Array (ROmA),
[2] ARGA Loop Pairing Criteria for Multivariable Systems
A. Balestrino, E. Crisostomi, A. Landi, and A. Menicagli ,2008
Absolute Relative Gain Array (ARGA),
Relative Normalized Gain Array (RNGA),
[3] RNGA based control system configuration for multivariable processesMao-Jun He, Wen-Jian Cai *, Wei Ni, Li-Hua XieJournal of Process Control 19 (2009) 1036–1042
Diagonal controller (decentralized control)
Dr. Ali Karimpour Feb 2017
Lecture 9
Next example, for which the RGA based loop pairing criterion gives an
inaccurate interaction assessment, are employed to demonstrate the
effectiveness of the proposed interaction measure and loop pairing criterion.
Example 9-6:
Consider the two-input two-output process:
26
Diagonal controller (decentralized control)
RGA=Diagonal pairing
RNGA =Off-diagonal pairing
To illustrate the validity of above results, decentralized controllers
for both diagonal and off-diagonal pairings are designed respectively based on
the IMC-PID controller tuning rules.
To evaluate the output control performance, we consider a unit step set-point
Change of all control loops one-by-one and the integral square error (ISE) is
used to evaluate the control performance.
Dr. Ali Karimpour Feb 2017
Lecture 9
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Diagonal controller (decentralized control)
The simulation results and ISE values are given in Figure. The results show
that the off-diagonal pairing gives better overall control system performance.
off-diagonal
diagonal
Dr. Ali Karimpour Feb 2017
Lecture 9
36
• Decoupling
• Decoupling by State Feedback
• Diagonal controller (decentralized control)
• Decoupling by Transfer Matrix (Nyquist-array methods)
Multivariable Control System Design(Decoupling, Diagonal controller and Nyquist-array method)
Dr. Ali Karimpour Feb 2017
Lecture 9
37
Nyquist-array methods
Nyquist-array methods ( Compensator structure )
We shall again assume that the plant's transfer function is square.
Suppose that a compensator is rational, invertible, and has all its poles and zeros in the
left half-plane (including the origin).
Theorem 9-2
Let K(s) be square, rational and invertible, and have all its poles and zeros in the open
left half-plane. Then
abccba KsKsKsKsKsKKsK )()()( and )()()(
aa KK and are permutation matrices, in other words, reorder the outputs or inputs,
)( and )( sKsK bb are products of elementary matrices
)( and )( sKsK cc are diagonal matrices, with rational, non-zero, and with poles and
zeros in the open left half-plane only.
Dr. Ali Karimpour Feb 2017
Lecture 9
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Nyquist-array methods
This theorem implies that compensator design can be split into two stages.
• In the first stage are used to make the return ratio diagonally dominant.)( and sKK ba
• The second stage begins when dominance has been achieved and consists of
designing a set of separate SISO compensators one for each loop.
At this stage no attention is paid to the remaining interactions in the system,
Except that Gershgorin bands of the return ratio replace SISO Nyquist loci.
Dr. Ali Karimpour Feb 2017
Lecture 9
39
Nyquist-array methods
Design can be pursued using either
• Direct Nyquist Arrays (DNA)
)()()()(0 sKsKKsGsQ cba
• Inverse Nyquist Arrays (INA)
Since Kc(s) post-multiplies the other transfer functions, and is diagonal, hence
column dominance of the G(s)KaKb(s) is not destroyed by Kc(s).
)()()()( 11111
0 sGKsKsKsQ abc
Since pre-multiplies the other transfer functions, and is diagonal,
hence row dominance of the is not destroyed by .
)(1 sKc
)()( 111 sGKsK ab
)(1 sKc
Therefore, when working with INA it is usual to try to achieve row dominance.
Therefore, when working with DNA it is usual to try to achieve column dominance.
)(sGaK)(sKb)(sKc
)(1 sG )(1
sKb
)(
1sKc
1
aK
There is the Ostrowski bands, which allows the designer to take into account the
effects of interactions between loops.
Dr. Ali Karimpour Feb 2017
Lecture 9
40
Nyquist-array methods
The direct Nyquist-array (DNA) method
The DNA has several advantages.
1- The designed compensator does not need to be inverted, and one consequence of this
is that any structure imposed by the designer, such as setting certain elements to zero,
is retained.
It also gives the designer more freedom in the choice of compensator, since its inverse
does not need to be realizable
2- The plant need not be square, since it need not have an inverse.
3- The inverse compensator designed by the INA method has right half-plane zeros,
particularly if it is designed semi-automatically by one of the methods described in
the next section.
However, there is no tool, such as the Ostrowski bands, which allows the designer
to take into account the effects of interactions between loops.
Dr. Ali Karimpour Feb 2017
Lecture 9
41
Nyquist-array methods
Achieving diagonal dominance
• Cut and try
• Perron-Frobenius theory
• Pseudo-diagonalization
It is based on some straightforward transformation.
Perron-Frobenius theory allows us to check whether a plant can be
made diagonally dominant by input and output scaling
A way of automatically generating compensators with a more general
structure.
Dr. Ali Karimpour Feb 2017
Lecture 9
42
Nyquist-array methods
Achieving diagonal dominance ( Cut and try )
It is sometimes possible to examine the display of a Nyquist array and observe that
some straightforward transformation will achieve diagonal dominance.
2323
223
1
23)(
s
s
s
ss
s
s
s
sG
Let
Its inverse is
ss
sssG
2
1)(ˆ
This is clearly neither row dominant nor column dominant anywhere on the Nyquist
contour.
ss
sssGKa
2
1
01
10)(ˆˆ
01
10ˆ
aK
1
2
ss
ss
This is clearly row dominant and column dominant anywhere on the Nyquist contour.
Dr. Ali Karimpour Feb 2017
Lecture 9
43
Nyquist-array methods
Achieving diagonal dominance ( Cut and try )
ss
sssGKa
2
1
01
10)(ˆˆ
01
10ˆ
aK
1
2
ss
ss
So that the compensated plant is represented by
23
2
23
2323
1
)(
s
s
s
ss
s
s
s
KsG a
01
10aK
Physically, this corresponds to nothing more than a re-ordering of the inputs (or a re-
assignment of inputs to outputs.
In this artificial example both the direct and the inverse array have been made equally
dominant, and the compensation required in each case is the same.
This is not usually true. It is quite possible for a particular compensator to make the
direct array dominant, but not the inverse array, and vice versa.
Dr. Ali Karimpour Feb 2017
Lecture 9
44
Nyquist-array methods
Achieving diagonal dominance ( Cut and try )
The ‘elementary matrices’ are supposed to represent simple transformations devised
by the designer.
In practice, it is rarely possible to make much progress by relying on being able to find
such transformations by ad hoc means.
10
21)(
ssKb
An alternative strategy is to try to diagonalize a system at one frequency, and hope
that the effect will be sufficiently beneficial over a wide range of frequencies.
If the system has no poles at the origin, then K=G-1(s) is a realizable (because constant)
compensator.
Fortunately, we already have an algorithm (ALIGN algorithm) for performing the
required approximation on other frequencies.
Dr. Ali Karimpour Feb 2017
Lecture 9
45
Nyquist-array methods
Achieving diagonal dominance ( Cut and try )
A plant has the inverse transfer function
2
115
3101
2
)(ˆ2
2
s
sss
ss
ss
sG
5.01
32)0(G
15.0
5.125.0)0(ˆ GKb
It is not diagonally dominant at low frequencies, since
The compensator
IGK b )0(ˆˆ
sasssGKb
44
18
14
29
)(ˆˆ
gives
which is obviously diagonally dominant, and it gives column, but not row, dominance at
high frequencies, since
Dr. Ali Karimpour Feb 2017
Lecture 9
46
Nyquist-array methods
Achieving diagonal dominance ( Perron-Frobenius theory )
Perron-Frobenius theory allows us to check whether a plant can be made diagonally
dominant by input and output scaling.
Theorem 9-3 (Mees, 1981):
If G is square and primitive, then there exist a diagonal matrix S such that
is diagonally dominate, if and only if
1~ SGSG
(I) 21
diagp GG
If (I) satisfied then X which achieve diagonal dominance is
nsssdiagS ,,..., 21
Where λp Perron-Frobenius eigenvalue and (s1, s2, … ,sn)T is left Perron-Frobenius
eigenvector of .1
diagGG
)(sG1S S
Dr. Ali Karimpour Feb 2017
Lecture 9
47
Nyquist-array methods
Achieving diagonal dominance ( Perron-Frobenius theory )
Output scaling is physically impossible, since the meaningful plant outputs
(which are variables such as velocity, or thickness of steel strip) cannot be affected by
mathematical operations.
But we can use
But we must be wary of falling into the trap of believing that this return ratio tells us
anything about interaction at the plant output.
The output variables may be interacting with each other to a considerable extent, and
this interaction may be being hidden by the measurement scaling S.
Dr. Ali Karimpour Feb 2017
Lecture 9
48
Nyquist-array methods
Achieving diagonal dominance ( Perron-Frobenius theory )
Fortunately, the Perron-Frobenius theory gives useful results, even if only pre-
compensation (input scaling) is allowed.
Theorem 9-4
If G is square and primitive, then there exist a diagonal matrix K(s) such that
is diagonally row dominate, if and only if
)()(~
sKsGG
(I) 21 GGabs diagp
If (I) satisfied then K(s) which achieve diagonal dominance is
)(,,...)(,)()( 21 skskskdiagsK n
Where λp Perron-Frobenius eigenvalue and (k1(s), k2(s), … ,kn(s))T is right Perron-
Frobenius eigenvector of .1 GGabs diag
)(sG)(sK
Dr. Ali Karimpour Feb 2017
Lecture 9
49
Nyquist-array methods
Achieving diagonal dominance ( Perron-Frobenius theory )
Everything else remains the same, except that if a dynamic compensator is used then
the elements of compensator must be chosen to have realizable inverses.
A drawback of using diagonal compensators
• Row dominance when using the DNA method
• Column dominance when using the INA method
This is exactly the opposite of what we would like, since further diagonal compensation,
for the purpose of ‘loop shaping’, may destroy the dominance which has been achieved.
Dr. Ali Karimpour Feb 2017
Lecture 9
50
Nyquist-array methods
Example 9-7 Consider the transfer function
12
2
10010
1
15
1
)5)(1(
4
)(
2 sss
s
sss
s
sG
10-2
10-1
100
101
102
0
0.5
1
1.5
2
2.5
3
3.5
4
Frequency (rad/sec)
Gain
(dB
)
The Perron-Frobenius eigenvalue of
)(ˆ)(ˆ 1 jGjGabs diag
is:
Its value is smaller than 4 dB ( i.e. λp<2 )
at all frequencies, so it is possible to obtain
column dominance by using a diagonal
compensator.
Achieving diagonal dominance ( Perron-Frobenius theory )
Dr. Ali Karimpour Feb 2017
Lecture 9
51
Nyquist-array methods
The second element of the Perron-Frobenius left eigenvector, when the first element
is fixed at 1 is:
10-2
10-1
100
101
102
-10
-5
0
5
10
15
20
25
30
Frequency (rad/sec)
Gain
(dB
)
31.0
898.4438.0)(ˆ
2
s
ssk
This matches the variation of the
eigenvector very well.
The inverse compensator
)(ˆ,1)(ˆ 2 skdiagsK
Leads to column dominant.
Achieving diagonal dominance ( Perron-Frobenius theory )
Dr. Ali Karimpour Feb 2017
Lecture 9
52
Nyquist-array methods
Achieving diagonal dominance ( Perron-Frobenius theory )
-20 -10 0 10 20-20
-10
0
10
20
Re
Im
-20 -10 0 10 20-20
-10
0
10
20
Re
Im
-20 -10 0 10 20-20
-10
0
10
20
Re
Im
-20 -10 0 10 20-20
-10
0
10
20
Re
Im
GK ˆˆNvquist array of
Dr. Ali Karimpour Feb 2017
Lecture 9
53
Nyquist-array methods
Achieving diagonal dominance ( Pseudo-diagonalization )
This can be done by choosing some measure of diagonal dominance, some compensator
structure, and then optimizing the measure of dominance over this structure.
We shall use the term pseudo-diagonalization for any such scheme, although the
term is often reserved for the particular scheme proposed by Hawkins (1972).
Hawkins assumed that inverse arrays are to be used, but his method can be applied
equally well to direct arrays.
If we have a plant G(s) and a constant compensator K, with Q(s)=G(s)K, then
Inividual elements of Q are given by
j
T
iij kjgjq )()(
N
k ji
jk
T
ik
N
k ji
kijkj kjgpjqpJ1
2
1
2
)()(
Hawkins proposed to minimize
subject to the constraint1jk
jth column of K
Otherwise it leads to kj=0
ith row of G
Dr. Ali Karimpour Feb 2017
Lecture 9
54
Nyquist-array methods
Achieving diagonal dominance ( Pseudo-diagonalization )
N
k ji
jk
T
ik
N
k ji
kijkj kjgpjqpJ1
2
1
2
)()(
Hawkins proposed to minimize
subject to the constraint1jk
Hawkins method may not prevent from being made small, as well as the
off-diagonal elements, so that diagonal dominance may not be obtained.
)( kjj jq
Suppose, however
N
k
kjjk
N
k ji
kijk
j
jqp
jqp
J
1
2
1
2
)(
)(
2
2
)(
)(
k
jk
T
jk
k ji
jk
T
ik
j
kjgp
kjgp
J
The solution is given by the multi frequency ALIGN algorithm. (Maciejowski (1989)).
Dr. Ali Karimpour Feb 2017
Lecture 9
55
Nyquist-array methods
Achieving diagonal dominance ( Pseudo-diagonalization )
Ford and Daly (1979) have extended this approach to dynamic compensators:
)(),...,(),...,()( 1 sksksksK mj
skksk jjj ...)( 0
j
T
iij jjq )()(
)(....)()()(
jgjjgjjgjT
i
T
i
T
i
T
i
j
j
j
k
k
...
0
k
jk
T
jk
k ji
jk
T
ik
j
jp
jp
J2
2
)(
)(
Vector
ijth element of G(s)K(s) is :
ith row of G
Dr. Ali Karimpour Feb 2017
Lecture 9
56
Nyquist-array methods
Achieving diagonal dominance ( Pseudo-diagonalization )
A realizable compensator is therefore obtained by dividing Kj(s) by any polynomial
of degree β (or greater).
Pseudo-diagonalization can be applied to either direct or inverse Nyquist arrays.
But a practical difficulty arises if a dynamic compensator is found for an inverse array:
its inverse needs to be realizable.
In particular, it can be viewed as an extension of the ALIGN algorithm, and can
therefore be applied in the context of the characteristic-locus method.
Pseudo-diagonalization have described in the context of Nyquist array methods,
it can clearly be applied whenever approximate inverses of frequency responses are
required.
Dr. Ali Karimpour Feb 2017
Lecture 9
57
Design example
0732.005750.1
6650.104190.4
000
00000.11200.0
000
,
6859.00532.102909.00
0130.18556.000485.00
00000.1000
0705.001712.00538.00
0000.101320.100
BA
000
000
000
,
00100
00010
00001
DC
the model has three inputs, three outputs and five states.
Example 9-8: Consider the aircraft model AIRC described in the following
state-space model.
DuCxy
BuAxx
Nyquist-array methods
Dr. Ali Karimpour Feb 2017
Lecture 9
58
Design example
• We shall attempt to achieve a bandwidth of about l0 rad/sec for each loop.
THE SPECIFICATION
• Little interaction between outputs.
• Good damping of step responses and zero steady-state error in the face of step
demands or disturbances.
• We assume a one-degree-of-freedom control structure.
Nyquist-array methods
Dr. Ali Karimpour Feb 2017
Lecture 9
59
Design example
PROPERTIES OF THE PLANT
• The time responses of the plant to unit step signals on inputs 1 and 2 exhibit very
severe interaction between outputs.
• The poles of the plant (eigenvalues of A) are
jj 1826.00176.0,03.178.0,0
so the system is stable (but not asymptotically stable).
• Thus this plant has no finite zeros, and we do not expect any limitations on
performance to be imposed by zeros. since
the number of finite zeros of the plant can be at most
)(2 CBrankmn 013.25
Nyquist-array methods
Dr. Ali Karimpour Feb 2017
Lecture 9
60
Nyquist-array methods
Design example Obtaining column dominance
10-3
10-2
10-1
100
101
102
-100
-50
0
50
100
150
Frequency (rad/sec)
Gai
n (d
B)
Column dominances of the plant: column 1 (solid curve), column 2 (dashed curve)
and column 3 (dotted curve).
)(
)(
jg
jg
jj
ji
ijcolumn dominance measure
column 1
Dr. Ali Karimpour Feb 2017
Lecture 9
61
Nyquist-array methods
We use pseudo-diagonalization (the algorithm of Ford and Daly (1979)) to obtain
column dominance, and apply it to one column at a time.
For this purpose, and in the rest of this design example, all frequency responses are
evaluated at a set of 50 frequency points, equally spaced on a logarithmic scale
between 0.001 and 100 rad/sec, except for a greater density of points in the region
of 0.18 rad/sec .
Applying pseudo-diagonahzation to the first column and optimizing over constant
compensator elements only, with uniform weighting on all frequencies, was not
successful.
Diagonal dominance was improved at low frequencies, where it was not needed, but
remained almost unchanged at high frequencies.
Frequencies above 0.1rad/sec were therefore weighted 10 times as much as lower
frequencies, but dominance was still not achieved above l rad/sec.
Design example Obtaining column dominance
Dr. Ali Karimpour Feb 2017
Lecture 9
62
Nyquist-array methods
This produced virtually perfect diagonal dominance, with a dominance measure less
than 10-6 at all frequencies.
33
54
43
1
1031.11005.4
1011.31083.1
1078.91052.1
)(
s
s
s
sk
The lower degree of dominance may lead a simpler compensator structure. Only one
element of the column may need to be dynamic, for example.
For the second column, it was again necessary to optimize over a first order dynamic
structure,
2
32
2
2
1040.30
1050.41074.5
1022.10
)(
s
s
s
sk
Again, almost perfect dominance was obtained everywhere.
Design example Obtaining column dominance
Dr. Ali Karimpour Feb 2017
Lecture 9
63
Nyquist-array methods
The third column proved to be the most difficult to compensate.
Optimizing over a first-order structure with high weighting on low frequencies lead to:
Reducing the weighting on the low frequencies gave no benefit at higher frequencies.
Design example Obtaining column dominance
Dr. Ali Karimpour Feb 2017
Lecture 9
64
Nyquist-array methods
So optimization over a second-order structure was attempted, with uniform weighting
at all frequencies.2
2313033 )( skskksk
This achieved dominance, except
in a narrow range of frequencies
near of frequencies near 0.2 rad/s.
The weighting on frequencies
between 0.1 and 1 rad/sec was
therefore increased to 10 times
as much as on other frequencies.
Design example Obtaining column dominance
Dr. Ali Karimpour Feb 2017
Lecture 9
65
Nyquist-array methods
1221
2223
1222
3
1052.11008.31086.2
1034.31058.21007.5
1014.11078.91021.4
)(
ss
ss
ss
sk
The design obtained for the third column of the compensator is
)(),(),()( 321 sksksksKb
10-3
10-2
10-1
100
101
102
-140
-120
-100
-80
-60
-40
-20
0
20
Frequency (rad/sec)
Gain
(dB
)
Design example Obtaining column dominance
Dr. Ali Karimpour Feb 2017
Lecture 9
66
Nyquist-array methods
Design example Loop compensation
The first two columns are so dominant that the Gershgorin circles, superimposed on the
(1,1) and (2,2) loci, cannot be distinguished from the loci themselves, and the
Compensation of the first two loops is no different from compensation of a SISO system.
?00
0?0
003980
)(sKc
10-3
10-2
10-1
100
101
102
-100
-80
-60
-40
-20
0
20
40
60
80
Frequency (rad/sec)
Gain
(dB
)
Response of (1, l) element with and without compensation.
72)log(20 k
Dr. Ali Karimpour Feb 2017
Lecture 9
67
Nyquist-array methods
Design example Loop compensation
The frequency response of the (2,2) element, shown as a Nyquist plot in Figure, is
essentially constant at -25dB (=0.0562) at all frequencies.
Response of (2,2) element before compensation.
?00
0174
0
003980
)(s
sKc
Dr. Ali Karimpour Feb 2017
Lecture 9
68
Nyquist-array methods
Design example Loop compensation
The response of the (3,3) element is shown as a Nyquist plot with its Gershgorin band
(computed column-wise), and its magnitude is shown in Bode form in Figures
Response of (3, 3) element before compensation.
10-3
10-2
10-1
100
101
102
-14
-13
-12
-11
-10
-9
-8
-7
Frequency (rad/sec)
Gain
(dB
)
Gain of (3, 3) element before compensation.
Dr. Ali Karimpour Feb 2017
Lecture 9
69
Nyquist-array methods
Design example Loop compensation
Suitable compensation of this element is obtained by first changing its sign, so that the
locus starts and ends on the positive real axis, and then inserting an integrator with
enough gain to add about l0 dB at l0rad/sec.
s
ssK c
6.3100
0174
0
003980
)(
10-3
10-2
10-1
100
101
102
-40
-20
0
20
40
60
80
Frequency (rad/sec)
Gain
(dB
)
Gain of (3, 3) element after compensation.
Dr. Ali Karimpour Feb 2017
Lecture 9
70
Nyquist-array methods
Design example Loop compensation
For this element the thickness of the Gershgorin band is significant, so we should check
that the band does not overlap 1 in order to be sure that our inference of closed-loop
stability is correct.
Response of (3,3) element after compensation,
with Gershgorin band, on a Nichols chart.
Dr. Ali Karimpour Feb 2017
Lecture 9
71
Nyquist-array methods
Design example Loop compensation
The compensation is not yet finished, because the product Kb(s)Kc(s) is not realizable:
each element in the first and third columns has one more zero than poles.
REATTZATTON OF THE COMPENSATOR
Dr. Ali Karimpour Feb 2017
Lecture 9
72
Nyquist-array methods
Design example Analysis of the design
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
Re
Im
M=3 dB-circle
Characteristic loci, with 3 dB M-circle.
Dr. Ali Karimpour Feb 2017
Lecture 9
73
Nyquist-array methods
Design example Analysis of the design
10-3
10-2
10-1
100
101
102
-40
-35
-30
-25
-20
-15
-10
-5
0
5
Frequency (rad/sec)
Gain
(dB
)
Largest and smallest closed-loop singular values (principal gains).
Dr. Ali Karimpour Feb 2017
Lecture 9
74
Nyquist-array methods
Design example Analysis of the design
0 0.5 1 1.5 2 2.5 3-0.2
0
0.2
0.4
0.6
0.8
1
Time (sec)
Am
plit
ude
Closed-loop step responses to step demand on output 1 (solidcurves),
output 2 (dashed curves) and output 3 (dash-dotted curves).
Dr. Ali Karimpour Feb 2017
Lecture 9
75
Nyquist-array methods
Design example Analysis of the design
0 1 2 3 4 5 6 7 8 9 10-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Impulse Response
Time (sec)
Am
plit
ude
Response to impulse disturbance on plant input 1.
Dr. Ali Karimpour Feb 2017
Lecture 9
76
Exercise 9-3: Use state feedback to decouple the following system and put the
poles of new system on s=-3.
xyuxx
110
001
00
10
01
6116
100
010
Exercise 9-2: Decouple following system and find the decoupled transfer function.
xyuxx
1000
0010
00
11
01
10
0100
0000
0001
0000
Exercises
Exercise 9-1: Mentioned in the lecture.
Dr. Ali Karimpour Feb 2017
Lecture 9
77
Exercises
Exercise 9-4: RGA of a 4 input, 4 output system is:
919.1900.0030.2215.0
910.1270.0314.3135.0
154.1286.0429.0011.0
164.0080.0150.0931.0
Suggest suitable pairing.
Dr. Ali Karimpour Feb 2017
Lecture 9
78
Exercises
Exercise 9-5: Consider following system.(Final)
21
54
01
)(
s
sssP
a) Show that P(s) is not diagonal column dominant.
b) Show that L can convert P(s) to diagonal column dominance.
12
01L
c) Derive controller such that its steady state error to step input is zero and open loop
bandwidth of each canal be 10 rad/sec.
s
Controllerans 520
010:
d) Derive step response.
Dr. Ali Karimpour Feb 2017
Lecture 9
79
References
• Control Configuration Selection in Multivariable Plants, A. Khaki-Sedigh, B. Moaveni, Springer Verlag, 2009.
References
• Multivariable Feedback Control, S.Skogestad, I. Postlethwaite, Wiley,2005.
• Multivariable Feedback Design, J M Maciejowski, Wesley,1989.
• http://saba.kntu.ac.ir/eecd/khakisedigh/Courses/mv/
Web References
• http://www.um.ac.ir/~karimpor
• تحليل و طراحی سيستم های چند متغيره، دکتر علی خاکی صديق
• Process Dynamics and Control, Seborg, Edgar, Mellichamp and Doyle, 2011.