7/30/2019 Multivariable Lec6
1/54
Multivariable ControlMultivariable Control
Ali Karim our
Assistant Professor
7/30/2019 Multivariable Lec6
2/54
Chapter 6
Chapter 6
Introduction to Decou lin Control and Uncertaint
Topics to be covered include:
Decoupling
Pre and post compensators and the SVD controller
ecoup ng y a e ee ac
Ali Karimpour July 2012
2
7/30/2019 Multivariable Lec6
3/54
Chapter 6
Introduction
BuAxx +=&BAsICsG 1)()( =
)()(.....)()()()()( 12121111 susgsusgsusgsy pp+++=
....................................................................................
....................................................................................
..... 22221212 pp
)()(.....)()()()()( 2211 susgsusgsusgsy pppppp +++=
output is controlled by more than one input.
Ali Karimpour July 2012
3
, ,
very difficult to control a multivariable system.
7/30/2019 Multivariable Lec6
4/54
Chapter 6
Topics to be covered
Pre and post compensators and the SVD controller
Decoupling by State Feedback
Diagonal controller (decentralized control)
Uncertainty in MIMO Systems
Ali Karimpour July 2012
4
7/30/2019 Multivariable Lec6
5/54
Chapter 6
Decoupling
Definition 6-1
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
)()()( sWsGsG ss =Decoupling
Ali Karimpour July 2012
5
. .
)()()( sKsWsK ss=)(sKs
7/30/2019 Multivariable Lec6
6/54
Chapter 6
Decoupling
1. We first design a compensator to deal with the interactions in G(s) and
sWsGsG = Decou lin
Dynamic decoupling s.frequencieallatdiagonalis)(sGs
1 ss ==
(s)l(s)GK(s)IslsK -s1havewe)()(byThen ==
It usuall refers to an inverse-based controller.
Steady-state decoupling diagonal.is)0(sG
1=
Approximate decoupling at frequency 0
s
possible.asdiagonalasis)( 0jGs
1
Ali Karimpour July 2012
6
0s
)(ofionapproximatrealais 00 jGG s forselectiongoodaisfrequency 0BW
7/30/2019 Multivariable Lec6
7/54
Chapter 6
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).
Ali Karimpour July 2012
7
Anot er common strategy, w c avo s most o t e pro ems ust ment one , s to
use partial (one-way) decoupling where Gs(s) is upper or lower triangular.
7/30/2019 Multivariable Lec6
8/54
Chapter 6
Topics to be covered
Pre and post compensators and the SVD controller
Decoupling by State Feedback
Diagonal controller (decentralized control)
Uncertainty in MIMO Systems
Ali Karimpour July 2012
8
7/30/2019 Multivariable Lec6
9/54
Chapter 6
Pre and post compensators and the SVD controller
The pre compensator approach may be extended by introducing a post compensator
)()()()( sWsGsWsGssps
=
The overall controller is then
)()()()( sWsKsWsK spss=
Ali Karimpour July 2012
9
7/30/2019 Multivariable Lec6
10/54
Chapter 6
Topics to be covered
Pre and post compensators and the SVD controller
Decoupling by State Feedback
Diagonal controller (decentralized control)
Uncertainty in MIMO Systems
Ali Karimpour July 2012
10
7/30/2019 Multivariable Lec6
11/54
Chapter 6
Decoupling by State Feedback
In this section we consider the decoupling of a control system in state space
representation.
DuCx
BuAxx
+=
+=&Let DBAsICsG += 1)()(Suppose 0|D|ifThen
diagonalsGsG 1)()(
( ) 11111111 )()()( ++=+= DBDCBDAsICDDBAsICsG
u n e case o =
)()()( tTrtKxtu +=Static state feedback
Ali Karimpour July 2012
11
ryu +=a c ou pu ee ac
Dynamic output feedback
7/30/2019 Multivariable Lec6
12/54
Chapter 6
Decoupling by State Feedback
Decoupling through state feedback
Cxy
BuAxx
=
+=&( ))()()(Suppose 1 trtFxEtu =
Cx
rBExFBEAx
=
+= 11 )(&haveThen we
The transfer function matrix is 111 )()( += BEFBEAsICsG)
We s a er ve n t e o ow ng t e con t on on G s un er w c t e system can e
decoupled by state feedback.
Ali Karimpour July 2012
12
7/30/2019 Multivariable Lec6
13/54
Chapter 6
Decoupling by State Feedback
Theorem 6-1 A system represented by
Cxy
BuAxx
=
+=&
with the transfer function matrix G(s) can be decoupled by state feedback of the form
( ))()()( 1 trtFxEtu =
if and only if the constant matrix E is nonsingular.
=
d
new
s
sG
0
)(
1
OFurthermore the new system is in the form:
1E
mds0
dAC 11
=
=
)(
0
0lim
.
.
12
sG
s
sEE
pd
d
sO
=F
.
.
2
2
Ali Karimpour July 2012
13
Ep
Proof: See Linear system theory and design Chi-Tsong Chen
pd
pAC
7/30/2019 Multivariable Lec6
14/54
Chapter 6
Decoupling by State Feedback
Example 6-1 Use state feedback to decouple the following system.
01010
xyuxx
=
+
=110
00
10
6116
100&
Solution: Transfer function of the system is
+++
+
+++
++
==
66
6116
6
6116
116
)()(
2323
2
1
s
sss
s
sss
ss
BAsICsG
++++ 6565 22 ssss
The differences in degree of the first row ofG(s) are 1 and 2, hence d1=1 and
]01[6116
66116
116lim23231 =
+++
++++
++= sss
ssss
sssEs
The differences in de ree of the second row ofG s are 2 and 1 hence d=1 and
Ali Karimpour July 2012
14]10[
65
6
65
6lim
222=
++
++
=
ss
s
sssE
s
7/30/2019 Multivariable Lec6
15/54
Chapter 6
Decoupling by State Feedback
=
10
01E
Solution (continue):
Now E s un tary matr x an c ear y nons ngu ar so ecoup ng y state ee ac s
possible and
=
=01011
d
dACF
2
( )
==
)()(01001
)()()( 1 trtxtrtFxEtu
The decoupled system is
rxrBExFBEAx
+
=+= 10
01
6116
000
)( 11&
xCxy
==
001
006116
Ali Karimpour July 2012
15Exercise 1: Derive the corresponding decoupled transfer function matrix.
7/30/2019 Multivariable Lec6
16/54
Chapter 6
Property of Decoupling by State Feedback
1- All poles of decoupled are on origin.
2- Decoupled system is:
ndd
decouple ssdiagsG= ...,,)( 1
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.
Ali Karimpour July 2012
16
7/30/2019 Multivariable Lec6
17/54
Chapter 6
Decoupling by State Feedback
Exercise 2: Decouple following system and find the decoupled transfer function.
100000
xyuxx
=
+
=1000
00
11
0100
0000&
Exercise 3: Use state feedback to decouple the following system and put the
poles of new system on s=-3.
xyuxx
=
+
=110
00110
01
100
010
&
Ali Karimpour July 2012
17
7/30/2019 Multivariable Lec6
18/54
Chapter 6
Topics to be covered
Pre and post compensators and the SVD controller
Decoupling by State Feedback
Diagonal controller (decentralized control)
Uncertainty in MIMO Systems
Ali Karimpour July 2012
18
7/30/2019 Multivariable Lec6
19/54
Chapter 6
Diagonal controller (decentralized control)
Another simple approach to multivariable controller design is to use a diagonal or
. .
Clearly, this works well ifG(s) is close to diagonal, because then the plant to be
,
K(s) may be designed independently.
However if off dia onal elements in G s are lar e then the erformance with
Ali Karimpour July 2012
19
decentralized diagonal control may be poor because no attempt is made to counteract
the interactions.
7/30/2019 Multivariable Lec6
20/54
Chapter 6
Diagonal controller (decentralized control)
e es gn o ecen ra ze con ro sys ems nvo ves wo s eps:
_
2_ The design (tuning) of each controllerki(s)
Ali Karimpour July 2012
20
7/30/2019 Multivariable Lec6
21/54
Chapter 6
Input-Output Pairing
Definition of RGA (Relative Gain Array)
uu +=Physical Meaning of RGA: Let
ijijij hg /=2221212 ugugy +=
Relative gain?
i
jiij u
yuyg
= or0inputsotherifandbetweenrelation
uk= ,
ijiij
u
yuyh
= or0outputsotherifandbetweenrelation
2121111 ugugy
=
+=1
212 u
gu =
ikyk
= ,0
121
12111 )( ug
ggy
+=
Ali Karimpour July 2012
21TGGGRGAG == )()(
22
7/30/2019 Multivariable Lec6
22/54
Chapter 6
Input-Output Pairing
Example: Let
==
1
1)( TGGG
2221212
2121111
ugugy
ugugy
+=
+=
=1 Open loop and closed loop gains are the same,so interactions has no effect.
=0 g11=0 so u1 has no effect on y1.
0
7/30/2019 Multivariable Lec6
23/54
Chapter 6
Input-Output Pairing
RGA property:
- .
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. Ifij() (0) exist andhave 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.
Ali Karimpour July 2012
23
6- If gijgij(1-1/ij) then the perturbed system is singular.
7- Changing two columns/rows of G leads to same changes to its RGA
7/30/2019 Multivariable Lec6
24/54
Chapter 6
Diagonal controller (decentralized control)
Example 6-2 Select suitable pairing for the following plant
= 7.04.85.154.16.52.10
)0(G
...
Solution: RGA of the system is
= 43.037.094.0
41.145.196.0
)0(
...
Ali Karimpour July 2012
24
7/30/2019 Multivariable Lec6
25/54
Chapter 6
Diagonal controller (decentralized control)
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 istaken into consideration.
Many improved approaches, RGA-like, have been proposed and described in all
process control textbooks, for defining different measures of dynamic loop
.
[1] D.Q. Mayne, The design of linear multivariable systems,Automatica, vol. 9, no. 2, pp.
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),
Ali Karimpour July 2012
25
,
[3] RNGA based control system configuration for multivariable processes
Mao-Jun He, Wen-Jian Cai *, Wei Ni, Li-Hua XieJournal of Process Control 19 (2009) 10361042
7/30/2019 Multivariable Lec6
26/54
Chapter 6
Diagonal controller (decentralized control)
Next example, for which the RGA based loop pairing criterion gives an
inaccurate interaction assessment, are employed to demonstrate the
e ect veness o t e propose nteract on measure an oop pa r ng cr ter on.
Example 6-3:
Consider the two-input two-output process:
RGA=Diagonal pairing= - agona pa r ng
To illustrate the validity of above results, decentralized controllers
forboth 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
Ali Karimpour July 2012
26
used to evaluate the control performance.
7/30/2019 Multivariable Lec6
27/54
Chapter 6
Diagonal controller (decentralized control)
The simulation results and ISE values are given in Fig. 3. The results show
that the off-diagonal pairing gives better overall control system performance.
off-diagonal
diagonal
Ali Karimpour July 2012
27
7/30/2019 Multivariable Lec6
28/54
Chapter 6
Topics to be covered
Pre and post compensators and the SVD controller
Decoupling by State Feedback
Diagonal controller (decentralized control)
Uncertainty in MIMO Systems
Ali Karimpour July 201228
7/30/2019 Multivariable Lec6
29/54
Chapter 6
Uncertainty in MIMO Systems
LQG Control: Optimal state feedback
( )
+=
0
dtRuuQzzJ TT
r
0and0,where >=== TTxz
The optimal solution for any initial state is
txtu r=
where
BRK T1=
Where X=XT0 is the unique positive-semidefinite solution of the algebraic
Riccati equation
Ali Karimpour July 2012290
1 =++
QMMXBXBRXAXA
TTT
7/30/2019 Multivariable Lec6
30/54
Chapter 6
Uncertainty in MIMO Systems
Robustness Properties
For an L R-controlled s stem if the wei htR is chosen to be dia onal then
( )( )11 += BAsIKIS r satisfies ( ) ,1)( jS
mikand ii ,...,2,1,5.00 =
7/30/2019 Multivariable Lec6
31/54
Chapter 6
Uncertainty in MIMO Systems
Example 6-4: LQR design of a first order process.
32 +=
ssG uxx
101
+
=&23 ++ ss
The cost function to be minimized is
[ ]xy 11=
dtRuyr +=0
.=
75.3816]-86.7008[1
==
XBRKT
r
( ) ibkA 9828.61596.7 =
( ) 0 allforstablebkAmikand ii ,...,2,1,5.00 =
7/30/2019 Multivariable Lec6
32/54
Chapter 6
Uncertainty in MIMO Systems
Example 6-5: Decoupling controller
=564721
)(2
sssG
ss
1
The pre compensator approach may be extended by introducing a post compensator
)(
2
20
176
87)(
76
87sG
s
ssG d=
+
+=
)()()()( sWsGsWsG ssps =
The overall controller is then
)()()()( sWsKsWsK spss=
=
=k
k
k
ksK
0
0
76
87
0
0
76
87)(
We have good stability margin in both channel.
k 0
Ali Karimpour July 201232
xerc se : er ve s a y marg n or eren va ue o
+
=k
sK0
)(
For k=1 so find the smallest that lead to instability. Repeat for k=2.
Chapter 6
7/30/2019 Multivariable Lec6
33/54
Chapter 6
Uncertainty in MIMO Systems
Type of uncertainty
.
Model structure and order are known, but (some)arameter values are uncertain.
)(
)(
ass
ksG
+
=
Dynamic (frequency-dependent) uncertainty or nonparametricuncertainty. unstructured uncertainty
There exists (some) erroneous or missing
dynamics. Usually unmodeled dynamics is in
Ali Karimpour July 201233
high frequencies.
Chapter 6
7/30/2019 Multivariable Lec6
34/54
Chapter 6
Uncertainty in MIMO Systems
Type of unstructured uncertainty
Ali Karimpour July 201234
Chapter 6
7/30/2019 Multivariable Lec6
35/54
Chapter 6
Uncertainty in MIMO Systems
Parametric uncertaintyNonparametric uncertainty
Example 6_6: Consider a plant with parametric uncertainty
maxmin0 )(1
)( = sGsG +psNow let
1,
2/)(
)(
1
)1(maxmin
maxmin
7/30/2019 Multivariable Lec6
36/54
p
Uncertainty in MIMO Systems
Parametric uncertaintyNonparametric uncertainty
Example 6_7: Consider a plant with two parametric uncertainty
3,,2 = kek
sG s
+ s
Ali Karimpour July 201236
Chapter 6
7/30/2019 Multivariable Lec6
37/54
p
Uncertainty in MIMO Systems
3,,21
)( +
=
kes
ksG sp
Consider additive uncertainty as:
+= ,1)();()()()( jsswsGsG Ap
Additive uncertainty can be representBy multiplicative one:
( ) ,1)(;)()(1)()( jsswsGsG Mp +=
Ali Karimpour July 201237
)()(
jGjwM =
Chapter 6
7/30/2019 Multivariable Lec6
38/54
Uncertainty in MIMO Systems
System without uncertainty
=
wPPz 1211
Ku =
NwwPKPIKPPz =+= 211
221211 )(
2221
),( KPFN l=
System with uncertaintystructure
u ou
uncertainty
Ali Karimpour July 20123838
Suitable forrobust
performance analysis
Chapter 6
7/30/2019 Multivariable Lec6
39/54
Uncertainty in MIMO Systems
System without uncertainty
=
wPPz 1211
Ku =
NwwPKPIKPPz =+= 211
221211 )(
2221
),( KPFN l=
S stem with uncertaint N structure
=
w
u
NN
NN
z
y
2221
1211
= yu
FwwNNINNz =+= 121
112122 )(
Ali Karimpour July 201239
),( = FF u
Chapter 6
7/30/2019 Multivariable Lec6
40/54
Robust Stability of Parametric Uncertain Systems
Robust stability in parametric uncertainty
Ali Karimpour July 201240
Chapter 6
7/30/2019 Multivariable Lec6
41/54
Robust Stability of Unstructured Uncertain Systems
structureNSystem with uncertaintySystem without uncertainty
Suitable forrobust performance analysis
FwIz =+= 1
Suitable fornominal performance analysis
NwwPKPIKPPz =+= 1
ionConfigurat
on roenera
Checking robuststability?
Ali Karimpour July 201241Suitable forcontroller design
Chapter 6
7/30/2019 Multivariable Lec6
42/54
Robust Stability of Unstructured Uncertain Systems
structureNSystem with uncertaintySystem without uncertainty
Suitable forrobust performance analysisSuitable fornominal performance analysis
FwIz =+= 1
If there is no uncertainty we
structureM
N11, N12, N21 and N22 are
stable
Ali Karimpour July 2012
42
Suitable forcontroller design
Suitable forrobust stability analysis11=
Chapter 6
7/30/2019 Multivariable Lec6
43/54
Robust Stability of Unstructured Uncertain Systems
structureM
Suitable forrobust stability analysisNS:Nis internally stable
RS: NS and F=F u(N,) is stable for any ||||1
Theorem: RS for unstructured(full) perturbation.
Assume that the nominal systemM(s) is stable (NS) and that the
perturbations (s) are stable. ThenThe M-structure is stable
( )
7/30/2019 Multivariable Lec6
44/54
Robust Stability of Unstructured Uncertain Systems
Suitable forrobust stability analysisSystem without uncertainty
12 wwp +=
= MuySystem with additive uncertainty
2
1
1 )( wGKIKwM+=
Robust stabilit condition: In
the case of |||| 1
11
7/30/2019 Multivariable Lec6
45/54
Robust Stability of Unstructured Uncertain Systems
System without uncertaintySuitable forrobust stability analysis
)( 12 wwIGGp +=
= MuySystem with multiplicativeinput uncertainty
21 ww +=
Robust stability condition: In
the case of |||| 1
1)( 1
7/30/2019 Multivariable Lec6
46/54
Uncertainty in MIMO Systems
Suitable forrobust stability analysisSystem without uncertainty
p 12
= MuySystem with multiplicative
2
1
1 )( wGKIGKwM+=
ou pu uncer a n y
Robust stability condition: In
the case of |||| 1
11
7/30/2019 Multivariable Lec6
47/54
Robust Stability of Unstructured Uncertain Systems
System without uncertaintySuitable forrobust stability analysis
1
12 )(= GwwIGGp
System with inverse additive uncertainty = Muy
21 ww +=
Robust stability condition: In
1)( 21
1
7/30/2019 Multivariable Lec6
48/54
Robust Stability of Unstructured Uncertain Systems
System without uncertaintySuitable forrobust stability analysis
112 )( = wwIGGp 112
7/30/2019 Multivariable Lec6
49/54
Robust Stability of Unstructured Uncertain Systems
System without uncertaintySuitable forrobust stability analysis
GwwIGp1
12 )( = 112
7/30/2019 Multivariable Lec6
50/54
Robust Stability of Unstructured Uncertain Systems
Suitable forrobust stability analysis
Perturbed PlantUncertainty M in M-structure
Additive uncertainty2
1
1 )( wGKIKwM+=12 wwGGp +=
Multiplicative input uncertainty2
1
1 )( GwGKIKwM+=)( 12 wwIGGp +=
Multiplicative output uncertainty2
1
1 )( wGKIGKw+=GwwIGp )( 12+=
Inverse additive uncertaint 1=1
=p
Inverse multiplicative input uncertainty2
1
1 )( wKGIwM+=( ) 112
= wwIGGp
Ali Karimpour July 2012
50
Inverse multiplicative output uncertainty2
1
1 )( wGKIwM+=( ) GwwIGp
1
12
=
Chapter 6structureM
7/30/2019 Multivariable Lec6
51/54
Robust Stability of Unstructured Uncertain Systems
System with coprime factor uncertainty Suitable forrobust stability analysis
ll NMG1=
)()( 1 NlMlp NMG ++=
[ ]MN = 11)( +
= lMGKII
M
Since there is no weight for uncertainty so the theorem is
Ali Karimpour July 2012
51
:
7/30/2019 Multivariable Lec6
52/54
Robust Stability of Unstructured Uncertain Systems
Remind Example 6-5: Decoupling controller
564721 ss kk 087087
+++ 25042232 ssss kk 076076
Exercise 4: Derive stability margin for different value of if
=k
sK0
)(
For k=1 so find the smallest that lead to instability.
)( += IGGpConsider system with multiplicative input uncertainty
) 1)( 1
7/30/2019 Multivariable Lec6
53/54
Robust Stability of Unstructured Uncertain Systems
Suitable forrobust stability analysis)GGKIK 1)(/1)( +
7/30/2019 Multivariable Lec6
54/54
Robust Stability of Unstructured Uncertain Systems
Exercise 5: Consider following block diagram. We have both input and output uncertainty.
a) Find the set of possible plants(G )
b) Find M and derive robust stability condition. ( )1and,1
io
Exercise 6: Assume we have derived the following detailed model:
Ali Karimpour July 2012
54
Suppose we chose G(s)=3/(2s+1) with multiplicative uncertainty. Derive suitable scaling
Matrix.