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Approximate Decoupling of Multivariable Control Systems A Time-domain Approach.

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Dev. Chem. Eng. Mineral Process. 13(3/4). pp. 361-3 70, 2005.
Approximate Decoupling of Multivariable
Control Systems: A Time-domain Approach
Wei-Jie Mao* and Shao Liu
National Lab of Industrial Control Technology, Institute of Advanced
Process Control, Zhejiang University, Hangzhou 310027, P.R.China
A new approximate decoupling method for linear time-invariant systems based on
the concept of energetic diagonal dominance is proposed in this paper. It is a timedomain approach and the design procedure is simpler than that of j-equencydomain approach. The conditions for the energetic decoupling with simultaneous
stabilizability are established in terms of linear matrix inequalities (LMIs). It is
very convenient to solve this problem using the powerfil LMI toolbox. The results
are derived first for proper systems with nonsingular D, and then extended to
strictly proper systems. Illustrative examples demonstrate that this method has
satisfactory decoupling pe~ormance.
Introduction
Decoupling technique for the design of multivariable control systems is of
considerable theoretical and practical importance. A great deal of interest has been
brought to this problem over recent decades. The aim of decoupling is that each
output of a multi-input multi-output system can be independently controlled by a
corresponding input. On the view of the depth of decoupling, it can be divided into
complete decoupling and approximate decoupling. A number of state-space
methods aimed at Morgan’s problem [e.g. 1-41 all belong to the complete
decoupling approach. Based on complete cancellation, the decoupling performance
of these methods will be impossibly achleved in the case of perturbation.
Furthermore, if an original system has zeros, pole-zero cancellation will take place
in the decoupling procedure. For instance, we consider the following linear timeinvariant system:
. According to [2],
L
it is found that
J
this system can be decoupl d by the following control law:
* Authorfor correspondence (wjmao@iipc.zju.edu.cn).
361
Wei-Jie Ma0 and Shao Liu
u ( t ) = Fx(t)+ Gv(t)
where F = [-2
-1
-4
0
0],G=[ 1
-1
-1
1.
0
1
{::I
The closed-loop transfer matrix is: H,,(s) = diag
-,-
Indeed, a 3rd-order system has been decoupled to a 2nd-order system. Only two
poles of the decoupled system can be determined. The lost pole (-1), which is fixed,
is merely the zero of the original system. As a result, if the original system has
zeros in right half plane, it can be impossibly decoupled with closed-loop stability.
However, the frequency-domain method [6, 71 belongs to the approximate
decoupling approach. The aim of this method is the diagonal dominance of the
controlled plant, not the diagonalization. Thus, avoiding the drawback of the
complete decoupling approach is possible. The key step of this method is how to
achieve the diagonal dominance by a certain pre-compensator. Until now practical
and efficient methods do not exist to acheve the diagonal dominance, except for the
pseudo-diagonalization method which is based on a single frequency and unlikely
to be extended to the multi-frequency case.
In this paper, we are looking for a new approximate decoupling method. Similar
to the concept of diagonal dominance [6,7], energetic diagonal dominance in timedomain framework is introduced. Thus, the proposed method is a time-domain
approach and the design procedure is simpler than that of Rosenbrock [6, 71 . The
problem to achieve energetic diagonal dominance is called as an energetic
decoupling problem (EDP) here. Our contribution consists in establishing the
conditions for the EDP with simultaneous stabilizability. All the results are derived
using the LMI approach, for which there are efficient algorithms such as LMI
toolbox in MATLAB.
Problem Formulation
Consider the linear time-invariant (LTI) system described by state-space model of
the following form:
X(t) = Ax(t) + Bu(t)
y ( t ) = Cx(t)+ Du(t)
...(1)
where x ( t ) E R" is the state vector; u(t) E R" is the input vector; y(z) E R" is the
output vector of the above system satisfying m I n ; A, B,C, D are constant real
matrices of suitable dimensions satisfying detD # 0 . Employing a control law of
the form:
u(t) = Fx(t)+ Gv(t)
with v ( t )E R" , the resultant system is:
362
...(2)
Approximate Decoupling of Multivariable Control Systems
x ( t ) = ( A + BF)x(t)+ BGv(t)
y ( t ) = (C + DF)x(t)+ DGv(t)
...(3)
Definition 1: The LTI system of Equation (3) with zero-initial condition is said to
be energetically diagonal row dominant 8
I,'j:(t)dt I I,'yi(t)dt
n
...(4)
V r 2 0,vj = vi,i = 1,2,--.,rn
where yii represents the part in thoutput caused by iIh input vi , jii represents the
part in f h output caused by the other inputs but thinput (denoted as Gi). Energetic
diagonal column dominance can be defined similarly by:
j,'j,'(t)ji(t)dt I Iiyf(t)dt
vz 2 0,vj = ~
(# i),i
j = 1,2,..-,m
... ( 5 )
where j i E R"-' represents the vector composed of all outputs in y but thoutput yi.
In contrast to diagonal dominance in the frequency-domain framework, the
concept of energetic diagonal dominance by Definition 1 is in time-domain
framework, and thus easy to understand. However, the inequalities of Equations (4)
and ( 5 ) are not easy to solve. They need to be transformed into more solvable
forms. For instance, Equation (4) can be transformed into:
jiyii(t)vi(t)dt2 aijiv,?(t)dt
...(6)
Iiji(t)dt I fliIiGr(t)Ci(t)dt,Vr 2 0,vj = vi,i = 1,2,...,m
...(7)
It is obvious that Equation (4) can be guaranteed by Equations (6) and (7) with
properly chosen positive constants ai,pi. Then, the EDP of LTI system (1) can be
summarized as: to find F E R""",G E R""" , such that, for the closed-loop system
of Equation (3), then Equations (6) and (7) are satisfied.
Remark 1. The inequalities in Equations (6) and (7) may be meaningless with
r + 00 . However, we validate them in this paper if ~im~;yri(r)vr(t)dr
and
r+-
Iiii(tvf
!!
I;v,?(tWt
exist. It is the same for inequalities of Equations (4) and (5).
j~iy(1)qI)dl
Remark 2. The decoupling perjormance of closed-loop system is dependent on the
prescribed positive constants a = [a, a2
am ,fl =
p, . p,,, , and
on the controller F,G . Thus, we denote the EDP as EDP(a,fl I F,G) in detail.
However, a,P may not be regarded as the final decoupling measure because of
the inequality feature of (6), (7). Final decoupling per$ormance will be determined
by the input-output responses of closed-loop system with designed controller.
p
b,
p
363
Wei-Jie Ma0 and Shao Liu
Note that the stability is the most important goal in the system synthesis, we
shall consider the stable version of the above problem (namely EDPS) in the
following.
Definition 2: The EDPS(a,P I F,G) of LTI system (1) is said to be solvable $
(1) there exist a,P and F,G satisfiing inequalities (6) and (7) with zeroinitial condition;
(2) the closed-loop system (3) is asymptotically stable.
Conditions for EDPS
If the LTI system (1) is open-loop stable, the control law (2) may be simplified as:
...(8)
u(t) = Gv(t)
Then the closed-loop system (3) becomes:
i ( t ) = Ax(t) + BGv(t)
y ( t ) = &(t) + DGv(t)
...(9)
Theorem 1. The EDPS (a,P I0,G) of LTI system (1) is solvable if there exist
matrices X i , q > 0 , i =1,2,.-.,m, such that:
X,A' + AX,
g:Br - C,X, 2a,I - (D,g,+ gfD:)
KA'+AK
Bg,
KC:
-pi+I iTD: < O
gfBT
c,<
where
D,i,
-dI
1
iiE Rmx("'-')represents the matrix composed of all columns in
...(11)
G but the ih
column g, , Ci E RIXnand D, E RIXmrepresent the rows of C and D respectively.
Proof. Consider the i" output of Equation (9),which can be divided into two
subsystems described as:
X(t) = Ax@)+ Bg,v,(t)
yii(t) = Cix(t)+ Digivi(t)
and:
I
m
x(t) = Ax(t)+ BCg,v,(t) = Ax@)+BiiCi(t)
1
364
j=l
jti
I"
Approximate Decoupling of Multivariable Control Systems
Define
quadratic h c t i o n s K ( t , x ) = x'(t)@(t)
q ( t , x ) = x'(t)Q.x(t)
,
e,Qi> 0 , for above subsystems respectively. It is easy to show that:
d
dr
M , ( t ) = -~ , ( t , x-) 2yii(t)vi(t)
+ 2aiv,?(t)
= x T ( t ) ( A T+4@ ) x ( t ) +
[
2x'(t)eBg,vi(t)- 2yii(t)v,(t)+2a,vf(t)
A T 4+ 4 A
eBgi - Cr
vi(t) g,?BT4-Ci 2ai1-(Digi
= x(t)]'[
d
dt
+ pl:+j;(t)
N i ( t ) = - (t,x) - pif;,?(t)i(t)
$11
$1
= xT(t)(ATQi
+QiA)x(r)+2~T(~)QiB~i~i(t)-pi~~,?(t)ti(f)+~~~ji:(r)
=[
A'Qi +QiA QiBii
if13'Qi
ci
-Ddi iIi
C:
i,?Dr][ pi ' y ,
Pi 'Yii
c-',
Set X i =
= Ql:' . Then it follows immediately from inequalities of Equations
(10) and (11) that M i ( t ) < O , N i ( t ) < O . Integrating both sides of the above
inequalities from 0 to 5 with initial condition x(0) = 0 , we obtain:
j,'Mi(t)dt= y(r,x(r))-jl(2yiivi- 2aiv,2)dt 5 o
j,'Ni(t)dt = q ( r , x ( s ) ) + j , ' ( ~ ; -/$?,?~~)dt
+jj
Io
Thus, for all
r 20:
j,'yiividt 2 aijOrv,?dt,
j,'jtdt 5 pij,';:Cidt
In addition, with v(t) = 0 , selecting the quadratic function V;,(t,x) or q . ( t , x ) as
Lyapunov hnctional candidate with nonzero-initial condition, we have:
<(t,x)
<o , q ( t , x ) <o
Therefore the condition of Equations (10) or (1 1) can guarantee the asymptotic
stability of system (9). This completes the proof of the theorem.
For the LTI system (1) with unstable system matrix A, a simple way to achieve
energetic decoupling may be described as: fKst, to stabilize LTI system (1) with
state feedback; then, to design input transformation with the above method. Due to
insufficient utilization of state feedback, it may result in conservative design. In the
following we consider the EDPS with simultaneous state feedback and input
transformation.
Theorem 2. The EDPS(a,PI F,G)of LTI system (1) is solvable if there exist
matrices Z and X > 0, such that:
365
Wei-Jie Ma0 and Shao Liu
X.4' + A X + Z T B T+ BZ
gTBT -C,X-D,Z
Bg, -XCT -ZTDT
2aiI-(Dig,+g,?DT)
M T+ AX + Z TBT + BZ
g;BT
CiX + DiZ
Bi,
XCT + 2'0:
-p/Z
$DT
- p,!I
[
where
Diii
...(12)
1
<O
...(13)
iiE Rmx(m-"represents the matrix composed of all columns in G but the ilh
column gi , CiE RIXnand D, E RIXmrepresent the rows of C and D respectively.
Furthermore, the state feedback matrix is:
...(14)
F=m-'
Proof. Applying Theorem 1 to the closed-loop system (3), this theorem follows
immediately from the definition of Z = FX and Xi= $ = X , i = 1,2,...,m
Extension to Strictly Proper Systems
Usually, D in the LTI system ( 1 ) is zero. In th~scase, the LTI system ( ) can be
described by state-space model of the following form:
X(t) = Ax(t)+ Bu(t)
Y ( t )= CxO)
..(15)
Let H(s) be the transfer matrix of (15). There are unique integers di, i = 1, 2, ..., m,
such that:
lims"'"H,(s) = E~
...(16)
I +m
where H,(s)is rh row of H ( s ) , and Ei is both finite and nonzero. Obviously, d , ,
i = 42;. .,m , are nonnegative and there results in the following lemma.
Lemma 1 [2]:Let d,, i=1,2;--,m begiven by Equation (16). Then:
di= {
min{j : C,A'B
f
0 , j = 0,1,...,n - 1)
n-1 if C,A'B=O forallj=O,l,...,n-l
...(17)
Based on Lemma 1 , the following LTI system:
H ( s ) = diagb''
+I,
..,sd-
)H(s)
can be described by state-space model of the following form:
366
...(18)
Approximate Decoupling of Multivariable Control Systems
If d e 6 # 0 , applying the results above to Equation (19), we find the control law (2)
such that, for the resulting closed-loop system HJs) from (19) with zero-initial
condition, energetic diagonal row dominance is achieved. Let H,,(s) be the
resulting closed-loop system from (15) with the same control law. Then:
H,,(S) = diag
Sd"""-1
{ l
Sd, + I } H , ( S )
Therefore, H,,(s) is of the same decoupling performance as
...(20)
pc,(s). However,
despite the stability of gc/(s), H,,(s) is an integrator system. What is still required
is a procedure for specifying the poles of H,,(s) while reserving the decoupling
performance. In fact, we can incorporate these two procedures. Let:
,
.(21)
where y,(j = 1,2,...,d, + 1,i = 1,2,...,m ) are selected by the requirement of closedloop system poles. fi(s) can be described by state-space model of the form:
( X ( t ) = Ax(t)
+ Bu(t)
If d e G # 0,applying the results of the above section to Equation (22), we will find
the control law (2) such that, for the resulting closed-loop system Ec,(s)from (22)
with zero-initial condition, energetic diagonal row dominance is achieved. Then,
with the same control law:
..(23)
Therefore, H,(s) is of the same decoupling performance as gc,(s). The stability
of H,(s) can be guaranteed by the appropriate selection of y, and the application
of the results of the above section.
367
Wei-Jie Ma0 and Shao Liu
Illustrative Examples
Example 1. Consider the LTI system as in (l), with:
In this example, A is unstable. By Theorem 2, choose a,= a2= Dl = p2 = 1 , then
we get the state feedback and input transformation matrices:
’=[
-4.6898
0.1132
[
8.1118 3.53211
-5.7065 -1.02781
,G=
- 0.373 1 7.0004
-2.9086 -3.0030
The step response of the resulting closed-loop system H,,(s) is shown in Figure 1.
Example 2. Consider the LTI system as in (15), with
This example is a variant of Example 1 with D = 0 . Note that C,B = [l 01f 0 ,
C2B= [l 1]+ 0 , thus d , = d, = 0 . Selecting y,, = y2, = 2 , we get i ( s ) described
by state-space model of the form (22) with:
Apply Theorem 2 to E(s) and choose a,= a, = p, = p2 = 1 , then we get the state
feedback and input transformation matrices:
- 3.4457
0.1528
] [
1.2026
- 5.0742 - 0.2349
,G =
-0.7155
-0.6899 -2.0727
0.28561
1.0993
The step response of the resulting closed-loop system H,,(s) is shown in Figure 2.
The poles of this closed-loop system are: eig(A + BF) = -1.4056, -0.8070 f 0.7334i.
It is obvious that none of the poles is fixed to -1, zero of the original system.
Therefore, this method avoids the pole-zero cancellation in the decoupling
procedure as stated for the example in the Introduction.
Conclusions
This paper has presented an energetic decoupling method, which regards the
reduction of energetic relationship as the main goal. In contrast to the complete
decoupling approach, this method avoids the sensitivity to the model perturbation
368
Approximate Decoupling of Multivariable Control Systems
Figure 1. Step response of Example1 with designed controller.
Time (secs)
0.2
Tim a (secs)
Time (ssca)
Figure 2. Step response of Example2 with designed controller.
and the pole-zero cancellation in the decoupling procedure. Conditions for the
proper linear time-invariant systems, as well as strictly proper linear time-invariant
systems, are obtained in terms of LMIs. A memory-less state feedback and input
transformation controller can be constructed by solving LMIs to guarantee both
stability and decoupling performance.
369
Wei-Jie Ma0 and Shao Liu
Acknowledgments
This work was supported by the National Natural Science Foundation of People's
Republic of China under grant 60004002, and by the Zhejiang Provincial Natural
Science Foundation of China under grant 602055.
References
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Falb, P.L., and Wolovich, W.A. 1967. Decoupling in the design and synthesis of multivariable
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Gilbert, E.G.1969. The decoupling of rnultivariable systems by state feedback. SIAM J. Control
Optimization, 16,83-105.
Morse, A.S., and Wonham, W.M. 1971. Status of noninteracting control. IEEE Trans. Automatic
Control, 16,568-581.
Descusse, J., Lafay, J.F., and Malabre, M. 1988. Solution to Morgan's problem. IEEE Trans.
Automatic Control, 33,732-739.
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7. Rosenbrock, H.H. 1974. Computer Aided Control system Design. Academic Press, New York.
8. Vafiadis, D., and Karcanias, N. 1997. Decoupling and pole assignment of singular systems: a
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and Control Theory. SIAM,Philadelphia, USA.
Received 15 November 2003; Accepted after revision: 9 September 2004.
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