International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 4 Number2–Aug 2012
Interaction reducer for closed-loop control of multivariable systems
M.Bharathi1 and C. Selvakumar+
1
Department of Electronics & Instrumentation Engineering, Bharath University,Selaiyur,
Chennai – 600 073, India.
+
Department of Instrumentation & Control Engineering, St. Joseph’s College of Engineering,
Chennai – 600 119, India.
Abstract: In process multi-input multi-output (MIMO) systems are observed that are dominated
by interactions between input and outputs. Process decouplers are synthesized to reduce these
interactions before controller design. In this work analytical expressions for closed-loop interactive
transfer functions are found and interaction reducers (IR) are designed (as process decouplers) for
closed-loop control to minimize the interactions between input/outputs of distillation column. Multiloop
PI controllers are tuned using different tuning rules and closed-loop systems with IRs are simulated. The
closed-loop results reveal that undesirable responses due to interactive transfer functions are made
negligible.
Key words: Multivariable control, PI control, Interaction, Sequential design, Decouplers
1. Introduction
Chemical and biochemical processes invariably exhibit large dead time characteristics and
process non-linearity, which makes the controller design complex. Moreover the existence of
noise and process interactions generally adds up to the problem. One of the objectives of
multivariable control is to maintain several controlled variables at independent set points so that
the process operates safely. The controller must be designed in such a way that it takes care of
load changes and will be capable of rejecting disturbances effectively. Interactions between input
and output cause a manipulated variable to affect more than one controlled variable and hence
MIMO processes are difficult to control.
The task of developing a satisfactory design procedure for multivariable PI type process
controllers remains a difficult problem. Niederlinski(1971) propose a generalized heuristic
method to tune PID controllers based on Zeigler-Nichols criteria. But as it is complex and yield
poor performance, it is rarely use. Multiloop PI controllers were designed using biggest log
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modulus tuning(BLT) for ten distillation ttcolumns (Luyben,1986) of different input output
structures. This BLT method was refined by Basuldo and Marchetti(1990) where each individual
loop was tuned with IMC rule to satisfy closedloop stability and performance.Chiu and Arkun
(1992), Shen and Yu (1994) and Shiu and Hwang(1998) proposed sequential design method for
controller design of multivariable processes that reduces the MIMO problem to a SISO problem.
This method of tuning is an efficient method indeed. A transfer function matrix or, equivalently,
a set of individual transfer function, is formulated that simplifies the design of controller.
Recently, after selecting of proper input output paring, various controller structures like
centralized, decentralized and decoupled were explored and tuning methods like IMC-PI and
BLT methods were used by (Selvakumar et.al,2006) to evaluate performances of 2x2, 3x3
distillation columns in terms of IAE. They also studied parametric sensitivity and related
robustness issues on the stability of closed loop system. Two schemes for iinverted decoupling
technique using internal model control tuning were proposed (Chen & Zhang 2007) to control
multivariable systems with delays and zeros.Decentralized robust PI controller was designed
(Labibi et al.2009 ) using closed-loop stability criteria for industrialboiler. Control structure
(configuration) was selected (He et.al., 2009) by interaction analysis using relative normalized
gain array (RNGA) criteria that uses both steady state and transient perspectives. Decoupling
control of electro pneumatic pressure system tank with different conditions are explained by
Sompol et.al(2009).
After isolating input and output pairs of a multivariable system, relative gain (ii) array (Bristol)
is analyzed to find out correct input-output pairing for effective control. If ii is far from unity,
then decentralized controller may not show good results, and it is preferred to use decoupler. By
the method of decoupling, we decouple the process and then design a suitable controller for
decoupled process. Thus there exist two relative gains : one for the process before decoupling (ii
far away from 1) and the other for the process after decoupling (ii=1).But, in the present method
of interaction reducer, interaction transfer functions (ITF) are found through sequential design
and then a block is designed and added to counter-act the ITF and, thus in this method, the above
mentioned problem can be avoided. The objective of the present article is to find out analytical
expressions for interaction transfer functions and hence derive interaction reducer and then use
PI control to study the closed-loop performance of the MIMO system.
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Thus the rest of the paper is arranged as follows: Principle of designing IR for a 2x2 system is
presented in section 2. Section 3 discusses simulation results of implementing an IR on WB
column. Comparison of performances with IR and decouplers are also brought for perusal in this
section. The conclusion is drawn at the end.
2. Design of Interaction Reducer’s transfer function
2.1 Closed-loop interactions
Figure 1. Schematic representation of a 2 X 2 multivariable system
For a 2x2 mimo system shown in Figure 1, one can easily find the sequential transfer functions
as
 
 G G
G22GC 2 
1 
as G11,CL  G11 1  12 21
  G11 1  1 

 G11G22 I  G22GC 2 
  11 (s )  
(1)
 
 G G
G11GC1 
1 
and G22,CL  G22 1  12 21
  G22 1  1 

 G11G22 I  G11GC1 
  22 ( s )  
(2)
Where suffix CL represents closed-loop. Using sequential design approach ( Shen, 1994), it is
easy to synthesize controllers Gc1 and Gc2 individually in a sequential manner. In this case, the
interactive transfer function can be found as
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y 
G12
G12,CL   1 

 u2  with loop1 closed 1  GC1G11
(3)
y 
G21
G21,CL   2 

 u1 with loop 2 closed 1  GC 2G22
(4)
Equations (3) and (4) gives interaction measure in closed loop sense. In order to reduce the
undesirable response, we need to minimize this interaction. The subsequent transfer functions are
developed in latter sections.
2.2 Decoupled Structure
A decoupler is a device that eliminates interaction between manipulated and controlled variables
by, in effect, changing all the manipulated variables in such a manner that only the desired
controlled variable changes.
This structure has additional elements called decouplers to compensate for the interaction
phenomenon. When RGA shows strong interaction then a decoupler is designed. But however
decouplers are designed only for orders less than 3 as the design procedure becomes more
complex as order increases. Different types of decoupling techniques (Deshpande and Ash,
1988) employed here are,
(i) Static decoupling, in which only steady state gains are considered in designing the
decouplers and are always physically realizable and easily implemented with the less process
information available. A disadvantage is that control loop interactions still exist during transient
conditions.
(ii) Dynamic decoupling, in which the complete process information is taken into account in
the decoupler design. But this may not be feasible always because of the presence of nonlinearities.
(iii) Partial decoupling, in which decouplers are designed for the loops where significant
interaction exists. It is preferable in control problems where one of the control variables is more
important than the other or where one of the process interactions is weak or absent. But it may be
less sensitive to modeling errors than complete decoupling.
By principle, in a two-variable interacting system, shown in Figure 1, we can write:
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Y1 (s)  G11 (s)M1 (s)  G12 (s)M 2 (s)
(5)
Y2 (s)  G 21 (s)M1 (s)  G 22 (s)M 2 (s)
The decoupler inputs are two fictitious manipulated variables u 1 and u2 , and its outputs are
the real manipulated variables M1 and M2 , as shown in Figure 2. We want to design the
decoupler so that u1 affects only Y1 and u2 affects only Y2.
Figure 2. Schematic of a conventional decoupler for a 2 X 2 system
From Figure 2. it can be seen that the decoupler equations can be written as
M1 ( s)  D11 ( s)u1 ( s)  D12 ( s)u2 ( s)
M 2 ( s)  D21 ( s)u1 ( s)  D22 ( s)u2 ( s)
(6)
For ease in building the decoupler, let us specify that D11(s)= D22(s) =1. In this case, the
decoupler equations become:
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 4 Number2–Aug 2012
M1 (s)  u1 (s)  D12 (s)u 2 (s)
(7)
M 2 (s)  D 21 (s)u1 (s)  u 2 (s)
Substituting Equation (7) into Equation (5) gives the equations for the process-decoupler
combination:
Y1  G11  G12 D 21 u1  G11D12  G12 u 2
Y2  G 21  G 22 D 21 u1  G 21D12  G 22 u 2
(8)
For complete decoupling, we want Y1 to be affected only by u1 and Y2 only by u2; that is,
After introducing the decouplers D12 ( s)  
G12 ( s)
G11 ( s)
and D21( s)  
G21( s)
in Equation(8)
G22 ( s)
becomes
Y1   G11  G12 D21  u1
(9)
Y2   G21  G22 D21  u2
But by introducing decouplers the interaction got reduced but the decouplers introduce another
loop that affects Y1 and Y2 as indicated by Dashed and Dash-Dot line in Figure 3. By
introducing the decoupler D12(s) the interaction of the loop 2 from loop1 got reduced but the
effective process gain of loop1 is halved. Similarly by introducing the decoupler D21(s) the
interaction of the loop1 from loop 2 got reduced but the effective process gain of loop2 is halved.
So decoupler produces an inhibiting or competing effect to the normal process path.
To overcome this, the proposed method introduces two more interaction reducers IR1 and IR2
along with D12(s) and D21(s) which reduces the interaction and but it does not affect its own
response. With these, the output (from Eqn.9) from propose decoupled structure becomes:
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Figure 3 2 X 2 Multivariable system with process decouplers
Y1  G11IR11  G12IR 21 u1  G11IR12  G12IR 22 u 2
(10a)
Y2  G 21IR11  G 22IR 21u1  G 21IR12  G 22IR 22 u 2
(10b)
In order to reduce interactions, Y1 should be affected only by u1 and Y2 should be affected only
by u2. Thus the modified decouplers (proposed interaction reducers) can be synthesized by
making the corresponding interactive terma null (=0) from which we obtain
 G IR 
IR11  1  12 21 
G11 

… (11a)


G12G 22

IR12   
 G11G 22  G 21G12 
… (11b)


G 21G11

IR 21   
 G11G 22  G 21G12 
… (11c)
 G IR 
IR 22  1  21 12 
G 22 

… (11d)
The output or the desired response from the above structure can be derived as
Y1   G11  IR1  u1
Y2   G21  IR2  u2
Y1  (G11)u1
… (12a)
Y2  (G 22 )u 2
… (12b)
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Thus, one can observe from Eqn (4) that, output can be retained to its desired value if
denominator of RHS of Eqn(4) were larger, i.e.,
y 
y 
Y1   1   IR 21 2 
 u1  CL
 u1 
(13)
In the above equation, LHS represents totaloutput and RHS is the sum of desirable (1st part) and
undesirable (2nd part) parts. This reveals that IR21 should have a very high gain for minimizing
interactions. Thus by multiplying IR12 with G12 and IR21 with G21 we get desired interaction
reducers. Thus the schematic diagram with interaction reducer becomes (Figure 3(b)). A close
look to Equation (13) and Figure 3(b) reveals that IR1 is proportional to D21 and IR2 is
proportional to D12. Thus it is possible to design these interaction reducers and implement them
as shown in Figure 3(c).
G11
U1
Y1
IR1
IR2
U2
Y2
G22
Figure 3.(b) Schematic representation of proposed Interaction reducers for multivariable
systems
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Y1
-
R1
Gp11
Gc1
IR1
Gp21
IR2
Gp12
R2
Y2
Gp22
Gc2
Figure 3.(c) Schematic of proposed control scheme with interaction reducers
2.3 Controller tuning
The PI controller is of multiloop structure and based on first order plus dead time (FOPDT)
model (K p e Ds / s  1) structure the parameters are tuned using following formula for IMCLaurent tuning
Kc 

I
K P   Dp

(14)
D 2p
 2  D 
p
I  p   
2 p 
D 


D 3p
3   Dp
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2 I
(15)
D 2p  I
    D p 

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(16)
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International Journal of Engineering Trends and Technology (IJETT) – Volume3 Issue 4 Number2–Aug 2012
For BLT tuning, K ci 
K ZNi
and  Ii  F ZNi where F varies from 2 to 5 and K ZNi and  ZNi
F
are Ziegler-Nichols tuning parameters. Sequential tuning is based on Shen & Yu (1994) method
and IMC tuning is based on following formula
Kc 
p
K p

and  i   p where   max 1.7,0.2 p

Results and discussion:
The present scheme was implemented on a WB to study closed-loop performance
through simulation. A unit step change was first applied to the set point of the first loop, and the
process responses were recorded, then another step change was applied to the second loop. The
PI controller parameters designed using BLT criteria are shown in table 1. A comparison of the
closed loop results (Figure 2 to Figure 5 as well as IAE values in Table 2) reveals that the
proposed method produces significantly less interactions when compared to those of
decentralized PI controllers and comparable with decoupled method and also the proposed
scheme for setpoint changes in product size, the set points are reached quickly and smoothly with
less overshoot and offset. It seen from Fig 2 (Y1) that closed-loop (with BLT tuning) response
with IR (Modified decoupler) scheme yields smoother response both in set point as well as load
change cases. This can be also observed from Table 2 where we see that IAE value with
modified decoupler (IR) scheme gives IAER=1.736 and IAEL=0.0302 that are least when
compared to that with other schemes , decoupler and decentralized . Table.2 gives the
comparison of the integral absolute error (IAE) value which is computed as the sum of the
absolute difference between the closed-loop process response and the final steady state value.
The performance of modified decoupler (using IR) gives a better performance. Figure 3 (with
sequential tuning) shows with decoupler scheme, the setpoint (Y2) response oscillates and
becomes unstable whereas with IR schemes. It is stable and takes less time settle as it is evident
from load response also. In Figure 4, with IMC tuning an IR scheme, least oscillations and fast
settling compared to other strategies, like decoupler and decentralized. At the same time if we
see Table 2, we find modified decoupler scheme (IR) produces least values of IAEs (both during
set point and load change).When a step change in Y2 is given, the responses obtained are shown
in Figure 5. This figure also reveals that with IR scheme, a better performance can be achieved.
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Figure 2 .Closed Loop response for a step change in Y1,set (with BLT Tuning
on WB Column) Decoupler(Solid line),Modified decoupler (Dashed
line),Decentralised (Dash-dot line)
Figure.3 Closed Loop response for a step change in Y2,set (with BLT Tuning
on WB Column ) Decoupler(Solid line),Modified decoupler (Dashed
line),Decentralised (Dash-dot line)
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Figure 4. Closed Loop response for a step change in Y1,set (with
Sequential Tuning on WB Column) Decoupler(Solid line),Modified
decoupler (Dashed line),Decentralised (Dash-dot line)
Figure 5 Closed Loop response for a step change in Y2,set (with
Sequential Tuning on WB Column ) Decoupler(Solid line),Modified
decoupler (Dashed line),Decentralised (Dash-dot line)
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WB
Table 1: Tuning results for various distillation column examples
BLT tuning
Sequential
Independent IMC
autotuning
tuning
0.375
0.868
0.737
K
c
I
-0.075
8.29
-0.0868
3.246
-0.103
17.2
Table 2: IAE value comparisons for a Wood and Berry(WB) column
Method
For step input Y1,set
For step input Y2,set
BLT Tuning Decoupler
3.477
0.1249
0.0001185
28.67
Sequential
Autotuning
Modified Decoupler
1.736
0.0302
0.0006164
15.82
Decentralised
3.31
14.67
3.049
28.41
Decoupler
6.231
0.04321
0.1413
12.36
Modified Decoupler
0.2922
0.0009452
0.0001073
6.177
Decentralised
0.5851
6.368
0.5676
12.34
Decoupler
3.634
0.06555
0.003942
15.71
1.823
0.006035
0.0007742
7.957
3.591
8.058
3.418
15.59
IMC Tuning Modified Decoupler
Decentralised
The present IR control scheme is used in closed-loop (with different PI tuning schemes,
BLT, IMC and Sequential autotuning) with WB column for robustness studies. There exists
trade-off between robustness and fast response. If anyone wants to have faster response then he
has to sacrifice on sacrifice on stability. Similarly, if anyone needs to have better stability then he
may to be satisfied with a sluggish response . In order to achieve desired performance under
model uncertainties, a good controller has to be robust. We analyze robust stability and compute
stability margins for the case in which parameters linearly in the state space matrices.
With variation on process parameters, the sensitivity of the closed-loop transfer function
is observed and results obtained. The sensitivity of the closed loop system is determined by
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perturbing parameters of state space system to +10%. Process for study is selected as WB
column. The controller is multiloop PI controller tuned with three different schemes, namely,
BLT, IMC and sequential autotuning. In the IMC case, tuning parameters are calculated based on
independent diagonal process transfer functions. With the IR scheme under sequential autotuning
case, the results show that there are enough scopes to increase the peak gain to make the system
response faster, or in other words, the present controller can accept more degree of perturbation
compared to other strategies. Thus it provides a wider stability margin.
Conclusion:
Analytical expressions for closed loop transfer function representing response with direct
input-output relation and with the interaction phenomena are achieved using sequential
identification techniques/method. The closed loop interaction transfer function (gij CL , with ij)
gives information regarding interactions present between input/output of the system. The present
method illustrates synthesizing a decoupler like block (interaction reducer, IR) to reduce the
interaction between input-output for MIMO systems. Internal model control based PID tuning
rules are developed an implemented on MIMO systems using sequential tuning criteria. The
present IR scheme with sequential autotuning is suggested to be roust and can be implemented
even for multivariable systems with strong interactions.
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