Design of multiloop controller for multivariable system using coefficient 2

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 4, May – June (2013), © IAEME
253
DESIGN OF MULTILOOP CONTROLLER FOR MULTIVARIABLE
SYSTEM USING COEFFICIENT DIAGRAM METHOD
M. Senthilkumar a
and S.Abraham Lincon b
Department of Electronics and Instrumentation Engineering, Annamalai University,
Annamalai Nagar, Tamilnadu, India
ABSTRACT
In this study the controller for coupled tank multivariable system is designed using
coefficient Diagram method. Coefficient Diagram Method is one of the polynomial methods
in control design. The controller design by CDM method is based on the choice of
coefficients of the characteristics polynomial of the closed loop system according to the
convenient performance such as equivalent time constant, stability indices and stability limit.
Controller is designed for the coupled tank system by using CDM method; it is shown that
CDM design is fairly stable and robust whilst giving the desired time domain system
performance.
Keywords: coefficient diagram method, coupled tank, multiloop, CDM-PI.
I. INTRODUCTION
Many processes in power plants, refinery process, aircrafts and chemical industries
are multivariable or multi-input multi-output (MIMO). The control of MIMO processes are
more complicated than SISO processes. The methodology used to design a controller for the
SISO process cannot be applied for MIMO process because of the interaction exhibit between
the loops. Many methods have been presented in the literature for control of MIMO process.
Proportional-integral-derivative (PID) or Proportional-Integral (PI) based controllers are used
very commonly to control TITO systems. Generalized Ziegler-Nichols method [2],
feedforward method [3], decentralized relay autotuner method [4 , ISTE optimization method
[5]] are among them. Usually two types of control schemes are available to control MIMO
processes. The first is decentralized control scheme or multiloop control scheme, where
single loop controllers are used (the controller matrix is a diagonal one). The second scheme
is a full multivariable controller (known as the centralized controller), where the controller
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ISSN 0976 - 6480 (Print)
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254
matrix is not a diagonal one. Multiloop controllers do not explicitly consider the decoupling
of the inter-loop interactions unlike full multivariable controllers.
In this work, the Coefficient Diagram Method (CDM) is used to design a controller
for MIMO process. CDM is now a well established approach to design controllers that
provide outstanding time domain characteristics in closed-loop (Manabe, 1994; Manabe,
1998S; Budiyono and Sutarto, 2004; Cahyadi, et al, 2004).Basically CDM is based on pole
assignment, where the locations of the closed-loop system are obtained using predetermined
templates. Although it has been demonstrated that the designs based on CDM has some
robustness, it is possible to show that some of the nice characteristics of the design can be lost
if large perturbations in the model of the system exist.
II. COEFFICIENT DIAGRAM METHOD
The design of controller is not a difficult except the robustness issue, if the
denominator and numerator of the transfer function of the system are determined
independently according to stability and response requirements. But this is also addressed by
coefficient diagram method [7].CDM is a polynomial algebraic method which uses
characteristics polynomial for controller design which also gives sufficient information
with respect to stability, response and robustness in a single diagram. When the plant
dynamics and the performance specifications are given, one can find the controller under
some practical limitations together with the closed-loop transfer function satisfactorily. As a
first step, the CDM approach specifies partially the closed-loop transfer function and the
controller, simultaneously; then decides on the rest of parameters by design. The parameters
are stability index ߛ௜ , equivalent time constant τ , and stability limit ߛ௜
‫כ‬
, which represent the
desired performance. The choice of the stability indices affect the stability and unsteady-state
behavior of the system, and can also be used for the robustness investigation. As for the
equivalent time constant, which specifies the response speed, hence the settling time [7]. The
basic block diagram of CDM control system is shown in Fig 1. Where y is the output, r is the
reference input, u is the controller signal and d is the external disturbance signal. N(s) and
D(s) are the numerator and Denominator of the plant transfer function. A(s) is the
denominator polynomial of the controller transfer function while F(s) and B(s) are called the
reference numerator and the feedback numerator polynomials of the controller transfer
function.CDM controller structure resembles to a Two Degree of Freedom (2DOF) control
structure because it has two numerators in controller transfer function. The system output is
given by
yሺ‫ݏ‬ሻ ൌ
ேሺ௦ሻிሺ௦ሻ
௉ሺ௦ሻ
‫ݎ‬ ൅
஺ሺ௦ሻேሺ௦ሻ
௉ሺ௦ሻ
݀ (1)
where P(s) is the characteristics polynomial of the closed loop system.
ܲሺ‫ݏ‬ሻ ൌ ‫ܣ‬ሺ‫ݏ‬ሻ‫ܦ‬ሺ‫ݏ‬ሻ ൅ ‫ܤ‬ሺ‫ݏ‬ሻܰሺ‫ݏ‬ሻ ൌ ∑ ܽ௜‫ݏ‬௜௡
௜ୀ଴ (2)
CDM needs some design parameters with respect to the characteristic polynomial
coefficient which are τ equivalent time constant, ߛ௜ stability index, and ߛ௜
‫כ‬
stability limit. The
relations between these parameters and the coefficients of the characteristic polynomial ( ai )
are shown in (3).

255
߬ ൌ
௔భ
௔బ
(3a)
ߛ௜ ൌ
௔೔
మ
௔೔శభ௔೔షభ
, ݅ ൌ 1~ሺ݊ െ 1ሻ, ߛ଴ ൌ ߛ௡ ൌ ∞ (3b)
ߛ௜
‫כ‬
ൌ
ଵ
ఊ೔షభ
൅
ଵ
ఊ೔శభ
(3c)
From eq. 3a-c, the coefficients ai can be written as
ܽ௜ ൌ
ఛ೔
∏ ఊ೔షೕ
ೕ೔షభ
ೕసభ
ܽ଴ୀܼ௜ܽ଴ (4)
The design parameters are substituted in eq (2) and the target characteristic
polynomial is obtained as
ܲ௧௔௥௚௘௧ሺ‫ݏ‬ሻ ൌ ܽ଴ ቈቊ∑ ቆ∏
ଵ
ఊ೔షೕ
ೕ
௜ିଵ
௝ୀଵ ቇ ሺ߬‫ݏ‬ሻ௜௡
௜ୀଶ ቋ ൅ ߬‫ݏ‬ ൅ 1቉ (5)
The equivalent time constant specifies the time response speed. The stability indices
and the stability limit indices affect the stability and the time response. The variation of the
stability indices due to plant parameter variation specifies the robustness.
Fig 1. standard block diagram of CDM
III. CONTROLLER DESIGN USING CDM
Most of the processes encountered in industry are described as FOPTD
‫ܩ‬௣ሺ‫ݏ‬ሻ ൌ
௞೛
ఛ೛௦ାଵ
݁ିఏ௦
(6)
Where kp is process gain τ is time constant and θ is time delay. Since the transfer
function of the process is of two polynomials, one is numerator polynomial N(s) of degree m
and other is the denominator polynomial D(s) of degree n where m n, the CDM controller
polynomial A(s) and B(s) of structure shown in Fig 1are represented by
‫ܣ‬ሺ‫ݏ‬ሻ ൌ ∑ ݈௜‫ݏ‬௜
and ‫ܤ‬ሺ‫ݏ‬ሻ ൌ ∑ ݇௜‫ݏ‬௜௤
௜ୀ଴
௣
௜ୀ଴ (7)
+
-
y(s)
F(s)
+
)s(D
)s(N
)s(A
1
B(s)
r(s)
CDM controller
d(s)

256
For the controller to be realized the condition p q must be satisfied. For a good performance
the degree of controller polynomial chosen is important. in this paper the controller
polynomial for FOPTD process with numerator Taylor’s approximation is chosen as
A(s)=s (8a)
B(s) =k1s+k0 (8b)
For computation of the coefficient of the controller polynomial in CDM pole
placement method is used. A feedback controller is chosen by pole-placement technique and
then, a feedforward controller is determined so as to match the steady-state gain of closed
loop system. According to this, the controller polynomials which are determined by Eq. 8a
and 8b are replaced in Eq. 2. Hence, a polynomial depending on the parameters ki and li is
obtained. Then, a target characteristic polynomial Ptarget(s) is determined by placing the
design parameters into Eq. 5. Equating these two polynomials.
A(s) D(s) + B(s) N(s) =Ptarget(s) (9)
Is obtained, which is known to be Diophantine equation. Solving these equations the
controller coefficient for polynomial A(s) and B(s) is found .The numerator polynomial F(s)
which is defined as pre-filter is chosen to be
‫ܨ‬ሺ‫ݏ‬ሻ ൌ ܲሺ‫ݏ‬ሻ ܰሺ‫ݏ‬ሻ⁄ |௦ୀ଴ ൌ ܲሺ0ሻ ܰሺ0ሻ⁄ (10)
This way, the value of the error that may occur in the steady-state response of the closed loop
system is reduced to zero. Thus, F(s) is computed by
‫ܨ‬ሺ‫ݏ‬ሻ ൌ ܲሺ‫ݏ‬ሻ ܰሺ‫ݏ‬ሻ⁄ |௦ୀ଴ ൌ 1 ݇௣⁄ (11)
IV. CDM-PI
The transfer function of conventional PI controller is
‫ܩ‬௖ሺ‫ݏ‬ሻ ൌ ݇௖ ቀ1 ൅
ଵ
்೔௦
ቁ (12)
The controller gain kc and integral time Ti are related with polynomial coefficient as k1=kc
and k0=kc/Ti . By using the CDM, the values of k1, and K0 can be designed as follows:
1) Find the equivalent time constant τ
2) The stability index γ2 = 2, γ1 = 2.5 [7] are used.
3) From eq. (2), derive the characteristic polynomial with the PI controller stated in eq (12)
and equates to the characteristic polynomial obtained from eq (14). Then the parameters k1
and K0 of the PI controller are obtained.
4) Set the pre-filter B,(s) = K0.
And Gf(s) is feed forward controller
‫ܩ‬௙ሺ‫ݏ‬ሻ ൌ
ிሺ௦ሻ
஻ሺ௦ሻ
ൌ
௞బ
௞భୱା௞బ
ൌ
ଵ
்೔௦ାଵ
(13)

257
Fig 2. equivalent block diagram of CDM
V. COUPLED TANK PROCESS
Coupled tank is used in Petro-chemical industries, paper making industries and water
treatment industries for processing chemicals or for mixing treatment .the control of level of
fluid in tanks are a challenging problem due to interactions between tanks and also serves as
a MIMO process. The schematic diagram of coupled tank process is shown in Fig 3. The
controlled variables are levels of tank1 (h1) and( h2). The levels of the tank are maintained
by manipulating the inflow to the tanks. βx is the valve ratio of the pipe between tank1 and
tank2, β1 and β2 are the valve at the outlet of tank1 and tank2 respectively. The mass balance
equation of the coupled tank process is
‫ܣ‬ଵ
ௗ௛భ
ௗ௧
ൌ ݇ଵ‫ݑ‬ଵ െ ߚଵܽଵඥ2݃‫ܪ‬ଵ െ ߚ௫ܽଵଶඥ2݃ሺ‫ܪ‬ଵ െ ‫ܪ‬ଶሻ (14)
‫ܣ‬ଶ
ௗ௛మ
ௗ௧
ൌ ݇ଶ‫ݑ‬ଶ ൅ ߚ௫ܽଵଶඥ2݃ሺ‫ܪ‬ଵ െ ‫ܪ‬ଶሻ െ ߚଶܽଶඥ2݃‫ܪ‬ଶ (15)
Where A1 and A2 are the cross sectional area of the tanks, a1 and a2 is the cross
sectional area of outlet pipe in tank1 and tank2 respectively, K1 and K2 are the gains of
pump1 and pump2 and g is the specific gravity. The parameters of the process and its
operating points are listed in Table I and Table II
Fig 3. schematic diagram of coupled tank process
GC(s) Gp(s)
+
-
r(s) y(s)
Gf(s)
d(s)
+
h1
Fin1 Fin2
h2
Fout1 Fout2
ββββx
ββββ1
ββββ2
A1 A2

258
Table .I Parameters of Coupled Tank Process
Table .II Operating Conditions of Coupled Tank Process
VI. SIMULATION AND EXPERIMENTAL RESULTS OF COUPLED TANK
PROCESS
The transfer function model for the coupled tank process is identified using reaction
curve method.
‫ܩ‬௣ሺ‫ݏ‬ሻ ൌ ቎
ଵ଺.ଽଽ ௘షభమ.ఴవೞ
ሺଶଵସ.଴ଷ௦ ା ଵሻ
଺.଺ଽ ௘షళమ.ఱళೞ
ሺଶ଴ସ.ଽଷ௦ ା ଵሻ
ଽ.ଶଷ ௘షయఱ.బభೞ
ሺଶହ଺.ସସ௦ ା ଵሻ
ଵଵ.ଷ଼ ௘షమఱ.బరೞ
ሺଵ଺ଽ.ଵହ௦ାଵሻ
቏ (16)
The controller parameter are obtained using the above identified model.The
multiloop PI controller for the coupled tank process is obtained using BLT method as for
loop1 Kc1=0.2080 ,Ki1=0.5826 and for loop2 Kc2=0.1309 , Ki2=0.6926.The multloop CDM-
PI controller is obtained with τ=25 and found to be Kc1=0.5300 ,Ki1=0.0590 and Kc2=0.7900
, Ki2=0.0880 for loop 1 and loop2 respectively The closed loop response of the coupled tank
process for a setpoint change in tank 1 from its operating value of 18.32 to 25 is shown in
Fig. 4and its coressponding interaction effect in tank2 is shown in Fig 5. Fig 6 shows the
closed loop reponse of tank2 for a setpoint change of 17 cm from its operating value and its
corresponding effect of interaction is shown in Fig 7.simulink oriented VDPID control
system is used for the real time control of coupled tank process.VDPID is high speed two
input two output interface card.
A1, A2 a1,a2, a12 β1 β2 βx
154 0.5 0.7498 0.8040 0.2245
u1 u2 h1 h2 k1 k2
2.5 2.0 18.32 12.23 33.336 25.002

259
Fig.4 closed loop response of coupled tank for Fig.7 closed loop interaction
response of setpoint change in tank1 coupled tank for setpoint change in tank2
Fig.5 closed loop interaction response of coupled Fig.8 closed loop response of coupled
tank tank for setpoint change in tank1 for setpoint change in tank1 (real time)
Fig.6 closed loop response of coupled tank for Fig.9 closed loop interaction response
of setpoint change in tank2 coupled tank for setpoint change in
tank1(real time)
0 100 200 300 400 500 600
11
12
13
14
15
16
17
18
Timein seconds
Levelincm
0 50 100 150 200 250 300 350 400 450 500
12
13
14
15
16
17
18
19
20
21
Time in seconds
Levelincm
BLT
CDM-PI
0 50 100 150 200 250 300 350 400 450 500
18
20
22
24
26
28
30
32
Time in seconds
Levelincm
BLT
CDM-PI
0 50 100 150 200 250 300 350 400 450 500
18.1
18.15
18.2
18.25
18.3
18.35
18.4
18.45
18.5
18.55
Time in seconds
Levelincm
BLT
CDM-PI
0 50 100 150 200 250 300 350 400 450 500
12.15
12.2
12.25
12.3
12.35
12.4
12.45
Time in seconds
Levelincm
BLT
CDM-PI
0 100 200 300 400 500 600
18
20
22
24
26
28
30
Timein seconds
Levelincm

260
Fig.10 closed loop response of coupled tank for Fig.11 closed loop interaction response of
setpoint change in tank2 (real time) coupled tank for setpoint chang in
tank2(real time)
Fig.12 Block diagram of multiloop CDM-PI control system
Table III Performance measure of controllers
controller
Setpoint change in tank1 Setpoint change in tank2
LOOP1 LOOP2 LOOP1 LOOP2
ISE IAE ISE IAE ISE IAE ISE IAE
BLT 803.97 277.94 O.485 9.44 1.69 15.31 286.44 15.383
CDM-PI 257.97 89.33 0.37 4.123 0.793 5.95 93.1 45.63
0 100 200 300 400 500 600
10
12
14
16
18
20
Timein seconds
Levelincm
0 100 200 300 400 500 600
15
20
25
Time inseconds
Levelincm
Gc1
Gc2
+
+
-
-
+
+
+
+
r1
y1
r2
y2
g11
g12
g21
g22
Gf1
Gf2

261
VII. CONCLUSIONS
In this paper Multiloop CDM-PI controller is designed for coupled tank process and
compared with the controller designed by Biggest Log Modulus method through simulation
and experimentation. The ISE and IAE are taken as performance indices. Results shows
supremacy of the multi loop CDM-PI controller and the ease in design of Multiloop
controller
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