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)

ISSN 0976 - 6499 (Online)

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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 �� , equivalent time constant τ , and stability limit �� , 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

y�� (1)

where P(s) is the characteristics polynomial of the closed loop system.

�� ∑ �� (2)

CDM needs some design parameters with respect to the characteristic polynomial

coefficient which are τ equivalent time constant, �� stability index, and �� stability limit. The

relations between these parameters and the coefficients of the characteristic polynomial ( ai )

are shown in (3).



255

� � �� (3a)

�� !��"� , $ � 1~�' ( 1�, �� ∞ (3b)

�� )*�"� � )*�!� (3c)

From eq. 3a-c, the coefficients ai can be written as

�� +�∏ *�"--�"�-.� ��/�� (4)

The design parameters are substituted in eq (2) and the target characteristic

polynomial is obtained as

�0�1230�� 45∑ 6∏ )*�"--�7)8�) 9 ��: ; � �� 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

=>�� ?@+@A) B7C (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

�� ∑ D��and �� ∑ E��F��>�� (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

G�� ⁄ |�� 0� ��0�⁄ (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

G�� ⁄ |�� 1 E>⁄ (11)

IV. CDM-PI

The transfer function of conventional PI controller is

=K�� EK L1 � )M�N (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

=O�� P�� ?�?�QA?� � )M�A) (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

�) RS�R0 � E)T) ( U)�)V2XY) ( UZ�):V2X�Y) ( Y:� (14)

�: RS R0 � E:T: � UZ�):V2X�Y) ( Y:� ( U:�:V2XY: (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

ββββ1ββββ2

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.

=>�� [)\.^^ 3"� ._`a

�:)b.�c A )� \.\^ 3"d .eda�:�b.^c A )�^.:c 3"fe.��a�:g\.bb A )� )).ch 3" e.�ia�)\^.)gA)�j (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 60011

12

13

14

15

16

17

18

Time in seconds

Le

ve

l in

cm

0 50 100 150 200 250 300 350 400 450 50012

13

14

15

16

17

18

19

20

21

Time in seconds

Level in

cm

BLT

CDM-PI

0 50 100 150 200 250 300 350 400 450 50018

20

22

24

26

28

30

32

Time in seconds

Level in

cm

BLT

CDM-PI

0 50 100 150 200 250 300 350 400 450 50018.1

18.15

18.2

18.25

18.3

18.35

18.4

18.45

18.5

18.55

Time in seconds

Level in

cm

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

Level in

cm

BLT

CDM-PI

0 100 200 300 400 500 60018

20

22

24

26

28

30

Time in seconds

Le

ve

l in

cm



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 60010

12

14

16

18

20

Time in seconds

Le

ve

l in

cm

0 100 200 300 400 500 60015

20

25

Time in seconds

Le

ve

l in

cm

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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