Design of Set-Point Filter and PID Controller in Double Feedback Loop for Conical Tank

Design of Set-Point Filter and PID Controller in Double Feedback Loop for Conical Tank

Sophiya. K. Jacob

PG Student

Department of Electronics and Instrumentation Engineering Karunya University, Coimbatore

Hepsiba. D

Assistant Professor

Department of Electronics and Instrumentation Engineering Karunya University, Coimbatore

Abstract:

The aim of this paper is to eliminate the peak overshoot in the response of a conical tank. To design a set-point filter, the following values are required: the peak overshoot values and the corresponding peak time. The controller of the inner loop in the double feedback is tuned by Zeigler Nichols method, the outer loop is tuned by internal model controller (IMC) based PID tuning rule. This method needs a tuning parameter which is obtained from the set-point filter. The time constant of the set-point filter is used to tune the internal model controller based PID tuning rule. The simultaneous usage of the set-point filter and the double feedback results in complete elimination of peak over shoot. Keywords: peak overshoot, PID, IMC, Set- point filter, Tuning.

1. Introduction:

   The objective of every industry is to measure and control its parameters. In most of the chemical industries, controlling the process parameters such as level, pressure, flow, temperature etc is very essential. For example, controlling the level of the dangerous liquids such as acids in Chemical Industries.PID controllers are the simplest and perfect controllers available today. Various tuning methods can be employed for designing the controller parameters. These parameters are obtained by designing the filter. The time constant of the filter is obtained by performing simple calculations. It requires only the peak overshoot value and the corresponding peak time. The conical tank is commonly used in process industries because its shape provides better drainage of viscous liquids. The nonlinear shape of the conical tank makes the level control a typical problem. The non linearity and varying area of cross section of a conical tank is a use

   the filter in the circuit, the amplitude of the oscillation can be reduced. The proposed method will give us the kp, ki, and kd values by using the time constant of the filter[5]. In double feedback loop the controller used can be P, PI, or PID. Among these the PID controller is the most effective one.

2. System description:

   1. Conical tank system:

      The conical tank level process which is a nonlinear process whose parameters vary with respect to the process variable is considered. At a fixed outlet flow rate the system is controlled and maintained at the desired level. The time constant and gain are the important variables which vary as a function of level in the chosen process.

      Figure 1: Experimental Setup of a Conical Tank level system

      The desired level h is maintained by manipulating the inlet flow rate q1 to the system. Thus h is the controlled variable and

      q1 is the manipulated variable.

   2. Block diagram:

      Figure 2: Block diagram of proposed method

      accumulation of input of output of

      total mass total mass total mass

      The double-feedback loop provides stability and

      time

      time

      time

      better performance. The stability is obtained

      from the inner loop. The outer loop and set-point filter can be used for good set-point tracking and reducing peak overshoot.

   3. Process model

      d V t q t q t

      dt 1 2 0

      Assume that the room temperature as well as the density of liquid is constant, =1=2.

      V 1 r2h

      The volume of cone 3

      r R h

      Where,

      H

      H (S)

      Q(S)

      mt

      s 1

      Where, time constant

      m hs2

      and process

      Figure 3: Process model of proposed system

      t

      t

      gain

      m 2 h

      t

      t

      c

      Where

      LT :Level transmitter.

      I/V :Current to voltage converter. V/I :Voltage to current converter. I/P :Current to pressure converter. DAQ :data acquisition card.

3. Mathematical modeling:

   The conical tank is the process considered which

   Specifications of conical tank: Height : 80 cm

   Volume : 33.5 litres Bottom Diameter : 7.62 cm Top Diameter : 36.62 cm

   Angle : 10deg

   Material : Stainless Steel

   20s

   20s

   The transfer function of the conical tank system

   H (s) 0.9363e

   is given in figure 4.

   obtained is

   Gp

   Q(s)

   86.982s 1

   Figure 4: conical tank

   :density of liquid in the tank Kg/cm3

   1 :density of liquid in the inlet stream Kg/cm3

   2 :density of liquid in the inlet stream Kg/cm3

   V :the total volume of the conical tank cm3

   qI :volumetric flow rate of inlet stream LPH q :volumetric flow rate of outlet stream LPH R :Maximum radius of the cone cm

   r :Radius of the cone at steady state cm H :Maximum height of the cone cm

   h :Height of the cone at steady state cm Using the law of conservation of mass,

4. Set point filter design[3]:

   Numerous methods are available to design a set-point filter which needs extensive calculations. Moreover, the existing techniques need information about the process parameters, controller setting values and are laborious. But the proposed method is simple and requires only the peak overshoot value and the peak time of the system response regardless of the type and order of the system with arbitrary PID parameters.

   Figure 5: Peak over shoot response of PID controller

   P1 : Closed loop Peak overshoot

   tp1 : Corresponding Peak time

   P2 : desired overshoot of closed loop tp2 : Corresponding Peak time

   Gain k = p1/0.6321

   0.9363e20s

5. IMC based PID controller tuning[2]:

Gc1 is a proportional controller, therefore Gc1=kc1.Gc2 is tuned by this tuning rule. The transfer function of the conical tank is given as

0.9363e20s

Gp

86.982s 1

The process transfer function is

Gp

86.982s 1

Let Gp1 is the inner loop transfer function.

The approximated transfer function is

Y s k eds

Gp

Gc1Gp

1 G G

U s s 1

c1 p

: time constant d : time Delay

f : time constant of set-point filter Introducing Set Point filter

Y s k eds * 1

5.45 fs 1e20s

Gp1

86.982s 1 5.45*e20s

Where K = kc1*kp =5.816*0.9363 =5.45

GD is the desired closed loop transfer function of the block diagram.

1/ s

s 1

k

fs 1

1

Y G

GC 2GC 2GP

1 GC1GP

Y s

s s 1

eds *

fs 1

B D 1 GC 2GC1GP

1 G G

Apply partial fraction

G GD 1 GC1GP

C1 P

GD

C 2 G G 1 G

GP1 1 GD

k ds 1 A B C

C1 P D

eds

s s 1 e * fs 1 s s 1 fs 1

Let GD is given by

GD xs 1 ys 1

A = K,

B 2

k

f

ed /

and

k

Where x= f (23.0482) and y=0 for the first order system. Whereas x=y=f for second order system.

c f 2 ed / f

f

Inverse Laplace Transform

Y(t) = K+ (k/-f) )[ f* ed-1/f -* d-1/]

At t = tp2 +d then Y(t) = p2 , take -d+1 = -t,

Here the tie constant of set-point filter is used as the tuning parameter to tune kc, ki, and kd values.

e20s

GD 23.0482s 1

If Y(t) = p2 then t becomes tp2

p2= K+ (k/-f) * [ -* e-tp2/ ]

Outer controller Gc2 is given by

15.96s 0.1835 e20s

Since e-tp2/f<< e-tp2/

f = *(K- p2 – K *e-tp2/)/ (K- p2) = 23.0482

Gc2

1 23.0482s e20s

Infinite series is given by, 6. Results and discussion:

1! 2!

1! 2!

Ds Ds2

e Ds 1 ………, Ds

15.96s 0.1835 1 20s 200s2

Gc2

1

1 23.0482s 1 20s 200s2

1.1835 4.04 200s2

s * 43.0482 200s

Gc2 can be written as Gc2 = (s)/s

2

2

s 1.1835 4.04s 200s

43.0482 200s

According to Laurent series[4]

s ……. 0 '0 ''0 …..

1! 2!

To find out kc, ki, and kd. (0)=0.02749,(0)=0.03388, (0)=9.606746

1 0.03388s 9.606746s2

Gc2 s * 0.02749 1!

2!

0.03388 1 0.811393152 141.7761s

s

The standard form of PID controller is given by

Gc2

kc 1

1

is

ds

Kc =0.03388

i = 1.23245

d = 141.7761

kd = kc* d

= 4.8034

ki kc

i

=0.02748

Figure 6: Responses of closed loop systems

The P, PI, PID controllers are designed using Zeigler Nichols tuning method[6]. In proportional controller the output is proportional

to the error signal. The response has offset error. The PI controller response is sluggish and the output is proportional to error signal as well as the integral of error signal. Even though PID controller is a robust controller, its response has peak overshoot. The set-point filter is designed and the filter time constant obtained is f = 23.0482. The IMC based PID[1] tuning rule is used to tune the parameters kc=0.03388, kd = 4.8034, ki =0.02748. The tuning parameter considered is the filter time constant f = 23.0482.

6.1 Comparison:

Table 1: comparison of performance measures

6.1.1 Explanation:

The peak overshoot of the response is zero by using double feedback loop with set-point filter. The proportional controller is having rapid response. The proportional integral controller provides large settling time compared to other controllers. The usage of set-point filter along with single closed loop will give more fast settling. The best settling time is obtained from the double feedback loop with set-point filter.

1. Conclusion and Future Work:

   Here the proposed method is a simple first order set-point filter and a new IMC based PID tuning rule for open loop low order processes. The tuning parameter that is used for tuning the PID controller is used as filter time constant. According to design objective, the peak overshoot as well as the settling time is reduced. Future work is real time implementation of PID

   controllers in double feedback loops for conical tank with set-point filters using LabVIEW. The controller can be used for disturbance rejection and performance under the presence of measurement noise.

2. References:

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2. V. Vijayan a, Rames C. Design of PID controllers in double feedback loops for SISO systems with set-point filters Pandab ISA Transactions 51 (2012) 514-52.

3. V. Vijayan ,Rames C. Pandab Design of a simple set point filter for minimizing overshoot for low order processes ISA Transactions 51 (2012) 271-276

.[4] Panda RC. Synthesis of PID tuning rule using the desired closed-loop response.Ind Eng Chem Res 2008; 47(22):868492.

1. Padmasree R, Chidambaram M. Set point weighted PID controller for unstable system. Chem Eng Commun 2005;192:113.

2. De Paor AnnraoiM, OMalley Mark. Controllers of ZieglerNichols type for unstable processes. Internat J Control 1989;49:127384.