PID Controller Design for An Interacting Tank Level

2016 2nd International Conference on Control, Instrumentation, Energy & Communication (CIEC)

PID Controller Design for An Interacting Tank Level Process with Time Delay Using MATLAB FOMCON Toolbox Deep Mukherjee School of Electronics Engineering KIIT University Bhubaneswar, Pin-751024 , India [email protected]

Palash Kumar Kundu

Apurba Ghosh

Department of Electrical Engineering Jadavpur University Kolkata, Pin- 700032, India [email protected]

PIȜDμ controller using multi objective GA which minimizes rise time and overshoot. Here Nelder mead optimization method has been proposed to tune fractional order PID controller [8] and integer order PID controller also using FOMCON toolbox for the current plant.

Abstract— This paper presents a new way to design PID controller for both integer order and fractional order with a time delay for a typical interacting cylindrical tank system using MATLAB FOMCON toolbox. Here, our work aims to study the performance characteristics of integer order and fractional order PID controller on the current integer order plant obtaining minimum objective function by Nelder – Mead optimization technique with different performance metrics ISE, ITSE and IAE. Next our work shows to make comparison between integer order PID controller based on AMIGO model performance and fractional order PID controller on time domain characteristics. The proposed method aims finally to analyze overall desired performance on fractional order PID controller by adding two extra degrees of freedom over the integer order PID controller with different performance criteria.

II. INTERACTING PROCESS A. Description A simplified interacting coupled tank system [6] is shown in Fig. 1 below consisting of two cylindrical tanks which are interconnected through manual control valve. Nowadays coupled tank [1] is required to control the level of fluid properly in process industry where the control of liquid level of tank is a major problem. Process industries require liquids to be pumped, stored in tanks and pumped to another tank but to get better regulation of flow cascading two tanks must be required. Generally there are two types of coupled tank model i.e., single input and multi input coupled tank but here multi input interacting process has been chosen to control the level of fluid in tank 2 by varying flow rate in tank 1 where liquid is supplied through control valve. Here in our process PID controller will control the water pump so that water in both tank in level as required. The main function of the controller is to maintain level set point at a given value and be able to accept new set point values dynamically and then to produce a control signal .

Keywords— Interacting Cylindrical Tank, AMIGO IO tuning, FO Tuning, Nelder-Mead optimization, FOMCON toolbox

I. INTRODUCTION Recently nowadays a real object is modeled and described more accurately and precisely by Fractional Order method over Integer Order method. By using two degrees of freedom of fractional order performance of classical PID controller [4], [7] is more improved and it provides greater degree of flexibility also. Basically fractional order provides an effective means of capturing the approximate, inexact nature of the real world. Recent time have wide applications of fractional integral and derivatives also in the theory of control of dynamical systems, where the controlled system is described by a set of fractional differential equations. Several researchers proposed various scheme to tune Fractional order controller using different optimization techniques. Podlubny [9] suggested more flexible structure given PIȜDμ by extended in tradition notion of PID controllers with the controller gains define the fractional differ-integrals as design variables and given several techniques for efficient tuning of such fractional order PIȜDμ controller. Maity [10] proposed FOPID tuning technique for stable minimum phase system with particle swarm optimization method. Cai, Pan & Du [11] tuned a PIȜDμ controller using multi-parent cross over evolutionary algorithm optimization method. Luo & Li [12] proposed PIȜDμ controller scheme with bacterial foraging oriented by particle swarm optimization method. Meng & Xue [13] proposed

978-1-5090-0035-7/16/$31.00©2016IEEE

Department of Electronics & Instrumentation UIT, University of Burdwan Burdwan, Pin- 713104, India [email protected]

FC

Inflow

C2

C1

q

LT R2

R1

Tank-1

h1

q2 h2 Outflow

Tank-2

Fig.1. A model of interacting coupled tank.

1

2016 2nd International Conference on Control, Instrumentation, Energy & Communication (CIEC)

In Fig. 1, q= volumetric flow rate into tank 1, q1=volumetric flow rate from tank 1 to tank2, q2= volumetric flow rate from tank 2, h1 & h2= height of tank 1 and tank 2, R1 & R2 = restriction element of tank 1 and tank 2 and c1 & c2= volume of tank 1 & tank 2.

Now the process is considered with a dead time model computed using 1st order pade approximation shown in (8). ‫ܩ‬ሺܵሻ ൌ 

(8)

B. Mathematical Modelling

and the step response of the process is shown in Fig. 2.

For tank 1, dh C1 1 = q1 − q dt

1.5

1 Amplitude

, ' q = ( h1 − h2 ) R1 ( h − h2 ) dh ∴ C1 1 = q1 − 1 R1 dt

(1)

For tank 2, dh h C 2 2 = q − q 2 , ' q2 = 2 dt R2 dh h − h2 h2 C2 2 = 1 − dt R1 R2

R2 C 2 Sh2 ( S ) + h2 ( S ) +

R R2 h2 ( S ) = 2 h1 ( S ) R1 R1

40 60 Time [sec]

80

100

III. DESIGN OF PID CONTROLLER FOR INTERACTING TANK LEVEL PROCESS A. IOPID Controller Here, for controlling the liquid level of tank AMIGO(Approximate m constrained integral gain optimization) model PID controller [2] has been used and this method is used to obtain desired tuning rules and attempt to find controller parameter with the objective of optimizing the load disturbance with a constraint on the maximum load disturbance to output sensitivity. This model consists in applying a set of equations to calculate the parameter of the controller in a similar way to the procedure used in zeigler Nichols technique. This tuning rule considers a controller described as shown in (9).

(3) (4)

On solving the above equation we get h2 ( S ) R2 (5) = q1 ( S ) R1C1 R2C 2 S 2 + ( R1C1 + R2 C1 + R2 C 2 ) S + 1 Tank specification: a. D ( Diameter) of both tank = 92 cm b. H (Height) = 300 mm c. q (maximum inflow to tank 1) = 20 cm3/sec d. C1 ( volume of tank 1) = 0.00948 cm3 e. C2(volume of tank 2) = 0.00475 cm3 f. R1 (Restriction element of tank 1) = 10800 min/cm2 g. R2(Restriction element of tank 2) = 10800 min/cm2 G( S ) =

U(t) = K(b‫ݕ‬ଵ ‫݌ݏ‬ሺ‫ݐ‬ሻ െ ‫ݕ‬ଵ ݂ሺ‫ݐ‬ሻሻ ൅ ݇ଵ ݅ ‫׬‬ሺ‫ݕ‬ଵ ‫݌ݏ‬ሺ߬ሻ െ ‫ݕ‬ଵ ݂ሺ߬ሻ ൅ ݇ௗ ቀ

஼ௗ௬ೞ೛ ሺ௧ሻ ௗ௧

െ

௬೑ ሺ௧ሻ ௗ௧

ቁ

(9)

where u is control variable, ysp is the set point, y the process output and ‫ݕ‬௙ is the filtered process variable. Basically filtered process provides smooth transition signal to the controller and do not influence disturbance rejection performance of a controller. It basically allows a controller to be tuned perfectly to reject noise. The transfer function is a first order filter with ଵ time constant T is give by . Parameters b and c are called ଵା௦் set point weights. Neglecting the filter of the process the feedback part of the controller has the transfer function is shown in (10).

Substituting the values of parameters for a real time process the overall system transfer function is shown in (6). 10800 (6) G(S ) = 5252 .2 S 2 + 256.28S + 10801

Generally a system behaves like a nonlinear in nature and delay which is used to make the system more practical, to observe the stability of the process. But if more delay is used then the system becomes unstable then in our system 5 sec less dead time has been considered to show the stability. The new transfer function with dead time is shown in (7). 10800e −5 S 5252.2 S + 256.28S + 10801

20

Fig. 2. Step response using pade approximation.

h2 ( S ) q1 ( S )

2

0

-1 0

Assuming q1 as input and h 2 as output for interacting tank level process, the transfer function is given by

G(S ) =

0.5

-0.5

(2)

Taking Laplace transform on both sides of (1), R1C1 Sh1 ( S ) + h1 ( S ) − h2 ( S ) = R1q1 ( S ) Taking Laplace transform on both sides of (2)

G(S ) =

ʹʹͷͲͲ‫ ݏ‬ଶ െ ʹ͹ͲͲͲ‫ ݏ‬൅ ͳͲͺͲͲ ͳͲͻͲͶʹǤͲͺ‫ ݏ‬ସ ൅ ͳ͵͸͸ͶǤͶͳ‫ ݏ‬ଷ ൅ ʹͺ͵ͻͶǤͻ‫ ݏ‬ଶ ൅ ʹ͹ʹͷͺǤʹͺ‫ ݏ‬൅ ͳͲͺͲͳ

‫ܩ‬௖ ൌ ݇ ቂͳ ൅

ଵ ௦்೔

൅ ‫ܶݏ‬ௗ ቃ

(10)

Here, AMIGO model has been preferred over ZieglerNichols(Z-N) method on current integer order process as Z-N method does not achieve the set point and therefore it is not possible to obtain time domain specifications using this method. The performance analysis of AMIGO method & Z-N method is shown in Fig. 3.

(7)

2

2016 2nd International Conference on Control, Instrumentation, Energy & Communication (CIEC)

which is calculated as the difference between the set point and the output. The following steps are taken to design PID controllers as shown below :

1.2 1

Amplitude

0.8

AMIGO method

0.6

Step1. Identified by Integer order model which is approximated with FOPDT model and the transfer function is given by K e-L / Ts+1 , where K= process gain, L= delay time and T= time constant.

0.4 Z-N method

0.2 0 -0.2 0

20

40 60 Time [sec]

80

100

Fig. 3. Performance analysis of AMIGO & Z-N method.

Step2. AMIGO tuning rule is based on KLT process model where,

It is based on approximation of process by simple FOPDT(first order plus dead time) model. The block diagram of IOPID controller is shown in Fig. 4.

+

e(t)

+ de(t ) KD dt

Σ

u(t)

Σ +

K I ³ e(t ) dt

-

ଵ

்

௄ ଴Ǥସ௅ା଴Ǥ଼்

௅

‫ܭ‬௖ ൌ ሺͲǤʹ ൅ ͲǤͶͷ ሻ

(11)

ܶ௜ ൌ

(12)

௅ା଴Ǥଵ் ଴Ǥହ௅்

‫ܮ‬

Process

ܶௗ ൌ

+

In order to use the PID controller with filtering , the rules are extended as follows: Kp= Kc, Ki = Kc/Ti and Kd = Kc . Td

Output

(13)

଴Ǥଷ௅ା்

Step3. Obtained parameters Kc, Ki& Kd Step 4. Obtained overall closed loop transfer function : G(s)= Gc(S)/[1+Gc(s) * Gp(s)]

Fig. 4. Block diagram of IOPID controller.

Using two point curve fitting method model parameters of the process are obtained. This method has been used for obtaining FOPDT model by Astrom and Hagglund. Static gain K is obtained( K= 1.009) from the ratio of steady state output deviation over the step change. The time delay is obtained(L= 4.314) from the intercept of tangent to the step response and time constant (T=0.06958) is obtained as difference between time delay and the time spent for the step response to reach a value of 0.63 of gain.

and the closed loop step response is shown in Fig. 6. 1.5

Response

1 0.5 0 -0.5 0

2

10

20

30

40

50 Time (S)

60

70

80

90

100

Fig. 6. Closed loop step response of IOPID controller before optimization.

1.5

Step5. Now applied Nelder Mead optimization method using FOMCON optimization toolbox with performance metric ISE, IAE, ITSE.

1 Amplitude

Set point

K Pe(t )

0.5

Step6. Obtained minimum objective function F(x) after iteration process and optimized value of Kp, KI & Kd.

0 Identified FOPDT Model -0.5 Original Model -1

0

10

20

30

40

50 Time

60

70

80

90

B. NELDER-MEAD Optimization Tuning rules of PID controller is based on Nelder Mead optimization method [5] which is more heuristic search method than fmincon (interior point or active set) optimization method that can converge to non-stationary points. It uses basically the concept of simplex for finding a minimum of a objective function f(x) of several variables where simplex is a triangle and it is defined by n+1 vertices. The initial simplex is created from an initial guess x1. The method compares function values at the three vertices of triangle which are considered as f(xG), f(xB) and f(xW). Here xG, xB and xW are considered as good to worst point of triangle. Simplex method goes through a process of reflection (R), expansion (E), contraction(C), reduction(S) until the function is minimized

100

Fig. 5. Identified FOPDT model.

As mentioned before tuning for ¼ decay ratio often leads to oscillatory responses and also this criterion is developed by considering the closed loop response only at two points(the first two peaks). Another approach is to introduce controller design relation based on a performance index that considers the entire closed loop response. Some of the performance indices are given below. Integral of absolute value of the error (IAE): IAE= œ |e(t) |dt Integral of the square value of the error (ISE): ISE= œ e2 (t) dt Integral of the time weighted square value of the error (ITSE): ITSE= œ t |e(t) |2dt, where t is the time and e (t) is the error

3

2nd International Conference on Control, Instrumentation, Energy and Communication, 2016 (CIEC16)

KP after iteration process. In a triangle (M) centrroid is calculated in all vertices except xw. a. Reflection (xR) = M + Į(M – xW), whhere Į= 1 b. Expansion(xE) = M + Ȗ(M – xW), Ȗ = 2 c. Contraction(xC) = xW + ȕ(M – xW), ȕ = 0.5 or -0.5 d. Reduction(xS) = (xS+ xB)/2 The complete iteration process to find out m minimum function using geometric method developing an algorrithm with figure is shown in Fig. 7.

+

+

Error

Σ Set point

+

KDS q

Σ

Derivative action

-

Proportional action Process

Output

Integra l action

+

KI Sq

Fig. 8. FOPID controllerr block diagram.

Optimization steps for designing off fractional order controller using FOMCON: Step1. Optimization method: Nelder-Mead with performance metric: ISE,IAE,ITSE. Step2. Gain: KP, KI, KD between 0 to 100 and Exponents: lam and mu between 0 to 1. Step3. Approximation method d: Oustalaup Recursive approximation. Step4. Obtained minimum objecctive function F(x) after iteration process and optimized paarameters Kp, KI , KD, lam and mu.

Fig. 7. Iteration process.

1. Order the vertices xB