Design of IMC based PID Controller for Coupled Tank System

Design of IMC based PID Controller for Coupled Tank System

Alok Prakash Electrical and Instrumentation department SLIET, Longowal, Sangrur, Punjab India-148106

Ashutosh Prasad Yadav Electrical and Instrumentation department SLIET, Longowal, Sangrur, Punjab India-148106

Raj Kumar Electrical and Instrumentation department SLIET, Longowal, Sangrur, Punjab India-148106

AbstractThe performance of the Proportional Integral Derivative (PID) controllers commonly used in process control industries depends upon the controller tuning parameters. In this paper an Internal Model Control (IMC) based PID controller is proposed to estimate controller tuning parameters in terms of single parameter, known as closed-loop time constant which provides improved performance and robustness to control system. An IMC based controller is designed and presented here for a coupled tank level control system which is of non-interacting type. The transfer function of the system is obtained from the equipment specifications. The obtained transfer function is approximated into first order plus delay time (FOPDT) model for the estimation of the IMC-PID controller tuning parameters in terms closed-loop time constant. The process is simulated in MATLAB/Simulink to record the closed-loop performance with IMC-PID based tuning parameters. The result are compared with the Ziegler-Nichols, Cohen-Coon and Tyreus-Luyben tuning methods in terms of time response characteristics and various performance Indices like Integral of Absolute Error (IAE), Integral Squared Error (ISE) and Integral Time Absolute Error (ITAE). The robustness is checked by incorporating uncertainties in the process. The results indicate PID controller tuned with IMC has better performance and robustness as compared to other tuning techniques.

Keywords Proportional Integral Derivative, Internal Model Control (IMC), First Order plus Delay Time Model, Closed-loop time constant, tuning, robustness.

1. INTRODUCTION

   PID Controllers are extensively employed in process control industries because of their relatively simple structure and design. Tuning technique is adopted for determining the proportional, integral and derivative constants of these controllers which depend upon the dynamic response of the plants. Ziegler-Nichols and Cohen-Coon [1-3] tuning methods are the most popular methods used in process control to determine the parameters of a PID controller. Although these methods are very old, they are still widely used because of their capability to achieve desired optimal performance for specific inputs with less tolerance to plant variations. PID controller tuned with these methods shows less robust results. A controller is said to be robust if it is insensitive to small changes in process or to inaccuracies in process model. Robustness can be defined as amount of Uncertainty in process parameters or inaccuracy in Process model that can be tolerated by controller before the closed-loop system becomes

   unstable [4-5]. In reality a, model is never perfect, so controllers must be designed to be robust (to remain stable even when the true plant characteristics are different from the model). Internal model control (IMC) based PID controller has gained attention because of its robustness and single tuning parameter selection [6-7].

   Maintaining the level at a desired state is an important and common task in all process industries. IMC based PID controller is developed in this paper to control the liquid level in the coupled tank system. Among the other tuning methods, IMC based PID controllers tuning methods has gained widespread acceptance in the process industries because of its easy in design and simple in understand, robustness and fast in real time applications.

2. INTERNAL MODEL CONTROL

   1. Internal Model Control Strategy

      Internal Model Control (IMC) has been presented by Garcia and Morari [6] which is developed upon Internal Model principle to combine the process model and external signal dynamics. The IMC controller is a model based procedure, where a process model is embedded in the controller, and is considered to be robust. Mathematically, robust means that the controller must perform to specification, not just for one model but also for a set of models [4]. The IMC controller design philosophy adheres to this robustness by considering all process model errors as bounded and stable. IMC Theory states that a perfect control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled.

      The IMC basic structure is shown in Fig.3 is characterized by a controller Gc(s), actual process or plant Gp(s) and predictive model of the plant Gp(s). d(s) is an unknown disturbance affecting the system. The manipulated input U(s) is introduced to both the process and its model. The process output is Y(s). d*(s) is the difference between the output of the actual process Gp(s) and process model Gp(s)which is the result of model mismatch and the disturbances; this is used by the internal model controller.

      Fig.3 IMC basic structure

      The design procedure of IMC involves factorization of the plant model Gp(s) as invertible Gp (s) and non- invertibleGp + (s) parts as shown in Eq. (1) by simple factorization or all pass factorization. Ideal IMC controller Gc(s) is the inverse of the invertible portion Gp (s) of the process model Gp(s).

      () = + () () (1)

      () = [ ()]1 (2)

      Gc(s) will be stable, but may not be proper.A low pass filter f(s) of the form of Eq. (3) is added to Gc(s) for making it proper, which also attenuates the effects of process model mismatch, which usually occurs at high frequency and provides good set point tracking.

      controller in Eq. (5) is compared to ideal PID Controller Eq.

      (6) to find out PID parameters (Kc,i,d) in terms of closed- loop time constant whose value is adjusted to give IMC-PID tuning.

      Fig.4 Feedback control structure

3. MATHEMATICAL MODELLING AND CONTROLLER DESIGN

A. Obtaining Transfer Function of the Process

The control objective in a coupled tank system is that a desired level of the liquid in tank is to be maintained when there is an inflow and outflow of water out of the tank respectively. The coupled tank system [8-9] is a multi-input multi output system (MIMO) consisting of two independent single-input single-output systems (SISO) with control voltage as input and water level as the output.

Consider the process consisting of two non-interacting liquid tanks in the Fig.1 here Load Changes in first tank affects the second tank but not the vice-versa. Qi is the volumetric flow rate into Tank1, Q is the volumetric flow rate

() = 1

( + 1)

(3)

from Tank 1 to Tank 2 and Qo is the volumetric flow rate out of Tank 2.Height of liquid level in Tank1 is H1 and in Tank 2 is H2.Both tanks are having same cross-sectional area A.Two

() = ()() (4)

Value of n is chosen to make Gc(s) proper or semiproper. is filter time constant or closed-loop time constant or filter tuning parameter whose value is adjusted to vary the speed of response of the closed-loop system. Gc(s) in

ball valves V1 and V2 having Hydraulic resistances R1 and R2 are connected at the outlet of each tanks.Vi is the control input voltage to pump.

Assuming linear resistance to flow, transfer function of the coupled tank system through mathematical modeling is

Eq. (4) is the final form of the IMC Controller.

() = 2() = 2

(7)

1. IMC based PID Controller

() (1 + 1)(2 + 1)

Although the Internal Model Control (IMC) procedure is simple but it cannot be implemented practically since most industries still uses the PID controller. So the IMC structure can be modified and rearranged to the form of a standard feedback control diagram or Conventional PID structure shown in Fig.4.

GPID(s) is standard feedback controller which is a function

where 1 = AR1 and 2 = AR2 are the time constants of Tank 1 and Tank 2 related to operating levels in the tank

Flow rate of the pump is related as:

() = () ; is pump constant relating to control voltage

Hence, overall transfer function of the process becomes

of plant model Gp(s) and IMC Controller Gc(s) shown in Eq. (5) which can be obtained by rearrangement of IMC basic structure Fig.3 to Feedback control structure Fig.4

() =

2()

()

= 2

(1 + 1)(2 + 1)

(8)

() = ()

1 () ()

2 + + 1

(5)

(6)

Here H2 is controlled variable and Vi is manipulated variable

() = [

]

In IMC based PID design procedure Gc(s) is made semi proper or even improper to give the resulting PID controller derivative action. A first or second order pade approximation is used if a process model has a time delay. The standard PID

coupled tank system is approximated into a FOPDT transfer function using the same method as:

The method is based on times, t1 and t2, which can be estimated from a step response curve (Fig.2), corresponding to the 35.3% and 85.3% response times, respectively.

Fig.2 FOPDT approximation curve

The time delay and time constant are then estimated from the following equations:

= 1.3t1 .29t1 (11)

= .67(2 1) (12)

The FOPDT Transfer function is given by:

( + 1)

(13)

FOPDT model of Coupled Tank System is represented as:

Fig.1 Two tank non-interacting process

Therefore, Obtained Transfer function of coupled two tank

() 2.9646

16.22 + 1

7.1 (14)

non-interacting level process using coupled tank parameters from Table 1 is

C. IMC based PID Controller Design

FOPDT transfer function of Coupled Tank System obtained in Eq. (8) is:

() = 2.9646 (11.2509 + 1)2

(9)

() = 2.9646 7.1

16.22 + 1

(15)

() = 2.9646

126.58272 + 22.5018 + 1

(10)

Process Model Gp(s) after first order pade approximation

Table 1 parameters of coupled tank system

B. FOPDT Approximation of process model

Industrial processes are of higher order so finding a real value of it is very difficult. The transfer functions of plants

() = 2.9646(1 1.35)

(16.22 + 1)(1 + 1.35)

Performing Simple factorization: Invertible part

() = 2.9646 (16.22 + 1)(1 + 3.5)

Non-invertible part

(16)

(17)

that can be approximately modelled by some definite transfer function. Sundaresan and Krishnaswamy [10] have proposed a simple method for fitting the dynamic response of higher order systems in terms of first order plus time delay transfer functions. The obtained second order transfer function of the

+ () = (1 3.5) (18)

Adding Filter with n=1 to make our Controller Improper in order to obtain an ideal PID Controller

(16.22 + 1)(1 + 3.5) 1

() = (

) (19)

B. Simulation Results for Performance

2.9646

Using Eq. (13) and (19)

1 +

The Controller performance is measured by calculating performance indices like ISE, IAE and ITAE and determining the time response characteristics like rise time(tr), settling time(ts) and peak overshoot(Mp) through closed-loop

(16.22 + 1)(1 + 3.5) () = 2.9646( + 3.5)

(20)

simulation in MATLAB/Simulink. Performance results for IMC-PID tuning were compared with ziegler-nichols, cohen- coon and tyreus-luyben tuning methods to see its

Again, by comparing Eq. (19) with ideal PID Controller Eq. (20), PID parameters are obtained in terms of closed- loop time constant which has been easily adjusted to tune the controller.

effectiveness. The Simulations performed for step changes in set-point and in the disturbance at t=100sec for different tuning methods. Simulation responses in Fig 9 and Fig. 10 shows set-point tracking and disturbance rejection capability of IMC-PID tuning in comparison with other tuning methods

= 6.6397

+ 3.5

(21)

used.

Table 3 Performance results for different tuning methods

= 19.72 (22)

= 2.9199 (23)

Thus we can easily do IMC-PID tuning by adjusting .Rivera et al. [7] recommend that > 0.8 because of the model uncertainty due to the pade approximation.

1. SIMULATION RESULTS

   Simulation results are presented to illustrate the effectiveness of IMC based PID Controller for coupled tank liquid level control system.

   A. Simulation Results for different values of

   Simulation was performed for IMC-PID tuning for different values of .Simulation response in Fig.5 shows the behavior of response with increase in value of .Results in Table 2 indicates the variation of Gain and Phase Margin with as Gain and Phase Margins are related to Robustness of the Controller.

   Table 2 PID parameters for different values of

   		= /	p> =	Gain Margin	Phase Margin
   0.9	0.6713	0.03404	1.901	2.7862	69.5402
   	0.6263	0.03175	1.8287	2.9864	70.9729
   1.1	0.5870	0.02977	1.7139	3.1865	72.1926
   1.2	0.5523	0.02801	1.6126	3.3866	73.2730
   1.3	0.5215	0.02644	1.5227	3.5866	74.2363

   Fig.5 simulation Response for Step input for various values

   Fig.6 simulation response for step input for different tuning methods

   Fig.7 simulation response of Integral of Absolute value of error (IAE) for different tuning methods

   Fig.8 simulation response of Integral square error (ISE) for different tuning methods

   Fig.9 Simulation response of different tuning methods for step change in disturbance

   Fig.10 Simulation response of different tuning methods for step change in set-point

   C. Simulation Results for Robustness testing

   The robustness testing of IMC tuned PID Controller was evaluated by incorporating uncertainty in the actual process by a factor of 20% and 25% in gain(), delay time() and time constant().Results in Table 4-9 and simulation responses Fig.11-16 were presented to show the robustness of IMC tuned PID Controller in comparison with other tuning techniques.

   Table 4 results with 20% change in gain()

   Specifications	20% change in gain()
   	IMC- PID	Ziegler- Nichols	Cohen- Coon	Tyreus- Luyben
   Rise time(sec)	10.4603	4.4962	4.4384	6.7058
   Settling time(sec)	18.8504	20.555	18.6782	65.9047
   Peak Overshoot (%)	0	40.6757	16.0683	0
   IAE	124	150	109.1	194.9
   ISE	1630	2220	1891	1730
   ITAE	597.2	1021	541	5435

   Fig.11 simulation response for step input for different tuning methods for 20% change in gain (K)

   Table 5 Performance results with 25% change in gain()

   Specifications	25% change in gain()
   	IMC- PID	Ziegler- Nichols	Cohen- Coon	Tyreus- Luyben
   Rise time(sec)	9.7258	4.2	4.0947	5.9553
   Settling time(sec)	17.5648	19.6499	17.4233	61.2076
   Peak Overshoot (%)	0	41.5147	17.0456	0
   IAE	119.1	147.3	108.1	187.1
   ISE	1617	2303	1970	1745
   ITAE	655.6	1080	621.5	5317

   Table 6 Performance results with 20% change in delay time()

   Specifications	20% change in delay time()
   	IMC-PID	Ziegler- Nichols	Cohen- Coon	Tyreus- Luyben
   Rise time(sec)	12.6976	5.9046	6.0346	9.3253
   Settling time(sec)	19.3810	50.9195	22.9978	84.0347
   Peak Overshoot (%)	0.5638	63.8238	28.7914	0
   IAE	150.2	249.5	152.7	233.3
   ISE	2021	3130	2318	2069
   ITAE	941.3	3376	1086	6924

   Fig.12 simulation response for step input for different tuning methods for 25% change in gain ()

   Fig.13 simulation response for step input for different tuning methods for 20% change in delay time ()

   Table 7 Performance results with 25% change in delay time()

   Specifications	25% change in delay time()
   	IMC- PID	Ziegler- Nichols	Cohen- Coon	Tyreus- Luyben
   Rise time(sec)	12.5172	5.9184	6.0447	9.1274
   Settling time(sec)	18.4789	52.1761	22.9321	83.5019
   Peak Overshoot (%)	1.2969	71.1971	33.1745	0
   IAE	153.2	280.3	164.4	233.4
   ISE	2097	3514	2495	2186
   ITAE	978.8	4211	1238	6840

   Fig.14 simulation response for step input for different tuning methods for 25% change in delay time ()

   Table 8 Performance results with 20% change in time constant()

   Specifications	20% change in Time Constant()
   	IMC- PID	Ziegler- Nichols	Cohen- Coon	Tyreus- Luyben
   Rise time(sec)	16.1031	6.9982	7.5528	15.9628
   Settling time(sec)	24.1089	45.5994	32.5747	86.0580
   Peak Overshoot (%)	2.3230	41.1355	15.7591	0
   IAE	170.1	199.1	140.2	233.8
   ISE	1853	2090	1713	1809
   ITAE	1735	2677	1231	6604

   Fig.15 simulation response for step input for different tuning methods for 20% change in time Constant ()

   Table 9 Performance results with 25% change in time constant ()

   Specifications	25% change in Time Constant()
   	IMC- PID	Ziegler- Nichols	Cohen- Coon	Tyreus- Luyben
   Rise time(sec)	16.5442	7.2542	7.8865	16.7595
   Settling time(sec)	44.8506	47.2821	34.1925	85.5761
   Peak Overshoot (%)	3.00	41.9782	16.4586	0
   IAE	176.7	207.8	145.6	233.9
   ISE	1886	2118	1725	1825
   ITAE	1943	2976	1339	6448

   Fig.16 simulation response for step input for different tuning methods for 25% change in time Constant ()

2. REAL TIME RESULTS

   Real time closed-loop responses obtained for IMC-PID tuning and Ziegler-Nichols tuning for Coupled-Tank System are shown in Fig.17-19.Step changes in set point is made at t=350sec for IMC-PID tuning to see its set-point tracking capability.

   Fig.17 Real time response for IMC based PID tuning

   Fig.18 Real time response for Ziegler-Nichols tuning

   Fig.19 Real time response for step changes in set-point

3. CONCLUSION

Simulation Results were presented to illustrate the IMC based PID tuning and to demonstrate its effectiveness, we considered Coupled Tank System for Liquid level Control. The four tuning methods IMC-PID, Ziegler-Nichols, Cohen-

Coon and Tyreus-Luyben are considered for PID Controller and are comparatively analyzed based on performance and robustness. From Table 3 it is evident that IMC based PID tuning provides better time response characteristics i.e. optimum settling time and reduced overshoot as compared to other tuning methods. Table 3 and Fig.6-8 also shows that IMC based PID tuning exhibits minimum Integral error criterias i.e. ISE, IAE and ITAE compared to other tuning methods. Simulation responses in Fig.9 and Fig.10 shows that IMC-PID tuning has better set-point tracking and disturbance rejection capability than other tuning methods. The Robustness of IMC tuned PID Controller was tested by incorporating uncertainty in the actual process by a factor of 20% and 25% in gain(), delay time() and time constant().Results in Table 4-9 and simulation responses in Fig.11-16 indicates that IMC based PID tuning shows robust performance in Comparison with other tuning techniques. It is evident from the robustness analysis that Gain Margin is related to the amount of gain uncertainty that can be tolerated, and the Phase Margin is related to the amount of delay time uncertainty that can be tolerated. Therefore we can say that Gain and Phase Margin indicates the Robustness of the Controller. The result can be found from Table 3 that as we increase the value of the Gain and Phase Margin values increases which indicates Robustness increases. Decreasing the value of makes the closed-loop response fast whereas increasing its value makes the closed-loop system more robust. Hence, the IMC based PID controller tuning has the advantage of using only a single tuning parameter () whose value is adjusted to achieve a clear trade-off between the closed loop performance and robustness.

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