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COMPARISON BETWEEN DIFFERENT METHODS FOR TUNING PID CONTROLLERS

FABIO CASTRILLÓN Department of Chemical Engineering, Universidad Pontificia Bolivariana,

Cq 1 # 70 – 01 Bq. 11. Medellín, P.O. Box 56006, Colombia. fcastrillon@upb.edu.co

MARISOL OSORIO Department of Electrical Engineering, Universidad Pontificia Bolivariana

Cq 1 # 70 – 01 Bq. 11. Medellín, P.O. Box 56006, Colombia. mosorio@upb.edu.co

RAFAEL VÁSQUEZ Department of Mechanical Engineering, Universidad Pontificia Bolivariana

Cq 1 # 70 – 01 Bq. 11. Medellín, P.O. Box 56006, Colombia. rafavasquez@asme.org

Tuning of Proportional-Integral-Derivative (PID) control algorithms is a common problem of the chemical processes industry that has not been completely studied yet. Despite there have been several authors working on the matter, this approach has been a bit subjective and a wider study including techniques comparison is required. In this work, some methodologies for PID controllers tuning are reviewed. First, several algorithms employed by industrial controllers are studied; then, their philosophy and their limits are discussed. A comparison among the different methods is made using a process for drying phosphate pebbles, and their behavior is evaluated by different criterions of performance and robustness. Finally, a statistical analysis of the variability of the results is made.

1. Introduction

Chemical processes show some characteristics that make difficult their regulation, for instance: multivariable interactions between controlled and manipulated variables, non-measurable state variables, non-measurable disturbances, uncertain and time-variable parameters, restrictions in manipulated variables, presence of dead time, etc, [1]. The most used algorithm, at the industrial level, is the called Proportional-Integral-Derivative (PID), [1], some of its advantages are:

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• It has been used for many years by the instrumentation and control engineers [3].

• Many advanced control algorithms can be reduced to a PID form [3], [4]. • Some reports suggests that there is no feedback algorithm that gives better

performance and robustness than a well tuned PID control, for processes with frequent and not measured changes in disturbance variables, except for those situations in which dead time predominates [3], [4], [5], [6], [7].

• It is mathematically simple [5]. In consequence, the study of the PID algorithm and its effects over the

control loop is very common.

2. The PID algorithm

The most studied structure is an ideal PID controller structure that has implementation restrictions. There are other algorithms that are widely used by different manufacturers, and it’s fundamental to recognize these structures and how their differences with respect to the ideal algorithm affect the determination of the tuning parameters. The most commonly used structures are the non-interacting ideal PID structure, the parallel ideal PID structure and the interacting PID structure, [8], [9], [10], [11].

2.1. Non-interacting ideal PID structure

1( ) 1⎛ ⎞

= + +⎜ ⎟⎝ ⎠

DI

Gc s Kc ss

ττ

. (1)

Where, Kc: Controller gain τD: Derivative time τI: Integral time Gc(s): Controller transfer function

2.2. Parallel ideal PID structure

( ) = + +ID

KGc s Kp K SS

. (2)

Where, KP: Proportional gain KI: Integral gain

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KD: Derivative gain Gc(s): Controller transfer function

2.3. Interacting PID structure

' 11( ) ' 1

' ' 1⎛ ⎞⎛ ⎞+

= +⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠D

I D

sGc s Kc

s sτ

τ ατ. (3)

Where, Kc’: Gain of the interacting controller τD’: Derivative time of interacting controller τI’: Integral time of the interacting controller α: Derivative time filter Gc(s): Controller transfer function

3. PID controllers tuning methods

Tuning is the adjusting of the feedback controller parameters to obtain a specified closed-loop response. Castrillón, [12], reviewed twenty-four different methods; eight of which were selected and used for the development of this work.

3.1. Ziegler-Nichols Method

Ziegler and Nichols developed a set of semi-empirical equations to tune PID controllers and obtained a closed loop response with decay ratio of 0.25. The ultimate gain, Kcu, and the ultimate period, Tu, are required in this method, [10], [13]. Table 1 contains the equations to adjust the parameters.

Table 1. Ziegler-nichols tuning equations

Proportio-nal Gain

Integral Time

Derivative Time Controller type

Kc’ τI’ τD’

Proportional Kcu/2 --- ---

Proportional – Integral Kcu/2.2 Tu/1.2 ---

Proportional – Integral – Derivative Kcu/1.7 Tu/2 Tu/8

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3.2. IAE Method for disturbances

The method was developed by López, Miller, Smith and Murrill at Louisiana State University, [14]. They proposed a set of equations that minimizes the integral of the absolute value of the error, using dead time, the time constant and the gain of the process, [9]. They assumed that the controlled variable is affected in the same way by both, the manipulated variable and the main disturbance variable. Table 2 contains the equations to adjust the parameters.

Table 2. IAE Method for disturbances tuning equations

Controller parameter a b

0⎛ ⎞= ⎜ ⎟⎝ ⎠

btaKcK τ

1.435 -0.921

01

⎛ ⎞= ⎜ ⎟⎝ ⎠

btaττ

τ 0.878 -0.749

0⎛ ⎞= ⎜ ⎟⎝ ⎠

b

Dt

aτ ττ

0.482 1.137

3.3. Tyreus-Luyben Method

This method was developed for control of chemical processes, assuming slow dynamics of the system, using classical frequency response methods. For its application the developers used the ultimate gain, and the ultimate period [15], [16]. This method gives more conservative settings (higher closed-loop damping coefficient) than the Ziegler – Nichols [16]. Table 3 contains the equations to adjust the parameters.

Table 3. Tyreus-Luyben tuning equations

Proportio-nal Gain

Integral Time

Derivative Time Controller type

Kc’ τI’ τD’

Proportional – Integral Kcu/3.22 2.2Tu ---

Proportional – Integral – Derivative Kcu/2.2 2.2Tu Tu/6.3

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3.4. Shinskey Method

Shinskey proposes a change to the IAE Method for disturbances, that allows to find a good relationship between the performance obtained with the design of the controller to compensate changes in the set-point, and the design to compensate disturbances; to do this a lead-lag unit is required in the set-point, and it is adjusted using the graphical procedure described in [17]. The dead time, time constant, gain and time constant attached to the disturbance variable are required [17].

3.5. Gain Margin Specification Method

Astrom and Hagglund proposed a methodology for PID controllers tuning, based on the gain margin specification, which can be considered to quantify the robustness of the control system, [3]. Table 4 contains the equations to adjust the parameters.

Table 4. Gain Margin Specification tuning equations

Controller parameter

Proportional Gain =KcuKcAm

Derivative Time 2

1=D

IWuτ

τ

3.6. Controller Synthesis Method

Martin, Corripio and Smith propose a method for PID controllers tuning, which requires the determination of the structure of the controller, having a model of the process and the closed loop response specification for the control system [8], [9], [10], [18]. For a first order plus dead time (FOPDT) closed-loop response, the equations are shown in Table 5, [10].

Table 5. Controller synthesis tuning equations

Controller parameter

Proportional Gain ( )0

' =+C

C

KK t

ττ

Integral Time ' =Iτ τ

Derivative Time 0' 2=Dtτ

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Here, τc, is the closed-loop desired time constant, and must be specified. Different criterions to choose τc could be found in [18], [19], [7], [20].

3.7. Method of Ciancone and Marlin

Marlin and Ciancone developed a set of rules for PID controllers tuning. The rules have to keep in mind multiple factors, such as: minimization of the IAE, prevention of the excessive variability of the manipulated variable and a 25% for the maximum change in the model parameters [20]. The controller parameters can be calculated using the graphical procedure proposed in [20].

3.8. Practical Rules of Tuning

These methods recommend the mode of the PID controller and suggest the values to adjust the controller parameters, depending of the variable to be controlled. These methods do not require the exact knowledge of the process dynamics [16], [21], [22].

4. Case of study

The comparison between the different methods is made using a process for drying phosphate pebbles, Figure 1, [10]. A table feeder transports the pebbles into the bed of the dryer. In this bed the pebbles are dried by direct contact with hot combustion gases. The controlled variable is the output moisture of the dryer, the manipulated variable is feeding speed of the pebbles and the main disturbance variable is input moisture.

Figure 1. Dryer of phosphate pebbles.

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To obtain the process model, a step change in the controller output was

made. There were obtained the transfer functions that relate the controlled variable with the controller output (4) and the controlled variable with the main disturbance variable (5), using the “two-point method”, proposed in [10].

1.234.69( )

1.90 1

− ⋅⋅=

⋅ +

seG ss

. (4)

0.86716.67( )

1.667 1

− ⋅⋅=

⋅ +

seGp ss

. (5)

A simulation of the system was made using Simulink™. Responses of the control system with set point and disturbances were observed. The input signals were of step kind, and saturation in the controller output and noise in the controlled variable, were considered. A parallel ideal structure for the PID controller was used, and its tuning parameters were adapted using transformation equations, reported in [9], [10], [19].

5. RESULTS

The maximum peak height (MPH), the IAE and the settling time (Ts), were used as performance indexes. The relations between the ultimate parameters of the process and the normal operation parameters were used to measure the robustness of the system, [9]. Figure 2 to Figure 9 contain graphics of results.

5.1. Ziegler-Nichols KP = 0.492, KI = 0.160, KD = 0.242

Figure 2 Deviation of the moisture regarding set-point and disturbance changes. Ziegler – Nichols

5.2. IAE disturbances KP =0.455, KI = 0.290, KD = 0.255

Figure 3. Deviation of the moisture regarding set-point and disturbance changes. IAE Disturbances

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5.3. Tyreus – Luyben KP =0.322, KI = 0.034, KD = 0.194

Figure 4. Deviation of the moisture regarding set-point and disturbance changes. Tyreus – Luyben

5.4. Shinskey KP =0.455, KI = 0.290, KD = 0.255

Figure 5. Deviation of the moisture regarding set-point and disturbance changes. Shinskey

5.5. Gain Margin KP = 0.220, KI = 0.141, KD = 0.059, Am=3

Figure 6. Deviation of the moisture regarding set-point and disturbance changes. Gain Margin

5.6. Synthesis KP = 0.436, KI = 0.173, KD = 0.203

Figure 7. Deviation of the moisture regarding set-point and disturbance changes. Synthesis

5.7. Ciancone – Marlin KP = 0.256, KI = 0.117, KD = 0.024

Figure 8. Deviation of the moisture regarding set-point and disturbance changes. Ciancone – Marlin

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5.8. Practical rules KP = 0.737, KI = 0.067, KD = 0.670

Figure 9. Deviation of the moisture regarding set-point and disturbance changes. Practical rules

5.9. Numerical results

To select each comparison index, a statistical analysis of the variability of the results was made using the variation coefficient.

Table 6. Performance index for set-point and disturbance changes

Set-point changes Disturbance changes Method * Best result, **Worst result MPH IAE Ts MPH IAE Ts Ziegler-Nichols 1.40 68.5 23.4 1.04 67.5 24.88 IAE disturbances 1.6** 70.04 15.02 1.04* 49.0* 16.81 Tyreus – Luyben 1* 158** 60** 1.04 316** 76** Shinskey 1.27 57.3* 14.33 1.04 49.08 37.42 Gain margin 1.22 71.42 13.02 1.15 95.34 16.4* Synthesis 1.35 58.35 13.8 1.04 62.16 17.86 Ciancone – Marlin 1.16 60.7 11.8* 1.2** 91.29 17.73 Practical rules N/A N/A N/A N/A N/A N/A Variation coefficient (%) 5.73 91.18 74.61 14.82 46.18 80.94

Table 7. Robustness index

Method * Best result, **Worst result tou/to Ku/K τ / τu

Ziegler-Nichols 1.56** 1.26** 1.29** IAE disturbances 1.58 1.36 1.41 Tyreus – Luyben 3.25* 1.66 1.74 Gain margin 2.24 2.88* 6.79* Synthesis 1.80 1.46 1.56 Ciancone – Marlin 2.32 2.32 5.59 Practical rules N/A N/A N/A Variation coefficient (%) 27.71 28.62 65.66

6. Conclusion

With regards to the performance of the controller, the integral of the absolute value of the error (IAE) for the set-point changes response was

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evaluated. Accordingly, the method that showed the best performance was the Shinskey method; conversely, the method that showed the worst results was Tyreus – Luyben.

In regards of the performance of the controller, the settling time (Ts) for the disturbance changes response was evaluated. Accordingly, the method that showed the best performance was the minimization of the IAE for disturbances; conversely, the method that showed the worst results was Tyreus – Luyben.

Regarding the robustness, the relation between the normal operation time constant and the ultimate time constant was evaluated. Accordingly, the best results were obtained with the gain margin specification method and the worst results with Ziegler-Nichols.

Both, the wide variety of rules for tuning PID controllers, and the continued research in this field, show the importance of these classical controllers, in spite of the existence of more advanced control algorithms

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