[ieee 2013 ieee international conference on control system, computing and engineering (iccsce) -...

A Symbolic PI Tuning Method for First Order

Systems with Time Delay

Uğur Yıldırım

Control Engineering Department

Istanbul Technical University

Istanbul, Turkey

[email protected]

Emre Dincel



Istanbul, Turkey

[email protected]

Mehmet Turan Söylemez



Istanbul, Turkey

[email protected]

Abstract—This paper presents a symbolic PI tuning method for

first orders systems with time delay based on Pade

approximation. The presented method is very easy to use like

well-known methods such as Ziegler-Nichols and Cohen-Coon.

Besides its easiness, the method performs a faster step response

and no overshoot. Furthermore, it provides a good robust

performance.

Index Terms—Process Control, First Order Systems, Time

Delay, Robust Control.

I. INTRODUCTION

First order systems are the mostly encountered processes in

the industry. Moreover, some high order systems are also

modeled as first order systems with a time delay in many cases

in order to simplify the design process. This model is suitable

for a wide range of industrial processes [1]. Furthermore, there

are several controller tuning methods in the literature for this

kind of systems.

For this kind of systems, low order controllers such as PI

and PID controllers are the most well-known and commonly

used controllers in industrial applications because of their

simple structure and acceptable robustness [2]. There are many

studies for tuning methods of these kinds of controllers such as

well-known Ziegler-Nichols method [3] which provides an S-

shape reaction curve in step responses of the systems. There

exist many studies on the improvement of this method [4], [5].

Cohen-Coon method [6] is another commonly-used tuning rule

for industrial processes. Smith predictor [7], [8], [9] and

internal model control [10] are also preferred in the industry,

but these methods have different control structures and usually

need an extra work to implement the controller. Some of these

methods have a poor performance for longer time delays.

In this study, a symbolic method for PI tuning of first order

stable systems with time delay is introduced. In section II, a

brief explanation of the method is given and choosing of the PI

controller parameters is shown. Robustness analysis through

simulation results are given in section III and it is shown that

the method has a good performance for different ratios of time

constant and time delay. Finally, conclusions are given in

section IV.

II. SYMBOLIC TUNING METHOD WITH PADE APPROXIMATION

Consider a closed loop control system with a PI controller

and a first order stable system with a time delay given in Fig. 1.

G(s)F(s)YUR E

+-

Fig. 1. Closed-loop control system.

Here, is the PI controller for which the transfer function is

given as

(1)

and is the first order system with a time delay which has a

step response as given in Fig. 2.

Fig. 2. Step response of the first order system with a time delay.

The transfer function of the system is given as

(2)

where, k is the proportional gain, τ is the time constant and L is

the time delay of the system. First order Pade approximation

for the time delay is given as below.

(3)

If the Pade approximation is used for the system given above,

the following transfer function is obtained.

uMaxyMax

0.63 yMax

0 L L

y t , u t

time

2013 IEEE International Conference on Control System, Computing and Engineering, 29 Nov. - 1 Dec. 2013, Penang, Malaysia

978-1-4799-1508-8/13/$31.00 ©2013 IEEE 325

(4)

The proportional gain of the controller can be chosen as

(5)

in order to cancel the pole of the system. In this case, the open

loop transfer function of the system is obtained as follows.

(6)

(7)

Negative root locus plot of the control system is shown in Fig.

3.

Fig. 3. Negative root locus plot of the control system.

The design point on the root locus of Fig. 3 can be chosen as

the breaking point in the left half plane. The breaking point can

be calculated as below.

(8)

(9)

The integrator gain of the controller can then be calculated as

follows.

(10)

(11)

Hence, the proportional gain of the PI controller becomes

(12)

and the transfer function of the PI controller can be written as

(13)

III. SIMULATIONS

To simulate the performance of the proposed method,

MATLAB-Simulink™ is used. Parameters of the first order

system with time delay are chosen as , and

to see the performance for different amount of

time delay. Well-known methods like Ziegler-Nichols and

Cohen-Coon are also simulated to make a comparison.

Parameters of each controller are given in Table 1 and time

domain characteristic of the controllers are given in Table 2.

Simulation results, which are obtained using MATLAB-

Simulink™, are shown in Fig. 4, Fig. 5 and Fig. 6. It should be

remarked that pure time delay elements (not the Pade

approximations) are used in the simulations. From the

simulations, it can be seen that the Cohen-Coon method has a

large overshoot and the Ziegler-Nichols method has a long

settling time for longer time delays. On the other hand,

proposed method almost has a critical damping and an

acceptable settling time.

TABLE I. PI CONTROLLER PARAMETERS

L / τ = 0.5 L / τ = 1 L / τ = 2

Method Kp Ki Kp Ki Kp Ki

ZN 1.8 1.0909 0.9 0.2727 0.45 0.0682

CC 1.695 2.0448 0.885 0.7777 0.48 0.3267

PM 0.6864 0.6864 0.3432 0.3432 0.1716 0.1716

TABLE II. TIME DOMAIN CHARACTERISTICS

L / τ = 0.5 L / τ = 1 L / τ = 2

Method Ts Mp% Ts Mp% Ts Mp%

ZN 4.87 0.181 18.22 0.0 52.74 0.0

CC 5.35 0.462 7.61 0.286 9.94 0.119

PM 3.25 0.0 6.51 0.0 12.95 0.0

Fig. 4. Step responses for L/τ=0.5 (ZN: green, CC: blue, PM: red).

2 2 4 61 L

3

2

1

1

2

3

j 1 L

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

Time (s)

y(t)


326

Fig. 5. Step responses for L/τ=1 (ZN: green, CC: blue, PM: red).

Fig. 6. Step responses for L/τ=2 (ZN: green, CC: blue, PM: red).

Another important issue in controller design is the robustness of the controller. Therefore simulations are done again under ±10% uncertainty of each parameter of the system and the results are obtained as shown in Fig. 7, Fig. 8 and Fig. 9. It is seen that the step response of the proposed method does not change considerably, while the other methods are settling too late or having a large overshoot.

Fig. 7. Step responses for L/τ=0.5 with ± 10% uncertainty (ZN: green, CC:

blue, PM: red).

Fig. 8. Step responses for L/τ=1 with ± 10% uncertainty (ZN: green, CC:

blue, PM: red).

Fig. 9. Step responses for L/τ=2 with ± 10% uncertainty (ZN: green, CC:

blue, PM: red).

IV. CONCLUSION

A symbolic PI tuning method based on the Pade

approximation is proposed for first order systems with a time

delay. The proposed method is as easy as some of the other

well-known tuning methods. However, it provides a faster

settling time without any overshoot. Moreover, the robust

performance of the proposed method has been examined

through simulations and found to be very satisfactory.

Designing controllers for higher order systems using a similar

strategy can be considered a part of the possible future work.

REFERENCES

[1] G. J. Silva, A. Datta, S. P. Bhattacharyya, PI Stabilization of First-Order

Systems with Time Delay, Automatica, vol. 37, pp. 2025-2031, 2001. [2] K. J. Åström, and T. Hägglund, PID controllers: Theory, design and

tuning. Research Triangle Park, NC: Instrument Society of America.

[3] J. G. Ziegler, N. B. Nichols, Optimum Setting for Automatic Controller, Transactions ASME, vol. 64, pp. 759-768, 1942.

[4] T. Hägglund, K.J. Åström, “Revisiting the Ziegler–Nichols tuning Rules

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[7] O. J. Smith, A Controller to Overcome Dead Time, ISA J., vol. 6, pp.

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0 2 4 6 8 10 12 14 16 18 200

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[9] D. Kaya, “Obtaining controller parameters for a new PI-PD Smith

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[10] D. E. Rivera, M. Morari, S. Skogestad, Internal Model Control: PID

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