[IEEE 2013 IEEE International Conference on Control System, Computing and Engineering (ICCSCE) -...

Modeling and Control of Under-Damped Second

Order Systems with Dead-Time and Inverse Response

Emre Dincel

Control Engineering Department

Istanbul Technical University

Istanbul, Turkey

[email protected]

Uğur Yıldırım



Istanbul, Turkey

[email protected]

Mehmet Turan Söylemez



Istanbul, Turkey

[email protected]

Abstract—Systems with inverse response are difficult to be

identified because of the existence of at least one zero at the right

half s-plane. However, it is important to obtain the transfer

function as accurately as possible for such systems to be able to

provide the desired performance using a controller. If the system

has dead-time, the design problem becomes more complicated.

This paper presents a new modeling method for under-damped

second order systems with an inverse response to overcome the

difficulties both in the analysis and design. Besides, a PID

controller design in discrete-time domain is also introduced to

provide a good performance in the closed-loop. The performance

of the proposed modeling and control technique is demonstrated

on an example using simulations.

Index Terms—System Identification, Inverse Response,

Process Control, Discrete-Time Control Systems.

I. INTRODUCTION

Systems with dead-time and inverse response is quite

common especially in process control systems such as tubular

exothermic reactor, drum boiler and distillation column [1],

[2]. If a step input is applied to such a system, initially output

of the system remains constant throughout dead-time and after

that it moves into the opposite direction to the desired final

value before moving towards the final value. In such systems,

dead-time is caused by the late measurement of the system

output, and the inverse response is caused by the right half

plane system zero. This non-minimum phase characteristic

leads to difficulties in analysis and controller design [3]. For

this reason, it provides a significant advantage in controller

design phase to know the system parameters as well as

possible.

In the literature, there are various studies on the systems

with inverse response to obtain the open-loop transfer function

using response of the system to various input signals or to

ensure simplicity for controller design. In some studies, during

the controller design, inverse response is assumed as a dead-

time by using Pade approximation [4] whereas in some studies,

the transfer function is obtained using numerical approaches

[5]. However, to the best of the authors’ knowledge, there is

not any proposed formula which gives a suitable transfer

function in an analytical way for under-damped second order

systems with inverse response and dead-time. Although an

under-damped response in the open-loop is not usual in

chemical processes, it can still be useful to be able to obtain the

transfer function of such systems, especially when only the

closed-loop system response can be measured (since such

systems can have oscillatory responses in the closed-loop), or

in the case of cascade control system with the primary loop

cutoff [6], [7].

In general, it is not possible to obtain the condition of

perfect control by any stable and causal controller for the

systems with inverse response since the designed controller

operates on the error which has wrong sign at the beginning

[3]. There are many studies available in the literature to

develop a convenient controller design including those based

on PID controllers [8], [9], [10], inverse response compensator

[11], and H∞ control theory [12], [13]. In the studies based on a

PID controller, large overshoots can be observed and in some

of these studies, the setting of controller parameters is

empirical. Besides, some presented methods increase the

complexity and cause difficulties during the tuning process. In

order to control such systems, PID control design in the z-

domain is considered in this paper. Despite the important

developments in control theory, majority of the processes in

industry are controlled by PID controllers [14]. In addition,

since the system has a dead-time, it is possible to take

advantage of discrete-time domain representation by

transferring infinite number of poles caused by dead-time in s-

domain to a finite number of poles in z-domain. The controller

provides better performance because no approximation is used

for dead-time. Furthermore, since most of the controllers are

implemented in computer based systems due to technological

advancements; discrete-time design method provides a

significant advantage. The resulting controller parameters are

also ready to be implemented in computer based systems such

as industrial computers or PLCs in this design method.

In this study, to be able to overcome the difficulties in

modeling and design, a new modeling method is proposed for

the under-damped second order systems with inverse response.

In addition, in order to achieve desired performance for such

systems, a PID controller design method in the z-domain is

given. The performance of the proposed modeling and control

techniques are demonstrated on an example using simulations.

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 329

II. MODELING OF THE SYSTEM

A. Identification of the System Parameters

The step response of an under-damped second order system

with dead-time and inverse response is given as in Fig. 1.

Fig. 1. Step response of the system.

The transfer function of such a system can be given as follows:

(1)

Here, five parameters, namely the damping ratio ( ), the

natural frequency ( ), the location of the zero ( ), the

parameter ( ) and the dead-time ( ), are needed to be obtained

using the step response of the system. It is possible to show

that these parameters can be calculated using the following

expressions as shown in the next subsection.

(2)

(3)

(4)

(5)

(6)

(7)

(8)

The parameters that need to be known in (2)-(8) are

illustrated in Fig. 2. Here, the unit step input is applied to the

system at the time of and the output of the control system

starts to take a non-zero value at the time of . At the points of

and , the output of the control system takes its minimum

value ( ) and maximum value ( ), respectively. The value of

is called the final value of the system. Finally, is the time

value in which the output of the control system intersects the

axis.

Hence, the system model is obtained using the unit step

response of the system. Proofs of the proposed formulas are

given in the next section.

Fig. 2. Parameters used to obtain the transfer function.

B. Proofs

In the proof of the proposed method, to avoid confusion the

dead-time is assumed to be zero without losing generality.

(9)

The time-domain response of the system in (9) can be given

as follows by the help of the Inverse Laplace Transform,

(10)

where

(11)

It is possible to find the points of and by equating the

derivative (10) with respect to time to zero. Hence, we get,

(12)

and the general solution is given as follows.

(13)

Therefore, we obtain the points of and in the Fig. 2 for

and , respectively. Then, if the expressions and

are put in (10), we obtain the points of and ,

respectively. The final value of the system response can also be

obtained as follows.

(14)

By rearranging the expressions for , and equations

(2) and (3) can be obtained under the assumptions ,

and .

In order to prove (4), it is enough to put the expressions for

and into (4). Finally, location of the system zero (6) is obtained by

equating (10) to zero and performing some simplifications.

III. CONTROLLER DESIGN FOR THE SYSTEM

In this study, a PID type controller is chosen to control the

given system since using PID type controllers is usually more

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Step Response

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Unit Step Response


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suitable for the type of systems under consideration. In order to

design a PID controller in the discrete-time domain, a root-

locus design technique is preferred. Firstly, the discrete-time

transfer function is required to be obtained using an appropriate

sampling time. After that, proposed control method explained

in this section can be applied on the discrete-time transfer

function of the system.

A typical root-locus plot of a second order system with an

inverse response and a dead-time is given in Fig. 3.

Fig. 3. Root-locus plot of the system.

The discrete PID controller structure is given as follows

(15)

As can be seen from this transfer function that there are two

controller poles whose locations are at and . In

addition, there are two zeros whose locations are to be

determined.

In the design, locations of the PID controller zeros are

usually chosen near the open-loop system poles and then the

controller gain is adjusted such that the closed-loop system

poles move to the break point. Hence, a critically damped

response can be provided. Fig. 4 shows this design procedure

on the root-locus plot.

After the calculation of PID parameters ( , and ), it is

possible to find the industrial PID coefficients so that the

designed controller can be implemented in industrial controller

such as a PLC. In general, PID controller parameters need to be

written in time format such as integral time constant ( ) and

derivative time constant ( ) in the PID control function blocks

of PLCs. In order to find these parameters, following PID

controller structure can be used.

(16)

It is possible to obtain the new controller parameters of , and as in (17) in terms of the PID controller parameters given in (15).

(17)

Fig. 4. A part of the root-locus plot for the system with a PID controller.

IV. SIMULATIONS & RESULTS

In this section, first of all, the system parameters are

identified and the transfer function of a system is found for a

given step response. After that the design procedures given in

Section 3 are applied. Successes of the methods are shown by

simulations.

A. System Identification

Fig. 5 shows the step response of a system with the transfer

function

(18)

Fig. 5. Step response of the given transfer function.

Parameters that need to be known, in order to be used in

proposed formulas, are read using the step response of the

system and are given in Table I.

-1 -0.5 0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

1.5

Root Locus

Real Axis

Imagin

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Axis

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-1

-0.5

0

0.5

1

Root Locus Editor for Open Loop 1 (OL1)

Real Axis

Imag A

xis

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Time (s)

Outp

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Step Response


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TABLE I. STEP RESPONSE PARAMETERS

Parameter Value Parameter Value

1 s -0.1895

2 s 0.3225

2.206 s 0.25

2.497 s

3.11 s

If these parameters are used in (2)-(8), then the transfer

function of the system is obtained as in (18). Therefore, it is

possible to say that with the proposed modeling formulas, such

systems are identified easily.

B. Controller Design

To be able to design a discrete PID controller, the open-

loop system transfer function in s-plane should be transformed

to its discrete counterpart. In this example, the discrete time

transfer function of the system is obtained as in (19) using a

zero order hold and by taking the sampling time as

seconds.

(19)

If the controller zeros are located on the open-loop system

poles and closed-loop system poles are moved to the breaking

point as shown in Fig. 4, the discrete PID controller is obtained

as

(20)

Finally, PID controller parameters ( , and ) can be

calculated by equating (15) to (20) as follows.

(21)

To calculate industrial PID parameters, (17) can be used.

(22)

The closed-loop unit step response of the system is given in

Fig. 6.

Fig. 6. Closed-loop step response with designed PID controller.

After the system settles its final value, step inputs of

amplitude are applied as an input disturbance and output

disturbance at the 15th and 30

th seconds, respectively. It can be

observed from the given figure that the system has no steady-

state error and perfect disturbance rejection in the steady state.

In addition, there is no overshoot, as expected. Variation of the

control signal with designed discrete PID controller is given in

Fig. 7.

Fig. 7. Variation of the control signal.

C. Robustness of the Identification Method

In general, real processes do not have perfect responses to

the input signals. Noisy output signals may cause difficulties

during the system identification phase. Therefore, the

identification method should provide a suitable performance

under the noise effects. In this section of the study, robustness

of the proposed identification method will be discussed using a

test case.

Unit step response of a process, with the noise whose

amplitude is unit, to the applied unit step input is given in

Fig. 8.

Fig. 8. Unit step response of a process.

In order to identify the given process by using the

proposed identification method, the step response parameters

which are already illustrated in Fig. 2 are determined using the

Fig. 8 approximately. These determined parameters are given

as in Table 2.

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Closed-Loop Response

Reference

Output

Input D.

Output D.

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Step Response of the Process


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TABLE II. APPROXIMATE STEP RESPONSE PARAMETERS

Parameter Value Parameter Value

1 s -0.35

3 s 1.15

4.2 s 0.75

5.6 s

9.5 s

If the parameters, which are read using the step response of the

system by taking approximate mean values into account, are

used in (2)-(8), the transfer of the system is obtained as

(23)

which is actually given by the transfer function of

(24)

Step response of the obtained transfer function is very similar

to the actual transfer function as seen in Fig. 9.

Fig. 9. Comparison of the real system and identified model.

It is possible to say that the proposed method gives a

suitable transfer function in case of the noisy process output.

Here, the important thing is that, the identification method is

not critically sensitive to the changes of the determined

parameters.

D. Robustness of the Control Method

Robustness of the designed controller is one of the

important criterions. If the controller does not have an

acceptable robustness to the parameter changes, it is not

possible to use it. Discrete PID controller structure is used to

control the systems under consideration because of the reasons

mentioned earlier. In this section, it is aimed to discuss

robustness of the designed discrete PID controller through the

example given in section 4.2. For this purpose, all system

parameters ( , , , and ) are changed and closed-

loop system responses are observed with the same PID

controller. Fig. 10 shows the response of the system to the

changes of parameters in 5-D parameter space. It is seen from

the figure that even if all system parameters change at the

same time, control system presents an acceptable performance

with the designed PID controller.

Fig. 10. Closed-loop system responses.

V. CONCLUSION

In this study, a new modeling method is presented for

under-damped second order systems with inverse response and

dead-time using step response. A set of formulas are proposed

to determine the system parameters. Additionally, a method for

designing a discrete PID controller is proposed. The validity of

the proposed formulas and the success of the proposed design

method are demonstrated using a case study. The robustness of

the identification and control methods are also considered in

this paper.

REFERENCES

[1] F. G. Shinskey, Robust process control, McGraw-Hill, New York, 1979.

[2] G. Stephanopoulos, Chemical Process Control: An Introduction to

Theory and Practice, Prentice-Hall: Englewood Cliffs, NJ, 1984. [3] C. Scali and A. Rachid, “Analytical design of Proportional-Integral-

Derivative controllers for inverse response processes”, Ind. Eng. Chem.

Res., vol. 37, pp. 1372-1379, 1998. [4] P. Chen, Y. Tang, Q. Zhang, W. Zhang, “New design method of PID

controller for inverse response processes with dead time”, IEEE International Conference on Industrial Technology, 2005, pp.1036-

1039.

[5] Q. Wang, X. Guo, Y. Zhang, “Direct identification of continuous time delay systems from step responses”, Journal of Process Control, 11

(2001), pp. 531–542.

[6] C. Huang and C. Chou, “Estimation of the underdamped second-order parameters from the system transient”, Industrial and Engineering

Chemistry Research, 33, 1994, pp. 174-176.

[7] G. P. Rangaiah and P. R. Krishnaswamy, “Estimating second-order dead time parameters from underdamped process transients”, Chemical

Engineering Science, 51, 1996, pp. 1149-1155.

[8] K. T. V. Waller and C. G. Nygårdas, “On inverse response in process control”, Ind. Eng. Chem. Fundam., vol. 14, pp. 221-223, 1975.

[9] I. L. Chien, Y. C. Cheng, B. S. Chen, C. Y. Chuang, “Simple PID

controller tuning method for processes with inverse response plus dead time or large overshoot response plus dead time”, Ind. Eng. Chem. Res.,

vol. 42, pp. 4461-4477, 2003.

[10] W. L. Luyben, “Tuning proportional-integral controllers for processes with both inverse response and deadtime”, Ind. Eng. Chem. Res., vol.

39(4), pp. 973-976, 2000.

[11] K. Iinoya and R. J. Altpeter, “Inverse Response in Process Control”, Ind. Eng. Chem., vol. 54, pp. 39-43, 1962.

[12] W. Zhang, X. Xu, Y. Sun, “Quantitative Performance Design for

Inverse-Response Processes”, Ind. Eng. Chem. Res., vol. 39, pp. 2056-2061, 2000.

[13] S. Alc´antara, C. Pedret, R. Vilanova, W. D. Zhang, “Analytical H∞ design for a Smith-type inverse-response compensator”, American

Control Conference, USA, pp. 1604-1609, June 2009. [14] K. J. Aström and T. Hagglund, Advanced Pid Control, ISA-The

Instrumentation, Systems, and Automation Society, 2006.

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