Time-Delay Approximation: Its Influence on the ...

Time-Delay Approximation: Its Influence on the Structure and Performance of the IMC-PI/PID Controller P.V. Gopi Krishna Rao, M.V. Subramanyam and K. Satyaprasad

Abstract Proportional integral (PI) and proportional integral derivative (PID) controllers have been at the heart of control engineering practice for several decades. 95 % of the controllers employed in the industry are PI/PID. In process control, one often encounters systems described by transfer functions with time delays, which become transcendental functions. The design of the controller demands the rational transfer function approximation of the time-delay term. This paper focuses on the effect of time-delay approximation techniques, viz. Taylor series expansion and Padé approximation, on the structure and performance of PI/ PID controllers designed with Internal Model Control (IMC). The performance of the PI/PID controllers was tested in simulation environment on various processes with time delay. For uniform comparison, the controllers were tuned to have a same robustness measure, in terms of maximum sensitivity (MS). The results indicate, irrespective of time-delay approximation considered, the controllers provide good set point tracking and poor disturbance rejection.



Keywords Time delay Padé approximation control PI/PID Disturbance Set point







 Taylor series  Internal model

1 Introduction Proportional integral (PI) and proportional integral derivative (PID) controllers have been at the heart of control engineering practice for several decades [1]. The use of the PI or PID controller is ubiquitous in industry. In process control P.V. Gopi Krishna Rao (&)  K. Satyaprasad Department of ECE, JNTUK, Kakinada 533003, India e-mail: [email protected] M.V. Subramanyam Shantiram Engineering College, Nandyal 518501, India  Springer India 2015 L.C. Jain et al. (eds.), Intelligent Computing, Communication and Devices, Advances in Intelligent Systems and Computing 308, DOI 10.1007/978-81-322-2012-1_60

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applications, more than 95 % of the controllers are of PI or PID type [1–3]. PID controllers can assure satisfactory performances with a simple algorithm for a wide range of processes [3, 4]. In process control, one often encounters systems, described by transfer functions with time delays [2]. If a dynamic system with time delay modeled as a time invariant linear system, its transfer function (rational function) becomes a transcendental function because of time delay [5]. For design and analysis of controllers, these delays are usually approximated by rational transfer functions [6]. This is usually carried out using delay approximation methods, viz. Taylor series expansion and Padé approximation. The internal model control provides a progressive, effective, natural, generic, unique, powerful, and simple framework for analysis and synthesis of control system performance [7–10]. The simplicity and enhanced performance of the IMC-based tuning rule, and the analytically derived IMC-PID tuning techniques have attracted the attention of the industrial users, in the past decade [9, 10]. The well-known IMC-PID tuning rule provides a clear compromise in the midst of closed-loop performance and robustness to model uncertainties, by only one userdefined tuning parameter k, which is directly related to the closed-loop time constant [6, 7, 9–12], but the method used for time-delay approximation influences the nature of the IMC—PI/PID controller, controller parameters, and performance. The organization of the paper is, Sect. 2 describes the basic design of IMC— PID controller and Sect. 3 discusses time-delay approximation techniques and the design of PI/PID controllers using IMC and time-delay approximation. Section 4 demonstrates the simulation results of the performance evaluation for set point tracking and disturbance rejection, and Sect. 5 draws the conclusions.

2 IMC: PID Design The widely used approximate or predictive models of the chemical process are the first-order process with time delay (FOPTD) (1). Garcia and Morari [11, 13] introduced internal model control; it is characterized as a controller where the process model is explicitly an integral part. The design process of IMC involves factorizing the predictive plant model GM ðsÞ as invertibleGM ðsÞand non-invertible GMþ ðsÞparts depicted in (2) and (3) by simple factorization or all-pass factorization [7, 11, 12, 14–16]. The IMC in (4) is the inverse of invertible part of the plant model GM ðsÞ. GM ðsÞ ¼

Kehs ss þ 1

GM ðsÞ ¼ GM ðsÞGMþ ðsÞ

ð1Þ ð2Þ

Time-Delay Approximation…

563

GM ðsÞ ¼

K ; GMþ ðsÞ ¼ ehs ss þ 1

ð3Þ

The IMC controller QðsÞ ¼ G1 M ðsÞGf ðsÞ

ð4Þ

where Gf ðsÞ is a low-pass filter of the form Gf ðsÞ ¼ 1=ð1 þ ksÞ used to physically realize the IMC controller. The IMC controller will take the form of ideal feedback controller of Fig. 2 with rearrangement of Fig. 1, expressed mathematically in terms of QðsÞ and GM ðsÞas (5) and (6). QðsÞ 1  QðsÞGM ðsÞ

ð5Þ

ðss þ 1Þ K ½ðks þ 1Þ  ehs 

ð6Þ

GC ðsÞ ¼

GC ðsÞ ¼

The feedback controller of (5) and (6) lacks the standard PI/PID form and rearranges the equations with approximation of the time delay in the process model with Taylor series expansion or Padé approximate, to obtain PID controller form of (7).

Fig. 1 Basic IMC structure

Fig. 2 Feedback control structure

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P.V. Gopi Krishna Rao et al.

  1 þ Td s GC ðsÞ ¼ KP 1 þ Ti s

ð7Þ

3 Time-Delay Approximation 3.1 Taylor Series The Taylor series expansion of the time-delay term ehs is [5] ehs ¼ 1 

hs ðhsÞ2 ðhsÞ3 ðhsÞ4 ðhsÞ5 ðhsÞn þ  þ  þ . . . þ ð1Þn 1! 2! 3! 4! 5! n!

ð8Þ

Considering the first two terms and truncation of other terms in (8) results in the first-order Taylor series (9). ehs ¼ ð1  hsÞ

ð9Þ

Switching (6) and (9) and comparison with (7) produces a PI controller with proportional gain KP and integral time Ti , represented in (10). KP ¼

s ; Ti ¼ s K ðks þ 1Þ

ð10Þ

3.2 Padé Approximation of Time Delay Another often used method for rational approximation of time-delay transfer function is application of Padé approximate [5, 16]. This method represents one of the most used and favorite approximations. It is based on the comparison of derivatives of the approximating and approximated functions in zero [17]. The Padé approximant enables to approximate more complex functions using rational function [6]. The expression for Padé approximation is [17–19]. ehs 

PðsÞ ¼

PðsÞ PðsÞ

n   X n ð2n  kÞ! k¼0

k

ð2nÞ!

ð11Þ ðhsÞk

ð12Þ

Rivera et al. [11] and Rao et al. [20] utilized the first-order (1/1) Padé approximate of (13), for design of the controller. Rearrangement of (13), (6), and (7) produces the PID controller parameters represented in (14).

Time-Delay Approximation…

565

  1  h2s ffi 1 þ h2s

ð13Þ

2s þ h h hs ; Ti ¼ s þ ; Td ¼ Kð2k þ hÞ 2 2s þ h

ð14Þ

hs

e

KP ¼

Lee et al. [21] utilized second-order (2/2) Padé approximate of (15) for design of PID controller cascaded with lead/lag filter, which is achieved from of (15), (6), and (16). The PID parameters and the filter coefficients are represented in (17). e

hs

 Gc ðsÞ ¼ KP

KP ðsÞ ¼

 h2 ffi

s2  h2s þ 1 s2 þ h2s þ 1

12 h2 12

1 þ Td s 1þ Ti s





ds2 þ cs þ 1 bs2 þ as þ 1

ð15Þ  ð16Þ

h h h hk h2 k ; Ti ¼ ; Td ¼ ; a ¼ ;b ¼ ; c ¼ s; d ¼ 0 2Kðk þ hÞ 2 6 2ðk þ hÞ 12ðk þ hÞ ð17Þ

4 Simulation Results Influence of time-delay approximation, viz. 1/1 Padé, 2/2 Padé approximations and Taylor series expansion, on structure and performance of PI/PID controllers for set point tracking and disturbance rejection on FOPTD systems with different h=sratio, was tested in simulation environment. The process models used by other researchers were considered for study. For uniform comparison, the PI/PID controllers were tuned to have a same robustness measure, in terms of maximum sensitivity (MS) using IMC technique. The closed-loop performance and robustness were evaluated using integral absolute error (IAE) criterion with step change in set point and load disturbance. Example 1: The 500-MW  superheated steam boiler system, with transfer function GðsÞ ¼ 0:7717e56:278s ð42:934s þ 1Þ [20] and h=s [ 1 was considered for study. The controllers were tuned to have the same robustness of MS ¼ 1:59 with the adjustment of single tuning parameter k. The simulation results of Figs. 3, 4, and the Table 1 indicate that the IMC-PI/PID controllers provide good set point tracking but poor disturbance rejection. It is also inferred from the responses that the controllers with Padé approximation perform better than Taylor series for set point tracking and disturbance rejection.

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Fig. 3 Responses for step change in set point input of example 1

Fig. 4 Response for step-load disturbance input of example 1

Table 1 Performance of IMC-PI/PID controller for set point and disturbance of example 1 Delay approximation

KP

1/1 Padé

0.9290

a

2/2 Padé

Taylor a ds2 þcsþ1

bs2 þasþ1

Ti

Td

k

MS

Set point Peak

71.073

16.998

0.3839

28.139

9.379

38.7

1.59

1

0.4955

42.934

–

56

1.59

1.042

42:934sþ1 ¼ 107:5435s 2 þ11:4656sþ1

71

1.59

0.998

Disturbance IAE 99.14 94.99 122.1

Peak

IAE

0.559

76.48

0.584

73.3

0.613

90.38

Time-Delay Approximation…

567

 Example 2: The lag-time-dominant model GðsÞ ¼ 100e1s ð100s þ 1Þ [22] with h=s ¼ 0:01 was considered for study. The controllers were designed to have same robustness of MS ¼ 1:4 by adjusting single tuning parameter k. The simulation results are represented in Figs. 5, 6, and the Table 2.

Fig. 5 Responses for step change in Set point input of example 2

Fig. 6 Response for step-load disturbance input of example 2

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Table 2 Performance of IMC-PI/PID controller for set point and disturbance of example 2 Delay approximation

KP

Ti

Td

k

MS

Set point Peak IAE

Disturbance Peak IAE

1/1 Padé

0.437

100.5

0.4975

1.8

1.4

1.004

2.812

2.131

229.4

0.0021

0.5

0.1667

1.36

1.4

1

2.381

2.243

237.5

0.3724

100

–

1.685

1.4

1

2.685

2.544

267.8

a

2/2 Padé

Taylor a ds2 þcsþ1

bs2 þasþ1

¼

100sþ1 0:0480s2 þ0:2881sþ1

5 Conclusions Simulation study was conducted on the IMC—PI/PID controllers, with conventional filter proposed by Rivera et al. The time-delay approximation method influences the design of the controller. First-order Taylor series expansion resulted in PI controller, 1/1 Padé approximate produced PID controller, and 2/2 Padé approximate resulted in PID controller cascaded with lead/lag filter. The performance evaluation of the controllers was carried out on the basis of the peak value of the response and integral performance criterion IAE, for set point tracking and disturbance rejection, on FOPTD with various h=s ratios. The PI/PID controllers were tuned to have the same robustness, in terms of maximum sensitivity (MS ), for uniform comparison. It was observed that irrespective of time-delay approximation, IMC-PID provided good set point tracking but poor disturbance rejection and the disturbance attenuation provided by the controllers with 1/1 Padé and 2/2 Padé approximates was better in comparison with the controller with Taylor series expansion. It is suggested to use PID controller cascaded with lead/lag filter for disturbance rejection, obtained with 2/2 Padé approximant with conventional IMC filter or 1/1 Padé approximant with improved IMC filter.

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