Study on the Relationships between Two Typical

Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003

ThMPI-2

Study on the Relationships between Two Typical Modeling Methods in Process Control Danying Gu, Tao Liu, Weidong Zhang Dept. of Automation, Shanghai Jiaotong University, Shanghai, 200030, P. R. China Tel/Fax:+86.21.62826946; Email: [email protected], [email protected] Abstract

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There are two typical modeling methods for PID design and tuning in process control：step response method and self-oscillation method. The former usually results in a first-order plus dead-time model and the later gives ultimate gain and ultimate frequency of controlled plant. Usually the two modeling methods are considered different. The aim of this paper is to bridge the gap between them. In this paper, some internal relationships between the two modeling methods are studied. Analytical mapping expressions between them are derived with time domain and frequency domain analysis method, respectively. This implies that PID design and tuning techniques need not be limited to either step response method or the self-oscillation method only. Thus, the designer can have more choices for PID controller design and tuning. Key words: process control; modeling; first-order plus dead-time model; PID controller; step response; self-oscillation

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Introduction

So far, many different methods have been developed for plant modeling. Among them, two typical modeling methods, step response method and self-oscillation method, are mostly used in process control practice at present (Yuwana, 1982). Step response method is based on open-loop step test and usually results in a first-order plus dead-time model Gp ( s) = K p e−θ s (τ s + 1) . The model describes a linear monotonic process quite well and is often sufficient for PID controller tuning (Hang and Chin, 1991; Halevi, 1991). Self-oscillation method is often referred to as the closed-loop trial-and-error identification procedure, i.e. a way to find the important process information: ultimate gain Ku and ultimate period Tu (that is ultimate frequency ωu = 2π Tu ) (Yuwana, 1982; Yu C.C., 1999). However, the above two types of modeling methods are usually considered different and used separately. Correspondingly, PID controller tuning methods based on them also have some limitations (Zhang W.D., 2002). For example, the well-known Ziegler- Nichols tuning method is only based on self- oscillation modeling method while Cohen-Coon tuning method is confined to step response method. In this paper, relationships between two typical modeling methods are studied. Analytical mapping expressions between [ Ku , Tu ] and [ K p ,τ ,θ ] are derived with time domain and frequency domain analysis method, respectively. Therefore, the plant model can be identified more accurately and effectively with optimal combination of [ Ku , Tu , K p ,τ ,θ ] obtained from step response modeling experiment and self-oscillation modeling experiment. This will help to acquire better PID controllers in most cases. 0-7803-7924-1/03/$17.00 ©2003 IEEE

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Analytical analysis

2.1 Time domain analysis Consider the block diagram of the unity feedback closed-loop system shown in Fig. 1. Here assume that the controlled process is stable and its transfer function is Gp ( s) = K p e−θ s (τ s + 1) . According to the typical procedure proposed by self-oscillation modeling method, the feedback controller in Fig. 1 is firstly set as a proportional controller with transfer function Gc ( s ) = K c . Secondly, the controller gain is increased gradually until the output of closed-loop system becomes oscillatory. The value of K c in this situation is just equal to the ultimate gain Ku . e

r +

Gc ( s )

u

G p (s )

y

Fig. 1.

Block diagram of a simple feedback control system

Suppose that the setpoint input is a step with magnitude A at t = 0 . Then the output of closed-loop control system can be written as Y ( s) =

A G p ( s )Gc ( s ) A Ke −θ s ＝ s 1 + G p ( s )Gc ( s ) s τ s + 1 + Ke −θ s

(1)

where K = K p Kc

For the convenience of calculation, let f ( s) =

Ke −θ s τ s + 1 + Ke −θ s

(2)

According to the Pade approximation theory (Cabannes, 1976), we can get f ( s) =

a2 s 2 + a1 s + a0 b2 s 2 + b1 s + 1

where K  a0 =  1+ K  Kθ (θ 2 + 5θτ + 8τ 2 )  a = − 1 2  2( Kθ + 4 Kθτ + 6 Kτ 2 + θ 2 + 4θτ + 6τ 2 )   Kθ 2 (θ 2 + 6θτ + 12τ 2 ) a2 = 2  12( Kθ + 4 Kθτ + 6 Kτ 2 + θ 2 + 4θτ + 6τ 2 )   ( Kθ 3 + 5 Kθ 2τ + 8 Kθτ 2 − θ 3 − 5θ 2τ − 12θτ 2 − 12τ 3 )  b1 = − 2( Kθ 2 + 4 Kθτ + 6 Kτ 2 + θ 2 + 4θτ + 6τ 2 )   θ ( Kθ 3 + 6 Kθ 2τ + 12 Kθτ 2 + θ 3 + 6θ 2τ + 18θτ 2 + 24τ 3 )  b2 = 12( Kθ 2 + 4 Kθτ + 6 Kτ 2 + θ 2 + 4θτ + 6τ 2 ) 

(3)

Through the inverse Laplace transformation, Eq.1 is transformed into

1 − qt  p0  q 2 + ω 2 + ω e Sin (ω t + Φ ) a y (t ) = A 2  b2  ( q 2 − ω 2 − p1 q + p 0 ) 2 + ω 2 ( p1 − 2 q ) 2  q2 + ω 2 

where

p1 =

     

(4)

a b2 b a1 1 , p0 = 0 , q = 1 , ω 2 = − 1 2 a2 b2 4b2 2b2 a2

Φ = a tan[

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ω ( p1 − 2 q) ω ] − a tan( ) q 2 − ω 2 − p1 q + p0 −q

On the self-oscillation condition, q is zero. Then the

ω in this situation is just equal to ω u . So we get: K θ 3 + 5θ 2τ + 12θτ 2 + 12τ 3 Ku = = (5) Kp K p (θ 3 + 5θ 2τ + 8θτ 2 ) θ ( Kθ 3 + 6Kθ 2τ + 12Kθτ 2 + θ 3 + 6θ 2τ + 18θτ 2 + 24τ 3 ) (6) Tu = 2π 12( Kθ 2 + 4 Kθτ + 6Kτ 2 + θ 2 + 4θτ + 6τ 2 )

2.2 Frequency domain analysis The self-oscillation method can be explained through studying the Nyquist diagram and the Nyquist criteria. We have the following equations when the closed-loop system in Fig.1 is oscillatory: ∠GH ( jω u ) = −θω u − a tan(τω u ) = −π  Ku K p  GH ( jω ) = =1 u  (τω u ) 2 + 1 

(7)

Considering K p is easily tested by step test, we can derive the following equations from Eq. 7:  ( K u K p ) − 1 Tu ( K u K p ) − 1 τ = =  ωu 2π   Tu Tu 2 θ = 2 − 2π a tan( ( K u K p ) − 1) 2

2

(8)

The above conclusion is strictly analytical, because its derivation procedure includes no approximation.

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Simulation example

In this section, the accuracy and effectivity of these two mapping expressions will be verified, and the similarity of them will be confirmed. Example: Choose a series of processes with different time delays to study. Calculate the exact [ Ku , Tu ] with the actual [ K p ,τ ,θ ] from Expression 2 (i.e. Eq. 8). Compare it with [ Ku , Tu ] directly calculated from Expression 1 ( i.e. Eq. 5 and Eq. 6 ). The result is listed Actual process Kp

τ

θ

1 1 1 1 1 1 1

1 1 1 1 1 1 1

0.2 0.4 0.6 0.8 1 1.5 2

in Tab. 1. It is obvious that the precision of Exp. 1 is a little worse than that of Exp. 2. The error produced in Exp. 1 is due to Pade approximation. Despite of the fact that Exp. 1 is not absolutely analytical, it is still a valuable result. When we have known [Kp ,τ ,θ ] from step test, [ Ku , Tu ] can directly derived from Exp. 1. It is much easier than substituting [ K p ,τ ,θ ] into Exp. 2 and solving these equation sets.

conclusions

The relationships between two typical modeling methods are studied in this paper. Mathematic expressions between [ Ku , Tu ] and [ K p ,τ ,θ ] are analytically figured out with time domain and frequency domain analysis methods, respectively. This bridges the gap between the two modeling methods and the corresponding PID tuning methods. Furthermore, the accurate values of  Ku , Tu , K p ,τ ,θ  are able to be simultaneously obtained from one modeling procedure, so that plant model could be more accurate and effective. When PID controllers are designed, we can have more choices on the design methods. The accuracy and effectiveness of the proposed relationships between two typical modeling methods have been demonstrated through the simulation example.

Acknowledgment This project is supported by the National Natural Science Foundation of China (69804007) and the National Key Technologies R&D Program in the Tenth Five-Year Plan (2001BA201A04).

References [1] Cabannes, H. Pade approximants method and its applications to mechanics, Berlin: Springer, 1976. [2] Halevi, Y. Optimal reduced order models with delay, Proc, 30th Conf. On Decision and Control. Brighton, England, 1991, 602-607. [3] Hang, C.C., & D.Chin. Reduced order process modeling in self-tuning control. Automatica, 1991, 27(3), 529-534. [4] Yu, C.C. Autotuning of PID Controllers――Relay feedback Approach. Springer-Verlag Berlin Herdelberg, NY, 1999. [5] Yuwana, M. & D.E. Seborg. A new method for on-line controller tuning. AIChE Journal, 1982, 28(3), 434-439. [6] Zhang, W.D., & D.Y. Gu. A Unified Approach to Design the RZN PID Controller for Stable and Unstable Processes with Time Delay, submitted to Automatica, 2002.

Tab.1 Comparison of identification results of two mapping expressions [ K u , Tu ] [ K u , Tu ] from Exp.1 from Exp.2 absolute relative absolute relative Ku Tu Ku Tu error error (%) error error(%) 8.5 0.74 8.08 0.42 4.95 0.72 0.02 2.41 4.59 1.40 4.35 0.24 5.31 1.35 0.05 3.51 3.29 2 3.11 0.18 5.39 1.92 0.08 4.22 2.64 2.56 2.50 0.14 5.18 2.44 0.12 4.87 2.26 3.09 2.14 0.12 5.19 2.92 0.17 5.44 1.76 4.32 1.68 0.08 4.77 4.04 0.28 6.41 1.52 5.49 1.45 0.07 4.31 5.08 0.41 7.42

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