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Chapter 6

PID Controller Design

PID (proportional integral derivative) control is one of the earlier control strategies [59].Its early implementation was in pneumatic devices, followed by vacuum and solid stateanalog electronics, before arriving at todays digital implementation of microprocessors.It has a simple control structure which was understood by plant operators and which theyfound relatively easy to tune. Since many control systems using PID control have provedsatisfactory, it still has a wide range of applications in industrial control. According to asurvey for process control systems conducted in 1989, more than 90 of the control loops wereof the PID type [60]. PID control has been an active research topic for many years;see themonographs [6064]. Since many process plants controlled by PID controllers have similardynamics it has been found possible to set satisfactory controller parameters from less plantinformation than a complete mathematical model. These techniques came about because ofthe desire to adjust controller parameters in situ with a minimum of effort, and also becauseof the possible difficulty and poor cost benefit of obtaining mathematical models. The twomost popular PID techniques were the step reaction curve experiment, and a closed-loopcycling experiment under proportional control around the nominal operating point.

In this chapter, several useful PID-type controller design techniques will be presented,and implementation issues for the algorithms will also be discussed. In Sec. 6.1, the pro-portional, integral, and derivative actions are explained in detail, and some variations of thetypical PID structure are also introduced. In Sec. 6.2, the well-known empirical ZieglerNichols tuning formula and modified versions will be covered. Approaches for identifyingthe equivalent first-order plus dead time model, which is essential in some of the PID con-troller design algorithms, will be presented. A modified ZieglerNichols algorithm is alsogiven. Some other simple PID setting formulae such as the ChienHronesReswick for-mula, CohenCoon formula, refined ZieglerNichols tuning, WangJuangChan formulaand ZhuangAtherton optimum PID controller will be presented in Sec. 6.3. In Sec. 6.4,the PID tuning formulae for FOIPDT (first- order lag and integrator plus dead time) andIPDT (integrator plus dead time) plant models, rather than the FOPDT (first-order plus deadtime) model, will be given. A graphical user interface (GUI) implementing hundreds ofPID controllers tuning formulae for FOPDT model will be given in Sec. 6.5. In Sec. 6.6, anoptimization-based design algorithm, together with a GUI for optimal controller design, is

183

Copyright 2007 by the Society for Industrial and Applied Mathematics.This electronic version is for personal use and may not be duplicated or distributed.

From "Linear Feedback Control" by Dingyu Xue, YangQuan Chen, and Derek P. Atherton.This book is available for purchase at www.siam.org/catalog.

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184 Chapter 6. PID Controller Design

given. In Sec. 6.7, some of the advanced topics on PID control will be presented, such asintegrator windup phenomenon and prevention, and automatic tuning techniques. Finally,some suggestions on controller structure selections for practical process control are pro-vided.

6.1 Introduction

6.1.1 The PID Actions

A typical structure of a PID control system is shown in Fig. 6.1, where it can be seen thatin a PID controller, the error signal e(t) is used to generate the proportional, integral, andderivative actions, with the resulting signals weighted and summed to form the controlsignal u(t) applied to the plant model. A mathematical description of the PID controller is

u(t) = Kp[e(t) + 1

Ti

t0

e()d + Td de(t)dt

], (6.1)

where u(t) is the input signal to the plant model, the error signal e(t) is defined as e(t) =r(t) y(t), and r(t) is the reference input signal.

The behavior of the proportional, integral, and derivative actions will be demonstratedindividually through the following example.

Example 6.1. Consider a third-order plant model given by G(s) = 1/(s + 1)3. If a pro-portional control strategy is selected, i.e., Ti and Td 0 in the PID control strategy,for different values of Kp, the closed-loop responses of the system can be obtained usingthe following MATLAB statements:

>> G=tf(1,[1,3,3,1]);for Kp=[0.1:0.1:1], G_c=feedback(Kp*G,1); step(G_c), hold on; endfigure; rlocus(G,[0,15])

The closed-loop step responses are obtained as shown in Fig. 6.2(a), and it can be seenthat when Kp increases, the response speed of the system increases, the overshoot of theclosed-loop system increases, and the steady-state error decreases. However when Kp islarge enough, the closed-loop system becomes unstable, which can be directly concludedfrom the root locus analysis in Sec. 3.4. The root locus of the example system is shown

u(t) plantmodel

e(t) y(t)

r(t)controller

..

..

..

..

..

..

..

..

..

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..............................................................

PID controller

disturbance d(t)

measure-ment noise

um

Figure 6.1. A typical PID control structure.

Copyright 2007 by the Society for Industrial and Applied Mathematics.This electronic version is for personal use and may not be duplicated or distributed.

From "Linear Feedback Control" by Dingyu Xue, YangQuan Chen, and Derek P. Atherton.This book is available for purchase at www.siam.org/catalog.

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6.1. Introduction 185

0 5 10 150

0.1

0.2

0.3

0.4

0.5

Step Response

Time (sec)

Am

plitu

de Kp = 1

Kp = 0.1

(a) closed-loop step response

Root Locus

Real Axis

Imag

inar

y A

xis

2.5 2 1.5 1 0.5 0 0.52.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

System: GGain: 8Pole: 2.36e005 + 1.73iDamping: 1.37e005Overshoot (%): 100Frequency (rad/sec): 1.73

(b) root locus

Figure 6.2. Closed-loop step responses.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Step Response

Time (sec)

Amplitude

Ti = 0.7

Ti =1.5

(a) PI control

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Step Response

Time (sec)

Amplitude

Td = 0.1

Td =1.5

(b) PID control

Figure 6.3. Closed-loop step responses.

in Fig. 6.2(b), where it is seen that when Kp is outside the range of (0, 8), the closed-loopsystem becomes unstable.

If we fix Kp = 1 and apply a PI (proportional plus integral) control strategy fordifferent values of Ti, we can use the following MATLAB statements:

>> Kp=1; s=tf(s);for Ti=[0.7:0.1:1.5]

Gc=Kp*(1+1/Ti/s); G_c=feedback(G*Gc,1); step(G_c), hold onend

to generate the closed-loop step responses of the example system shown in Fig. 6.3(a). Themost important feature of a PI controller is that there is no steady-state error in the stepresponse if the closed-loop system is stable. Further examination shows that if Ti is smallerthan 0.6, the closed-loop system will not be stable. It can be seen that when Ti increases,the overshoot tends to be smaller, but the speed of response tends to be slower.

Copyright 2007 by the Society for Industrial and Applied Mathematics.This electronic version is for personal use and may not be duplicated or distributed.

From "Linear Feedback Control" by Dingyu Xue, YangQuan Chen, and Derek P. Atherton.This book is available for purchase at www.siam.org/catalog.

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186 Chapter 6. PID Controller Design

Fixing both Kp and Ti at 1, i.e., Ti = Kp = 1, when the PID control strategy is used,with different Td , we can use the MATLAB statements

>> Kp=1; Ti=1; s=tf(s);for Td=[0.1:0.2:2]

Gc=Kp*(1+1/Ti/s+Td*s); G_c=feedback(G*Gc,1); step(G_c), hold onend

to get the closed-loop step response shown in Fig. 6.3(b). Clearly, when Td increases theresponse has a smaller overshoot with a slightly slower rise time but similar settling time.

In practical applications, the pure derivative action is never used, due to the derivativekick produced in the control signal for a step input, and to the undesirable noise amplifica-tion. It is usually replaced by a first-order low pass filter. Thus, the Laplace transformationrepresentation of the approximate PID controller can be written as

U(s) = Kp1 + 1

Tis+ sTd

1 + sTdN

E(s). (6.2)The effect of N is illustrated through the following example.

Example 6.2. Consider the plant model in Example 6.1. The PID controller parametersare Kp = 1, Ti = 1, and Td = 1. With different selections of N, we can use the MATLABcommands

>> Td=1; Gc=Kp*(1+1/Ti/s+Td*s); step(feedback(G*Gc,1)), hold onfor N=[100,1000,10000,1:10]

Gc=Kp*(1+1/Ti/s+Td*s/(1+Td*s/N)); G_c=feedback(G*Gc,1); step(G_c)endfigure; [y,t]=step(G_c); err=1-y; plot(t,err)

to get the closed-loop step response with the approximate derivative terms as shown inFig. 6.4(a). The error signal e(t) when N = 10 is shown in Fig. 6.4(b). It can be seen thatwith N = 10, the approximation is fairly satisfactory.

6.1.2 PID Control with Derivative in the Feedback Loop

From Fig. 6.4(b), it can be seen that the