[IEEE Fifth International Conference on Hybrid Intelligent Systems (HIS'05) - Rio de Janeiro, Brazil...

Assessing Fuzzy and Neural Approaches for a PID Controller Using UniversalModel

Rodrigo R. SumarDepartment of Automation and SystemsFederal University of Santa CatarinaBox 476, Zip Code 88094-900Florianópolis, SC, [email protected]

Antonio A. R. CoelhoDepartment of Automation and SystemsFederal University of Santa CatarinaBox 476, Zip Code 88094-900Florianópolis, SC, [email protected]

Leandro dos Santos CoelhoAutomation and Systems LaboratoryPontifical Catholic University of Paraná

Imaculada Conceição, 1155, Zip Code 80215-901Curitiba, PR, Brazil.

[email protected]

Abstract

Despite the popularity, the tuning aspect of PIDcontrollers is a challenge for researches and plantoperators. Various control design methodologies have beenproposed in the literature such as auto-tuning, self-tuningand pattern recognition. The main drawback of thesemethodologies in the industrial environment is the numberof tuning parameters to be selected. In this paper, thedesign of a PID controller, based on the universal modelof the plant, is derived. The design characteristic has onlyone parameter to be tuned. This is a good idea from theviewpoint of plant operators. Fuzzy and neural approachesare used for designing and assessing the efficiency of thePID control in nonlinear plants.

1 Introduction

In spite of many efforts the PID (Proportional-

Integral-Derivative) controller continues to be the main

component in industrial control systems, included in the

following forms: embedded controllers and programmable

logic controllers, distributed control systems. From

the viewpoint of simplicity and effectiveness, the PID

controller represents a good solution for controlling many

practical applications. However, the control literature still

showing different tuning structures for the PID control in

order to overcome complex dynamics and limitations [1].

Some difficult problems in industry are presented in this

environment to be answered or understood, not from the

enthusiastic form neither from the absence of knowledge of

plant operators or engineers, as the tuning aspects of a PID

controller [2].

Since nineties, many methodologies for setting the gains

of PID controllers have been proposed. In this context there

are classical (Ziegler/Nichols, Cohen/Coon, Abas, pole

placement and optimization) and advanced techniques

(minimum variance, gain scheduling and predictive) [1, 13,

4]. Some disadvantages of these control techniques for

tuning PID controllers are: i) excessive number of rules

to set the gains, ii) inadequate dynamics of closed-loop

responses, iii) difficulty to deal with nonlinear processes,

and iv) mathematical complexity of the control design

[12]. Therefore, it is interesting for academic and

industrial communities the aspect of tuning PID controllers,

especially with a reduced number of parameters to be

selected and a good performance to be achieved when

dealing with complex processes [4].

This paper discusses the intelligent tuning and design

aspects of a PID controller based on the universal model

of the plant where there is only one design parameter to be

selected. The proposed controller has the same structure

of a conventional three-term PID controller. The design is

based on a version of the generalized minimum variance

control of Clarke and Gawthrop [7]. Simulation tests

are shown for nonlinear plants (continuous stirred tank

reactor and heat exchanger). Fuzzy and neural methods

Proceedings of the Fifth International Conference on Hybrid Intelligent Systems (HIS’05) 0-7695-2457-5/05 $20.00 © 2005 IEEE

are included in order to improve the design of the PID

controller.

This paper is organized as follows. The design idea of

the PID controller, based on the universal model, is derived

in section 2. Tuning aspects of the proposed approach is

shown in section 3. Applications and conclusions are given

in sections 4 and 5, respectively.

2 Control Design

In this section a PID control design is proposed based on

an alternative representation for dynamic systems [9, 10].

The system output can be approximated by the universal

model defined as follows:

y(k + d) =N∑

i=0

Δiy(k) + (d − 1)bN∑

i=0

Δiu(k − 1)

+b

{u(k − 1) −

N−1∑i=0

Δiu(k − 2)} (1)

where isΔ the backward difference, d is the dead time, y(k)is the output, u(k) is the input, and y(k) is an estimate ofthe system output.

It can be observed that the universal model is simple

and has only one parameter, b, which contains the system

information. Considering Park et al.[9] approach thedimension of the universal model, N , can be chosen. Theparameter b can be derived from the previous measurementsof the system, i.e., if the values of y(k),

∑Ni=0 y(k − 1) and

ΔNu(k − 1 − d) are given, then, from equation (1), b iscalculated as follows:

b =

{y(k) −

N∑i=0

Δiy(k − 1)}

ΔNu(k − 1 − d)(2)

Using the universal model, equation (1), and the

generalized minimum variance control law, equation (3), it

is possible to obtain the desired PID structure as follows [7]:

u(k) =1

Q(z−1){Λ(z−1)yr(k) − [Γ(z−1)y(k + d)]

}(3)

u(k) =1

H(z−1)

[Λ(z−1)yr(k) − Γ(z−1)

N∑i=0

Δiy(k)

]

(4)

where yr(k) is the reference, Λ(z−1) and Γ(z−1) areweighting polynomials tuned by the operator and

H(z−1) = Q(z−1)+

+Γ(z−1)b{[

1 + (d − 1)N∑i=0

Δi

]z−1 −

N−1∑i=0

Δiz−2

}(5)

Next, N = 2, d = 1 and Γ(z−1) = Λ(z−1) = 1 areapplied to equations (4) and (5). Since Q(z−1) is free to bechosen, then it is possible to tuneQ(z−1) = q0Δ+Q(z−1),(in order to guarantee offset elimination for setpoint and

load changes), where Q(z−1) is a polynomial defined as

Q(z−1) = Γ(z−1)b{[1 + (d − 1)

N∑i=0

Δi

]z−1 −

N−1∑i=0

Δiz−2

}(6)

Therefore, the digital PID control law, that has the same

structure of a conventional PID controller, is given by

Δu(k) =1q0

{yr(k) − [3 − 3z−1 + z−2]y(k)

}(7)

where q0 is the single parameter to be tuned, it can penalize

the control effort and it can determine the closed-loop

dynamic of the system. For applications, it is desirable

to have a systematic approach for obtaining a satisfactory

value of q0. Similar PID control design was proposed by

Cameron and Seborg [5] with, at least, four parameters to

be tuned. Fig. (1) shows the I+PD structure that can be

related to the control structure of the equation (7). The

I+PD structure presents advantages over the conventional

PID structure. The I+PD algorithm is implementing the

proportional (P) and derivative (D) parts with the output

and, therefore, avoiding the ringing state for step setpoint

changes in the overall control input.

� ��

�

�

��

�

��

�

��

��

��

Figure 1. I+PD control system.

The representation of the equation (7) can be written to

the standard PID form through

Kc = Ki = Kd = 1/q0

where Kc, Ki and Kd are the proportional, integral and

derivative gains, respectively.


3 Tuning Aspects of q0

The tuning procedure of q0 for the PID controller can be

done from different control techniques. For the case studies

of this paper fuzzy and neural methods are implemented.

3.1 Fuzzy Approach

In this approach the tuning of the controller is done using

fuzzy logic issues [6]. In order to manipulate information of

the dynamic system, it is necessary to use e(k) as input andthe variation of e(k), denoted by Δe(k). Then, the fuzzytuner is designed with a rule base that has two linguistic

input variables, E and DE, with respect to the crisp inputvariables e(k) and Δe(k). A linguistic output variable, O,with respect to the crisp parameter q0(k), is also included.The control law has the following form:

Δu(k) =1

q0(k){yr(k) − [3 − 3z−1 + z−2]y(k)

}(8)

Fig. (2) shows the idea of the fuzzy tuning based on the

Eq. (8).

��

� ��

��

��

��

Figure 2. PID control system with fuzzytuning.

Linguistic sets, shown in Table (1) and membership

functions, shown in Fig. (3), are defined by the linguistic

variables E, DE and O in the universes of discourse U , Vand W , respectively. In Fig. (3) the scale factors Se, Sde,and So are defined by using the Nelder-Mead optimizationmethod [8], and esc = Sc/(number of output membershipfunctions). Table (2) presents the rule base of the fuzzy

tuner.

3.2 Neural Approach

Neural networks (NNs) have been used in a vast

variety of different control structures and applications,

serving as controllers and as process models or parts

of process models. They have been used to recognize

and forecast disturbances, to detect and diagnose faults,

to combine data from partially redundant sensors, to

Table 1. Fuzzy Sets.NE ZE PE

–––

e(k) is negative e(k) is zero e(k) is positive

NDEZDE PDE

–––

�e(k) is negative

�e(k) is zero

�e(k) is positive

ZO PTO PSO PMOPBO

–––––

q0(k) is zero q0(k) is positive tiny q0(k) is positive small q0(k) is positive medium q0(k) is positive big

Table 2. Rule base for the fuzzy tunner. �e(k) e(k)

NDE ZDE PDE

NE PMO PBO PMO

ZE PPO PPO PPO

PE PMO PBO PMO

�

��

��

��

��

�

��

��

�

��

�

�

��

��

�

��

��

�

Figure 3. Membership functions of fuzzytuner.

perform statistical quality control, and to adaptively

tune conventional controllers such as PID controllers.

Many different structures of NNs have been used, with

feedforward networks, radial basis functions, and recurrent

networks being the most popular.


A NN presents the following features: generalization,

learning, fault tolerance, adaptability, inherent parallelism

and abstraction. Thus, the major advantage of NN

learning is the ability to accommodate poorly modelled and

nonlinear dynamical systems.

Fig. (4) shows the neural network idea to update q0(k)with respect equation (8).

��

��

��

��

��

��

��

� ��

� � ��

Figure 4. PID control system with neuraltuning.

The adaptive neural model shown in Fig. (4) for tuning

the PID control parameter was proposed by Wang et al.[11]. The neuron output q0(k) can be derived as

q0(k) = q0(k − 1) + Kn

n∑i=1

wixi(k)

n∑i=1

|wi|(9)

where, xi(k) (i = 1, 2, , n) denote the neuron inputs;Kn > 0 is the neuron proportional coefficient; wi are the

connection weights of each xi(k) and are determined withlearning rule or optimization algorithms. In this paper there

are three inputs of neural (n = 3) defined as x1(k) = yr(k),x2(k) = e(k) and x3(k) = Δe(k).

4 Simulation Results

This section presents the experiments and the tuning

procedures described in section 3 for the digital PID

controller. The procedures were applied to a heat exchanger

and a wind tunnel.

4.1 Heat Exchanger Application

The heat exchanger model, Fig. (5), was obtained

from [3], which have identified the heat exchanger as a

Hammerstein Model as shown in equations (10) and (11)

x(k) = u(k) − 1.32u2(k)+0.76u3(k)−2.17u4(k)(10)

y(k) =−6.5306z−1 + 5.5652z−2

1 − 1.608z−1 + 0.6385z−2x(k) (11)

where x(k) is the static nonlinearity, u(k) is the processflow rate and y(k) is the process exit temperature. Theprocess input is constrained between [0, 1]. The controlobjective is to keep the process output as close as possible

to the reference. The nonlinear characteristics of the heat

exchanger is shown in Fig. (6).

Steam

Cold Water Inlet

Hot Water Outlet

Condensate

Figure 5. Heat exchanger plant.

Figure 6. Nonlinear characteristic of a heatexchanger.

Responses for servo essays are given in Figs. (7) and

(8). For analysis of this behavior, the reference signal is

changing such as: yr = 25 (from sample 1 to 32), yr =45 (from sample 33 to 67) and yr = 15 (from sample 68to 100). For the fuzzy tuner the scale factors are Se =0.0474, Sde = 39.0474, So = 91.3176, esc = 18.2635and for the neural tuner the parameters are w = [−0.0524×103, −1.731 × 103, 1.7812 × 105] andKn = 0.7382.Results from Figs. (7) and (8) illustrate that the servo

behavior of the PID system with intelligent tuning is good

with a small control variation and zero steady-state error.

Fig. (9) show the behavior of the adaptive parameter.

4.2 Experimental System Application

Practical test for the PID control algorithm is assessed

in a small wind tunnel plant implemented at Federal

University of Santa Catarina. For more information visit the


Figure 7. Servo response of the heatexchanger.

Figure 8. Control action of the heatexchanger.

Figure 9. Behavior of q0 for the heatexchanger.

web-site at http://lcp.das.ufsc.br. Wind tunnelsystem is shown in Fig. 10.

The wind tunnel process is implemented over an

Figure 10. Experimental plant.

aluminum base and consists of a 9 cm wide and 60 cm

long air duct having on its left extremity a DC motor with

a working range from zero to 24 volts. On the opposite

extremity there is another DC motor (tachometer) driven by

the air flow made by the first motor (the voltage arising

from the rotation is proportional to the air flow). The

transmission circuit utilizes several operational amplifiers

and ranging the output voltage of the measurement motor

from zero to 5 volts while the actuator circuit works at the

range from zero to 24 volts and employs a power transistor

bridge. It is possible to insert a disturbance in the process

variable by switching a key that will add a step voltage

to the output signal. This practical prototype, besides

the nonlinear behavior, presents dead-time, resonant and

turbulent characteristics, so that it can be used as a tangible

real proof for evaluating PID design techniques in difficult

situations.

Figs. (11) and (12) show output and input signals

when the plant is submitted to a servo (setpoint changing

from 3 to 1 to 3 volts) and regulatory tests (disturbance

applied around 60 seconds with magnitude of 0.6 volts).

For the fuzzy tuner the scale factors are Se = 1 × 10−4,

Sde = 0.993, So = 9, esc = 1.8 and for the neuraltuner the parameters are w = [1.1104 × 103, 3.4376 ×103, −3.2836×103] andKn = 5.2661×10−4. The system

is controlled in a satisfactory way.

Figure 11. Servo response of the tunnelprocess.


Figure 12. Control action of the tunnelprocess.

Results from Figs. (11) and (12) illustrate that the servo

behavior of the PID system with intelligent tuning is good

with a small control variation and zero steady-state error.

Fig. (13) shows the behavior of the adaptive parameter.

Figure 13. Behavior of q0 for the tunnelprocess.

5 Conclusion

In this paper a digital control with a conventional PID

structure was developed. The design method is based

on the universal plant representation. The proposed PID

controller structure is very interesting from the viewpoint

of operators because the control design presents only one

parameter to be tuned. Simulation tests for nonlinear

plant have indicated that the PID controller with fuzzy

and neural tuning performed very well and, therefore,

demonstrating the successful application of the intelligent

tuning approaches.

The paper had shown only the step responses to setpoint

changes and good disturbance rejection properties can be

obtained.

Further works include multivariable, disturbance and

time-delay applications.

References

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[2] K. J. Åström and T. Hägglund. The future of PID control.

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control of hammerstein and wiener models using genetic

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[4] C. W. Alexander and R. E. Trahan. A comparison of

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[12] W. K. Wojsznis and T. L. Blevins. Evaluating PID adaptive

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