Multiobjective evolution based fuzzy PI controller design for nonlinear systems

Multiobjective evolution based fuzzy PI controller design for nonlinear systems

ARTICLE IN PRESS Engineering Applications of Artificial Intelligence 19 (2006) 157–167 www.elsevier.com/locate/engappai Multiobjective evolution base...

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ARTICLE IN PRESS

Engineering Applications of Artificial Intelligence 19 (2006) 157–167 www.elsevier.com/locate/engappai

Multiobjective evolution based fuzzy PI controller design for nonlinear systems$ G.L.O. Serra, C.P. Bottura Control and Intelligent Systems Laboratory, State University of Campinas, 400, 13083-852, Campinas-SP, Brazil Received 28 June 2005; accepted 1 August 2005 Available online 19 September 2005

Abstract This work proposes a gain scheduling adaptive control scheme based on fuzzy systems, neural networks and genetic algorithms for nonlinear plants. A fuzzy PI controller is developed, which is a discrete time version of a conventional one. Its data base as well as the constant PI control gains are optimally designed by using a genetic algorithm for simultaneously satisfying the following specifications: overshoot and settling time minimizations and output response smoothing. A neural gain scheduler is designed, by the backpropagation algorithm, to tune the optimal parameters of the fuzzy PI controller at some operating points. Simulation results are shown to demonstrate the efficiency of the proposed structure for a DC servomotor adaptive speed control system used as an actuator of robotic manipulators. r 2005 Elsevier Ltd. All rights reserved. Keywords: Neural–genetic-fuzzy systems; Adaptive control; Multiobjective optimization

1. Introduction Adaptive control is intrinsically related to the identification problem and its motivating idea is sufficiently attractive: a controller that modifies itself based on the controlled plant behavior, in order to satisfy some design specifications (A˚stro¨m and Wittenmark, 1995; Ioannou and Sun, 1996; Chalam, 1987). These efforts started in the 1960s. In 1980s adaptive control research progressed mainly in terms of the following directions: self-tuning, model reference, and gain scheduling controls. Particularly the gain scheduling problem has been subject of a great deal of research, in both theoretical and practical viewpoints (Korba et al., 2003; Rugh and Shamma, 2000). This renewed interest probably stems from the development of new $ A preliminary version of this paper was accepted in 2004 IEEE International Symposium on Intelligent Control. Corresponding author. Tel.: +55 19 37883852; fax: +55 19 32891395. E-mail address: [email protected] (G.L.O. Serra).

0952-1976/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engappai.2005.08.003

techniques and softwares, which allow a more rigorous and systematic treatment of the gain scheduling problem. The classical approach to this problem consists essentially on repeated design synthesis, which is associated with some scheduling strategy connecting locally designed controllers. This quick improvement opened the door for a large range of applications. However, the area of complex numerically research based on adaptive control algorithms continues and the reliance on crisp numerical operations presents a dichotomy if compared with intelligent behavior (Sinha and Gupta, 1996). The need of intelligent systems has grown in the past decade due to the increasing demand of humans and machines for better performance. One of the reasons for this increasing demand is that we are passing through an era of information explosion, information globalization, and consequently increasing competition. The information explosion has put huge constraints on the time at which decisions have to be made. Areas such as knowledge discovery, data mining, and soft computing are manifestations of intelligent systems and knowledge in general. The four most used

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intelligent methodologies in the 90s are symbolic knowledge based systems (e.g. expert systems), artificial neural networks, fuzzy systems, and genetic algorithms. Under the banner of real world applications, these methodologies have been used in various industrial applications such as telecommunications, quality control, command and control systems, process control systems, flight control systems, financial systems, data prediction, and modeling. These applications, however, highlighted methodological limitations which encouraged their hybridization (Bottura and Serra, 2003; Vellasco et al., 2001; Melin and Castillo, 2001). In this work, a computational strategy directed more toward intelligent behavior is employed as a tool within the gain scheduling adaptive control technique, exploiting the strengths of fuzzy systems, neural networks, and genetic algorithms (Bottura and Serra, 2004; Goonatilake and Khebbal, 1995; Dillon and Khosla, 1997). Simulation results are shown to demonstrate the efficiency of the proposed structure for a DC servomotor adaptive speed control system used as an actuator of robotic manipulators. 2. Neuro genetic fuzzy control system The key element in adaptive control is the controller parameters adjustment mechanism. Among the types of parameters adjustment techniques are: gain scheduling, model reference adaptive control, self-tuning regulators, and dual control (A˚stro¨m and Wittenmark, 1995; Ioannou and Sun, 1996; Chalam, 1987). How the dynamics of a plant change with the operating conditions due to nonlinearities in many situations, it is possible to find measurable variables correlated with these changes in the plant. These variables can be used to modify the controller parameters and to accommodate changes in plant gain. This approach is called gain scheduling and is illustrated in Fig. 1. The diagram presents two loops: an ordinary feedback loop composed by the plant and the controller and another loop

that adjusts the controller parameters based on known a priori operating conditions. The classical gain scheduler consists of a look-up table and an appropriate logic for detecting an operating condition and choosing the corresponding values of the controller parameters from the table. The advantage of gain scheduling is that the controller gains can be changed as quickly as the auxiliary measurements respond to plant changes. Frequent and rapid changes of the controller gains, however, may lead to instability; therefore, there is a limit at how often and how fast the controller parameters can be changed. One of the disadvantages of classical gain scheduling is that the adjustment mechanism of the controller gains is precomputed offline and, therefore, provides no feedback to compensate for incorrect schedules. Unpredictable changes in the plant dynamics may lead to deterioration of performance or even to complete failure. Another possible drawback of classical gain scheduling are the high design and implementation costs that increase with the number of operating conditions. To overcome these problems for PI control, an alternative neuro genetic fuzzy control structure with neural gain scheduling is proposed, as shown in Fig. 2. A fuzzy PI controller is developed, which is a discrete time version of a conventional one. Its data base as well as the constant PI control gains are optimally designed by using a genetic algorithm for simultaneously satisfying the following specifications: overshoot and settling time minimizations, and output response smoothing. Hence, the optimization problem is a multiobjective one, from which results an optimal fuzzy PI controller. A neural gain scheduler is designed, by backpropagation algorithm, to tune the optimal parameters of the fuzzy PI controller at some operating points. This structure has the following characteristics: improves loop performance, works with an intelligent tuner (neural network), updates all seven fuzzy PI tuning parameters, provides interpolation between operating conditions. A detailed explanation of the system is presented in sequel. 2.1. Discrete fuzzy PI controller The output of an analog PI controller, in the s domain, is given by   Ki uPI ðsÞ ¼ K p þ EðsÞ, s

Gain Scheduler Controller Parameters Operating Conditions Output

Reference Controller

Plant

Fig. 1. Classical gain scheduling adaptive control system.

where Kp and Ki are the proportional and integral gains, respectively, and E(s) is the Laplace transform of the tracking error signal. Applying the bilinear transformation s¼

2 z1 , Tzþ1

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Neural Gain Scheduler Feedforward Neural Network

Backpropagation Algorithm

On-line tuning Training Ku, Ki, Kp, L1 , L 2 , m1 , m 2 Error err(k)

Reference ref(k) +

_

Fuzzy PI Controller

Control

Nonlinear plant

u pi (k)

Output y(k)

Rate of Change of the error dt_err(k) Ku, Ki, Kp, L1 , L 2 , m1 , m 2 Evolution

Optimal parameters

Genetic Algorithm

Fig. 2. Neuro gain scheduling genetic fuzzy control.

where T40 is the sampling period, results the following discrete form for the PI controller output equation   K iT K iT þ uPI ðzÞ ¼ K p  EðzÞ, 2 1  z1 where E(z) is the z-transform of the tracking error signal. Considering K dp ¼ K p  K i T=2 and K di ¼ K i T, and taking the inverse z-transform, we obtain

eðkÞ  eðk  1Þ , T

edc ðkÞ ¼ eðkÞ, Ddu ðkÞ ¼ K dp edr ðkÞ þ K di edc ðkÞ,

(1)

uPI ðkÞ ¼ uPI ðk  1Þ þ K du Ddu ðkÞ.

ð2Þ

Fig. 3 shows the control system scheme with the discrete fuzzy PI controller. The study of this controller in this configuration is very important for practical application in computer assisted control based on data acquisition system. The fuzzification was performed taking into consideration the fuzzy PI control law given in (3). The input and output membership functions of the fuzzy PI controller are shown in Fig. 4(a), (b), and (c). The fuzzy PI controller has two inputs, the error signal e~dc ðkÞ ¼ K di edc ðkÞ with ec d ¼ refðkÞ  yðkÞ and the rate of change of the error signal e~dr ðkÞ ¼ K dp edr ðkÞ, and has a single output Ddu ðkÞ as shown in Fig. 4, where the constants L1 40 and L2 40 must be determined. L1 and L2 are used in the input and output membership functions to provide more flexibility in obtaining the optimal parameters. To improve the transient response of the closed loop control system, we propose the insertion of

Dividing (1) by T, we have   uPI ðkÞ  uPI ðk  1Þ d eðkÞ  eðk  1Þ ¼ Kp þ K di eðkÞ T T and

Hence, replacing the term multiplying T in (2) with a fuzzy control action, results the fuzzy PI control law uPI ðkÞ ¼ uPI ðk  1Þ     eðkÞ  eðk  1Þ þ K di eðkÞ , þ K du K dp T

edr ðkÞ ¼

results the fuzzy PI control law

uPI ðkÞ  uPI ðk  1Þ ¼ K dp ½eðkÞ  eðk  1Þ þ K di TeðkÞ.

uPI ðkÞ ¼ uPI ðk  1Þ     eðkÞ  eðk  1Þ þ K di eðkÞ . þ T K dp T

where Kpd, Kid, and Kud are the control gains to be determined. By letting

(3)

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edc(k)

ref(k) +

K di

+ -

~d e r (k)

z -1

1 ___

edr (k)

T

∆d u (k)

~d e c(k)

-

Fuzzy Controller

u (k) K ud

PI

+

+

Nonlinear Plant

y(k)

z -1

K dp

Fig. 3. Fuzzy PI control system. The fuzzy PI controller has two linguistic inputs (error e~dc ðkÞ, rate of change of the error e~dr ðkÞ) and one linguistic output (control Ddu ðkÞ).

1

zero (z)

negative (n)

positive (p)

form: DilðAðxÞÞ ¼ Ar ðxÞ,

0.5

with 0opo1. 0

−L1

(a) 1

0

L1

zero (z)

negative (n)

positive (p)

0.5 0

− L1

(b)

1

negative (n)

0

L1

zero (z)

positive (p)

0.5 0

(c)

−L2

0

L2

Fig. 4. Membership functions of the fuzzy PI controller: (a) error; (b) rate of change of the error; (c) control.

the concentration and dilation operators in the membership functions as defined in sequel. Definition. (Concentration). By concentration is meant the operation of making the fuzzy sets membership functions become more concentrated around the point with higher membership degree. It is made by the following concentration operator

The following control rules are defined for the fuzzy PI controller based on the membership functions already mentioned: ðR1 Þ : IF e~dc ¼ n

AND

e~dr ¼ n

THEN

Ddu ¼ n,

ðR2 Þ : IF e~dc ¼ n

AND

e~dr ¼ z

THEN

Ddu ¼ n,

ðR3 Þ : IF e~dc ¼ n

AND

e~dr ¼ p

THEN

Ddu ¼ z,

ðR4 Þ : IF e~dc ¼ z

AND

e~dr ¼ n

THEN

Ddu ¼ n,

ðR5 Þ : IF e~dc ¼ z

AND

e~dr ¼ z

THEN

Ddu ¼ z,

ðR6 Þ : IF e~dc ¼ z

AND

e~dr ¼ p

THEN

Ddu ¼ p,

ðR7 Þ : IF e~dc ¼ p

AND

e~dr ¼ n

THEN

Ddu ¼ z,

ðR8 Þ : IF e~dc ¼ p

AND

e~dr ¼ z

THEN

Ddu ¼ p,

ðR9 Þ : IF e~dc ¼ p

AND

e~dr ¼ p

THEN

Ddu ¼ p.

These nine rules can be explained as follows: Rule R1 means that if the output of the closed loop control system is above the set-point and the error rate of change is decreasing then the control signal must be decreasing too. The rules R2 to R9 are similarly analysed. In the defuzzification step, the centroid formula is employed to defuzzify the incremental control of the fuzzy control law Ddu ðkÞ as follows: Pn Ai ðxÞai d Du ðkÞ ¼ Pi¼1 . n i¼1 Ai ðxÞ

ConðAðxÞÞ ¼ Ap ðxÞ, with p41.

2.2. PI control multiobjective evolutionary optimization

Definition. (Dilation). The dilation operator has the opposite effect of the concentration one and is obtained by modifying the membership function in the following

For many real-world decision making problems there is a need for simultaneous optimization of multiple objectives and, in practice, some of these objectives may be conflicting. Hence, such multiobjective optimization

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problems require separate techniques, which are very different from the standard optimization techniques for single objective optimization. Multiobjective evolutionary optimization seeks to optimize the parameters of a fitness vector function, obtained according to some objectives. Formally, it can be expressed by . min max y ¼ f ðxÞ ¼ ðf 1 ðxÞ; f 2 ðxÞ; . . . ; f n ðxÞÞ; x

x

x ¼ ðx1 ; x2 ; . . . ; xm Þ 2 X ; y ¼ ðy1 ; y2 ; . . . ; yn Þ 2 Y ;

subject to

where x is the decision vector, X is the parameter space, y is the objective vector, Y is the objective space. Assuming, without loss of generality, a minimization problem, the following definitions apply: Definition. (Dominated solution). A solution or decision vector is dominated if there exist a feasible solution y not worse than x on all coordinates, i.e., for all objectives fi ði ¼ 1; . . . ; kÞ f i ðxÞpf i ðyÞ

for all 1pipk.

Definition. (Nondominated solution). A solution or decision vector is nondominated if the corresponding objective vector cannot be improved in any dimension without degradation in another. These vectors are optimal in the Pareto sense and the decision vectors denoted as Pareto optimal or, the so-called, Pareto optimal set or Pareto optimal front. Thus, considering x1, x2AX, x1 is said to dominate x2 (or x1  x2 ) iff 8i 2 f1; 2; . . . ; ng : f i ðx1 ÞXf i ðx2 Þ ^ 9j 2 f1; 2; . . . ; ng : f i ðx1 Þ4f i ðx2 Þ. 2.2.1. The weighted-sum approach Among the current evolutionary approaches to multiobjective optimization, in this work, the weighted-sum approach will be used. The n objectives f 1 ; . . . ; f n are weighted by user-defined positive coefficients w1 ; . . . ; wn and added together to obtain a scalar measure of cost for each individual. This measure can then be used as the basis for selection, e.g., proportional, tournament, or based on ranking. This approach is widely known, intuitive, simple to implement and is the most popular. Formally, we have, f ðxi Þ7!

n X

wk f k ðxi Þ,

k¼1 n

f : < ! <. The setting of the weighting coeficients wk is generally dependent on the problem instance, and not just on the problem class. Thus, the initial combination of weights usually needs to be finely tuned in order to lead to satisfactory compromise solutions.

161

The multiobjective genetic algorithm used to optimize the control gains ðK u ; K p ; K i Þ, the constants L1 and L2, and the modifiers m1 and m2 also present the following characteristics (Goldberg, 1989; Michalewicz, 1996): Floating point codification, Candidate length: 7, Population length: 20, Selection: Elitist (it keeps the better candidate and the others are chosen randomly), as well as arithmetic crossover operator, non-uniform mutation operator and fitness vector function. 2.2.2. Arithmetic crossover operator Combines linearly two vectors. If the vectors S tv and t S w are chosen to crossover at the tth step, then their sons will be S tþ1 ¼ aS tw þ ð1  aÞS tv , v 1 t S tþ1 w ¼ aS v þ ð1  aÞS w ,

where a is a random number in [0,1]. 2.2.3. Non-uniform mutation operator If S tv ¼ hv1 ; . . . ; vm i is a chromosome and the element vk is selected for this mutation, in the domain ½l k ; uk ; the result is a vector S tþ1 ¼ hv1 ; . . . ; v0k ; . . . ; vm; ijk2f1;...;ng ; and v ( for randp0:5; vk þ Dðt; uk  vk Þ v0k ¼ vk þ Dðt; vk  l k Þ for rand40:5; where r is a random number in [0,1], T is the maximum number of generations, and b is a parameter that determines the degree of non-uniformity. 2.2.4. Fitness vector function For the general control problem, the optimization of a different number of systems performances is desired. The following simultaneous performance specifications (the objectives) are adopted in this work: 1. Overshoot minimization:   1  w1 , min K p ;K i ;K u ;L1 ;L2 ;m1 ;m2 1 þ kref  maxðyÞk 2. Settling time minimization:   1  w2 , min K p ;K i ;K u ;L1 ;L2 ;m1 ;m2 1 þ kerrðt ¼ ti =t ¼ tf Þk 3. Output transient response smoothing: ( w3 for ev p0; min ev ¼ K p ;K i ;K u ;L1 ;L2 ;m1 ;m2 0 for ev 40; P with 3i¼1 wi ¼ 1: The variable ref is the reference (set-point) signal, y is the output of the closed loop system, err is the error signal, ti and tf are the initial and final time, ev is the

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error rate of change, and wi fi¼1;2;3g are the weights defined by the user. This problem is a multiobjective optimization one and the loop performance can be improved with the obtained optimal fuzzy PI controllers (Serra, 2003; Tang et al., 2001).

the scheduling variable for soft changes of these parameters between the operating conditions, as shown in Fig. 5, and, as a consequence, better performance of the loop control; the output neural gain scheduler maps a relation of the form

2.3. Neural gain scheduler

pi ðkÞ ¼ f~ðyðkÞÞ, f~ : < ! <7 .

p ¼ ½p1 ; p2 ; . . . ; pn , r ¼ ½r1 ; r2 ; . . . ; rn , with pi ¼ ½K iu ; K ip ; K ii ; Li1 ; Li2 ; mi1 ; m12 Ti¼1;2;...;n : The output yðkÞ of the system at the kth sampling instant is used as

Σ

Input y(k)

Σ Input layer

f

The training scheme of the neural gain scheduler is shown in Fig. 6. In this scheme, the training is done in the usual form, based on the chosen set-point ri and corresponding optimal fuzzy PI controller parameters pi by the back-propagation algorithm. After that, the obtained neural gain scheduler is used to tune these parameters on-line according to Fig. 2.

3. Computational results To demonstrate the efficiency and characteristics early presented (see Section 2) for adaptive speed control of a DC servomotor, used as an actuator of robotic manipulators, with the proposed structure, in this section some results will be presented. 3.1. Nonlinear plant The plant to be controlled consists in a DC (Direct Current) servomotor (12 V), which can be used as an actuator of robotic manipulators. Once the objective is speed control of the DC servomotor, a model describing

Σ

f

Σ

f

Σ

f

Σ

f

Σ

f

Σ

f

Σ

f

Σ

f

Σ

f

Σ

f

Σ

f

f

Σ Σ

f f

First hidden layer

Σ Σ

Σ

f

Ku

Σ

f

Ki

Σ

f

Kp

Σ

f

L1

Σ

f

L2

Σ

f

m1

Σ

f

m2

Output

In artificial neural network design, for a given application, the structure cannot be very small for accuracy reasons, neither be very big due to generalization (Lau, 1991; Miller et al., 1990; Sinha and Gupta, 1996). In this application, the neural network presents the following characteristics, Fig. 5. Feedforward, one input layer (one input), two hidden layers (with eight neurons in each layer), one output layer (with seven outputs), sigmoid activation function, and backpropagation algorithm. The neural gain scheduler maps all the fuzzy PI controllers designed in Section 2.1, according to the selected operating conditions, in the sense of implementing the gain scheduling technique and providing soft interpolation of the optimal fuzzy PI controller parameters for all operating conditions. The neural gain scheduler must represent a nonlinear relation among all of the operating conditions r 2
f f

Output layer

Second hidden layer

Fig. 5. Neural gain scheduler: a multilayer feedforward neural network of one input, two hidden layers with eight neurons in each layer, and seven outputs, with sigmoid activation function for all layers.

ARTICLE IN PRESS G.L.O. Serra, C.P. Bottura / Engineering Applications of Artificial Intelligence 19 (2006) 157–167 pi ^ pi _

Neural Network

4

4

3.5

+ 3.5

output of the tachogenerator (Volts)

ri

4.5

163

3

Fig. 6. Training scheme of the neural gain scheduler.

input (Volts)

ei

2.5 2 1.5

plant to be identified

1

plant model

3

2.5

2

1.5

1

ξ(κ) 0.5

0.5

+ DC Servomotor

Power Amplifier

+

Tachogenerator 0

0

2

4

6

0

0

2

time (seconds)

Identification Algorithm

^ y(k)

6

Fig. 8. Open-loop response of the DC servomotor. The oscillations of the plant output consist in the noise from the tachogenerator quantized by the A/D (Analog/Digital) converter.

COMPUTER

u(k)

4

time (seconds)

y(k)

(b1, b0, a1, a0) -

+ 100

D/A

A/D 10-5

MSE

Fig. 7. Block diagram of the identification problem.

the relation between the voltage level applied to it and its speed response is needed. An identification algorithm was developed and implemented to do this task, Fig. 7, and a discrete second order linear time invariant model b1 z þ b0 GðzÞ ¼ 2 , z þ a1 z þ a0 where xðkÞ is the disturbance, u(k) is the input voltage, ^ y(k) and yðkÞ are the actual and estimated outputs of the DC servomotor, was obtained (Serra, 2001). In Fig. 8 the input voltage of 4 V and the respective real output from the speed sensor (tachogenerator) and the estimated output are shown. The identified model has the following form: GðzÞ ¼

0:082360z  0:030469 . z2  0:370435z  0:570980

However, this model does not take the uncertainties, nonlinearities from the power amplifier and from the tachogenerator into consideration. As a consequence, operating conditions different from the one used in the identification can put the system in an instability limit. The control response may be underdamped or go to instability, indicating a robustness problem. The follow-

10 -10

10 -15

10 -20 0

2

4

6

8

10

12

14

16

18

Epochs

Fig. 9. Neural training by backpropagation. The curves from the epoch 2 to 14 and from the 14 to 19 correspond to a local minimum and to the convergence to the global minimum, respectively.

ing nonlinear model is used to describe this kind of behavior yðkÞ ¼ 0:37044yðk  1Þ þ 0:57098yðk  2Þ þ 0:08236u4 ðk  1Þ  0:03047u4 ðk  2Þ þ xðkÞ, where k is the sample time, u is the input of the plant, y is the output of the plant, and x is a unknown noise. The fuzzy PI controllers are obtained off-line, by the multiobjective genetic algorithm, according to the three design specifications presented in Section 2.2.4, for four operating conditions, with 500 generations, a crossover

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probability of 0.98, and a mutation probability of 0.35, as follows (a) Operating condition 1 (set-point ¼ 1)—Obtained optimal parameters: K u ¼ 0:2377, K p ¼ 0:8326, K i ¼ 0:2040, L1 ¼ 11:7643, L2 ¼ 2:7926, m1 ¼ 0:9252, m2 ¼ 1:4487. Fitness ¼ 0:9621. (b) Operating condition 2 (set-point ¼ 3)—Obtained optimal parameters: K u ¼ 0:1705, K p ¼ 0:8193, K i ¼ 0:1656, L1 ¼ 7:0902, L2 ¼ 8:5811, m1 ¼ 0:9582, m2 ¼ 1:7502. Fitness ¼ 0:9186. Ku

Kp

0.25

1

0.2

0.9

0.15

0.8

0.1

0.7 Ki

L1

12

0.25

10 0.2

8 6

0.15 L2

m1

10

1

5

0.95 0.9

0 m2 2.5

0

2 4 6 8 10 Operating Conditions (Set-points)

2 1.5 1

0

2 4 6 8 10 Operating Conditions (Set-points)

Fig. 10. Neural interpolation: ‘‘3’’—fuzzy parameters; ‘‘-’’—neural output. 3.5

(c) Operating condition 3 (set-point ¼ 7)—Obtained optimal parameters: K u ¼ 0:1002, K p ¼ 0:9845, K i ¼ 0:1775, L1 ¼ 9:5716, L2 ¼ 4:0926, m1 ¼ 0:9938, m2 ¼ 2:2785. Fitness ¼ 0:8804. (d) Operating condition 4 (set-point ¼ 10)—Obtained optimal parameters: K u ¼ 0:128256, K p ¼ 0:7747, K i ¼ 0:1518, L1 ¼ 9:2470, L2 ¼ 8:3267, m1 ¼ 0:9738, m2 ¼ 2:0488. Fitness ¼ 0:8265. Fig. 9 shows the neural network training performance related to the mapping of operating condition/optimal parameters. The mean square error was reduced from 23.3715 to 3:0895  1022 , within 19 epochs, which is a good result for this application. The proposed neural gain scheduling guarantees a smooth transition among different operating conditions as shown in Fig. 10. Fig. 11 shows the neuro genetic fuzzy control system tracking performance from set-point 1 to 3, with the presence of noise in the tachogenerator output (mean ¼ 0 and variance ¼ 0.01), and on-line tuning mechanism of the fuzzy PI controller on the Kp and Ki parameters. Fig. 12 shows the performance with setpoint 7 þ 2 sinðtÞ, on the same noise situation, but at different operating conditions from the presented to the neural gain scheduler. Figs. 13 and 14 show the performance of a classical adaptive gain scheduling PI control, with the same set-points and noise situation presented to the proposed structure. In this case, ten PI controllers of the form uðkÞ ¼ uðk  1Þ þ nðkÞðb0 errðkÞ þ b1 errðk  1ÞÞ, 3 control input

3

2.5

2.5 2

2

1.5

1.5

1

0

1

output set-point

0.5 0

10

20

30

0.5

0

10

seconds

20

30

seconds

0.85

0.26 Kp

Ki 0.24

0.84 0.22 0.83

0.2 0.18

0.82 0

10

20 seconds

30

0.16

0

10

20 seconds

Fig. 11. Neuro genetic fuzzy PI control system output tracking.

30

ARTICLE IN PRESS G.L.O. Serra, C.P. Bottura / Engineering Applications of Artificial Intelligence 19 (2006) 157–167 10

3.5

8

3

165

control input

2.5 6 2 4 1.5 2 0

1

output set-point 0

10

20

30

0.5

0

10

seconds

20

30

seconds

1.05

0.26

1

0.24

Ki

0.95 0.22 0.9 0.2 0.85 0.18

0.8 Kp 0.75

0

10

20

30

0.16

0

10

seconds

20

30

seconds

Fig. 12. Neuro genetic fuzzy PI control system output tracking.

3.5

0.4

2.5

0.35

2

0.3

1.5

0.25

1 output set-point

0.5 0

gain scheduler

0.45

3

0

20

40

0.2 60

seconds

0.15

0

20

40

60

seconds

5

controller

4 3 2 1 0

0

20

40

60

seconds

Fig. 13. Classical gain scheduling PI control output tracking.

where errðkÞ is the error signal, b0 and b1 are the controller parameters designed by a gain and phase margins based technique (Ho et al., 1995), for the same operating range presented to the proposed structure. The classical gain scheduler nðkÞ is used to change as

quickly as the auxiliary measurements respond to the output of the plant, based on a look-up table and on an appropriate logic for detecting an operating condition change modifying the PI controller parameters using the table controller gains. It can be seen that the classical

ARTICLE IN PRESS G.L.O. Serra, C.P. Bottura / Engineering Applications of Artificial Intelligence 19 (2006) 157–167

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10

0.22 gain scheduler 0.2

8

0.18 6

0.16

4

0.14 0.12

2 0

output set-point 0

20

40

0.1 60

0.08

seconds

0

20

40

60

seconds

10

controller

8 6 4 2 0

0

20

40

60

seconds

Fig. 14. Classical gain scheduling PI control output tracking.

operating conditions and fuzzy PI controllers, the performance was much better. Fig. 15 shows the tracking response of a fixed conventional discrete time PI controller when the set-point is changed from 1 to 3. Clearly, a fixed PI controller presents a very poor performance in this application.

6 output set-point 5

4

3

4. Conclusions and future works 2

4.1. Conclusions

1

0

0

10

20

30 seconds

40

50

60

Fig. 15. Classical PI controller output tracking.

gain scheduling PI control system tracks the output, Figs. 13 and 14, but its performance is inferior to the one of the neural gain scheduling, Figs. 11 and 12, with larger errors and slower responses, because of the changes on the PI controller parameters according to the operating conditions. To obtain better results with this classical structure, it is necessary to increase the number of PI controllers, and consequently the operating conditions, which can be a tedious task. On the other hand, for the proposed structure, using only four

In this work, a neuro genetic fuzzy control structure is proposed to implement a gain scheduling adaptive PI control scheme. A fuzzy PI controller was developed and its data base as well as control gains were optimized by a multiobjective genetic algorithm so that the following specifications were simultaneously satisfied: overshoot and settling time minimizations, and output response smoothing. A neural gain scheduler was designed by the backpropagation algorithm to tune the optimal parameters of the fuzzy PI controller at some operating points. From the simulation results with the proposed control structure, we can conclude that

 

it improves the loop performance because of the multiobjective evolutionary optimization; it works as an intelligent tuner updating all seven fuzzy PI tuning parameters and providing interpolation among operating conditions because of the neural gain scheduler;

ARTICLE IN PRESS G.L.O. Serra, C.P. Bottura / Engineering Applications of Artificial Intelligence 19 (2006) 157–167



it achieves trajectory tracking in spite of noise and is efficient in the control of a nonlinear plant with noise because of its genetic fuzzy nature.

4.2. Future works The presented scheme is adequate for nonlinear SISO plants. The generalization for MIMO plants as well as for other controllers structures should be considered.

Acknowledgements The authors would like to thank CAPES and CNPq for financial support of this work. References A˚stro¨m, K.J., Wittenmark, B., 1995. Adaptive Control, second ed. Addison-Wesley, Boston. Bottura, C.P., Serra, G.L.O., 2003. An optimal knowledge based PI controller. Advances in Intelligent Systems and Robotics: Frontiers in Artificial Intelligence and Applications, vol. 101. IOS Press, The Netherlands, pp. 197–206. Bottura, C.P., Serra, G.L.O., 2004. Neural gain scheduling multiobjective genetic fuzzy PI control. Proceedings of the IEEE International Symposium on Intelligent Control, pp. 483–488, CD-ROM(ISBN: 0-7803-8634-5) SaM07.5. Chalam, V.V., 1987. Adaptive Control Systems: techniques and applications. Marcel Dekker, New York. Dillon, T., Khosla, R., 1997. Engineering intelligent hybrid multiagent systems. Kluwer Academic Publisher, Boston.

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