Controlling a complex electromechanical process on the basis of a neurofuzzy approach

Future Generation Computer Systems 21 (2005) 1083–1095

Controlling a complex electromechanical process on the basis of a neurofuzzy approach Rodolfo E. Haber a,b,∗ , J.R. Alique a , A. Alique a , R.H. Haber c b

a Instituto de Autom´ atica Industrial (CSIC), km 22, 800 N-III, La Poveda, 28500 Madrid, Spain Escuela Polit´ecnica Superior, Universidad Aut´onoma de Madrid, Ciudad Universitaria de Cantoblanco, Ctra. de Colmenar Viejo, km 15, 28049 Madrid, Spain c Departamento de Control Autom´ atico, Universidad de Oriente, Santiago de Cuba, Cuba

Available online 9 April 2004

Abstract This paper shows the viability of implementing a control strategy based on the internal-model control paradigm, which is a useful synergy of a dynamic ANN trained from real-life data and used to predict process output and a fuzzy-logic control (FLC) that enhances the control system’s overall performance. A force control problem involving a complex electromechanical system, represented here by the machining process, is considered as a case study. The main goal is to control a single-output variable, cutting force, by changing a single-input variable, feed rate. The proposed neurofuzzy-control (NFC) scheme consists of a dynamic model using ANNs to estimate process output, and a fuzzy-logic controller (FLC) with the same static gain as the inverse model to determine the control inputs (feed rate) necessary to keep the cutting force constant. Four approaches, the fuzzylogic controller (FLC), the direct inverse controller based on ANNs (DIC-NN), the internal-model controller (IMC-NN) and a neurofuzzy controller (NFC), are simulated and their performances are assessed in terms of several performance measurements. The results demonstrate that the NFC strategy provides better disturbance rejection than the IMC-NN and the FLC for the cases analyzed. © 2004 Elsevier B.V. All rights reserved. Keywords: Fuzzy control; Neural networks; Internal-model control; Machining process

1. Introduction Nowadays, certain electromechanical processes are so rife with complexity and uncertainty as to make what is known as intelligent systems technology a feasible ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (R.E. Haber). 0167-739X/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.future.2004.03.008

option to classic formal description. Fuzzy-logic (FL) is one such intelligent technique, and it has proven useful in control and industrial engineering as a very practical optimizing tool. Through FL, control systems can be invested with the verbally expressed experience of a trained operator. On the other hand, artificial neural networks (ANNs) are probably the one other most widely used artificial intelligence technique in model identification and control system design. ANNs are suitable

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for the identification and control of nonlinear plants [11,17], due basically to their excellent capability for modeling any nonlinear function to a desired degree of accuracy [9]. Of all the different classes of networks, feedforward neural networks and particularly multilayer perceptrons (MLPs) are most frequently used for nonlinear control. Combining different AI approaches in the control field has yielded interesting and efficient applications [10,21]. This paper pursues the same line and presents some results regarding a control strategy that is a useful synergy of a dynamic neural network trained from real-life data and used to predict process output and a fuzzy-logic control (FLC) that enhances the overall performance of the closed-loop system. Several schemes have been proposed for the neural control of nonlinear systems [8]. Internal-model control (IMC) is a well-established approach to controller design in which the process model is explicitly used in the control system design procedure [3,16]. Lightbody and Irwin [15] have shown how MLPs are used for providing controllers on the basis of the IMC principle. An internal model is linearized and an inverse controller is obtained via Kalman’s method. Recently, Kambhampati et al. have suggested the utilization of recurrent neural networks (i.e., Hopfield neural networks) in the IMC strategy, but once again the system needs to be linearized around an equilibrium point [12]. In general, the inversion of nonlinear models is not an easy task, and analytical solutions may not exist, so solutions have to be found numerically. Another issue is that the inversion of the process model may lead to unstable controllers when the plant has unstable zeros. Fortunately, there are several strategies for obtaining the inverse model so that the nonlinear performance can be fully exploited in order to cope with a complex plant [2]. Indeed, ANNs can be used to produce an efficient control scheme based on the IMC principle. The IMCbased ANN (IMC-NN) consists in training a network to learn the process’ dynamics. Another ANN can be trained to learn the inverse dynamics so that it can be used as a nonlinear controller [18]. The main issue is how to capitalize the robustness of an FLC without making design difficult. If we can synthesize an FLC with a static gain equal to the static gain of the inverse model, and if the control system is stable with this controller, we obtain the offset-free control for

constant setpoints and output disturbances [20]. The neurofuzzy control (NFC) scheme proposed here is inspired by IMC-based ANNs [5]. The inverse of the dynamic ANNs is replaced in the forward path by a fuzzy controller, so as to improve disturbance-rejection capability. In this paper a complex electromechanical process, the machining process, is used as the test bed [6]. The electrical portion of the system includes dc and ac rotational motors, amplifiers, sensors and other components. The mechanical portion includes the rigid structure and the body with its different shafts and gears and its reducer. The main goal is to implement machining-process optimization through controlling a single-output variable, the cutting force, by changing a single-input variable, the feed rate. The effectiveness of the NFC scheme is demonstrated through simulations. The comparison of an FLC, direct inverse control based on ANNs (DIC-NN), an IMC-NN and an NFC is assessed using several performance criteria on the basis of the given simulation results. The NFC can be inferred to make the control system more effective in disturbance rejection. This paper is organized as follows. Section 2 addresses some issues related with the ANNs considered in this work. Section 3 introduces the machining process models used in simulations. Section 4 explains the fuzzy approach in a simple feedback system to control the cutting force. Section 5 presents the design of an IMC based on ANNs. Section 6 addresses the tuning of the fuzzy controller and the NFC system’s design. Section 7 shows the comparison of the four control systems based on simulation results. The final section draws a number of conclusions.

2. Neural network background Consider a nonlinear system with uk ∈ Rm inputs and yk ∈ Rp (m ≥ p) outputs. Let us assume that the system can be modeled exactly by the following onehidden-layer feedforward network: yk+1 = Wo tanh(Wx xk + Wu uk + bx ) + bo

(1)

where xk ∈ Rn is given by T

T T , . . . , yk−ny , uTk−1 , . . . , uTk−nu ] xk = [ykT , yk−1

(2)

R.E. Haber et al. / Future Generation Computer Systems 21 (2005) 1083–1095

with n = (ny + 1)p + num, Wo ∈ Rp×l , Wx ∈ Rl×n , Wu ∈ Rl×m , bx ∈ Rl×1 and bo ∈ Rp×1 , where bx is the bias vector in the hidden layer, bo is the bias vector in the output layer, Wx , Wu are input weight matrices and tanh represents the hyperbolic tangent function. For the sake of simplicity, let us consider the singleinput single-output (SISO) nonlinear system (yk ∈ R1 ). The identification can be viewed as the determination T of the mapping from the set zN = [ u y ] to the set of possible weights (parameters) θˆ so that the network can produce a prediction yˆ k+1 as close as possible to the actual output yk+1 . ˆ o ) vec(W ˆ x ) vec(W ˆ u )bˆ x bˆ o ]T θˆ = [vec(W

(3)

Using a prediction-error identification method, the weights are calculated as θˆ = arg min(J1 (θ, zN ))

(4)
N

where J1 (θ, zN ) = (1/2N) t=1 (y(t) − yˆ (t|θ))T (y(t)− yˆ (t|θ)) The Levenberg-Marquardt training method can be used to determine the weights in the network [6,10]. The derivative of the prediction ψ(k|θ) = ∂yˆ (k|θ)/∂θ is the key component in the implementation of the training method as well as the Hessian ∂yˆ 2 (k|θ)/∂θ 2 . From the two general modeling structures available (i.e., series–parallel and parallel identification schemes), we can select the parallel model shown in Eq. (5) that expresses the approximation of the nonlinear process by the function Gm (·) in terms of the past inputs and the past outputs of the model, the nonlinear autoregressive-moving average (NARMA) model: ˆ yˆ (k + 1|θ) = Gm (φ(k), θ) = Gm (ˆyk , yˆ k−1 , . . . , yˆ k−ny+1 , uk , . . . , ˆ uk−nu+1 , θ)

(5)

Using Eq. (1) to form the parallel model of Eq. (5), the model can be rewritten as Eqs. (6) and (7).  ny  φk(q) = tanh  ai(q) yˆ k−i+1 i=1

+

nu  j=1

 bj(q) uk−j+1 + bx(q,0) 

(6)

yˆ (k + 1|θ) =

p 

Wo(k,q) φk(q) + bo(k,0)

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(7)

q=1

Once again we will consider a single hidden MLP with hyperbolic tangent units in the hidden layer and a single-output linear neuron.

3. The machining process The characteristics of the machining process as a complex electromechanical process severely limit the use of classic mathematical tools for modeling and control [1]. The dynamics of the milling process (cuttingforce response F to changes in feed rate f) can be approximately modeled using a second-order differential equation. The following linear model is suggested in Ref. [14]: GLU (z) =

0.019z + 0.017 − 1.75z + 0.77

z2

(8)

Another model obtained using second-order differential equations [19] is GRS (z) =

0.052z + 0.04 z2 − 1.42z + 0.45

(9)

The structure of a first-order cutting-force process including cutting speeds and nonlinear depth-of-cut effects is proposed [13]; thus GLA (z) =

0.11 0.65 0.63 a f z − 0.85

(10)

The approximate models in Eqs. (8)–(10) provide a characterization of the dynamic behavior of the machining process that helps to investigate and analyze machine-tool performance and limitations. In order to obtain the best workpiece surface quality and reduce the in-process time, the cutting force should be kept constant during machining (direct relationship between cutting force and material-removal rate). When there are disturbances such as variable depth of cut in the workpiece, this requirement can only be met under control. Therefore, an approximate mathematical model can be used to design new controllers for the aim of process optimization. GP = {GLU (z), GRS (z), GLA (z)} represents the machining process from the classical viewpoint. Eqs. (8)–(10) are only valid over a narrow range; hence

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programmed), whereas the spindle speed is considered constant and preset by the operator.   eT = KE F KCE 2 F , u = [GC f ]

(11)

where KE , KCE and GC are scaling factors for inputs (error and change in error) and output (change in feed rate), respectively. We will consider a set of rules consisting of linguistic statements that link each antecedent with its respective consequent, having the following syntax: IF F is positive AND 2 F is positive THEN Fig. 1. Overall diagram of a typical machining and control scheme for process optimization.

they cannot trespass certain limits in representing the process’ complexity and uncertainty. However, they do provide a rough characterization of the dynamic behavior of the machining process and are used in Section 7 for simulations. The full diagram of a typical machining task and a control scheme for process optimization is shown in Fig. 1. The computerized numerical control (CNC) performs important tasks involving the velocity-control loop for the feed rate, fr , and the spindle speed, sr ), and the interpolator (i.e., spatial-position control loop of the T cutting tool OP = [ xP yP zP ] ). The variables OPo , so and fo represent the internal references for the CNC control loops, which for f may differ from the dynamic values generated by the cutting-force controller.

4. A fuzzy approach to controlling cutting force The milling process is a complex one, but the control task can be solved by applying a controller that is also nonlinear and whose design does not require any analytical model of the plant: a fuzzy-logic control (FLC) [4]. In the design of an FLC as detailed in Ref. [7], the input variables included in the error vector e are the cutting-force error (F in N) and the change in cuttingforce error (2 F in N). Error and change in error are two variables commonly used in fuzzy control. The manipulated (action) variable we selected is the feed rate increment (F in percentage of the initial value

f is Positive Big During normal manual operation, the operator selects a constant feed rate for the whole machining path, corresponding to the points of the workpiece that produce maximum cutting force (conservative criterion). An efficient operator, however, should adjust the feed rate in real-time according to the cutting parameters to improve the metal-removal rate. By relating operator experience with a suitable variable whose dynamic behavior represents the status of the process (i.e., cutting force), we can establish the control rules. So, on the basis of keeping a constant cutting force, when the force increases (e.g., due to increased depth of cut), the feed rate should be reduced. On the other hand, when the force decreases due to cutting air, the feed rate should be increased to maximize the metal-removal rate. The controller output is inferred by means of the compositional rule. The Sup-Product compositional operator was selected for the compositional rule of inference. For instance, applying the T2 norm (product) and applying S1 s-norm (maximum) yields Eq. (12): m×n

µ(F, 2 F, f ) = S1 [T2 [µFi (F ), µ2 Fi (2 F ), i=1

µfi (f )]]

(12)

where T2 represents the algebraic product operation and S1 represents the union operation (max), m × n = 49 rules. The crisp controller output used to change the machine-table feed rate is obtained by defuzzification employing the center of area (COA) method defined as
i µR (fi )fi f =
(13) i µR (fi )

R.E. Haber et al. / Future Generation Computer Systems 21 (2005) 1083–1095

Fig. 2. The control scheme based on the IMC method.

where f is the crisp value of fi for a given crisp input (Fi , 2 Fi ). The strategy used to compute f determines what type of fuzzy regulator is to be used. In this case, it is a PI regulator: f (k) = f (k − 1) + f (k)

(14)

5. An IMC system based on neural networks A block diagram of an IMC system from the classical viewpoint is depicted in Fig. 2. All disturbances are considered to take place in the process output. In the figure GM denotes a model of the process, G M is an approximate inverse of GM and GF is a low-pass filter. Construction of the IMC system consists of two stages: (i) selection of a controller (usually the inverse model) to achieve perfect control and (ii) the introduction of a filter. The inclusion of a filter, GF , reduces the highfrequency gain and hence improves the robustness of the system. The filter also smoothes out noisy/rapidly changing signals, reducing the transient response of the controller. A common choice from classical literature is the low-pass filter. Using the z-transform, GF (z) is given by GF (z) =

1 − k2 z − k1

(15)

where k1 , k2 are design parameters and usually k1 = k2 . 

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Indeed, in IMC, robustness is considered explicitly in the design, although it can be adjusted by choosing filter GF accordingly. A similar kind of scheme can be implemented using an ANN in what is called the IMC-NN. First an ANN is trained to learn the dynamics of the process and is therefore given known input and output data sets. So, one of the neural network models developed in Ref. [6] is selected as a basis for IMC control. The dynamic equation can be described in reduced notation by Fˆ (t) = GM (F, f)

(16)

where GM is an unknown function to be identified, F is the cutting force exerted during the removal of metal chips and f is the relative feed speed between tool and worktable. Cast in vector form, f and F are the input and output respectively defined as F = [F (t − 1), . . . , F (t − n)] and f = [f (t − 1), . . . , f (t − m)], t is the discrete time instant and n, m ∈ Z. A successful identification scheme should insure Fˆ (t) values as close as possible to those of F(t) (actual output). The training algorithm was developed using MATLAB. The topology was initially chosen as follows: one input f and one output Fˆ , a linear activation function at the output and one hidden layer using the hyperbolic tangent for the activation function. The type of model was selected using a priori knowledge of the milling process and the types of models considered in previous work. Data obtained from actual machining operations were used for training. An ANN having four inputs, four neurons in the hidden layer and one neuron in the output layer was selected. The dynamic equation can be described in reduced notation by Fˆ (t + 1) = GM (f (t)f (t − 1)F (t)F (t − 1))

(17)

where GM ={wf , Wf }, wf = {Wu , Wx , bx }, Wf = {W0 , b0 }:

−0.0094  0.0081  wf =   0.0122 −0.0056

0.0019 0.0084 −0.0068 0.0167

0.0226 −0.0075 −0.0202 −0.0103

 Wf = −28.7834

13.8937

47.0386

−0.0158 −0.0031 0.0245 0.0140 54.5429

 2.2530 −6.2151    −2.5668 

(18)

−4.1976 378.4564



(19)

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The inverse model was obtained on the basis of generalized training [18]. Therefore, the network is trained off-line to minimize J(θ) =

M 

(f (t) − fˆ (t))

2

(20)

t=1

Another ANN is trained to learn the inverse dynamics of the process and to work as a nonlinear controller. The dynamic equation of the inverse model can be described in reduced notation by f (t) = G M (F (t + 1)F (t)f (t − 1)f (t − 2)) where, G M {W0 , b0 }:

(21)

= {wi , Wi }, wi = {Wu , Wx , bx }, Wi =



residuals (i.e., the difference between prediction and actual process output) and feed rate, as well as the autocorrelation function of residuals allow us to determine the actual contribution of the feed rate to the cutting force (suitability of model structure) as well as to verify whether the residuals are independent of past inputs or not. Additionally, several error-based performance indices were used to evaluate the forward model and the inverse model, defined as SSE =

−0.0004 0.0141 −0.0027

−0.0003 −0.0076 0.0017

−0.0042 −0.0213 0.0070

 − 0.3665 −18.1896    − 1.7947 

0.0029

−0.0006

−0.0010

−0.0048

− 0.7328

Wi = −157.9236 2.6474

94.7554

129.0931

459.2642

The bias vectors (bX , b0 ) are represented by the last columns of Eqs. (18), (19), (22) and (23). The result of the forward model for predicting cutting force in real-time is shown in Fig. 3a. The behavior of the inverse model is depicted in Fig. 3b. The auto-correlation and cross-correlation functions are depicted in Fig. 3c. The cross-correlation function of

(y(t) − yˆ (t|θ))T (y(t) − yˆ (t|θ))

(24)

i=1

0.0022  0.0332  wi =   0.0026 

N 

(22)



2 σFPE =

2 σUNV =

(23)

SSE(N + p) N −p SSE N −p

(25)

(26)

Fig. 3. (a) Comparison between plant and model outputs; (b) comparison between actual control signal and prediction; (c) correlation functions.

R.E. Haber et al. / Future Generation Computer Systems 21 (2005) 1083–1095 2 σSBC = 2 σGCV =

N + (ln(N) − 1)p SSE N −p N N SSE

(27) (28)

(N − p)2

2 where SSE is the sum of squared errors, σFPE is the 2 is the unbiased estimate final prediction error, σUNV 2 of variance, σGCV is the generalized cross validation, 2 σSBC is Schwarz’s Bayesian information criterion, p is the effective number of weights and N is the total number of samples. 2 2 , σ2 2 (σUEV , σFPE GCV , σSBC ) = (686.67, 707.44, 708.09, 860.54) was yielded for the forward model. 2 2 , σ2 2 (σUEV , σFPE GCV , σSBC ) = (709.13, 730.59, 731.25, 888.69) was yielded for the inverse model. Note that the ANN only provides an approximation to the behavior of the actual plant. Therefore, a filter (15) was used before the controller G M in the forward path to compensate for plant model mismatch with k1 = 0.735 and k2 = 0.905.

6. Optimal tuning of the fuzzy controller The back-propagation of error is applied for tunˆ E, K ˆ CE ] corresponding to the input scaling ing θˆ = [K

1089

factors of the fuzzy block that will replace the inverse model (21). Details are shown in Fig. 4. The goal is therefore the  optimal setting of input  ˆ CE to ensure that the overall ˆ E, K scaling factors K system follows the reference signal Fr (t) closely. Indeed, if Eq. (21) actually describes the inverse dynamic of the plant, there will be a perfect cancellation ˆ E, K ˆ CE ] and therefore we should attempt to find θˆ = [K ∼ such that fNN (t) = fFLC (t). However, the trained network (Eqs. (22) and (23)) will have certain inaccuracies, and we cannot be sure of the true performance of the final closed-loop system. On the basis of the recursive training algorithm described in Section 2 and assuming the following error in the output of the fuzzy controller en (t) =

∂F (t) et (t) ∂f (t − 1)

(29)

where et (t) = Fr (t) − F (t), F(t) is the output of the system and Fr (t) is the output of the reference model. Using the forward model in Eq. (17), we can estimate the Jacobians and therefore ∂F (t) ∼ ∂Fˆ (t) = ∂f (t − 1) ∂f (t − 1)

Fig. 4. Scheme for tuning scaling factor using back-propagation and the inverse model.

(30)

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ˆ E, K ˆ CE ]; (b) behavior of Mpt criterion with regard to θˆ = [K ˆ E, K ˆ CE ]. Fig. 5. (a) Behavior of ITAE criterion with regard to θˆ = [K

The reader can find more details about the applied algorithm in Ref. [18]. An important issue is therefore the stopping conˆ E, K ˆ CE ]. Indeed, dition to avoid overestimating θˆ = [K Eq. (4) is a version of an integral of square error (ISE) criterion, which is not usually very sensitive. The best course is therefore to use the integral of time multiplied by the absolute value of error (ITAE) criterion to optimize the transient response and therefore to penalize lengthy transients.  T J2 = te(t) dt (31) 0

The ITAE criterion (31) is selected to obtain smaller overshoots and oscillations, which are quite harmful for the cutting tools used in machining. The scheme for the optimal tuning of the FLC block on the basis of ITAE is depicted in Fig. 4. The reference model using the z-transform GR (z) is given by 0.11 GR (z) = z − 0.89

(32)

The initial parameters of the FLC (i.e., scaling factors) were θ = [KE , KCE ] = [10.2, 5.1]. Eqs. (17)–(19) were selected as the forward model GM . For the sake of simplicity, Eq. (10) was considered tobe GP .  ˆ E, K ˆ CE = After 150 iterations, we obtain K [5.69, 29.9], corresponding to a minimum J2 = 80.1. Fig. 5a shows the behavior of J2 versus the estimate of ˆ CE ]. The overshoot was also minimized, as ˆ E, K θˆ = [K

Fig. 6. Time response of F, Fˆ , Fr .

expected (see Fig. 5b). Finally, the time response of all three schemes is shown in Fig. 6.

7. Simulations and results Models (8)–(10) were used in a feedback-control structure as depicted in Figs. 7 and 8. The fuzzy-control surface has the same static gain as the inverse neural model G M . This static gain was obtained by adjusting the scaling factors (KE , KCE , GC) (see Eq. (11)).

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Fig. 7. Fuzzy-control scheme for the machining process.

The parameters (i.e., scaling factors) corresponding to the FLC (see Fig. 7) were: (KE , KCE , GC) = (10.0, 5.1, 0.09)

(33)

After applying the procedure described in Section 6, the scaling factors obtained were: (KE , KCE , GC) = (5.69, 29.9, 0.09)

(34)

In the sequel, Eq. (34) was applied to the fuzzy block in the neurofuzzy scheme. Simulations were run based on linearized plant models (8)–(10) representing approximate process models and using the control schemes depicted in Figs. 7 and 8. In order to analyze the disturbance-rejection capabilities of the control system, additive noise plus the influence of unmodeled dynamics were considered in

Fig. 8. (a) DIC scheme; (b) IMC scheme; (c) NFC diagram.

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Fig. 9. Closed-loop response without noise: (a) DIC-NN; (b) FLC; (c) IMC-NN; (d) NFC.

order to estimate the dynamics that can be expected in real-time applications. The following additive noise is assumed to corrupt the output:

represented by

d(t) = 0.1(sin 8t + sin 12t + sin 23.66t + sin 35.49t)

where G∗ (z) is an ideal process model represented by Eqs. (8)–(10). Various performance indices, such as the integral absolute errors (IAE), integral square errors (ISE) and integral of time per absolute errors (ITAE), were

GP (z) = G∗ (z)

(35) Now, the more realistic model of the process, including unmodeled multiplicative dynamics plus (35), is

0.095 + d(z) z − 0.904

(36)

Table 1 Summary of the comparison among the strategies analyzed Model

Criterion ISE

(8) (9) (10) (8)–(31) (9)–(31) (10)–(31)

IAE

ITAE

Mpt (%)

FLC

IMC

NFC

FLC

IMC

NFC

FLC

IMC

NFC

FLC

IMC

NFC

+ + + − − −

+ + + + + +

++ + ++ ++ + ++

+ + + − − −

− − + + ++ +

+ − + ++ + ++

+ + ++ − − −

− − + + + ++

+ − − ++ + ++

++ ++ + −− −− −−

++ ++ + − − −

++ ++ + + + +

(++) Excellent; (+) good; (−) bad; (−−) very bad.

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Fig. 10. Closed-loop response in presence of disturbances: (a) DIC-NN; (b) FLC; (c) IMC-NN; (d) NFC.

calculated in order to evaluate the simulation results. The overshoot, Mpt , was also computed. A comparison of DIC-NN, FLC, IMC-NN and NFC, with and without the influence of unmodeled dynamics and disturbances, is depicted in Figs. 9 and 10. DIC-

NN displays the worst performance. The control signal generated by the inverse model is highly oscillatory and causes the system to exhibit oscillation. Fig. 11 shows the error-performance indices for both cases analyzed. First let us consider the

Fig. 11. Comparison of the four control schemes (a) without noise and (b) with disturbances.

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error-performance indices for the ideal (disturbancefree) case. The FLC has the simplest control scheme and requires less computational resources than the other schemes. Likewise, on the basis of the ITAE and IAE criteria, the FLC performs very well with minimum values for ITAE and IAE criteria. However, the NFC approach yields the best performance vis-`a-vis the ISE criterion. The situation changes significantly if disturbances are taking place. The FLC, IMC-NN and NFC are all able to regulate the process in the presence of noise. Despite its intrinsic robustness, the FLC gives a better transient response but more error and a bigger overshoot, and therefore the worst performance after the DIC-NN. On the other hand, there is no great difference between the IMC-NN and the NFC on the basis of the IAE and ISE criteria (slightly better for the NFC). However, the NFC behaves very well according to the ITAE criterion (see Table 1).

8. Conclusions The neurofuzzy-control (NFC) scheme proposed in this paper is inspired by the nonlinear IMC; the inverse model represented by a dynamic ANN is replaced by a fuzzy controller in order to improve disturbancerejection capability. A dynamic ANN is trained to learn the process dynamics and another ANN is trained to learn the inverse dynamics. An FLC with the same static gain as the inverse model is used in the forward path of the control loop. This study shows the viability of implementing a control strategy based on the internal-model control paradigm, which is a useful synergy of a dynamic ANN trained from real-life data and used to predict process output and a fuzzy-logic control (FLC) that enhances the control system’s overall performance. For the electromechanical process under study, the simulation tests show that the NFC performs better than the FLC and the IMC-NN in the presence of noise. Severe disturbances and additive noise make the simulation more realistic and useful for process optimization. In this paper the ANN was trained off-line using actual real-time data, but in future both on-line training and the application of the NFC in real-time will be incorporated in order to enable adaptation on-line.

References [1] S.T.S. Bukkapatnam, A. Lakhtakia, S.R.T. Kumara, Analysis of sensor signals shows turning on a lathe exhibits low-dimensional chaos, Phys. Rev. E 52 (3) (1995) 2375– 2387. [2] R. Carotenuto, An iterative system inversion technique, Int. J. Adapt. Control Signal Process 15 (2001) 85–91. [3] G.C. Goodwin, S.F. Graebe, M.E. Salgado, Control System Design, Prentice Hall, Englewood Cliffs, NJ, 2001. [4] R.E. Haber, A. Alique, J.R. Alique, R. Haber-Haber, S. Ros, Current trend and future developments of new control systems based on fuzzy logic and its application to high speed machining, Revista Metalurgia Madrid 38 (2002) 124–133. [5] R.E. Haber, J.R. Alique, Nonlinear internal model control using neural networks: applications to machining processes, in: Neural Computing and Applications, Springer-Verlag, London, 2004, doi:10.1007/s00521-003-0394-8. [6] R.E. Haber, J.R. Alique, A. Alique, R.H. Haber, Nonlinear internal model control using neural networks and fuzzy logic: application to an electromechanical process, in: Proc. ICCS’03, Lecture Notes in Computer Science, vol. 2657, Springer, 2003, pp. 351–360. [7] R.E. Haber, G. Schmitt-Braess, R.H. Haber, A. Alique, J.R. Alique, Using circle criteria for verifying asymptotic stability in PI-like fuzzy control systems. An application to the milling process, IEE Proc. Control Theory Appl. 150 (6) (2003) 619–627. [8] M.T. Hagan, H.B. Demuth, O. De Jes´us, An introduction to the use of neural networks in control systems, Int. J. Robust Nonlinear Control 12 (2002) 959–985. [9] K. Hornik, M. Stincheombe, H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (1989) 359–366. [10] S.J. Hung, R.J. Lian, A hybrid fuzzy logic and neural networks algorithm for robot motion control, IEEE Trans. Ind. Electron. 44 (3) (2000) 217–229. [11] K.J. Hunt, D. Sbarbaro, R. Zbikowski, P.J. Gawthrop, Neural networks for control systems—a survey, Automatica 28 (6) (1992) 1083–1112. [12] C. Kambhampati, R.J. Craddock, M. Tham, K. Warwick, Inverse model control using recurrent networks, Math. Comput. Simul. 51 (2000) 181–199. [13] R. Landers, A. Ulsoy, Model-based machining control, ASME J. Dyn. Syst. Meas. Control 122 (3) (2000) 521–527. [14] L.K. Lauderbaugh, A. Ulsoy, Model reference adaptive force control in milling, ASME J. Eng. Ind. 111 (1989) 13–21. [15] G. Lightbody, G.W. Irwin, Nonlinear control structures based on embedded neural systems models, IEEE Trans. Neural Networks 8 (3) (1997) 553–567. [16] M. Morari, E. Zafiriou, Robust Process Control, Prentice Hall, Englewood Cliffs, NJ, 1989. [17] O. Nelles, Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Logic, SpringerVerlag, Berlin, 2000. [18] M. Norgard, O. Ravn, N.K. Poulsen, L.K. Hansen, Neural Networks for Modelling and Control of Dynamics Systems, Springer-Verlag, London, 2000.

R.E. Haber et al. / Future Generation Computer Systems 21 (2005) 1083–1095 [19] S.J. Rober, Y.C. Shin, Control of cutting force for milling processes using an extended model reference adaptive control scheme, J. Manuf. Sci. Eng. 118 (1996) 339–347. [20] I. Rivals, L. Personnaz, Nonlinear internal model control using neural networks: application to processes with delay and design issues, IEEE Trans. Neural Networks 11 (1) (2000) 80–90. [21] L. Wang, Y. Frayman, A dynamically-generated fuzzy neural network and its application to torsional vibration control of tandem cold rolling mill spindles, Eng. Appl. Artif. Intell. 15 (6) (2003) 541–550. Rodolfo E. Haber was born in Santiago the Cuba, Cuba, in 1969. He received the BE degree with first class honors in automatic control engineering from the Universidad de Oriente (UO), in 1992. In 1995, he was the recipient of PhD grant supported by the Spanish Office for Scientific Cooperation (AECI) in Spain. He received an Excellent cum Laude PhD in industrial engineering from Technical University of Madrid (UPM), in 1999. He joined with the Spanish Council for Scientific Research (CSIC) in 1999 working in several research and development projects. In 1999, he also joined with the Computer Science Department at the Universidad Autonoma de Madrid (UAM) teaching in several graduates courses. He has published several technical papers in specialized journals and chapters of books. His research interests include control theory and applications, hardware–software solutions, soft-computing techniques, classical and adaptive control, supervisory control, and complex electromechanical processes. Jos´e R. Alique received the BSc degree in physics from the University Complutense of Madrid (UCM), Spain, in 1969. He received the MSc and PhD degree of physics sciences from International Institute of Philips, North Holland, in 1971, and University Complutense, in 1973, respectively. From 1969 to 1970, he was assistant professor of physics and automation at UCM. He has been working in automation since 1972 in

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the Spanish Council for Scientific Research (CSIC). From 1991 to 1996 he was the director of the Technology Transfer Office in the CSIC. From 1996 to 2001, he was the vice director of R&D projects in the Ministry of Science and Technology. He is currently a researcher in Instituto de Automatica Industrial (CSIC) and head of the Computer Science Department. His research interests include automation of machining processes, numerical controls, robotics systems, and intelligent modeling and control.

Angel Alique received the BSc and PhD degree in physics from University Complutense of Madrid in 1972 and 1979, respectively. He is currently a researcher at Instituto de Automatica Industrial (CSIC). He has published several technical papers on supervisory fuzzy control and modeling of machining processes. He is currently the main researcher of two projects funded by CICYT. His research interests include fuzzy modeling and control, robotics systems and intelligent control.

Rodolfo H. Haber graduated in electrical engineering from the University of Oriente (UO, Santiago de Cuba, Cuba) in 1964, and obtained a PhD in technical cybernetics at Prague Technical University (CVUT), Czech Republic, in 1976. Since 1966, he is the professor of automatic control theory (and related subjects), including linear and non-linear control theory, undergraduate and postgraduate levels, at the Automatic Control Department, Electrical Engineering Faculty, UO. During 1993–1994 he joined the Industrial Automatics Institute (CSIC), Madrid, Spain, as an invited researcher. He is honorary member of the Cuban national permanent commission for PhD degrees granting in automatics and computers science fields. He is also representing his faculty in two Latin-American networks under the program CYTED. His main research area is fuzzy control.