Abstract:
A robust Artificial Neural Network controller is proposed for the motion control of a magnetic disk drive voice coil motor (voice coil motor). The neural controller is used to approximate the nonlinear functions (actuator electromechanical dynamics) of the voice coil motor while having on line training. One main advantage of this approach, when compared with standard adaptive control, is that complex dynamical analysis is not needed. Using this design, not only strong robustness with respect to uncertain dynamics and non-linearities can be obtained, but also the output tracking error between the plant output and the desired reference can asymptotically converge to zero. Additionally, standard offline training, utilizing training vectors to stimulate the voice coil motor, is not required.

Description:
FIELD OF THE INVENTION 
     The present invention is generally related to the field of Artificial Neural Network based control systems. The invention provides a method of Artificial Neural Network training without the need of an offline learning phase and training vectors. 
     BACKGROUND OF THE INVENTION 
     Moore&#39;s law has long been cited as the prime mover in the information technology revolution. Now this revered law may have to share the spotlight with another, equally important phenomenon: the amazing growth of areal densities in the storage industry. 
     Driven by the increasingly rich content of files, easy access to Internet downloads and data collected by corporate web sites (not to mention the pervasive reluctance among users to delete old e-mail files), the need for storage capacity is growing at rates estimated as high as 100% per year. 
     Luckily for corporate space planners, areal density is keeping pace with the need.  Data Storage Magazine  reports annual increases of 60 w, with 100% increases projected in the near future. Maintaining such increases in a highly price-competitive environment puts tremendous pressure on every aspect of disk drive technology. Given the relentless market pressure to increase areal density by at least 60% annually and the problems associated with increasing linear bit densities (bpi), considerable interest has been focused on radial track density (tpi). 
     Magnetic disk drive motion control design has been dominated by classical control techniques. While this tradition has produced many effective control systems, recent developments in radial track density and micro-actuators, have forced a growing interest in robust, nonlinear and artificial intelligence control. 
     Disk drives store data on constantly spinning disks made of aluminum or glass and coated on both sides with a thin film of magnetic material. Magnetic read/write heads store data on both sides of the disk by magnetizing very small areas on the disk surface in closely spaced, concentric tracks. A positioning device called an actuator or, voice coil motor, moves the heads rapidly from one track to another by the servo control system. An example of such Prior Art disk drive architecture is disclosed, for example in C. Denis Mee,  Magnetic Recording , McGraw Hill Inc., 1988, incorporated herein by reference. 
     The expected increase in track density places an extra burden on the servo control system, which must hold the off-track motion of the heads within increasingly tighter limits for errorless reading and writing. This limit, known as track mis-registration (TMR), amounts to about 10% of the total track width. At today&#39;s average track density of 25000 tracks per inch, the TMR budget is approximately four micro-inches. This implies eliminating as much as possible, any disturbance that might cause the head to move off track. 
     The major causes of off-track disturbances include Non-repeatable runout from the spindle bearing, Residual vibrations due to actuator modes, Servo writer errors, Nonlinear friction effects from the VCM bearings, Slipped disks, leading to repeatable runout, and Casting warpage. 
     The servo bandwidth frequency provides a measure of how well the control system will dampen the effects of off-track disturbances. 
     Voice coil motor control can be classified into two basic problems: tracking a reference trajectory, (seeking) and track following. Several linear controllers as well as nonlinear controllers have been proposed in the Prior Art for solving these problems. The main idea behind these control systems is to achieve suitable bandwidth to obtain the track density required by the marketplace. 
     Artificial Neural Networks are known in the art. Although Artificial Neural Networks have been around since the late 1950&#39;s, it wasn&#39;t until the mid-1980&#39;s that algorithms became sophisticated enough for general applications. Today, Artificial Neural Networks are being applied to an increasing number of real-world problems of considerable complexity. They are good pattern-recognition engines and robust classifiers, with the ability to generalize in making decisions about imprecise input data. They offer ideal solutions to a variety of classification problems such as speech, character and signal recognition, as well as functional prediction and system modeling where the physical processes are not understood or are highly complex. 
     Artificial Neural Networks may also be applied to control problems, where the input variables are measurements used to drive an output actuator, and the network learns the control function. The advantage of Artificial Neural Networks lies in their resilience against distortions in the input data and their capability of learning. They are often good at solving problems that are too complex for conventional methods and are often well suited to problems that people are good at solving, but for which traditional methods are not. 
     In its most general form, an Artificial Neural Network-is a machine that is designed to model the way in which the brain performs a particular task. Artificial neural networks are collections of mathematical models that emulate some of the observed properties of biological nervous systems and draw on the analogies of adaptive biological learning. The key element of the Artificial Neural Network paradigm is the novel structure of the information processing system. It is composed of a large number of highly interconnected processing elements that are analogous to neurons and are tied together with weighted connections that are analogous to synapses. 
     Learning in biological systems involves adjustments to the synaptic connections that exist between the neurons. This is true of Artificial Neural Networks as well. Learning typically occurs by example through training, or exposure to a training set of input/output data where the learning algorithm iteratively adjusts the connection weights. These connection weights store the knowledge necessary to solve specific problems. 
     There are many different types of Artificial Neural Networks. Some of the more common include the multilayer perceptron which is generally trained with the back-propagation of error algorithm, learning vector quantization, radial basis function, Hopfield, and Kohonen, to name a few. Some Artificial Neural Networks are classified as feedforward while others are recurrent depending on how data is processed through the network. 
     Another way of classifying Artificial Neural Network types is by their method of learning, as some Artificial Neural Networks employ supervised training while others are referred to as unsupervised or self-organizing. Supervised training is analogous to a student being guided by an instructor. Unsupervised algorithms essentially perform clustering of the data into similar groups based on the measured attributes or features serving as inputs to the algorithms. This is analogous to a student who derives the lesson totally on his or her own. Artificial Neural Networks can be implemented in software or in specialized hardware. 
     FIG. 1 illustrates the general architecture of a two layer artificial neural network. The left layer represents the input layer, in this case with three inputs nodes  110 ,  120 , and  130  receiving inputs X 1  through X 3 . The middle layer is called the hidden layer, with five nodes  140 ,  150 ,  160 ,  170 ,  180 , and  190 . It is this hidden layer which performs much of the work of the network. The output layer in this case has one node  190  outputting signal Y 1  representing output values determined from the inputs. 
     Each node  140 ,  150 ,  160 ,  170 , and  180  in the hidden layer may be fully connected to the input nodes  110 ,  120 , and  130 . That means what is learned in a hidden node is based on all the inputs taken together. This hidden layer is where the network “learns” interdependencies in the model. FIG. 2 provides some detail into what goes on inside a hidden node. 
     As illustrated in FIG. 2, a weighted sum  210  may be performed as follows: X 1  times W 1  plus X 2  times W 2  and so on through X 3  and W 3 . This weighted sum is performed for each hidden node and each output node and is how interactions are represented in the network. Each summation is then transformed using activation function  220  before the value is passed on to the next layer. The activation function translates the weighted input into a value which may be used as an input to a next step or portion of the neural network. 
     Examples of activation functions include a linear function  230 , a clipped linear function  240 , and a Gaussian distribution  250 . Of course, other types of functions may be applied in activation function block  220 . However, in general, neural networks work best with activation functions with limited dynamic ranges. 
     In the human brain, information is passed between the neurons in form of electrical stimulation along the dendrites. If a certain amount of stimulation is received by a neuron, it generates an output to all other connected neurons and so information takes its way to its destination where some reaction will occur. If the incoming stimulation is too low, no output is generated by the neuron and the information&#39;s further transport will be blocked. A further description of the operation of neural networks may be found in S. Haykin,  Neural Networks: A Comprehensive Foundation . Macmillan, NY, 1994, incorporated herein by reference. 
     Neural nets try to simulate the human brain&#39;s ability to learn. Unlike the biological model, an artificial neural network generally has an unchangeable structure, built of a specified number of neurons and a specified number of connections between them, which have certain values (weights). What may change during the learning process are the values of those weights. Incoming information exceeds a specified threshold value of certain neurons that pass the information to connected neurons or prevent further transportation along the weighted connections. The value of a weight will be increased if information should be transported and decreased if not. While learning different inputs, the weight values are changed dynamically until their values are balanced, so each input will lead to the desired output. 
     The training of a neural net results in a matrix that holds the weight values between the neurons. Once a neural net had been trained correctly, it will be able to find the desired output to a given input that had been learned, by using these matrix values. 
     For control engineers, the approximation capability of artificial neural networks is usually used for system identification. However there is very little about the use of Artificial Neural Networks in closed loop controllers that yield guaranteed performance. A problem with the use of Artificial Neural Networks in control applications is the uncertainty of how to initialize the Artificial Neural Network weights, which leads to the necessity of offline tuning. See, e.g., T. Yamada, Remarks on a learning type self-tuning neural network controller, Int. Conf. Adv Robotics, 1993, pp 43-48, and F. C. Chen, Adaptive Control of nonlinear systems using Neural networks, IEEE Control Systems Magazine, vol 55 no. 6, pp. 1299-1317, 1992, both of which are incorporated herein by reference. 
     Additionally, the processing requirements of Artificial Neural Networks for a voice coil motor control system, until recently, has been prohibitive. New high-performance 32-bit processors such as the ARM7 and ARM9 logic cores used in the Cirrus Logic 3Ci platform, the industry&#39;s most-advanced mixed-signal system-on-a-chip for magnetic hard disk drives, are playing a major role in enabling the recent developments in motion control. With these processors and associated support peripherals, sophisticated algorithms from modern control theory, nonlinear control and AI control can be implemented in realtime, and the effective control bandwidth can be pushed to higher limits. 
     Traditional controller design usually involves complex mathematical analysis and yet has many difficulties in controlling highly nonlinear plants. These nonlinearities are due in part to VCM bearing friction and electro-mechanical properties of the motor and power driver. The use of the learning ability of a neural network helps control design to be rather flexible, especially where plant dynamics are complex and highly nonlinear. This is a distinct advantage over traditional methods. 
     Artificial Neural Network voice coil motor controllers are known in the art (e.g., Khan, U.S. Pat. No. 5,471,381, issued Nov. 28, 1995 and incorporated herein by reference). However, such conventional Artificial Neural Network controllers may have to be trained such that they learn the characteristics of the plant or the plant and controller combination. Learning typically occurs by example through training, or exposure to a training set of input/output data where the learning algorithm iteratively adjusts the connection weights. 
     These connection weights store the knowledge necessary to solve specific problems. However, the use of Artificial Neural Networks in a control system application has limitations with respect to its practicality. The general learning phase as illustrated in FIG. 2 is very broad. To achieve maximal generalized learning, rigorous learning in the form of using many sets of training patterns may be required. 
     In order to train a neural network to perform some task, the weights of each unit may need to be adjusted in such a way that the error between the desired output and the actual output is reduced. This process requires that the neural network compute the error derivative of the weights (EW). In other words, it must calculate how the error changes as each weight is increased or decreased slightly. The back propagation algorithm is the most widely used method for determining the EW. 
     The back-propagation algorithm is easiest to understand if all the activation functions in the network are linear. The algorithm computes each EW by first computing the EA, the rate at which the error changes as the activity level of a unit is changed. For output units, the EA is simply the difference between the actual and the desired output. 
     To compute the EA for a hidden unit in the layer just before the output layer, all the weights between that hidden unit and the output units to which it is connected are first identified. Those weights may then be multiplied the EAs of those output units and add the products. This sum equals the EA for the chosen hidden unit. 
     After calculating all the EAs in the hidden layer just before the output layer, the EAs of other layers may be computed in like fashion the EAs for other layers, moving from layer to layer in a direction opposite to the way activities propagate through the network. This is what gives back propagation its name. Once the EA has been computed for a unit, it is straightforward to compute the EW for each incoming connection of the unit. The EW is the product of the EA and the activity through the incoming connection. 
     Note that for non-linear units, the back-propagation algorithm includes an extra translation step. Before back-propagating, the EA must be converted into the EI, the rate at which the error changes as the total input received by a unit is re changed. 
     Due to the complexities of off-line training and weight initialization, it would therefore be desirable to have an Artificial Neural Network based control system that does not require this specialized learning phase and have a stable system when the weight are initialized to zero. Thus, a need exists in the Prior Art for an artificial neural network voice coil motor controller system which does not require a learning phase in order to operate properly. 
     SUMMARY OF THE INVENTION 
     An Artificial Neural Network control system is derived using VCM control techniques. This means that the neural network weights are tuned online, implying that no offline learning is needed. Additionally, constant stimulus of the plant is not required for the Artificial Neural Network to learn online as is true for some adaptive systems as well as Real Time Recurrent Learning. The control system ensures good performance during the initial period if the neural network weights are initialized to zero. Tracking performance is guaranteed using a Lyapunov approach, even though there do not exist ideal weights, such that the neural net perfectly reconstructs the required nonlinear function. 
     The control system is comprised of an Artificial Neural Network incorporated into a VCM dynamical system, where the structure comes from some error notations standard in VCM control. Unlike adaptive VCM control, where a regression matrix must be computed from the dynamics of the structure, the basis functions for the Artificial Neural Network controller can be derived from the physics of the VCM structure. 
     The present invention uses an Artificial Neural Network controller in solving the tracking problems associated with the nonlinear properties of the voice coil motor of a magnetic disk drive. The present invention provides a method and apparatus for training an Artificial Neural Network based controller in the normal mode of operation, (i.e., without the need for a specialized learning phase). The resulting Artificial Neural Network control system weight update algorithm overcomes many of the inherent difficulties associated with the standard Artificial Neural Network control architectures. 
     The approach of the present invention uses the error states of the system to modify the weights. As the error states approach zero, the weight changes approach zero. The present invention is particularly novel when compared to conventional supervised learning methods which use training sets of input-output pairs and gradient-descent methods for weight update algorithms. Examples of such conventional supervised learning methods are disclosed, for example, in B. Kosko,  Neural Networks and Fuzzy Systems , 1992, and C. S. George Lee,  Neural Fuzzy Systems , 1996, both of which are incorporated herein by reference. 
     Additionally, the control system of the present invention is stable when the initial weights are set to zero. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a diagram illustrating general Artificial Neural Network architecture. 
     FIG. 2 is a diagram illustrating the architecture of a typical neuron. 
     FIG. 3 is a functional block diagram of the Artificial Neural Network controller, modeling the response of magnetic disk drive voice coil motor in the preferred embodiment of the present invention. 
     FIG. 4 is a graph illustrating tracking output of the plant of the preferred embodiment Artificial Neural Network controller in addition to two standard (prior art) linear controllers. 
     FIG. 5 is a graph illustrating tracking error of the preferred embodiment ANN controller in addition to two standard (prior art) linear controllers. 
     FIG. 6 is a functional block diagram of the Artificial Neural Network controller, as applied to an actual magnetic disk drive voice coil motor in the preferred embodiment of the present invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The control system of the present invention comprises an Artificial Neural Network incorporated into a voice coil motor dynamical system, where the structure comes from some error notations standard in voice coil motor control. Unlike adaptive voice coil motor control, where a regression matrix may be computed from the dynamics of the structure, the basis functions for the Artificial Neural Network controller can be derived from the physics of the voice coil motor structure. 
     The model for the voice coil motor is disclosed, for example, in Franklin, Powell and Workman,  Digital Control of Dynamic Systems , 1990, which is incorporated herein by reference and is given by: 
     
       
           K   t   I   c   =J x″+K   d   x′+H ( x″,x′,x )  (1) 
       
     
     
       
           U   c ( t )= L   c   I′   c   +R   c   I   c   +K   bemf   x′   (2) 
       
     
     Where: 
     K t  is the torque constant matrix, 
     J is the inertia matrix, 
     K d  is the viscous damping, 
     H is the unmodeled, nonlinear voice coil motor dynamics such as bearing friction and stiction, and high frequency resonances, 
     x″ is the acceleration state, 
     x′ is the velocity state, 
     x is position, 
     U c  is the current command, 
     I c  is the coil current, 
     L c  is the coil inductance, 
     R c  is the coil resistance, and 
     K bemf  represents voice coil motor back-emf constant. 
     For any given voice coil motor design, the physical properties are bounded by design and manufacturing specifications. The control objective is to provide a controller for the magnetic disk drive voice coil motor dynamics given by equations (1) and (2) based on inexact knowledge of the voice coil motor dynamics. To accomplish this, the voice coil motor tracking error e(t) is first defined as: 
     
       
           e ( t )= x   d ( t )− x ( t )  (3) 
       
     
     Where x d (t) denotes the desired position trajectory. It is assumed that x d  and its derivatives are bounded. Modifying the basic tracking error e(t) to include the error velocity, defines: 
     
       
           g ( t )=ρ 1   e ′( t )+ρ 2   e ( t )  (4) 
       
     
     where ρ 1  and ρ 2  are scalar gains. 
     Using equations (1) and (4) the following equation can be derived: 
     
       
           Jg′NN   1   −K   d   g−K   t   I   c   (5) 
       
     
     Where NN 1  is a highly nonlinear function, representing the unmodeled mechanical modes of the voice coil motor. 
     A part of the controller design is a first Artificial Neural Network to estimate coil current I c  of the voice coil motor, represented by I est . Equation (5) may be rewritten as: 
     
       
           Jg′=NN   1   −K   d   g−K   t   I   est   +K   t   I   err   (6) 
       
     
     Where I err  is the estimation error, which is the difference between the actual current and the estimated current. 
     The next feature of the controller design is a second Artificial Neural Network which will generate coil current control signal U c  such that I err  is minimized. To achieve this a dynamics equation in terms I err  needs to be derived as follows: 
     
       
         I err   =I   est   −I   c   (7) 
       
     
     Differentiating (7) and using equation (2) yields: 
     
       
           L   c   I′   err   =NN   2   −U   c   (8) 
       
     
     Where NN 2  represents the unmodeled, nonlinear electro-mechanical functions of the voice coil motor and associated power driver. 
     The role of the two Artificial Neural Networks in this design is to approximate the two complicated nonlinear functions NN 1  and NN 2 . Notice that no dynamical analysis is needed as in typical adaptive control schemes. Thus, the controller of the present invention may be applied to many types of voice coil motor systems without modifications. 
     Refer back to FIG. 1, which illustrates the general architecture of a two layer artificial neural network. By collecting all the Artificial Neural Network weighting values W ij  into matrices of weights W T , the equation for the Artificial Neural Network output becomes: 
     
       
           Y=W   T θ( x )  (9) 
       
     
     With the vector of activation functions: 
     
       
         θ( x )=[θ( x   1 ), . . . , θ( x   n )] T   (10) 
       
     
     As disclosed in G. Cybenko,  Approximation by superpositions of a sigmodial function , Math. Contr. Signals Syst., vol 2, no. 4, pp 303-314, 1989 (incorporated herein by reference), the Artificial Neural Network function approximation property states that a general nonlinear function C(x) can be approximated by: 
       C ( x )= W   T θ( x )+φ( x )  (11) 
     The value φ(x) is called the Artificial Neural Network functional approximation error. The estimate of C(x) may be given by: 
     
       
           {overscore (C)} ( x )= {overscore (W)}   T θ( x )  (12) 
       
     
     where {overscore (C)} is the estimate of the ideal Artificial Neural Network weights that is generated by an online weight tuning algorithm. 
     In designing an Artificial Neural Network to estimate the motor current I c , the following equation may be used: 
     
       
           I   est   [N{overscore (N)}   1   +k   2   g]/k   1   (13) 
       
     
     Where N{overscore (N)}={overscore (W)}θ 1 , and k 1  and k 2  are scaling constants. The weight estimates may be generated by the online weight tuning algorithm discussed below. Substituting equation (13) into equation (6) yields: 
     
       
           Jg″K   d   g =( I−K   t   /k   1 )( {overscore (W)}   T θ 1 )+φ 1   +W   1     err     T θ 1   −K   t   /k   1   k   2   g+K   t   I   err   (14) 
       
     
     
       
         Where  W   1     err     T θ 1   W   1   T θ 1−{overscore (W)}   1 θ 1   (14) 
       
     
     For the design of the control effort, U c , the following equation may be selected: 
     
       
           U   c   =N{overscore (N)}   2   +k   3   I   err   (15) 
       
     
     Where N{overscore (N)} 2 ={overscore (W)} 2 θ 2 , k&gt;0. The weight estimates may be generated by an online weight tuning algorithm. Combining equation (15) and equation (8) yields: 
     
       
           L   c   I′   err   ={overscore (W 2     err     T )}θ   2 +φ 2   −k   3   I   err   (16) 
       
     
     Let the desired position signal x(t) and it&#39;s derivatives be bounded. The control effort is given by equations (13), and (15). In order to allow online weight training, the weights may be updated according to the following equations: 
     
       
         Δ {overscore (W)}   1 β 1 θ 1   g   T   −k   w β 1 ∥([ g   T   I   err   T ] T )∥ {overscore (W)}   1   (17)  
       
     
     
       
         Δ {overscore (W)}   2 =β 2 θ 2   g   T   −k   w β 2 ∥([ g   T   I   err   T ] T ) ∥ {overscore (W)}   2   (18) 
       
     
     Where β 1 =β 1   T , β 2 =β 2   T  are positive constant matrices, and k w  is a positive scalar constant. Notice that the weight tuning algorithm uses the system error states g and I err . As the error states approach zero, the weight changes approach zero. This is novel when compared to conventional supervised learning methods which use training sets of input-output pairs and gradient-descent methods for weight update algorithms. 
     Thus, in the present invention, no matter what values are chosen for initial weight values (random numbers or even all zeros) the system will remain stable and train on-line to adjust the weight values. By carefully selecting scaling constants k 1 , k 2 , and k 3 , such that the system will be initially stable, no off-line training is required in the present invention. 
     FIG. 3 is a functional block diagram of the Artificial Neural Network controller, modeling the response of the magnetic disk drive voice coil motor in the preferred embodiment of the present invention. The diagram of FIG. 3 illustrates the characteristics of a hard disk drive as separate elements to aid in the understanding of how these elements affect disk drive is performance. In addition, the diagram of FIG. 3 illustrates how the present invention may be modeled using computer software to simulate operation of the Artificial Neural Network controller of the present invention. 
     Digital to analog converter (DAC) and power driver  335  receives digital control input signals including coil current command Uc, for the voice coil motor and outputs an analog control signal (coil current) Ic. Coil current signal Ic represents the current signal fed to the voice coil motor to control actuation of a read/write head control arm on a disk drive. 
     Block  340  (represented here schematically as an amplifier) modifies coil current signal Ic by multiplying the coil current signal by the torque constant matrix, K t  divided by the inertia matrix J. Torque contrast matrix Kconverts the current signal Ic to torque. When divided by inertia matrix J, the resulting value represents an acceleration value for the voice coil motor and read/write head arm system. 
     The output of block  240  is fed to subtractor  345  with which subtracts from the output of block  240 , the output from nonlinear friction model  310 . Nonlinear friction model  310  may store parameters, a matrix, or function representing the nonlinear friction effects of the system such as from voice coil motor bearings, slipped disks and the like (e.g., nonlinear friction and stiction). The resultant signal therefor represents the acceleration value as modified by nonlinear friction values. 
     The output of subtractor  345  may be fed to block  350  which may emulate the high frequency resonances of the read/write head arm, as represented by the variable H in the equations given above. The output of block  350  may then be fed to integrators  355  and  365  to double integrate the resulting signal to produce a position reference. Note that the output of integrator  355  (a velocity value) may be used as an input signal to DAC and power driver  335  and Nonlinear friction model  310 . 
     Demodulator and ADC  365  receives the resulting double integrated position signal from integrator  360  and demodulates the signal and converts it into digital form as signal X (position) for further processing. The output of Demodulator and ADC  365  may then be summed in adder  370  with reference position signal Xd to produce tracking error signal e. Reference signal Xd may represent the position of a target track, for example. 
     Error signal e may be fed to error filter  375  (See equation 4), the output of which may be fed to neural networks  380  and  390 . The output of error filter  375  is used to update the weighting values of neural networks  380  and  390 . Neural network  380  may receive other input signals including the position reference signal X output from Demodulator and ADC  365 , position reference signal Xd, voice coil motor current signal Ic and error signal e. 
     Neural network  390  may receive the output of error filter  375  as well as other input signals including the position reference signal X output from Demodulator and ADC  365 , position reference signal Xd, as well as the output of neural network  380 , being passed through amplifier  385  which multiplies the output of neural network  380  by constant k 3 . 
     The output of neural network  390  is the coil current command signal Uc which in turn feeds DAC and power driver  335 . As can be readily seen, the system is a closed loop, and the outputs of the neural networks are fed back into their inputs to produce a stable and adaptive system which does not require off-line training. In the Prior art, a system of training vectors (sample inputs and ‘correct’ outputs) would first be used to train the neural networks such that the networks would “know” the correct response ahead of time. In the present invention, however, no such training is required. 
     FIGS. 4 and 5 illustrate a simulation of a 50 K tpi magnetic disk drive VCM. FIGS. 4 and 5 show the results with a PID compensator, an H-infinity controller and finally the Artificial Neural Network controller. As it can be seen in FIGS. 4 and 5, the voice coil motor with a highly nonlinear friction model cannot be controlled well enough using a simple PID controller. The H-infinity controller does much better at attempting to follow the reference, but there is still a significant amount of tracking error. 
     The Artificial Neural Network controller shows excellent performance as the tracking error goes to a very small value. Each Artificial Neural Network may have a 4-input, 5-hidden, and 1-output neuron architecture. The Artificial Neural Network activation functions may comprise bipolar sigmoids. 
     Compared with conventional linear control techniques, the artificial neural net VCM controller of the present invention shows excellent tracking performance. Unlike conventional Artificial Neural Network controllers, this approach does not require off-line training or constant excitation for online training. Additionally, this Lyapunov stable control algorithm does not require exact knowledge of the voice coil motor dynamics. In fact, perfect knowledge of the voice coil motor parameters may be unattainable, (e.g. bearing friction is very difficult to model by conventional techniques). 
     FIG. 6 is a functional block diagram of the Artificial Neural Network controller, as applied to an actual magnetic disk drive voice coil motor in the preferred embodiment of the present invention. Most of the elements of the diagram of FIG. 6 are the same as illustrated in FIG.  3  and thus need not be discussed in further detail here. 
     FIG. 6 represents an embodiment of the present invention as actually implements in a hard drive system, represented here by plant  660 , as opposed to FIG. 3, which illustrates elements representing aspects of (or simulating aspects of) the hard drive system. FIG. 6 also illustrates how weighting values are applied from equations (17) and (18) to the first and second neural networks. 
     While the preferred embodiment and various alternative embodiments of the invention have been disclosed and described in detail herein, it may be apparent to those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope thereof.