Abstract:
Linear PID controllers have a transfer function that resembles the frequency response of a notch filter. The PID parameters, K P , K I , and K D  (proportional, integral, and derivative gains, respectively) can be extracted from the parameters of a linear notch filter. The linearized modes of scanning probe microscope (SPM) actuators have frequency responses that resemble those of simple second order resonance. Reasonable feedback control can be achieved by an inverse dynamics model of the resonance. A properly parameterized notch filter can cancel the dynamics of a resonance to give good closed-loop response.

Description:
BACKGROUND 
     A proportional-integral-derivative controller (PID controller) is a control loop feedback mechanism commonly used in industrial control systems, e.g. a Scanning Probe Microscope (SPM). A PID controller attempts to correct the error between a measured process variable and a desired setpoint by calculating and then outputting a corrective action that can adjust the process accordingly. 
       FIG. 1  shows a block diagram of a scanning probe microscope. The input to the PID controller is the error signal from the surface sensor, and the controller output is used to move an actuator, e.g. Z actuator, that ultimately controls the relationship between the probe and surface. 
     The input to the PID controller is the error signal and the output of the controller is the voltage or current used to drive the actuator. The responsiveness of the controller is set with the coefficients K P , K I , and K D  as shown below: 
                   u   =         K   P     ⁢   e     +         K   I       T   I       ⁢     ∫     e   ⁢     ⅆ   t           +       K   D     ⁢     T   D     ⁢       ⅆ   e       ⅆ   t                   Eq   .           ⁢   1               
T I  represents the integration time and T D  represents the differentiation time, e.g. the amount of time that the integral and the differentiation take place over. It is common, but certainly not necessary to set these to a common value, T. Alternately, their values can be included in the calculation of K I  and K D , respectively.
 
     The PID controller calculation involves three separate components; the Proportional component, the Integral component and Derivative component. The proportional component determines the reaction to the current error, the integral component determines the reaction based on the current and all previous errors and the derivative component determines the reaction based on the rate by which the error is changing. The weighted sum of the three components is output as a corrective action to a control element. 
     By adjusting constants in the PID controller algorithm, the PID can provide individualized control specific to process requirements including error responsiveness, overshoot of setpoint and system oscillation.  
     SUMMARY 
     Linear PID controllers have a transfer function that can be set to resemble the frequency response of a notch filter. For these settings they have high gain at low frequency (due to the integrator), high gain at high frequency (due to the differentiator, and a minimum in between. Practical implementations of a PID use filtering to roll off the gain at very high frequencies. 
     The PID parameters, K P , K I , and K D  (proportional, integral, and derivative gains, respectively) can be extracted from the parameters of a linear notch filter. 
     The main linearized mode of scanning probe microscope (SPM) actuators have frequency responses that can be modeled as a simple second order resonance. 
     Reasonable feedback control can be achieved by using an inverse dynamics model of the resonance. A properly parameterized notch filter can adjust the dynamics of a resonance to give good closed-loop response. This enables control at closed-loop bandwidths beyond the frequency of the main linearized mode of the SPM actuator. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a block diagram for a scanning probe force microscope system (PRIOR ART). 
         FIG. 2  shows a block diagram according to the invention. 
         FIG. 3  illustrates a process flow chart according to the invention. 
         FIG. 4  illustrates a process flow chart for continuous-time model generation. 
         FIGS. 5A and 5B  illustrate an actuator response from a 3-wire measurement. 
         FIG. 6  illustrates a block diagram corresponding to continuous model generation. 
         FIGS. 7A and 7B  illustrate a closed loop frequency response from r 1  to y. 
         FIGS. 8A and 8B  illustrate the open loop frequency response from r 1  to y extracted from a closed loop measurement. 
         FIGS. 9A and 9B  illustrate the frequency response measurement from e to u C  of a compensator. 
         FIGS. 10A and 10B  illustrate the actuator frequency response function extracted from a closed loop measurement.  
         FIGS. 11A and 11B  illustrate a curve fit model for a second order system. 
         FIG. 12  illustrates a process flow chart for discrete model generation. 
         FIGS. 13A and 13B  illustrate the frequency response beyond the main resonant frequency. 
         FIGS. 14A and 14B  illustrate the projected closed loop frequency response. 
         FIG. 15  illustrates a block diagram corresponding to discrete model generation. 
         FIG. 16  is a block diagram of a parameter estimator in discrete time. 
         FIG. 17  is a block diagram of parameter estimator in discrete time where the signals into the parameter estimator are filtered to emphasize selected frequency ranges. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a block diagram of one type of scanning probe force microscope (SPM) system, e.g. a scanning sample design, where the sample is moved and the cantilever base is kept stationary. Other designs may move the cantilever in addition to or in place of moving the sample. There are three axes of motion for a typical SPM system. Any one of these may have a feedback loop controlling it. In  FIG. 1 , the X and Y axes are open loop and the Z axis is closed-loop. Furthermore, all three axes are moved by a 3 degree of freedom piezo actuator. It is important to know that other actuation methods are possible, such as having a MEMS actuator in the Z direction and having closed-loop control on the X and/or Y axes. 
     While the invention will be described with respect to Z axis control, the concepts can be easily extended to any of the axes of motion of a SPM. 
       FIG. 2  illustrates a block diagram according to the invention. A scanning probe microscope  10  is controlled by a controller block  12 . A controller design block  14  adjusts the operation of the controller block  12 . 
     Within the controller block  12 , a summer  20  receives a reference signal and a tip deflection signal. The output of the summer  20  is received by a PID controller  22  which generates a controller output signal. The PID controller  22  further receives an  input from the controller design block  14 . Filtering may be added to mitigate the effects of system dynamics, e.g. higher frequency resonances of the cantilever. 
     The controller block  12  receives a reference signal from the microscope  10  and design parameters from the controller design block  14 . The controller forms an error signal and uses a PID design to generate a controller output that is used to control the microscope  10 . 
     The controller design block  14  includes a notch filter model  16  and a system identifier  18 . The system identifier  18  receives signals from the controller block  12  and the microscope  10 . The output of the system identifier  18  are parameters corresponding to the actuator resonance. These parameters are used to design a notch filter by the notch filter model  16 . The notch filter model  16  generates a set of PID parameters that are transmitted to the controller block  12 . 
       FIG. 3  illustrates a process flow chart according to the invention. In step  102 , a model for the scanning probe microscope dynamics is generated. In step  104 , filter parameters that shape selected dynamics of the model are chosen. In step  106 , a notch filter is generated using the filter parameters. In step  108 , the notch filter is encoded as PID parameters. In step  110 , the PID parameters are implemented in a PID controller to control the scanning probe microscope. 
     For step  106 , one method would be to select the gain of the notch filter so that the overall magnitude of controller response is at a desired value for a given frequency. Alternatively, the gain of the notch filter can be selected so that the open loop magnitude of the combined controller and actuator response is unity at a desired frequency. 
     The model for the scanning probe microscope dynamics may be a continuous-time, a sampled-data, or a discrete-time model. These models can often be well represented as a filter, although they represent physical devices. The controller models can also be represented as filters. The term filter can be used to represent a model of a physical system, a controller, or an implementation of a controller model. In a continuous-time model, the signals are continuously fed into the filter elements, which in turn continuously process them. Such filters are commonly generated using analog circuitry. In a discrete-time model, the signals are measured at discrete  sample points and converted to digital signals through the use of mixed signal circuits such as an analog to digital converter (ADC). The processing of these signals is also done using digital circuitry, such as a microprocessor, a digital signal processor (DSP), or a field programmable gate array (FPGA). Some of the processed signals may then be sent back to the physical system via digital to analog converters (DACs). A sampled data model is more general than a discrete-time model in that it may include systems where one or more of the signals are sampled, but the processing may be done using analog circuitry. These definitions are well known to those skilled in the art. For the purposes of this disclosure, the models to be discussed will be either continuous-time models or discrete-time models. However, it will be clear to those skilled in the art that the invention can be applied to more general systems such as sampled-data systems or hybrid systems where the physical system contains continuous time and discrete time components. 
     It is well known to those skilled in the art that controllers may be designed using continuous-time models and implemented discretely using so-called discrete equivalents. Alternately, controllers may be implemented directly in analog form, or they may be designed using discrete-time methods from discrete-time models of the system in question. 
     The continuous-time model may be generated in a multitude of ways, two of which can be seen from  FIG. 4 . In step  112 A, the model is generated by measuring an open-loop frequency response measurement of the scanning probe microscope. Alternatively, in step  112 B, the model is generated by measuring a closed-loop frequency response of the scanning probe microscope. In step  114 , a corresponding actuator response of the scanning probe microscope dynamics is extracted. In step  115 , the actuator model is extracted. 
     For a closed-loop measurement, the actuator response may be measured using a 3-wire measurement (illustratively shown in  FIGS. 5A and 5B ). In a 3-wire measurement, a signal is injected into the loop at a convenient location, e.g. r in  FIG. 6 , and signals from the controller output and actuator, e.g. u c  and y in  FIG. 6  are measured. If y is not available, it can be reconstructed from e if we know the value of r. Alternately, the closed-loop response of the system (illustratively shown  in  FIGS. 7A and 7B ) can be measured, e.g. 2-wire measurement from r 1  to y in  FIG. 6 , and this can be unwrapped to reveal the open-loop response (illustratively shown in  FIGS. 8A and 8B ). A frequency response measurement of the compensator (illustratively shown in  FIGS. 9A and 9B ), e.g. from e to u C  in  FIG. 6 , can be divided out to reveal the actuator frequency response function (illustratively shown in  FIGS. 10A and 10B ). 
     The frequency response functions may be computed in several ways. The system can be stimulated with white or colored noise or with a chirped sine signal. From any of these, the frequency responses can be calculated using DFT or FFT based methods. Alternately, the system may be stimulated using a method known as swept-sine or sine-dwell. A sinusoidal signal at a single frequency is injected into the system for an extended time. Once the system has reached steady state, various system outputs are measured. These measured outputs are mixed with a related sinusoidal signal, e.g. input signal, using both in-phase (0 degrees phase) and quadrature (+/−90 degrees phase) signals. These signals are then integrated to yield demodulated signals from which the complex response at the input frequency is obtained (and from which the magnitude and phase can be extracted). By doing this at a desired set of frequencies, a frequency response function (FRF) can be extracted. FRFs generated using swept sine methods typically have better signal to noise ratios (SNR) than FFT based FRFs. 
     After the frequency response function (FRF) is generated for the frequencies of interest, a parametric model of the actuator is generated. This can be done using curve fitting. A general curve fit model may be used. Alternately, restricting the order of the curve fit model to a second order system may be used when since the actuator&#39;s linear response is second order. The models may be obtained from the full FRF or from the magnitude response alone. This is illustratively shown in  FIGS. 11A and 11B . 
     The discrete-time model can be generated as shown in  FIG. 12 . In step  116 , the response model with physical parameters is discretized. In step  118 , the system response is measured. In step  120 , the measurements are fit to the discrete time model from step  116 . This results in a set of values for the physical parameters from step  116 , which are used in step  122  to generate the notch filter and the PID controller  parameters. Steps  118  and  120  may be accomplished in the time or frequency domain. 
     The notch filter and PID parameters can be directly generated from either the continuous-time or discrete-time model. The filter parameters correspond to a notch filter with a gain K, center frequency ω 0  and quality factor Q C . The resulting PID controller enables the system to be controlled at frequencies beyond the main resonant frequency (shown in  FIGS. 13A and 13B ). The projected closed-loop frequency response function is shown in  FIGS. 14A and 14B . 
     The resulting PID controller may be implemented using either analog or digital circuitry. 
     Next, while generating the model for the inventive concept will be illustrated for a continuous-time model (as in  FIG. 6 ) and for a discrete-time model (as  FIG. 15 ) for PID parameter generation, it will be apparent to one skilled in the art that the methods may be combined. 
     Continuous-time Model Using Frequency Response 
     The SPM model is generated (step  102 ) by measuring the frequency response function of the SPM and extracting the frequency response function of the actuator. The frequency response function of the actuator is fit to a second order transfer function model. Next, the gain, resonant frequency, and quality factor (K, w 0 , Q) of the resonance from the second order fit are derived. This is done by matching terms, e.g. 
                           H   ⁡     (   s   )       =       ⁢     1         A   o     ⁢     s   2       +       A   1     ⁢   s     +     A   2                     =       ⁢       1     A   0           s   2     +         A   1       A   0       ⁢   s     +       A   2       A   0                       =       ⁢         K   A     ⁢     ω   n   2           s   2     +         ω   n       Q   A       ⁢   s     +     ω   n   2                       Eq   .           ⁢   2               
where the three transfer functions have terms that can be extracted from a curve fit or a direct on-line adaptation. The third term is in terms of resonant parameters. 
 
     
       
         
           
             
               
                 
                   
                     
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                       n 
                     
                     = 
                     
                       
                         
                           A 
                           2 
                         
                         
                           A 
                           0 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Q 
                       A 
                     
                     = 
                     
                       
                         1 
                         
                           A 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             A 
                             0 
                           
                           ⁢ 
                           
                             A 
                             2 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       K 
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                         A 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     Next, as in steps  104  and  106 , an inverse filter based on (K A , ω n , Q A ), e.g. a notch filter with a gain at K, center frequency ω 0  and quality factor αQ C . Typically Q C , is equal to αQ A , where 0&lt;α≦1. Often ω 0  is close to or equal to ω n . By picking the center frequency of the notch equal to the center frequency of the resonance, the notch dip is positioned at the maximum value of the resonance. α=1 corresponds to complete cancellation of the resonance (for an idealized notch filter with no poles) where 0&lt;α&lt;1 allows the notch to be broader and not be so sensitive to small changes in ω 0 . The analog actuator model may be optionally discretized to yield a digital transfer function model, e.g. P 1 (z), P 2 (z), or P 3 (z). 
     Next, as in step  108 , the linear notch filter is mapped into the PID gains: K P , K I , and K D . An idealized PID (without derivative filtering) is described as: 
                           C   ⁡     (   s   )       =       ⁢       U   ⁡     (   s   )         E   ⁡     (   s   )                     =       ⁢       K   P     +       K   I     Ts     +       K   D     ⁢   Ts                   =       ⁢           K   D     ⁢   T     s     ⁡     [       s   2     +         K   P       K   D       ⁢     s   T       +       K   I         K   D     ⁢     T   2           ]                   =       ⁢       K       ω   0   2     ⁢   s       ⁡     [       s   2     +         ω   0       Q   C       ⁢   s     +     ω   0   2       ]                     Eq   .           ⁢   4               
In this equation, for simplification T=T I =T D  although similar results can be obtained without this simplification.
 
From the above,
 
     
       
         
           
             
               
                 
                   
                     
                       K 
                       D 
                     
                     = 
                     
                       K 
                       
                         
                           ω 
                           0 
                           2 
                         
                         ⁢ 
                         T 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       K 
                       I 
                     
                     = 
                     KT 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       K 
                       P 
                     
                     = 
                     
                       K 
                       
                         
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                           0 
                         
                         ⁢ 
                         
                           Q 
                           C 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
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                   5 
                 
               
             
           
         
       
     
     A PID with a first order derivative filter is described as:  
                             C   ⁡     (   s   )       =       ⁢       U   ⁡     (   s   )         E   ⁡     (   s   )                     =       ⁢       K   P     +       K   I     Ts     +       K   D     ⁢     Ts     Ts   +     a   1                     ⁢     
     ⁢             C   ⁡     (   s   )       =       ⁢         (       K   P     +     K   D       )     ⁡     [       s   2     +             K   P     ⁢     a   1       +     K   I         (       K   P     +     K   D       )       ⁢     s   T       +         K   I     ⁢     a   1           (       K   P     +     K   D       )     ⁢     T   2           ]         s   ⁡     (     s   +       a   1     T       )                     =       ⁢       K   ⁡     [       s   2     +         ω   0       Q   C       ⁢   s     +     ω   0   2       ]           ω   0   2     ⁢     s   ⁡     (     s   +       a   1     T       )                           Eq   .           ⁢   6               
Those skilled in the art will recognize that higher order filtering can be used, either on the derivative term alone or on the entire controller.
 
From the above,
 
     
       
         
           
             
               
                 
                   
                     
                       
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                     = 
                     
                       K 
                       
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                         2 
                       
                       
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                         1 
                       
                     
                   
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                       P 
                     
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                         ( 
                         
                           
                             1 
                             
                               
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                   7 
                 
               
             
           
         
       
     
     Next, as in step  110 , the PID gains are mapped into a practical implementation. Practical PID loops consider the effects of high frequency noise on the signals and thus include additional filtering of high frequency signals. The design considerations include integrator wind-up for which anti-windup methods are known to those skilled in the art. 
     Discrete Model 
     In one discrete model embodiment, as in step  110 , the digital frequency response of the system is measured. As in step  112 , the digital frequency response of the motor is extracted. As in step  114 , the model parameters are fit to yield a digital transfer function model of the motor, e.g. P 1 (z), P 2 (z), or P 3 (z). 
     In another discrete model embodiment (step  102 ), the response may be characterized as 
                     P   ⁡     (   s   )       =             K   A     ⁢     ω   o   2           s   2     +     2   ⁢     ζ   A     ⁢     ω   o     ⁢   s     +     ω   o   2         ⁢           ⁢   where   ⁢           ⁢     ζ   A       =     1     2   ⁢     Q   A                   Eq   .           ⁢   8               
The poles of the filter are at 
 
 s=ω   o (−ζ A ±√{square root over (ζ A   2 −1)})  Eq. 9
     If ζ A &lt;1 then the poles are a complex pair. If ζ A =−1 then, the poles are real and identical. If ζ A &gt;1, then the poles are real and distinct. For a resonant structure ζ A &lt;1. For ζ A &lt;1, s=ω o (−ζ A ±j√{square root over (1−ζ A   2 )}), where j is √{square root over (−1)}. For pole zero mapping, the poles at s p  are mapped to z p =e s     P     τ . Finite zeroes at s z  are mapped to z z =e s     z     τ . Zeroes at s=∞ are mapped to z ∞ =−1. The gain of the digital model is chosen to match the gain of the analog model at a critical frequency. Often, this is the DC gain, but it could be the resonant frequency.   

     Using pole zero mapping 
                       P   1     ⁡     (   z   )       =         K   1     ⁢         ω   o   2     ⁡     (     z   +   1     )       2           z   2     -     2   ⁢       ⅇ       -     ω   o       ⁢     T   s     ⁢     ζ   A         ⁡     (     cos   ⁡     (       ω   o     ⁢     T   s     ⁢       1   -     ζ   A   2           )       )       ⁢   z     +     ⅇ       -   2     ⁢     ζ   A     ⁢     ω   o     ⁢     T   s                     Eq   .           ⁢   10               
Alternately, one zero may be positioned at z=−∞ and one at z=−1,
 
                       P   2     ⁡     (   z   )       =         K   2     ⁢       ω   o   2     ⁡     (     z   +   1     )             z   2     -     2   ⁢       ⅇ       -     ω   o       ⁢     T   s     ⁢     ζ   A         ⁡     (     cos   ⁡     (       ω   o     ⁢     T   s     ⁢       1   -     ζ   A   2           )       )       ⁢   z     +     ⅇ       -   2     ⁢     ζ   A     ⁢     ω   o     ⁢     T   s                     Eq   .           ⁢   11               
The DC gain of the analog model may be matched by P(0)=K, by setting
 
                                 ⁢       K   A     =       ⁢       P   1     ⁡     (   1   )                     =       ⁢         K   1     ⁢         ω   o   2     ⁡     (   2   )       2         1   -     2   ⁢       ⅇ       -     ω   o       ⁢   T   ⁢           ⁢     ζ   A         ⁡     (     cos   ⁡     (       ω   o     ⁢     T   s     ⁢       1   -     ζ   A   2           )       )         +     ⅇ       -   2     ⁢     ζ   A     ⁢     ω   o     ⁢     T   s                           Eq   .           ⁢   12                             ⁢       K   A     =       ⁢       P   2     ⁡     (   1   )                     =       ⁢         K   2     ⁢       ω   o   2     ⁡     (   2   )           1   -     2   ⁢       ⅇ       -     ω   o       ⁢     T   s     ⁢     ζ   A         ⁡     (     cos   ⁡     (       ω   o     ⁢     T   s     ⁢       1   -     ζ   A   2           )       )         +     ⅇ       -   2     ⁢     ζ   o     ⁢     ω   A     ⁢     T   s                           Eq   .           ⁢   13                       So   ⁢           ⁢     K   1       =       ⁢       K   A     (       1   -     2   ⁢       ⅇ       -     ω   o       ⁢     T   s     ⁢     ζ   A         ⁡     (     cos   ⁡     (       ω   o     ⁢     T   s     ⁢       1   -     ζ   A   2           )       )       ⁢   z     +     ⅇ       -   2     ⁢     ζ   A     ⁢     ω   o     ⁢     T   s             4   ⁢     ω   o   2         )                 =       ⁢       K   2     2                   Eq   .           ⁢   14               
Another embodiment to match the measured response is to allow some other zero in the discrete model. 
 
                         P   3     ⁡     (   z   )       =         K   3     ⁢       ω   o   2     ⁡     (     z   +   1     )       ⁢     (     z   -   b     )           z   2     -     2   ⁢       ⅇ       -     ω   o       ⁢     T   s     ⁢     ζ   A         ⁡     (     cos   ⁡     (       ω   o     ⁢     T   s     ⁢       1   -     ζ   A   2           )       )       ⁢   z     +     ⅇ       -   2     ⁢     ζ   A     ⁢     ω   o     ⁢     T   s               ,     
     ⁢           ⁢       where   ⁢           -   1     &lt;   b   &lt;   1             Eq   .           ⁢   15                         P   3     ⁡     (   1   )       =       ⁢       K   A     ⇒     K   3                   =       ⁢       K   A     (       1   -     2   ⁢       ⅇ       -     ω   o       ⁢     T   s     ⁢     ζ   A         ⁡     (     cos   ⁡     (       ω   o     ⁢     T   s     ⁢       1   -     ζ   A   2           )       )         +     ⅇ       -   2     ⁢     ζ   A     ⁢     ω   o     ⁢     T   s             2   ⁢     (     1   -   b     )         )                   Eq   .           ⁢   16               
The motor response can be measured in one of several ways. Other variations include measuring the analog frequency response of the system, extracting the analog frequency response of the motor.
 
     In another discrete model embodiment, as in step  110 , the samples of the inputs and outputs of the motor are measured in discrete time, e.g. {u(k), u(k−1), . . . , u(k−n)}, where {y(0), y(1), . . . , y(k−n)} are related by 
                       Y   ⁡     (   z   )         U   ⁡     (   z   )         =       ⅇ     -   jΔτ       ⁢         b   o     +       b   1     ⁢     z     -   1         +       b   2     ⁢     z     -   2             1   +       a   1     ⁢     z     -   1         +       a   2     ⁢     z     -   2                       Eq   .           ⁢   17               
where Δτ is time delay.
 
     Those skilled in the art will also recognize that there are many design methodologies that can generate the response of a notch filter. These methods include—but are not limited to—state-space methods and optimization methods such as H 2  and H ∞  design. 
     In another illustrative example, the model is generated using Discrete-Time, Time Domain Identification of SPM Dynamics. The discrete time filter parameters are generated from time response measurements by running a second order model of the actuator in parallel with the measured system. The inputs and the output of the system and the model can be compared. The model adjusted to minimize some cost criterion of the error, e.g. on-line adaptation. 
     Digital filters can be represented as transfer functions in the z transform operator, z: 
                       Y   ⁡     (   z   )         U   ⁡     (   z   )         =           b   0     ⁢     z   n       +       b   1     ⁢     z     n   -   1         +       b   2     ⁢     z     n   -   2         +   …   +     b   n           z   n     +       a   1     ⁢     z     n   -   1         +       a   2     ⁢     z     n   -   2         +   …   +     a   n                 Eq   .           ⁢   18               
or equivalently in the unit delay operator, z −1 : 
 
                       Y   ⁡     (   z   )         U   ⁡     (   z   )         =         b   0     +       b   1     ⁢     z     -   1         +       b   2     ⁢     z     -   2         +   …   +       b   n     ⁢     z     -   n             1   +       a   1     ⁢     z     -   1         +       a   2     ⁢     z     -   2         +   …   +       a   n     ⁢     z     -   n                       Eq   .           ⁢   1     ⁢   9               
It is worth noting that the transfer functions in Equations 18 and 19 are not unique. One could easily multiply the numerator and denominator by the same numbers to yield equivalent transfer functions. However, this representation has an advantage in that the coefficient of the output term. y(k), in the equations below is 1.
 
     This gets implemented in a filter as:
 
 y ( k )=− a   1   y ( k− 1)− a   2   y ( k− 2)− . . . − a   n   y ( k−n )+ b   0   u ( k )+ b   1   u ( k− 1)+ . . . + b   n   u ( k−n )  Eq. 20
 
Alternately, we can use a direct form filter which reduces storage requirements. (There was an extra figure for this if you want to use it.)
 
 d ( k )=− a   1   d ( k− 1)− a   2   d ( k− 2)− . . . − a   n   d ( k−n )+ u ( k )  Eq. 21
 
 y ( k )= b   0   d ( k )+ b   1   d ( k− 1)+ . . . + b   n   d ( k−n )
 
When digital filters are implemented in an adaptive scheme, the Z transform is no longer applicable as the coefficients are varying. However, the unit delay operator is still valid. To avoid confusion, it is common for those skilled in the area to replace the unit delay operator z −1  with the equivalent unit delay operator, q −1 . Since the latter is not associated with the Z transform (which is not valid when the coefficients are varying) confusion is avoided.
 
     If the system model is unknown then an estimate can be generated using:
 
ŷ( k )=−â 1   y ( k− 1)−â 2   y ( k− 2)− . . . −â n   y ( k−n )+{circumflex over (b)} 0   u ( k )+{circumflex over (b)} 1   u ( k− 1)+ . . . +{circumflex over (b)} n   u ( k−n )  Eq. 21
 
For simplicity, the noise free case will be discussed here, but those skilled in the art will recognize that equivalent results are available when the system has noise.
 
     If one compares the measured output to the output of the system model one gets: 
 
ε( k )= y ( k )−ŷ( k )  Eq. 23
 
ε( k )= y ( k )−[â 1   y ( k− 1)−â 2   y ( k− 2)− . . . −â n   y ( k−n )+{circumflex over (b)} 0   u ( k )+{circumflex over (b)} 1   u ( k− 1)+ . . . +{circumflex over (b)} n   u ( k−n )]  Eq, 22
 
This can be rewritten as:
 
ε( k )= y ( k )−φ T ( k ){circumflex over (θ)}( k )  Eq. 25
 
Where
 
φ T ( k )=[− y ( k− 1),− y ( k− 2), . . . , − y ( k−n )., u ( k ), u ( k− 1), . . . ,  u ( k−n )] and
 
{circumflex over (θ)} T ( k )=[â 1 ,â 2 , . . . , â n , {circumflex over (b)} 0 , {circumflex over (b)} 1 , . . . , {circumflex over (b)} n )].
 
The parameters in {circumflex over (θ)} T (k)=[â 1 , â 2 , . . . , â n , {circumflex over (b)} 0 , {circumflex over (b)} 1 , . . . +{circumflex over (b)} n )] can be adjusted through a variety of schemes with the goal of having them converge to the true model of the system, {circumflex over (θ)} T (k)=[â 1 , â 2 , . . . , â n , {circumflex over (b)} 0 , {circumflex over (b)} 1 , . . . +{circumflex over (b)} n )]. Commonly used algorithms well known to those skilled in the art include methods that minimize the mean squared error,  e 2 (k) , such as Least Mean Squares (LMS), Recursive Least Squares (RLS), or algorithms that are modifications of these (such as Filtered-X LMS).
 
     In  FIG. 15 , a discrete-time model of the system is shown. In  FIG. 16 , the pieces of the block diagram that need to be replicated for doing system identification are shown. What is important to note, is that the identification block diagram of  FIG. 16  is fed by all the available inputs and outputs of the system. If the system is run in closed-loop, then the estimation algorithm is fed the closed-loop quantities. If it is run in open loop, then the estimation algorithm is fed the open loop quantities. 
     Depending upon the system configuration, different input and output signals will be available. However, the goal of the estimation algorithm is to match {circumflex over (P)} to P, or equivalently {circumflex over (θ)} to θ. It will be known to those skilled in the art, that the signals entering the parameter estimator may be filtered, as shown in  FIG. 17 , so as to emphasize certain frequencies and de-emphasize others. 
     We can use information about the SPM actuator to simplify this procedure. For example, in one embodiment the identification is run while the system is in open-loop, with r 1 (k) and u C (k) are both 0. Thus, the signals to be measured are u(k) and  e(k) (or their filtered versions), from which y(k) can be derived. In this case the system and the model are both stimulated with u(k)=r 2  (k) and the model output is compared to the measured output. In another embodiment, the system is run in closed-loop and the dynamics may be stimulated from a variety of signals such as r 1 (k) or r 2 (k). 
     For generating the discrete-time model of the main dynamics of a SPM actuator, we can restrict our model to be a second order discrete-time model. That is, the n would be 2 in Equations 18-22 and 24. Thus Equations 19 and 20 become 
                       Y   ⁡     (   z   )         U   ⁡     (   z   )         =         b   0     +       b   1     ⁢     z     -   1         +       b   2     ⁢     z     -   2             1   +       a   1     ⁢     z     -   1         +       a   2     ⁢     z     -   2                     Eq   .           ⁢   23               
and
 
 y ( k )=− a   1   y ( k− 1)− a   2   y ( k− 2)+ b   0   u ( k )+ b   1   u ( k− 1)+ b   2   u ( k− 2)  Eq. 24
 
respectively.
 
Furthermore as r 1 (k) enters into the calculation for both y(k) and ŷ(k), e(k) and ê(k) can be used in the generation of  ε 2  (k) .
 
     Next, the parameters are matched to the Discrete-Time Resonance Model. With the second order discrete-time model identified, the parameters in Equation 210 can be matched against the discrete resonances, P 1 (z), P 2 (z), or P 3 (z). PID parameters can then be matched to the extracted parameters, ω 0 , ζ A , and K A  as described earlier.