Patent Publication Number: US-10317856-B2

Title: Control system with response time estimation

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATION 
     This application is a continuation of U.S. patent application Ser. No. 13/794,683 filed Mar. 11, 2013, now U.S. Pat. No. 9,395,708, the entire disclosure of which is incorporated by reference herein. 
    
    
     BACKGROUND 
     The present disclosure relates generally to the field of feedback controllers. The present disclosure relates more specifically to systems and methods for determining an appropriate sampling rate for a feedback controller. 
     Feedback controllers are used to control a wide variety of systems and processes. Typically, a feedback controller receives a measured value of a controlled variable (e.g., a feedback signal) and adjusts an input provided to a control device based on the measured value. The object of feedback controllers is to adjust the input provided to the device in a way that maintains a controlled variable at a desired setpoint. 
     Many feedback controllers respond to a feedback signal based on one or more control parameters. One control parameter frequently used in feedback control processes is a proportional gain (i.e., the proportional term, the gain, etc.). Feedback controllers typically apply the proportional gain as a multiplier to an error signal (e.g., a difference between a setpoint and a feedback signal) in determining an input to provide to the controlled system or process. In addition to the proportional gain, feedback controllers can use other control parameters such as an integral term (e.g., in a proportional-integral (PI) controller) and/or a derivative term (e.g., in a proportional-integral-derivative (PID) controller, etc.). 
     For dynamic systems in which conditions outside of the control loop affect the controlled variable or where an aspect of the control loop is variably imperfect, the optimal control parameters may also be dynamic. Accordingly, some feedback controllers automatically adjust the control parameters (e.g., the controller is “tuned”) based on observed behavior of the system. Some feedback controllers include adaptive tuning algorithms that automatically adjust the control parameters during normal operation. Such adaptive tuning algorithms can provide for improved performance relative to other tuning strategies. 
     The rate at which measurements are collected from the controlled system or process (e.g., the sampling rate) can affect the operation of an adaptive tuning algorithm. If the sampling rate is too fast, the feedback controller may tune improperly and the proportional gain may be too small. If the sampling rate is too slow, the performance of the feedback controller may suffer. It is often challenging to determine an appropriate sampling rate for a feedback control system. 
     SUMMARY 
     One implementation of the present disclosure relates to a method for adaptively and automatically adjusting a sampling rate in a feedback control system. The method includes monitoring an error signal for a disturbance event, evaluating the error signal in response to the disturbance event to estimate a time constant of a control process, and determining a sampling rate for use in the feedback control system based on the estimated time constant. The error signal may be based on a difference between a setpoint and a feedback signal and the feedback signal may be received from the control process. 
     In some embodiments, monitoring the error signal for the disturbance event includes identifying the disturbance event as at least one of a setpoint change and a load disturbance. If the disturbance event is identified as a setpoint change, evaluating the error signal in response to the disturbance event may include tracking an area bounded by the error signal in response to the setpoint change and dividing the tracked area by a magnitude of the setpoint change to determine the estimated time constant of the control process. If the disturbance event is identified as a load disturbance, evaluating the error signal in response to the disturbance event may include tracking a magnitude of the error signal in response to the load disturbance, determining a time at which the magnitude of the error signal reaches a maximum in response to the load disturbance, and subtracting a time at which the magnitude of the error signal begins to increase in response to the load disturbance from the time at which the magnitude of the error signal reaches the maximum to determine the estimated time constant of the control process. 
     In some embodiments, determining a sampling rate for use in the feedback control system includes selecting a sampling interval which is a fraction of the estimated time constant. In more specific embodiments, determining a sampling rate for use in the feedback control system includes selecting a sampling interval between one-fortieth of the estimated time constant and the estimated time constant. 
     In some embodiments, the method further includes comparing the determined sampling rate with a previous sampling rate used by the feedback control system and updating the previous sampling rate to the determined sampling rate if the comparison reveals that one or more updating criteria are met. The updating criteria may include updating the previous sampling rate to the determined sampling rate if the determined sampling rate is at least twice the previous sampling rate or no greater than half the previous sampling rate. 
     Another implementation of the present disclosure is a method for estimating a time constant of a process under closed loop feedback control. The method includes monitoring an error signal for a disturbance event, determining whether the disturbance event qualifies as a setpoint change or a load disturbance, and evaluating the error signal in response to the determination of a setpoint change or a load disturbance to estimate a time constant of a control process. The error signal may be based on a difference between a setpoint and a feedback signal and the feedback signal may be received from the control process. 
     Evaluating the error signal in response to a setpoint change may include tracking an area bounded by the error signal in response to the setpoint change and dividing the tracked area by a magnitude of the setpoint change to determine the estimated time constant of the control process. Evaluating the error signal in response to a load disturbance may include tracking a magnitude of the error signal in response to the load disturbance, determining a time at which the magnitude of the error signal reaches a maximum in response to the load disturbance, and subtracting a time at which the magnitude of the error signal begins to increase in response to the load disturbance from the time at which the magnitude of the error signal reaches the maximum to determine the estimated time constant of the control process. 
     Another implementation of the present disclosure is a feedback control system for operating a control process. The feedback control system includes a process controller having a processing circuit configured to monitor an error signal for a disturbance event, evaluate the error signal in response to the disturbance event to estimate a time constant of a control process, and determine a sampling rate for use in the feedback control system based on the estimated time constant. The error signal may be based on a difference between a setpoint and a feedback signal and the feedback signal may be received from the control process. 
     In some embodiments, monitoring the error signal for the disturbance event includes identifying the disturbance event as at least one of a setpoint change and a load disturbance. If the disturbance event is identified as a setpoint change, evaluating the error signal in response to the disturbance event may include tracking an area bounded by the error signal in response to the setpoint change and dividing the tracked area by a magnitude of the setpoint change to determine the estimated time constant of the control process. If the disturbance event is identified as a load disturbance, evaluating the error signal in response to the disturbance event may include tracking a magnitude of the error signal in response to the load disturbance, determining a time at which the magnitude of the error signal reaches a maximum in response to the load disturbance, and subtracting a time at which the magnitude of the error signal begins to increase in response to the load disturbance from the time at which the magnitude of the error signal reaches the maximum to determine the estimated time constant of the control process. 
     In some embodiments, determining a sampling rate for use in the feedback control system includes selecting a sampling interval no greater than the estimated time constant. In more specific embodiments, determining a sampling rate for use in the feedback control system includes selecting a sampling interval approximately one-tenth of the estimated time constant. 
     In some embodiments, the processing circuit is further configured to compare the determined sampling rate with a previous sampling rate used by the feedback control system and update the previous sampling rate to the determined sampling rate if the comparison reveals that one or more updating criteria are met. The updating criteria may include updating the previous sampling rate to the determined sampling rate if the determined sampling rate is at least twice the previous sampling rate or no greater than half the previous sampling rate. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a closed-loop control system including a proportional integral (PI) controller and a plant, illustrating the application of a setpoint r and a load disturbance d to the closed-loop system, according to an exemplary embodiment. 
         FIG. 2  is a block diagram of an adaptive feedback control system including an adaptive feedback controller, a time constant estimator configured to estimate a time constant τ p  for the plant based on an error signal e, and a sampling rate adjustor configured to determine a sampling rate h for the adaptive feedback controller based on the estimated time constant τ p , according to an exemplary embodiment. 
         FIG. 3  is a graph of the error signal e used by the adaptive feedback control system of  FIG. 2  in response to a step change in the setpoint r, according to an exemplary embodiment. 
         FIG. 4  is a graph of the error signal e used by the adaptive feedback control system of  FIG. 2  in response to a step change in the load disturbance d, according to an exemplary embodiment. 
         FIG. 5A  is a graph of a feedback signal obtained from a controlled process as a function of time for several closed-loop control systems, illustrating the expected performance advantages of the adaptive feedback control system of  FIG. 2 , according to an exemplary embodiment. 
         FIG. 5B  is a graph of the sampling rate as a function of time as may be determined and used by the sampling rate adjustor of  FIG. 2 , according to an exemplary embodiment. 
         FIG. 5C  is a graph of the controller gain parameter over time as may be determined and used by the adaptive feedback controller of  FIG. 2  having an adaptively updated sampling rate, as well as the controller gain parameter determined by an adaptive feedback controller with a fixed sampling rate, illustrating the expected performance advantage of adaptively updating the sampling rate, according to an exemplary embodiment. 
         FIG. 5D  is a graph of the integral time parameter over time as may be determined and used by the adaptive feedback controller of  FIG. 2  having an adaptively updated sampling rate, as well as the integral time parameter determined by an adaptive feedback controller with a fixed sampling rate, illustrating a performance advantage of adaptively updating the sampling rate, according to an exemplary embodiment. 
         FIG. 6  is a detailed block diagram of the process controller of  FIG. 2 , according to an exemplary embodiment. 
         FIG. 7  is a flowchart of a process for adaptively adjusting a sampling rate in a feedback control system, according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Before turning to the figures, which illustrate the exemplary embodiments in detail, it should be understood that the disclosure is not limited to the details or methodology set forth in the description or illustrated in the figures. It should also be understood that the terminology is for the purpose of description only and should not be regarded as limited. 
     Referring generally to the figures, systems and methods for adaptively adjusting a sampling rate in a feedback control system are shown, according to various exemplary embodiments. The systems and methods described herein automatically estimate a time constant for a controlled process or system and automatically set the sampling rate based on the estimated time constant. The time constant is estimated by analyzing an error signal (e.g., a difference between a setpoint and a feedback signal received from the controlled system). 
     Estimating the time constant includes determining whether the system is subject to a setpoint change or a load disturbance. If the system is subject to a setpoint change, the time constant is estimated by integrating the error signal (e.g., numerically, analytically, etc.) to determine an area under the error curve. The area under the error curve may be divided by the magnitude of the setpoint change to determine the estimated time constant. If the system is subject to a load disturbance, the time constant may be estimated by determining a time at which the error signal reaches an extremum (e.g., a minimum or a maximum) in response to the load disturbance. The time value at which the load disturbance begins may be subtracted from the time value at which the error signal reaches the extremum to determine the estimated time constant. The estimated time constant may be used to automatically and adaptively set the sampling rate for the feedback control system, without manual observation or adjustment. 
     The systems and methods described herein may be incorporated into an existing feedback controller (e.g., a proportional-integral (PI) controller, a proportional-integral-derivative (PID) controller, a pattern recognition adaptive controller (PRAC), etc.) or supplement an existing feedback control system. Advantageously, the adaptively determined sampling rate may improve the adaptive feedback controller&#39;s performance in determining optimal control parameters (e.g., a proportional gain, an integral time, etc.) for the controlled process or system. 
     Referring now to  FIG. 1 , a block diagram of a closed-loop control system  100  is shown, according to an exemplary embodiment. System  100  may be a building management system or part of a building management system (e.g., a HVAC control system, a lighting control system, a power control system, a security system, etc.). System  100  may be a local or distributed control system used to control a single building, a system of buildings, one or more zones within a building. In some implementations, system  100  may be a METASYS® brand control system as sold by Johnson Controls, Inc. System  100  is shown to include a PI controller  102 , a plant  104 , a subtractor element  106 , and a summation element  108 . 
     Plant  104  may be a system or process monitored and controlled by closed-loop system  100  (e.g., a control process). Plant  104  may be a dynamic system (e.g., a building, a system of buildings, a zone within a building, etc.) including one or more variable input devices (e.g., dampers, air handling units, chillers, boilers, actuators, motors, etc.) and one or more measurement devices (e.g., temperature sensors, pressure sensors, voltage sensors, flow rate sensors, humidity sensors, etc.). In some implementations, plant  104  may be a zone within a building (e.g., a room, a floor, an area, etc.) and control system  100  may be used to control temperature within the zone. For example, control system  100  may actively adjust a damper position in a HVAC unit (e.g., an air handling unit (AHU), a variable air volume (VAV) box, etc.) for increasing or decreasing the flow of conditioned air (e.g., heated, chilled, humidified, etc.) into the building zone. 
     Plant  104  may receive an input from summation element  108  which combines a control signal u with a disturbance signal d. In some embodiments, plant  104  may be modeled as a first-order plant having a transfer function 
                   G   p     ⁡     (   s   )       =         K   p       1   +       τ   p     ⁢   s         ⁢     e     -   Ls           ,         
where τ p  is the dominant time constant, L is the time delay, and K p  is the process gain. In other embodiments, plant  104  may be modeled as a second-order, third-order, or higher order plant. Plant  104  may produce a feedback signal y in response to control signal u and disturbance signal d. Feedback signal y may be subtracted from setpoint r at subtractor element  106  to produce an error signal e (e.g., e=r−y).
 
     PI controller  102  is shown receiving error signal e from subtractor element  106 . PI controller  102  may produce a control signal u in response to the error signal e. In some embodiments, controller  102  is a proportional-integral controller. PI controller  102  may have a transfer function 
                   G   c     ⁡     (   s   )       =         K   c     ⁡     (     1   +       T   i     ⁢   s       )           T   i     ⁢   s         ,         
where K c  is the controller gain and T i  is the integral time. Controller gain K c  and integral time T i  are the control parameters which define the response of PI controller  102  to error signal e. That is, controller gain K c  and integral time T i  control how PI controller  102  translates error signal e into control signal u. In some embodiments, K c  and T i  are the only control parameters. In other embodiments, different control parameters (e.g., a derivative control parameter, etc.) may be used in addition to or in place of control parameters K c  and T i .
 
     Still referring to  FIG. 1 , in the Laplace domain, the error signal e(s) may be expressed in terms of the setpoint r(s) as 
               e   ⁡     (   s   )       =       r   ⁡     (   s   )       ⁢       1     1   +         G   c     ⁡     (   s   )       ⁢       G   p     ⁡     (   s   )             .             
Error signal e(s) may be expressed in terms of the disturbance signal d(s) as
 
               e   ⁡     (   s   )       =       -     d   ⁡     (   s   )         ⁢       G   p     ⁡     (   s   )       ⁢       1     1   +         G   c     ⁡     (   s   )       ⁢       G   p     ⁡     (   s   )             .             
The common term in both expressions for the error signal may be rewritten as
 
     
       
         
           
             
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     This expression can be simplified by assuming that K p  K c ≈1 and that T i ≈τ p . The simplified closed-loop transfer function may then be expressed as 
     
       
         
           
             
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     Referring now to  FIG. 2 , a block diagram of a closed-loop system  200  is shown, according to an exemplary embodiment. System  200  is shown to include a process controller  201  having an adaptive feedback controller  210 , a time constant estimator  214 , and a sampling rate adjustor  216 . Adaptive feedback controller  210  may be a pattern recognition adaptive controller (PRAC), a model recognition adaptive controller (MRAC), or any other type of adaptive tuning or feedback controller. PRAC controllers are described in greater detail in U.S. Pat. Nos. 5,355,305, 5,506,768, and 6,937,909, as well as other resources. 
     Adaptive feedback controller  210  may include a proportional-integral (PI) controller, a proportional-derivative (PD) controller, a proportional-integral-derivative (PID) controller, or any other type of controller which generates a control signal in response to a feedback signal, an error signal, and/or a setpoint. Adaptive feedback controller  210  may be any type of feedback controller (e.g., PRAC, MRAC, PI, etc.) which adaptively adjusts one or more controller parameters (e.g., a proportional gain, an integral time, etc.) used to generate the control signal. Adaptive feedback controller  210  is shown to include a PI controller  202  and an adaptive tuner  212 . 
     PI controller  202  may be the same or similar to PI controller  102  described in reference to  FIG. 1 . For example, PI controller  202  may be a proportional-integral controller having a transfer function 
                 G   c     ⁡     (   s   )       =           K   c     ⁡     (     1   +       T   i     ⁢   s       )           T   i     ⁢   s       .           
PI controller  202  may receive an error signal e from subtractor element  206  and provide a control signal u to summation element  208 . Summation element  208  may combine control signal u with a disturbance signal d and provide the combined signal to plant  204 . Elements  206 ,  208 , and plant  204  may be the same or similar to elements  106 ,  108 , and plant  104  as described in reference to  FIG. 1 .
 
     Adaptive tuner  212  may periodically adjust (e.g., calibrate, tune, update, etc.) the control parameters used by PI controller  202  in translating error signal e into control signal u. The control parameters determined by adaptive tuner  212  may include a controller gain K c  and an integral time T i . Adaptive tuner  212  may receive control signal u from PI controller  202  and adaptively determine control parameters K c  and T i  based on control signal u (e.g., as described in the aforementioned U.S. patents). Adaptive tuner  212  provides the control parameters K c  and T i  to PI controller  202 . 
     Still referring to  FIG. 2 , system  200  is further shown to include a time constant estimator  214 . Time constant estimator  214  determines a dominant time constant τ p  for plant  204 . The estimated time constant τ p  is then used by sampling rate adjuster  216  to adaptively determine an appropriate sampling rate h for PI controller  202 . Time constant τ p  may also be used to predict the response of plant  204  to a given control signal u. Time constant estimator  214  is shown receiving setpoint r as well as error signal e. In other embodiments, time constant estimator  214  may receive only error signal e or may calculate error signal e based on setpoint r and feedback signal y (e.g., e=r−y). Time constant estimator  214  may determine the dominant time constant τ p  based on error signal e, setpoint r, and/or other inputs received from various components of control system  200 . In some embodiments, time constant estimator  214  may estimate a time constant based on control signal u (e.g., in a feed forward, model predictive control, and/or open loop control system). In some embodiments, control signal u may be used in place of or in addition to error signal e in estimating a time constant. 
     The process used to estimate time constant τ p  may depend on whether system  200  is subject to a setpoint change or a load disturbance. A setpoint change is an increase or decrease in setpoint r. A setpoint change may be instantaneous (e.g., a sudden change from a first setpoint value to a second setpoint value) or gradual (e.g., a ramp increase or decrease, etc.). A setpoint change may be initiated by a user (e.g., adjusting a temperature setting on a thermostat) or received from another controller or process (e.g., a supervisory controller, an outer loop cascaded controller, etc.). 
     If system  200  is subject to a setpoint change, time constant estimator  214  may estimate time constant τ p  by integrating the error signal e (e.g., numerically, analytically, etc.) to determine an area A under the error curve. Time constant estimator  214  may then divide the area under the error curve A by a magnitude of the setpoint change α to determine the estimated time constant 
     
       
         
           
             
               
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     A load disturbance is an uncontrolled input applied to plant  204 . For example, in a temperature control system for a building, the load disturbance may include heat transferred through the external walls of the building or through an open door (e.g., during a particularly hot or cold day). The load disturbance may be measured or unmeasured. In some embodiments, time constant estimator  214  receives a signal (e.g., a status indicator, a process output, etc.) from adaptive feedback controller  210  indicating whether system  200  is subject to a setpoint change or a load disturbance. In other embodiments, time constant estimator  214  determines whether a setpoint change or load disturbance has occurred by analyzing the error signal e and/or setpoint r. 
     If system  200  is subject to a load disturbance, time constant estimator  214  may estimate the time constant τ p  by determining a time t ex  at which the error signal e reaches an extremum (e.g., a minimum or a maximum) in response to the load disturbance. Time constant estimator  214  may subtract a time at which the load disturbance begins t d  from the time at which error signal e reaches an extremum t ex  in response to the load disturbance to determine the estimated time constant τ p  (e.g., r p =t ex −t d ). The systems and methods used to estimate the time constant τ p  in response to a setpoint change and load disturbance are described in greater detail in reference to  FIG. 3  and  FIG. 4  respectively. 
     In some embodiments, time constant estimator  214  determines time constant τ p  in response to an identified load disturbance or setpoint change. Advantageously, time constant estimator  214  may determine time constant τ p  in real time (e.g., immediately after sufficient data has been collected to perform the aforementioned calculations). For example, time constant estimator  214  may monitor error signal e for a sign change (e.g., positive to negative, negative to positive, zero crossings, etc.). Upon the occurrence of a load disturbance or setpoint change, the error signal e may experience a “zero crossing” and increase or decrease until reaching an extremum. PI controller  202  may attempt to reduce the magnitude of error signal e by manipulating control input u. Upon reaching a steady-state, error signal e may experience another “zero crossing” as the error signal e approaches and/or crosses zero. Time constant estimator  214  may use the zero crossings as time boundaries for calculating the area under error signal e or for calculating a time difference between t d  and t ex . 
     In other embodiments, time constant estimator may determine τ p  using historical disturbance data. In some embodiments, time constant estimator  214  is local to the controlled process or system for which the time constant τ p  is estimated. In other embodiments, time constant estimator may be implemented remotely (e.g., in the “cloud,” on a supervisory level, etc.). Time constant estimator  214  may be part of PI controller  202 , adaptive tuner  212 , or implemented separately from such components. 
     In some embodiments, time constant estimator  214  determines whether a disturbance (e.g., a load disturbance or setpoint change) exceeds a significance threshold before proceeding with the time constant estimation process. For example, time constant estimator  214  may compare a magnitude of the error signal e in response to the disturbance with a threshold value. The magnitude of the error signal e may be a maximum or minimum magnitude after the occurrence of a disturbance. In some embodiments, the threshold value is a noise threshold (e.g., an upper noise threshold, a lower noise threshold, a noise band, etc.). The threshold value may be pre-defined (e.g., retrieved from memory, specified by a user, etc.) or automatically determined based on steady-state measurements obtained from the controlled system or process. In other embodiments, the threshold value is an area threshold (e.g., an area under the error curve), a time threshold, or a combination thereof. For example, time constant estimator  214  may compare the area under the error curve in response to a disturbance with a threshold area value. If the integrated area exceeds the threshold value, time constant estimator  214  may determine that the disturbance is significant. Time constant estimator  214  may proceed with the time constant estimation process if a disturbance is determined to be significant. In some embodiments, time constant estimator  214  does not proceed to estimate the time constant τ p  if a disturbance is not determined to be significant. In other embodiments, time constant estimator  214  does not determine the significance of disturbances and/or estimates τ p  regardless of a determined significance. 
     Still referring to  FIG. 2 , system  200  is further shown to include a sampling rate adjustor  216 . Sampling rate adjustor  216  may receive the estimated time constant τ p  from time constant estimator  214  and calculate a sampling rate h based on the estimated time constant τ p . In some embodiments, sampling rate h defines a sampling interval or sampling frequency used by PI controller  202  to obtain feedback measurements from the controlled system or process (e.g., plant  204 ). In other embodiments, sampling rate h defines a rate at which the control signal u is adjusted by PI controller  202 . In further embodiments, sampling rate h defines a rate at which control parameters K c  and T i  are updated by adaptive tuner  212 . 
     In some embodiments, sampling rate adjustor  216  may set sampling rate h to a value between one-fiftieth of the estimated time constant τ p  and the estimated time constant 
                 τ   p     ⁡     (       e   .   g   .     ,         τ   p     50     ≤   h   ≤     τ   p         )       .         
In more specific embodiments, the sampling rate h may be set to a value between one-fortieth of the estimated time constant τ p  and the estimated time constant
 
                 τ   p     ⁡     (       e   .   g   .     ,         τ   p     40     ≤   h   ≤     τ   p         )       .         
In some embodiments, the sampling rate h may be set to a value between one-twentieth of the estimated time constant τ p  and one-third of the estimated time constant
 
                 τ   p     ⁡     (       e   .   g   .     ,         τ   p     20     ≤   h   ≤       τ   p     3         )       .         
In an exemplary embodiment, the sampling rate h may be set to a value approximately equal to one-tenth of the estimated time constant
 
                 τ   p     ⁡     (       e   .   g   .     ,     h   ≈       τ   p     10         )       .         
In some embodiments, sampling rate adjustor  216  may be combined with time constant estimator  214 , PI controller  202 , or adaptive tuner  212 .
 
     In some embodiments, sampling rate adjustor  216  compares the calculated sampling rate h with a current or previous sampling rate h 0 . If the calculated sampling rate h differs significantly from h 0 , the sampling rate may be updated to the recently calculated value h. In some embodiments, the current sampling rate h 0  is updated to the new sampling rate h if the new sampling rate h is greater than or equal to twice the current sampling rate (e.g., h≥2h 0 ). In other embodiments, the current sampling rate h 0  is updated to the new sampling rate h if the new sampling rate h is less than half the current sampling rate (e.g., h≤0.5h 0 ). In further embodiments, sampling rate adjustor  216  may update the current sampling rate h 0  if the difference between the current sampling rate h 0  and the calculated sampling rate h exceeds a difference threshold (e.g., |h 0 −h|&gt;threshold) or if the ratio between h 0  and h exceeds a ratio threshold 
               (       e   .   g   .     ,         h   0     h     &gt;   threshold       )     .         
The updated sampling rate h may be communicated to adaptive feedback controller  210 .
 
     Advantageously, a properly set sampling rate h (e.g., a sampling rate which is a function of the process time constant τ p ), may provide improved stability and control functionality for adaptive feedback controller  210 . Adaptively and automatically determining sampling rate h may eliminate the need for human intervention (e.g., a trial and error approach or a rough estimation of the proper sampling rate) to properly configure a wide variety of control systems. 
     Referring now to  FIG. 3 , a graph  300  illustrating the time response of error signal e to a change in setpoint r is shown, according to an exemplary embodiment. Graph  300  shows error signal e as a function of time in response to an step increase of magnitude α in setpoint r. In some embodiments, the output y of plant  204  is a continuous variable unable to instantaneously increase or decrease. Because the error signal e is defined as the difference between setpoint r and output y (e.g., e=r−y), the error signal e may suddenly increase from a first value (e.g., e=0) immediately before the setpoint change to a second value immediately after the setpoint change. The second value may equal the magnitude α of the setpoint change. PI controller  202  may respond to the increase in error signal e by adjusting the control signal u applied to plant  204 . Such adjustment may cause error signal e to continuously decrease from the magnitude α of the setpoint change to a steady state value (e.g., e=0). 
     Still referring to  FIG. 3 , time constant estimator  214  may estimate the time constant τ p  for plant  204  based on the response of error signal e to a step change in setpoint r. As was previously mentioned, in the Laplace domain the error signal e(s) may be expressed in terms of the setpoint r(s) as 
                 e   ⁡     (   s   )       =       r   ⁡     (   s   )       ⁢     1     1   +         G   c     ⁡     (   s   )       ⁢       G   p     ⁡     (   s   )                 ,     
     ⁢   where                 1     1   +         G   c     ⁡     (   s   )       ⁢       G   p     ⁡     (   s   )             =           τ   p     ⁢   s           τ   p     ⁢   s     +     e     -   Ls           .           
A step change of magnitude α in the setpoint produces the error signal
 
               e   ⁡     (   s   )       =         a   ⁢           ⁢     τ   p             τ   p     ⁢   s     +     e     -   Ls           .           
The area under the error curve in graph  300  may be determined by integrating the error signal e(s) (e.g.,
 
               e   ⁡     (   s   )       ⁢       (       e   .   g   .     ,       I   ⁡     (   s   )       =       1   s     ⁢     e   ⁡     (   s   )             )     .           
In a stable system, the area A under the curve can be expressed as the difference in the areas at t=∞ and t=0. That is,
 
             A   =           lim     s   →   0       ⁢     {     sI   ⁡     (   s   )       }       -       lim     s   →   ∞       ⁢     {     sI   ⁡     (   s   )       }         =     a   ⁢           ⁢       τ   p     .               
From this expression, it is apparent that the time constant τ p , depends only on the magnitude α of the setpoint change and the area A under the error curve.
 
     Time constant estimator  214  may track the area A under the error curve in response to a setpoint change and estimate the time constant τ p  , according to the equation 
               τ   p     =       A   a     .           
The area A under the error curve may be tracked by multiplying the magnitude of the error signal e at each time step k by the duration of an interval between time steps
 
               (       e   .   g   .     ,     A   =       ∑     k   =   1     n     ⁢           ⁢           e   k     ⁡     (   t   )       ·   Δ     ⁢           ⁢     t   k             )     .         
In some embodiments, the magnitude α of the setpoint change may be estimated based on the magnitude of the error signal e immediately after the setpoint change. In other embodiments, the magnitude α of the setpoint change may be determined by analyzing setpoint r or may be received as an input from adaptive feedback controller  210 .
 
     Still referring to  FIG. 3 , in some embodiments, time constant estimator  214  determines a time delay L for the controlled system or process (e.g. plant  204 ). The time delay L may represent an interval between the time at which an input is applied to a controlled system and the time at which the input begins to take effect. Time constant estimator  214  may determine time delay L by analyzing the response of error signal e to a setpoint change. For example, graph  300  shows the error signal e instantaneously changing to the magnitude α of the setpoint change immediately after the setpoint change occurs. The error signal e may maintain this value until after time delay L has passed and the dynamics of the system begin to take effect. Time constant estimator  214  may estimate the time delay L by subtracting a time at which the setpoint change occurs from the time at which the magnitude of the error signal e begins to decrease (e.g., in response to an adjusted control input u). In some embodiments, time constant τ p , and time delay L may be separately determined. In other embodiments, time constant τ p  and time delay L may be combined into a single variable (e.g., an average residence time, a dominant time constant, etc.) 
     Referring now to  FIG. 4 , a graph  400  illustrating the time response of error signal e to a load disturbance is shown, according to an exemplary embodiment. The load disturbance may be applied to plant  204  as an uncontrolled and/or unmeasured input. The load disturbance may cause a change in the output y received from plant  204  and consequently in error signal e (e.g., e=r−y). In some embodiments, the output y of plant  204  is a continuous variable unable to instantaneously increase or decrease. If the setpoint r is held constant, a load disturbance may cause a continuous increase or decrease in the error signal e (e.g., rather than the instantaneous increase or decrease caused by a setpoint change). Graph  400  shows the error signal e continuously decreasing in response to a load disturbance. The error signal e is shown decreasing from an initial value (e.g. e=0) to a minimum value e min . In other embodiments, the load disturbance may cause an increase in the error signal from an initial value (e.g., e=0) to a maximum value e max . 
     In the Laplace domain, the response of the error signal e to a step change of magnitude α a in the load disturbance may be represented by the equation 
                 e   ⁡     (   s   )       =           aK   p     ⁢     τ   p           (         τ   p     ⁢   s     +   1     )     ⁢     (         τ   p     ⁢   s     +     e     -   Ls         )         ⁢     e     -   Ls           ,         
where K p  is the process gain and L is the time delay of the controlled system or process (e.g., plant  204 ). When a load disturbance occurs, the dynamics of the system do not begin to affect the error signal e until after the time delay L. Accordingly, the time delay L may be neglected and the response of the error signal e to a step change of magnitude α in the load disturbance may be represented by the equation
 
     
       
         
           
             
               e 
               ⁡ 
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 
                   
                     aK 
                     p 
                   
                   ⁢ 
                   
                     τ 
                     p 
                   
                 
                 
                   
                     ( 
                     
                       
                         
                           τ 
                           p 
                         
                         ⁢ 
                         s 
                       
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                       1 
                     
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                   2 
                 
               
               . 
             
           
         
       
     
     In the time domain, the error signal e may be expressed as 
               e   ⁡     (   t   )       =       aK   p     ⁢     τ   p     ⁢       te       -   t       τ   p         .             
The derivative of this expression is
 
                 e   .     ⁡     (   t   )       =       aK   p     ⁢     τ   p     ⁢         te       -   t       τ   p         ⁡     [     1   -     t     τ   p         ]       .             
An extremum (e.g., a maximum or minimum) of the error signal is found at time t ex  when ė(t)=0. The extremum of the error signal e in response to a load disturbance may occur at a time approximately equal to the time constant τ p  after the disturbance has taken effect. Time constant estimator  214  may determine time constant τ p  by subtracting a time at which the magnitude of error signal e begins to increase t d  in response to a load disturbance from a time at which error signal e reaches an extremum t ex  in response to the load disturbance (e.g., τ p =t ex −t d ).
 
     Referring now to  FIGS. 5A-5D , a series of graphs illustrating the expected advantages of adaptively updating sampling rate h are shown, according to an exemplary embodiment. In  FIGS. 5A-5D , a simulated first order plant is provided with a control signal u determined by three different types of controllers—a proportional integral (PI) controller having optimal controller parameters (PI opt ), a pattern recognition adaptive controller (PRAC) having a fixed sampling rate h (PRAC fixed ), and a PRAC with an adaptively adjusted sampling rate h (e.g., using the systems and methods described in reference to  FIGS. 2-4 ) (PRAC adaptive ). 
     Referring specifically to  FIG. 5A , a graph  501  is shown illustrating the resultant measured variable y as a function of time when the simulated plant is independently subjected to a series of setpoint changes and load disturbances. The setpoint r is shown by line  510  and the performance of PI opt , PRAC fixed , and PRAC adaptive  are shown by lines  520 ,  530 , and  540  respectively. As can be seen from  FIG. 5A , PRAC adaptive  (line  540 ) demonstrates a superior ability to track setpoint r (e.g., control the measured variable y to equal the setpoint) when compared with PRAC fixed  (line  530 ). The performance of PRAC adaptive  may even exceed the performance of PI opt  (line  520 ). 
     Referring to  FIG. 5B , a graph  502  is shown illustrating the sampling rate h as a function of time for PRAC adaptive . The sampling rate h is shown starting at a first value at t=0. At approximately t=250 seconds, the sampling rate h is adjusted to a lower value. Referring again to  FIG. 5A , at approximately this same time (e.g., t≈250 seconds), the performance of PRAC adaptive  significantly improves. Notably, PRAC adaptive  is shown to adjust the sampling rate h only once throughout the simulation. The single sampling rate adjustment may suggest that time constant τ p  estimation (e.g., as described in reference to time constant estimator  214 ) and subsequent sampling rate adjustment (e.g., by sampling rate adjustor  216 ) accurately capture the true time constant τ p  of the simulated plant. 
     Referring to  FIGS. 5C and 5D , graphs  503  and  504  are shown illustrating the controller gain parameter K c  and integral time parameter T i  (e.g., determined by adaptive tuner  212 ) as a function of time for PRAC fixed  and for PRAC adaptive . In  FIG. 5C , line  550  illustrates the controller gain parameter K c  for PRAC fixed  and line  560  illustrates the gain parameter K c  for PRAC adaptive . In  FIG. 5D , line  570  illustrates the integral time parameter T i  for PRAC fixed  and line  580  illustrates the integral time parameter T i  for PRAC adaptive . At approximately t=250 seconds (i.e., the same time that the sampling rate h is adjusted), PRAC adaptive  compensates by adjusting the controller gain K c  from an initial value to a lower value and by adjusting the integral time T i  from an initial value to a lower value. Notably, PRAC adaptive  is shown to adjust the controller gain K c  and integral time T i  only one time throughout the simulation. This single adjustment may suggest that once the sampling rate h is properly set, PRAC adaptive  can accurately determine the appropriate controller gain K c  and integral time T i  without requiring subsequent correction. 
     Referring now to  FIG. 6 , a block diagram of process controller  201  is shown, according to an exemplary embodiment. Process controller  201  is shown to include a communications interface  610  and a processing circuit  620 . Communications interface  610  may include any number of jacks, wire terminals, wire ports, wireless antennas, or other communications adapters, hardware, or devices for communicating information (e.g., setpoint r information, error signal e information, feedback signal y information, etc.) or control signals (e.g., a control signal u, etc.) Communications interface  610  may be configured to send or receive information and/or control signals between process controller  201  and a controlled system or process (e.g., plant  204 ), between process controller  201  and a supervisory controller, or between process controller  201  and a local controller (e.g., a device, building, or network specific controller). Communications interface  610  may be configured to send or receive information over a local area network (LAN), wide area network (WAN), and/or a distributed network such as the Internet. Communications interface  610  can include communications electronics (e.g., receivers, transmitters, transceivers, modulators, demodulators, filters, communications processors, communication logic modules, buffers, decoders, encoders, encryptors, amplifiers, etc.) configured to provide or facilitate the communication of the signals described herein. 
     Processing circuit  620  is shown to include a processor  630  and memory  640 . Processor  630  can be implemented as a general purpose processor, an application specific integrated circuit (ASIC), one or more field programmable gate arrays (FPGAs), a group of processing components, or other suitable electronic processing components. 
     Memory  640  (e.g., memory device, memory unit, storage device, etc.) is one or more devices (e.g., RAM, ROM, Flash memory, hard disk storage, etc.) for storing data and/or computer code for completing or facilitating the various processes, layers and modules described in the present disclosure. Memory  640  may be or include volatile memory or non-volatile memory. Memory  136  may include database components, object code components, script components, or any other type of information structure for supporting the various activities and information structures described in the present application. According to an exemplary embodiment, memory  640  is communicably connected to processor  630  via processing circuit  620  and includes computer code for executing (e.g., by processing circuit  620  and/or processor  630 ) one or more processes described herein. Memory  640  is shown to include a time constant estimation module  642 , a sampling rate adjustment module  644 , and an adaptive feedback control module  646 . 
     Time constant estimation module  642  may be configured to perform the functions of time constant estimator  214  as described in reference to  FIG. 2 . Time constant estimation module  642  may receive an error signal e, a setpoint signal r, and or a feedback signal y. Time constant estimation module  642  may be configured to monitor the error signal e and estimate a time constant τ p  for a controlled system or process based on the error signal e. In some implementations, time constant estimation module  642  may determine whether the controlled system is to a setpoint change or a load disturbance (e.g., by receiving a signal from adaptive feedback control module  646 , by analyzing the error signal e, etc.). 
     If the system is subject to a setpoint change, time constant estimation module  642  may estimate the time constant τ p  by determining an area under the error curve (e.g., defined by error signal e). The area under the error curve may be divided by the magnitude of the setpoint change to determine the estimated time constant τ p . If the system is subject to a load disturbance, the time constant τ p  may be estimated by determining a time at which the error signal e reaches an extremum (e.g., a minimum or a maximum) in response to the load disturbance. Time constant estimation module  642  may subtract the time value at which the load disturbance begins from the time value at which the error signal e reaches an extremum to determine the estimated time constant τ p . The error signal e may be monitored for zero crossings (e.g., a sign change from positive to negative or negative to positive) to determine the time at which a load disturbance begins. Time constant estimation module  642  may communicate the estimated time constant τ p  to sampling rate adjustment module  644 . 
     Sampling rate adjustment module  644  may be configured to perform the functions of sampling rate adjustor  216  as described in reference to  FIG. 2 . Sampling rate adjustment module  644  may receive the estimated time constant τ p  from time constant estimation module  642  and calculate a sampling rate h based on the estimated time constant τ p . In some embodiments, sampling rate adjustment module  644  may set the sampling rate h to a value approximately equal to one-tenth of the estimated time constant τ p  (e.g., 
                 τ   p     ⁡     (       e   .   g   .     ,     h   ≈       τ   p     10         )       .         
In some embodiments, sampling rate adjustment module  644  compares the calculated sampling rate h with a current or previous sampling rate h 0 . If the calculated sampling rate h differs significantly from h 0  (e.g., h is less than half h 0  or greater than twice h 0 ), sampling rate adjustment module may update the sampling rate to the recently calculated value h. The updated sampling rate h may be communicated to adaptive feedback control module  646 .
 
     Adaptive feedback control module  646  may be configured to perform the functions of PI controller  202  and adaptive tuner  212  as described in reference to  FIG. 2 . Adaptive feedback control module  646  may include the functionality of a pattern recognition adaptive controller (PRAC), a model recognition adaptive controller (MRAC), or any other type of adaptive tuning or feedback controller. Adaptive feedback control module  646  may receive an error signal e representing a difference between a feedback signal y and a setpoint r. Adaptive feedback control module  646  may calculate a control signal u for a controlled process or system based on the error signal e. The control signal u may be communicated to the controlled process or system via communications interface  610 . 
     Referring now to  FIG. 7 , a flowchart of a process  700  for adaptively adjusting a sampling rate in a feedback control system is shown, according to an exemplary embodiment. Process  700  is shown to include monitoring an error signal for a disturbance event (step  702 ). The error signal e may be a difference between a setpoint r and a feedback signal y received from a control process or system. In some embodiments, step  702  includes determining whether a disturbance in the error signal e (e.g., a deviation in the error signal from a zero error steady state) qualifies as a disturbance event. Step  702  may involve comparing a magnitude of the error signal e or an area bounded by the error signal e to a threshold value. In some embodiments, the threshold value may be a noise threshold (e.g., measurement noise, process noise, combined noise, etc.). If the magnitude or area exceeds the threshold value, the disturbance may be classified as a disturbance event. In some embodiments, the remaining steps of process  700  (e.g., steps  704 - 710 ) are only performed if a detected disturbance qualifies as a disturbance event. In other embodiments, steps  704 - 710  are performed for all detected disturbances. 
     Step  702  may include identifying the disturbance event as at least one of a setpoint change and a load disturbance. In some embodiments, the disturbance event may be identified via a status indicator or signal received from an adaptive feedback controller. For example, a PRAC may output a signal indicating whether the control system is subject to a setpoint change or a load disturbance. In other embodiments, the disturbance event may be identified by analyzing the error signal e. For example, a setpoint change may result in an instantaneous change in the error signal e whereas a load disturbance may result in continuous increase in the magnitude of the error signal e. 
     Still referring to  FIG. 7 , process  700  is shown to further include evaluating the error signal in response to the disturbance event to estimate a time constant for a control process (step  704 ). If the disturbance event is identified as a setpoint change in step  702 , evaluating the error signal e may involve tracking an area A bounded by the error signal e in response to the setpoint change. The area A bounded by the error signal may be an area between an error curve (e.g., a graphical representation of the error signal as a function of time) and the time axis (e.g., a “zero error” line extending along the time axis). The area A bounded by the error signal may be an area above the time axis (e.g., if the error curve includes positive error values) or an area below the time axis (e.g., if the error curve includes negative error values). The area A bounded by the error signal may be tracked analytically (e.g., by expressing the error signal as an equation, integrating the equation, A=∫ 0   x  e(t)dt, etc.) or numerically (e.g., by multiplying a magnitude of the error signal by a duration of a time step, adding the resultant products, 
               A   =       ∑     k   =   1     n     ⁢           ⁢           e   k     ⁡     (   t   )       ·   Δ     ⁢           ⁢     t   k           ,         
etc.).
 
     In some embodiments, the area A bounded by the error signal e may be tracked during a tracking interval. The tracking interval may begin at the time of the setpoint change and end at a time when the error signal e crosses the time axis (e.g., a “zero crossing”). In other embodiments, the tracking interval may end at a time when a magnitude of the error signal |e| crosses a magnitude threshold. The magnitude threshold may be a non-zero error value at which it may be determined that the control system has adequately responded to the disturbance event. In some embodiments, the magnitude threshold may be a noise threshold (e.g., measurement noise, process noise, etc.) for the controlled system or process. 
     If the disturbance event is identified as a setpoint change in step  702 , step  704  may further include dividing the area A bounded by the error signal by a magnitude α of the setpoint change to determine the estimated time constant of the control process 
               (       e   .   g   .     ,       τ   p     =     A   a         )     .         
In some embodiments, the magnitude of the setpoint change may be obtained by analyzing the setpoint signal (e.g., subtracting a setpoint value before the setpoint change from a setpoint value after the setpoint change). In other embodiments, the magnitude α of the setpoint change may be obtained by analyzing the error signal e (e.g., determining the magnitude of the instantaneous increase in the magnitude of the error signal). In further embodiments, the magnitude α of the setpoint change may be received from another process or system.
 
     Still referring to step  704 , if the disturbance event is identified as a load disturbance in step  702 , evaluating the error signal in response to the disturbance event may include tracking a magnitude of the error signal |e| in response to the load disturbance and determining a time at which the magnitude of the error signal |e| reaches a maximum in response to the load disturbance. The maximum magnitude of the error signal |e| max  may be an extremum (e.g., a maximum value e max  or minimum value e min ) in response to the load disturbance. Unlike setpoint changes, load disturbances may cause the magnitude of the error signal |e| to increase continuously (e.g., non-instantaneously) until reaching a maximum |e| max  at a time t ex  after the disturbance occurs. If the disturbance event is identified as a load disturbance in step  702 , step  704  may involve subtracting a time t d  at which the magnitude of the error signal begins to increase in response to the load disturbance from the time at which the magnitude of the error signal reaches the maximum t ex  to determine the time constant of the control process (e.g., τ p =t ex −t d ). 
     Still referring to  FIG. 7 , process  700  is shown to further include determining a sampling rate for use in the feedback control system based on the estimated time constant (step  706 ). The sampling rate h may be a sampling period (e.g., an interval between samples) or a sampling frequency (e.g., a rate at which samples are obtained). In some embodiments, the sampling rate h may be used to select a subset of measurements from a feedback signal y for use in determining a control signal u to be applied to a controlled process. In other embodiments, the sampling rate h may specify the interval at which the control signal u is updated. In some embodiments, step  706  may involve selecting a sampling rate h between one-fortieth of the estimated time constant τ p  and the estimated time constant 
                 τ   p     ⁡     (       e   .   g   .     ,         τ   p     40     ≤   h   ≤     τ   p         )       .         
In a more specific embodiment, step  706  may involve selecting a sampling rate h approximately one-tenth of the estimated time constant
 
     
       
         
           
             
               
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     Still referring to  FIG. 7 , in some embodiments, process  700  may further include comparing the determined sampling rate h with a previous sampling rate h 0  used by the feedback control system (step  708 ) and updating the previous sampling rate h 0  to the determined sampling rate h if the comparison reveals that one or more updating criteria are met (step  710 ). The updating criteria may prevent the previous sampling rate h 0  from being updated if the change would be minimal or insignificant. For example, the updating criteria may include updating the previous sampling rate h 0  to the sampling rate h determined in step  706  if the determined sampling rate h is at least twice the previous sampling rate (e.g., h≥2h 0 ). In some embodiments, the previous sampling rate h 0  may be updated to the sampling rate h determined in step  706  if the determined sampling rate h is no greater than half the previous sampling rate (e.g., h≤0.5h 0 ). In further embodiments, steps  708  and  710  may involve updating the previous sampling rate h 0  if the difference between the previous sampling rate h 0  and the sampling rate h determined in step  706  exceeds a difference threshold (e.g., |h 0 −h|&gt;threshold) or if the ratio between h 0  and h exceeds a ratio threshold 
     
       
         
           
             
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     In some implementations, the systems and methods described herein may be used in a closed-loop feedback control system. When implemented in a closed-loop system, a disturbance event may be detected by monitoring an error signal (e.g., for a load disturbance or a setpoint change). The error signal may be based on a feedback signal received from a control process. In other implementations, the described systems and methods may be used in a feed-forward control system, and open loop control system, a model predictive control system, a cascaded control system, and/or any other type of automated control system. When implemented in non-feedback control systems, the disturbance event may be detected by monitoring a control signal (e.g., provided to a control process), a feed-forward signal (e.g., received from a feed-forward estimator), a sensor signal (e.g., from a sensor monitoring a variable other than the controlled variable), or any other type of signal (e.g., calculated or measured) communicated between one or more components of the control system. The monitored signal may be analyzed as described above (e.g., by time constant estimator  214 ) for estimating a time constant and determining a sampling rate for use in the non-feedback control system. 
     The construction and arrangement of the systems and methods as shown in the various exemplary embodiments are illustrative only. Although only a few embodiments have been described in detail in this disclosure, many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.). For example, the position of elements may be reversed or otherwise varied and the nature or number of discrete elements or positions may be altered or varied. Accordingly, all such modifications are intended to be included within the scope of the present disclosure. The order or sequence of any process or method steps may be varied or re-sequenced according to alternative embodiments. Other substitutions, modifications, changes, and omissions may be made in the design, operating conditions and arrangement of the exemplary embodiments without departing from the scope of the present disclosure. 
     The present disclosure contemplates methods, systems and program products on any machine-readable media for accomplishing various operations. The embodiments of the present disclosure may be implemented using existing computer processors, or by a special purpose computer processor for an appropriate system, incorporated for this or another purpose, or by a hardwired system. Embodiments within the scope of the present disclosure include program products comprising machine-readable media for carrying or having machine-executable instructions or data structures stored thereon. Such machine-readable media can be any available media that can be accessed by a general purpose or special purpose computer or other machine with a processor. By way of example, such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to carry or store desired program code in the form of machine-executable instructions or data structures and which can be accessed by a general purpose or special purpose computer or other machine with a processor. Combinations of the above are also included within the scope of machine-readable media. Machine-executable instructions include, for example, instructions and data which cause a general purpose computer, special purpose computer, or special purpose processing machines to perform a certain function or group of functions. 
     Although the figures show a specific order of method steps, the order of the steps may differ from what is depicted. Also two or more steps may be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. Likewise, software implementations could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various connection steps, processing steps, comparison steps and decision steps.