Patent Publication Number: US-10324424-B2

Title: Control system with response time estimation and automatic operating parameter adjustment

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATION 
     This application is a continuation-in-part 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 is a control system for a plant. The control system includes a controller and a sensor. The controller is configured to estimate a response time of the plant and adjust a sampling rate based on the estimated response time. The response time is a parameter that characterizes a response of the plant to a disturbance. The sensor is configured to receive the adjusted sampling rate from the controller, collect samples of a measured variable from the plant at the adjusted sampling rate, and provide the samples of the measured variable to the controller. 
     In some embodiments, the controller is configured to use the samples of the measured variable to generate and provide an input to the plant. In some embodiments, the response time includes at least one of a dominant time constant, a bandwidth, and an open loop response time of the plant. 
     In some embodiments, the controller is configured to detect the disturbance in the control system and evaluate a signal affected by the disturbance to estimate the response time of the plant. In some embodiments, the controller is configured to use the adjusted sampling rate to generate samples of the signal affected by the disturbance. 
     In some embodiments, the signal affected by the disturbance includes at least one of the measured variable and a function of the measured variable. In some embodiments, the signal affected by the disturbance includes at least one of an input provided to the plant and a function of the input provided to the plant. 
     Another implementation of the present disclosure is a method for monitoring and controlling a plant. The method includes estimating a response time of the plant. The response time is a parameter that characterizes a response of the plant to a disturbance. The method further includes adjusting a sampling rate based on the estimated response time, collecting samples of a measured variable from the plant at the adjusted sampling rate, and using the samples of the measured variable to generate and provide an input to the plant. 
     In some embodiments, the method includes detecting the disturbance in a control system for the plant and evaluating a signal affected by the disturbance to estimate the response time of the plant. In some embodiments, the signal affected by the disturbance includes at least one of the measured variable and a function of the measured variable. In some embodiments, the signal affected by the disturbance includes at least one of the input provided to the plant and a function of the input provided to the plant. 
     In some embodiments, the method includes using the adjusted sampling rate to generate samples of the signal affected by the disturbance. In some embodiments, the response time includes at least one of a dominant time constant, a bandwidth, and an open loop response time of the plant. 
     Another implementation of the present disclosure is a control system for a plant. The control system includes a disturbance detector configured to detect a disturbance in the control system and a response time estimator configured to evaluate a signal affected by the disturbance to estimate a response time of a plant. The response time is a parameter that characterizes a response of the plant to the disturbance. The system includes a fault detector configured to detect a fault in the control system based on the estimated response time of the plant and an adaptive controller configured to generate and provide an input to the plant. 
     In some embodiments, the fault detector is configured to detect the fault by comparing the estimated response time to a previous response time for the plant and determining that a fault has occurred in response to the estimated response time deviating from the previous response time by a predetermined amount. 
     In some embodiments, the system includes an operating parameter calculator configured to adjust an operating parameter used by the adaptive controller based on the estimated response time. In some embodiments, adjusted operating parameter includes at least one of a controller gain and an integral time. In some embodiments, the response time includes at least one of a dominant time constant, a bandwidth, and an open loop response time of the plant. 
     In some embodiments, the system includes a sampling rate adjustor configured to adjust a sampling rate based on the estimated response time. In some embodiments, the system includes a sensor configured to collect samples of a measured variable from the plant at the adjusted sampling rate and provide the samples of the measured variable to the adaptive controller. 
    
    
     
       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. 
         FIG. 8  is a block diagram of a control system configured to estimate the response time of a plant and use the response time to update operating parameters used by a process controller, according to an exemplary embodiment. 
         FIG. 9  is a block diagram illustrating the process controller of  FIG. 8  providing a sampling rate to a wireless sensor, which uses the sampling rate to collect and transmit samples of a measured variable to the process controller, according to an exemplary embodiment. 
         FIG. 10  is flowchart of a process for automatically adjusting the sampling rate used by a wireless sensor in a control system, according to an exemplary embodiment. 
         FIG. 11  is a flowchart of a process for detecting faults in a 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. 
     Overview 
     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. 
     Control Systems 
     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
 
               1     1   +         G   c     ⁡     (   s   )       ⁢       G   p     ⁡     (   s   )             =       1     1   +           (       K   p     ⁢     K   c       )     ⁢     (     1   +       T   i     ⁢   s       )           T   i     ⁢     s   ⁡     (     1   +       τ   p     ⁢   s       )           ⁢     e     -   Ls             .           
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 adjustor  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 τp 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 a 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 τp 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., τ 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     50     ≤   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≥2 h 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≤0.5 h 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. 
     Response Time Graphs 
     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 a 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 a 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 a 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 a 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   .     ,       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 a 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 a 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 a 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 a 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 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 a in the load disturbance may be represented by the equation
 
     
       
         
           
             
               e 
               ⁡ 
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 
                   
                     aK 
                     p 
                   
                   ⁢ 
                   
                     τ 
                     p 
                   
                 
                 
                   
                     ( 
                     
                       
                         
                           τ 
                           p 
                         
                         ⁢ 
                         s 
                       
                       + 
                       1 
                     
                     ) 
                   
                   2 
                 
               
               . 
             
           
         
       
     
     In the time domain, the error signal e may be expressed as 
               e   ⁡     (   t   )       =       aK   p     ⁢     τ   p     ⁢   t   ⁢           ⁢       e       -   t       τ   p         .             
The derivative of this expression is
 
                 e   .     ⁡     (   t   )       =       aK   p     ⁢     τ   p     ⁢   t   ⁢           ⁢         e       -   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 ).
 
Performance Graphs
 
     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. 
     Process Controller 
     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  640  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   .     ,     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 . 
     Adaptive Sampling Rate Adjustment 
     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 
                           
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                             t 
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                         t 
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                 , 
                 
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               ) 
             
             . 
           
         
       
     
     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 a 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 a 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 a 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≥2 h 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≤0.5 h 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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                   threshold 
                 
                 , 
                 
                   etc 
                   . 
                 
               
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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. 
     Using Response Time to Detect Faults and Adjust Operating Parameters 
     Referring now to  FIG. 8 , a block diagram of another control system  800  is shown, according to an exemplary embodiment. In some embodiments, control system  800  is a feedback control system. In other embodiments, control system  800  can be 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. Control system  800  is shown to include a process controller  801 , a wireless sensor  802 , and a plant  804 . Process controller  801  may be configured to monitor and control plant  804 . When a disturbance occurs in control system  800 , process controller  801  can monitor a signal affected by the disturbance to estimate a response time of plant  804 . Process controller  801  can use the estimated response time of plant  804  to determine an appropriate sampling rate, detect faults, and/or adjust operating parameters used by process controller  801 . In some embodiments, process controller  801  provides the determined sampling rate to wireless controller  802 . Wireless controller  802  can use the sampling rate to collect and transmit data samples to process controller  801 . 
     Process controller  801  is shown to include a communications interface  810  and a processing circuit  820 . Communications interface  810  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  810  may be configured to send or receive information and/or control signals between process controller  801  and a controlled system or process (e.g., plant  804 ), between process controller  801  and a supervisory controller, or between process controller  801  and a local controller (e.g., a device, building, or network specific controller). Communications interface  810  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  810  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  820  is shown to include a processor  830  and memory  840 . Processor  830  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  840  (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  840  may be or include volatile memory or non-volatile memory. Memory  840  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  840  is communicably connected to processor  830  via processing circuit  820  and includes computer code for executing (e.g., by processing circuit  820  and/or processor  830 ) one or more processes described herein. 
     Process controller  801  is shown to include a data collector  842 . Data collector  842  can receive a data signal and/or data samples via communications interface  810 . In some embodiments, data collector  842  samples a signal affected by a disturbance. For example, data collector  842  is shown receiving a signal affected by a disturbance. The signal affected by a the disturbance can be a feedback signal from plant  804  (e.g., feedback signal y), a setpoint signal (e.g., setpoint signal r), an error signal e (e.g., e=r−y), a control signal provided to plant  804  (e.g., input signal u), or any other signal in control system  800 . The signal affected by the disturbance can be any signal that changes when a disturbance occurs in control system  800 . The change can be a direct result of the disturbance (e.g., a change in feedback signal y) or an indirect result of process controller  801  reacting to the disturbance. For example, the signal affected by the disturbance can be an input signal u provided to plant  804  to compensate for a detected disturbance. 
     In some embodiments, data collector  842  samples the signal affected by the disturbance at a sampling rate received from sampling rate adjustor  856 . The sampling rate can be automatically adjusted based on the estimated response time of plant  804 . The data samples generated by data collector  842  can be provided to adaptive controller  850  for use in determining an appropriate input signal u for plant  804 . In some embodiments, data collector  842  receives data samples from wireless sensor  802 . The data samples from wireless sensor  802  can be collected and/or transmitted at the sampling rate determined by sampling rate adjustor  856 . 
     Still referring to  FIG. 8 , process controller  801  is shown to include a disturbance detector  844 . Disturbance detector  844  can detect a disturbance in control system  800 . In some embodiments, the disturbance is a setpoint change or a load disturbance. Disturbance detector  844  can detect a setpoint change or a load disturbance by monitoring the signal affected by the disturbance. In some embodiments, disturbance detector  844  detects the disturbance using a status indicator or signal received from adaptive controller  850 . For example, controller  850  may output a signal indicating whether control system  800  is subject to a setpoint change or a load disturbance. In some embodiments, the disturbance includes a user input (e.g., a user-specified setpoint change) and disturbance detector  844  detects the disturbance by monitoring the user input. 
     In some embodiments, disturbance detector  844  detects the disturbance by analyzing the error signal e in a feedback control system. 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. When implemented in non-feedback control systems, disturbance detector  844  can detect the disturbance by monitoring a control signal provided to plant  804 , 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 control system  800 . Disturbance detector  844  can notify response time estimator  846  when a disturbance is detected in control system  800   
     Response time estimator  846  may be configured to perform the functions of time constant estimator  214  as described in reference to  FIG. 2 . For example, response time estimator  846  can evaluate the signal affected by the disturbance to estimate the response time of plant  804 . The response time of plant  804  is a parameter that characterizes the response of plant  804  to the disturbance. For example, the response time of plant  804  can be the time constant τ p  (as previously described), a dominant time constant of plant  804 , a bandwidth of plant  804 , an open loop response time, or any other parameter that characterizes the time response of plant  804  to a disturbance. In some embodiments, response time estimator  846  estimates the response time of plant  804  in response to a detected disturbance in control system  800  (e.g., in response to receiving a disturbance notification from disturbance detector  844 ). Response time estimator  846  can evaluate the signal affected by the disturbance in the same way that time constant estimator  214  evaluates the error signal e. 
     In some embodiments, response time estimator  846  determines whether control system  800  is subject to a setpoint change or a load disturbance (e.g., by receiving a signal from adaptive controller  850 , by analyzing the signal affected by the disturbance, etc.). If the system is subject to a setpoint change, response time estimator  846  may estimate the response time by integrating the signal affected by the disturbance. For example, response time estimator  846  can determine an area under a curve defined by the signal affected by the disturbance. Response time estimator  846  can divide the area under the curve by the magnitude of the setpoint change to determine the estimated response time. 
     If the system is subject to a load disturbance, response time estimator  846  can estimate the response time by determining a time at which the signal affected by the disturbance reaches an extremum (e.g., a minimum or a maximum) in response to the load disturbance. Response time estimator  846  may subtract the time value at which the load disturbance begins from the time value at which the signal reaches an extremum to determine the estimated response time. Response time estimator  846  can monitor the signal affected by the disturbance for threshold value crossings (e.g., a sign change from positive to negative or negative to positive) to determine the time at which a load disturbance begins. Response time estimator  846  may communicate the estimated response time to operating parameter calculator  848 , fault detector  852 , sampling rate adjustor  856 , or store the estimated response time in a response time database  854 . 
     Sampling rate adjustor  856  may be configured to perform the functions of sampling rate adjustor  216  as described in reference to  FIG. 2 . Sampling rate adjustor  856  may receive the estimated response time from response time estimator  846  and calculate a sampling rate h based on the estimated response time. In some embodiments, sampling rate adjustor  856  sets the sampling rate h to a value approximately equal to one-tenth of the estimated response time. In some embodiments, sampling rate adjustor  856  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 adjustor  856  may update the sampling rate to the recently calculated value h. 
     In some embodiments, the updated sampling rate h is a minimum acceptable sampling rate. The minimum acceptable sampling rate may be a function of the estimated response time (e.g., one-tenth of the estimated response time) and may indicate the minimum rate at which data samples can be collected from plant  804  to ensure adequate control. In some embodiments, sampling rate adjustor  856  provides the updated sampling rate h to data collector  842 . Data collector  842  can use the updated sampling rate h to adjust the rate at which data collector  842  collects samples of the signal affected by the disturbance. In some embodiments, sampling rate adjustor  856  provides the updated sampling rate h to wireless sensor  802 . Wireless sensor  802  can use the updated sampling rate h to adjust a rate at which wireless sensor  802  collects and/or transmits data samples to process controller  801 . 
     Still referring to  FIG. 8 , process controller  801  is shown to include an operating parameter calculator  848 . Operating parameter calculator  848  may be configured to perform the functions of adaptive tuner  212 , as described with reference to  FIG. 2 . In some embodiments, operating parameter calculator  848  uses the estimated response time of plant  804  to calculate values of one or more operating parameters used by adaptive controller  850 . For example, operating parameter calculator  848  can calculate values for controller parameters such as the controller gain K c  and the integral time T i  as a function of the estimated response time. In some embodiments, operating parameter calculator  848  adjusts an initial value of each operating parameter based on the estimated response time, as described with reference to  FIGS. 5A-5D . Several examples of how operating parameter calculator  848  can calculate controller operating parameters based on the estimated response time are described in detail in U.S. Pat. Nos. 5,355,305, 5,506,768, and 6,937,909. The entire disclosure of each of these patents is incorporated by reference herein. 
     Adaptive controller  850  may be configured to perform the functions of PI controller  202  as described in reference to  FIG. 2 . Adaptive controller  850  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 controller  850  can be a feedback controller, a feedforward controller, a model predictive controller, or any other type of controller that generates an input signal for plant  804 . Adaptive controller  850  may receive the data samples from data collector  842  and/or wireless sensor  802 . Adaptive controller  850  may use the data samples in combination with the operating parameters calculated by operating parameter calculator  848  to generate the input signal u for plant  804 . The input signal u may be communicated to plant  804  via communications interface  810 . 
     Still referring to  FIG. 8 , process controller  801  is shown to include a fault detector  852  and a response time database  854 . Response time database  854  may store a history of the response times determined by response time estimator  846 . Each time a new response time is estimated for plant  804 , the new response time can be stored in response time database  854 . The response time of plant  804  may be expected to remain relatively constant in the absence of a fault. A significant change in the response time may indicate that a fault has occurred. For example, faulty building equipment may cause plant  804  to react more slowly to a disturbance, which can result in a longer response time. 
     Fault detector  852  can use the history of response times for plant  804  to detect faults in control system  800 . For example, fault detector  852  can compare a current response time generated by response time estimator  846  with a previous response time from response time database  854 . If the current response time is significantly different from the previous response time, fault detector  852  may determine that a fault has occurred. In some embodiments, fault detector  852  calculates a difference between the current response time and the previous response time and compares the difference with a threshold value. If the difference exceeds the threshold value, fault detector  852  may determine that a fault has occurred. In some embodiments, fault detector  852  calculates a ratio of the current response time to the previous response time. If the ratio is less than a minimum threshold value or greater than a maximum threshold value, fault detector  852  may determine that a fault has occurred. 
     Referring now to  FIG. 9 , a portion of control system  800  is shown in greater detail, according to an exemplary embodiment. System  800  is shown to include process controller  801  and wireless sensor  802 . Process controller  801  can provide wireless sensor  802  with a sampling rate. The sampling rate can be determined automatically by process controller  801  based on the estimated response time of plant  804 , as described with reference to  FIG. 8 . Wireless sensor  802  can use the sampling rate from process controller  801  to collect and transmit samples of a measured variable to process controller  801 . For example, wireless sensor  802  can collect and/or transmit samples of the measured variable at the sampling rate provided by process controller  801 . Wireless sensor  802  can provide the samples of the measured variable to process controller  801  as a feedback signal (e.g., feedback signal y). 
     Wireless sensor  802  is shown to include a sensor  908 , a low-power microcontroller  902 , and a wireless radio chip  910 . Sensor  908  can measure a variable of interest and provide measured data values to low-power microcontroller  902 . Sensor  908  may be a temperature sensor, humidity sensor, enthalpy sensor, pressure sensor, lighting sensor, flow rate sensor, voltage sensor, valve position sensor, load sensor, resource consumption sensor, and/or any other type of sensor capable of measuring a variable of interest in control system  800 . In some embodiments, sensor  908  is powered by a battery within wireless sensor  802 . Sensor  908  may collect data samples at a regular sampling interval. In some embodiments, the sampling interval is defined by the sampling rate received from process controller  801 . For example, if the sampling rate received from process controller  801  is one sample per minute, sensor  908  may collect a temperature measurement in a particular zone of a building every minute. 
     Low-power microcontroller  902  can generate a message containing a value of the measured variable and can provide the message to process controller  801 . Low-power microcontroller  902  may be any controller component capable of processing data. For example, microcontroller  902  may include a processing circuit containing a processor capable of receiving, processing, and outputting data. In some embodiments, microcontroller  902  may contains memory capable of storing data. In other embodiments, microcontroller  902  may not include a memory. Low power microcontroller  902  is shown to include a wireless transmission timer  904  and a message generator  906 . 
     Transmission timer  904  may be configured to monitor and control timing of wireless transmissions from sensor  802  to process controller  801 . Transmission timer  904  may be configured to identify or determine a transmission interval and/or a transmission rate for wireless sensor  802  based on the sampling rate received from process controller  801 . For example, transmission timer  904  may cause low power microcontroller  902  to transmit data samples to process controller at the sampling rate provided by process controller  801 . In some embodiments, transmission timer  904  determines a transmission interval that optimizes (e.g., minimizes) the power consumption of wireless sensor  802 . The optimal transmission interval may be subject to a sampling rate constraint. The sampling rate constraint may define a minimum acceptable sampling rate, which may be provided by process controller  801 . Transmission timer  904  can use the sampling rate from process controller  801  to determine the optimal rate at which to collect and/or transmit data samples to process controller  801 , subject to the sampling rate constraint. 
     Message generator  906  may be configured to generate a messages containing samples of the measured variable. In some embodiments, each message contains one sample of the measured variable. In other embodiments, each message contains multiple samples of the measured variable. For example, message generator  906  can package multiple samples of the measured variable into a single message to further reduce the number of messages transmitted to process controller  801 . Message generator  906  may provide the messages containing the data samples to process controller  801  at the transmission interval determined by transmission timer  904 . Low-power microcontroller  902  provides the messages to wireless radio chip  910 , which wirelessly transmits the message to process controller  801 . 
     Process controller  801  is shown to include a wireless radio chip  912 , a message parser  914 , and adaptive controller  850 . Wireless radio chip  912  receives messages from wireless radio chip  910  of wireless sensor  802  and provides the messages to message parser  914 . Message parser  914  extracts data values from the message and provides the data values to adaptive controller  850 . Adaptive controller  850  uses the data values as inputs to a control algorithm to generate a control output (e.g., a control signal for equipment of plant  804  that operate to affect the measured variable). In some embodiments, process controller  801  includes some or all of the features described with reference to  FIG. 8 . 
     In some embodiments, wireless radio chips  910  and  912  communicate with each other using a wireless communications protocol (e.g., ZigBee, WiFi, Bluetooth, NFC, etc). In some embodiments, other communications interfaces and components may be included, such as a wired connection. Wireless radio chips  910  and  912  may contain transceivers capable of transmitting and receiving data through an antenna. Wireless radio chips  910  and  912  may be different chips and may use different hardware while using the same wireless communications protocol. Wireless radio chips  910  and  912  may operate using any frequency range, such as RF. Chips  910  and  912  may use frequencies outside of the RF range, and may not be radio chips. In some embodiments, chips  910  and  912  may communicate using other frequency ranges, such as IR. Chips  910  and  912  may utilize any communications interface, and are not limited to those specifically enumerated. 
     In some embodiments, wireless sensor  802  and process controller  802  include some or all of the components described in U.S. patent application Ser. No. 14/989,740 titled “Systems and Methods for Extending the Battery Life of a Wireless Sensor in a Building Control System” and filed Jan. 6, 2016, the entire disclosure of which is incorporated by reference herein. 
     Sampling Rate Adjustment and Fault Detection Processes 
     Referring now to  FIG. 10 , a flowchart of a process  1000  for automatically adjusting the sampling rate used by a wireless sensor in a control system is shown, according to an exemplary embodiment. Process  1000  can be performed by one or more components of control system  800 , as described with reference to  FIG. 8 . 
     Process  1000  is shown to include estimating a response time of a plant (e.g., plant  804 ) by evaluating a signal affected by a disturbance (step  1002 ). In some embodiments, step  1002  is performed by response time estimator  846 . The response time of the plant  804  is a parameter that characterizes the response of plant  804  to the disturbance. For example, the response time of plant  804  can be the time constant τ p  (as previously described), a dominant time constant of plant  804 , a bandwidth of plant  804 , an open loop response time, or any other parameter that characterizes the time response of plant  804  to a disturbance. In some embodiments, step  1002  is performed in response to a detected disturbance in control system  800  (e.g., in response to receiving a disturbance notification from disturbance detector  844 ). 
     In some embodiments, step  1002  includes determining whether the control system is subject to a setpoint change or a load disturbance (e.g., by receiving a signal from adaptive controller  850 , by analyzing the signal affected by the disturbance, etc.). If the system is subject to a setpoint change, step  1002  may include estimating the response time by integrating the signal affected by the disturbance. For example, step  1002  can include determining an area under a curve defined by the signal affected by the disturbance. Step  1002  can include dividing the area under the curve by the magnitude of the setpoint change to determine the estimated response time. 
     If the system is subject to a load disturbance, step  1002  can include estimating the response time by determining a time at which the signal affected by the disturbance reaches an extremum (e.g., a minimum or a maximum) in response to the load disturbance. Step  1002  can include subtracting the time value at which the load disturbance begins from the time value at which the signal reaches an extremum to determine the estimated response time. Step  1002  can include monitoring the signal affected by the disturbance for threshold value crossings (e.g., a sign change from positive to negative or negative to positive) to determine the time at which a load disturbance begins. 
     Process  1000  is shown to include adjusting a sampling rate based on the estimated response time (step  1004 ). In some embodiments, step  1004  is performed by sampling rate adjustor  856 . In some embodiments, step  1004  includes setting the sampling rate h to a value approximately equal to one-tenth of the estimated response time. In some embodiments, step  1004  includes comparing 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 ), step  1004  may include updating the sampling rate to the recently calculated value h. In some embodiments, the updated sampling rate h is a minimum acceptable sampling rate. The minimum acceptable sampling rate may be a function of the estimated response time (e.g., one-tenth of the estimated response time) and may indicate the minimum rate at which data samples can be collected from plant  804  to ensure adequate control. 
     In some embodiments, step  1004  includes providing the updated sampling rate h to a data collector (e.g., data collector  842 ). The data collector can use the updated sampling rate h to adjust the rate at which the data collector collects samples of the signal affected by the disturbance. In some embodiments, step  1004  includes providing the updated sampling rate h to a wireless sensor (e.g., wireless sensor  802 ). The wireless sensor can use the updated sampling rate h to adjust a rate at which the wireless sensor collects and/or transmits data samples to process controller  801 . 
     Process  1000  is shown to include collecting samples of a measured variable from the plant at the adjusted sampling rate (step  1006 ). In some embodiments, step  1006  is performed by wireless sensor  802 . For example, wireless sensor  802  can receive the adjusted sampling rate from process controller  801  and use the adjusted sampling rate to collect samples of the measured variable (e.g., collecting samples of the measured variable at the adjusted sampling rate). In some embodiments, wireless sensor  802  uses the adjusted sampling rate to optimize the power consumption of wireless sensor  802 . For example, wireless sensor  802  may perform an optimization process to minimize the power consumed by measuring and/or transmitting data samples to process controller  801 . Step  1006  can include using the adjusted sampling rate as a constraint on the optimization process to ensure that the measured variable is sampled and/or transmitted at least as frequently as the adjusted sampling rate. 
     Process  1000  is shown to include using the samples of the measured variable to generate and provide an input to the plant (step  1008 ). In some embodiments, step  1008  is performed by adaptive controller  850 . Step  1008  may include using the samples of the measured variable to generate an input signal u for plant  804 . In some embodiments, the samples of the measured variable are used in combination with one or more operating parameters (e.g., controller gain K c , integral time T i , etc.) to generate the input signal u. For example, adaptive controller  850  can apply the samples of the measured variable as inputs to a transfer function 
             (       e   .   g   .     ,         G   c     ⁡     (   s   )       =         K   c     ⁡     (     1   +       T   i     ⁢   s       )           T   i     ⁢   s           )         
to generate the input signal u.
 
     In some embodiments, step  1008  includes using the estimated response time of plant  804  to calculate values of one or more operating parameters used by adaptive controller  850 . For example, step  1008  can include calculating values for controller parameters such as the controller gain K c  and the integral time T i  as a function of the estimated response time. In some embodiments, step  1008  includes adjusting an initial value of each operating parameter based on the estimated response time, as described with reference to  FIGS. 5A-5D . Several examples of how controller operating parameters can be adjusted or calculated based on the estimated response time are described in detail in U.S. Pat. Nos. 5,355,305, 5,506,768, and 6,937,909. 
     Process  1000  is shown to include using the adjusted sampling rate to generate samples of a signal affected by the disturbance (step  1010 ). In some embodiments, step  1010  is performed by data collector  842 . For example, data collector  842  can monitor the signal affected by the disturbance and collect samples of the signal affected by the disturbance at the adjusted sampling rate. In some embodiments, the signal affected by the disturbance is a feedback signal from plant  804  or an error signal based on the feedback signal (e.g., a difference between a feedback signal and a setpoint). In some embodiments, the signal affected by the disturbance is the measured variable and/or a function of the measured variable. In some embodiments, the signal affected by the disturbance is an input signal provided to the plant  804  and/or a function of the input signal provided to the plant. 
     In some embodiments, process  1000  returns to step  1002  after performing step  1010 . For example, the samples of the signal affected by the disturbance generated in step  1010  can be evaluated in step  1002  to estimate the response time of plant  804 . Process  1000  can be repeated iteratively to update the estimated response time, adjust the operating parameters, and provide updated control signals to plant  804  based on the adjusted operating parameters. 
     Referring now to  FIG. 11 , a flowchart of a process  1100  for detecting faults in a control system is shown, according to an exemplary embodiment. Process  1100  can be performed by one or more components of control system  800 , as described with reference to  FIG. 8 . 
     Process  1100  is shown to include detecting a disturbance in a control system (step  1102 ). In some embodiments, step  1102  is performed by disturbance detector  844 . Step  1102  may include detecting a setpoint change or a load disturbance by monitoring a signal affected by the disturbance. In some embodiments, step  1102  includes detecting the disturbance using a status indicator or signal received from adaptive controller  850 . For example, controller  850  may output a signal indicating whether control system  800  is subject to a setpoint change or a load disturbance. In some embodiments, the disturbance includes a user input (e.g., a user-specified setpoint change) and step  1102  includes detecting the disturbance by monitoring the user input. 
     In some embodiments, step  1102  includes detecting the disturbance by analyzing the error signal e in a feedback control system. 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. In non-feedback control systems, step  1102  can include detecting the disturbance by monitoring a control signal provided to plant  804 , 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 control system  800 . 
     Process  1100  is shown to include estimating a response time of a plant by evaluating a signal affected by the disturbance (step  1104 ). In some embodiments, step  1104  is performed by response time estimator  846 . The response time of the plant  804  is a parameter that characterizes the response of plant  804  to the disturbance. For example, the response time of plant  804  can be the time constant τ p  (as previously described), a dominant time constant of plant  804 , a bandwidth of plant  804 , an open loop response time, or any other parameter that characterizes the time response of plant  804  to a disturbance. In some embodiments, step  1104  is performed in response to a detected disturbance in control system  800  (e.g., in response to step  1102 ). 
     In some embodiments, step  1104  includes determining whether the control system is subject to a setpoint change or a load disturbance (e.g., by receiving a signal from adaptive controller  850 , by analyzing the signal affected by the disturbance, etc.). If the system is subject to a setpoint change, step  1104  may include estimating the response time by integrating the signal affected by the disturbance. For example, step  1104  can include determining an area under a curve defined by the signal affected by the disturbance. Step  1104  can include dividing the area under the curve by the magnitude of the setpoint change to determine the estimated response time. 
     If the system is subject to a load disturbance, step  1104  can include estimating the response time by determining a time at which the signal affected by the disturbance reaches an extremum (e.g., a minimum or a maximum) in response to the load disturbance. Step  1104  can include subtracting the time value at which the load disturbance begins from the time value at which the signal reaches an extremum to determine the estimated response time. Step  1104  can include monitoring the signal affected by the disturbance for threshold value crossings (e.g., a sign change from positive to negative or negative to positive) to determine the time at which a load disturbance begins. 
     Process  1100  is shown to include detecting a fault in the control system based on the estimated response time (step  1106 ). In some embodiments, step  1106  is performed by fault detector  852 . The response time of plant  804  may be expected to remain relatively constant in the absence of a fault. A significant change in the response time may indicate that a fault has occurred. For example, faulty building equipment may cause plant  804  to react more slowly to a disturbance, which can result in a longer response time. 
     Step  1106  may include using a history of response times for plant  804  (from response time database  854 ) to detect faults in control system  800 . For example, step  1106  may include comparing the current response time generated in step  1104  with a previous response time from response time database  854 . If the current response time is significantly different from the previous response time, step  1106  may include determining that a fault has occurred. In some embodiments, step  1106  includes calculating a difference between the current response time and the previous response time and comparing the difference with a threshold value. If the difference exceeds the threshold value, step  1106  may include determining that a fault has occurred. In some embodiments, step  1106  includes calculating a ratio of the current response time to the previous response time. If the ratio is less than a minimum threshold value or greater than a maximum threshold value, step  1106  may include determining that a fault has occurred. 
     Configuration of Exemplary Embodiments 
     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.