PID controller theory -- proportional??ntegral??erivative controller (PID controller) (3)

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   proportional??ntegral??erivative controller (PID controller) (3)
PID controller theory

From Wikipedia

This section describes the parallel or non-interacting form of the PID controller. For other forms please see the Section "Alternative notation and PID forms".

The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). Hence:

where P_out, I_out, and D_out are the contributions to the output from the PID controller from each of the three terms, as defined below.

Proportional term

The proportional term makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant K_p, called the proportional gain.
The proportional term is given by:

Where
       *  P_out: Proportional term of output 
       * K_p: Proportional gain, a tuning parameter 
       * e: Error = SP - PV 
       * t: Time or instantaneous time (the present) 

A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (See the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive (or sensitive) controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances.
In the absence of disturbances, pure proportional control will not settle at its target value, but will retain a steady state error that is a function of the proportional gain and the process gain. Despite the steady-state offset, both tuning theory and industrial practice indicate that it is the proportional term that should contribute the bulk of the output change.

Figure. Plot of PV vs time, for three values of Kp (Ki and Kd held constant)

Integral term

The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain, K_i.
The integral term is given by:

I_out = 

Where

I_out: Integral term of output 
K_i: Integral gain, a tuning parameter 
e: Error = SP - PV 
?: Time in the past contributing to the integral response 
The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction). For further notes regarding integral gain tuning and controller stability, see the section on loop tuning.

Figure. Plot of PV vs. time, for three values of Ki (Kp and Kd held constant)

Derivative term

The rate of change of the process error is calculated by determining the slope of the error over time (i.e. its first derivative with respect to time) and multiplying this rate of change by the derivative gain K_d. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, K_d.
The derivative term is given by:

Where

D_out: Derivative term of output 
K_d: Derivative gain, a tuning parameter 
e: Error = SP - PV 
t: Time or instantaneous time (the present) 
The derivative term slows the rate of change of the controller output and this effect is most noticeable close to the controller setpoint. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, differentiation of a signal amplifies noise and thus this term in the controller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large.

Figure. Plot of PV vs time, for three values of Kd (Kp and Ki held constant)

Summary

The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining u(t) as the controller output, the final form of the PID algorithm is:

and the tuning parameters are

K_p: Proportional gain - larger K_p typically means faster response since the larger the error, the larger the Proportional term compensation. An excessively large proportional gain will lead to process instability and oscillation. 
K_i: Integral gain - larger K_i implies steady state errors are eliminated quicker. The trade-off is larger overshoot: any negative error integrated during transient response must be integrated away by positive error before we reach steady state. 
K_d: Derivative gain - larger K_d decreases overshoot, but slows down transient response and may lead to instability due to signal noise amplification in the differentiation of the error. 

Edited by control - 17 Dec 2008 at 23:16

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control Members Profile Send Private Message Find Members Posts Add to Buddy List Newbie Joined: 16 Dec 2008 Posts: 12	Quote Reply Topic: PID controller theory -- proportional??ntegral??erivative controller (PID controller) (3) Posted: 17 Dec 2008 at 21:55
	proportional??ntegral??erivative controller (PID controller) (3) PID controller theory From Wikipedia This section describes the parallel or non-interacting form of the PID controller. For other forms please see the Section "Alternative notation and PID forms". The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). Hence: where P_out, I_out, and D_out are the contributions to the output from the PID controller from each of the three terms, as defined below. Proportional term The proportional term makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant K_p, called the proportional gain. The proportional term is given by: Where * P_out: Proportional term of output * K_p: Proportional gain, a tuning parameter * e: Error = SP - PV * t: Time or instantaneous time (the present) A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (See the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive (or sensitive) controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances. In the absence of disturbances, pure proportional control will not settle at its target value, but will retain a steady state error that is a function of the proportional gain and the process gain. Despite the steady-state offset, both tuning theory and industrial practice indicate that it is the proportional term that should contribute the bulk of the output change. Figure. Plot of PV vs time, for three values of Kp (Ki and Kd held constant) Integral term The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain, K_i. The integral term is given by: I_out = Where I_out: Integral term of output K_i: Integral gain, a tuning parameter e: Error = SP - PV ?: Time in the past contributing to the integral response The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction). For further notes regarding integral gain tuning and controller stability, see the section on loop tuning. Figure. Plot of PV vs. time, for three values of Ki (Kp and Kd held constant) Derivative term The rate of change of the process error is calculated by determining the slope of the error over time (i.e. its first derivative with respect to time) and multiplying this rate of change by the derivative gain K_d. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, K_d. The derivative term is given by: Where D_out: Derivative term of output K_d: Derivative gain, a tuning parameter e: Error = SP - PV t: Time or instantaneous time (the present) The derivative term slows the rate of change of the controller output and this effect is most noticeable close to the controller setpoint. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, differentiation of a signal amplifies noise and thus this term in the controller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large. Figure. Plot of PV vs time, for three values of Kd (Kp and Ki held constant) Summary The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining u(t) as the controller output, the final form of the PID algorithm is: and the tuning parameters are K_p: Proportional gain - larger K_p typically means faster response since the larger the error, the larger the Proportional term compensation. An excessively large proportional gain will lead to process instability and oscillation. K_i: Integral gain - larger K_i implies steady state errors are eliminated quicker. The trade-off is larger overshoot: any negative error integrated during transient response must be integrated away by positive error before we reach steady state. K_d: Derivative gain - larger K_d decreases overshoot, but slows down transient response and may lead to instability due to signal noise amplification in the differentiation of the error. Edited by control - 17 Dec 2008 at 23:16