P, I, D, PI, PD, PID Mode

Proportional Control (P Mode)

Proportional control is a pure gain adjustment acting on the error signal to provide the driving

input to the process. Implementation of this mode requires a circuit which has a response

given by

p = K p e p  + p0

Kp = Proportional gain (controller transfer function)

Ep = error in percent of variable range

Po = controller output with no error (bias)

If we consider both the controller output and error to be expressed in terms of voltage, we see that the above equation is simply a summing amplifier. The op-amp circuit is given in the following figure.

Integral Control (I Mode)

Integral control is implemented through the introduction of an integrator. Implementation of this mode requires a circuit which has a response given by

U(t) K E (t) dt P(0)

Where:       U =controller output (0 - 100%)
                  KI = Integration gain (s-1)
                  Ep = error in percent of variable range
                  P(0) = controller output at time t=0

If we consider both the controller output and error to be expressed in terms of voltage, we see that the above equation is simply an integrator and an inverted-summing amplifier.

In this case, the analog electronic equation for the output voltage is

V K V dt V(0)

Where:      Vo = output voltage
                 KI = 1/RC = Integration gain (s-1)
                 RC = integration time in seconds
                 VE = error in percent of full scale range
                 V(0) = Initial output voltage

If KI is too large, the output rises so fast that overshoots of the optimum setting occur and cycling is produced. The integration time constant RC determines the rate at which controller output increases when the error is constant. The initial controller output V(0) is relatively unimportant because the integration output floats at values determined by error history.

Derivative Control(D MODE)

The derivative of the process error is calculated by determining the slope of the error over 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:

Derivative action predicts system behavior and thus improves settling time and stability of the system.An ideal derivative is not causal, so that implementations of PID controllers include an additional low pass filtering for the derivative term, to limit the high frequency gain and noise.Derivative action is seldom used in practice though - by one estimate in only 20% of deployed controllers because of its variable impact on system stability in real-world applications.

PROPOTIONAL DERIVATIVE CONTROL(PD MODE)

This mode involves the serial use of proportional and derivative mode and this modes has many industrial applications.The analytical expression for PD mode is :

                                                                                                  p = K p e p  + K p  K Dde pdt+ p0

PROPORTIONAL INTEGRAL DERIVATIVE CONTROL(PID) 

PID control adds a predictive element to the control response. In addition to the proportional and integral calculation, the derivative or slope of the control response will be computed. This calculation will have the effect of dampening a control response that is returning to setpoint so quickly that it will overshoot the setpoint.

PID is a precision process control response and is not always required for HVAC applications. The routine application of PID control to every control loop is labor intensive and its application should be selective.

                                         p = K p e p  + K p  K I  ∫ e p dt + K p  K Dde p+ pI (0)