Digital PID Controller

What is PID Controller?

A PID controller is an instrument used in industrial control applications to regulate temperature, flow, pressure, speed, and other process variables. PID (proportional integral derivative) controllers use a control loop feedback mechanism to control process variables and are the most accurate and stable controller.

What is a Digital PID Controller?

For example, if application related to temperature measurement then,

A digital PID controller reads the sensor signal, from a thermocouple or RTD and connects the measurement to engineering units, such as degree Fahrenheit or Celsius, that are then displayed in a digital format.

How it works?

In the digital control mode, the error signal is first sampled and the controller output is computed numerically through a digital processor.

Now Controller output for a continuous-type P-I-D controller:

The above equation can be discretized at small sampling interval T0 as shown:

Discretization of Error Signal

Taking the first-order derivative,

and using rectangular integration, we can approximate as:

Now replacing the derivative and integral terms in Eqn. (1), one can obtain,

The above algorithm is known as Position algorithm. But the major problem here is that the error values at all the time instants are to be stored (or at least the second term of the r.h.s of Eqn. (2) at each instant have to be stored). An alternative approach known as Velocity algorithm can be obtained as follows.

From (2), one can write the error signal at the (k-1)th instant as:

Subtracting Eqn. (3) from (2), we can have:

where, 

The above algorithm is known as Velocity algorithm. The major advantage of this algorithm is that it is of recursive type. It calculates the incremental output at each sample instant. As a result, it requires only to store three previous values: e(k), e(k-1), and e(k-2). Besides, it has got several other advantages also those are elaborated below:

1. Bumpless Transfer

During the transfer from manual to auto mode, it is desired that the input command to the process should not change suddenly. In Position algorithm, due to the difference between the setpoint and the output variable, it is always possible that the existing error will wind up, and the value of ∑e(k) being large when switching from manual to auto mode takes place. This will cause a large change in the input u(k) in the auto mode. But in the velocity algorithm, this will be prevented, since it provides an only incremental change in input [u(k)-u(k-1)]. This will lead to bumpless transfer.

2. Prevention of Integration Windup

If there is a sudden change in setpoint, the error will increase continuously to take the value of ∑e(k)  in position algorithm to a large value. Afterward, even if the error reduces to zero, or changes sign, ∑e(k) will take a large time to come to zero, or change sign, resulting in integration windup. But in velocity algorithm, as soon as the error changes sign, the term corresponding to the integration T0/T1 e(k-1)(in Eqn. (4)) will change significantly. Thus even if the actuator is saturated, it will come back to linear range within one sampling period.

3. Protection against Computer Failure

Another advantage of velocity algorithm is its ability to protect the process in case of computer failure. In case there is a failure of the computer, there will be no increment or decrement of the control input, and it will retain the last value before the computer failure, thus preventing process failure. However, there are certain pitfalls of velocity algorithm also. In case of the presence of noise in the measurement of the error value at a particular sampling instant, the controller will immediately act, taking it to be a signal. But in position algorithm, the integration term ∑e(i) where i= 0 to (k-1) will prevent such quick action. Some times, a digital filter with low pass characteristics is used to filter out the unwanted noise before it reaches the controller input.

That's it for this blog.

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