# Evaluating different PID forms

## A theoretical basis for effects of different proportional-integral-derivative (PID) forms on idealized line 1st & 2nd order systems.

December 21, 2021
Courtesy: Wunderlich-Malec

#### Learning Objectives

• The impact of different proportional-integral-derivative (PID) forms can be evaluated using idealized models.
• Critical indicators are integrated square error (ISE), critical dampening, quarter amplitude dampening stability and derivative of the output squared.
• There are infinite combinations of P&ID that provide constant amplitude dampening with differing ISEs.

Real processes have nonlinearities, stiction, deadtime, hysteresis, etc. The general impact of different proportional-integral-derivative (PID) forms can be evaluated using the idealized models. The critical performance indicators are integrated square error (ISE), critical dampening, quarter amplitude dampening stability and derivative of the output squared.

The dependent PID form is used where proportional (K) acts on all three terms, I units are minutes/repeat and D units are repeats per minute. See the simplified block diagram (Figure 1).

Unit step changes in the setpoint and load are used to calculate these indicators. P represents the process. For the A equation on a set point changes, P,I&D act on error. For the B equation P&I act on error and D acts on measurement. For the C equation, I acts on error and P&D act on measurement. See the formulas for calculating the various indicators (Figures 2a and 2b).

Figure 1: Example of a simplified block diagram for PID. Courtesy: Wunderlich-Malec

### Pure integrating process for a level loop

For a load step change, derivative has no effect on the ISE. In the case of B&C equations derivative increases the ISE. Since most level loops are tuned for load response rather than setpoint response derivative isn’t generally used and an appropriate change in K will give the same set point change result as having some derivative.

In surge applications the goal is to minimize the change in the controlled flow rate given a change in the load flow. Derivative is 0 in this discussion. One performance indicator that can be used to measure this is output derivative squared. The formula for this is K*I+T) .

Reducing the reset time lowers the ISE, but increases the output derivative squared. Similarly increasing the proportional gain reduces the ISE but increase the output derivative squared.  Another indicator is the maximum overshoot due to a step load change. Increasing K reduces the overshoot, but increases the output derivative squared. Reducing the reset time lowers the overshoot but increases the output derivative squared.

Integral doesn’t affect the setpoint step change ISE in the case of A & B equations. If the A or B equations are used, there will be a proportional and derivative kick (A equation) or a proportional kick (B equation) to the output on a set point change.  There is no difference in the load ISE for either of the PID forms.

There are an infinite number of combinations of P,I&D that provide constant amplitude dampening with differing ISEs. This is true for higher order processes. If the objective is to achieve quarter amplitude dampening C1 = 5.385572. The I range between critical dampening and quarter amplitude dampening is significant (~21.5:1).

Figures 2a and 2b: Example of calculating the various PID indicators. Courtesy: Wunderlich-Malec

### First order lag for flow loop

Many of the same above comments apply to a first order lag loop. In general, if a loop is part of cascade logic one would use either the A or B equation. Again, in the case of a load change derivative has no effect. If I = T and D = 0 the A & B equation ISEs simplify to I/2K.

Let’s looks at a second order process such as a temperature loop.

The equations are complex. I appears as a squared term in the numerator.  Derivative improves the ISE for second order processes. Note the condition for an infinite ISE matches the stability result from a Routh-Hurwitz matrix.  If I = T1+T2 and D =T1* T2 /( T1+T2 ) the A  equation ISE also simplifies to I/2K.

Brian Philbin, senior project manager, Wunderlich-Malec. Edited by Chris Vavra, web content manager, Control Engineering, CFE Media and Technology, cvavra@cfemedia.com.