# PID controller

Topics: PID controller, Control theory, Settling time Pages: 13 (1030 words) Published: July 22, 2014
PID Controller Tuning: A Short Tutorial
Jinghua Zhong
Mechanical Engineering, Purdue University

Spring, 2006

Outline
This tutorial is in PDF format with navigational control. You may press SPACE or →, or click the buttons in the lower right corner to move to the next slide. Clicking on the outlined
items will take you directly to that section.
Goals and Objectives
What are we going to learn?
Introduction
What is a PID controller?
Why do we want to learn the PID Controller?
Tuning Rules
How does the PID parameters aﬀect system dynamics?
The Ziegler-Nichols tuning rule

What are we going to learn?

The goal of the tutorial is for you to learn about the PID
controller and a few basic tuning rules of it. After taking this lesson, you will be able to
1. relate PID controller parameters to step response
characteristics of the controlled system, and
2. apply the famous Ziegler-Nichols tuning method to come
up with an initial set of working PID parameters for an
unknown system.

What is a PID controller?
A PID controller is a simple three-term controller. The letters P, I and D stand for:
P - Proportional
I - Integral
D - Derivative
The transfer function of the most basic form of PID controller, as we use in ME475, is
C (s) = KP +

KI
KD s 2 + KP s + KI
+ KD s =
s
s

where KP = Proportional gain, KI = Integral gain and KD =
Derivative gain.

PID Controller structure

In this tutorial, we assume the controller is used in a
closed-loop unity feedback system. The variable e denotes the tracking error, which is sent to the PID controller. The control signal u from the controller to the plant is equal to the
proportional gain (KP ) times the magnitude of the error plus the integral gain (KI ) times the integral of the error plus the derivative gain (KD ) times the derivative of the error.
u = KP e + KI

edt + KD

de
dt

Why learn the PID controller?
Because PID Controllers are everywhere! Due to its simplicity and excellent if not optimal performance in many applications, PID controllers are used in more than 95% of closed-loop
industrial processes.1 It can be tuned by operators without
extensive background in Controls, unlike many other modern
controllers that are much more complex but often provide only marginal improvement. In fact, most PID controllers are tuned on-site. Although we are learning all the theories in ME475 to design the controller, the lengthy calculations for an initial guess of PID parameters can often be circumvented if we

know a few useful tuning rules. This is especially useful when the system is unknown.
1

Astrom K. J. and Hagglund T. H., “New tuning methods for PID controllers”, Proceedings of the 3rd European Control Conference, 1995

How do the PID parameters aﬀect system
dynamics?
We are most interested in four major characteristics of the
closed-loop step response. They are
1. Rise Time: the time it takes for the plant output y to rise beyond 90% of the desired level for the ﬁrst time.
2. Overshoot: how much the the peak level is higher than
3. Settling Time: the time it takes for the system to
4. Steady-state Error: the diﬀerence between the
steady-state output and the desired output.

How do the PID parameters aﬀect system
dynamics?
The eﬀects of increasing each of the controller parameters KP , KI and KD can be summarized as
Response Rise Time
KP
Decrease
KI
Decrease
KD
NT

Overshoot Settling Time S-S Error
Increase
NT
Decrease
Increase
Increase
Eliminate
Decrease
Decrease
NT

NT: No deﬁnite trend. Minor change.
You may want to take notes of this table. It will be useful in the later part of the lesson.

How do we use the table?
Typical steps for designing a PID controller are
1. Determine what characteristics of the system needs to be
improved.
2. Use KP to decrease the rise time.
3. Use KD to reduce...