# PID controller

**Topics:**PID controller, Control theory, Settling time

**Pages:**13 (1030 words)

**Published:**July 22, 2014

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

the steady state, normalized against the steady state.

3. Settling Time: the time it takes for the system to

converge to its steady state.

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...

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