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PID Control. Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo. Outline of Today’s Lecture. Review Margins from Nyquist Plots Margins from Bode Plot Non Minimum Phase Systems Ideal PID Controller Proportional Control

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Pid control

PID Control

Professor Walter W. Olson

Department of Mechanical, Industrial and Manufacturing Engineering

University of Toledo


Outline of today s lecture

Outline of Today’s Lecture

  • Review

    • Margins from Nyquist Plots

    • Margins from Bode Plot

    • Non Minimum Phase Systems

  • Ideal PID Controller

  • Proportional Control

  • Proportional-Integral Control

  • Proportional-Integral Derivative Control

  • Ziegler Nichols Tuning


Margins

Margins

  • Margins are the range from the current system design to the edge of instability. We will determine

    • Gain Margin

      • How much can gain be increased?

      • Formally: the smallest multiple amount the gain can be increased before the closed loop response is unstable.

    • Phase Margin

      • How much further can the phase be shifted?

      • Formally: the smallest amount the phase can be increased before the closed loop response is unstable.

    • Stability Margin

      • How far is the the system from the critical point?


Gain and phase margin definition nyquist plot

Gain and Phase Margin DefinitionNyquist Plot

-1


Gain and phase margin definition bode plots

Gain and Phase Margin DefinitionBode Plots

Magnitude, dB

0

Positive Gain Margin

w

Phase, deg

-180

Phase Margin

w

Phase Crossover Frequency


Stability margin

Stability Margin

  • It is possible for a system to have relatively large gain and phase margins, yet be relatively unstable.

Stability

margin, sm


Non minimum phase systems

Non-Minimum Phase Systems

  • Non minimum phase systems are those systems which have poles on the right hand side of the plane: they have positive real parts.

    • This terminology comes from a phase shift with sinusoidal inputs

    • Consider the transfer functions

      • The magnitude plots of a Bode diagram are exactly the same but the phase has a major difference:


Another non minimum phase system a delay

Another Non Minimum Phase SystemA Delay

  • Delays are modeled by the function which multiplies the T.F.


Proportional integral derivative controller

Proportional-Integral-DerivativeController

Based on a survey of over eleven thousand controllers in the refining, chemicals and pulp and paper industries, 97% of regulatory controllers utilize PID feedback.

L. Desborough and R. Miller, 2002 [DM02].

  • PID Control, originally developed in 1890’s in the form of motor governors, which were manually adjusted

  • The first theory of PID Control was published by a Russian (Minorsky) who was working for the US Navy in 1922

  • The first papers regarding tuning appeared in the early 1940’s

    • Today. there are several hundred different rules for tuning PID controllers (See Dwyer, 2003, who has cataloged the major methods)

  • While most of the discussion is about the “ideal” PID controller, there are many forms of the PID controller


Pid control1

PID Control

  • Advantages

    • Process independent

      • The best controller where the specifics of the process can not be modeled

    • Leads to a “reasonable” solution when tuned for most situations

    • Inexpensive: Most of the modern controllers are PID

    • Can be tuned without a great amount of experience required

  • Disadvantages

    • Not optimal

    • Can be unstable unless tuned properly

    • Not dependent on the process

    • Hunting (oscillation about an operating point)

    • Derivative noise amplification


The ideal pid controller

The Ideal PID Controller

  • The input/output realtionship for the PID Controller is the Integral-Differential Equation

  • The ideal PID controller has the transfer function

  • Structurally it would look like

+

+

+

-

+


The ideal pid controller1

The Ideal PID Controller

  • The system transfer function is

+

+

+

-

+


Proportional control

Proportional Control

Y(s)

R(s)

+

+

+

-

+


Proportional control1

Proportional Control


Proportional controller

Proportional Controller

kp=7.2


Proportional controller1

Proportional Controller

  • With kp=7.2,

  • We could reduce the error with a prefilter:

}

Error

Also: the response time is poor

+

+

+

+

+

Y(s)=a(s)

R(s)

0.139

-1


Proportional integral controller

Proportional – Integral Controller

  • Most controllers using this technology are of this form:

  • This reacts to the system error and reduces it

Y(s)

R(s)

+

+

+

-

+


Proportion integral control

Proportion-Integral Control

  • Applying PI control to the F-16 Elevator,

Response time

improved with

no error


Proportional integral derivative control

Proportional- Integral-DerivativeControl

  • The derivative component is rarely used.

  • Reduces overshoot

  • May slows the response time depending on the system

  • Sensitive to noise

  • For the F-16 Elevator


Pid tuning

PID Tuning

  • Tuning is the choosing of the parameters kd, ki, and kp, for a PID Controller

  • The oldest and most used method of tuning are the Ziegler-Nichols (ZN) methods developed in the 1940’s.

  • The first method is based on the assumption that the process without its feedback loop performs with a 1st order transfer function, perhaps with a transport delay

  • The second method assumes that a higher order system has dominant poles which can be excited by gain to the point of steady oscillation

  • In order to establish the constants for computing the parameters simple tests are performed of the process


Ziegler nichols pid tuning method 1 for first order systems

Ziegler-Nichols PID Tuning Method 1 for First Order Systems

  • A system with a transfer function of the formhas the time response to a unit step input:

  • This response might also be generated from a higher order system that is has high damping.


Ziegler nichols pid tuning method 1 for first order systems1

Ziegler-Nichols PID Tuning Method 1 for First Order Systems

  • The advice given is to draw a line tangent to the response curve through the inflection point of the curve.

    • However, a mathematical first order response doesn’t have a point of inflection as it is of the form (at no place does the 2nd derivative change sign.) My advice: place the line tangent to the initial curve slope

    • Adjust the gain K of the system by multiplying compensator by 1/K

Rise

Time

T

Lag L


Ziegler nichols pid tuning method 1 for first order systems2

Ziegler-Nichols PID Tuning Method 1 for First Order Systems

  • For this example,

Rise

Time

T

Lag L


Ziegler nichols pid tuning method 2 for unknown oscillatory system

Ziegler-Nichols PID TuningMethod 2 for Unknown Oscillatory System

  • The form of the transfer function unknown but the system can be put in steady oscillation by increasing the gain:

  • Increase gain, K, on closed loop system until the gain at steady oscillation, Kcr, is found

  • Then measure the critical period, Pcr

  • Apply table for controller constants and multiply by system gain 1/K

1

Cycle

Pcr


Ziegler nichols pid tuning method 2 example

Ziegler-Nichols PID TuningMethod 2 Example

k=1000

k=1500

k=1750

k=2000

1

Cycle

Pcr

17.8

Kcr=1875

1.0


Ziegler nichols pid tuning method 2 example1

Ziegler-Nichols PID TuningMethod 2 Example

1

Cycle

Pcr

17.8

Kcr=1875

1.0


Summary

Summary

  • Ideal PID Controller

  • Proportional Control

  • Proportional-Integral Control

  • Proportional-Integral Derivative Control

  • Ziegler-Nichols Tuning

Next Class: PID Controls Continued


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