Loop transfer function
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Loop Transfer Function. Imaginary. -B( i w ). Plane of the Open Loop Transfer Function. -1 is called the critical point. -1. B(0). Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo. Real. Stable. B( i w ). Unstable.

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Loop Transfer Function

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Loop transfer function

Loop Transfer Function

Imaginary

-B(iw)

Plane of the Open Loop

Transfer Function

-1 is called the

critical point

-1

B(0)

Professor Walter W. Olson

Department of Mechanical, Industrial and Manufacturing Engineering

University of Toledo

Real

Stable

B(iw)

Unstable


Outline of today s lecture

Outline of Today’s Lecture

  • Review

    • Partial Fraction Expansion

      • real distinct roots

      • repeated roots

      • complex conjugate roots

  • Open Loop System

  • Nyquist Plot

  • Simple Nyquist Theorem

  • Nyquist Gain Scaling

  • Conditional Stability

  • Full Nyquist Theorem


Partial fraction expansion

Partial Fraction Expansion

  • When using Partial Fraction Expansion, our objective is to turn the Transfer Functioninto a sum of fractions where the denominators are the factors of the denominator of the Transfer Function:Then we use the linear property of Laplace Transforms and the relatively easy form to make the Inverse Transform.


Case 1 real and distinct roots

Case 1: Real and Distinct Roots


Case 1 real and distinct roots example

Case 1: Real and Distinct RootsExample


Case 2 complex conjugate roots

Case 2: Complex Conjugate Roots


Case 3 repeated roots

Case 3: Repeated Roots


Heaviside expansion

Heaviside Expansion


Loop nomenclature

Loop Nomenclature

Disturbance/Noise

Reference

Input

R(s)

Error

signal

E(s)

Output

y(s)

Controller

C(s)

Plant

G(s)

Prefilter

F(s)

Open Loop

Signal

B(s)

Sensor

H(s)

+

+

-

-

The plant is that which is to be controlled with transfer function G(s)

The prefilter and the controller define the control laws of the system.

The open loop signal is the signal that results from the actions of the

prefilter, the controller, the plant and the sensor and has the transfer function

F(s)C(s)G(s)H(s)

The closed loop signal is the output of the system and has the transfer function


Closed loop system

Closed Loop System

Error

signal

E(s)

Output

y(s)

Input

r(s)

Controller

C(s)

Plant

P(s)

Open Loop

Signal

B(s)

-1

+

+


Open loop system

Note: Your book uses L(s) rather than B(s)

To avoid confusion with the Laplace transform, I will use B(s)

Open Loop System

Error

signal

E(s)

Output

y(s)

Input

r(s)

Controller

C(s)

Plant

P(s)

Open Loop

Signal

B(s)

Sensor

-1

+

+


Open loop system nyquist plot

Open Loop SystemNyquist Plot

Error

signal

E(s)

Output

y(s)

Input

r(s)

Controller

C(s)

Plant

P(s)

Open Loop

Signal

B(s)

Imaginary

B(-iw)

Plane of the Open Loop

Transfer Function

Sensor

-1

-1

B(0)

Real

+

+

B(iw)

-1 is called the

critical point


Simple nyquist theorem

Simple Nyquist Theorem

Error

signal

E(s)

Output

y(s)

Input

r(s)

Imaginary

Controller

C(s)

Plant

P(s)

-B(iw)

Plane of the Open Loop

Transfer Function

Open Loop

Signal

B(s)

-1 is called the

critical point

-1

B(0)

Sensor

-1

Real

Stable

B(iw)

Unstable

+

+

Simple Nyquist Theorem:

For the loop transfer function, B(iw), if B(iw) has no poles in the right hand side, expect for simple poles on the imaginary axis, then the system is stable if there are no encirclements of the critical point -1.


Example

Example

  • Plot the Nyquist plot for

Im

-1

Re

Stable


Example1

Example

  • Plot the Nyquist plot for

Im

-1

Re

Unstable


Nyquist gain scaling

Nyquist Gain Scaling

  • The form of the Nyquist plot is scaled by the system gain

  • Show with Sisotool


Conditional stabilty

Conditional Stabilty

  • While most system increase stability by decreasing gain, some can be stabilized by increasing gain

  • Show with Sisotool


Full nyquist theorem

Full Nyquist Theorem

  • Assume that the transfer function B(iw) with P poles has been plotted as a Nyquist plot. Let N be the number of clockwise encirclements of -1 by B(iw) minus the counterclockwise encirclements of -1 by B(iw)Then the closed loop system has Z=N+P poles in the right half plane.

  • Show with Sisotool


Summary

Summary

  • Open Loop System

  • Nyquist Plot

  • Simple Nyquist Theorem

  • Nyquist Gain Scaling

  • Conditional Stability

  • Full Nyquist Theorem

Im

-1

Re

Unstable

Next Class: Stability Margins


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