# Loop Transfer Function - PowerPoint PPT Presentation

1 / 19

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Loop Transfer Function

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## 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

• 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

• 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.

### 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

Error

signal

E(s)

Output

y(s)

Input

r(s)

Controller

C(s)

Plant

P(s)

Open Loop

Signal

B(s)

-1

+

+

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

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

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 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

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

• Plot the Nyquist plot for

Im

-1

Re

Stable

### Example

• Plot the Nyquist plot for

Im

-1

Re

Unstable

### Nyquist Gain Scaling

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

• Show with Sisotool

### Conditional Stabilty

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

• Show with Sisotool

### 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

• Open Loop System

• Nyquist Plot

• Simple Nyquist Theorem

• Nyquist Gain Scaling

• Conditional Stability

• Full Nyquist Theorem

Im

-1

Re

Unstable

Next Class: Stability Margins