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

2. 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

3. Partial Fraction Expansion
When using Partial Fraction Expansion, our objective is to turn the Transfer Function into 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.

4. Case 1: Real and Distinct Roots

5. Case 1: Real and Distinct Roots Example

6. Case 2: Complex Conjugate Roots

7. Case 3: Repeated Roots

8. Heaviside Expansion

9. Loop Nomenclature Disturbance/Noise Reference Error Input signal+ -
Output
y(s)
Error
signal
E(s)
Open Loop
Signal
B(s)
Plant
G(s)
Sensor
H(s)
Prefilter
F(s)
Controller
C(s)
Disturbance/Noise
Reference
Input
R(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

10. Closed Loop System Error signal Input Output E(s) r(s) y(s) Controller
C(s)
Plant
P(s) +
Open Loop
Signal
B(s) -1

11. Open Loop System Error signal Input Output E(s) r(s) y(s) Controller
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)
Input
r(s)
Output
y(s)
Controller
C(s)
Plant
P(s) +
Open Loop
Signal
B(s)
Sensor -1

12. Open Loop System Nyquist Plot
Error
signal
E(s) +
Output
y(s)
Open Loop
Signal
B(s)
Plant
P(s)
Controller
C(s)
Input
r(s)
Sensor -1
Imaginary
B(-iw)
Plane of the Open Loop
Transfer Function -1
B(0)
Real
B(iw)
-1 is called the
critical point

13. Simple Nyquist Theorem
Error
signal
E(s) +
Output
y(s)
Open Loop
Signal
B(s)
Plant
P(s)
Controller
C(s)
Input
r(s)
Sensor -1 -1
Real
Imaginary
Plane of the Open Loop
Transfer Function
B(0)
B(iw)
-1 is called the
critical point
Stable
Unstable
-B(iw)
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.

14. Example -1 Im Re
Plot the Nyquist plot for
Stable

15. Example -1 Im Re
Unstable
Plot the Nyquist plot for

16. Nyquist Gain Scaling
The form of the Nyquist plot is scaled by the system gain
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17. Conditional Stabilty
Whlie most system increase stability by decreasing gain, some can be stabilized by increasing gain
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18. 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.
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19. Summary Open Loop System Nyquist Plot Simple Nyquist Theorem-1 Im Re
Unstable
Open Loop System
Nyquist Plot
Simple Nyquist Theorem
Nyquist Gain Scaling
Conditional Stability
Full Nyquist Theorem
Next Class: Stability Margins