1. Stability Margins Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Stability Margins

2. Outline of Today’s Lecture
Review
Open Loop System
Nyquist Plot
Simple Nyquist Theorem
Nyquist Gain Scaling
Conditional Stability
Full Nyquist Theorem
Is stability enough?
Margins from Nyquist Plots
Margins from Bode Plot
Non Minimum Phase Systems

3. 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

4. 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

5. 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.

6. Nyquist Gain Scaling
The form of the Nyquist plot is scaled by the system gain

7. Conditional Stabilty
Whlie most system increase stability by decreasing gain, some can be stabilized by increasing gain
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8. Definition of Stable
A system described the solution (the response) is stable if that system’s response stay arbitrarily near some value, a, for all of time greater than some value, tf.

9. 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.

10. Determination of Stability from Eigenvalues
Continuous Time
Discrete Time
Unstable
Stable
Asymptotic Stability

11. Is Stability Enough?
If not Why Not?

12. 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?

13. Gain and Phase Margin Definition Nyquist Plot-1

14. Example
Using Matlab command
nyquist(gs)

15. Example Here the gain from the previous plot has been
multiplied by
The result is that
stability is about to be
lost

16. Example
Using Matlab command
nyquist(gs)

17. Gain and Phase Margin Definition Bode Plots
Magnitude, dB
Positive Gain Margin w
Phase, deg
-180
Phase Margin w
Phase Crossover Frequency

18. Example
Using Matlab command
bode(gs)

19. Example
Again, stability is about to
be lost.

20. Example
Using Matlab command
bode(gs)

21. Note
The book does not plot the Magnitude of the Bode Plot in decibels.
Therefore, you will get different results than the book where decibels are required.
Matlab uses decibels where needed.

22. Stability Margin
It is possible for a system to have relatively large gain and phase margins, yet be relatively unstable.
Stability
margin, sm

23. 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:

24. Another Non Minimum Phase System A Delay
Delays are modeled by the function which multiplies the T.F.

25. Summary Is stability enough? Margins from Nyquist Plots
Margins from Bode Plot
Non Minimum Phase Systems
Next Class: PID Controls