Professor Walter W. Olson Department of Mechanical, Industrial
and Manufacturing Engineering University of Toledo Stability
Margins Slide 2 Outline of Todays 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 Slide 3 Loop Nomenclature Reference Input R(s) + - 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 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 Slide 4 Open
Loop System + + Output y(s) Error signal E(s) Open Loop Signal B(s)
Plant P(s) Controller C(s) Input r(s) Note: Your book uses L(s)
rather than B(s) To avoid confusion with the Laplace transform, I
will use B(s) Sensor Slide 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 Simple Nyquist Theorem: For the loop
transfer function, B(i ), if B(i ) 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. Real Imaginary Plane of the Open Loop Transfer Function
B(0) B(i ) -1 is called the critical point Stable Unstable -B(i )
Slide 6 Nyquist Gain Scaling The form of the Nyquist plot is scaled
by the system gain Slide 7 Conditional Stabilty Whlie most system
increase stability by decreasing gain, some can be stabilized by
increasing gain Show with Sisotool Slide 8 Definition of Stable A
system described the solution (the response) is stable if that
systems response stay arbitrarily near some value, a, for all of
time greater than some value, t f. Slide 9 Full Nyquist Theorem
Assume that the transfer function B(i ) with P poles has been
plotted as a Nyquist plot. Let N be the number of clockwise
encirclements of -1 by B(i ) minus the counterclockwise
encirclements of -1 by B(i )Then the closed loop system has Z=N+P
poles in the right half plane. Slide 10 Determination of Stability
from Eigenvalues Continuous TimeDiscrete Time Unstable Stable
Asymptotic Stability Slide 11 Is Stability Enough? If not Why Not?
Slide 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? Slide 13 Gain and Phase Margin
Definition Nyquist Plot Slide 14 Example Using Matlab command
nyquist(gs) Slide 15 Example Here the gain from the previous plot
has been multiplied by 3.2359 The result is that stability is about
to be lost Slide 16 Example Using Matlab command nyquist(gs) Slide
17 Gain and Phase Margin Definition Bode Plots Positive Gain Margin
Phase Margin -180 0 Phase, deg Magnitude, dB Phase Crossover
Frequency Slide 18 Example Using Matlab command bode(gs) Slide 19
Example Again, stability is about to be lost. Slide 20 Example
Using Matlab command bode(gs) Slide 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. Slide 22 Stability Margin It is
possible for a system to have relatively large gain and phase
margins, yet be relatively unstable. Stability margin, s m Slide 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: Slide 24 Another Non Minimum Phase
System A Delay Delays are modeled by the function which multiplies
the T.F. Slide 25 Summary Is stability enough? Margins from Nyquist
Plots Margins from Bode Plot Non Minimum Phase Systems Next Class:
PID Controls
