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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
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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
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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.
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Case 1: Real and Distinct Roots
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Case 1: Real and Distinct Roots Example
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Case 2: Complex Conjugate Roots
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Case 3: Repeated Roots
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Heaviside Expansion
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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
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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
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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
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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
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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.
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Example -1 Im Re Plot the Nyquist plot for Stable
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Example -1 Im Re Unstable Plot the Nyquist plot for
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Nyquist Gain Scaling The form of the Nyquist plot is scaled by the system gain Show with Sisotool
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Conditional Stabilty Whlie most system increase stability by decreasing gain, some can be stabilized by increasing gain Show with Sisotool
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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
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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
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