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

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 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.

Case 1: Real and Distinct Roots

Case 1: Real and Distinct Roots Example

Case 2: Complex Conjugate Roots

Case 3: Repeated Roots

Heaviside Expansion

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

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

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

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 Real Imaginary Plane of the Open Loop Transfer Function B(0) B(i  ) -1 is called the critical point B(-i  )

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  )

Example Plot the Nyquist plot for Im Re Stable

Example Plot the Nyquist plot for Im 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(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. Show with Sisotool

Summary Open Loop System Nyquist Plot Simple Nyquist Theorem Nyquist Gain Scaling Conditional Stability Full Nyquist Theorem Next Class: Stability Margins Im Re Unstable