MECH 466 - Lecture6-Stability-2009W

# Mech466 automatic control 13 200809 course roadmap

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Unformatted text preview: ol ??? 13 2008/09 Course roadmap Modeling Analysis Laplace transform Transfer function Models for systems • electrical • mechanical • electromechanical Linearization Time response • Transient • Steady state Frequency response • Bode plot Stability • Routh-Hurwitz Routh• Nyquist MECH466 : Automatic Control 14 Routh-Hurwitz criterion Design This is for LTI systems with a polynomial This polynomial denominator (without sin, cos, exponential etc.) cos, It determines if all the roots of a polynomial It Design specs Root locus lie in the open LHP (left half-plane), lie halfor equivalently, have negative real parts. or Frequency domain PID & Lead-lag LeadDesign examples It also determines the number of roots of a It polynomial in the open RHP (right half-plane). halfIt does NOT explicitly compute the roots. It No proof is provided in any control textbook. No Matlab simulations & laboratories 2008/09 MECH466 : Automatic Control 15 2008/09 MECH466 : Automatic Control 16 4 Polynomial and an assumption Routh array Consider a polynomial Consider From the given polynomial Assume Assume If this assumption does not hold, Q can be factored as If where The following method applies to the polynomial The 2008/09 MECH466 : Automatic Control 17 Routh array (How to compute the third row) 2008/09 MECH466 : Automatic Control 2008/09 MECH466 : Automatic Control 18 Routh array (How to compute the fourth row) 19 2008/09 MECH466 : Automatic Control 20 5 Routh-Hurwitz criterion Example 1 Routh array The number of roots in the open right half-plane halfis equal to the number of sign changes in the first column of Routh array. 2008/09 MECH466 : Automatic Control 21 Two sign changes in the first column 2008/09 Example 2 Two roots in RHP MECH466 : Automatic Control Example 3 (from slide 14) Routh array Routh array Always same! No sign changes in the first column 2008/09 22 Always same! No roots in RHP MECH466 : Automatic Control No sign changes in the first column 23 2008/09 No roots in RHP MECH466 : Automatic Control 24 6 Examples Simple & important criteria for stability All roots in open LHP? 1st order polynomial Yes / No 2nd order polynomial Yes / N...
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## This document was uploaded on 03/05/2014 for the course MECH 466 at The University of British Columbia.

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