Stability

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Unformatted text preview: Automatic Control - 1 ( EPM 381) Automatic Prof. Dr. Hamdy El-Goharey Section 5: Stability complex GH-plane complex s-plane mapping s ⇒ GH(s) Im{GH(jω)} jω σ Re{GH(jω)} -1 RHS Copyright © 2007 by Hamdy S. K. El-Goharey. All rights reserved 12/13/2008 HSK - EPM 381 - Stability 1 Learning objectives • To state the definition of stability • To understand the Routh-Hurwitz stability criterion • To understand the concept of the Nyquist stability criterion • To calculate the gain and phase margins of the system from the frequency response • To establish the relationship between the Bode plots and stability analysis 12/13/2008 HSK - EPM 381 - Stability 2 Stability Definitions • Bounded Input Bounded Output Stability: A system is BIBO stable if, for every bounded input, system BIBO stable the output remains bounded with increasing time (all the all system poles must lie in the left half of the s-plane). system • Marginal Stability: A system is marginally stable if some of the poles lie system on the imaginary axis, while all others are in the imaginary while LHS of the s-plane. Some inputs may result in the LHS plane. output becoming unbounded with time. 12/13/2008 HSK - EPM 381 - Stability 3 Stability Analysis • To test the stability of a Linear Time Invariant To test the (LTI) system we need only to examine the poles of the system, i.e. the roots of the c/c equation. c/c • Methods are available for testing for roots with Methods positive real parts, which do not require the positive which actual solution of the ch...
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