Lecture_F11_ECH157_stability - and then analyze the...

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Stability properties of steady state The stability properties of the steady state of a system of linear equations ˙ x = Ax (1) can be determined by examining the matrix A. If the matrix A has all its eigenvalues on the left half of the complex plane (i.e., A is a Hurwitz matrix) then the steady state (which is the origin in this particular case) of the system is stable; if A has at least one eigenvalue on the right half of the complex plane, the steady state is unstable. For a system of nonlinear equations ˙ x = f ( x ) (2) the stability properties of its steady state can be studied by linearizing the nonlinear equations first
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Unformatted text preview: and then analyze the eigenvalues of the Jacobian. If the Jacobian is a Hurwitz matrix, then the steady state of the nonlinear system is stable; if at least one of the eigenvalues of the Jacobian is on the right half of the complex plane, then the steady state is unstable; if the Jacobian has some of its eigenvalues on the imaginary axis and the other eigenvalues on the left half of the complex plane, then linearization fails to determine the stability of the steady state. For more information, you are referred to Chapter 7 in the textbook as well as the Appendix....
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This note was uploaded on 01/28/2012 for the course ECH 157 taught by Professor Palagozu during the Fall '08 term at UC Davis.

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