lecture 20 stability
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MAE 506: Advanced System Modeling, Dynamics and Control
Lecture 20 – Stability Analysis
Ch. 6 in text
Spring Berman Fall 2014
Phase Portraits • Drawn from the vector field of a dynamical system • Each point represents a possible system state, x • An arrow shows the velocity at state x • xeq = equilibrium point / stationary point / critical point
x
x =x1x2
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f1(x1, x2 )f2 (x1, x2 )
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x = f(xeq ) = 0
K. J. Astrom and R. M. Murray, Feedback Systems, 2011 (online)
K. J. Astrom and R. M. Murray, Feedback Systems, 2011 (online)
Example: Inverted Pendulum
c = viscous friction coefficient (system has damping) u = input = force applied at the pivot State variables:
Nonlinear, time-invariant, 2nd-order system
No control input (u = 0) à Equilibrium points are:
n even: pendulum up n odd: pendulum down
Phase Portrait of Uncontrolled, Damped Inverted Pendulum
K. J. Astrom and R. M. Murray, Feedback Systems, 2011 (online)
Nonlinear Systems Can Exhibit Rich Behaviors
• Limit cycles (closed trajectory in state space)
• Multiple equilibria • Stability about an equilibrium point is dependent on initial
conditions • States can go to infinity in finite time • Small changes in initial conditions may lead to widely diverging outcomes: system is deterministic but not predictable (chaos)
Electronic oscillator model
Lorenz attractor (a set of chaotic solutions of the Lorenz system)
Dynamical Behavior of LTI Systems
x =Ax
"Phase plane nodes" by Maschen - Own work. Licensed under Creative Commons Zero, Public Domain Dedication via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Phase_plane_nodes.svg#mediaviewer File:Phase_plane_nodes.svg
xy!
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A BC D
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Characteristic polynomial:
à
Dynamical Behavior of LTI Systems
Dynamical Behavior of LTI Systems
Dynamical Behavior of LTI Systems
Phase Portraits: Unforced Inverted Pendulum
K. J. Astrom and R. M. Murray, Feedback Systems, 2011 (online)
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