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)