lec3 - Outline Motivation Phase Portraits Singular Points...

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1 ME6402, Nonlinear Control Systems • Motivation • Phase Portraits • Singular Points • Constructing Phase Portraits • Phase Portrait of Linear Systems Outline What is Phase Plane Analysis (PPA)? ± Phase plane analysis is a graphical method for studying 2nd order systems. ± PPA shows the free response of the system in the phase-plane for different initial conditions. ± PPA is an old method; it was introduced over a century ago by mathematicians such as Poincare.

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2 ME6402, Nonlinear Control Systems Useful Properties of the PPA ± It provides a visual ‘signature’ for a nonlinear system ± It can handle ‘hard’ nonlinearities ± Applicable to many practical control systems often approximated 2nd order ± Disadvantage : Limited to 2nd order systems Phase Portrait The phase portrait of a 2nd order system is the graph of the free response of the 2nd vs. 1st state for different initial states. = = ) , ( ) , ( 2 1 2 2 2 1 1 1 x x f x x x f x & & x 1 x 2 The state-space (x1-x2 plane) is called the phase-plane
3 ME6402, Nonlinear Control Systems A Mass-Spring System Example = = 1 2 2 1 x x x x & & State-Equations : Singular Points Singular points : Equilibrium points in the phase- plane, i.e., states that satisfy 0 ) , ( ) , ( 2 1 2 2 1 1 = = x x f x x f Bifurcation Example : 0 3 = + α + + x x x c x & & &

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4 ME6402, Nonlinear Control Systems Bifurcation Example Constructing Phase Portraits ± The commonly used methods are ± Analytical (limited applicability) ± Isoclines Method (graphical) ± Computer Numerical Solution
5 ME6402, Nonlinear Control Systems

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lec3 - Outline Motivation Phase Portraits Singular Points...

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