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Unformatted text preview: 16.30/31, Fall 2010 — Recitation # 13 Brandon Luders December 6, 2010 In this recitation, we tie the ideas of Lyapunov stability analysis (LSA) back to previous ways we have demonstrated stability but also show that LSA can prove stability in cases where linear stability analysis cannot. 1 MassSpringDamper, Linear Spring Consider one of the simplest secondorder systems, the massspringdamper system: All motion is in the horizontal direction and denoted by x , where x is positive when the mass is moving to the right. The mass has a. . . , well, mass of m > 0. It is hooked to a damper with damping coeﬃcient c > 0; it exerts a force of − cx ˙. It is also hooked to a linear spring with spring constant k > 0; it exerts a force of − kx . Finally, a constant force F > is applied to the mass in the + x direction; note that is a constant force, not a system input. Thus this this is an autonmous system, of the form x ˙ = f ( x ) for some choice of x . Recall from your early statics and dynamics courses that you can derive the system dy namics by summing the forces and setting F = ma : mx ¨ = − cx ˙ − kx + F F = mx ¨ + cx ˙ + kx ⇒ We now get to the point of this recitation: prove that this is a stable system. This is an obvious conclusion, but there are many, many ways to prove it and part of the point of this recitation is to tie these methods together. Here are just some of the ways we can do that: 1. Solve for the trajectory x ( t ), using the above differential equation, and show that it converges to some equilibrium. 2. Derive a linearized firstorder system x ˙ = A x , and show that all eigenvalues are in the lefthand plane. 3. Use the Lyapunov stability theorem. Let’s consider each of these cases in turn. 1. The solution x ( t ) to this secondorder differential equation is the sum of the homogeneous solution and the particular solution: x ( t ) = x h ( t ) + x p ( t ) ....
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 Fall '04
 EricFeron
 Stability theory, Lyapunov, Lyapunov Stability Theorem

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