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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 Mass-Spring-Damper, Linear Spring Consider one of the simplest second-order systems, the mass-spring-damper 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 first-order system x ˙ = A x , and show that all eigenvalues are in the left-hand plane. 3. Use the Lyapunov stability theorem. Let’s consider each of these cases in turn. 1. The solution x ( t ) to this second-order 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
- Stability theory, Lyapunov, Lyapunov Stability Theorem