autonomous_2009_10_26_01_2up

autonomous_2009_10_26_01_2up - 10 1 Autonomous Linear...

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Unformatted text preview: 10 - 1 Autonomous Linear Dynamical Systems S. Lall, Stanford 2009.10.26.01 10. Autonomous Linear Dynamical Systems • Linear dynamical systems • Phase plane • RLC circuits • Chemical reactions • Markov chains • Higher-order linear systems • Mechanical systems • Linearization 10 - 2 Autonomous Linear Dynamical Systems S. Lall, Stanford 2009.10.26.01 Autonomous linear dynamical systems continuous-time autonomous LDS has form ˙ x = Ax • x ( t ) ∈ R n is called the state • n is the state dimension or (informally) the number of states • A is the dynamics matrix (system is time-invariant if A doesn’t depend on t ) picture ( phase plane ): x 1 x 2 x ( t ) ˙ x ( t ) = Ax ( t ) 10 - 3 Autonomous Linear Dynamical Systems S. Lall, Stanford 2009.10.26.01 Example 1: Autonomous linear dynamical systems ˙ x =- 1 0 2 1 x-2-1.5-1-0.5 0.5 1 1.5 2-2-1.5-1-0.5 0.5 1 1.5 2 10 - 4 Autonomous Linear Dynamical Systems S. Lall, Stanford 2009.10.26.01 Example 2: Autonomous linear dynamical systems ˙ x =- . 5 1- 1 . 5 x-2-1.5-1-0.5 0.5 1 1.5 2-2-1.5-1-0.5 0.5 1 1.5 2 10 - 5 Autonomous Linear Dynamical Systems S. Lall, Stanford 2009.10.26.01 Linear circuit- u + R C L Let x 1 be the voltage across C and x 2 be the current through L , then u = Rx 2 + x 1 + L ˙...
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This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.

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autonomous_2009_10_26_01_2up - 10 1 Autonomous Linear...

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