autonomous_2009_10_26_01

autonomous_2009_10_26_01 - 10 - 1 Autonomous Linear...

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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
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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 )
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10 - 3 Autonomous Linear Dynamical Systems S. Lall, Stanford 2009.10.26.01 Example 1: Autonomous linear dynamical systems ˙ x = b - 1 0 2 1 B x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
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10 - 4 Autonomous Linear Dynamical Systems S. Lall, Stanford 2009.10.26.01 Example 2: Autonomous linear dynamical systems ˙ x = b - 0 . 5 1 - 1 0 . 5 B x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
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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 ˙ x 2 x 2 = C ˙ x 1 and hence ˙ x = Ax + Bu where A = b 0 1 /C - 1 /L - R/L B B = b 0 1 /L B
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0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4
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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 - 10 - 1 Autonomous Linear...

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