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week8review_2011_05_22_06

# week8review_2011_05_22_06 - 8-1 Review Session for Week 8...

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8 - 1 Review Session for Week 8 Scott Hsieh, Stanford 2011.05.22.06 Review Session 8 Autonomous linear dynamical systems as problem setup Review of solution techniques Practice problems

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8 - 2 Review Session for Week 8 Scott Hsieh, Stanford 2011.05.22.06 The Autonomous LDS The fundamental description of the autonomous LDS in discrete time is x ( t + 1) = Ax ( t ) By iterating, solutions are of the form x ( t ) = A t x (0) In continous time, we have: ˙ x ( t ) = Ax ( t ) We know from lecture that solutions are of the form x ( t ) = e At x (0) An alternative (engineering) approach to thinking about the continuous-time problem is to discretize the problem into small time intervals. Using the definition of the derivative, we have for very small Δ t that x ( N Δ t ) = (1 + A Δ t ) N x (0)
8 - 3 Review Session for Week 8 Scott Hsieh, Stanford 2011.05.22.06 The Matrix Exponential For these two solutions to be consistent, we need that e A = lim k →∞ (1 + A/k ) k (This matches the most common definition of e from scalar math.) While equivalent, the matrix exponential is usually defined by its power series, not by this limit. In analogy with

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week8review_2011_05_22_06 - 8-1 Review Session for Week 8...

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