l15 - 16.06 Lecture 15 Solutions of State Space Dierential...

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16.06 Lecture 15 Solutions of State Space Differential Equations John Deyst October 8, 2003 Today’s Topics 1. General solution of state space D.E.s 2. Quanser example solution for constant input. 3. Stability 1
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Complete Solution of the State D.E.s Thus far we have obtained the homogeneous solution for the state as But we need the total solution to for non-zero inputs. We assume a solution of the form where f ( t ) is some, as yet undetermined, vector valued function of time. First we differentiate and substitute our assumed solution 2
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Referring back to our original differential euqation it is apparent that we must have or, using Property 1 of state transition matrices Now integrate both sides from time zero to time t so Referring back to the assumed solution we know that so 3
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and hence so, using Property 2 of state transition matrices The integral on the right is the vector/matrix equivalent of the scaler convolution integral.
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