lec7_print

# Linear matrix differential equation integrating both

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Linear Matrix Differential Equation Integrating both sides from time 0 to t , we obtain e - At X ( t ) - IX (0) = Z t 0 e - At BU ( t ) dt (6) Solving for X ( t ), we get X ( t ) = e At . X (0) + e At . Z t 0 e - At BU ( t ) dt (7) Srinivas Palanki (USA) Solution of Linear Differential Equations 5 / 15

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Linear Matrix Differential Equation The solution of the differential equation is given by: X ( t ) = e At . X (0) + e At . Z t 0 e - At BU ( t ) dt X ( t ) is a vector of dimension n e At is an n × n matrix ; thus e At . X (0) is a vector of dimension n B is an n × m matrix and U is a vector of dimension m ; thus e At . Z t 0 e - At BU ( t ) dt is a vector of dimension n If we can calculate e At , we can solve for X ( t ) . Srinivas Palanki (USA) Solution of Linear Differential Equations 6 / 15
Analytical Calculation of e At e At as a Finite Sum If A is an n x n matrix and P A ( λ ) = λ n + a 1 λ n - 1 + ... + a n - 1 λ + a n is its characteristic polynomial , then e At = Ψ 0 ( t ) I + Ψ 1 ( t ) A + ... + Ψ n - 1 A n - 1 (8) where Ψ 0 ( t ) , Ψ 1 ( t ) , ..., Ψ n - 1 ( t ) are scalar functions of time. The functions Ψ i ( t ) can be computed from the eigenvalues of the matrix A and for n = 2 , 3, this computation can be done easily as shown in the next slide. Srinivas Palanki (USA) Solution of Linear Differential Equations 7 / 15

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Analytical Calculation of e At Analytical Calculation of e At when n = 2 Suppose the eigenvalues of the matrix A are λ 1 and λ 2 .
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