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Unformatted text preview: L e c t u r e 1 6 Although numerical integration can solve nonlinear equations, the solutions are not easy to understand, even after postprocessing. The approach is to find ways to approximate the nonlinear equations with linear equations. This course presents: (i) Constant speed linearization (ii) Small signal perturbation The benefit of having linear equations is that students have already done courses on linear systems and can solve first and second order linear equations. From this background, the course will develop the methods of solving Norder linear equations expressed in a matrix form u B x A dt x d ] [ ] [ + = (1a) for ) ( x o x = (1b) where [A] and [B] are constant NxN and NxM matrices, x is a Norder statevector and u is an Morder input vector. Revision using scalar example—constant input Let q be a scalar state variable and U be a constant value input in the equation to be solved: bU aq dt dq + = (2) where ) ( q q = It is assumed that t ss q q t q + = ) ( (3) Substituting (3) in (2) bU q q a dt q q d t ss t ss + + = + ) ( ) ( (4) Equation (4) is broken up as: bU aq dt dq ss ss + = (5) and t t aq dt dq = (6) Steadystate solution Since U is a constant, the solution of (5) can be a constant also or SS ss Q q = (7) Since = dt dQ ss , from (5) U a b Q SS − = (8) Homogeneous solution Taking the Laplace Transform of (6), ) ( ) ( = − s q a s t (9) The characteristic equation of (9) is ) ( = − a s ( 1 ) Therefore, at t Ce q = ( 1 1 ) where C is a constant of integration which is evaluated from the initial condition. Substituting (7) and (11) in (3) at Ce U a b t q + − = ) ( ( 1 2 ) At t=0 , (12) becomes C U a b q + − = ( 1 3 ) Solving for C in (13) at at Ce q e U a b t q ) 1 ( ) ( + − − = (14) Revision using scalar example—sinusoidal input Repeat the solution of bu aq dt dq + = for the case where t Q t P u ω ω sin cos + = (15) The homogeneous solution can be reused, i.e at t Ce q = ....
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 Spring '09
 BonTekOoi
 Derivative, Elementary algebra, Constant of integration, Nonlinear system

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