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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Lecture 25 Laplace–domain solution of the State equations In the previous lecture, we derived the following state–space model for the uncompen- sated 2.004 Tower: q ˙ ( t ) = Aq ( t ) + b w ( t ) , [dynamics–equation of motion] (1) y ( t ) = cq ( t ) , [output or observation equation] (2) where the state vector is q 1 ( t ) x 1 ( t ) q ( t ) = = . (3) q 2 ( t ) v 1 ( t ) and the system matrix and input vector, respectively, are 1 A = , b = . (4) − k 1 /m 1 − b 1 /m 1 1 /m 1 The wind force (disturbance) is denoted as w ( t ) while the output vector c might be (1 0) or (0 1), depending on whether one wishes to define the tower displacement or velocity, respectively, as output; or c might be any linear combination of displacement and velocity such as (0 . 1 . 9). In this lecture, we will solve the state equations in the Laplace domain and show how we can obtain the transfer function from the state equations. Before studying these notes, you are advised to make sure that you are thoroughly familiar with matrices. You can find a Math supplement on matrices in the 2.004 Stellar website. Derivation of the transfer functions for position and velocity Before we knew about state space, we would have obtained the transfer functions for position or velocity, respectively, directly from the equation of motion as follows. First we would have Laplace–transformed the equation of motion to obtain m 1 s 2 X 1 ( s ) + b 1 sX 1 ( s ) + k 1 X 1 ( s ) = W ( s ) . (5) If the desired output were the displacement x 1 ( t ), then the transfer function would be found directly as X 1 ( s ) 1 /m 1 = . (6) W ( s ) s 2 + ( b 1 /m 1 ) s + ( k 1 /m 1 ) This is of the form ω 2 K n , where s 2 + 2 ζω n s + ω n 2 1 k 1 ω n = (7) m 1 b 1 ζ = (8) 2 √ k 1 m 1 1 K = . (9) k 1 This is our familiar 2 nd –order system which is undamped if ζ = 0, underdamped if < ζ < 1, critically damped if ζ = 1, and overdamped if ζ > 1....
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