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MIT6_241JS11_lec09

MIT6_241JS11_lec09 - 6.241 Dynamic Systems and Control...

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6.241 Dynamic Systems and Control Lecture 9: Transfer Functions Readings: DDV, Chapters 10, 11, 12 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology March 2, 2011 E. Frazzoli (MIT) Lecture 9: Transfer Functions Mar 2, 2011 1 / 13

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Asymptotic Stability (Preview) We have seen that the unforced state response ( u = 0) of a LTI system is easily computed using the A matrix in the state-space model: x [ k ] = A k x [0] , or x ( t ) = e At x (0) . A system is asymptotically stable if lim t + x ( t ) = 0, for all x 0 . Assume A is diagonalizable, i.e., V 1 AV = Λ, and let r = Vx be the vector of model coordinates. Then, r i [ k ] = λ k i r i [0] , or r i ( t ) = e λ i t r i (0) , i = 1 , . . . , n . Clearly, for the system to be asymptotically stable, | λ i | < 1 (DT) or Re ( λ i ) < 0 (CT) for all i = 1 , . . . , n . It turns out that this condition extends to the general (non-diagonalizable) case. More on this later in the course. E. Frazzoli (MIT) Lecture 9: Transfer Functions Mar 2, 2011 2 / 13
(Time-domain) Response of LTI systems summary Based on the discussion in previous lectures, the solution of initial value problems (i.e., the response) for LTI systems can be written in the form: k 1 y [ k ] = CA k x [0] + C A k i 1 Bu [ i ] + Du [ t ] i =0 or t y ( t ) = C exp( At ) x (0) + C exp( A ( t τ )) Bu ( τ ) d τ + Du ( t ) . 0 However, the convolution integral (CT) and the sum in the DT equation are hard to interpret, and do not offer much insight. In order to gain a better understanding, we will study the response to elementary inputs of a form that is particularly easy to analyze: the output has the same form as the input. very rich and descriptive: most signals/sequences can be written as linear combinations of such inputs.

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