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One such way is to use the Laplace transform. It is not difficult to see that the Laplace transform of eAt , denoted by L eAt , is L eAt = sI − A −1 Hence, eAt is the inverse Laplace transform of Is − A −1 . eAt = L−1 sI − A −1 Let us show the computation in the following example. 1 Let 1 0 0 2 A= then s 0 1 0 s−1 0 − = 0 s 0 2 0 s−2 ⎡ ⎤ 1 −1 0 s−1 0 ⎢ ⎥ sI − A −1 = = ⎣ s−1 1 ⎦ 0 s−2 0 s−2 sI − A = The inverse Laplace Transform can be calculated as eAt = L−1 sI − A −1 = Let us consider another example.
Such a transfer function model is most suitable for linear time-invariant systems with a single input and a single output. If the system to be controlled is nonlinear, or time-varying, or has multiple inputs or outputs, then it will be difficult, if not impossible, to model it by a transfer function. Therefore, for nonlinear, time-varying, or multi-input–multi-output systems, we often need to use state space representation to model the systems. The state variables of a system are defined as a minimum set of variables such that the knowledge of these variables at any time t0 , plus the information on the input subsequently applied, is sufficient to determine the state variables of the system at any time t > t0 .
IF) If there exists no x0 = 0 such that y t = CeAt x0 = 0 for all t ≥ 0, then ∀x0 = 0 ∃t ≥ 0 y t = CeAt x0 = 0 ⇒ ∀x0 = 0 ∃t ≥ 0 CeAt x0 ⇒ ∀x0 = 0 ∃ > 0 T CeAt x0 > 0 CeAt x0 0 ⇒ ∀x0 = 0 ∃ > 0 x0T T CeAt CeAt x0 dt > 0 T 0 CeAt dt x0 > 0 In other words, there exists > 0 such that the matrix M = 0 CeAt T CeAt dt is positive definite. In particular, M −1 exists. Using this result, we can deduce x0 from y t = CeAt x0 as follows. y t = CeAt x0 ⇒ CeAt T y t = CeAt T CeAt x0 ⇒ ⇒ ⇒ 0 0 0 CeAt T y t dt = CeAt T CeAt x0 dt 0 CeAt T y t dt = 0 CeAt T y t dt = M ⇒ x0 = M −1 CeAt T CeAt dt x0 x0 CeAt T y t dt 0 Hence, the system A C is observable.
Airbus A320 SOP 09After Landing