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By A. W. Babister

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0 0 ... 0 0 ... 0 1 ... 0 0 ... 1 Àa0 3 2 3 b0 7 Àa1 7 6 b 7 7 6 1 7 Àa2 7 6 . 7u(t); x(t0 ) ¼ x0 x(t) þ 7 6 . 7 7 4 . 5 7 5 bnÀ1 Àan (2:14) z(t) ¼ [0 0, . . 8. 8 in their ‘‘transpose’’ form we can obtain the observable canonical form: 2 0 0 x_ ¼ 4 1 0 0 1 3 2 3 À4 7 À6 5x þ 4 4 5u(t); À4 2 y ¼ [0 0 1]x 32 Flight Mechanics Modeling and Analysis The observability matrix of this system is obtained as Ob ¼ obsv(A,C) (in MATLAB) 2 0 Ob ¼ 4 0 1 3 0 1 1 4 5 4 10 Since the rank (Ob) ¼ 3, this state-space system is observable, and hence the name for this canonical form.

X1 0 þ u(t); z(t) ¼ [1 x2 1 x 0] 1 x2 ! 10. 10 without the Bu term is called the homogeneous equation. 6 Let the state-space model of a system have the following matrices [2]:    À2 0 1   A ¼  1 À2 0 ;  1 1 À1    1   B ¼  0 ; 1 C ¼ [2 1 À 1] Obtain the TF of this system as well as the Bode diagram. 9). We see that the Bode diagram of this seemingly three degrees of freedom (3DOF) system looks like that of a first-order system. 0. 0 and the (denominator) complex pole, effectively the frequency response degenerates to that of a TF with reduced order.

T is sufficient to determine the behavior of the system for all (future) T ! t. In order to arrive at the solution to an n-order differential equation completely, we must prescribe n initial conditions at the initial time ‘‘t0’’ and the forcing function for all (future times) T ! t0 of interest: n quantities are required to establish the system ‘‘state’’ at t0. However, t0 can be any time of interest, so one can see that n variables are required to establish the state at any given time. Let a second-order differential equation be given as d2 z(t) dz(t) þ a1 þ a0 z(t) ¼ u(t) dt 2 dt (2:8) Let z(t) !

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Aircraft Stability and Control by A. W. Babister


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