PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL4(13)

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      Because the given plant is not in phase-variable form, we have to do a transfor-
mation and express it in the phase-variable form. Then, do the full-state feedback
design in this new transformed space, Finally, do a reverse transformation to obtain
the full-state feedback law in the original state-space.
    The phase~variable form is given by
Z = ApZ +'BpU
(6.343)
where Z is the new state vector and matrices Ap and Bp are in the phase-variable
form
 Z-. PX
Ap = PAP-I
  Bp = PB
(6.344)
(6.345)
(6.346)
    The state vector x was defined in Eq. (6.183). The transformation matrix P is
constructed as follows.
    1) Obtain the lateral directional controllability matrix Qc given by
Qc-[B AB' A2B A3B A4B]      (6,347)
2) Form matrices Pi to Ps as follows:
            Pi-[0 0 O 0 l]Qci
                                             P2 = Pi A
                                                                P.3 = PI A2
                                               P4 = Pl A3
                                 P5 =

Then the phase-variable forms Ap and Bp are given by
                                 Ap = PAP-i
                                   Bp = PB
(6.348)
(6.349)
(6.350)
(6.351)
(6.352)
(6.353)
(6.354)
(6.355)
610           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
     For the general a\riation airplane, the lateral-directional'controllability matrix is
given by
           0   5.0  -4.0  -36.0   162.0
           0   3.0  -33.0  212.0  -1664.0
   Qc- 3.0 -33.0 212.0 -1664.0 14,376.0
          O   -5.0  3.0   30.0   -115.0
           -5.0  3.0   30.0 . -115.0   511.0
so that
        0.0000  -0.6953 -0.0717 -0.0369 -0.2549
       -3.8467  0.3386  0.0499  -4.2354 -0.0312
 Qci= -1.1859 0.0187  0.0063 -1.2253 -0.0147
       -0.7785  0.0926  0.0131  -0.8208 -0.0046
        -0.0815  0.0113  0.0016  -0.0867 -0.0003
        -0.0815  0.0113  0.0016  -0.0867 -0.0003
       -0.0070 -0.0148 -0.0022   0   -0.0014
     P -   0.0320   -0.0013   0.0046      0      0.0031
        -0.0692  0.0058  -0.0415   0   -0.0241
        0.5791  -0.0126  0.3650    0   -0.0039
       0.0000  1.0000   0.0000   0-0000   0.0000
       0.0000  0.0000   1-0000   0.0000   0.0000
 Ap= 0.0000 0.0000  0.0000  1.0000  0.0000
       0.0000  0.0000  0.0000   0.0000   1.0000
　
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