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┃     .l....                                   ┃
┗━━━━━━━━━━━━━━━━━━━━━━━┛
o      0.5
1-5     2     2.5     3     3.5     4
     Mach Number
   b)
Fig. 4.33    C,r.g and Cmq for the aircraft of Example 4.10.
parameter k appearing in Eq. (4.554) was chosen equal to 0.8. The planform
efficiency parameter e was evaluated using Eq. (4.478) based on exposed aspect
rat"io At.We have r = 3 deg,  zw  =  1.27 rn, and span b = 17.3228 m. For ALE  = 45
deg, A = 0.25, and Ae = 2.6893, we obtain from Fig. 4.25, (pciplk)cL =O.M =O =
-0.225. Similarly, the val_ues at other Mach numbers are obtained and the data are
curve :fitted to obtain the following expression:
( p~   ),,=o(;) = 0.3708 M3 _ 0.6662 M2+0.3128 M - 0.2325lrad
    For supersonic speeds, Datcod data are used. Because the Datcom7 method is
quite involved, the details are not presented in the text. The calculated values are
curve fitted to obtain the following expression:
(C   ) = -0.3806 M3 +3.422 M2 _ 10.1458 M +10.1346/rad
The contribution of the vertical tail was determined using Eq. (4.545), where the
lift-curve slope ay was evaluated at supersonic speeds using the methods discussed
in Chapter 3 and as explained in Example 3.8.
"
      ~
520           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
and
                   K -[ki k2  . . kn]              (5,266)
The full-state feedback law in the transformed z-space is given by
or, in the original state-space,
u - -Kz +r(t)
(5.267)
                                            u - -K Px + r(t)                                 (5.268)
so that the given system with full-state feedback is given by
x - Ax + Bu
  = (A - KPx)x+ Br(t)
(5.269)
(5.270)
    6) Perform a simulation to verify the design.
     The advantages of expressing the given plant in the phase-variable form is that
equations fort~ie gains k: are uncoupled and kr can be easily obtained as given
i:Eq. (5.265). However, if the plant is not controllable, then it is not possible to
represent it in the phase-variable form. For such a case, the above design procedure
remains same except for the fact the equations for k, will be coupled. Then the
gains ki have to be obtained by solving the n coupled algebraic equations.
5- t0-10   Dual Phase-Variable Form
The state-space representation, which is in the form,
where
x-
A-
x = Ax + BU
-an_i 1 o 0 . 0
-an_2 0 1 0 , o
    .         .
-a2  O O . 1 0
-ai  .    .  .  1
 -ao   .  .  .  -  .
B-
(5.271)
(5.272)
is said to be in dual phase-variable form. Similar to the phase-variable form, the
elements of the first column of the matrix A in dual phase-variable form constitute
the coefficients of the characteristic equation as follows:
s"  + a,r_isn-l  + an~2Sn-2 + ... + a} s + ao  - O                  (5.273)
Furthermore, this form of representation of the system matrix A is ver)r useful in
the design of state observers, which we will be discussing a little later.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       521
5.10-11   Conversion of Transfer Function Form to Dual
          Phase-Variable Form
     We willillustrate this method with the help of an example. Consider once again
the system (see Fig. 5.41) given by
       G(s)= k+~b++b+a+)b3     (5.274)
　
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