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               d
-CmuU- C"dcld +Cma)A.+T:(l,.,dd -Cnq,,)AO-C,mStA8e (6.3)
    For the study of airplane response, it is convenient to express Eqs. (6.1-6.3) in
the state-space form as follows:
du   1
d    = ~[(C.,, + glCz")u + (Cxcy + €iCzcr)Aa + [CrqCI + gi(m t  + CzqCI)] q
+ (Cx0 + 91'CzO)AO + (Cx8, + 91Cz8,)A8e]
(mi - Czd CI)
(6.4)
[Cz"u + Cza Aa + (m i  + Czq CI )q + Cz0 AO + Cz8e A8e]
(6.5)
dq   l
d    =  I~ [(C "" + g2Czu)U + (Cm , + g2CZa)Aa + [Cmq CI + g2(m I + Czq Cl )] q
+ g2Cz0 AO + (CmaFe + 92CzS.) A8e]
where
Let
dA9
dr -q
(6.6)
(6.7)
(6.8)
(6.9)
XI -U    x2-Act   X3-q    X4-AO        (6.10)
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL         539
Then, Eqs. (6.4-6.7) can be expressed in the state-space form as
where
and

a13 :=
X -.AX + BU
     ali  a12 013  /
A= a:,i a22 a2
    a31  a32 a33  t
    a41  a42 a43 t:
aii ==
Cxu + gl Czu
mi
a12 -
Cxq Cl + €i (mi + Czq Cl)
            Ttl I + CzqCI
a23 - -
              mi  - Czcr Cl
         Cmu + g2Czu
a31 :-. -
     Iyl
a33 -
Cxa, + gl Cza
a22 - -
   m]
a24 - -
   mi
mi
Czct
- Cza CI
Cz8
- CI Czci,
         Cmcr + 92Cza
a32 - -
      l,,l
Cmq CI + g2(1Tll + Czq Cl)
  2m
mi  ::: -
  'I       p Uo S
For free response, U - O so that
a34 = g-IC
CI = 2U
     1),
/),l = ~p U2Sc
X - AX
A solution to Eq. (6.13) can be obtained in the usual way by assuming
X = XoeA,
(6.12)
(6.13)
(6.14)
l:l

   mc
        Cx0 + gl Cz0
a14 - -
     mi
a21= Cz"
        mi - Czacl
041-0  a42-0  a43-1   a14-0
bi=Cx8,+91Cz8, b2-' Cz8,
                m]                    mi - ci Czd
         b3 = Cnae + g2C28e
                  y           b4 -0
 Cx&ci
gl  := -
       mi - Czotci
               Cmd CI
g2 -. ~
       mi - CZaCl
540
so that
and
or
PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
X = XoleA'
Xt,AeAr _ AXoeXr -: O
(AI - A)Xo = O
where I is the identity matrix
            100

          000
     For nontrivial solutions, the determinant of (AI - A) must be zero
                                               IAI - Al = 0
(6.15)
(6.16)
(6.17)
(6.18)
(6.19)
An expansion of the determinant in Eq. (6.19) results in a fourth-order algebraic
equation of the form
where
A8A4 + B8A3 + C8A2 + D8A + E8 - O
A8 = mil).i(mi - CzaCI)
(6.20)
(6.21)
Ba  = rTIi (-l}.i Czcr - Cmq CI [lTl.l  - Cza CIl - IT11Cma,CI)
       - Cx,,l).i(mi - CzaCI) - Cxd,CICzulyl                       (6.22)
C8 = ITI,I (Cza Cmq Cl  - tTZl Cma! - Czq CmerC21)
           - Cxu(-Iyl Czat - Cmq CI UTI1  - Cza CIl - itTIl Cma,CI )
　
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