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response, i.e., the desired characteristic equation is given by
Sn + dn _lSn -1  + ... + d] s + do - 0                           (5.263)
5) Equate coefficients of the two charactenstic equations
so that
dn -i  - an -i + kn, . - ', di  -. ai + k2, do - ao + ki                   (5.264)
ki  - do - ao, k2 = di - ai, . .., kn  - dn -i  - an -1                    5.265)
t)
1~
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)
Rewrite this in the following form:
                     k+ 2k+ 2k
           G(s)-.-\     -        (5.275)
            l -+b+~+b
y(s)
= r(s~                        (5.276)
or
     y(s)(l+b +b +b)=f(s)(k+ k+ k)     (5.277)
Then,
                            y(s) = 1 [-b,y(s) + kf (s) + 1 ([-(s)kai - b2y(s)]
                                                      s
　
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