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(5.207)
(5.208)
 (
            !
t
i
!
.l
tL
512           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
[2  }] [::] =A[::]
(5.209)
which is of the form Au - A,u. This equation is of the same form as Eq. (5.199)
 with A, as the eigenvalue of the matrix A. Furthermore, we recognize that the above
matrix A is th~ same as matrix A in Eq. (5.196), which has the eigenvalues of
-1 and 3 and eigenvectors of [-1]2  1)T and [1  2lT. This means, with \. = -1,
UI - -1/2, and U2 -. 1, we get the first solution as
XI = _~e-r     X2 = e-t
and, with A. - 3, ui - 1, and U2 = 2, we get the second solution as
XI -. e3r
Therefore, the general solution is given by
X2 = 2e3t
XI = _~e-r +e3r      X2 = e-'+2e3r
(5.210)
(5.211)
(5.212)
  Thus, we observe that the eigenvalues determine the nature of the transient
response and the eigenvectors determine the amplitude of this response.
     The concept of controllability is linked to the question of whether the input u
affects or controls the variation of each one of the state variables x,. If it does,
then we say that the given system is controllable. On the other hand, if any of
the state variables are not influenced by the input u, then the system is said to be
uncontrollable. Alternatively, if we can take the system from a given initial state
x(0) to a specified final state x(t f ) using the available control u, then the system
is said to be controllable. If not, the system is uncontrollable.
     The given nth order linear, time-invariant system
x - Ax + Bu
   y = Cx
is said to be controllable if the matrix
Qc-[B AB A2B ... Al-IB]
(5.213)
(5.214)
(5.215)
is nonsingular or has full rank n. The matrix Qc iS called the controllability
matrix.l-3
     The concept of observability is related to the question whether each one of the
state variables' affects or controls the variation of the output y. If the answer to
this question is yes, then the system is said to be observable. If not, the system is
<I:r:f:<
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW       513
unobservable. Thus, for an unobservable
all of the output variables.
system, the input does not affect some or
   A given linear, time-invariant system in the state-space form is said to be ob-
servable if the matrix
Qo -
(5.216)
is nonsingular or has the full rank n. The matrix Qo is called the observability
matrix.l-3
Let us suppose that we have a state-space representation in the form
where
A-
x - Ax + Bu
0
O
l
.    .
0
0
0
           '        -an - I
(5.217)
(5.218)
(5.219)
(5.220)
Equation (5,217) with matrices A and B given by Eqs. (5,219) and (5.220) is called
the phase-variable form of state Eq. (5.171). The advantage of the phase- variable
lr
 l\
   !
 -
 {
!
j;'

1  0
O  1
0  O
  .        .
  .       .

514
PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
form is that there is a minimum amount of coupling between the state variables.
 For example,  given all  the initial conditions, xi(0), X2(0), - . . , Xn (O),  we  can  first
solve xi(0) : X2(0) and obtain xi(At) by an integration over a small time step
  At. Similarly, we can get X2(At), X3(At),. - . ,  x" (At) by successively solving the
 other equations.
    The phase-variab/e form of representation has another advantage that the ele-
 ments of the last row constitute the coefficients of the characteristic equation as
follows:
A(s) = s   +an _lSn-l.+...+alS +00 = 0                (5.221)
This property is useful in the design of compensators using the pole-placement
method, which we will discuss a little later.
5.10.6   Conversion of Differential Equations to
          Phase-Variable Form
    Let the given dynamical system be represented by the following linear differ-
　
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