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                                     +1 [ka2-(s) - b3y(s)l)]                                               (5.278)
                      s
Let
                                     SX3(S) = ka2r(s) - b3y(s)                         (5.279)
                                                    SX2(S) - kair (s) - bzy(s) + X3(S)                                (5.280)
                                               SX1(S) - -bi y(s) + X2(S) + kr (s)                             (5.281)
so that
                                              y(s) = xi(s)                                 (5.282)
Talang the inverse Laplace transforms, we obtain the desired dual phase-variable
  b::-",l[x:l+[::lkr,t) (5.283,
and
                                            y(t) = xi(t)                                (5,284)
'I"'!

  'i

 

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522            PERFORMANCE, STABiLITY, DYNAMICS, AND CONTROL
5.10.12 ObserverDesign
     The pole-placement design method requires that all the state variables are accu-
rately measured and are available for feedback.lf this requirement is met and the
 system is controllable, then a complete control over all the eigenvalues is possible.
A problem arises if some or all of the states are not actually measured or are not
 avaYi:lable for state feedback. An obvious solution would be to add more sensors that
 can measure the missing states. However, this approach may not always be feasible
and often can be quite expensive. The other option is to estimate the unavailable
states using a subsystem called a state observer. An observer that estimates all the
states, including t<ose that are actually measured, is called a full-state observer,
 and one that estimates only those states that are not measured is called a reduced-
state observer. Here, we will discuss the procedure for the design of a full-state
observer.
      The design of an observer is based on the knowledge of a mathematical model
of the plant, input(s), and output(s). The basic idea is to make the estimated states
as close to the actual states as possible, but the problem is that all the actual states
are not available for comparison. However, we do know the output of the given
 plant, and we can compare it with the estimated output of the observer. The design
objective is then to drive the error between the actual and estimated outputs to zero
　
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