56
given by (4-21) is exponentially stable. This completes the proof.
In order to interpret Theorem 4.2 we need certain continuity
"fe *Hp
conditions to hold. That is, F (t) and G (t) should be continuous
functions of the reference r (t) and the disturbance w (t). Then
assumptions (i) and (ii) are realistic since both iiF (t) Fn^ and
iiG (t) G II.. will be small whenever nr (t)ii and iiw (t) it are
small. Furthermore, because F (t) and G (t) are often periodic due to
'ic "fc
the periodicity of x (t), u (t), and w (t), and will be,
* o
respectively, the maximum values that nF (t) F n.. and
* o
nG (t) G n.. assume over one period.
As a final point, note that if in condition (iii) of Theorem 4.2
the word "stabi.1 izable" is replaced by "controllable" then the feedback
gains Kj and l<2 can be selected to arbitrarily assign the eigenvalues of
the system (4-21). This may, in turn, make it possible to obtain a
large ratio of a/m with suitably chosen feedback gains. Then, provided
that nK^, Kg]ii- does not become too large, L will increase and hence
larger reference and disturbance signals will be allowed. Note also
that if the input enters into the nonlinear system by a linear time-
invariant mapping, will be zero so that increasing the ratio a/m will
always increase L.
We now give a rather lengthy example which makes use of many of the
results obtained so far for the nonlinear servomechanism problem. In
this example, simulated test results are provided to show the per
formance of the control algorithm. Also, simulations are provided which
show the consequence of using a controller based on linear servo
mechanism theory.