Nonlinear Control Sy.. - Free

9.10. SOME REMARKS ABOUT CONTROL DESIGN 249
Figure 9.5: Feedback Interconnection used in example 9.2.
A bit of work shows that the feedback system of Figure 9.5 can be redrawn as shown in Figure
9.6, which has the form of the standard setup of Figure 9.4. The control design problem can
now be restated as follows. Find C such that
(i) stabilizes the feedback system of Figure 9.6, in some specific sense, and
(ii) reduces the effect of the exogenous signal d on the desired output z. D
A lot of research has been conducted since the early 1980s on the synthesis of controllers that
provide an "optimal" solution to problems such as the one just described. The properties
of the resulting controllers depend in an essential manner on the spaces of functions used
as inputs and outputs. Two very important cases are the following:
L2 signals:
In this case the problem is to find a stabilizing controller that minimizes the L2 gain
of the (closed-loop) system mapping d -+ z. For linear time-invariant systems, we saw
that the L2 gain is the H,,. norm of the transfer function mapping d -4 z, and the
theory behind the synthesis of these controllers is known as H, optimization. The
H. optimization theory was initiated in 1981 by G. Zames [99], and the synthesis of
H... controllers was solved during the 1980s, with several important contributions by
several authors. See References [25] and [100] for a comprehensive treatment of the
H. theory for linear time-invariant systems.