Nonlinear Control Sy.. - Free

6.5. CLOSED-LOOP INPUT-OUTPUT STABILITY 169
H(34
IIHII.
Figure 6.6: Bode plot of H(jw)I, indicating the IIHII. norm of H.
Definition 6.11 We will denote by feedback system to the interconnection of the subsystems
H1 and H2 : Xe -* Xe that satisfies the following assumptions:
(i) e1, e2i yl, and y2 E Xe for all pairs of inputs ul, u2 E Xe .
(ii) The following equations are satisfied for all u1, u2 E Xe :
el = ul - H2e2 (6.23)
e2 = u2 + H1e1. (6.24)
Here ul and u2 are input functions and may represent different signals of interest such as
commands, disturbances, and sensor noise. el and e2 are outputs, usually referred to as
error signals and yl = H1e1, Y2 = H2e2 are respectively the outputs of the subsystems H1
and H2. Assumptions (i) and (ii) ensure that equations (6.23) and (6.24) can be solved
for all inputs U1, U2 E Xe . If this assumptions are not satisfied, the operators H1 and H2
do not adequately describe the physical systems they model and should be modified. It
is immediate that equations (6.23) and (6.24) can be represented graphically as shown in
Figure 6.7.
In general, the subsystems H1 and H2 can have several inputs and several outputs.
For compatibility, it is implicitly assumed that the number of inputs of H1 equals the
number of outputs of 112, and the number of outputs of H1 equals the number of inputs
of H2. We also notice that we do not make explicit the system dimension in our notation.
For example, if X = ,C,,,, the space of functions with bounded absolute value, we write
H : Xe --> Xe (or H : £ooe -* L.,) regardless of the number of inputs and outputs of the
system.
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