Page 893 - The Mechatronics Handbook

0066_Frame_C30 Page 4 Thursday, January 10, 2002 4:43 PM

Lemma 30.1 (Two Norm of a Stable System)
Consider a causal stable LTI strictly proper system F = [A, B, C]. It follows that

||F|| 2 = + = CL c C H = B L o B (30.5)
H
L R )
H ||f || 2 (
where L c is the system controllability gramian and L o is the system observability gramian. The controllability
gramian
∞ H
def
L c = ∫ e BB e t d (30.6)
H A t
At
0
is the unique symmetric (at least) positive semi-definite solution of the algebraic Lyapunov equation
H H
AL c + L c A + BB = 0 (30.7)

L c is positive definite if and only if (A, B) is controllable. The observability gramian

∞ H
def
L o = ∫ e A t C Ce d t (30.8)
H
At
0
is the unique symmetric (at least) positive semi-definite solution of the algebraic Lyapunov equation
H
H
A L o + L o A + C C = 0 (30.9)
L o is positive definite if and only if (A, C) is observable.

Comment 30.4 (HH 2 Norm May Mislead—LL ∞∞ ∞ ∞ Norm Is Important)
2 2
It is important to note that the H /L norm (or energy) of a function may be very small, while the function
itself may be very large in amplitude. Consider a tall thin pulse, for example. This observation is critical
because there are many important cases in which we are very concerned with the height of a function—
more so than its energy. A good example of this comes from classical Nyquist stability theory [2,8].
Nyquist taught us that the peak magnitude of the sensitivity function S = 1/(1 + L) associated with a
standard negative feedback loop is very important in terms of the feedback loop’s stability robustness. A
large sensitivity means that the Nyquist plot comes close to the critical −1 point—implying that a small
perturbation (or unanticipated modeling error) may cause the closed loop system to go unstable. To
assist us with this fundamental issue we may use frequency dependent weighting functions, but what we
∞ ∞
really need is a norm that directly addresses such concerns. This motivates the so-called H and L
∞ ∞
norms as well as H /L control theory [4,11].
Comment 30.5 (Computation of HH 2 Norm in MATLAB)
2
The H norm of a system F = [A, B, C, D] may be computed using the following MATLAB command
sequence:
lc = lyap (a, b∗b’)
twonorm = sqrt (trace(c∗lc∗c’))
or
lo = lyap (a’, c∗c’)
twonorm = sqrt (trace(b’∗lo∗b)).

©2002 CRC Press LLC