On feedback control systems with saturable drives and noisy measurements

We present some intuitive results concerning the dynamic performance of a basic feedback control system (FBCS), as constrained by a peak-limited drive and noisy measurements. We use a frequency domain approach based on the signal-to-noise ratio (
S/N
) properties of the control loop rather than on the usual small signal transfer characteristics. The drive and noise constraints are introduced explicitly into the equations governing the
S/N
, and the minimum requirements for active feedback control are expressed as minimum requirements on the
S/N
at appropriate points in the control loop. Thus, an intuitively derived requirement that the
S/N > 1
in the output of the FBCS leads to an upper hound on the realizable control bandwidth ω
ch
, such that
\omega_{ch} \leq \frac{L^{2}|G(\omega_{ch})|^{2}}{4\Phi_{n}}
where
L
is the peak drive level,
\Phi_{n}
, is the amplitude of the power density. spectrum of the measurement noise (assumed "white"), and
|G(\omega_{ch})|
is the frequency response of the plant. This is a hard upper bound on the maximum realizable ω
ch
, independent of design considerations. We also find that the nonlinear interaction of signal and noise in a peak limited drive generates the most severe requirements for active feedback control in these basic systems, namely that the
S/N > 1.2
at the input to the drive. This leads to a minimum ω
ch
, such that
\omega_{ch} \leq \frac{L^{2}|G(\omega_{ch})|^{2}}{4\Phi_{n}}(\frac{a_{nch}}{a_{n}})^{2}
where a
n
is the noise at the input to the drive and a
nch
is that part of a
n
that falls within ω
ch
. As (10) suggests, maximizing the dynamic performance of a FBCS calls for a low-pass filter at the input to the drive to minimize the above band part of a
n
thereby making
a_{nch}/a_{n} \rightarrow 1
. The implications of these results are illustrated by two designs. One is a classic linear design using a familiar form of equalizing network. An
S/N
analysis shows an
a_{nch}/a_{n} \cong 0.10
, indicating that this design developes only 10 percent of the inherent dynamic tracking potential for the system. The primary reason is that the requirements for attenuating a
n
for high dynamic performance potential, and the phase advance required for stability are not compatible in linear systems where the relationship between the shape of the attenuation-versus frequency and the phase-versus frequency characteristics are controlled by the minimum attenuation and phase relations for linear networks. The other is a nonlinear system using a first generation fast time model prediction scheme for stabilization. It is chosen to demonstrate stabilization by nonlinear techniques which are not subject to the tight coupling between attenuation and phase. The scheme also illustrates, some of the possibilities that nonlinear strategies can provide for improving the dynamic behavior by using various bits of knowledge about the system to good advantage. This nonlinear system shows an
a_{ncb}/a_{n}
of 0.4 or 40 percent.