Page 79 - Anatomy of a Robot

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64 CHAPTER TWO
HOW MANY VARIABLES CAN BE
CONTROLLED AT THE SAME TIME?
Practically speaking, the LMS algorithm can handle an arbitrary number of simultane-
ous variables. However, as the number of variables increases, the danger of interactions
increases drastically. The primary danger is that unknown interactions between the vari-
ables will throw off the calculations and destabilize the control system. This often shows
up in the math if the variables are not completely independent. In our example, the
derivative of X1 with respect to X2 may not truly be zero, or vice versa. This would
greatly compromise the stability of the stepping iterations. As a general rule, try not to
use a single control system to handle too many variables at the same time. Two to four
variables is a good place to stop.

WHAT IS THE EQUIVALENT FOR
STEADY STATE ERROR WHEN USING
MULTIPLE VARIABLES?
First of all, where multiple variables exist, be aware it’s entirely possible the system will
never come to a steady state. However, it is possible for the digital calculations to set-
tle into a completely stable and quiet solution. Such a solution would have X(t) stable
and equal to Xd(t).
However, with certain minimal step sizes, it may not be possible to converge on a
quiet solution. Think for a minute of a system at 9, seeking 10, with a back and forth
minimal step size of 2. The system will likely bounce back and forth from 9 to 11 and
back to 9 forever. A carefully designed control algorithm can avoid such a problem, but
we leave it up to the reader to work this out.

HOW DO YOU EVALUATE THE RELATIVE STATE
OF THE CONTROL SYSTEM? HOW FAR IS IT
FROM THE OPTIMAL CONTROL STATE?
WHAT IS THE ERROR SIGNAL?
For an LMS system, you can track the size of the cost function. All the terms in the
sum are positive, squared numbers. The magnitude can be used as a measure of the
state of the system. We clearly want it to be small. Further, the first derivative of the
cost function should be quiet. The relative noise level of the cost function is a meas-
ure of the volatility of the system and it can be used to indicate disruptions at the inputs
of the system.