Page 76 - Anatomy of a Robot

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What equivalent exists for a “steady state error” in a system with multiple vari-
ables? CONTROL SYSTEMS 61
How do we evaluate the relative state of the control system? How far is it from the
optimal control state? What is the error signal?
How can we alter the design of the system to affect its performance?
Let’s look at the first question.

HOW WILL WE DESIGN THE MULTIPLE VARIABLE
SYSTEM? WHAT FRAMEWORK WILL IT HAVE?

Let’s assume for simplicity’s sake that we are trying to design a control system to control
just two variables at the same time: X1 and X2 (perhaps position and velocity). The fol-
lowing discussion can be generalized to n variables (X1, X2, X3 ...Xn) on the reader’s
own time. We can call the combination of the variables X1 and X2, the vector X.
Let’s assume that the desired state of the two control variables is as follows:
X1 X1d
X2 X2d
We can call the desired state of vector X, the vector Xd.
If computers are used in the control system, the computer periodically finds a way to
change X based on the value of Xd. In such a control system, we speak of computations
executed at periodic, sequential times labelled t 1, t, t 1, and so on. We use the fol-
lowing notation:

X(t 1) shows the values of X at the previous computation time.
X(t) shows the values of X at the present computation time.
X(t 1) shows the values of X at the next computation time.
Similarly, Xd(t) represents the time series of values for Xd.
To compute the next value of X1, for instance, the computer will look at the previ-
ous and present values of both X1 and X2 and determine which way to change X1 in
an incremental way. The same computation is done for X2. Done properly, X1 and X2
will slowly track the desired values. But how do we go about finding the iteration?
Iteration is a process of repeating computations in a periodic manner toward some par-
ticular goal. Usually, an iteration equation governs the process of iteration. The follow-
ing is a general-purpose iteration equation that is often used in robots. X(t) is computed
by iteration by taking values at time t and iterating to the next value at time t 1:

X1t 12 X1t2 S1t2 1d C1X1t22>d X1t22