  In the semiconductor manufacturing batch processes, each step is a
complicated physiochemical batch process; generally it is difficult to perform
measurements online or carry out the measurement for each run, and hence there
will be delays in the feedback of the system. The effect of the delay on the
stability of the system is an important issue which needs to be understood.
Based on the exponentially weighted moving average (EWMA) algorithm, we propose
two kinds of controllers, EWMA-I and II controllers for single product process
and mixed product process in semiconductor manufacturing in this paper. For the
single product process, the stabilities of systems with both controllers which
undergo different kinds of metrology delays are investigated. Necessary and
sufficient conditions for the stochastic stability are established.
Routh-Hurwitz criterion and Lyapunov's direct method are used to obtain the
stability regions for the system with fixed metrology delay. By using
Lyapunov's direct method, the stability region is established for the system
with fixed sampling metrology and with stochastic metrology delay. We also
extended the theorems of single product process to mixed product process. Based
on the proposed theorems, some numerical examples are provided to illustrate
the stability of the delay system.
