Summary: 88 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. FEBRUARY 1977
parameters.) Also. theinputmustretain this propertyforall time. If
theseconditions, intuitively reasonableforadaptiveidentification.are
fulfilled, thenthe lower boundin (3.2) holds.while theupperbound
reflects boundedness of up(.).
A common procedure to ensure fulfillment of those requirements is to
take up(.)to be a finite sumof sinusoids or periodic signals. In this way.
up(.)is periodic, or almost periodic, and if there are sufficient different
frequencieswithin up( .), the persistently exciting condition holds for
V ( - ) .
A-111 Origin of (3.4)
An alternative approach to the above (useful because, as it turns out,
integratorsaresaved) is developed in. e.g.,  and .The model. this
time. partly in Laplace transform notation and neglecting the transform
of exponentially decaying quantities. is
Y,(s)=B`(sI-A)-'Bw,(s), A + A ' = - I .
One can shoa: that Y,,,(s)= Yp(s)if and only if I,(t)=k,, 12(t)=k2 for
two constant n-vectors k1,k2determinedby and determining the plant