 
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 (  ) .
A111 Origin of (3.4)
An alternative approach to the above (useful because, as it turns out,
integratorsaresaved) is developed in. e.g., [4] and [9].The model. this
time. partly in Laplace transform notation and neglecting the transform
of exponentially decaying quantities. is
i= I
 .
Y,(s)=B`(sIA)'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 nvectors k1,k2determinedby and determining the plant
