Persistency of Excitation and Overparametrization

Abslmct-In an overparametrized model reference adaptive control & r ) = dt)r(') system, the measured signal n(t) cannot he persistently exciting (PE) I + rT(r )r ( t ) for any hounded reference input r ( t ) . The controller parameter vector for model-plant transfer function matching lies in a linear variety in the where = y ( t ) ~ y , , 3 ( t ) p ( t ) { ( t ) ~ , ( ~ ) p ( r ) ~ ( r l , { ( t , = parameter space and its estimate converges to this variety asymptotically w,nls)a~t) , W ~ ( r ) = [ a ~ ( s ) / , i ( s ) U , a T ( s ) , ' , ~ ( S ) y , y ] , a(s) = [ I , s, if r ( t ) has enough frequencies. In the presence of disturbances or un, , , s n 2 T modeled dynamics, the PE Propem Can no longer be used to achieve bit'rary mon,c and stable p o ~ S n o m i a ~ of degree 8 1 and yrn ( 0 is the , 1 , R r = [ S T , Or, 03], 8 , . R?tR"'. BicR1, A(s) is an ar-