Persistency of Excitation and Overparametrization in by wxv15919

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									254                                                                             IEEE TRANSACTIONS OU AUTOMATIC
                                                                                                     CONTROL,                               VOL 35. NO 2 . FEBRUARY 1990

  Persistency of Excitation and Overparametrization in                                     The MRAC scheme for the plant (2.1) in the case where k, = 1 is
           Model Reference Adaptive Control                                              given as follows:

                      G . TAO    AND   P. A. IOANNOU
                                                                                                                       U(t)   = 8r(t)U(r)   +T(t)                       (2.2)


   Abslmct-In an overparametrized model reference adaptive   control                                        & r ) = - dt)r(')                                           (2.3)
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
formodel-plant  transfer
                       function    matching lies a linear variety in the where
                                                in                                           = y(t)         y , , 3 ( t ) p ( t ) { ( t ) - ~ , ( ~ ) p ( r ) ~ ( r {l (, t , =
                                                                                                                   ~




parameterspaceanditsestimate      converges to this variety asymptotically w , n l s ) a ~ t ) , ~ ( r ) = [ a ~ ( s ) / , i a(Ts s)) U~,( S ) y , y ] , a ( s ) = [ I , s,
                                                                                             W                                      ( ,',
             enough frequencies. In the presence of disturbances or un-
if r ( t ) has                                                                  ,
                                                                                   ,s n - 2 1T , Rr = [ S T , Or, 03], 8 , . R?tR"'. BicR1, A(s) is
                                                                                            ,   ,
                                                                                                                                                                 an ar-
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
robustness unless no near-overparametrization exists.                                   the
                                                                           output ofreference              model

                                 I. INTRODLCTlo\
                                                                                                                                                I
   One of the basicassumptions in model reference adaptive control                                           yrn = W r n ( 5 ) r :   W,,is) = -                         (2.4)
(MRAC) is that an upper bound for the order of the modeled pari of
                          ..
                                                                                                                                              D,is)
the plant is known [ I ] . This assumption is used for developing the con-
troller structure and adaptive law and for proving that the output of the where D m @ ) an arbitrary and stable polynomial of degree n* and r ( r )
                                                                                                is
                                                                          is a uniformly bounded reference signal with i ( t ) bounded.
plant tracks the output of the reference model asymptotically with time
                                                                             It has been shown i n [ l ] that if the MRAC scheme (2.2)-(2.4) is ap-
and that all signals in the adaptive loop are uniformly bounded, in the
                                                                          plied to the plant ( 2 , 1 ) , then all the signals in the closed-loop plant are
case when no modeling errors are present in the plant. In addition, under
                                                                          bounded and y ( t ) tracks y m ( r )asymptotically with time for any contin-
the assumption that the order of the plant is known exactly rather than the
                                                                          uous uniformly bounded reference signal r ( t ) .
upper bound and no plant zero-pole cancellations exist, a necessary and
                                                                             If, however. instead of assumption A2 the following more restrictive
sufficient condition for exponential convergence of parameter and track-
                                                                         assumption that:
ing error to zero is that the measured vector signal r(r) is persistently
                                                                             A?': Z o ( s ) .Ro(s)are relatively coprime and the plant order n is known
exciting (PE) [ 2 ] . The PE property of { ( t ) is guaranteed by selecting the
                                                                          exactly. i.e., A = n is used, then it is shown in 121, [ 5 ] that for a special
reference input signal to have a certain number of frequencies [3]. The
                                                                          class of reference signals r ( t ) referred to as sufficiently rich signals,
                          is
PE property of j-(/) used to establish robustness in the presence of
                                                                          the convergence of the signals in theclosed-loop plant (2.1)-(2.4)is
bounded disturbances and unmodeled dynamics in [4]-[7].
                                                                          exponential and the controller parameterR ( f ) converges to 8' , the desired
   In this note we show that the PE property of { ( t ) in MRAC can no
                                                                          vector for which the transfer function of the closed-loop plant with the
longer be achieved if the plant is overparametrized or if the plant has
                                                                          desired controller matches that of the reference model. The signal r(t)
zero-pole cancellations. Consequently, the assumption of minimality and
                                                                          is termed sufficiently rich for the closed-loop plant (2.1)-(2.4) if it ha5
exact knowledge of the order of the modeled part of the plant used
                                                                          spectral lines [3] at frequencies wI , ' ,a\, N 2 2n - I . Such a signal
                                                                                                                                      '
in [ 3 ] is crucialfor the results of [3] tohold. A similarproblem was
                                                                          guarantees that the internal vector { ( I ) is PE provided assumption A2' is
also indicated in [8] for the pole assignment control case. We show
                                                                          satisfied. When AZ'is violated, i.e.. Zo(s), R o ( s )have common factors
that under overparametrization the parameters for model-plant transfer
                                                                          and/or the upper bound A > n is used in the design of the MRAC scheme.
function matching lie in alinear variety. If the reference input signal
                                                                          the following theorem establishes that r ( t ) cannot be PE.
has enough frequencies. the estimated parameters will converge to this
                                                                              Theorem 2.1: If Ro(s)and Z,(s) are not relatively coprime ormnd
variety In the presence of modeling errors, however, such as hounded
                                                                          A > n in ( 2 . 2 ) , (2.3), then { ( t ) cannot be PE for any bounded reference
disturbances, the parameters may drift to infinity along the linear variet)
                                                                          input signal r ( t ) .
independent of the number of frequencies in the reference input signal.
In the case of exact parametrization but close zero-pole cancellation, the       Proof: Define C m ( f ) = H ( s ) r ( t ) . where H ( s ) = [ u T ( s ) , '
                                                                                                                      1l,
                                                                          .i(s)Go-'(s), a T ( s ) ~ h ( s ) r W ~ ( s 1 :it can be shown that r ( f ) =
signal vector r ( t ) is shown to be PE with a low level of excitation. Such
                                                                          H ( s ) ( d r ( t ) w ( t + r ( t ) ) . where
                                                                                                     )                             = O ( t ) - R* is the parameter
a low level of excitation may lead to poor rate of convergence and to
                                                                          error and that { ( t ) - {,,,(r) i L 2 . Therefore, a necessary and suf-
loss of robustness with respect to modeling errors.

                   11. PE FOR OVERPAKAVETRIZED PLARTS
                                                                           li6,                     x-'"
                                                                          ficient condition for r ( t ) to be                PE     is that {,(I)  is PE 131, i.e.,
                                                                                              i ; n ( t ) r ; ( t ) d t 2 13,~1, 2 0 for some 6,. 0 > 0.
                                                                                                                               V5
                                                                             If Zo(s),Ro(5) are not coprimeor ii > n. then there exist poly-
                                                                                                                                                      ,
                                                                          nomials A @ ) ,E ( 5 ) . of degree A - 2 , A - 1. respectively, such that
   Let us consider the following plant:
                                                                          A ( s l R o l s )+ E(s)Za(s) = 0. i.e., there exists a nonzero constant
                                                                          vector c E R"'-' such that c'H(s) = 0, M t C . Define Rcw =
                                                                          lim7+,         1,T f a r{ , r , ( t ) r T t ( / ) d f , > 0: it can be shown 131 that
                                                                                                                                Vs
                                                                          if r ( t ) has spectral lines at frequencies wI ,. . , w y . then R;," =
where R n ( s )and Z o ( s ) are monic polynomials.                       I>=l    lR(w,),'HCiw,)HT(-jW,) {,,,ct) is PE iff Rr,n is nonsingular.
                                                                                                                           and
                                                                             Sincethe
                                                                                   for                      above c and any A' > 0, C * R ; c =       ~
   A stable MRAC scheme can be designed for the plant (2. I ) by using
                                                                           ; R ( o ~ ~ ] ~ ~ ~ c = H (it o ~ ~ ) ' that Rr., is singular.Hence. {,,,(t)
                                                                                                              * 0, j follows ~
the following assumptions.
                                                                       notis       PE, which implies that { ( r ) is not PE.                                  CP
   AI: Zo(s) is a Hurwitz polynomial.
                                                                             In the following section we examine the behavior of the parameter
   A2: An upper bound A on the plant order n is known.
                                                                          estimates generated by theadaptive law (2.3) when assumption A2' is
   A3: The relative degree ' of Go(s) is known.
                                 n
                                                                          violated.
   A4: The sign of the high frequency gain k, of Go ( 5 ) is known.
IEEE TRANSACTIONS
  CONTROL.
 AUTOMATIC
        ON                                                       VOL 35. KO. 2 . FEBRU.4RY 1990                                                                                                 255

   The model-plant transfer function matching equation is given as                                   the level of PE of ( ( t ) has to be large relative to the bound of the dis-
                                                                                                     turbances in order to establish boundedness of the signals in the MRAC
                 X ( s ) R o ( s )+Yls)Zo(s)= .A(sIZo - (f)Dm(f)                          (3.1)
                                                                                                     system. A similar result was also established for the case of unmodeled
where
sn - I
          X ( S ) = si-'                         - xIs + x", Y ( S ) = y i - l
                                  + x~-2si'-' + " .
         + . . . + y l s + y o are some polynomials whose coefficients depend
                                                                                                     dynamics in a localsense in (51 and in a global sense with modified
                                                                                                     adaptive laws in 161. The high level of PE of ( ( I ) was established by
on the desired controller parameten. By examining the solution of (3. I )                            using assumption A2' for the modeled part of the plant and by choosing
we can establish the following theorem.                                                              the class of reference signals r ( f )appropriately. In this section we show
   Theorem 3.1: The solution { X ( s ) ,Y ( s ) } of (3.1) exists and the                            that forrobustness, assumption A2' ha5 tobe satisfied in a"strong"
coefficients of X @ ) ,Y ( s ) lie in the linear variety C = yo                               +      sense, i.e.. near-overparametrization should be avoided.
{ x E R2"-'IxrH(s) = 0, vs E C } of dimension I + q for some                                            Consider the following plant:
    E R2fl-l , Furthermore, if the reference input r ( f )has spectral lines
at wI , . , , . a v , with M 2 2ii - I - q - I, then the parameter vector B ( f )
converges into 6: asymptotically with time.
     Proof: Equation (3.1) is equivalent to the following equation:
                                 +
                   x ( s ) R ~ ( s ) Y(s)Zo(s) ' i ( s ) t o ( s ) D , ( ~ )
                                             =                                            (3.2)        where a > 0, b > 0, l c l << min(l, a ) , 0 < p << 1 and %(s), &(s) are
                                                                                                                                                               z0
where Ro(s) = s"-'
                          -
                                  -                                                                    monic relatively coprime polynomials with (s) being Hurwitz, and no
                                      r , _ i - I s " - ' - - i + ' ' ' + ro. z,,(s) = 5'"' t zero-pole cancellations in G o @ )
                                      +                                                                                                        when e f O , p > 0.
                  + afi.,-,-ls"n-'-2z0. If+ we. + U O , defines ) Z ~ ( s ) D ,[ (I=,s ) b ( Since.) /cR o ( s )and consider the terms due to the neglected tas p as un-
Z,n-,--ISm-'-l              " '                                   express ~ \ (                                                                  one could model Go(s)
                                                                                                                       , p are sufficiently small.                                                 =
SR-"-/-l                                                       "                       x =
                                                                                                          Zo(s               )                                                       ,
x, - 2 , . . . , X U , y n - 1 , ' ' ,yo]r. b = [ I , u,q-" - I - ? ,       . . , a ~ ] ' . then solv- modeled dynamics. Let us assume,however,one is not aware of the
ing (3.2) for X ( s ) and Y ( S ) is equivalent to solving Ax = b for x. smallness o f t , p or wants to be over cautious with the order of Go(s) and
where A E R12n-/lX:tt is the Sylvester matrix [SI consisting of the co-
                                                                                                       consider Go(s) as given in (4.1) for the designof the MRAC scheme. The
efficients of &[s), Zo(s). If Zo(s) and &(s) are relatively coprime, it following theorem establishes that the level of PE of ( ( t ) corresponding
can be shown that the matrix A has rank fr + n - I , and all the solutions to the plant (4.1) cannot be larger than O ( F ' )+ O ( i ) tO(pIt1).
of Ax = b can be expressed as                                                                              Theorem 4.1: Let the MRAC scheme (2.2)-(2.4) be designed for the
                                                                                                       full-order plant ( 4 . 1 ) .then the level of PE of ( ( 0 in the closed-loop plant
                x=[l,.i.rlr:                ~=xti*clx'+...+c/.qx'+~                              (3.3)
                                                                                                       ( 4 , l ) . (2.2)-(2.4) cannot be larger than O($) + O ( c 2 )       +O(&lEl)for any
for some linearly independent vectors X I ,. . . ,i' 0 E R2"' , VC; E R bounded reference input signal r ( r ) .
                      -
for i = 1,. . . ,I q . where x" E R2"' satisfies A y o = b where yo =                                                                                                                      +
                                                                                                               Proof: Since t ( t ) = H ( s ) r ( t ) , f ( t ) = H ( s ) ( @ ( t ) w ( t ) r ( t ) ) ,
[ I , (x.')']'. Hence,all-thesolutions of (3.2) lie in alinear variety of and ( ( t ) - (,,,(f) E L 2 . it is sufficient to show that the level of PE of
dimension I + q in R'", characterized by (3.3). It can be shown that { , , , ( f )cannot be larger than O($) O(c2 O(p1c ). It follows that
x r H ( s ) = 0 iff X ( s ) R " ( s )+ Y ( s ) Z " ( s )= 0. therefore, the linear variety H ( s ) = H I(s)- H z @ ) , where
                                                                                                                                                       +        1  -



( 3 . 3 ) c a n b e e x p r e s s e d a s , 6 : = y " + { x E R ' " ' ~ . u T H ( s ) = O , ~ s ~ C } . It can be shown that there exists a nonzero constant vector c E R2"-I .
   Define crn = f?(s)r. ( ( 1 1 = H(s)(4r(r)a(rl ( r ) ) .where H ( s )= where n is the order of the plant (4. I ) . such that $ H I (s) = (1 for all
                                                                     +r
[ a r ( s ) i h ( s ) C ~ ' ( s ~ b T ( s ) / . ~ ( s ) b(s) = ([sl ) , , s ' , . . . , s " - ' - ' ] ~ , s t C . Assume that r ( t ) has N frequencies, then for this c, it follows
                                                          ]'W~ ,~
                             ,
then we have that ( t ) - r ( t ) E L2.
                              (                                                                           that:
   Next we will show that if r ( t ) has 2ii - I - q - 1 or more frequencies,
then ( ( I ) is PE by contradiction. If!(?) has M ( M 2 2 n - I - q - 1)
frequencies and ((f) is not PE, then crn( t ) is not PE either. Hence, there
                                                                                                                                          ,=I
exists anonzero vector i. t R>"-'-q-' such that trR I. = P" =
1~R(w,)l~l?'fi~uj)~2            =-o. i.e.. C ' k u w , ) , = 0. for b = 1,. . , M . for any N > 0. If e f O , p f 0 , and r ( t ) has IV = 2n - 1 or more fre-
                                                                               I
Hence, cr[aT(s)Ro(s)2              b'(s)Z~!s)]' = O,.Vs E C , i.e.. there exist quencies, then t ( t ) is PE. But (4.2) shows that the level of persistent
nonzero polynomials A b ) and B(s) with DE(s) = n - I - I , (?Ah) = excitation of r m ( t )cannot be larger than O ( p ' ) + O ( E ' )- O ( p l c l ) . T'?
ii - 2 , such that A(s)Rq(s)+B(s)&(s) = 0 , d s E C , which contradicts                                      Theorem 4.1 establishes that when the plant is near-overpara-
the fact thateRo(s)and Zo(s) are relatively coprime and 8iio(s)= n - I .                                  metrization. Le.. A2' is satisfied only in a "weak" sense, the high level
Therefore,      r,,,(f)  is PE, so is ( ( 0 .                                                             of PE of ( ( t ) which is crucial for robustness can no longer be achieved
   Using the fact that ( ( r ) is PE if r ( f ) has 2R - I - 9 - 1 or more by manipulating the reference input signal r ( l ) . For example, if a general
frequencies. we have that r ( t ) is not PE only i n l + q directions in R2"-' . bounded disturbance is present in the plant input or output whose bound
characterized by c- for those c's such that ? H ( s ) = 0, Vs E C . Since is O(I), then for /€I, p < 0(1)theboundedness of thesignals in the
                                                                                                                                      <
lim,-xRr4(t) = 0 131 and R , , ! ; C               =         IR(w,),'Huw,)H'(-jw,), it closed-loop MRAC system (4. I ) , (2.2)-(2.4) cannot be established for
follows that lim,-x dist(B(f), C )= 0.                                                           G V any class of bounded reference input signals r ( t ) .The disturbances may
   Therefore, Theorem 3.1 demonstrates that under overparametrization cause parameters to drift to infinity [9] even though r ( t ) has a sufficient
the controller parameters will converge to a linear variety such that the number of frequencies in a similar way as in the case of nonrich reference
model-plant matching condition is satisfied provided that the reference input signals.
input signal r ( t ) is rich enough.
                                                                                                                                v. D ~ s c ~ AhD~ C~ NoC L\L S l O \ S
                                                                                                                                                  ~   O
                                                                         WETRI
                    IV. PE F O R N E ~ R - O V E R P ~ R A P L ~ N T SZ E D                                  In this note the problem of overparametrization of the plant in MRAC
     The PE property of the signal vector f ( t ) is not only crucial for param-                     is analyzed.Overparametrization may arise when anupper bound for
eter convergence but also for robustness in the presence of disturbances.                            the order of the plant is used in the design of the MRAC or when the
unmodeled dynamics when no modifications are used in the adaptive law                                transfer function of the modeled part of the plant has common zeros and
( 2 . 3 ) . In [4] it was shown that in the presence of bounded disturbances                         poles. It is shown that under overparametrization. the PE property of the
256                                                                          IEEE TRANSACTIOUS ON VOL.
                                                                                                  AUTOMATIC
                                                                                                   CONTROL.                                35. NO. 1.1990
                                                                                                                                                      FEBRUARY

measurement vector      r(r)
                          which is crucial for theexponential convergence              B. D.0. .Anderson. "Exponentla1 stablllr) of llnear equations arlbing in adaptive
                                                                                       identlficatlon." IEEE Trans AuromolConrr . bo1 AC-22. 1977.
of the parameter and tracking error can no longer be established for any
bounded reference input r ( t ) . Thedesiredcontrollerparameter      vector            S. Bojd and S S . Sdstp, "Necessaryand aufficlenr conditionsforparameter
                                                                                       convergence in adaptwe control." Auromatim. vol. 21. no. 6. 1986
0' for model-plant transfer function matching is no longer a point in the              K. S Narendra and A . M . Annasu'amy. "Robust adaptive control In the presence
parameterspace but lies in a linear variew S . When r ( r ) has enough                 of boundeddtsturbances." IEEE Trans AutomarConrr . ~ 1 AC-31.no 4 .     .
frequencies the estimated parameters converge asymptotically into C ,    In            1986.
                                                                                       E. 0 D Anderson. R.  R. Bitrnead.        C. R. Johanson. P. V . KokotoLr.   R. L
the presence of bounded disturbances,however,parameter drift to in-                    Kosut. I. M . Y.Mareek.L. Prdly. and B. D Rledle. SrabrlrrJ of Adaptive
finity along the linear variety cannot be excluded for any bounded r ( t ) .           Sprems: Pusslvir.I' and Awraglng Anolysis. Cambrldge.MA: M.I.T.Press,
The same robustness problem also arises when the modeled part of the                   1986.
plant is near-overparametrization due to some near zero-pole cancella-                 P. A. loannou and G . Tao, "Dominant rlchness and improvement of performance
tions. In this case the PE property of the vector { ( t )can be established            ofrobust ddaplive control."Dep.Elec.Eng.-Syat..        U n i v . Southern Calif.. Lus
                                                                                       Angeles, Rep. 04-01-87: also in Auromorica. lo be published.
by choosing r ( i ) appropriately.However, the level of PE maynot be                   J. Krause. 4 I . Athans, S. Sastry, and L Valavanl. "Robustnesi studles rn adaptwe
large enough relative to the bound for the modeling error and therefore                control." in Proc 22nd IEEE Conf. Decision Contr , San Anton~o.T X . Dec.
signal boundedness cannot be established.                                               1983.
                                                                                        L. Xla. I. B. Moore. and M. Gevers. "On adaptive estmatlon and pole awgnment
                                   REFERENCES                                          of over-paramemzed systems.'' Inr 3. Adapt~ve       Conrr Signa/ Processing. voI
 [I]   K. S . Narendra. Y I r l . Lin. and L. S. Valavanl, "Stable      controller
                                                                 adaptive               I, 1987
       deslgn. Pan 11, Proof of stabiliv." IEEE Trans Aulomar Conrr , \01. AC-25.      P. A. loannou and P. V . Kokotovic,"Instablltyanalksirand            Improvement of
       1980                                                                             robustness of adaptive control." Aulomaflco, vol. 20, no. 5 . 1984

								
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