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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. 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