Regulation of Linear Systems with Unknown Additive Sinusoidal Sensor Disturbances

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1 Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 8 Regulation of Linear Systems with Unknown Additive Sinusoidal Sensor Disturbances Riccardo Marino Giovanni L. Santosuosso Patrizio Tomei Dipartimento di Ingegneria Elettronica, Università di Roma Tor Vergata, Via del Politecnico, 33 Roma, ITALIA. marino@ing.uniroma.it santosuosso@ing.uniroma.it tomei@ing.uniroma.it Abstract: Known linear stabilizable and detectable systems, which are allowed to be non minimum phase, are considered:the problem of tracking unknown output reference trajectories andrejectingunknown inputdisturbances whenthe outputtracking erroris affectedby unknown additive sensor disturbances is addressed. All the exogenous signals to be tracked and/or to be rejected are assumed to be the sum of sinusoids: only upper bounds on their numbers are supposed to be known, along with a set in which the output disturbance frequencies may range. A constructive algorithm is proposed to drive the regulation error exponentially to zero. The regulation strategy includes an on-line detector of the number of excited frequencies and exponentially converging estimates of the exosystems parameters. An example containing a variable number of frequencies is worked out and simulated. Keywords: Adaptive Regulation, Sinusoidal disturbances, Adaptive Observers.. INTRODUCTION AND PROBLEM STATEMENT A major goal in feedback control system design is the rejection of unknown disturbances. The output regulation theory for linear systems establishes the internal model principle see Francis and Wonham 976: the feedback control which rejects a disturbance modelled by a linear exosystem must incorporate the exosystem itself. The exosystems for sinusoidal disturbances contain their unknown frequencies: in this case adaptive internal models are to be resorted to, as shown in Nikiforov 998. Typically periodic disturbances are acting on the system input see Bodson 5 for a recent survey in applications suchas active suspensions designin Landau et. al. 5, disk drives speed regulation in Liu and Yang 4, eccentricity compensation in De Wit and Praly, active noise control in Kuo and Morgan 996, Wu and Bodson 3, feedback control vibrations in helicopters in Bittanti and Moiraghi 994, Ariyur and Krstic 999: in these cases they can be viewed as matching disturbances in the tracking error dynamics. A different control problem arises when sinusoidal sensor disturbances act additively on the measured output. The stabilization problem for linear systems in this case has been addressed and solved in Serrani 6 andmarino, SantosuossoandTomei 8 see also applications in Wu and Bodson 3, Zarikian and Serrani 7. The aim of this paper is to combine the two problems above: given a linear stabilizable and detectable system, which is allowed to be non minimum phase, we solve the regulation problem concerned with tracking unknown This work was supported by the Ministero dell Universià e della Ricerca, under grant PRIN. reference trajectories andrejecting unknown inputdisturbances in the presence of unknown disturbances acting additively on the measured output, when all these signals are generated byunknown exosystems with simple eigenvalues on the imaginary axis. The class of linear time invariant systems { ẋ = Ax + bu + Pwt ;; x = x ẇ t = Rw t ; w t R r, w t = w t e = cx qw t ; is considered, with state x R n, control input u R; output e R to be regulated to zero; the signals qw t and Pw t, respectively to be tracked and to be rejected, are both generated by the unknown linear exosystem ẇ t = Rw t, with state w t R r. The measurable output yt = et + δt, y R is the sum of the output et of system and the disturbance δt generated by an unknown linear exosystem ẇ δ = R δ w δ, w δ R r δ, w δ = w δ, 3 δ = q δ w δ. We address the regulation problem formulated as follows. Problem.. Consider the extended system -3, with known A,b,c, unknown P, q, q δ, R, R δ where the orders of R and R δ are known and the spectra σr, σr δ contain only simple eigenvalues on the imaginary axis. Design a dynamic feedback controller { Ψ = FΨ,y 4 u = HΨ,y from the output y, with state Ψ R nc, suitable initial condition Ψ = Ψ R nc and FΨ,y, HΨ,y suitable functions, such that in the closed loop system -4, the output variable et of system tends exponentially to /8/$. 8 IFAC 4.38/876-5-KR-.48

2 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 zero as t goes to infinity and the state variables xt,ψt are bounded for any t, for any initial conditions x R n, w t R r, w δ R r δ, of the extended system -3. We assume the following hypotheses to hold: H The pair A,b is stabilizable, i.e.: ranka λi n,b < n implies Reλ <. H The pair A,c is detectable, i.e.: rank A T λi n,c T < n implies Reλ <. A λin b H3 rank = n+ for any eigenvalue λ of c the matrix R. The conditions H-H3, were established in Francis and Wonham 976 to be necessary and sufficient for the solution of the regulator problem arising when δ =, R and r are known. In particular, by virtue of H- H3 there exists a unique matrix Γ R n R r and a unique vector γ R r which solve the regulator equations ΓR = AΓ+bγ+P and cγ = q. The pair Γ,γ generates the signals x r = Γw t and u r = γw t which are the references for x and u respectively, since ẋ r = Ax r +bu r and cx r = qw t. The transformation x = x Γw t yields an error system { x = A x + bu u r 5 y = c x + δ with a scalar reference input u r = γw t which satisfies the matching condition given by the reference exosystem ẇ t = Rw t and a scalar output disturbance δt = q δ w δ generated by the disturbance exosystem ẇ δ = R δ w δ. H4 The set of the eigenvalues of R δ is disjoint with respect the set of the eigenvalues of A. Condition H4, introduced in Serrani 6, is shown in Marino, Santosuosso and Tomei 8 to be necessary along with H and H for the solution of the stabilization problem of the system with a controller from the disturbed output cx + δt. H5 The unknown eigenvalues of R δ belong to the set S δ = {±jω : κ i ω κ i, i [,N]} where κ < κ < κ <... < κ N < κ N are suitable known numbers and the eigenvalues of R do not belong to S δ. In this paper we prove that hypotheses H-H5 are sufficient for the solution of Problem... REGULATION ALGORITHM In this section we construct a dynamic controller that drives regulation error et in exponentially to zero. By virtue of H5 the eigenvalue λ = may belong either to R or to R δ. For ease of exposition we assume that λ = belongs to R. In this case let {, ±iω,...,±iω M } and {±i ω,...,±i ω M } be the eigenvalues of R and R δ respectively, with M + r, M r δ M, M are known integers, ω h R +, h [,M], ω j R +, j [, M ]. Notice that u r t and δt may contain respectively only m and m harmonics oftheexosystems, with m M, and m M. If we assume without loss of generality that the first harmonics ω h, h =,... m and ω j, j =,... m of the exosystems appear in u r and δ respectively, by setting θ [θ,θ,...θ m ] T and θ [ θ, θ,... θ m ] T where s m +θ s m +...+θ m = m h= s+ω h, and s m + θ s m θ m = m j= s + ω j, respectively, then u rt and δt can be expressed as the outputs of some reduced order unknown exosystems with states w R m+, w R m, defined { as [ ẇ = Am+,θ,...,,θ m, T ] C m+ w, u r = C m+ w, w R m+ 6 { w = [ A m, θ,...,, θ m T ] C m w, δ = C m w, w R m 7 where as in the rest of the paper, given a positive integer j, the canonical matrix A j R j R j, and the canonical row vector C j R j are [ ] Ij A j =, C j = [ ] j, j Z + 8 j j with I j denoting the identity matrix of order j. Let the rank of the observability matrix of the couple A,c be v n; then there is a suitable known coordinate change [ ] xu x = T u R n v x O x, o x o R v 9 operating a Kalman decomposition of the system x = A x with output c x into the unobservable and observable parts with vector states x u R n v, x o R v, respectively. The system 5 by virtue of 9, becomes ẋ u = A u x u + A uo x o + b u u u r ẋ o = [A v ac v ]x o + b o u u r ; y = C v x o + δ where system with state x u R n v, known matrices A uo R n v R v, A u R n v R n v, A u is Hurwitz since by H the pair A,c is detectable known vector b u R n v is unobservable from the output y = C v x o + δ and system, 6, 7 with state [ x T o,w T, w T] T R v R m+ R m by virtue of H-H5 is observable from the output y = C v x o + δ. The row vectors C v, C r are defined as in 8 with v, r, respectively, in place of j and a R v, b o R v are known vectors. Consider the linear filter { x o = [A v fc v ] x o + f ay + b o u, ỹ = y C v x o with state x o R v, inputs u R, y R, available output ỹ = y C v x o R, where a = [a,a,...a v ], b o = [b,b,...b v ] are defined in and f = [f, f,..., f v ] T with f i R +, i v are design parameters such that p f s = s v + f s v f v s + f v 3 has all its roots with negative real part. By setting x o = x o x o, from,, we obtain the error dynamics { x o = [A v fc v ] x o + a fδ b o u r 4 ỹ = C v x o + δ = y C v x o. Defining m T m + m, v T v + m + m +, and ϑ = [ϑ, ϑ,...,ϑ mt ] T so that s mt + ϑ s mt ϑ mt = m h= s + ω m h j= s + ω j, the autonomous system 4, 6, 7 with dimension v T is observable from the available output ỹ = C v x o +δ by virtue of H- H5; hence it is transformed into the observer canonical form ζ = A vt ζ f m T [m T ]ỹ ϑ i fi [m T ]ỹ; 5 i= ỹ = C vt ζ with state ζ R vt, constant vectors f i [m T ] R vt, i [,m T ] given by 43

3 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 f [m T ] = [f,...,f v,,,,...,,,] T, f [m T ] = [,,f,...,f v,,,...,,] T,.. f m [m T ] = [,,,...,,,f,...,f v,] T. 6 Let O m, m θ, θ be the observability matrix of system 4 6, 7 and Θ mt ϑ be the observability matrix of system 5; by hypotheses H-H5 they are both nonsingular. The coordinate transformation from [ x T o,w T, w T] T R vt to ζ R vt is ζ = [ Θ m T ϑo m, m θ, θ ] [ x T o,w T, w T] T 7 where the matrix [ Θ m T ϑo m, m θ, θ ] is well defined by construction. Set d [mt ] = [ d,..., d vt ] T vt R where d i R +, i v T, are positive real numbers such that all the roots of p d s = s vt + d s vt + + d vt have negative real part. Define as in Marino and Tomei 99 the filters ξ i R vt, µ i R, i m T ξ i = [ A vt d ] [m T ]C vt ξi [, I vt ] f i [m T ]ỹ; µ i = C vt ξ i, i m T. 8 According to Marino and Tomei 99, the transformation ζ [ = ζ ξi ϑ i T ] T, mapping the state, m+ m i= vector ζ R vt into a new state vector ζ R vt, transforms system 5 into an adaptive observer form [ ζ = A vt ζ f [m T ]ỹ + d [m T ] ỹ = C vt ζ ] µ T ϑ, 9 where µ = [ µ, µ,... µ mt ] T ; defining the vector function z m, m R v+ as z m, m θ, θ, ξ,..., ξ mt, ζ = [,I v+,] Adj [ O m, m θ, θ ] Θ mt ϑ ζt + ξ i ϑ i m+ m the state x o R v of system along with the reference input u r = C m+ w can be expressed as xo = u r xo i= + z m, mθ, θ, ξ..., ξ mt, ζ det O m, m θ, θ. The equality above requires the inversion of a parameter dependent mapping, so that if m and m are unknown the online estimation of these integers along with the vectors θ, θ is required; this is described in the following.. Identification of the adaptive observer form system. In this subsection we describe an estimation strategy for the state ζ and the parameters ϑ j j m T of system 9 in adaptive observer form which is a function of the maximum number of excited sinusoids M T M + M. We follow the strategy in Marino and Santosuosso 7, pages 355, 356. First, three cascaded filters are introduced to detect the number of frequencies that are excited. The first filter with state η = [ η, η,... η MT +] T R MT +, initial condition η R MT +, input ỹt given in 4, output ν = [ν, ν,...ν MT +] T is 44 η j = η j+, j M T +, η MT + = M T + j= ᾱ 3 j+mt η j + ỹ ν = [ η MT +, η MT,..., η 4, η ] T where the design parameters ᾱ i, i M T +, are such that the polynomial p α s = s MT + + ᾱ s MT + + ᾱ s MT... + ᾱ MT + 3 has all its roots with negative real part. The vector νt R MT + is the input to the second filter { Ω = c Ω + c νν T, Ω, i = det Ω i /i 4, i M T + with state Ω R MT + R MT +, i M T +, symmetric and positive semi-definite initial condition Ω, outputs it, i M T +, where Ω i R i R i denotes the matrix collecting the first i i entries of Ω and c, c are positive design parameters. The outputs it of filter 4 are the inputs of the third filter with state χ = χ,...,χ MT T, where χ j = [σ j j + ψχ j ]χ j + σ j MT +, j [,M T ] 5 in which χ j > ; σ j j and σ j MT + with j M T are suitable class K functions. The function ψχ j depends on a design parameter χ R + and is defined as ψχ j = if χ j χ ; ψχ j = 4χ j χ /χ j if χ χ j χ and ψχ j = if χ χ j. Let c 3 be a positive design parameter; define the residuals: β i = if i > c 3 χ i, i [,M T ] : i 6 β i = if i c 3 χ i. c 3 χ i By the arguments in Marino and Santosuosso 7 page 355, Lemma 3., it is shown that all the states of the filters, 4, 5 and the residuals 6 are bounded, and that lim t β i t = exponentially for i m T, while lim t β i t = exponentially for m T + i M T. Next we design an adaptive observer of system 9 that estimates the exosystem s parameters ϑ j j m T, according to the guidelines in Marino and Santosuosso 7. To this purpose we define the diagonal matrix Ūt R v+mt R v+mt with entries Ūi,it = for i v i v and Ūi,it = β k t, with k = for v + i v + M T. Consider the vectors βt = [ β t,..., β MT t ] and d β t defined from 6 as { β t = β t; βmt t = β MT t β i t = β i t β i+ t for i M T d β t = β t d [] β MT t d [MT ] 8 where in 8 the entries of the constant vectors d [i] R v+i, i M T are design parameters such that the polynomials s v+i + [ s v+i, s v+i,..., ] d [i] are Hurwitz. The matrix Ūt and the vector d β t are tools to construct a generalization of the filters 8 that are adaptive with respect to the unknown number m T : they are defined { as ξ i = Ū Av+MT d β tc v+mt ξi [, I v+mt ] f i [M T ] ỹ } c 4 Iv+MT Ū ξi ; µ i = β i C v+mt ξi ; i M T 9

4 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 with state variables ξ i R v+mt, i M T, arbitrary initial conditions ξ i R v+mt, where c 4 R + is a positive design parameter and f i [M T ], i M T are defined according to 6 with M T in place of m T. We consider now an observer for system 9 which is adaptive with respect to the unknown number m T of exosystem s frequencies [ ζ = U A v+mt + ζ + k ỹ ζ f [M T ]ỹ ] MT +d β i= µ i θ i c 5 {I v+mt + U} ζ; ϑ i = γ i µ i ỹ ζ γ i β i ϑ i ; i M T 3 with state ζ R v+mt +, ϑ = [ ϑ, ϑ,..., ϑ ] T MT R M T, arbitrary initial conditions ζ R v+mt +, ϑ R MT, in which ζ = C v+mt + ζ and f [M T ] is defined according [ to 6 ] with M T in place of m T. In system 3 Ut = Ū, d β =, the vector dβ kt v+mt + R defined as kt = A v+mt + + λi [ v+mt +, d ] T T β t with λ R + any positive design parameter and c 5 R +, γ i, γ i R +, with i M T, are any positive design parameters. Let j be an integer such that j [,M T ]; consider the subvectors ξ [j] i = [I v+j,] ξ i, ζ[j] = [I v+j+,] ζ, ϑ[j] = [I j,] ϑ. 3 By following the arguments in Marino and Santosuosso 7, page 356, Lemma 3., it is shown that the states of the systems 9, 3 are bounded and the partition in 3 obtained by setting j = m T complies with the following properties. Claim. i The vectors ξ i with i M T, ζ and ϑ are [mt ] bounded; ii The vectors ξ i ξ i, with i m T, ζ ζ [mt ], ϑ ϑ [mt ], tend exponentially to zero; iii the last M T m T entries of the vectors ξ i with i M T, the last M T m T entries of the vector ζ, the last M T m T entries of the vector ϑ tend exponentially to zero.. Identification of the exosystems structure In this section we describe an algorithm to obtain from ϑt along with the residuals βt suitable estimates of θ R m and θ R m along with their dimensions m and m respectively. To this purpose, given ϑ = [ ϑ, ϑ,..., ϑ ] T MT in 3, consider the polynomial s MT + ϑ ts MT ϑ MT t = M T s jt 3 j= with M T complex roots jt C j M T, and define the vector ˆωt = [ˆω, ˆω,..., ˆω MT ] T R MT, ˆω sort[ ] t,..., MT t, 33 obtained by sorting in descending magnitude order the square roots of the euclidean norms of the complex numbers jt C, j M. In order to determine if each angular frequency estimate ˆω j t, j m T, is associated to the input reference u r t or to the output disturbance δt, consider the piece-wise continuous time functions l t,...l MT t, l t,... l MT t, defined as { lh t = β h t if ˆω h t / S δ l h t = if ˆω h t S δ h M T 34 { lj t = β j t if ˆω j t S δ l j t = if ˆω j t / S δ j M T 35 where S δ is the known subset of R + to which the angular frequencies δt belong, while β, β,..., β MT are defined in 6. Define iteratively the vectors V i R MT, V i R MT, with i = M T,...,, as follows: V MT = V MT = [,,...,,] T ; V i t = [ A MT + ˆω i tl ] iti MT Vi t, M T i ; V i t = [ A MT + ˆω i t l ] i ti MT Vi t, M T i. It can be shown that the first m entries of V t tend to θ and the first m entries of V t tend to θ. Continuous estimates of these constant parameter vectors are given by the filters with states ˆθ R M, ˆ θ R M respectively, ˆθ = c 6 ˆθ [IM,] V t, ˆ θ = c 6 ˆ θ [I M,] V t, 36 with suitable initial conditions ˆθ R M, ˆ θ R M, where c 6 is a constant design parameter. The functions MT h= l ht and M T h= l h t can be shown to define suitable estimates of the positive numbers m and m. Continuous estimates of these integers are given by the filters with states ˆm R and ˆ m R defined as ˆm = c 6 [ ˆmt MT l h t] h= 37 ˆ m = c 6 [ ˆ mt MT l h t] h= with suitable initial conditions ˆm R and ˆ m R, where c 6 is the positive design parameter in 36. Let h, j be integers such that h [,M T ], j [,M T ] and consider a partition of the vectors ˆθt R M, ˆ θ R M into subvectors whose dimension depends on the integers h, j as follows: { ˆθ[h] = [I h,] ˆθ, ˆθ[h] R h, h [,M] ˆ θ [j] = [I j,] ˆ θ, ˆ θ[j] R j, j [, M ].. 38 The vector functions in 36 and 37 defined above can be shown to be bounded, and by construction see Marino and Tomei 99 comply with the following property: Claim. i: The functions ˆmt, ˆ mt, ˆθt, ˆ θt in 37 and 38 are bounded. ii: lim t ˆmt = m, lim t ˆ mt = m, lim ˆθ[m] t t = θ and m] lim t ˆ θ[ t = θ. iii: the last MT m entries of ˆθt and the last M T m entries of ˆ θt tend exponentially to zero..3 Compensator design In this section we describe the control law constructed after the estimation of the state ζ R vt of the system 9 and the unknown parameter vector ϑ R mt along with the vectors θ R m, θ R m via the estimates ˆθt R M, ˆ θt R M. The crucial step in the controller design is 45

5 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 obtaining estimates of the state x o R v of system and of the reference input u r given by. We follow the adaptive saturation approach in Marino and Santosuosso 7 to the problem in this note. In particular, let s C z : R R be the continuous and piece-wise linear even scalar function defined as: s C z = for z /4 and s C z = for z 3/4; define the M + functions κ h t = s C h ˆmt for h M and the M + functions κ j t = s C j ˆ mt for j M. Consider the M + M + scalar filters with states p h t, h M, and p j t, j M, defined as ṗ h = c 7 [ c 8 p h + κ h + π arctanc 9 ỹ ζ ], p j = c 7 [ c 8 p j + κ j + π arctanc 9 ỹ ζ ], 39 driven by the inputs κ h t, κ j t, h M, j M, along with the estimation error ỹ ζ, where c 7, c 8, and c 9 are positive design parameters. It can be shown that if p h >, p j > then all p h t, p j t with h m, j m are greater than a positive lower bound, while p m t and p m t both tend exponentially to zero as t goes to infinity. The filters in 39 are tools to estimate x o, u r given by via the adaptive saturation algorithm see Marino and Santosuosso 7 if det det Ôh,j > p h + p j, Υ hj = Ôh,j det Ôh,j p h + p j if det Ôh,j p h + p j, 4 ˆxo xo M M = + κ û r h κ j Υ hj ẑ h,j. h=j= where ẑ h,j = z h,j ˆθ [h], ˆ θ[j], ξ[h+j] [h+j],..., ξ m T, ζ [h+j] and Ô h,j = O h,j ˆθ [h], ˆ θ[j] are obtained from 7 and with h and j in place of m, m respectively, and ˆθ [h], ˆ θ[j], ξ [h+j],..., ξ[h+j] m T, ζ[h+j] are defined via 3, 38, in place of θ, θ, ξ,..., ξ m+ m, ζ. The task of the positive signals p h t, p j t is to avoid the singularities in which det Ôh,jt =, while the functions κ h, κ j select the correct disturbance estimate of x,u r given by with indices m, m among all combinations Υ hj ẑ h,j with h [,M], j [, M ]. Since by hypothesis H the system is stabilizable and by virtue of H the matrix A u is Hurwitz, then the couple A o,b o is stabilizable and there exists a suitable row vector k c R v such that the matrix A o + b o k c is Hurwitz. The overall compensating control law is u = k cˆx o + û r 4 Previous arguments lead to the following result. Proposition.. Consider system -3. If assumptions H-H5 are satisfied, then Problem. is solvable any initial condition x R n, w t R r, w δ R r δ via the control law 4-4 with dynamics,, 4, 5, 9, 3, 36, 37, 39, for any initial condition ofthedynamic compensator suchthat Ω, χ i >, i [,M + M ], p h >, p j >, h [,M], j [, M ], for any admissible choice of the design parameters, which are: any positive real numbers f, f,,...f v, and ᾱ, ᾱ,...ᾱ M+ M+, such that the polynomials p f s in 3 and p α s in 3 respectively, are Hurwitz; any vectors d [i] R v+i, i M+ M such that the polynomials s v+i + [ s v+i, s v+i,..., ] d [i] are Hurwitz; any class K functions σ j j and σ j MT + with j M + M; any positive real numbers c j R +, with j 9, γ i R +, γ i R +, i M + M, χ R +, λ R +, and any row vector k c R v such that the matrix A o + b o k c is Hurwitz. Proof. It is a consequence of the Claims, along with the arguments in Marino and Santosuosso 7, Marino, Santosuosso and Tomei AN EXAMPLE Consider the system [ ] [ ] 3 ẋ = x + u u r e = [ ]x; y = e + δ, 4 with state x = [x, x ] T R, control input u R. System 4 has non minimum phase and unstable unforced dynamics, that are stabilized via the state feedback control law u = k c x with k c = [,]. The output y = x + δ is available for measurement, where u r t is an input reference and δt is a disturbance. The regulation task is to drive the state x to zero. We consider two operating conditions, in particular for < t we set u r t =, δt = sin.5t and for < t we set u r t = sin.5t, δt = sin t + sint. We assume that S δ is the set of all angular frequencies ω.7. We construct a controller to regulate the output trajectory x t to zero via the algorithm proposed in this note, by setting M =, M =, so that the extended system -3 has order 9. We simulate the algorithm for < t choosing the numerical values of the constant design parameters as follows: in system we choose f = [5,6], in system we set the parameters ᾱ, ᾱ,..., ᾱ 8, so that p α s = s 8 + ᾱ s ᾱ 7 s + ᾱ 8 is the polynomial whose roots coincide all with the number 3/. In system 4 we let c =, c = 4, in system 5 we set σ =.5arctan.5, σ =.5arctan5, σ =.5arctan 3, σ = σ = σ 3 = 5arctan 3, χ = 6, in expression 6 we set c 3 = 4, in 8 we set d [], d [], d [], d [3] so that the roots of the polynomials s +h + [ s +h, s h,..., ] d [h] with h =,,,3 coincide with the number 3/ and in system 9 we set c 4 =. In system 3 we set c 5 =, λ =, γ = 5 ; γ = 5 5, γ 3 = 5 ; γ = γ = γ 3 = 5; in system 36 and 37 we set c 6 =, in system 39 we set c 7 =, c 8 = 5, c 9 = 3. The compensator dynamics have been simulated starting from zero initial conditions except for χ i = 6, i [,,3] and p j = p h = 5, j [,] and h [,]. The simulation results are reported in Figures -3. In Figure the input u r t along with disturbance δt are plotted. In Figure the time histories of the estimates ˆθ t, ˆ θ t, ˆ θ t of the parameters θ, θ, θ, are described. For < t in the reference exosystem 6 there aren t unknown parameters while in the disturbance exosystem 7 we have θ =.5, so that ˆθ t, ˆ θ t.5 and ˆ θ t. For < t the exosystem 6 has the 46

6 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 8 A B Fig.. In A: the reference input u r t; in B: the disturbance δt. A B C Fig.. In A: the estimate ˆθ t; in B: the estimate ˆ θ t, in C the estimate ˆ θ t. A B Fig. 3. In A: the control input ut; in B: the system undisturbed output et = x t. unknown parameter θ =.5, while in the exosystem 7 there are the parameters θ = 5, θ = 4; as a consequence ˆθ t.5, ˆ θ t 5, ˆ θ t 4. Figure 3 reports the control input ut that drives exponentially to zero the regulation error et in all three operating conditions. REFERENCES Ariyur K. B. and M. Krstic, Feedback attenuation and adaptive cancellation of blade vortex interaction on a helicopter blade element, IEEE Trans. Contr. Syst. Technology, Vol. 7, N. 5, pp , Sept Bittanti, S. andl. Moiraghi, Active controlofvibrations in helicopters via pole assignment techniques, IEEE Trans. Contr. Syst. Technology, vol., N. 4, pp , Dec Wu, B. and M. Bodson, Direct adaptive cancellation of periodic disturbances for multivariable plants, IEEE Trans. on speech and Audio Processing, vol., No 6. pp , Nov. 3. Bodson, M., Rejection of periodic disturbances of unknown and time-varying frequency, Int. J. of Adapt. Contr. and Sig. Proc., Vol. 9, N. -3, pp , March- April 5. De Wit, C.C. and L. Praly, Adaptive eccentricity compensation, IEEE Trans. Contr. Syst. Technology, Vol. 8, N. 5, pp , Sept.. Francis, B. and W. Wonham, The internal model principle of control theory, Automatica, Vol., N. 5, pp , Sept Kuo, S. and Morgan D, Active noise control systems: algorithms and DSP implementations, Wiley, New York, 996. Marino R. and P. Tomei, Global adaptive observers for nonlinear systems via filtered transformations, IEEE Trans. Automat. Contr., Vol. 37, N. 8, pp , Aug. 99. Marino, R., G. L. Santosuosso, and P. Tomei, Robust adaptive compensationofbiasedsinusoidaldisturbances with unknown frequency, Automatica, Vol. 39, N., pp , Oct. 3. Marino, R. and G. L. Santosuosso, Regulation of linear systems with unknown exosystems of uncertain order, IEEE Trans. Automat. Contr., Vol. 5, N., pp , Feb. 7. Marino, R., G. L. Santosuosso, and P. Tomei, Output feedback stabilization of linear systems with unknown additive output sinusoidal disturbances, to appear in the European Journal of Control, Vol. 4, N., 8. Nikiforov, V., Adaptive nonlinear tracking with complete compensation of unknown disturbances, European Journal ofcontrol, Vol. 4, N., pp. 3-39, 998. Landau, I. D. A., Constantinescu, and D. Rey, Adaptive narrow band disturbance rejection applied to an active suspension - an internal model principle approach, Automatica, Vol. 4, N. 4, pp , Apr. 5. Liu, J. and Y. Yang, Stability of the frequency adaptive control technique and its application to compact disk drives, Contr. Eng. Practice, Vol. 3, N. 5, pp , May 5. Serrani, A. Rejection of harmonic disturbances at the controller input via hybrid adaptive external models, Automatica, Vol. 4, N., pp , Nov. 6. Zarikian G, and A. Serrani, Harmonic disturbance rejection in tracking control of Euler-Lagrange systems: an external model approach, IEEE Trans. Contr. Syst. Technology, Vol. 5, N., pp. 8-9, Jan