Amine Othmane
email: amine.othmane@l2s.centralesupelec.fr

Joachim Rudolph
email: j.rudolph@lsr.uni-saarland.de

Hugues Mounier
email: hugues.mounier@l2s.centralesupelec.fr

Analysis of the parameter estimate error when algebraic differentiators are used in the presence of disturbances

Keywords: Algebraic differentiators, parameter estimation, least squares algorithm, time-varying linear observers

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Analysis of the parameter estimate error when algebraic differentiators are used in the presence of disturbances

INTRODUCTION

The problem of identifying parameters of a dynamic process from its operating data is an important task and is still receiving considerable attention. Among the various techniques that where developed, regression models combined with a least squares estimator have attracted much attention. In [START_REF] Durbin | Estimation of parameters in timeseries regression models[END_REF] for example, this approach is developed for discrete time dynamical systems. However, in the presence of disturbed measurements, the least squares analysis may lead to inconsistent estimates as discussed in detail for example in [START_REF] Wald | The fitting of straight lines if both variables are subject to error[END_REF]. Different approaches have been proposed in the literature in order to overcome this problem. The instrumental variable approach for example, first developed in [START_REF] Reiersøl | Confluence analysis by means of lag moments and other methods of confluence analysis[END_REF]; [START_REF] Durbin | Errors in variables[END_REF]; [START_REF] Young | An instrumental variable method for real-time identification of a noisy process[END_REF], can be seen as a slight variation of the least squares solution of the linear regression overcoming the problem. However, as noted in [START_REF] Young | Parameter estimation for continuoustime models -a survey[END_REF], the major problem with this approach is the generation of the instrumental variables themselves.

For the continuous-time estimation of process parameters using the least squares approach, the knowledge of the time derivatives of the measured signals is required and in the presence of disturbances represents an additional challenge. In [START_REF] Young | Parameter estimation for continuoustime models -a survey[END_REF], a state variable filter is proposed which simultaneously filters the signals and provides filtered time derivatives which replace the exact but unknown derivatives.

In [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]; [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], a family of numerical differentiators was introduced. The time derivative of a measured signal can be approximated using these estimators, which are time-invariant filters and operate on the non-differentiated signal. Their stability is implicitly guaranteed from their finite impulse response. In [START_REF] Liu | Analyse d'erreurs d'estimateurs des dérivées de signaux bruités et applications[END_REF], it was shown that the algebraic differentiators outperform recent numerical algorithms based on higher order sliding modes and high-gain observers. For a systematic configuration of the five filter parameters the methods presented in [START_REF] Kiltz | Parametrization of algebraic numerical differentiators to achieve desired filter characteristics[END_REF]; [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF] can be applied such that desired properties of the amplitude spectrum of the derivative estimator are achieved. Two methods were presented: The first one allows the elimination of harmonic disturbances. In the second one, the derivative estimators are approximated as low-pass filters of arbitrary large order. Then, the parameters can be calculated from the desired cutoff frequency and the desired filter order.

The contribution of this article is to analyze the parameter estimate error when algebraic differentiators are used in the context of continuous-time parameter estimation using linear regression with a least squares approach. Bounds for the error stemming from the disturbances and the derivative approximation are derived using recent results from [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF]. High frequency measurement disturbances are considered. It is shown that the rate of convergence to zero of the bias stemming from these disturbances can be tuned using the parameters of the algebraic differentiators. Furthermore, an observer with assignable error dynamics and delayed measurements for the online estimation of the parameters using the approximated derivatives is presented and compared with the recursive solution of the least squares optimization problem in a simulation.

This article is structured as follows. In section 2, the notation used in the work is summarized. The algebraic differentiators introduced in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]; [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] are recalled in section 3. The results from [START_REF] Kiltz | Parametrization of algebraic numerical differentiators to achieve desired filter characteristics[END_REF]; [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF] are used to analyze their behavior in the presence of high frequency disturbances and error bounds are derived. In section 4, the use of these filters in the context of parameter estimation is discussed. First, the use of a least squares optimization problem is introduced and analyzed. Then, an observer with assignable error dynamics is proposed. In section 5, a simulation is performed and the effectiveness of the proposed approaches is shown.

NOTATION

For a real-valued function f defined on R, the supremum norm of f is defined as

f ∞ = sup t∈R |f (t)|. For S ⊂ R it holds that f ∞,S = sup t∈S |f (t)|.
The convolution of two real-valued functions f and g is written f * g. A function f is said to be of class C n , for n a non-negative integer, if the derivatives of f up to the order n exist and are continuous.

In order to specify a C n function on a domain I ⊆ R, the notation C n (I) is adopted. For a C n function f , with n a non-negative integer, f (i) , for i = 0, . . . , n, denotes its i-th derivative and f (0) is the function f itself. Alternatively, Newtons notation for derivatives is also adopted for the first and the second derivative. The gamma function is denoted as Γ. The orthogonal Jacobi polynomial of degree i associated with the weight function

w (α,β) (τ ) = (1 -τ ) α (1 + τ ) β , τ ∈ [-1, 1] 0, else,
with the real parameters α, β > -1, is denoted P (α,β) i . The definitions for class K and KL functions are borrowed from [START_REF] Khalil | Nonlinear systems[END_REF].

ALGEBRAIC DIFFERENTIATORS

Time domain interpretation of algebraic differentiators

The algebraic derivative estimation methods introduced in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]; [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] were initially derived using differential algebraic manipulations of truncated Taylor series. Later works [START_REF] Liu | Analyse d'erreurs d'estimateurs des dérivées de signaux bruités et applications[END_REF]; [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF]) derived these filters using an approximation theoretical approach that yields a straightforward analysis of the filter characteristics, especially the estimation delay. Using this approach, the estimate of the n-th order derivative of a signal t → y(t) denoted ŷ(n) can be approximated as

ŷ(n) (t) = T 0 g (n) T (τ )y(t -τ ) dτ, (1)
with the filter kernel

g T (t) = 2 T N i=0 P (α,β) i (ϑ) P (α,β) i 2 w (α,β) (ν(t)) P (α,β) i (ν(t)) .
In the latter equation

• ν(t) = 1 -2 T t, • x = x,
x is the norm induced by the inner product

x, y = 1 -1 w (α,β) (τ )x(τ )y(τ ) dτ,
• N is the degree of the polynomial approximating the signal y (n) in the time window [t -T, t],

• T is the filter window length,

• and ϑ parameterizes the estimation delay as described in (3).

This approach yields a straightforward analysis of the estimation delay δ t and the degree of exactness1 γ which are given as [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF])

γ = n + N + 1, if N = 0 ∨ ϑ = p (α,β) N +1,k n + N, otherwise, (2)
δ t = α+1 α+β+2 T, if N = 0 1-ϑ 2 T, otherwise, (3) with p
(α,β)

N +1,k the k-th zero of the Jacobi polynomial of degree N + 1. In the sequel, h T is a function defined as

h T (t) = g T (t), if t ∈ [0, T ], 0, otherwise. (4)

Error analysis

Let ȳ be a disturbed measurement of y such that ȳ(t) = y(t) + η(t),

(5) where η represents the additive disturbance. The estimated derivative ŷ(n) is corrupted by three sources of errors [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]; [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]; [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF]):

• The error e d stemming from the delay δ t and defined as e d (t) := y (n) (t -δ t ) -y (n) (t). • The polynomial approximation yields a smoothing of fast changes in the signal. This error is denoted e a and is defined as e a (t) := (h T * y)(t) -y (n) (t -δ t ). • The error due to the disturbance corrupting the measurements and denoted e n . It is defined as

e n := h (n) T * y -h (n) T * ȳ = h (n) T * η.
The errors e a and e n are of interest in this work and are studied in the following propositions. Proposition 1. [START_REF] Liu | Analyse d'erreurs d'estimateurs des dérivées de signaux bruités et applications[END_REF]; [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF]). Let y ∈ C n+m+1 (I), with m ∈ N and I ⊆ R, γ and h T the degree of exactness defined in (2) and the filter given in (4), respectively. Assume there exists

M n ∈ R * + such that y (n+p+1) ∞,I
M n with p = min{m, γ -n}. Then, there exists a positive scalar M such that the approximation error satisfies

e a ∞,I M p! T 2 p+1 M n ,
for T → 0. Proposition 2. Let ȳ given in (5) be a disturbed measurement of y ∈ C n+1+p (I), p, n ∈ N, I ⊆ R and h T the filter kernel defined in (4). The disturbance is assumed to be integrable and bounded by σ, i.e., σ = sup t∈I η(t).

Then, for all times t ∈ I the error e n stemming from the disturbance is bounded such that

e n ∞,I σ T n+1 Q, with Q = T n+1 T 0 g (n) (τ ) dτ < ∞ independent of T . Proof.
The error stemming from the disturbance is given as e n (t) = (h

(n) T * y)(t) -(h (n) T * ȳ)(t) = (h (n) T * η)(t) = T 0 g (n) T (t -τ )η(τ ) dτ. Note that g (n) (t) = 2 2n+1 T n+1 w (α-n,β-n) (ν(t))•    N i=0 (i + n)! P (α,β) i (ϑ) i! P (α,β) i 2 P (α-n,β-n) i+n (ν(t))    , with ν(t) = 1 -2 T t. Let Q = T n+1 T 0 g (n) (τ )
dτ < ∞ which is independent of T . Thus, the error can be bounded by

|e n (t)| = T 0 g (n) T (t -τ )η(τ ) dτ σ T 0 g (n) T (τ ) dτ = σ T n+1 Q, for all times t. Hence, the error satisfies e n ∞,I σ T n+1 Q.
The error bounds for more general stochastic processes are derived in [START_REF] Liu | Analyse d'erreurs d'estimateurs des dérivées de signaux bruités et applications[END_REF]. However, it is stressed that these are only conservative bounds and do not take into account the low pass effect of these filters analyzed in detail in [START_REF] Kiltz | Parametrization of algebraic numerical differentiators to achieve desired filter characteristics[END_REF]; [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF].

The estimation error in the presence of high frequency disturbances is now analyzed using the results from [START_REF] Kiltz | Parametrization of algebraic numerical differentiators to achieve desired filter characteristics[END_REF]; [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF]. For this purpose, a disturbance model introduced in [START_REF] Astolfi | Sensitivity to high-frequency measurement noise of nonlinear high-gain observers[END_REF] for the analysis of the sensitivity of High-Gain observers to high frequency disturbances is used. The model is given as ẇ(t) = Sw(t), η(t) = P w(t), (6) where the matrices S and P take the form

S = blockdiag(S 1 , . . . , S m ), S i = 0 ω i -ω i 0 , P = ((0 p 1 ), (0 p 2 ), . . . , (0 p m )), with ω i ∈ R >0 , p i ∈ R, i = 1, . . . , m, m ∈ N,
and the parameter ∈ (0, 1) will be taken small in the forthcoming analysis. The initial conditions are denoted w 0 (t 0 ) = w 0 . System (6) can be conveniently seen as a generator of m > 0 harmonics at frequencies wi > 0, i = 1, . . . , m. The disturbance is then given as η(t) = m i=1 p i cos( ωi t + φ i ) where φ i = arccos(w 0,i,2 ). The next proposition proposes a bound for the error stemming from a high frequency disturbance generated by the model given in (6). Proposition 3. Let ȳ given in (5) be a disturbed measurement of y ∈ C n (I), n ∈ N, I ⊆ R, and h T the filter kernel defined in (4). Assume the disturbance to be generated by the model in (6). Introduce ω > = max i∈{1,...,m} ω i and ω < = min i∈{1,...,m} ω i . Then, for → 0 and min{α, β} > n the following holds for the error stemming from the disturbance:

• If N = 0 and α = β, there exists a function κ 1 ∈ KL such that 0 e n ∞,I κ 1 1 T , ω > ,
• otherwise there exist two functions κ 2 , κ 3 ∈ KL such that

κ 2 1 T , ω < e n ∞,I κ 3 1 T , ω > .
Proof. As observed in earlier references (see for instance [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]; [START_REF] Kiltz | Parametrization of algebraic numerical differentiators to achieve desired filter characteristics[END_REF]; Kiltz (2017)), the algebraic differentiator in ( 1) is a linear time-invariant filter. Thus, the response of the filter to a sinusoidal signal with amplitude u and angular frequency ω 0 is a sinusoidal signal with an identical frequency and amplitude U = uω n 0 G T (ω 0 ), where G T denotes the Fourier transform of g T . It is shown in (Kiltz, 2017, chapter β) where µ = min{α + 1, β + 1},

3) that for |ω| → ∞ the function G T satisfies 1 (ωT ) µ G -|G T (ω)| 1 (ωT ) µ G + , with G + = r (α,β) + s (α,β) G -= r (α,β) -s (α,
r (α,β) = N i=0 c (α,β) i Γ(µ + κ + i) P (µ-1,µ+κ-1) i (χϑ), s (α,β) = N i=0 (-1) i c (α,β) i Γ(µ + i) P (µ-1,µ+κ-1) i (χϑ), c (µ,κ) i = (2µ + κ + 2i -1)Γ(2µ + κ + i -1), χ = 1, β ≥ α, -1, α > β, , κ = |α -β| .
The model ( 6) generates a sinusoidal disturbance signal η(t) = m i=0 p i cos( ωi t + φ i ). For notational simplicity, let ω = (ω 1 , . . . , ω m ). From the above considerations it follows that

|e n (t)| m i=0 w i n p i G T ω i m i=0 w i n |p i | G T ω i , and, for → 0, b l (T, ω) e n ∞,I b u (T, ω), where b l (T, ω) = T -µ G - m i=0 |p i | w i n-µ , b u (T, ω) = T -µ G + m i=0 |p i | w i n-µ .
Using ω > and ω < it follows that

b l (T, ω) ≥ mT -µ ω < n-µ G -p < =: κ 2 1 T , ω < b u (T, ω) mT -µ ω > n-µ G + p > =: κ 3 1 T , ω > ,
with p < = min i∈{1,...,m} |p i | and p > = max i∈{1,...,m} |p i |. Thus,

κ 2 1 T , ω < e n ∞,I κ 3 1 T , ω > .
It is straightforward to see that for µ > n the functions κ 2 , κ 3 ∈ KL. For N = 0 and α = β it holds that

κ 2 1 T , ω < = 0 κ 3 1 T , ω > = 4 µ Γ(µ + 1/2) √ πT µ ω µ > =: κ 1 1 T , ω > .

PARAMETER IDENTIFICATION USING ALGEBRAIC DIFFERENTIATORS

Least squares parameter estimation

Introductory example Consider the system ẋ(t) = ax(t) + bu(t), (7) where a, b ∈ R are unknown parameters, u is a known function of time and y(t) = x(t) + η(t) is a disturbance corrupted measurement of the state. Assume η(t) is generated by ( 6). The estimation of the parameters a and b using a least squares approach and the algebraic differentiators presented above is now discussed.

Assume the filter in (1) is used to estimate ẏ and denote the estimate as ŷ. Taking into account the estimation delay δ t , equation ( 7) can then be rewritten as ŷ(t) = ay(t -δ t ) + bu(t -δ t ).

For the purpose of convergence analysis of the algorithms the latter equation is rewritten as x(t) = ax(t -δ t ) + bu(t -δ t ) + e(t), with e(t) = η(t) + η(t -δ t ), where η represents the output of the filter when applied to the disturbance signal. Note

that defining θ = [a b] T , H(t) = [x(t -δ t ) u(t -δ t )] T , and φ(t) = x(t) -e(t) yields φ(t) = H T (t)θ.
Assume that for all t, t 0 ≥ 0 satisfying t 0 < t the function

ψ(t) = t t0 e λ(τ -t) H T (τ )H(τ ) dτ, ( 8
)
with λ > 0, is non-zero. Let θ(t) be the solution of the minimization problem

min θ t t0 e λ(τ -t) H T (τ )θ -φ(τ ) T H T (τ )θ -φ(τ ) dτ.
Then,

θ(t) = 1 ψ(t) t t0 H(τ )φ(τ ) dτ = 1 ψ(t) t t0 H(τ ) x(τ ) dτ θ * (t) - 1 ψ(t) t t0 H(τ )e(τ ) dτ θ(t)
, i.e., the measurement disturbance yields a bias θ in the estimation of the parameters. A method where this bias vanishes is presented in the sequel.

In contrast to the approach presented above, where the filter was only used for the estimation of the numerical derivative of the measurement, it is applied to all signals in the system to get ŷ(t) = aŷ(t) + bû(t) (9) and equivalently

x(t) = ax(t) + bû(t) + ê(t), (10) with ê(t) = η(t) + η(t). Using the same arguments as in the proof of Proposition 3 it can be shown that the error satisfies

e ∞ κ 1 T , ω > = c 1 T -µ ω > 1-µ 1 + ω > -1
with ω > = max i∈{1,...,m} ω i , p > = max i∈{1,...,m} p i , c 1 some constant depending on the filter parameters and the disturbance amplitude, and µ = min{α + 1, β + 1}. Note that lim →0 κ ( ω> ) = 0 when µ > 1. Estimating the parameters a and b from (10) using a least squares approach yields a bias θ satisfying θ

∞,I = sup t∈I 1 ψ(t) t t0 H(τ )e(τ ) dτ sup t∈I 1 ψ(t) t t0 |H(τ )| |e(τ )| dτ κ 1 T , ω > sup t∈I 1 ψ(t) t t0 |H(τ )| dτ = κ θ 1 T , ω > ,
for all t ∈ I = [0 t * ], with t * bounded and strictly positive. If the input u is bounded, it can be concluded that lim

→0

κ θ 1 T , ω> = 0, i.e., the identification bias stemming from the disturbances vanishes when → 0. The convergence speed of the error to zero can be adjusted using µ = min{α + 1, β + 1}.

In the previous analysis only the effect of the disturbance on the estimation was taken into account. However, applying the algebraic differentiators also results in an error stemming from the polynomial approximation of the signals in the filter window. This is analyzed in the forthcoming paragraph.

For notational simplicity the disturbance is assumed to be identical to zero, i.e., η(t) = 0 for all t ≥ 0. Define e x (t) = x(t-δ t )-x(t) and e u (t) = u(t-δ t )-û(t). Recalling the estimation equation ( 9)

x(t) = ax(t) + bû(t)

used earlier, yields ẋ(t -δ t ) = ax(t -δ t ) + bu(t -δ t ) + ėx (t) -ae x (t) -be u (t) eg(t)
.

Thus, using the algebraic differentiators yields an estimation bias e g (t) compared to the use of the exact values. Assume that x ∈ C mx+2 (I), u ∈ C mu+1 (I), I ⊆ R, and that these functions are bounded. Let e θ (t) = θ(t) -θ(t),

H(t) = [x(t) u(t)] T and ψ(t) = t t0 e λ(τ -t) H T (τ )H(τ ) dτ
with t 0 < t. Assume ψ(t) = 0 for all times t ∈ I. The estimation error due to the use of the algebraic differentiators satisfies

e θ ∞,I = sup t∈I 1 φ(t) t t0 H(τ )e g (τ ) dτ sup t∈I 1 φ(t) t t0 |H(τ )| |e g (τ )| dτ e g ∞,I sup t∈I 1 φ(t) t t0 |H(τ )| dτ.
From Proposition 1 it follows that

e g ∞,I = O T 2 p+1
, with p = min{m x , m u , γ -1}. Thus, the estimation error due to the use of the algebraic differentiators satisfies

e θ ∞,I O T 2 p+1 .
Generalization The observation made with the simple first order system in (7) can be generalized for higher order systems as stated in the following proposition. Proposition 4. Let n and m be positive integers, u ∈ C m (I), I ⊆ R, a known signal, h T the filter defined in (4), x a C n (I) function and ȳ = x + η a disturbed measurement where the disturbance η is generated by the model in ( 6) with the parameter . Introduce ŷ(i) = h

T * ȳ and assume that u and x satisfy the linear relationship

x (n) (t) + n-1 i=0 a i x (i) (t) = m i=0 b i u (i) (t), a i , b i ∈ R. (11)
Let λ be a positive scalar and define

H x (t) = x(t), . . . , x (n-1) (t), u(t), . . . , u (m) (t) T , H ȳ (t) = ŷ(t), . . . , ŷ(n-1) (t), û(t), . . . , û(m) (t) T , θ = [a 0 , . . . , a n-1 , b 0 , . . . , b m ] T , ψ x (t) = t t0 e λ(τ -t) H T x (τ )H x (τ ) dτ, ψ ȳ (t) = t t0 e λ(τ -t) H T ȳ (τ )H ȳ (τ ) dτ.
Assume the matrices ψ x (t) and ψ ȳ (t) are invertible for all t ∈ I. Denote by θ x (t) and θ ȳ (t) the solution of the following optimization problems

min θ t t0 e λ(τ -t) H T x (τ )θ -x (n) (τ ) 2 dτ, min θ t t0 e λ(τ -t) H T ȳ (τ )θ -ŷ(n) (τ ) 2 dτ,
respectively. Assume further that lim t→∞ θ x (t) = θ. The estimated parameters using the measurements satisfy

θ x -θ ȳ ∞,I κ g T 2 p+1 + κ 1 T , ω > ,
with p = min{m, n, γ -1} and κ g , κ class K ∞ and class KL functions, respectively.

Proof. The proof follows from the considerations presented in the previous example.

Observer with assignable error dynamics

Consider the general model in ( 11) and the notation introduced in Proposition 4. First, for simplicity, assume that x (i) (t), i = 0, . . . , n and u (i) (t), i = 0, . . . , m, are known for all times t. Introduce θ(t) = [a 0 , . . . , a n-1 , b 0 , . . . , b m ] T . The functions θ, H x and x (n) satisfy θ(t) = 0

x (n) (t) = H T x (t)θ(t). Let z(t) =      x (n) (t) x (n) (t -δ 1 )
. . .

x (n) (t -δ κ )      =      H T x (t) H T x (t -δ 1 ) . . . H T x (t -δ κ )      =C(t) θ(t), κ = n + m,
with the delays δ i , i ∈ {1, . . . , κ} chosen such that det(C(t)) is not identically zero for all times t. The parameters can be estimated using the observer θ

(t) = L(t) C(t) θ(t) -z(t) , θ(0) ∈ R,
with L(t) the solution of the equation

L(t)C(t) = (det(C(t))) 2 A, if |det(C(t))| < d ρA, otherwise, (12)
where A is an arbitrary matrix whose eigenvalues lie in the left half plane, and d and ρ are two strictly positive parameters. The asymptotic convergence of the estimates to the true parameters can be shown using the Lyapunov function V (t) = e T θ (t)e θ (t), with e θ (t) = θ(t) -θ(t). When only the measurement y(t) = x(t) + η(t), with the additive disturbance η(t) satisfying (6), is available, an algebraic differentiator can be used and the parameters can be estimated by

θ(t) = L(t) Ĉ(t) θ -ẑ(t) , θ(0) ∈ R, with Ĉ(t) =      H T ȳ (t) H T ȳ (t -δ 1 ) . . . H T ȳ (t -δ κ )      , ẑ(t) = (h (n) T * z)(t),
and L(t) the solution of (12) when the function C is replaced by Ĉ. The estimate error derived in the last section for the least squares approaches can be derived for this observer in a similar manner.

EXAMPLE: PARAMETER ESTIMATION OF A 4-TH ORDER LINEAR SYSTEM

Consider the 4-th order linear system defined as

ξ(t) =    0 1 0 0 0 0 1 0 0 0 0 1 0 γ 1 γ 2 γ 3    ξ(t) +    0 ζ 1 ζ 2 ζ 3    u(t)
with the output y(t) = ξ 1 (t). The numerical values of the parameters are summarized in Tab. 1.

Table 1. Numerical values of the parameters used in the simulation.

γ 1 γ 2 γ 3 ζ 1 ζ 2 ζ 3
-16 -15.4 -1.2 0.5 -0.6 -0.1

The input and measurement disturbance are chosen to be a superposition of harmonics such that

u(t) = 6 i=1 u i i sin ω i t + sin(πt) , η(t) = 5 i=0 a i sin(ω i t + φ i ),
with u i = 3, ω = 10, a i ∈ (0, 0.005) and ω i ∈ [500, 5000] for i ∈ {0, . . . , 5}. The frequencies, amplitudes, and shifts of the disturbance and the input are chosen randomly using a uniform distribution. The signal-to-noise ratio is equal to 36 dB. In order to be able to compare the observer to the least squares estimator, the estimation of the parameters starts only for times greater than the highest delay used in the observer. The numerical values of the parameters of the algebraic differentiator are N = 2, T = 1.5, α = 5, β = 7.5, and ϑ is chosen to be the greatest zero of the Jacobipolynomial P (α,β) N +1 . The parameters d and ρ of the observer are chosen to be equal to 10 and 1, respectively, and A is the diagonal matrix A = -diag ([[2.4, 8, 3.2, 4, 0.8, 8]]) and the delays are δ 1 = 0.1, δ 2 = 0.33, δ 3 = 0.53, δ 4 = 0.87, δ 5 = 1.47. The least squares optimization problem is solved recursively to show the applicability of the approach in the context of online parameter estimation. The parameter λ = 0.98. The observer is implemented using the backward Euler method. The sampling period is equal to 0.01.

The simulation results given in Fig. 1 show clearly that the estimated parameters converge to the true parameters. During the transient phase, the results of the observer show less oscillations and overshoots than those of the least squares estimator. Varying its parameter λ introduced in (8) does not significantly affect this observation.

CONCLUSION

The use of algebraic differentiators in the context of asymptotic parameter estimation has been discussed. The estimation error when a least squares approach is used has been analyzed using recent results from [START_REF] Kiltz | Algebraische Ableitungsschätzer in Theorie und Anwendung[END_REF]. More specifically, the bias stemming from high frequency disturbances has been considered. It has been shown that it decreases with higher disturbance frequencies. The parameters of the filters can be used to tune the convergence rate of the bias to zero. For the online estimation, an observer with assignable error dynamics has also been proposed and compared to the recursive solution of the least squares problem. The observer shows better results in the simulation. Further research will be conducted on the analysis of the error stemming from the discretization of the approaches presented here. A systematic methodology for the choice of the delays must also be discussed. depicts the estimation results with the recursive least squares approach and when the proposed observer is used. The true parameters are given by .

Fig. 1 .

 1 Fig. 1. The estimated parameters:

The degree of exactness was introduced in Kiltz (2017) as the polynomial degree up to which the derivative estimation is exact. If γ =

for example, the first and second time derivatives of a polynomial signal of degree two are exact up to an estimation delay.

This work is supported by the "ADI 2018" project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-0


Romeo Ortega
email: ortega@lss.supelec.fr

Vladislav Gromov

Emmanuel Nuño
email: emmanuel.nuno@cucei.udg.mx

Anton Pyrkin

Jose Guadalupe Romero

Parameter Estimation of Nonlinearly Parameterized Regressions: Application to System Identification and Adaptive Control

Keywords: Adaptive control, systems identification, nonlinear control

We propose a solution to the problem of parameter estimation of nonlinearly parameterized regressions-continuous or discrete time-and apply it for system identification and adaptive control. We restrict our attention to parameterizations that can be factorized as the product of two functions, a measurable one and a nonlinear function of the parameters to be estimated. Another feature of the proposed estimator is that parameter convergence is ensured without a persistency of excitation assumption. It is assumed that, after a coordinate change, some of the elements of the transformed function satisfy a monotonicity condition. The proposed estimators are applied to design identifiers and adaptive controllers for nonlinearly parameterized systems, which are traditionally tackled using overparameterization and assuming persistency of excitation.

INTRODUCTION AND LITERATURE REVIEW

It is well known that nonlinear parameterizations are inevitable in any realistic practical problem [START_REF] Ljung | System Identification: Theory for the User[END_REF][START_REF] Nelles | Nonlinear System Identification[END_REF][START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF]. Unfortunately, designing adaptive (identification or control) algorithms for nonlinearly parameterized systems is a difficult poorly understood problem (Pyrkin et al., 2014). See [START_REF] Ortega | Parameter Estimation of Nonlinearly Parameterized Regressions without Overparameterization nor Persistent Excitation: Application to System Identification and Adaptive Control[END_REF] for an extensive review of the literature. Some results for gradient estimators have been reported in the literature for convexly or monotonically parameterized continuous-time (CT) systems. A very important drawback of these approaches is that the monotonicity or convexity conditions are imposed on functions that depend, not only on the parameters, but also on external signals, e.g., time or the system state. This renders the verification of the condition very hard to carry out. One of the main contributions of this paper is to overcome this limitation.

To the best of the authors' knowledge no developmentssimilar to the ones done for CT-have been reported for case of nonlinearly parameterized discrete-time (DT) regressions that, in spite of its great practical importance, have attracted less attention in the identification and adaptive control community. One of the objectives of our paper is to contribute, if modestly, towards the development of estimation algorithms for DT NPRE. In particular, we provide solutions to the, essentially open, problems of direct and indirect adaptive pole-placement control (APPC) without overparameterization nor persistency of excitation (PE) requirements. 1 It should be pointed out that a solution to the direct APPC problem using overparameterization, hence requiring some excitation conditions, has been reported in [START_REF] Pyrkin | A Globally convergent direct adaptive poleplacement controller for nonminimum phase systems with relaxed excitation assumptions[END_REF].

The dependence of the convexity or monotonicity assumption on external signals mentioned above happens even 1 We recall that a bounded vector signal Ω ∈ R q is said to be PE if there exist δ ∈ R >0 such that t+T t Ω(τ )Ω (τ )dτ ≥ δIq for some T ∈ R >0 and all t ∈ R ≥0 in CT, or k+K j=k+1 Ω(j)Ω (j) ≥ δIm, ∀k ∈ N ≥0 , for some K ∈ N >0 , with K ≥ m, in DT.

in the case when the uncertain terms appear as products of a function of the unknown parameters times a known function-the so-called, factorized mappings, that is NPRE of the form y = ΩS(θ), with S : R q → R p , with p > q. Although in this case it is possible to define the extended parameter vector θ a := S(θ) to obtain a linear parametrization, overparametrization suffers from the following well-known shortcomings [START_REF] Ljung | System Identification: Theory for the User[END_REF][START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF].

(S1) Performance degradation, e.g., slower convergence, due to the need of a search in a larger parameter space.

(S2) More stringent conditions imposed on the reference signals to ensure the PE requirement needed for convergence of the new parameters.

(S3) Inability to recover the true parameters-except for injecting mappings.

(S4) Conservativeness introduced when incorporating prior knowledge in restricted parameter estimation.

In this paper we propose a parameter estimator for monotonic, factorized NPRE that achieves the following objectives.

(O1) It does not rely on overparameterization.

(O2) Imposes the monotonicity property directly to the function S(θ).

(O3) Ensures parameter convergence without the stringent PE requirement.

CT estimators for NPRE with factorizable mappings that avoid overparameterization and rely on monotonicity have been reported in (Liu et al., 2011, Section 3) and(Aranovskiy et al., 2017, Section III). In [START_REF] Liu | On adaptive control of nonlinearly parameterized nonlinear systems: Towards a constructive procedure[END_REF] neither the second nor the third objectives above are achieved. On the other hand, in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] these objectives are achieved, via the use of a dynamic regressor extension and mixing (DREM) estimator. As is well known, the main feature of DREM is that it generates, out of a q-dimensional regression equations, one scalar equation for each of the q unknown parameters. Another important feature of DREM is that parameter convergence is ensured without assuming PE.

In this paper, we also use DREM to derive both, CT and DT, parameter estimators. We obtain simpler and stronger results than [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] due to the following three key modifications.

(M1) Generate the extended regressor matrix using the linear time-varying (LTV) operators first introduced in (Kreisselmeier, 1977).

(M2) Directly apply the "mixing" operation-that is the multiplication by the adjugate of the extended matrix-to generate the scalar regressions.

(M3) Incorporate the possibility of adding a change of coordinates to the original parameters to satisfy the required monotonicity property.

Notation. R >0 , R ≥0 , N >0 and N ≥0 denote the positive and non-negative real and integer numbers, respectively.

For n ∈ N >0 we define the set n := {1, 2, . . . , n}. For x ∈ R n , we denote |x| 2 := x x. CT signals s : R ≥0 → R are denoted s(t), while for DT sequences s : N ≥0 → R we use s(k) := s(kT s ), with T s ∈ R >0 the sampling time. When a formula is applicable to CT signals and DT sequences the time argument is omitted. The action of an operator H : L ∞ → L ∞ is denoted H[u](t), while for an operator H : ∞ → ∞ we use H[u](k). With i ∈ N >0 we define q ±i u(k) := u(k ± i), and p i [u](t) := d i u dt i . All mappings and reference signals are assumed smooth. Given a function F : R n → R we define ∇F := ∂F ∂x .

MONOTONIZABLE NONLINEARLY PARAMETERIZED FACTORIZED REGRESSIONS

Problem formulation

In many system identification and adaptive control applications one is confronted with the problem of estimation of the parameters appearing in a NPRE of the form y = ΩS(θ) + ε (1) where y ∈ R n , Ω ∈ R n×p are measurable signals, θ ∈ R q is a constant vector of unknown parameters, S : R q → R p , with p > q,

(2) and ε is a (generic) exponentially decaying term. The task is to identify on-line the parameters θ, out of the measurements of y and Ω. Remark 2.1. The NPRE (1) is a particular case of the more general regression y(•) = H(•, θ), with (•) = t in CT or k in DT. Although in the factorized case it is possible to introduce extra parameters to obtain a linear parametrization, overparametrization suffers from the well-known shortcomings S1-S5 mentioned in the Introduction.

Key monotonicity assumption

Similarly to [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF][START_REF] Liu | On adaptive control of nonlinearly parameterized nonlinear systems: Towards a constructive procedure[END_REF] the key property of the parameterization that we will exploit is P -monotonicity, which is defined a follows.

Definition 2.1. Given a positive definite matrix P ∈ R q×q , a mapping L : R q → R q is strongly P -monotone if and only if there exists a constant ρ ∈ R >0 such that

(a -b) P [L(a) -L(b)] ≥ ρ|a -b| 2 > 0, ∀a, b ∈ R q , a = b.
(3) Lemma 2.1. A sufficient condition for a differentiable mapping L : R q → R q to be strictly P -monotone is P ∇L + (∇L) P ≥ ρI q > 0.

(4) Assumption 2.1. Consider the mapping S(θ). There exists:

(i) a bijective mapping D : R q → R q , θ → η with right inverse D I : R q → R q , η → θ; (ii) a permutation matrix T ∈ R p×p and; (iii) a positive definite matrix P ∈ R q×q

such that P ∇W(η)C + C[∇W(η)] P ≥ ρI q > 0, (5)
where

W(η) := S(D I (η)), C := I q | 0 q×(p-q) T. (6)

222

In words, the construction associated with Assumption 2.1 proceeds as follows. First, introduce a bijective coordinate change for the parameters θ, namely η = D(θ), with inverse θ = D I (η). Second, write the original mapping S(θ) in terms of the parameters η via the definition of the new mapping W(η) := S(D I (η)). Third, assuming that these mapping contains q elements that are "good"term to be defined below-place them at the top with the permutation matrix T and select them with the fat matrix I q | 0 q×(p-q) . Whence, define the new "good" mapping G : R q → R q as G(η) := CW(η).

(7) Observing that ∇G = ∇WC , and invoking Lemma 2.1, the condition (5) ensures that this "good" mapping is strongly P -monotonic. For future reference we rewrite this condition in terms of the "good" mapping as

P ∇G(η) + [∇G(η)] P ≥ ρI q > 0. ( 8
)
Using the definitions above in the NPRE (1) we obtain the new NPRE in terms of the parameters η as y = ΩW(η) (9)

GENERATION OF SCALAR NPRE VIA DREM

Continuous-time case

Proposition 1. Consider the NPRE (9) for CT signal. Fix λ > 0 and define the signals

Ẏ (t) = -λY (t) + Ω (t)y(t) Φ(t) = -λΦ(t) + Ω (t)Ω(t) Y(t) = Cadj{Φ(t)}Y (t) ∆(t) = det{Φ(t)}. (10) The q scalar NPRE hold Y i (t) = ∆(t)G i (η), i ∈ q ⇔ Y(t) = ∆(t)G(η). ( 11
)

Discrete-time case

Proposition 2. Consider the NPRE (9) for DT sequences. Fix 0 < α < 1 and define the signals

Y (k) = -αY (k -1) + Ω (k -1)y p (k -1) Φ(k) = -αΦ(k -1) + Ω (k -1)Ω(k -1) Y(k) = Cadj{Φ(k)}Y (k) ∆(k) = det{Φ(k)}. (12) The q scalar NPRE hold Y i (k) = ∆(k)G i (η), i ∈ q ⇔ Y(k) = ∆(k)G(η). ( 13
(t) = ΓP ∆(t)[Y(t) -∆(t)G(η(t))], (14)
with Γ ∈ R q×q , Γ > 0 the adaptation gain.

(i) The norm of the parameter estimation vector η(t) := η(t) -η is monotonically non-increasing, that is,

|η(t 2 ) ≤ |η(t 1 )|, ∀t 2 ≥ t 1 ∈ R ≥0 . (15) (ii) The following implication holds ∆(t) / ∈ L 2 ⇒ lim t→∞ |η(t)| = 0.
Remark 4.1. Convergence in all parameter estimators-as well as in state observers-can only be ensured under some kind of excitation conditions [START_REF] Ljung | System Identification: Theory for the User[END_REF]. In particular, for standard gradient and least-squares estimators this property is encrypted in the well known PE requirement of the regressor. As it has been shown in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] convergence of DREM estimators can be ensured without requiring PE and replacing it, instead, by the assumption ∆(t) / ∈ L 2 , which is necessary and sufficient for parameter convergence for linear regression equation. Proposition 3 shows that this condition is sufficient for NPRE of the form (1) with a P -"monotonizable" regressor S(θ). Remark 4.2. We have assumed that the mapping G(η) is strongly P -monotonic. This can be relaxed to strictly Pmonotonic adding some further assumptions on ∆(t).

(k + 1) = η(k) + γP ∆(k) 1 + κ∆ 2 (k) [Y(k) -∆(k)G(η)], ( 17
)
with γ > 0 the adaptation gain selected such that the constant 2 σ := 2γρ -γ 2 ν 2 λ 2 max {P } > 0, (18) and the constant κ verifying κ ≥ max{1, σ}.

P1 The norm of the parameter estimation error η(k

) := η(k) -η is monotonically non-increasing, that is, |η(k 2 ) ≤ |η(k 1 )|, ∀k 2 ≥ k 1 ∈ N ≥0 . (20) P2 The following implication is true ∞ i=0 1 + (κ -σ) ∆ 2 (k) 1 + κ∆ 2 (k) = 0 ⇒ lim k→∞ |η(k)| = 0.
P3 The following implication is true lim

k→∞ ∆(k) =: ∆(∞) = 0 ⇒ lim k→∞ |η(k)| = 0.
Remark 4.3. Similarly to the observation made in Remark 4.1 the sufficient conditions for parameter convergence of Properties P2 and P3 should be interpreted as excitation requirements imposed on ∆(k). Notice that the condition of Property P3 is sufficient to ensure ∆(k) / ∈ 2 and necessary for it to be PE.

2 Clearly, for any positive ρ, ν and P > 0, this condition is satisfied with γ < 2ρ ν 2 λ 2 max {P } .

APPLICATION TO CT NONLINEARLY PARAMETERIZED NONLINEAR SYSTEMS

Direct adaptive control of a general class of CT nonlinear systems

Consider CT systems described by the state equations ẋ(t) = F (x(t), u(t)) + R(x(t))S(θ) (21) where x(t) ∈ R n is the measurable state, u(t) ∈ R m , with n ≥ m, is the control signal, the mappings F : R n × R m → R n , R : R n → R n×p and S : R q → R p are known with p > q, and θ ∈ R q is a constant vector of unknown parameters. Assumption 5.1. There exists a mapping β : 22) has a globally exponentially stable (GES) equilibrium at a desired value x ∈ R n .

R n × R q → R m , such that the system ẋ(t) = F (x(t), β(x(t), θ)) + R(x(t))S(θ) =: f (x(t)) (
The control objective is then to design a parameter estimator such that the (certainty-equivalent) adaptive control u = β(x(t), θ(t)) ensures the asymptotic convergence lim inf x(t) = x , (23) with all signals bounded. A fist step in the design is the derivation of NPRE (1) for the system (21). This is easily obtained applying to (21) the stable, LTI filter

H(p) = 1 p+λ and defining y(t) := pH(p)[x](t) -H(p)[F (x, u)](t) Ω := H(p)[R(x)](t), (24)
and ε(t) is the solution of H(p)[ε](t) = 0. A state-space realization of ( 24) is given by ż

(t) = -λ(z(t) + x(t)) -F (x(t), u(t)) Ω(t) = -λΩ(t) + R(x(t)) y(t) = z(t) + x(t).
(25) Proposition 5. Consider the nonlinearly parameterized, nonlinear system (21) satisfying Assumptions 2.1 and 5.1. Let the adaptive control be given by u(t) = β(x(t), D I (η(t))), together with the parameter estimator (10), ( 14) and ( 25). If ∆(t) / ∈ L 2 (23) holds with all signals bounded.

Adaptive tracking of Euler-Lagrange systems

System dynamics and adaptive control problem formulation. We consider n q degrees-of-freedom (dof), possibly underactuated, EL systems with generalized coordinates q(t) ∈ R nq and control vector u(t) ∈ R m , m ≤ n q , whose dynamics is described by M (q)q + C(q, q) q + ∇U(q) = G(q)u, (26) where M : R nq → R nq×nq is the generalized inertia matrix, which is positive definite and assumed to be bounded, U : R nq → R the potential energy function, G : R nq → R nq×m the input matrix and C : R nq × R nq → R nq×nq represents the Coriolis and centrifugal forces matrix, which satisfies the key skew-symmetry property z [ Ṁ (q) -2C(q, q)]z, ∀z ∈ R nq , (27) holds. See [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF] for additional details on this model and many practical examples.

Assumption 5.2. Given a desired bounded trajectory for the state vector (q (t), q (t)) ∈ R nq ×R nq . Define the state tracking error col(q, q) := col(q -q , q -q ). There exists a mapping β : R nq × R nq × R q × R ≥0 → R m , such that the system M (q)q + C(q, q) q + ∇U(q) = G(q)β(q, q, θ, t), has an error dynamics q q = f (q, q, t)

whose origin is GES.

The control objective is then to design a parameter estimator such that the (certainty-equivalent) adaptive control u = β(q, q, θ, t) ensures global asymptotic tracking, that is, lim inf col(q(t), q(t)) = 0, (28) with all signals bounded.

Derivation of the regression equation.

A fist step in the design is the derivation of the NPRE (1) for the system (26)-which was already reported in [START_REF] Slotine | Composite adaptive control of robot manipulators[END_REF]. Towards this end, we introduce the following parameterization of the inertia matrix M (q) and the potential energy U(q)

M (q) = i=1 m i (q)S m i (θ), U(q) = r j=1 U j (q)S U j (θ) (29)
with known matrices m i : R nq → R nq×nq and functions U j : R nq → R and known functions S m i (θ), S U j (θ) : R q → R of the unknown physical parameters θ ∈ R q . We group together all functions S m i (θ), S U j (θ) in a single vector mapping S : R q → R p as

S(θ) := col(S m 1 (θ), • • • , S m (θ), S U 1 (θ), • • • , S U r (θ)) ∈ R p , (30)
where p := + r > q. We are in position to present the following. Proposition 6. The regressor matrix Ω :

R nq × R nq → R nq×p Ω(q, q) := H(p)          
pm 1 (q) q -1 2 ∇q( q m 1 (q) q) . . . pm (q) -1 2 ∇q( q m (q) q)

∇U 1 (q) . . .

∇Ur(q)           , (31)
is such that the EL system (26) satisfies the NPRE y = Ω(q, q)S(θ) (32) where y := H(p) [G(q)u] , (33) with θ and S(θ) defined via ( 29) and (30).

Main stabilization result.

We are now in position of present the main result of this subsection. Proposition 7. Consider the EL system (26) with NPRE (32) verifying Assumptions 2.1 and 5.2. Let the adaptive control be given by u = β(q, q, D I (η), t) (34) together with the parameter estimator (10), ( 14), ( 33 In what follows we present two well-known choices of β(q, q, θ, t) for fully actuated systems, i.e., m = n q , and prove that they satisfy the key GES Assumption 5.2

• The Computed Torque Controller in the known parameter case is given by β(q, q, θ, t) = M (q)[q -K 1 q -K 2 q] + C(q, q) q + g(q), resulting in the LTI closed-loop system q + K 1 q + K 2 q = 0, that, obviously, has a GES equilibrium at the origin for all positive definite control gains K 1 , K 2 ∈ R nq×nq .

• The Slotine-Li Controller in the known parameter case is given by [START_REF] Slotine | Adaptive manipulator control: a case study[END_REF] β(q, q, θ, t) = M (q)q r + C(q, q) qr + g(q) + K 1 s, (35) where we defined the signals qr := q -K 2 q, s := q + K 2 q.

(36) The closed-loop system is then M (q) ṡ + [C(q, q) + K 1 ]s = 0, q + K 2 q = s, that-as indicated in (Ortega et al., 1998, Remark 4.5)-has an GES equilibrium at the origin. Remark 5.1. To the best of our knowledge, the proof of global stability of the adaptive version of the computed torque scheme proposed above is the first one reported in the literature.

Verifying Assumption 2.1 on a 2-DOF robot manipulator.

The equations of motion of the robot are given by ( 26) with

M (q) = S 1 (θ) + 2S 2 (θ) cos(q 2 ) S 3 (θ) + S 2 (θ) cos(q 2 ) S 3 (θ) + S 2 (θ) cos(q 2 ) S 3 (θ)
and S(θ

) := S m (θ) S U (θ))
. In the following lemma we verify Assumption 2.1 for the mapping S(θ). Lemma 5.1. Consider the vector θ ∈ R 4 >0 and the mapping S : R 4 >0 → R 5 >0 given by ( 38). Assume the bounds

θ 1 ≤ θ M 1 , θ m 2 ≤ θ 2 ≤ θ M 2 , θ m 4 ≤ θ 4 . (39) The mapping D : R 4 >0 → R 4 >0 η = D(θ) = col(θ 1 , θ 2 , θ 2 θ 4 , θ 1 (θ 3 + θ 4 )), with right inverse D I : R 4 >0 → R 4 θ = D I (η) = col(η 1 , η 2 , η 4 η 1 - η 3 η 2 , η 3 η 2 ), (40)
verifies Assumption 2.1 with

T =     0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0     , P = diag{a, a, 1, 1} for any a ≥ 1 4θ m 4 θ M 2 + (θ M 1 ) 2 θ m 2 .
Adaptive Slotine-Li control of the 2-DOF robot manipulator.

In this subsubsection we present in detail the adaptive controller of Proposition 7 with the Slotine-Li scheme for the 2-DOF robot manipulator. Simulation results showing that the proposed scheme largely outperforms the classical one relying on overparameterization may be found in [START_REF] Ortega | Parameter Estimation of Nonlinearly Parameterized Regressions without Overparameterization nor Persistent Excitation: Application to System Identification and Adaptive Control[END_REF].

To derive the NPRE (32) we invoke ( 29) and ( 37) and define

m 1 := 1 0 0 0 , m 2 (q 2 ) := cos(q 2 ) 2 1 1 0 , m 3 := 0 1 1 1 U 1 (q) := g[1 + sin(q 1 + q 2 )], U 2 (q 1 ) := g[1 + sin(q 1 )].
Thus, the regressor matrix (32) takes the form

Ω(q, q) = H(p) Ω 11 Ω 12 Ω 13 Ω 14 Ω 15 Ω 21 Ω 22 Ω 23 Ω 24 Ω 25
where Ω 11 = p q1 , Ω 12 = p cos(q 2 )(2 q1 + q2 ), Ω 13 = p q2 , Ω 14 = Ω 24 = g cos(q 1 + q 2 ), Ω 15 = g cos(q 1 ), Ω 21 = Ω 25 = 0, Ω 22 = p cos(q 2 ) q1 + sin(q 2 )( q2 1 + q1 q2 ) and Ω 23 = p( q1 + q2 ). The known parameter version of the Slotine-Li controller (35) may be parameterized as β(q, q, θ, t) := W (q, q, t)S(θ) + K 1 s, with the matrix

W (q, q, t) := W 11 W 12 W 13 W 14 W 15 W 21 W 22 W 23 W 24 W 25 , with W 11 = qr1 , W 12 = cos(q 2 )(2q r1 +q r2 )-sin(q 2 )( q2 qr1 + ( q1 + q2 ) qr2 ), W 13 = qr2 , W 14 = W 24 = Ω 14 , W 15 = Ω 15 , W 21 = W 25 = Ω 21 , W 22 =
cos(q 2 )q r1 + sin(q 2 ) q1 qr1 and W 23 = qr1 + qr2 , where qr and s are defined in (36). In its standard version [START_REF] Slotine | Adaptive manipulator control: a case study[END_REF], to get a linear parametrization, the adaptive implementation is obtained estimating the vector S := S(η), yielding β(q, q, Ŝ, t) := W (q, q, qr , qr ) Ŝ + K 1 s. The parameter estimator is given as Ṡ := -ΓW (q, q, t)s, that, as shown in [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF], yields a globally stable closed-loop system and ensures global tracking of the desired references.

In the proposed approach we estimate directly θ, that is, the adaptive control is β(q, q, θ, t) := W (q, q, qr , qr )S( θ) + K 1 s, with the parameter estimator ( 10), ( 14), ( 33) and

(31), combined with θ = D I (η), where the mapping D I (•) is given in (40).

Figure 1 shows the results of the simulations of the DREMbased and the standard schemes, from which we can observe that the trajectory tracking and the parameter estimation capabilities of our proposal clearly outperforms those of the classical adaptive controller.

6. APPLICATION TO NONLINEARLY PARAMETERIZED DT SYSTEMS

Adaptive Pole Placement Control of LTI Systems

We are interested in this subsection in the problem of APPC of LTI DT system represented by it pulse transfer function A(q -1 )y p (k) = B(q -1 )u(k), (41) where the polynomials

A(q -1 ) = 1 + a 1 q -1 + • • • + a n A q -n A , B(q -1 ) = b 0 + b 1 q -1 + • • • + b n B q -n B ,
are coprime, with a known upperbound on their order, say v, but with unknown coefficients a i , b i . The pole-placement problem consists of designing a controller L(q -1 )u(k) + P (q -1 )y p (k) = r(k) (42) such that the closed-loop system takes the form y p (k) = B(q -1 ) Am(q -1 ) r(k), where r(k) is a bounded external signal and A m (q -1 ) = 1 + a m 1 q -1 + • • • + a m n Am q -n Am , is a desired closed-loop polynomial whose roots are inside the unit circle. That is, the controller relocates the poles of the system in a desired position but preserves the open-loop zeros. For a lucid exposition of this problem see (Goodwin and Sin, 1984, Section 5.3) and [START_REF] Pyrkin | A Globally convergent direct adaptive poleplacement controller for nonminimum phase systems with relaxed excitation assumptions[END_REF] for a review of the recent literature.

6.2 Obstacles for the adaptive implementation.

Computing (41) in closed-loop with (42) we get y p (k) = B(q -1 ) A(q -1 )L(q -1 ) + B(q -1 )P (q -1 ) r(k).

Hence, to achieve the objective, we need to verify the Bezout equation A(q -1 )L(q -1 ) + B(q -1 )P (q -1 ) = A m (q -1 ). ( 44) As is well-known (Goodwin and Sin, 1984, Theorem 5.3.1), selecting n Am := 2v -1, there exists unique polynomials L(q -1 ) and P (q -1 ), both of order (v-1), solutions of (44). Indeed, it is possible to show that (44) admits a matrix representation S(a i , b i )η = col(a m 0 , a m 1 , . . . , a m 2v-1 ), (45) where η := col(l 0 , l 1 , . . . , l v-1 , p 0 , p 1 , . . . , p v-1 ) (46) and S(a i , b i ) ∈ R 2v×2v -called the Sylvester matrix-is linearly dependent on the coefficients a i , b i , and is full rank if and only if A(q -1 ) and B(q -1 ) are coprime.

It is well-known that the adaptive version of the previous controller, called APPC, suffers from serious drawbacks [START_REF] Goodwin | Adaptive observers with exponential rate of convergence[END_REF][START_REF] Pyrkin | A Globally convergent direct adaptive poleplacement controller for nonminimum phase systems with relaxed excitation assumptions[END_REF]. In its indirect version-that is when we estimate the parameters of the plant a i , b i and then compute from them, via the solution of (45), the parameters of the controller l i , p i -the problem is that the Sylvester matrix with the estimated parameters âi (k), bi (k) may loose rank during the transient behavior. Although this phenomenon can be avoided adding parameter projections, the prior knowledge required to implement this efficiently is never available in practice and relies on the availability of PE, see (Pyrkin et al., 2019, Section 1).

On the other hand, in its direct version the estimation of the controller parameters involves a NPRE. Indeed, applying (44) to the output of the plant y p (k) we get L(q -1 )A(q -1 )yp(k) + P (q -1 )B(q -1 )yp(k) = Am(q -1 )yp(k) ⇔ B(q -1 )[L(q -1 )u(k) + P (q -1 )yp(k)] = Am(q -1 )yp(k). ( 47)

The known parameter version of the direct pole-placement controller may be written in the LRE form u(k) + η ψ(k) = r(k) where we have used the fact that L(q -1 ) is monic and defined ψ(k) := col(y p (k), . . . , y p (k -v + 1), u(k -1), . . . , u(k -v +1)) ∈ R 2v-1 , with η, as defined in (46), contains the unknown coefficients of the polynomials L(q -1 ) and P (q -1 ). A direct adaptive implementation of this controller takes then the form u(k) + η (k)ψ(k) = r(k), where η(k) denotes the estimates of η. The difficulty of designing an estimator for the controller parameters η is due to the fact that, in terms of η, (47) defines a parameterization of the form

B(q -1 )[u(k) + η ψ(k)] = A m (q -1 )y p (k) =: y(k), (48)
which is bilinear because the polynomial B(q -1 ) is unknown.

In the next two subsubsections we show that using the results reported in the paper it is possible to overcome the two obstacles mentioned above. To simplify the presentation we illustrate this fact with simple representative examples, that can be easily extended to the general case.

DREM-based indirect APPC.

Consider the LTI DT system y p (k + 1) + θy p (k) = u(k) + θ 3 u(k -1), (49) where, to ensure the coprimeness assumption, θ = ±1. Fixing a dead-beat objective, e.g., A m (q -1 ) = 1, and selecting L(q -1 ) = l 0 + l 1 q -1 and P (q -1 ) = p 0 + p 1 q -1 the Bezout equation ( 44) takes the form

(1 + θq -1 )(l 0 + l 1 q -1 ) + q -1 (1 + θ 3 q -1 )(p 0 + p 1 q -1 ) = 1. (50)
The latter can be rewritten as

   1 0 0 0 θ 1 1 0 0 θ θ 3 1 0 0 0 θ 3       l 0 l 1 p 0 p 1    =    1 0 0 0    , (51)
whose solution is l 0 = 1, p 1 = 0 and 1 1 θ θ 3

l 1 p 0 = -θ 0 , (52)
which corresponds to

l 1 p 0 = 1 θ 3 -θ -θ 4 θ 2 .
Hence, the known-parameter controller (42) takes the form

u(k) = - 1 θ 3 -θ θ 2 y p (k) -θ 4 u(k -1) + r(k) (53)
and yields the desired closed-loop system y p (k) = q -1 (1 + θ 3 q -1 )r(k).

Obviously, the system admits an NPRE of the form (1) with y p (k) := y p (k) -u(k -1), Ω(k) := -y p (k -1) u(k -2) (54)

and S(θ) := θ θ 3 . If we overparametrize the NPRE and estimate the vector S ∈ R 2 the controller parameters are computed from

1 1 Ŝ1 (k) Ŝ2 (k) l1 (k) p0 (k) = -Ŝ1 (k) 0 , (55)
which yields the adaptive controller

u(k) = - 1 Ŝ2 (k) -Ŝ1 (k) Ŝ2 1 (k)yp(k) -Ŝ1 (k) Ŝ2 (k)u(k -1) + r(k). (56)
Clearly, the controller computation has a singularity on the line Ŝ1 (k) = Ŝ2 (k). On the other hand, if we estimate θ, the adaptive version of (53) has a singularity only at the points θ(k) = ±1. Simulation results illustrating these facts may be found in [START_REF] Ortega | Parameter Estimation of Nonlinearly Parameterized Regressions without Overparameterization nor Persistent Excitation: Application to System Identification and Adaptive Control[END_REF].

CONCLUSIONS

It has been shown that the DREM procedure can be used to estimate the parameters of a CT or DT NPRE of the form (1), provided the "monotonizability" Assumption 2.1 holds and some weak excitation conditions-encrypted in the scalar signal ∆-are satisfied. The applicability of the method has been illustrated with several classical examples.

We are currently pursuing the following research avenues.

(R1) The highly attractive parameterization of EL systems proposed in [START_REF] Slotine | Composite adaptive control of robot manipulators[END_REF] seems to yield a nonidentifiable NPRE. A rigorous proof of this claim is yet to be established.

(R2) Although the DREM estimator has a few tuning gains, e.g., the filter constants (λ for CT, and α for DT) and the adaptation gain γ, their impact on the transient behavior is hard to predict-see Subsubsection 5.2.5 in [START_REF] Ortega | Parameter Estimation of Nonlinearly Parameterized Regressions without Overparameterization nor Persistent Excitation: Application to System Identification and Adaptive Control[END_REF]. A more thorough analysis of the sensitivity of the design vis-à-vis these coefficients is yet to be derived.

(R3) Although avoiding overparameterization to handle NLPRE seems, in principle, a sensible objective, it is not clear under which conditions this approach is really more convenient. Particularly considering that this is, until now, only applicable to "monotonizable" NLPRE. The results presented in the paper shows that there are practically important examples where this is the case.

(R4) The verification of the conditions of Proposition 1 is carried out in our examples via direct inspection. A deeper understanding of the underlying structural features of the mapping S(θ) under which this is possible would be highly desirable. It seems that such a study should appeal to principles of differential algebra.

  4.2 Discrete-time case Assumption 4.1. The mapping G(η) satisfies the Lipschitz condition |G(a) -G(b)| ≤ ν|a -b|, ∀a, b ∈ R q , (16) for some ν > 0. Proposition 4. Consider the DT NPRE (13) with G(η) satisfying (8) of Assumption 2.1 and Assumption 4.1. Propose the DT parameter estimator η

  ) and (31). If ∆ / ∈ L 2 we have that (28) holds with all signals bounded.

Fig. 1 .

 1 Fig. 1. Simulation results for the DREM-based adaptive scheme and the classical adaptive Slotine-Li.

ACKNOWLEDGMENT

This work is supported by the Russian Science Foundation grant (project N o 19-19-00403).