Sanjar John M Campbell
email: jmaxwellcampbell@gmail.com

Sanjar Abrarov

An Analogue of Ramanujan's Master Theorem

Keywords: Ramanujan's Master Theorem, Infinite series, Closed form, Pi formula, Harmonic number Mathematics Subject Classification (2010) 33C75, 33C20, 65B10

come

Introduction

Many of Ramanujan's well-known formulas are based upon the evaluation of moments for Maclaurin-type power series. In particular, Ramanujan's Master Theorem

∞ 0 x s-1 ∞ i=0 (-x) i i! f (i) dx = Γ (s)f (-s) (1)
was frequently used by Ramanujan to construct closed-form evaluations of definite integrals and infinite series [START_REF] Amdeberhan | Ramanujan's master theorem[END_REF][START_REF] Berndt | The quarterly reports of S. Ramanujan[END_REF]. In the case whereby the series in ( 1) is hypergeometric, we obtain an evaluation of the following form [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF].

∞ 0 x s-1 p F q c 1 , c 2 , . . . , c p d 1 , d 2 , . . . , d q -x dx = Γ (s) Γ (c 1 -s) • • • Γ (c p -s)Γ (d 1 ) • • • Γ (d q ) Γ (c 1 ) • • • Γ (c p )Γ (d 1 -s) • • • Γ (d q -s)
There are many analogues and variants of classical versions of Ramanujan's Master Theorem [START_REF] Bagis | A General Method for Constructing Ramanujan-Type Formulas for Powers of 1/π[END_REF][START_REF] Ding | Ramanujan's master theorem for symmetric cones[END_REF][START_REF] Ólafsson | Ramanujan's master theorem for the hypergeometric Fourier transform associated with root systems[END_REF][START_REF] Ólafsson | Ramanujan's master theorem for Riemannian symmetric spaces[END_REF], and there are a number of different formulations of this theorem which all provide evaluations for the moments of sufficiently well-behaved functions defined in terms of power series, as in the identity whereby

∞ 0 x -n-1 ∞ k=0 (-1) k a(k)x k dx = - π sin πn a(n).
Given the wide-ranging applications of Ramanujan's Master Theorem, we are inspired to construct new variants and analogues of this result. Instead of dealing specifically with moments of power series, we consider the problem of constructing Ramanujan-like identities for integrals over products consisting of a parametric power of an input function with another input mapping given as a factor in the integrand. Our present article is inspired in large part by recent research that has dealt with the use of symbolic evaluations for moment-like integrals to construct new closed-form evaluations for infinite series and definite integrals. New results on the construction of Ramanujan-like series for 1 π are given in [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF] that are based on the evaluation of the moments of the elementary function ln(1-x 2 ) √ 1-x 2 on the domain [0, 1), and the integration technique from [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF] is heavily utilized in [START_REF] Campbell | New series involving harmonic numbers and squared central binomial coefficients[END_REF][START_REF] Campbell | Series containing squared central binomial coefficients and alternating harmonic numbers[END_REF]. A somewhat similar approach is given in [START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF], in which the moments of the elementary expression arcsin(x) ln(x) on (0, 1] are used to prove new rational approximations for constants involving 1 π , as in the intriguing equality

4G -12 ln 2 + 6 π = ∞ n=0 2n n 2 H 2n 16 n (2n -1) 2 ,
letting G denote Catalan's constant. In [START_REF] Sofo | Evaluation of integrals with hypergeometric and logarithmic functions[END_REF], the evaluation of moments of the form

1 0 x 2k-1 • ln x • 3 F 2 1 2 , 2, 2 p+3 2 , p+4 2
x 2k dx is explored. Integrals of the form b a

x n ln m (sin(x)) dx are applied in [START_REF] Orr | Generalized Log-sine integrals and Bell polynomials[END_REF] to establish some new results on the log-sine integral function, and a variety of moment formulas are used in [START_REF] Orr | Generalized rational zeta series for ζ(2n) and ζ(2n + 1)[END_REF] to construct new evaluations for series involving the Riemann zeta function. For example, symbolic formulas for moments of the following forms are applied in [START_REF] Orr | Generalized rational zeta series for ζ(2n) and ζ(2n + 1)[END_REF], motivating a full exploration on the application of the class of moments in Lemma 1 below in the construction of new infinite series evaluations.

πz 0 x p cot(x) dx 2πz 0 x p • Cl m (x) dx z 0 x p ψ(x) dx
A formula for integrals involving x ln 2m-1 x 1-x is used in [START_REF] Sofo | Integrals of logarithmic and hypergeometric functions[END_REF] in the symbolic computation of some harmonic summations, and a variety of new integral evaluations and series evaluations are proved in [START_REF] Campbell | An Algorithm for Trigonometric-Logarithmic Definite Integrals[END_REF] by substituting logarithmic functions into Maclaurin series expansions and integrating term-by-term; this illustrates the utility of exploring variants of (1) that make use of non-standard moments such as logarithmic moments, as elaborated below.

The crux of the technique from [START_REF] Campbell | An Algorithm for Trigonometric-Logarithmic Definite Integrals[END_REF] is based on the observation that logarithmic moments for rational expressions of the form 1 1±x n often may be expressed in a simple way with a zeta-type function and a gamma-type expression as a factor, so that if logarithmic expressions are substituted into the Maclaurin series for trigonometric or hyperbolic functions, if we integrate term-by-term, we often obtain a series involving the Riemann zeta function that Mathematica can evaluate directly. Through the use of non-standard moments such as logarithmic moments, the "signature method" given by Lemma 1 below, which is something of an analogue of Ramanujan's Master Theorem, often leads us to one-line proofs of new integral evaluations, such as

1 0 (x -1) 2 x(x + 1) ln 2 (x) dx = 12 ln(A) -1 - 4 ln(2) 3 , (2)
letting A denote the Glaisher-Kinkelin constant. We present interesting applications of non-standard moments in the upcoming section through the evaluation of difficult series containing harmonic-type numbers, using Lemma 1, and by using non-standard moments to evaluate new classes of 3 F 2 (1) series. Letting e 1 (x) and e 2 (x) be functions on a given domain (a, b) whereby the parameters for this interval are not necessarily finite, the crux of our strategy for generating and classifying Ramanujan-like series and integral formulas involves the evaluation of definite integrals of the form b a (e 1 (x)) n e 2 (x) dx for n ∈ N 0 . For the purposes of this article, our strategy may be summarized with the following observation.

Lemma 1 Letting (e 1 (x)) n e 2 (x) be integrable on (a, b) for n ∈ N 0 , letting

I b a (e 1 , e 2 )(n) = I(n) = b a (e 1 (x)
) n e 2 (x) dx for n ∈ N 0 , and writing

e 3 (x) = ∞ n=-∞ x n f (n), (3)
then the equality

b a e 3 (e 1 (x))e 2 (x) dx = ∞ n=-∞ I(n)f (n) (4)
holds, provided that the series in (3) and both sides of (4) are well-defined.

Proof This follows immediately by inputting e 1 (x) into both sides of (3), multiplying both sides of resultant equation by e 2 (x), integrating both sides of the resulting equality over (a, b), and then reversing the order of summation and integration.

Often, if a state-of-the-art computer algebra system (CAS) can evaluate one side of (4), without being able to provide a symbolic evaluation for the other side of the equation, then we obtain a new integral or series formula. The expression (e 1 , e 2 , e 3 ) a,b,f (or more compactly, (e 1 , e 2 , e 3 ), if the choice of a, b, and f is somehow tacit or obvious) is thus something of a "proof signature" for the evaluation in (4). This approach is reminiscent of, and is partly inspired by, the one-line proofs afforded by the famous Wilf-Zeilberger method from [22]. To illustrate the utility of Lemma 1, we offer an informal one-line proof of the below formulation of the famous Basel problem.

Basel problem: Prove that ζ(2) = π 2 6 .

Proof: sin(x), 1, sin -1 (x) 2 .

Explicitly, we have that

I(2n) = π 0 (sin(x)) 2n dx = √ πΓ (n+ 1 2 ) Γ (n+1) and sin -1 (x) 2 = 1 2 ∞ n=1 (2x) 2n n 2 2n n , so that π 3 12 = 1 2 ∞ n=1 4 n √ πΓ ( n+ 1 2 ) Γ (n+1) n 2 ( 2n n ) . Equivalently, ζ(2) = π 2 6
. This illustrates that the integration method given by Lemma 1 provides us with an efficient and novel way of significantly simplifying proofs of known closed-form formulas.

As noted above, our present article is inspired in part by the integration methods given in [START_REF] Campbell | An Algorithm for Trigonometric-Logarithmic Definite Integrals[END_REF][START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF][START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF]. For a sequence (s n ) n∈N , the fundamental lemma from [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF] deals with the integration of infinite series of the form

∞ n=0 (-1) n 1 2 n s n • x 2n ln 1 -x 2 √ 1 -x 2 (5)
to produce new Ramanujan-like series for 1 π involving harmonic numbers. For example, we find that the new evaluation

∞ n=1 2n n 2 H n 16 n (2n -1) = 8 ln(2) -4 π (6)
provided in [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF] follows immediately from the main lemma from [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF] simply by letting s n = 1 in (5), and applying a suitable integration operator to [START_REF] Borwein | On an intriguing integral and some series related to ζ(4)[END_REF]. So, in other words, the elegant evaluation provided in ( 6) is equivalent to the proof signature

x, ln 1 -x 2 √ 1 -x 2 , 1 -x 2 . ( 7
)
This illustrates how Lemma 1 may be regarded as a vast generalization of the integration method from [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF], and this leads us to the following question.

Question 1 How can we obtain new classes of Ramanujan-like formulas, with the use of variants of the proof signature in (7)?

The above proof of Lemma 1 is reminiscent of some of Ramanujan's work concerning the application of his Master Theorem. For example, as noted in [START_REF] Berndt | The quarterly reports of S. Ramanujan[END_REF], letting φ denote a reasonably well-behaved function, the following corollary of Ramanujan's Master Theorem may be proved by writing cos(nx) as a Maclaurin series, switching the order of summation and integration, and then applying the Master Theorem, term-by-term.

Corollary 1 The identity ∞ 0 ∞ k=0 φ(k)(-x) k k! cos(nx) dx = ∞ k=0 φ(-2k -1)(-n 2 ) k holds [3].
Given the interesting Ramanujan-like series results from [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF], and given how minor alterations in the corresponding proof signatures for these results provide us with new classes of series that cannot be computed symbolically with current CAS software, this strongly motivates us to continue to explore the application of our Ramanujan-like integration method from Lemma 1.

Lemma 1 may be regarded as something of a heuristic tool that motivates the development of new results on closed-form evaluations based on the use of creative combinations of functions in the integrand in [START_REF] Berndt | Ramanujan's notebooks. Part IV[END_REF]. We illustrate this idea with a variety of different examples, starting with a variant of the integration methods from [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF][START_REF] Campbell | Series containing squared central binomial coefficients and alternating harmonic numbers[END_REF][START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF] for constructing binomial-harmonic series for 1 π .

New harmonic summations

The fundamental lemma from Section 1 is very useful in the exploration of variants of the integration technique from [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF]. Through a clever application of the symbolic form for the moments of the mapping α

: [0, 1) → R whereby x → ln(1-x 2 ) √ 1-x 2 ,

a very general technique for constructing binomial-harmonic summations for 1

π is put forth in [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF]. It seems natural to consider the symbolic evaluation of moments on variations of the aforementioned transformation α. Since Mathematica is able to directly evaluate the series

∞ n=1 2n n 4n 2n H n 2 6n (n + 1) = 8 3 2 - √ 2 ln(64) π , (8)
in consideration of the recent research from [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF][START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF] concerning the evaluation of series involving binomial products of the form 2n n 4n 2n , we are inspired to try to generalize [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF]. From the results in [START_REF] Campbell | New series involving harmonic numbers and squared central binomial coefficients[END_REF][START_REF] Campbell | Series containing squared central binomial coefficients and alternating harmonic numbers[END_REF][START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF], it would seem that as a natural place to start, consideration of the problem of evaluating series of the form

∞ n=1 2n n 4n 2n H n 2 6n (n + z)
for z ∈ Z >0 would be suitable. However, as elaborated below, the problem of evaluating

∞ n=1 2n n 4n 2n H n 2 6n (n + 2) ( 9
)
turns out to be surprisingly difficult, compared with how Mathematica easily provides a closed-form formula for the series in [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF]. If we apply a re-indexing argument to express the summation in [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF] in terms of (9), we obtain the recalcitrant p F q series

4 F 3 1 4 , 3 4 , 1, 1 2, 2, 3 1
as well as the difficult harmonic series

∞ n=0 1 64 n 2n n 4n 2n H n (n + 1) 2 .
Through a direct application of the signature method given by the above lemma, we obtain the following closed-form evaluation for the seemingly recalcitrant series in [START_REF] Campbell | Series containing squared central binomial coefficients and alternating harmonic numbers[END_REF].

Theorem 1 The equality

∞ n=1 1 64 n 2n n 4n 2n H n n + 2 = 256 105 + 128 √ 2 105π - 304 √ 2 ln(2) 35π holds. Proof x 2 , ln(1-x 2 ) √ x 4 √ 1-x 2 , 4 √ 1-x(x+4)-4 x 2
.

Direct applications of Lemma 1 also provide us with the following new results, through the use of moments of the form 1 0

x m ln(1-x 2 ) 4 √ 1-x 2 dx. ∞ n=1 1 64 n 2n n 4n 2n H n n + 3 = 16384 10395 + 4352 √ 2 3465π - 21136 √ 2 ln(2) 3465π ∞ n=1 1 64 n 2n n 4n 2n H n n + 4 = 262144 225225 + 795776 √ 2 675675π - 356656 √ 2 ln(2) 75075π ∞ n=1 1 64 n 2n n 4n 2n H n n + 5 = 67108864 72747675 + 21632512 √ 2 19840275π - 94907536 √ 2

ln(2) 24249225π

There are many natural ways of going about generalizing the series evaluations listed above, as suggested in the symbolic form for the below series, which Mathematica 11 cannot evaluate.

Theorem 2 The evaluation

∞ n=0 1 27 n 2n n 3n n H n 3n -4 = 252 √ 3 ln(3) -279 √ 3 128π holds. Proof x 2 , ln(1-x 2 ) 3 √ x 3 √ 1-x 2 , 1 4 3 √ 1 -x(3x + 1) .
Using similar signature tuples, we obtain the following new results.

∞ n=0 1 27 n 2n n 3n n H n 3n -5 = 360 √ 3 ln(3) -207 √ 3 500π ∞ n=0 1 27 n 2n n 3n n H n 3n -7 = 28476 √ 3 ln(3) -33363 √ 3 21952π ∞ n=0 1 27 n 2n n 3n n H n 3n -8 = 69480 √ 3 ln(3) -43461 √ 3 128000π
Remark 1 The above results can be generalized using the main lemma from our article together with the identity whereby

1 0 ln(1-x k )x α (1-x k ) 1/m dx evaluates as Γ m-1 m Γ α+1 k H -1 m -H -k+m+mα km kΓ m-1 m + α+1 k
for parameters k, m, and α satisfying the following conditions:

1 m < 1, (α + k) > -1
, and (k) > 0. For example, if we note that the expression

ln 1 -x k • ∞ i=0 1 m i (-x k ) i (1 -x k ) 1/m
reduces to ln 1 -x k , through the signature method we find that

∞ i=0 (-1) i Γ i + 1 k H i+ 1 k -1 m Γ (1 + i)Γ 1 + i + 1 k -1 m Γ 1 -i + 1 m may be evaluated as km sin π m H 1 k + H -1 m π .
Series involving products of harmonic-type numbers are among the central objects of study in our article, since Lemma 1 seems to naturally given rise to simple ways of evaluating seemingly recalcitrant series of this form, especially when used in conjunction with known generating functions for sequences involving harmonic-like numbers. The study of series involving products of harmonic numbers is a very interesting area [START_REF] Furdui | Series involving products of two harmonic numbers[END_REF]. The unexpected [START_REF] Furdui | Series involving products of two harmonic numbers[END_REF] nonlinear harmonic series evaluation n∈N

H n n 2 = 17π 4 360 ( 10
)
had been discovered experimentally by Enrico Au-Yeung, and was later proved in [START_REF] Borwein | On an intriguing integral and some series related to ζ(4)[END_REF]. The identity in [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF] is generalized in [START_REF] Furdui | Series involving products of two harmonic numbers[END_REF], in which an evaluation for summations of the form

∞ n=1 H n n • H n+k n + k
is proved. It appears that there have not previously been any known summations for 1 π containing products of harmonic-type numbers. This motivates us to apply Lemma 1, together with integration techniques as in [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF][START_REF] Campbell | Series containing squared central binomial coefficients and alternating harmonic numbers[END_REF][START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF], to construct new series of this form. First, our strategy is to exploit the closed-form evaluation for the ordinary generating function for ( 2n n H n : n ∈ N) given in [START_REF] Chen | Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers[END_REF], as suggested in the proof signature for the new evaluation given below. Our use of the moments of ln(x) √ 1 -x 2 with regard to the following proof signature is directly inspired by the integral operator T ln,arcsin from [START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF].

Theorem 3 The evaluation ∞ i=0 2i i 2 H i H 2i 16 i (i + 1) = 16G -16 ln 2 (2) -16 ln(2) π + 4 + π -8 ln(2) holds. Proof x 2 , ln(x) √ 1 -x 2 , ln √ 1-x+1 2 √ 1-x √ 1-x .
Remark 2 Through the use of the above proof signature, one would need to also evaluate the series

∞ n=0 2n n 2 H n 16 n (n + 1) 2 ,
which was recently evaluated in closed form in [START_REF] Campbell | New series involving harmonic numbers and squared central binomial coefficients[END_REF]. Also, we need the evaluation of the series

∞ n=0 2n n 2 H 2n 16 n (n + 1) 2 ,
given in [START_REF] Campbell | Series containing squared central binomial coefficients and alternating harmonic numbers[END_REF] in the application of the following proof signature to prove the below theorem.

Theorem 4 The evaluation

∞ n=1 2n n 2 H 2n H 2n 16 n (n + 1) = π -4 ln(2) + 4 + 8G -8 ln(2) -12 ln 2 (2) π holds. Proof x 2 , √ 1 -x 2 ln(x), ln √ 1-x+1 2(1-x) √ 1-x .
The following result is especially useful.

Theorem 5 The equality

∞ n=1 2n n 2 H 2 n 16 n (n + 1) = 64 ln 2 (2) -32G π - 10π 3 + 16 ln(2) (11) holds.
Proof holds.

x 2 , √ 1 -x 2 ln 1 -x 2 , ln 1 2 1 √ 1-x +1 √ 1-x . Corollary 2
Proof This follows immediately from the preceding corollary together with Theorem 4.

Applying the following proof signature using the main lemma from our present article is quite involed, and requires a number of 1 π series from [START_REF] Campbell | New series involving harmonic numbers and squared central binomial coefficients[END_REF] and [START_REF] Campbell | Ramanujan-like series for 1 π involving harmonic numbers[END_REF], but we omit details for the sake of brevity.

Theorem 6 The series

∞ n=1 2n n 2 H 2 16 n (n + 2)
is equal to

28 9 + 64 ln(2) 9 - 50π 27 + 16 -128G -128 ln(2) + 320 ln 2 (2) 9π . Proof x 2 , x 2 √ 1 -x 2 ln 1 -x 2 , ln √ 1-x+1 2 √ 1-x √ 1-x .
We may evaluate series of the form

∞ n=1 2n n 2 H 2 n 16 n (n + z)
for z ∈ Z >0 in a recursive fashion, and we may similarly generalize the preceding theorems.

As discussed above, it is interesting how minor modifications to the tuple in [START_REF] Campbell | New series involving harmonic numbers and squared central binomial coefficients[END_REF] can lead to very different evaluations, compared to the rational approximation for 8 ln(2)-4 π afforded to us by [START_REF] Campbell | An Algorithm for Trigonometric-Logarithmic Definite Integrals[END_REF]. For example, using the signature

x, ln (1 -x) √ 1 -x , 1 -x 2 , (12)
we find that the sum

∞ i=0 4 i 2i i H 2i+ 1 2 (2i -1)(4i + 1) 4i 2i is equal to 2 9 7 -8 √ 2 + 3 5 √ 2 -1 ln(2) + 6 √ 2 ln √ 2 -1 .
This formula is noticeably different compared to the formulas provided by the signature in [START_REF] Campbell | New series involving harmonic numbers and squared central binomial coefficients[END_REF]. As suggested in the below theorem, further manipulations of the final entry in the tuple in [START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF] lead us to interesting new results.

Theorem 7 The series

∞ i=0 2i i 4i 2i 4 i H 2i+ 1 2 (2i -1) 2 is equal to - 2 9 83 -88 √ 2 + 9 10 √ 2 -3 ln(2) + 36 √ 2 ln √ 2 -1 . Proof x, ln(1-x) √ 1-x , - √ 1-x 2 +3x sin -1 (x) √ 1-x .
The above theorem is interesting because modern CAS software cannot compute this sum, and it seems that integral formulas for generalized harmonic numbers also cannot be applied to evaluate the series in Theorem 7. Moreover, it appears that there is not very much mathematical literature on series involving binomial quotients of the following form.

2i i 4i 2i , i ∈ N 0 ( 13
)
Theorem 7 is vastly generalized through the following new integration result.

Lemma 2 For a sequence (f n : n ∈ N), the series

∞ n=1 2n n 4n 2n • (-4) n H 2n+ 1 2 4n + 1 • f n is equal to -2 1 0 ∞ n=0 -1 2 n f n x 2 -1 2n ln(x) dx- √ π ln(2) ∞ n=1 2 2n f n Γ n + 1 2 Γ 1 2 -n Γ 2n + 3 2 -2f 0 ,
provided that the above expressions are all well-defined.

Proof This follows immediately from the symbolic evaluation of the antiderivative of (x 2 -1) 2n ln(x), for a parameter n.

For example, if we simply let

f n = (-1) n 4n + 1 (2n -3) 2
for elements n in N 0 , then we find that the infinite series This evaluation is interesting because if a re-indexing argument is applied to obtain the series in Theorem 7, this would require the evaluation of the following very difficult p F q series derived from hypergeometric sums involving binomial quotients as in [START_REF] Chen | Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers[END_REF]. Mathematica 11 is not able to evaluate these difficult expressions, emphasizing our previous point on the lack of mathematical resources on hypergeometric series involving quotients of the form in (13). The above evaluation for the series in ( 14) is also equivalent to the following proof signature:

∞ n=1 2n n 4n 2n 4 n H 2n+ 1 2 (2n -3) 2 (
x 2 -1, ln(x), 42x 3 sin -1 (x) + 23 1 -x 2 x 2 + 1 -x 2 ,
illustrating the versatility of the signature method. Through direct applications of Lemma 2, we obtain the following new results, as well as many extensions of the evaluation from Lemma 1 given by the tuple provided in [START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF].

∞ n=1 2n n 4n 2n 4 n H 2n+ 1 2 (2n -5) 2 = 312851954224 √ 2 -279413817016 11345882625 + 2042776 ln(2) -5609648 √ 2 ln(2) -2804824 √ 2 ln 2 - √ 2 297675 ∞ n=1 2n n 4n 2n 4 n H 2n+ 1 2 (2n -7) 2 = 892093651361104 √ 2 -786424570733896 30466216155375 + 4655127896 ln(2) -12916662064 √ 2 ln(2) -6458331032 √ 2 ln 2 - √ 2 676350675
A similar approach leads us to the following evaluation.

Theorem 8 The evaluation

∞ i=0 4i 2i 2i i • H i+ 1 2 4 i (2i + 1) 2 = 16 -8 √ 2 + 4 3 √ 2 -2 ln(2) -8 √ 2 ln 1 + √ 2 holds. Proof x, ln(1-x 2 ) √ 1-x 2 , √ x + 1 .
It should be noted that deriving the evaluation presented in Theorem 8 using the above proof signature, together with the master lemma from Section 1, is nontrivial, although the proof signature

x, ln 1 -x 2 √ 1 -x 2 , √ x + 1 ( 15
)
provides us with a very succinct summary of how to go about this derivation. The complexity of the following rigorous proof of Theorem 8 compared with the informal proof signature displayed in ( 15) is illustrative of the power and versatility of the integration method put forth in our article.

Detailed proof of Theorem 8: Through a direct application of Lemma 1 with regard to the proof signature from [START_REF] Furdui | Series involving products of two harmonic numbers[END_REF], we find that

π ln(2) - 16 15 3 F 2 1, 1, 3 2 7 4 , 9 4 1 ( 16
)
must be equal to

∞ n=1 1 2 n -2 -1-n nπΓ (n) H n 2 + ln(4) Γ 1 + n 2 2 . ( 17
)
Mathematica is not able to evaluate the 3 F 2 series displayed in ( 16), even through the use of the FunctionExpand command. The evaluation of hypergeometric series with quarter-integer parameters was recently explored in [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF][START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF], and it turns out that a clever variant of some of the proof techniques from [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF][START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF] can be applied to obtain a simple closed-form evaluation for [START_REF] Janous | Around Apéry's constant[END_REF]. One of the key observations from [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF] that is used in the closed-form evaluation of a 3 F 2 series with quarter-integer parameters is that Pochhammer products of the form 2n + 1 4 i 2n + 3 4

i can be expressed in a very natural way in terms of central binomial coefficients of the form 4i 2i , so that integral formulas for binomial expressions of this form can be substituted into the appropriate hypergeometric summand in order to express the corresponding series as a definite integral. With regard to the lower parameters of the 3 F 2 (1) series from ( 16), we have that

7 4 i 9 4 i = 1 15 64 -i (4i + 1)(4i + 3)(4i + 5)(2i)! 4i 2i , which shows us that 3 F 2 1, 1, 3 2 7 4 , 9 4 1 = ∞ i=0 15 16 i (2i + 1) (4i + 1)(4i + 3)(4i + 5) 4i 2i . (18)
Through the use of the beta function, we find that the equality

1 4i 2i = 1 0 (4i + 1)(1 -t) 2i t 2i dt (19)
holds for i ∈ N 0 . Replacing the factor 1 ( 4i 2i )

in the summand in [START_REF] Ólafsson | Ramanujan's master theorem for the hypergeometric Fourier transform associated with root systems[END_REF] with the definite integral in [START_REF] Ólafsson | Ramanujan's master theorem for Riemannian symmetric spaces[END_REF], and reversing the order of summation and integration, we find that the summations in [START_REF] Ólafsson | Ramanujan's master theorem for the hypergeometric Fourier transform associated with root systems[END_REF] are equal to the following seemingly recalcitrant integral.

1 0 1 256 ((-1 + t) 2 t 2 ) 5/4 15 -12 4 (-1 + t) 2 t 2 + 3 + 4 (-1 + t) 2 t 2 tan -1 2 4 (-1 + t) 2 t 2 + 3 -4 (-1 + t) 2 t 2 tanh -1 2 4 (-1 + t) 2 t 2 dt
Amazingly, Mathematica is able to provide a simple closed-form evaluation for the above integral, yielding the following elegant evaluation.

3 F 2 1, 1, 3 2 7 4 , 9 4 1 = - 15 8 √ 2 tanh -1 2 √ 2 3 -4 .
Using the above evaluation, and by bisecting the series in [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF], we obtain the desired result.

Using the signature method, we can easily derive many natural variants of the above theorem that cannot be evaluated otherwise, as suggested below.

Theorem 9 The evaluation

∞ n=0 4n 2n 2n n • H 2n 4 n (2n + 1)(4n -1) = √ 2 ln(2) + 4 ln(2) + 4 ln √ 2 -1 3 holds. Proof x 2 -1, ln 1 -x 2 , 4 √ x + 1 cos 1 2 tan -1 ( √ x) .
Through a simple application of Lemma 1, we obtain a very general method of evaluating difficult series involving alternating harmonic numbers and inverses of central binomial coefficients, as suggested by the following theorems.

Theorem 10 Letting φ denote the Golden Ratio, we have that the equation

∞ n=0 (-1) n+1 H 2n (2n + 1) 2n n = 1 6 - 1 5 √ 5 π 2 -ln(φ) ln 2 + √ 5 holds. Proof (1 -x)x, ln(1 -x), 1 x+1 . Theorem 11 The series ∞ n=0 (-1) n H 2n (n + 1) 2n n is equal to 4ζ(3) 5 + ln √ 5 + 2 -8 ln 2 (φ) -3 ln 3 (φ) 2 + 2 75 π 2 3 √ 5 + 5 ln √ 5 -2 . Proof (1 -x)x, ln(1 -x), ln(x+1) x -2 x+1 .
It is remarkable how such intricate symbolic forms come out of such simple proof signatures, and are numerically equivalent to such simple and elegant harmonic series. The study of the evaluation of series containing harmonic-type numbers and inverses of binomial coefficients of the form 2n

n is an interesting subject in its own right.

The new evaluation provided below is inspired by the nonlinear harmonic sums presented in [START_REF] Borwein | On an intriguing integral and some series related to ζ(4)[END_REF][START_REF] Furdui | Series involving products of two harmonic numbers[END_REF].

Theorem 12 The equality ∞ i=1 H i H i+ 1 2 (2i + 1)(2i + 3) = 7ζ(3) 4 + π 2 12 + 2 ln 2 (2) -2 ln(2) - 1 4 π 2 ln(2) (20) holds
. Proof x, √ 1 -x ln(1 -x), ln √ 1-x+1 2 √ 1-x √ 1-x .
Without having already known that the above proof signature may be used to prove the above theorem, it would not be obvious at all as to how to evaluate the series in this theorem. We find that partial fraction decomposition cannot be applied in any kind of trivial way to evaluate this series, since if we rewrite the summation in [START_REF] Orr | Generalized Log-sine integrals and Bell polynomials[END_REF] as

∞ i=1 H i H i+ 1 2 1 2(2i + 1) - 1 2(2i + 3) ,
we cannot rewrite the above expression as two separate series based on the above decomposition, since neither of these series is convergent. Similarly, we find that current CAS software cannot evaluate the generating function obtained by replacing one of the four main factors in the summand in [START_REF] Orr | Generalized Log-sine integrals and Bell polynomials[END_REF] with x i . For example, it is not at all clear as to how to evaluate ∞ i=1 H i x i (2i + 1)(2i + 3) so as to be able to make use of a suitable integral formula for the sequence (H i+ 1 2 : i ∈ N 0 ). Using variants of the above proof signature, we obtain many similar results. For example, using Chen's generating function G(H n C n ; x) for (H n C n : n ∈ N 0 ) presented in [START_REF] Chen | Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers[END_REF], we find that the following theorem holds.

Theorem 13

The series

∞ n=1 H n H n+ 1 2 (2n + 1)(2n + 3)(2n + 5) is equal to 7ζ(3) 16 - 1 3 + π 2 72 + ln 2 (2) 3 - ln(2) 36 - 1 16 π 2 ln(2). Proof x, √ 1 -x ln(1 -x), G(H n C n ; x) .
To prove an analogue of the above theorem for products of the form H 2i H i+ 1 2 , we at first need a few preliminary results. Lemma 3 The evaluation

∞ n=0 H 2n (2n -3)(2n -1) 2 = 1 96 -63ζ(3) -20 -6π 2 + 56 ln(2) (21)
holds.

Proof x 2 -1, ln(x), 1 Remark 3 The above evaluation is interesting in its own right, and in a forthcoming article, we explore in full generality the evaluation of series with summands given by products of even-indexed harmonic numbers and rational expressions. A closed-form evaluation for the equivalent series n∈N0 H 2n (2n -1)(2n + 1) 2 is given in [START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF], through a simple application of the integral operator T ln,arcsin from [START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF]. Simple variants of the above proof signature may be used to obtain the following results.

∞ n=0 H 2n (2n -5)(2n -1) 2 = 1 960 -315ζ(3) -86 -30π 2 + 272 ln(2) ∞ n=0 H 2n (2n -7)(2n -1) 2 = - 7ζ(3) 32 - π 2 48 + 2103 ln(2) -601 11340 ∞ n=0 H 2n (2n -9)(2n -1) 2 = - 21ζ(3) 128 - 311 8640 - π 2 64 + 173 ln(2) 1260 ∞ n=0 H 2n (2n -11)(2n -1) 2 = - 21ζ(3) 160 - 39277 1485000 - π 2 80 + 37693 ln(2) 346500
As an alternative way of proving the above lemma, we may apply partial fraction decomposition to the rational component of the summand in [START_REF] Orr | Generalized rational zeta series for ζ(2n) and ζ(2n + 1)[END_REF], obtaining the expresion

1 2 ∞ n=0 H 2n (2n -3)(2n -1) - 1 2 ∞ n=0 H 2n (2n -1) 2 . (22)
A proof signature providing a closed-form evaluation for the former sum is

x 2 , ln(1 -x) x , 1 4 3x 3/2 tanh -1 √ x -3x - √ x tanh -1 √ x ,
whereas an equivalent version of the latter sum from ( 22) was introduced in [START_REF] Janous | Around Apéry's constant[END_REF], and a highly simplified proof of this evaluation was later given in [START_REF] Campbell | An integral transform related to series involving alternating harmonic numbers[END_REF].

Lemma 4 The equality

∞ n=0 H 2n (2n + 1)(2n + 3) 2 = 1 32 -7ζ(3) + 56 -3π 2 -24 holds.
Proof This follows from Lemma 3, through the use of a re-indexing argument.

The series Lemma 4 arises through an application of the proof signature given below. With regard to this signature, we are implicitly making use of the generating function for (H 2n 2n n : n ∈ N 0 ) from [START_REF] Chen | Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers[END_REF].

( √ 1-x+1))-ln(1-x) √ 1-x .

New results on hypergeometric series

Many new results concerning the evaluation of hypergeometric series involving quarter-integer parameters were recently introduced in [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF][START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF]. The techniques in [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF][START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF] for evaluating series of this form mainly rely upon the use of Wallistype integrals for central binomial coefficients, as well as the use of Fourier-Legendre series. Our Ramanujan-like integration technique from Lemma 1 is very powerful, so it seems natural to try to make use of this lemma to derive new results inspired by [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF][START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF]. Many of the main results from [START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF] are given by closed-form evaluations for ratios of 3 F 2 series, as in the interesting identity whereby

π 4 = 3 F 2 -η, 1 2 , 1 3 2 , 2 + η -1 3 F 2 1 2 , 1 2 , 1 + η 1, 2 + η 1
for η > -1 proved in [START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF]. The following new identity, which is of interest in its own right, does not from the 3 F 2 quotients from [START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF], and may be used to derive very interesting corollaries through the use of a lemma from [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF]. As discussed in [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF], the symbolic computation of 3 F 2 (1) series is an interesting subject with many applications, and there are no known general formulas for summations of this form. Since 3 F 2 (1) series have important applications in combinatorics, mathematical physics, and countless other areas [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF][START_REF] Wimp | Irreducible recurrences and representation theorems for 3 F 2 (1)[END_REF], this strongly motivates the exploration of the master lemma introduced in this article in the evaluation of 3 F 2 (1) series that would otherwise seem to have no possible closed-form expression.

Theorem 15 The identity

4Γ 5 2 -a Γ (a) π 3/2 = 3 F 2 3 2 , 1 2 -a, 3 2 -a 1, 5 2 -a 1 3 F 2 -1 2 , 1 -a, a 1 2 , 3 2 1 (23)
holds for a > 0.

Proof sin 2 (x), cos(x), 3 F 2 -1 2 , 1 -a, a x .

In [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF], it is noted that Pochhamer products given by consecutive quaterinteger parameters, i.e., products of the form for n ∈ Z and any i ∈ N 0 . As discussed in [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF], this often allows us to make use of Wallis' integral formula for central binomial coefficients to explicitly evaluate p F q series with consecutive quarter-integer parameters as in [START_REF] Sofo | Integrals of logarithmic and hypergeometric functions[END_REF]. However, it appears that the problem of evaluating hypergeometric series containing parameters in (Z/4) \ (Z/2) that do not form Pochhammer products as in [START_REF] Sofo | Integrals of logarithmic and hypergeometric functions[END_REF] can be much more difficult, thus motivating the interesting corollary provided below.

Theorem 17 The identity holds.

Proof Let a = b = 1 4 with regard to Theorem 17.

Conclusion

Our paper introduces an integration method that is very general but also very powerful, and that seems to "unite" many different kinds of proofs for series/integral evaluations. In a sense, Ramanujan's Master Theorem can be thought of as a special case of Lemma 1. Through the use of the master lemma introduced in this article, simple combinations of input functions lead us to unexpected results as in the evaluations presented in Section 2. It seems likely that this lemma will continue to lead to surprising new results in classical analysis-related fields. The main goal of our article is to show how new infinite series that are seemingly recalcitrant can be easily proved using Lemma 1, and to illustrate how this lemma provides us with a convenient and systematic way of generating and classifying evaluations for series that can be proved using moment identities as applied term-by-term with respect to a given generating function. The results given in Section 2 are new and highly nontrivial, in that the new series in this section cannot be proved using any "obvious" or standard summation techniques. The one-line proofs in this article are instructive in that these proofs show us how 3-tuples of elementary functions can "generate" new Ramanujan-like formulas, and how experimentation with the entries in these tuples can lead to great generalizations and variations of these formulas.

  64 i (2n -1)!! |n| j=1 (4i + sgn(n)(2j -1)) sgn(n)

1 -

 1 3)(2a -1)(2a + 1) holds for a > 0 and b > 0.Proof sin 2 (x), cos(x), 3 F 2

  The equality

	Proof This follows immediately from Theorem 5 and Theorem 3.
	Corollary 3 The equality
	∞ n=1	2n n 16 n (n + 1) 2 H 2 2n	=	4 -8G + 36 ln 2 (2) -24 ln(2) π	+ 4 -	4π 3	+ 4 ln(2)
	∞ n=1	2n n 16 n (n + 1) 2 H n H 2n	= 4 -	7π 3	+ 8 ln(2) +	48 ln 2 (2) -16G -16 ln(2) π
	holds.

√x+ 1(2x -1) + √ x sinh -1 ( √ x) .

Acknowledgements

The authors would like to thank Dr. Brendan Quine for some useful discussions regarding some of the results introduced in this article.

holds.

Proof Letting a = 1 4 with regard to [START_REF] Sofo | Evaluation of integrals with hypergeometric and logarithmic functions[END_REF], we find that

1 .

We see that the hypergeometric expression in the left-hand side of the above equality contains a Pochhammer product of the form indicated in 24. Applying the lemma from [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF], we see that

with Mathematica 11 unable to symbolically evaluate the right-hand side of [START_REF] Wimp | Irreducible recurrences and representation theorems for 3 F 2 (1)[END_REF]. Through the use of a standard formulation of Wallis' integral identity, we see that

and this provides us with the desired result.

The evaluation provided in the above corollary is especially interesting, since it is not otherwise obvious as to how to evaluate the 3 F 2 series from Corollary 4. We see that the lemma from [START_REF] Campbell | Hypergeometry of the Parbelos[END_REF] cannot be applied, since there are three quarter-integer parameters in the left-hand side of ( 25), and a Pochhammer product of the form in [START_REF] Sofo | Integrals of logarithmic and hypergeometric functions[END_REF] does not appear. The hypergeometric function x produces a very complicated expression that cannot be simplified, so Wallistype functions cannot be applied. As in the case with the main 3 F 2 (1) series under consideration from [START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF], it is easily seen that Watson's theorem on the evaluation of series of the form

cannot be applied directly to prove the above Corollary, and this is also the case for Clausen's product

A similar approach leads us to the following new result.

Corollary 5

The evaluation

Proof This may be proved by letting a = 3 4 in the above theorem, and by mimicking the preceding proof.

Experimentation with the proof signature from Theorem 15 leads us to many new results of a similar flavor, as suggested below. From the variety of results on 3 F 2 quotients from [START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF], as in the intriguing evaluation

for m ≥ 0 and η > -1 introduced and proved in [START_REF] Campbell | On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre series expansions[END_REF], we are inspired to develop new results in the same vein.

Theorem 16 The identity

x .

Corollary 6 The evaluation The above corollaries show us how the master lemma from Section 1 can be used to provide us with a very powerful technique for evaluating new 3 F 2 (1) series. Theorem 15 may be greatly generalized as follows.


Romain Postoyan
email: romain.postoyan@univ-lorraine.fr.

Qinghua Zhang
email: qinghua.zhang@inria.fr.

Local adaptive observer for linear time-varying systems with parameter-dependent state matrices

We present an adaptive observer for linear timevarying systems whose state matrix depends on unknown parameters. We first assume that the state matrix is affine in these parameters. In this case, the proposed observer generates state and parameter estimates, which exponentially converge to the plant state and the true parameter, respectively, provided a persistence of excitation condition holds and the unknown parameters lie in a neighborhood of some known nominal values. Hence, some prior knowledge on the unknown parameters is required, but not on the state. We then modify the adaptive observer and its convergence analysis to systems whose state matrix is smooth, instead of being affine, in the unknown parameters. The convergence is approximate, and no longer exponential, in this case. An example is provided to illustrate the results, for which the required distance between the unknown parameters and their nominal values is investigated numerically.

I. INTRODUCTION

The objective of this work is to estimate both the state and the parameters of the system ẋ(t) = A(t, θ)x(t) + B(t)u(t) y(t) = C(t)x(t),

where 1 x(t) ∈ R nx is the state, u(t) ∈ R nu is the input, y(t) ∈ R ny is the output, θ ∈ R n θ is the vector of unknown constant parameters, t ≥ 0 is the time, and n x , n u , n y , n θ ∈ Z >0 . In other words, we aim at designing an adaptive observer for system [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF]. This problem arises in many practical applications, including electrochemical batteries [START_REF] He | State-of-charge estimation of the lithium-ion battery using an adaptive extended Kalman filter based on an improved Thevenin model[END_REF] or mechanical systems [START_REF] Döhler | Vibration monitoring by eigenstructure change detection based on perturbation analysis[END_REF] to cite a couple of examples, as soon as we need to estimate on-line both the unmeasured state and unknown parameters of a plant for monitoring purposes for instance.

In the majority of works on adaptive observers, the state matrix is assumed to be fully known contrary to [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF], and the unknown parameters enter in the system through an additional term of the form Φ(t, u(t), y(t))θ where Φ is known. Solutions for this case can be found in e.g., [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF], [START_REF] Besanc ¸on | On adaptive observers for state affine systems[END_REF], [START_REF] Marino | Nonlinear control design[END_REF], [START_REF] Besanc ¸on | Adaptive observer design for discrete time LTV systems[END_REF]- [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF]. The fact that A depends on θ in (1) is a major difference with these references, which makes the problem difficult because of the cross-terms involving θ and x(t). There are nevertheless results in the literature, which propose adaptive observers for systems with partially unknown state matrix A. In [START_REF] Kresselmeier | Adaptive observers with exponential rate of convergence[END_REF], several designs are given for single-input single-output linear time-invariant (LTI) systems. In [START_REF] Ljung | Analysis of a general recursive prediction error identification algorithm[END_REF], it is explained how to optimally fit a discrete-time LTI model to any given output signals. In [START_REF] Pinto | State and parameter estimation based on switched observers -an LMI approach[END_REF], a switched approach is proposed for LTI systems based on linear matrix inequalities (LMI) conditions. More recently, a generic approach for the state and parameter estimation of nonlinear systems was presented in [START_REF] Chong | Parameter and state estimation of nonlinear systems using a multi-observer under the supervisory framework[END_REF] using a multi-observer set-up. On the other hand, adaptive observers for systems with known state matrices with an additive term of the form Φ(x)θ (see e.g., [START_REF] Besanc ¸on | On adaptive observers for systems with state and parameter nonlinearities[END_REF], [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF], [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], [START_REF] Farza | Adaptive observer design for a class of nonlinear systems. Application to speed sensorless induction motor[END_REF]) are relevant in this context when A is affine in θ, as a simple manipulation brings system (1) to the right form.

In this work, we propose an approach, which relies on a simple idea. We essentially assume that we know a stateobserver for system [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF] for some known nominal parameter value θ 0 . Because θ = θ 0 , the idea is to modify the original state-observer and design an adaptation law to estimate θ by relying on similar techniques as in [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF]. Indeed, by adding and subtracting A(t, θ 0 )x(t) in the first equation in [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF], we obtain ẋ(t) = A(t, θ 0 )x(t) + B(t)u(t) + [A(t, θ) -A(t, θ 0 )] x(t). Assuming A(t, θ) is affine in θ we can write A(t, θ)x(t) -A(t, θ 0 )x(t) = Λ(t, x(t))(θ -θ 0 ), where Λ(t, x(t)) is a time-varying matrix linear in its second argument; this is explained in more detailed in the sequel. Hence, ẋ(t) = A(t, θ 0 )x(t) + B(t)u(t) + Λ(t, x(t))(θ -θ 0 ). Denoting by x the state estimate of the adaptive observer to be designed and x = x -x the associated state estimation error, we derive ẋ(t) = A(t, θ 0 )x(t) + B(t)u(t) + Λ(t, x(t))(θ -θ 0 ) + Λ(t, x(t))(θ -θ 0 ), since Λ is linear in its second argument. Now Λ(t, x(t)) is known since x is the state estimate we generate. The term Λ(t, x(t))(θ -θ 0 ) is therefore of the form Φ(t)(θ -θ 0 ) where Φ(t) is known. We are almost back to the same class of systems as in [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF] where the unknown parameter is θ -θ 0 instead of θ, but this is fine since we know θ 0 . The difference with [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF] is that we have to deal with the extra term Λ(t, x(t))(θ -θ 0 ). It appears that when ||θ -θ 0 || is small, so is Λ(t, x(t))(θ -θ 0 )x(t) so that stability of the estimation error may be ensured. The point of these developments is to show that after some simple manipulations, we can write the problem in a similar, though different form, as in [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF]. Still, the design of the adaptive observer is not a straightforward application of [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF] and requires appropriate modifications as well as specific assumptions. The exponential convergence of the proposed adaptive observer is guaranteed provided a persistence of excitation condition holds and ||θ -θ 0 || is small enough. Such an excitation condition is needed in almost all works on adaptive observers. The fact that ||θ -θ 0 || has to be small means that the design is local in the parameter estimate, but we are free to initialize the state estimate as we wish. It can be noted that many estimation or identification schemes are also local, see [START_REF] Ljung | Analysis of a general recursive prediction error identification algorithm[END_REF] for instance and all the works on extended Kalman filters, see e.g., [START_REF] Bonnabel | A contraction theory-based analysis of the stability of the deterministic extended Kalman filter[END_REF], [START_REF] Krener | The convergence of the extended Kalman filter[END_REF], [START_REF] Reif | An EKF-based nonlinear observer with a prescribed degree of stability[END_REF]. Afterwards, the results are extended to the case where A is only smooth in θ, and not necessarily affine. The adaptive observer and its analysis are modified accordingly and an approximate convergence property is ensured in this case.

Compared to [START_REF] Kresselmeier | Adaptive observers with exponential rate of convergence[END_REF], our results apply to multi-input multioutput linear time-varying systems, as opposed to singleinput single-output LTI systems, and compared to [START_REF] Pinto | State and parameter estimation based on switched observers -an LMI approach[END_REF], we address LTV systems and our approach relies on different type of conditions (no LMI). The closest work in spirit, although it is written in discrete-time for LTI models, is the one in [START_REF] Ljung | Analysis of a general recursive prediction error identification algorithm[END_REF]. A key difference with [START_REF] Ljung | Analysis of a general recursive prediction error identification algorithm[END_REF] is that we do not need to use projection techniques to maintain the parameter estimate in some given set, and the estimates we generate cannot be trapped in the border of the given set due to projections, but always converge to the true parameter provided, again, that ||θ -θ 0 || is small enough and a persistence of excitation holds. On the other hand, the adaptive observer is less computationally demanding than the supervisory observer presented in [START_REF] Chong | Parameter and state estimation of nonlinear systems using a multi-observer under the supervisory framework[END_REF], which requires a large number of observers to work, and we ensure the asymptotic convergence of the estimation errors as opposed to a practical property in [START_REF] Chong | Parameter and state estimation of nonlinear systems using a multi-observer under the supervisory framework[END_REF]. Finally, compared to [START_REF] Besanc ¸on | On adaptive observers for systems with state and parameter nonlinearities[END_REF], [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF], [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], [START_REF] Farza | Adaptive observer design for a class of nonlinear systems. Application to speed sensorless induction motor[END_REF]: (i) we do not require any particular triangular or block-diagonal structure of the system ([3], [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], [START_REF] Farza | Adaptive observer design for a class of nonlinear systems. Application to speed sensorless induction motor[END_REF]); (ii) we provide results for the case where the state matrix is not affine in θ in which case the results in [START_REF] Besanc ¸on | On adaptive observers for systems with state and parameter nonlinearities[END_REF], [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF], [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF] do not apply; (iii) we do not rely on high-gain techniques ( [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], [START_REF] Farza | Adaptive observer design for a class of nonlinear systems. Application to speed sensorless induction motor[END_REF]), which are not known to be very sensitive to noise measurements; (iv) the observer is of smaller dimension, which is relevant for computational reasons ( [START_REF] Besanc ¸on | On adaptive observers for systems with state and parameter nonlinearities[END_REF], [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], [START_REF] Farza | Adaptive observer design for a class of nonlinear systems. Application to speed sensorless induction motor[END_REF]). Also, we believe that the way we approach the problem has its own interest, which can be relevant for further extensions.

A similar problem is addressed in [START_REF] Döhler | Vibration monitoring by eigenstructure change detection based on perturbation analysis[END_REF]. First, exponential convergence is ensured in our case when A is affine in θ as opposed to an ultimate boundedness property in [START_REF] Döhler | Vibration monitoring by eigenstructure change detection based on perturbation analysis[END_REF]. Second, the proposed solution is computationally lighter, since [START_REF] Döhler | Vibration monitoring by eigenstructure change detection based on perturbation analysis[END_REF] requires the use of both a state-observer for the nominal parameter values and of an adaptive observer, while we only need the latter. Some proofs are omitted for space reasons, these can be found in [START_REF] Postoyan | Local adaptive observer for linear timevarying systems with parameter-dependent state matrices[END_REF].

Notation and terminology. Let R := (-∞, ∞), R ≥0 := [0, ∞), R >0 := (0, ∞), Z ≥0 := {0, 1, 2, . . .}, and Z >0 := {1, 2, . . .}. The notation (x, y) stands for [x , y ] , where x ∈ R n , y ∈ R m , and n, m ∈ Z >0 . The identity matrix of size n ∈ Z >0 is denoted I n or simply I when the dimension is clear from the context. Given two matrices

Q, R ∈ R n×n , we write Q ≤ R if R -Q is positive semi-definite, n ∈ Z >0 .
The symbol stands for symmetric blocks in matrices. A time-varying matrix M (t) ∈ R n×m is said to be bounded if there exists c ≥ 0 such that ||M (t)|| ≤ c for any t ≥ 0,

where ||M (t)|| := sup x∈R m \{0} ||M (t)x||
||x|| , || • || standing for the Euclidean norm when the argument is a vector. We write

N (x) = O(||x|| 2 ) for N : R n → R m with n, m ∈ Z >0 , when there exists c ≥ 0 such that ||N (x)|| ≤ c||x|| 2 for any x ∈ R n . The set of piecewise continuous functions from R ≥0 to R n is denoted PC(R n ), n ∈ Z >0 . Let p ∈ R n and δ ∈ R >0 ∪ {∞}, the closed ball of radius δ centered at p is denoted B(p, δ), with B(p, δ) = R n when δ = ∞.

II. ASSUMPTIONS

The matrices A(t, θ), B(t) and C(t) in ( 1) are assumed to be continuous and bounded with respect to the time t. We make the next assumption on how A(t, θ) depends on θ.

Assumption 1: The matrix A(t, θ) is affine in θ, in the sense that there exist matrices

A 0 (t), . . . , A n θ (t) ∈ R nx×nx such that, for any t ≥ 0, A(t, θ) = A 0 (t) + n θ i=1 A i (t)θ i , where θ = (θ 1 , . . . , θ n θ ).
When Assumption 1 is not satisfied, we may redefine the vector of unknown parameters to enforce it, by eventually over-parameterizing the original system. This assumption will be relaxed in Section III-C. It is important to note that the matrices A 0 (t), . . . , A n θ (t) in Assumption 1 are known, as these can be directly derived from the expression of A(t, θ). These matrices are continuous and bounded with respect to the time since so is A(t, θ).

Our design implicitly relies on the knowledge of a state observer for system (1) if θ would be known. More precisely, we make the following assumption.

Assumption 2: There exist δ 1 ∈ R >0 ∪ {∞} and K : R ≥0 × B(θ, δ 1 ) → R nx×ny , which is continuous and bounded, such that for any θ 0 ∈ B(θ, δ 1 ) the origin of the system ẋ(t) = [A(t, θ 0 ) -K(t, θ 0 )C(t)] x(t) is uniformly globally exponentially stable.

Assumption 2 essentially means that we know a set where parameter θ lies and, if we knew θ, we would be able to design a state-observer for the corresponding system. More precisely, Assumption 2 states that there exists a neighborhood of parameter θ, corresponding to B(θ, δ 1 ), such that we can design a standard (Luenberger) state observer with gain K(t, θ 0 ) for any θ 0 in this set, whose solutions uniformly, globally, and exponentially converge to solutions to [START_REF] Besanc ¸on | Remarks on nonlinear adaptive observer design[END_REF]

when θ = θ 0 . To see it, let ẋ(t) = A(t, θ 0 )x(t) + B(t)u(t) + K(t, θ 0 )(y(t) -C(t)x(t)). If θ = θ 0 , then the estimation error system with variable x = x -x is ẋ(t) = [A(t, θ 0 ) -K(t, θ 0 )C(t)] x(t) as in Assumption 2.
A necessary and sufficient condition for Assumption 2 to hold is provided next.

Lemma 1: Under Assumption 1, Assumption 2 holds if and only if there exists a continuous and bounded mapping K * : R ≥0 → R nx×ny such that the origin of ẋ(t) = [A(t, θ) -K * (t)C(t)] x(t) is uniformly globally exponentially stable for the (unknown) true parameter value θ.

Lemma 1 involves θ, which we do not know. The following lemma provides exploitable conditions for the satisfaction of Assumption 2.

Lemma 2: Suppose the following holds.

(i) Assumption 1 holds.

(ii) There exists δ 1 ∈ R ≥0 ∪ {∞} such that the pair (A(t, θ 0 ), C(t, θ 0 )) is uniformly completely observable 2 for any θ 0 ∈ B(θ, δ 1 ). Assumption 2 is verified with K(t, θ 0 ) = P (t, θ 0 )C(t) R -1 (t) where P (t, θ 0 ) is the solution to Ṗ (t, θ 0 ) = P (t, θ 0 )A(t, θ 0 ) + A(t, θ 0 )P (t, θ 0 ) -P (t, θ 0 )C(t) R -1 (t)C(t)P (t, θ 0 ) + Q(t) with P (0) = P 0 , where P 0 is any symmetric, positive definite matrix, and R(t), Q(t) are any symmetric, positive definite, continuous and bounded matrices.

When the matrices A and C in (1) are time-invariant, it suffices to have the pair (A(θ 0 ), C(θ 0 )) observable for any θ 0 with ||θ -θ 0 || ≤ δ 1 for some δ 1 ∈ R >0 ∪ {∞} to ensure Assumption 2. Indeed, in this case, Bass-Gura formula 3 leads to a gain K(θ 0 ), which is continuous in θ 0 . The same applies when L(θ 0 ) is selected to minimize a linear quadratic cost, i.e. when L(θ 0 ) = P (θ 0 )C R -1 and P (θ 0 ) is the solution to the Riccati equation P (θ 0 )A(θ 0 ) + A(θ 0 )P (θ 0 )-P (θ 0 )C R -1 CP (θ 0 )+Q. It suffices to select Q and R symmetric and positive definite to ensure the continuity of P (θ 0 ) in θ 0 according to Proposition 1 in [START_REF] Faibusovich | Algebraic Riccati equation and symplectic algebra[END_REF], which in turn implies the continuity of K(θ 0 ) in θ 0 .

We finally make the next boundedness assumption on system (1).

Assumption 3: There exists a hyper-rectangle

X = {x = (x 1 , . . . , x nx ) ∈ R nx : x i ∈ [c i , c i ]} where c i ≤ c i ∈ R, i ∈ {1, . . . , n x }, a non-empty set of inputs M u ⊂ PC(R nu ),
and a non-empty set of initial conditions S X ⊂ R nx , such that any solution x to system (1) initialized in S X with input u ∈ M u satisfies x(t) ∈ X for all t ≥ 0.

Assumption 3 means that the solutions to (1) lie in a known compact set, which can always be embedded in a closed hyper-rectangle, whenever its input lies in set M u . Note that there is no restriction of the "size" of X .

III. MAIN RESULT A. Design and analysis

Like in [START_REF] Shim | Semi-global observer for multi-output nonlinear systems[END_REF], we introduce the element-wise saturation function σ X : R nx → R nx defined as σ X (x) := (σ 1,X (x 1 ), . . . , σ nx,X (x nx ))

with σ i,X (x i ) = x i when x i ∈ [c i , c i ], σ i,X (x i ) = c i when x i ≤ c i and σ i,X (x i ) = c i when x i ≥ c i , for any x i ∈ R, where X , c i , c i come from Assumption 3, i ∈ {1, . . . , n x }.
Based on Assumptions 1-2 and inspired by [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF], we propose the adaptive observer

ẋ(t) = A(t, θ 0 )x(t) + B(t)u(t) + Λ(t, σ X (x(t))) θ(t) + K(t, θ 0 ) + γΥ(t)Υ (t)C (t)Σ(t) ×(y(t) -C(t)x(t)) θ(t) = γΥ (t)C (t)Σ(t)(y(t) -C(t)x(t)) Υ(t) = [A(t, θ 0 ) -K(t, θ 0 )C(t)] Υ(t) + Λ(t, σ X (x(t))) θ(t) = θ(t) + θ 0 , (2)
where θ(t) ∈ R n θ is the estimate of θ -θ 0 , θ(t) ∈ R n θ is therefore the estimate of θ, Υ(t) ∈ R nx×n θ , 2 See [START_REF] Silverman | Controllability, observability and stability of linear systems[END_REF] for a definition of uniform complete observability as well as conditions to ensure it. 3 See Chapter 4 in [START_REF] Kailath | Linear systems[END_REF] for instance.

Λ(t, z) := [A 1 (t)z . . . A n θ (t)z] ∈ R nx×n θ for z ∈ R nx with A 1 (t), . . . , A n θ (t) coming from Assumption 1, Σ(t) :
R → R ny×ny is any symmetric, positive definite, bounded, continuous matrix, γ ∈ R >0 , and X ⊂ R n comes from Assumption 3. We have two degree of freedom when designing (2): Σ(t) and γ. The former has to be designed to verify a persistence of excitation condition stated next. In practice, we typically take it diagonal. Parameter γ tunes the convergence speed of θ, the bigger γ, the faster the convergence, but the more sensitive to noise. Remark 1: The use of the saturation function σ X in ( 2) is essential to guarantee the boundedness of Υ in the proof of the next theorem, which is needed to establish the desired convergence property. This is the main difference with [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF] in terms of design. Other differences reside in the assumptions we rely on.

The next theorem ensures that the state and the parameter estimates provided by (2) exponentially converges to x and θ, provided a persistence of condition holds and ||θ -θ 0 || is sufficiently small. Its proof is provided in Section III-B.

Theorem 1: Consider systems ( 1) and ( 2) and suppose the following holds.

(i) Assumptions 1-3 hold.

(ii) Let δ 2 ∈ R >0 ∪ {∞}, x0 ∈ R nx , θ0 ∈ R n θ , Υ 0 ∈ R nx×n θ ,
there exist α, T > 0, which depend on x0 , θ0 , Υ 0 , such that, for any x(0) ∈ S X and input u ∈ M u , any θ 0 such that ||θ -θ 0 || ≤ δ 2 , the solution x, θ, Υ to system (2) initialized at (x 0 , θ0 , Υ 0 ) with ||θ 0 -θ|| ≤ δ 2 , verifies for any t ≥ 0,

αI ≤ t+T t Υ (τ )C (τ )Σ(τ )C(τ )Υ(τ )dτ. (3)
There exist δ ∈ (0, min{δ 1 , δ 2 }) and c 1 , c 2 > 0 , which depend on X , M u , x0 , θ0 , Υ 0 , such that, if ||θ -θ 0 || ≤ δ, the solution (x, θ, Υ) to (2) initialized at (x 0 , θ0 , Υ 0 ) and any solution to system (1) with x(0) ∈ S X and input u ∈ M u , verify ||(x(t)-x(t), θ(t)-θ)|| ≤ c 1 e -c2t ||(x(0)-x0 , θ(0)θ)|| for any t ≥ 0. Item (ii) of Theorem 1 is a persistence of excitation condition on the Υ-system, like in [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF]. It actually is a uniform persistence of excitation property, see Definition 2 in [START_REF] Loría | Uniform exponential stability of linear timevarying systems: revisited[END_REF], as that the constants α and T are independent of θ 0 , which is essential. Indeed, if the bound δ 2 , and thus δ, on the norm of θ -θ 0 would depend on θ 0 , there will be no guarantee that we can always select θ 0 sufficiently close to θ such that ||θ -θ 0 || ≤ δ. That is also the reason why we assume the existence of the gain K(t, θ 0 ) for any θ 0 ∈ B(θ, δ 1 ) in Assumption 2, and not for a given θ 0 , otherwise δ 1 in Assumption 2 may depend on θ 0 and the same issue would arise. On the other hand, δ depends on the initial conditions of observer [START_REF] Besanc ¸on | On adaptive observers for state affine systems[END_REF]. It is possible to ensure that δ is uniform over these initial conditions over given sets, when (3) holds for any initial conditions in these sets.

Property ( 3) is difficult to verify as it involves any θ 0 in a set, and any solution initialized in S X with input u ∈ M u . Even if we could verify it, that would not mean that the estimates converge to the true values, as we also need for that ||θ -θ 0 || to be sufficiently small, which is something we cannot check in practice. It is important to note that this difficulty is generic to any local estimation schemes, including works on extended Kalman filters where the initial conditions x 0 and x0 have to be close to each other (among other conditions), see e.g., [START_REF] Bonnabel | A contraction theory-based analysis of the stability of the deterministic extended Kalman filter[END_REF], [START_REF] Krener | The convergence of the extended Kalman filter[END_REF], [START_REF] Reif | An EKF-based nonlinear observer with a prescribed degree of stability[END_REF]. The only way to avoid it is when we have the property that the observation error y(t) -C(t)x(t) converging to zero implies that the estimation errors x(t) -x(t) and θ -θ(t) also converging to zero, see Section III.A in [START_REF] Postoyan | A multi-observer approach for the state estimation of nonlinear systems[END_REF]. In practice, we can check numerically on-line whether the inequality in (3) holds for a given θ 0 and a given solution x to (1), in which case the non-satisfaction of (3) indicates that the obtained estimate are not valid.

The fact that we need to know a set where the unknown parameters lie is a valid assumption in various applications, like electrochemical batteries for which the parameters are accurately known when the battery is produced, but slowly vary with time. Then, the interest of the adaptive observer is to track these slow variations for monitoring purposes. If we only have some vague knowledge about θ, we may overcome this issue by employing a multi-observer architecture as explained in [START_REF] Postoyan | A multi-observer approach for the state estimation of nonlinear systems[END_REF].

B. Proof of Theorem 1

We introduce the estimation errors x := x -x and θ := θ -θ 0 -θ. Let x(0) ∈ R nx , t ≥ 0 for which the solution to (1), ( 2) is defined, and θ 0 ∈ B (θ, min{δ 1 , δ 2 }) where δ 1 , δ 2 come from Assumptions 2 and item (ii) of Theorem 1, respectively. In view of ( 1) and [START_REF] Besanc ¸on | On adaptive observers for state affine systems[END_REF],

ẋ(t) = A(t, θ)x(t) + B(t)u(t) -A(t, θ 0 )x(t) -B(t)u(t) -Λ(t, σ X (x(t))) θ(t) -[K(t, θ 0 ) +γΥ(t)Υ (t)C (t)Σ(t) (y(t) -C(t)x(t)), (4
) we add and subtract A(t, θ 0 )x(t) and we obtain

ẋ(t)=[A(t, θ 0 ) -K(t, θ 0 )C(t)] x(t) + [A(t, θ) -A(t, θ 0 )] x(t) -Λ(t, σ X (x(t))) θ(t) -γΥ(t)Υ (t)C (t)Σ(t)(y(t) -C(t)x(t)). (5)
In view of Assumption 1 and the definition of

Λ(•, •) given after (2), [A(t, θ) -A(t, θ 0 )] x(t) = Λ(t, x(t))(θ -θ 0 ).
We now restrict our attention to the case where x(0) ∈ S X with u ∈ M u . Hence, x(t) ∈ S X and x(t) = σ X (x(t)) in view of Assumption 3 and the definition of σ, respectively.

Thus [A(t, θ) -A(t, θ 0 )] x(t) = Λ(t, σ X (x(t)))(θ -θ 0 ). Consequently, in view of (5), ẋ(t)=[A(t, θ 0 ) -K(t, θ 0 )C(t)] x(t) +Λ(t, σ X (x(t)))(θ -θ 0 ) -Λ(t, σ X (x(t))) θ(t) -γΥ(t)Υ (t)C (t)Σ(t)(y(t) -C(t)x(t)). (6)
We add and subtract Λ(t, σ X (x(t)))(θ -θ 0 ) and we obtain, by exploiting the linearity of Λ(•, •) in its second argument and using

θ(t) = θ -θ 0 -θ(t), ẋ(t)=[A(t, θ 0 ) -K(t, θ 0 )C(t)] x(t) +Λ(t, σ X (x(t)) -σ X (x(t)))(θ -θ 0 ) +Λ(t, σ X (x(t))) θ(t) -γΥ(t)Υ (t)C (t)Σ(t)(y(t) -C(t)x(t)). (7)
Like in [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF], we introduce η := x -Υ θ. We deduce from the last equation and (2),

η(t)=[A(t, θ 0 ) -K(t, θ 0 )C(t)] x(t) +Λ(t, σ X (x(t)) -σ X (x(t)))(θ -θ 0 ) +Λ(t, σ X (x(t))) θ(t) -γΥ(t)Υ (t)C (t)Σ(t)(y(t) -C(t)x(t)) -[(A(t, θ 0 ) -K(t, θ 0 )C(t)) Υ(t) + Λ(t, σ X (x(t)))] × θ(t) + Υ(t) γΥ (t)C (t)Σ(t)(y(t) -C(t)x(t)) =[A(t, θ 0 ) -K(t, θ 0 )C(t)] η(t) +Λ(t, σ X (x(t)) -σ X (x(t)))(θ -θ 0 ). (8
) On the other hand, in view of (2) and since

x(t) = η(t) + Υ(t) θ(t), θ(t) = -γΥ (t)C (t)Σ(t)C(t)x(t) = -γΥ (t)C (t)Σ(t)C(t) Υ(t) θ(t) + η(t) . (9)
The dynamics of the (η, θ)-system in ( 8), ( 9) is the same as in ( 14)-( 15) in [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF] up to the perturbative term Λ(t, σ X (x(t))-σ X (x(t)))(θ -θ 0 ) in [START_REF] Faibusovich | Algebraic Riccati equation and symplectic algebra[END_REF]. Because of the latter, we can no longer invoke cascade arguments to conclude about the stability of the (η, θ)-system as in [START_REF] Zhang | Adaptive observer for multi-input-multi-ouput (MIMO) linear time-varying systems[END_REF]. We will use small-gain arguments instead, and the small-gain condition will be enforced by imposing θ -θ 0 to be small, which explains why the convergence property we will guarantee is local and not global (in general).

Let (x, θ, Υ) be the solution to system (2) initialized at (x 0 , θ0 , Υ 0 ). We define P : R ≥0 × R n θ → R nx×nx as the solution to -Ṗ (t, θ 0 ) = P (t, θ 0 ) [A(t, θ 0 ) -K(t, θ 0 )C(t)]+ [A(t, θ 0 ) -K(t, θ 0 )C(t)] P (t, θ 0 ) + I, with P (0, θ 0 ) = P 0 and P 0 is symmetric and positive definite. Since θ 0 belongs to the closed ball B(θ, δ 1 ), Assumption 2 holds and A(t, θ 0 ) -K(t, θ 0 )C(t) is continuous and bounded, Lemma 1 in [START_REF] Loría | Uniform exponential stability of linear timevarying systems: revisited[END_REF] ensures that P (t, θ 0 ) exists for all t ≥ 0, is continuously differentiable, bounded, symmetric, positive definite and there exist a P , a P > 0, which are independent of θ 0 , such that a P I ≤ P (t, θ 0 ) ≤ a P I for any t ≥ 0. We define V : (P, η) → η P η. Let4 t ≥ 0, in view of [START_REF] Faibusovich | Algebraic Riccati equation and symplectic algebra[END_REF] and the definition of P (t, θ 0 ) (we omit the time and θ 0 dependencies of the solutions),

V (P, η) = -||η|| 2 + 2η P Λ(t, σ X (x) - σ X (x))(θ -θ 0 ), where x = x -η -Υ θ.
The definition of Λ(•, •) and the fact that the matrices A 1 (t), . . . , A n θ (t) are bounded according to Assumption 1 imply that there exists a constant a Λ > 0 such that ||Λ(t, z)|| ≤ a Λ ||z|| for any z ∈ R nx . Consequently, since P ≤ a P I,

V (P, η) ≤ -||η|| 2 +2a P a Λ ||η||•||σ X (x)-σ X (x)||×||θ -θ 0 ||. From the definition of σ, ||σ X (x)-σ X (x)|| ≤ n x ||x-x|| = n x ||x|| ≤ n x ||η|| + n x ||Υ|| • || θ||. Thus, V (P, η) ≤ -||η|| 2 + 2n x a P a Λ ||η|| 2 ||θ -θ 0 || +2n x a P a Λ ||η|| • ||Υ|| • || θ|| • ||θ -θ 0 || = -[1 -2n x a P a Λ ||θ -θ 0 ||] ||η|| 2 +2n x a P a Λ ||η|| • ||Υ|| • || θ|| ×||θ -θ 0 ||. (10)
We now investigate the θ-system in [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF]. We need the next claim for that purpose.

Claim 1: There exists a continuous function ν : R nx×n θ → [1, ∞), which is independent of θ 0 , such that for any continuous function x : R ≥0 → R nx , the corresponding solution Υ to the Υ-system in (2) initialized at Υ 0 is such that ||Υ(t)|| ≤ ν(Υ 0 ) for any t ≥ 0. Proof of Claim 1. The result follows from Assumption 2, Lemma 1 in [START_REF] Loría | Uniform exponential stability of linear timevarying systems: revisited[END_REF] and the fact that Λ(t, σ X (x)) is bounded uniformly with respect to θ 0 ∈ B(θ, min{δ 1 , δ 2 }). The fact that the image of ν is in [1, ∞) can always be ensured by adding 1 to any bound on ||Υ(t)||.

The matrix γΥ (t)C (t)Σ(t)C(t)Υ(t) is continuous and bounded since so are C(t) by assumption, Σ(t) by design, and Υ(t) according to Claim 1 for the boundedness. Therefore, since θ ∈ B(θ, δ 2 ), item (ii) of Theorem 1 is assumed to hold, Σ(t) and C(t) are bounded and Claim 1 applies, according to Lemmas 1 and 5 in [START_REF] Loría | Uniform exponential stability of linear timevarying systems: revisited[END_REF], the solution P (t) to -˙ P (t) = -γ P (t)Υ (t)C (t)Σ(t)C(t)Υ(t) -γΥ (t)C (t)Σ(t)C(t)Υ(t) P (t) + I, with P (0) = P 0 and P 0 is symmetric and positive definite, exists for all t ≥ 0, is continuously differentiable, bounded, symmetric, positive definite and there exist a P , a P > 0, which are independent of θ 0 , such that a P I ≤ P (t) ≤ a P I for any t ≥ 0. Let V : ( P , θ) → θ P θ. Let t ≥ 0, in view of [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF] and the definition of P (t), ˙ V ( P , θ) = -|| θ|| 2 -2γ θ P Υ (t)C (t)Σ(t)C(t)η. By following similar steps as above, we derive [START_REF] He | State-of-charge estimation of the lithium-ion battery using an adaptive extended Kalman filter based on an improved Thevenin model[END_REF] where a C > 0 is a constant such that ||C (t)Σ(t)C(t)|| ≤ a C for all t ≥ 0, which exists since C(t) and Σ(t) are bounded.

˙ V ( P , θ) ≤ -|| θ|| 2 + 2γa P a C || θ|| • ||Υ|| • ||η||,
Define U : (P, P , η, θ) → V (P, η)+µ V ( P , θ) with µ > 0 a constant, which will be selected below. In view of ( 10) and [START_REF] He | State-of-charge estimation of the lithium-ion battery using an adaptive extended Kalman filter based on an improved Thevenin model[END_REF], for any t ≥ 0,

U (P, P , η, θ) ≤ -[1 -2n x a P a Λ ||θ -θ 0 ||] ||η|| 2 +2n x a P a Λ ||Υ|| • || θ|| • ||η|| • ||θ -θ 0 || -µ|| θ|| 2 + 2µγa P a C || θ|| • ||Υ|| • ||η||. ( 12
) According to Claim 1, ||Υ|| ≤ ν(Υ 0 ), thus U (P, P , η, θ) ≤ -[1 -2n x a P a Λ ||θ -θ 0 ||] ||η|| 2 +2n x a P a Λ ν(Υ 0 )|| θ|| • ||η|| • ||θ -θ 0 || -µ|| θ|| 2 + 2µγa P a C ν(Υ 0 )|| θ|| • ||η|| = -(||η||, || θ||), M(||η||, || θ||) , (13)
   1 -2nxaP aΛ||θ -θ0|| > 0 [1 -2nxaP aΛ||θ -θ0||] µ -nxaP aΛν(Υ0)||θ -θ0|| + µγa P aC ν(Υ0) 2 > 0. (14)
The first inequality above is ensured as long as ||θ -θ 0 || is sufficiently small, in particular we assume that ||θ -θ 0 || < 1 4n x a P a Λ so that 1 -2n x a P a Λ ||θ -θ 0 || > 1 2 . The second inequality also holds with ||θ -θ 0 || sufficiently small, by taking µ < 1 2 γa P a C ν(Υ 0 ) 2 ; note that the denominator is well-defined since ν(Υ 0 ) ≥ 1 in view of Claim 1. As a result, there exists δ ∈ (0, min{δ 1 , δ 2 }), which is independent of θ 0 , such that ||θ -θ 0 || ≤ δ implies the existence of ε > 0 such that, for t ≥ 0, U (P, P , η, θ) ≤ -ε||(η, θ)|| We study the case where A(t, θ) is smooth in θ but not necessarily affine in θ, as required in Assumption 1. Let t ≥ 0, θ ∈ R n θ and denote A(t, θ) = [a ij (t, θ)] (i,j)∈{1,...,nx} 2 . The Taylor expansion of each a ij with respect to θ gives

a ij (t, θ) = a ij (t, θ 0 ) + ∂a ij ∂θ (t, θ 0 )(θ -θ 0 ) + ρ ij (t, θ, θ 0 ) where 5 ρ ij (t, θ, θ 0 ) = O(||θ -θ 0 || 2 ). As a result, A(t, θ) = A(t, θ 0 ) + n θ i=1 A i (t, θ 0 )(θ i -θ 0,i ) + R(t, θ, θ 0 ), (15)
where θ = (θ 1 , . . . , θ n θ ), θ 0 = (θ 0,1 , . . . , θ 0,n θ ),

A k (t, θ 0 ) := ∂aij ∂θ k (t, θ 0 ) (i,j)∈{1,...,nx} 2 for k ∈ {1, . . . , n θ }, R(t, θ, θ 0 ) := [ρ ij (t, θ, θ 0 )] (i,j)∈{1,...,nx} 2 .
In view of (15), we modify the observer in (2) as

ẋ(t) = A(t, θ 0 )x(t) + B(t)u(t) + Λ(t, θ 0 , σ X (x(t))) θ(t) + K(t, θ 0 ) + γΥ(t)Υ (t)C (t)Σ(t) ×(y(t) -C(t)x(t)) θ(t) = γΥ (t)C (t)Σ(t)(y(t) -C(t)x(t)) Υ(t) = [A(t, θ 0 ) -K(t, θ 0 )C(t)] Υ(t) + Λ(t, θ 0 , σ X (x(t))) θ(t) = θ(t) + θ 0 , (16)
where Λ(t, θ 0 , z) := A 1 (t, θ 0 )z . . . A n θ (t, θ 0 )z for z ∈ R nx . The other matrices and parameters are selected as in [START_REF] Besanc ¸on | On adaptive observers for state affine systems[END_REF].

Proposition 1: Consider systems (1) and ( 16). Suppose the following holds.

(i) Assumptions 2 and 3 hold.

(ii) Item (ii) of Theorem 1 holds along the solutions to (1) and ( 16). There exist δ, c 1 , c 2 , c 3 > 0, which depend on X , M u , x0 , θ0 , Υ 0 such that, if ||θ -θ 0 || ≤ δ, the solution (x, θ, Υ) to (16) initialized at (x 0 , θ0 , Υ 0 ) and any solution to system (1) with x(0) ∈ X and input u ∈ M u are such that ||(x(t) -x(t), θ(t) -θ)|| ≤ c 1 e -c2t ||(η(0), θ(0))|| + c 3 ||θ -θ 0 || 2 .

As already mentioned, Assumption 2 implies that we know a state observer for system (1) when θ = θ 0 . If we would implement this classical (non-adaptive) observer on system (1) with θ = θ 0 , the state estimate would converge to x up to an error of the order of ||θ -θ 0 ||, and, trivially, the parameter estimation error would be θ -θ 0 . Proposition 1 shows that these properties can be improved by employing the adaptive observer [START_REF] Loría | Uniform exponential stability of linear timevarying systems: revisited[END_REF], which provides estimates with errors of the order of ||θ -θ 0 || 2 , after a sufficiently long time. We are not able to ensure the asymptotic convergence of the estimates of the true values, contrary to Section III-A, because of the perturbative term R(t, θ, θ 0 ) in [START_REF] Ljung | Analysis of a general recursive prediction error identification algorithm[END_REF].

IV. ILLUSTRATIVE EXAMPLE

Consider the mass-spring system with two elements depicted in Figure 1 with masses m 1 = 100 and m 2 = 100, spring stiffness k 1 = 5, k 2 = 10 and θ > 0 being the stiffness of the middle spring, which is assumed to be unknown, and b 1 = 0.1 and b 2 = 0.4 are the damping coefficients. The system is of the form of 6 (1). Assumption 1 holds 6 and Assumption 2 is also verified as (A, C) is observable as θ = 0. u(t) = 10 sin(10t) for any t ≥ 0, Assumption 3 is satisfied since the infinity norm of the input is bounded and A is Hurwitz. We have designed the adaptive observer as in (2) with γ = 10 3 , Σ(t) = 10 3 and K such that the eigenvalues of A -KC are (-1, -1.5, -2, -2.5). We have run simulations with θ = 15, and different values of θ 0 , x(0), x(0), θ(0) and Υ(0) were considered. Simulations suggest that the convergence of the adaptive observer is independent of the initial conditions of x, x, θ and Υ: only θ 0 seems to matter. Also, the asymptotic convergence of the estimation errors is seen whenever θ 0 ∈ [θ -11, θ + 75], which means that θ 0 does not need to be very "close" to θ for the observer to work for this example.

V. CONCLUSION

We presented an adaptive observer for linear time-varying system whose state matrix A(t, θ) depends on unknown parameters. When A(t, θ) is affine in θ, the proposed scheme ensures the exponential convergence of the estimates to the true values provided some initial guess of the unknown parameter is sufficiently closed to the latter and a persistence of excitation holds. When A(t, θ) is only smooth in

with M 11 := 1 -

 111 2n x a P a Λ ||θ -θ 0 ||, M 12 := -n x a P a Λ ν(Υ 0 )||θ -θ 0 ||µγa P a C ν(Υ 0 )and M 22 := µ. Matrix M is positive definite if and only if

Fig. 1 .

 1 Fig. 1. Coupled mass-spring system.

  2 . Since min{a P , µa P }||(η, θ)|| 2 ≤ U (P, P , η, θ) ≤ (a P + µa P )||(η, θ)|| 2 , we deduce that (η(t), θ(t)) exponentially converges to 0 as t → ∞. More precisely, there existc 1 , c 2 > 0, which depend on X , M u , x0 , θ0 , Υ 0 , such that ||(η(t), θ(t))|| ≤ c 1 e -c 2 t ||(η(0), θ(0)||. Moreover, since ||x(t)|| ≤ ||η(t)|| + ||Υ(t)|| • || θ(t)|| ≤ ||η(t)|| + ν(Υ 0 )|| θ(t)||according to the definition of η and Claim 1, we deduce that there exist c 1 , c 2 > 0, which depend on X , M u , x0 , θ0 , Υ 0 , such that ||(x(t), θ(t))|| ≤ c 1 e -c2t ||(x(0), θ(0))||, which ends the proof.

C. When A(t, θ) is not affine in θ

We can take any t ≥ 0 here as the solutions we consider are defined for all positive times since x(t) ∈ X and in view of (2).

We can write that ρ ij (t, θ, θ 0 ) = O(||θ -θ 0 || 2 ) even though ρ ij depends on the time t, since A(t, θ) is assumed to be bounded with respect to the time, see Section II.

The detailed expressions of the matrices can be found in[START_REF] Postoyan | Local adaptive observer for linear timevarying systems with parameter-dependent state matrices[END_REF].θ, a modified version of the observer has been proposed, which ensures the approximate convergence to zero of the estimation errors.