Mai Gehrke

In Memoriam

Bjarni Jónsson

Canonical extensions -an algebraic approach to Stone duality

This note aims to highlight some of the conceptual contributions to duality theory made by Bjarni Jónsson through the theory of canonical extensions.

Introduction

In the fall of 1987 Bjarni Jónsson invited me to apply for a postdoctoral position at Vanderbilt University. It was clear from our conversation at the time of the interview that he was very interested in exploring the content and meaning of Stone and Priestley duality for bounded distributive lattices. As this was the topic of my PhD, I was very enthusiastic about the subject. By the time I came to Vanderbilt the following fall, Bjarni had received a preprint from Robert Goldblatt, which was to become Goldblatt's 1989 paper Complex Varieties [START_REF] Goldblatt | Varieties of complex algebras[END_REF]. As a first task, Bjarni asked me to lecture in the seminar on this preprint. I knew nothing about Boolean Algebras with Operators (the name alone seemed to me so long and clunky), nor about modal logic, and the results in the paper were many and difficult. I am afraid I gave some extremely uninspired talks drudging through the paper supplying very little insight. Bjarni sat in the front row and slept through a couple of months of such talks, waking up at the end of each session to ask the most pertinent and challenging questions. Nevertheless I was not able to see or understand any of it really. By late November, Bjarni changed tack and handed me a short write-up entitled Twenty Four Functors, with a cubic diagram of functors, some more defined than others, asking me if I would study these. Now this seemed very exciting to me and I understood the task immediately. I worked diligently for the next 18 months handing off quite a substantial packet of work in the spring of 1990 as my postdoc was ending.

The next thing I heard from Bjarni came to my office at New Mexico State University in the spring of 1994 in a brown manila envelope. It was an essentially complete manuscript of our 1994 paper on canonical extensions of distributive lattices with operators [START_REF] Gehrke | Bounded Distributive Lattices with Operators[END_REF]. I only knew this topic from my talks about Goldblatt's paper and, at first, I could not imagine what this might have to do with me. Nevertheless, as I started reading, I recognized many of the results from my twenty four functors work, including what was now phrased as the definition of the canonical extension of a bounded distributive lattice and the fact that canonical extension acted compositionally on order preserving maps whose canonical extensions are Scott continuous.

Feeling embarrassed by having a paper with the famous Bjarni Jónsson on a topic I knew nothing about, I started studying BAOs and DLOs in earnest and this is when our close collaboration, which lasted through the early 2000's, really started. Robert Goldblatt's contribution to this volume [START_REF] Goldblatt | Canonical extensions and ultraproducts of polarities[END_REF] is also about canonical extensions and it begins with an excellent historical overview of the subject, which I highly recommend for its comprehensive and succinct analysis of the development of the theory of canonical extensions. He uses a metaphor of Bjarni's of an 'acorn that has grown into a forest'. I have been working in this large and beautiful forest most of the time since my collaboration with Bjarni. Even my more recent work, e.g. with van Gool on sheaf representations [START_REF] Gehrke | Sheaves and duality[END_REF] or with Grigorieff and Pin on formal language theory [START_REF] Gehrke | Duality and Equational Theory of Regular Languages[END_REF][START_REF] Gehrke | Stone duality, topological algebra, and recognition[END_REF], owes very much to the algebraic view of topological duality, which Bjarni imparted to me through the work on canonical extensions.

This contribution is non-technical. Neither am I giving a historical overview of the development of the subject as this is done in [START_REF] Goldblatt | Canonical extensions and ultraproducts of polarities[END_REF]. What I would like to do with this paper is to highlight some of the conceptual contributions to duality theory made by Bjarni through the theory of canonical extensions. The paper is organized around four sections: Section 2 on the pointfree, algebraic approach to duality theory provided by Jónsson-style canonical extensions, Section 3 on canonical extension of maps as upper and lower semicontinuous envelopes, Section 4 on the functoriality of canonical extension, and finally Section 5 on the decomposition of Cartesian products into Boolean products and ultraproducts (also the fundamental ingredient in the famous Jónsson's Lemma) and its consequence in duality theory.

Canonical extension as pointfree Stone duality

A topological space is a pair, (X, O), where X is a set and O is a collection of subsets of X satisfying certain properties. In the pointfree study of topology, the set X is discarded and just the complete lattice O is kept. If the topological space is sufficiently separated (sober), then the points can be recovered from the lattice of opens. However, a characteristic feature of topology is precisely the tension between the discrete world of points and the only finite-intersections-closed collection of opens. The canonical extension point of view of topology corresponds to getting rid of the points without getting rid of the complete lattice P(X). Thus instead of studying the pair (X, O), one studies the embedding O → P(X). Further, in the case of Stone duality for Boolean algebras, since O is generated by the dual Boolean algebra B, the data O → P(X), in turn, amounts to giving an embedding B → P(X). This latter formulation brings the entire duality within the setting of lattice theory, making the proper translation of additional structure such as operations on the Boolean algebra more transparent. The key insight needed here is that this embedding may be uniquely characterized among the completions of B.

We and we want to transport the forgetful functor U across the dualities. Here Clop and St are the functors of Stone duality for Boolean algebras, assigning to a Boolean algebra its Stone space and to a Boolean Stone space its Boolean algebra of clopen sets, whereas the functors P and At are the functors of the discrete or Birkhoff duality between sets and complete and atomic Boolean algebras assigning to a set its powerset and to a complete and atomic Boolean algebra its set of atoms. Thus the dual of the forgetful functor U : BStone → Set from Boolean spaces to sets is transported by duality to the functor BA → BA + which sends a Boolean algebra B to [P • U • St](B), that is, the powerset of the set of ultrafilters of B. However, we want an intrinsic algebraic characterization rather than the one using the detour through Stone spaces and sets.

This problem was solved by Jónsson and Tarski in their famous paper [START_REF] Jónsson | Boolean algebras with operators I[END_REF] on Boolean Algebras with Operators (BAOs). They introduced what they called the perfect extension of a Boolean algebra. The name was later changed to the canonical extension, see [START_REF] Goldblatt | Canonical extensions and ultraproducts of polarities[END_REF] for more details. The notation for the canonical extension of A also changed from A + to A σ . More recently, through work with H. A. Priestley, we believe it more apt to denote the canonical extension of a Boolean algebra or bounded distributive lattice by A δ as the sigma and pi extensions are equal for objects. Incidentally, the translation of the forgetful functor BA + → BA, to a functor Set → BStone is given by the Stone-Čech compactification. In fact, these functor pairs, U : BA + BA : ( ) δ and U : BStone Set : β, are very nice categorical adjunctions which are dual to each other.

The key to the power of the Jónsson-Tarski canonical extension is precisely that one no longer needs to travel around the square. They gave an intrinsic characterization of the unit η B : B → B δ of the adjunction as a completion of B which is 'compact' and 'separating'. This description is a powerful tool for working with duality theory in an algebraic form and may be seen as a form of pointfree topology well before it appeared elsewhere. However, the 'separating' property refers to the existence of sufficiently many atoms in the extension, so to show it requires the Axion of Choice and thus the Jónsson-Tarski characterization does not satisfy the objective of later pointfree topology. That being said, as we observed later in joint work with Harding [START_REF] Gehrke | Bounded lattice expansions[END_REF], the separation property may be replaced by a density property making the canonical extension available in an Axiom-of-Choice-free setting. More details of this Jónsson-style pointfree topology (in the setting of distributive lattices and Heyting algebras) may be found in [7, Section 2].

Twenty four functors

Thus, what Bjarni asked me to do in the fall of 1988, was to study the following diagram.

BStone Set

Stone POS

BA BA + DL DL + U β U ( ) - ( ) σ U bulletitem
The back square should be a bounded distributive lattice version of the front one just discussed above. DL + is a bit harder to describe than BA + . It consists of those complete lattices which are completely distributive and have enough completely join irreducible or, because of the complete distributivity, this is equivalent to having enough completely join prime elements, see also [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Theorem 2.1] for other equivalent characterizations. Stone is the category of Stone or spectral spaces with proper maps [START_REF] Stone | Topological representations of distributive lattices and Brouwerian logics[END_REF]. The functors between the back and the front are inclusion and forgetful functors and Booleanization (which on the topological level corresponds to the patch topology construction). For example, the dual of the forgetful functor from DL + to DL is an ordered version of the Stone-Čech compactification first studied by Banaschewski in [START_REF] Banaschewski | Remarks on dual adjointness[END_REF]. While studying this cube, I rediscovered the results of Banaschewski and many more things about these twenty four functors and the categories on which they are defined. One problem that had not been solved before was the problem 'parallel' to the one solved by Jónsson-Tarski. Namely giving an intrinsic characterization of the dual of the forgetful functor Stone → POS, which sends the Stone dual of a distributive lattice to its underlying poset of points in the specialization order (formulated in terms of the Priestley duality this is even simpler as it just forgets the topology of the space while retaining the order component). For a duality theorist this was really quite an easy exercise and I never realized that I had done much of anything even though, in later years, Bjarni always asked me how I figured out what the canonical extension of a bounded distributive lattice should be. I always just told him it was easy given his cubic diagram and the Boolean case as input.

Naturality of the extension of additional operations

A key point, in the work of Jónsson and Tarski as well as in our subsequent work, is that canonical extension is not just applied to bounded lattice homomorphisms but to a much bigger clone of functions. The original construction in Jónsson-Tarski of the sigma extension [24, Definition 2.1 and Theorem 2.3] applies to arbitrary order preserving maps. There are however several problems with this:

• It is not general enough as we cannot extend order-reversing operations -or even worse -compositions, such as x → x ∨ ¬x, which may be neither order preserving or reversing. • It is too general because on the level of arbitrary order-preserving maps sigma extension is not functorial as it does not commute with composition. This point is a serious hindrance as it makes inductive proofs very difficult if not impossible. • The sigma extension, and its dual (known as the pi extension), are in general just two among (infinitely) many possible extensions.

The first two points are serious practical problems but Bjarni also cared very much about the third point, which he brought up often in our discussions. Even though he did not advocate the perspective of category theory per se, it has been my experience, again and again, in working with Bjarni, that it was centrally important for him that an object of study should be uniquely determined by an abstract property rather than just an ad hoc construction. Interestingly (and probably not surprisingly for mathematicians), the solution to the third bullet point was in the end closely connected to improvement on the two first issues. As a first step, we realized that sigma extension is functorial on the clone of all (necessarily order preserving) maps whose sigma extensions are Scott continuous: Theorem 3.1. [14, Theorem 4.3] Let g : A →B be order preserving and f :

B →C be such that f σ is Scott continuous. Then (f • g) σ = f σ • g σ .
This, along with the fact that the sigma extension of any operator is Scott continuous, is what allowed us to give a very simple (inductive) proof of the Jónsson-Tarski canonicity theorem in the setting of bounded distributive lattices with operators [14, Theorem 4.6].

Topology on canonical extensions

In the summer of 1998, while I was visiting Bjarni and Harriet at their cabin in Minnesota, we finally unblocked the question of generalizing beyond order preserving maps by describing in which sense the sigma and pi extensions of maps are also natural (or 'canonical'). The answer involves putting topology on canonical extensions. Once we realized this, Bjarni was in quite a state of puzzlement for a while: He told me that it had been suggested to him already in the late 1940s that 'adding topology' might be the way to understand extension of maps. However, viewing canonical extensions as topological algebras does not work and had abandoned the idea. One has to consider distinct topologies on the domains and codomains of operations, and, as we shall see, this explains the great complexity of questions of canonicity.

The answer comes from a classical concept from real analysis! Given a topological space (X, O), a complete lattice L and a function f : A → L defined on a dense subset A ⊆ X, the upper and lower semicontinuous envelopes of f are defined by

lim(f )(x)= { f [U ∩ A]|x ∈ U ∈ O} lim(f )(x)= { f [U ∩ A]|x ∈ U ∈ O}.
Under mild assumptions, which are satisfied whenever L is the canonical extension of a distributive lattice, these are, respectively, the greatest function (in the pointwise order) which is below f on A and which is upper semicontinuous and the least function which is above f on A and which is lower semicontinuous [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Theorem 2.11]. Upper and lower semicontinuity refer to the two halves of the interval topology on the poset L. Definition 3.2. Let P be a poset. The upper, respectively, lower topology on P is generated by the complements of principal downsets, respectively, upsets, and the interval topology is the join of these two topologies. That is,

ι ↑ = <(↓p) c | p ∈ P >, ι ↓ = <(↑p) c | p ∈ P >, and ι = ι ↑ ∨ ι ↓ .
A function from a topological space to P is upper, respectively, lower semicontinuous provided it is continuous when viewed as a function into the topological space (P, ι ↑ ), respectively, (P, ι ↓ ).

The standard topology of the reals is the interval topology for the usual order on the reals. However, for the interval topology, a 'best continuous approximant' of a function does not exist in general. The fact that such canonical upper and lower envelopes exist for the two halves of the interval topology often plays an important role in analysis. With the appropriate topology on the domain one may see the sigma and the pi extensions of a map between lattices as the upper and lower semicontinuous envelopes of the map, respectively. Definition 3.3. Let A be a bounded distributive lattice and η : A → A δ its canonical extension. An element x ∈ A δ is called a filter element provided it is in the complete meet closure of A in A δ , while an element y ∈ A δ is called an ideal element provided it is in the complete join closure of A in A δ . The strong topology on A δ is generated by the sets of the form [x, y] = {u ∈ A δ | x ≤ u ≤ y} with x ranging over filter elements of A δ and y ranging over ideal elements. This topology is not akin to the interval topology since the basic opens are closed intervals.

One can show that A δ equipped with the interval topology is a Priestley space (in fact, for these lattices, the interval topology is equal to the double Scott topology and to the Lawson topology and this topology is the unique topology for the given order making it into a Priestley space, see the discussion leading up to [13, Theorem 2.8] and the further references given there). The strong topology on A δ contains the interval topology properly (for A infinite) thus it follows that the strong topology on A δ makes it into a totally order disconnected space which is not compact. Further, one can show that η : A → A δ embeds A in the space as its set of isolated points and the image is topologically dense in A δ , again see [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Theorem 2.8]. This is very convenient as it implies that any function with domain A is continuous with respect to the strong topology on A δ and has unique upper and lower semicontinuous extensions to A δ . This is central in showing that lim(f ) and lim(f ) are always extensions of f . For further details, see [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Section 2]. See also [START_REF] Gehrke | A View of Canonical Extension[END_REF]Section 4] for details on how this works for lattices that are not necessarily distributive.

Canonical extensions of maps, at least if they are operators or dual operators, are of interest because they allow us to define Stone duals of these maps. In particular, consider a bounded distributive lattice A with an additional nary operation f which is a normal operator. That is, f preserves finite joins in each coordinate. In this situation, one can show that the sigma extension of f , obtained as f σ = lim(f ), is a complete operator. That is, it is completely join preserving in each coordinate. Invoking the connection with Stone duality as discussed in Section 2.1, we know that A δ ∼ = U(X, ≤) where X is the dual space of A with its specialization (or Priestley) order. Now identifying A δ with U(X, ≤) and using the fact that f σ is a complete operator, it follows that f σ is completely determined by its restriction to the tuples whose coordinates are principal upsets ↑x for x ∈ X; these being the completely join irreducible elements of A δ . Thus, defining an (n+1)-ary relation on X by (y, x 1 , . . . , x n ) ∈ R if and only if ↑y ≤ f σ (↑x 1 , . . . , ↑x n ), one may show that f σ is given by inverse image under R, i.e.

f σ : (U 1 , . . . , U n ) → R -1 [U 1 × • • • × U n ].
It follows that we can recapture f from the relation R. In the case of Boolean algebras with operators, this is the Jónsson-Tarski duality, exploited in their work on canonical extensions [START_REF] Jónsson | Boolean algebras with operators I[END_REF] in order to obtain set representations of various classes of BAOs, and foreshadowing Kripke semantics of modal logic about a decade before it was introduced. For further details, see [START_REF] Goldblatt | Canonical extensions and ultraproducts of polarities[END_REF] in this volume and [START_REF] Goldblatt | Mathematical modal logic: a view of its evolution[END_REF]Section 3.3].

Syntactic conditions for canonicity

An important issue concerning canonical extension is which properties are preserved when moving from a lattice or Boolean algebra with additional operations to its canonical extension. When a property is preserved, it is said to be canonical. It follows that the dual of a canonical property is purely order theoretic and thus independent of topology. Accordingly, if a class is given by canonical properties then canonical extension provides concrete set-theoretic representation within the class.

An important algebraic condition closely related to canonicity is the question of when canonical extension for maps commutes with composition of maps. In fact, as we will explain below, in order to be able to give inductive proofs of canonicity for large classes of varieties, it is particularly important to identify whole clones (classes containing all and closed under composition) with this property.

Consider the diagrams

A B A δ B δ C D C δ D δ h f g k h σ f σ g σ k σ
Given that the first diagram commutes, we want to be able to conclude that the second diagram also commutes. An obvious way to prove this is to show that

g σ f σ = (gf ) σ = (kh) σ = k σ h σ .
Here the second equality holds by hypothesis. The question is therefore, when canonical extension distributes over composition, i.e., under what conditions it is the case that (gf ) σ = g σ f σ ? The topological description of sigma (and pi) extensions clearly exhibits the problem with composition since the topology on the domain (the strong topology) and on the codomain (the Scott (or dual Scott) topology) are different thus causing problems when we want to compose. In fact the Scott and dual Scott topologies are contained in the strong topology. Nevertheless, the topological description of sigma (and pi) extensions, points us towards two special cases where things go better:

(1) when the extension is continuous with respect to the Scott (or dual Scott) topology on both the domain and codomain; (2) when it is continuous with respect to the strong topology on both the domain and codomain.

The situation (1) covers the case of operators as discussed above, while (2) is essential for dealing with join or meet preserving or reversing maps. Combining these two special cases in various ways leads to Sahlqvist-type theorems. This algebraic approach to Sahlqvist theorems, using algebraic induction and based on the properties of basic operations and the syntactic shape of the terms involved, has been studied extensively, starting with Bjarni's paper [START_REF] Jónsson | On the canonicity of Sahlqvist identities[END_REF]. See among others [START_REF] Gehrke | A Sahlqvist theorem for distributive modal logic[END_REF][START_REF] Dunn | Canonical extensions and relational completeness of some substructural logics[END_REF][START_REF] Conradie | Unified Correspondence[END_REF][START_REF] Conradie | Constructive Canonicity for Lattice-Based Fixed Point Logics[END_REF].

Functoriality and Bjarni's PH property

As we have seen above, the view of canonical extensions of maps as upper and lower topological envelopes leads to a natural dichotomy, depending on whether one considers maps that are continuous with respect to a weaker topology on the domain or with respect to a stronger one on the codomain. Also, as discussed above, the first option, which leads one to focus on sigma extensions that are Scott continuous, may be seen to be responsible for the original Jónsson-Tarski canonicity result as well as the positive version from [START_REF] Gehrke | Bounded Distributive Lattices with Operators[END_REF] and the subsequent work on Sahlqvist theorems. The second choice, of imposing continuity with respect to a stronger topology on the codomain, leads to Bjarni's PH (as in 'Preserving Homomorphisms') property, discussed in this section, and through that to the model theoretic results on canonicity, which we will discuss in Section 5.

The PH property comes about when we ask whether canonical extension is functorial. First of all, there is quite a bit of good news. Canonical extension is functorial even on the category of bounded (not-necessarily-distributive) lattices with homomorphisms. In fact, the upper and lower envelopes of homomorphisms agree thus providing the unique doubly continuous extensions, which are in fact complete homomorphisms. Furthermore, these extensions are continuous both with respect to the weaker topology on both domain and codomain and with respect to the stronger topology on both. In fact, in the setting of the category of bounded distributive lattices with their homomorphisms (and also for Boolean algebras), canonical extension is not only a functor but a reflection into the category DL + (respectively BA + ). However, if we are interested in distributive lattices or Boolean algebras with additional operations, so called Distributive Lattice Expansions (DLEs), then canonical extension is not functorial in general. This may be seen with a simple example due to Bjarni: Let L be a bounded chain with a bounded subchain A so that both A and its complement are dense in L (e.g. the unit interval with A the set of rational points). Further, consider as a unary operation on L the characteristic function of A. Then the embedding of A equipped with the constant function 1 into this algebra does not extend to a DLE homomorphism as it does not commute with the sigma extensions of the additional operations.

The fact that the above counterexample is provided by an embedding is no coincidence: by the First Isomorphism Theorem, every homomorphism between abstract algebras factors into a quotient map followed by an embedding and one may show that the extensions of surjective homomorphisms are always homomorphisms. Thus, if functoriality of canonical extension is violated, it is violated by the associated embedding. The characteristic insight of Bjarni is that this allows us to see functoriality as a local property at an algebra: a DLE has the PH property provided the canonical extension of the inclusion of each of its subalgebras is a subalgebra inclusion. Then it follows from the factorization of morphisms and the fact that quotients are well behaved that canonical extension on a category of DLEs with their homomorphisms is functorial if and only if each individual algebra in the class has the PH property.

In our paper [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF], we showed that large classes of DLEs have the property PH. In particular, any DLE whose basic operations are monotone, that is, preserve or reverse order in each coordinate (it may preserve in some and reverse in others), has the PH property. Also, any DLE whatsoever which lies in a finitely generated variety of DLEs has the PH property. Much of this analysis goes through for lattices but things are more complicated -mainly because the various topologies (in particular, the one generated by the principal upsets of completely join irreducible elements, the Scott topology, and the upper topology) no longer are equal. However, some results are provided in [START_REF] Gehrke | Bounded lattice expansions[END_REF][START_REF] Gehrke | A View of Canonical Extension[END_REF] and it was studied in the PhD thesis of Jacob Vosmaer [START_REF] Vosmaer | Logic, Algebra and Topology. Investigations into canonical extensions, duality theory and point-free topology[END_REF].

Boolean products and model theoretic conditions for canonicity

The most fundamental model theoretic result about algebras is Birkhoff's variety theorem. In a formula this may be stated as follows

V(K) = HSP (K).
Here K is a class of similar algebras and V(K) is the class of all algebras satisfying all the equations simultaneously satisfied by all the algebras in K. H, S, and P are the operators on classes of similar algebras taking all homomorphic images, isomorphic copies of subalgebras, and isomorphic copies of Cartesian products, respectively. An important fact that Bjarni often has used to great advantage in his work, is that one can further decompose the operator P into P = P B P µ , where P B takes all possible Boolean products of algebras from a class K and P µ takes all possible ultraproducts of algebras from K. This decomposition comes about as follows. Let

h : A ∼ = Π i∈I B i
be a direct decomposition of an algebra A, let X be the Boolean space of all ultrafilters on I, that is, X = β(I), the Stone-Čech compactification of the discrete space on I. For each x ∈ X let C x = Π i∈I B i /x. That is, C x is the ultraproduct obtained from the product Π i∈I B i by considering two tuples as equivalent if they are equal on a subset of I which belongs to the ultrafilter x. For a ∈ A and x ∈ X, let h (a)(x) = h(a)/x. Then the map

h : A → Π x∈X C x
is a Boolean product decomposition of A. A weak Boolean product decomposition is simply a sheaf representation over a Boolean space, and thus a Boolean product decomposition is a special sheaf representation over a Boolean space. For more details see [START_REF] Burris | A course in universal algebra[END_REF] or more specifically [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Section 3.3].

In [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF], we obtained a canonical extension formulation of a result on Boolean products from my PhD dissertation: The canonical extension of a weak Boolean product decomposition is the full Cartesian product of the canonical extensions of the factors [START_REF] Gehrke | The order structure of Stone spaces and the TD separation axiom[END_REF]. That is, This result, combined with the decomposition P = P B P µ yields a number of model theoretic results on the PH property, on canonicity, and on sheaf representations. The first of these are in [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]. In particular, we showed that every distributive lattice with additional operations (DLE) that is either monotone (i.e. each additional operation is either order preserving or reversing in each coordinate) or smooth (i.e. the sigma and pi extensions of each additional operation agree) has the property PH, see [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Theorem 3.24 and Theorem 3.26]. We showed that the property PH is preserved by a number of constructions on monotone DLEs such as homomorphic images [13, Theorem 3.12], subalgebras [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Theorem 3.11], binary joins [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Theorem 3.27], and weak Boolean products [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Theorem 3.16]. Finally, we showed that a direct product has the PH property if and only if all the ultraproducts on it have it, and thus ultraproducts is the one badly behaved construction in the lot. This leads to a number of canonicity results [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Section 4.1], including what one may see as an algebraic version of one of the main results of [START_REF] Goldblatt | Varieties of complex algebras[END_REF].

The line of inquiry into model theoretic conditions for canonicity started, at least for Bjarni and me, with Robert Goldblatt's paper [START_REF] Goldblatt | Varieties of complex algebras[END_REF], which, in 1989, as Hilary Priestley has put it 'made the wall come down' between the model theory of modal logic and the study of canonicity via lattices and duality. Since then a great number of contributions in this direction have been made, such as [START_REF] Goldblatt | Erdös graphs resolve Fine's canonicity problem[END_REF][START_REF] Gehrke | MacNeille completions and canonical extensions[END_REF][START_REF] Hodkinson | Hybrid formulas and elementarily generated modal logics[END_REF], the most recent of which is Goldblatt's contribution to this volume [START_REF] Goldblatt | Canonical extensions and ultraproducts of polarities[END_REF]. For further discussion and a historical overview we refer the reader to [START_REF] Goldblatt | Canonical extensions and ultraproducts of polarities[END_REF].

A somewhat different direction of generalization of this work is the duality theoretic investigation of sheaf representations of universal algebras and DLEs in particular. The fact that the canonical extension of a weak Boolean product is the Cartesian product of the canonical extensions of the factors [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF]Theorem 3.15] may be seen as a first step in this direction, while [START_REF] Gehrke | Sheaves and duality[END_REF] is the most recent contribution in this still active direction.

Theorem 5 . 1 .

 51 [START_REF] Gehrke | Bounded Distributive Lattice Expansions[END_REF] Theorem 3.15] Supposeg : B → Π x∈X C xis a weak Boolean product decomposition of the DLE B. If all the induced homomorphisms g x : B → C x are preserved by canonical extension, theng : B σ Π x∈X C σ xwhere, for all b ∈ B and x ∈ X, g (b)(x) = g σ x (b). Of course a similar result holds for canonical extensions using pi extensions.

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No.670624).


Qinghua Zhang
email: qinghua.zhang@inria.fr

Fouad Giri
email: fouadgiri@unicaen.fr

Tarek Ahmed-Ali
email: tarek.ahmed-ali@ensicaen.fr

Regularized Adaptive Observer to Address Deficient Excitation

Keywords: Adaptive observer, regularization, persistent excitation, linear time varying (LTV) systems, joint state-parameter estimation

Adaptive observers are recursive algorithms for joint estimation of both state variables and unknown parameters. Usually some persistent excitation (PE) condition is required for the convergence of adaptive observers. However, in practice, it may happen that the PE condition is not satisfied, because the available sensor signals do not contain sufficient information for the considered recursive estimation problem, which is ill-posed. To remedy the lack of PE condition, inspired by typical methods for solving ill-posed inverse problems, this paper proposes a regularized adaptive observer for general linear time varying (LTV) systems. Two regularization terms are introduced in both state and parameter estimation recursions, in order to preserve the state-parameter decoupling transformation involved in the design of the adaptive observer. Like in typical ill-posed inverse problems, regularization implies an estimation bias, which can be reduced by using prior knowledge about the unknown parameters.

INTRODUCTION

Dynamic systems in various engineering fields are often described in state-space form, with only part of the state variables directly accessible through sensor instruments. State estimation is a thus a common task for different engineering purposes. Moreover, some model parameters may be unknown a priori, due to variations in the production of system components, or because of some evolution reflecting aging or degradation of the underlying dynamic system. Adaptive observers are recursive algorithms for joint estimation of both state variables and unknown parameters, based on available sensor measurements. The design of adaptive observers is, since a few decades, an active research area. Early contributions about linear time invariant (LTI) systems go back to the seventies [START_REF] Kreisselmeier | Adaptive observers with exponential rate of convergence[END_REF][START_REF] Ioannou | Adaptative systems with reduced models[END_REF], then some classes of nonlinear systems are studied (see e.g. [START_REF] Bastin | Stable adaptive observers for nonlinear time varying systems[END_REF][START_REF] Marino | Nonlinear control design. Information and system sciences[END_REF][START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF][START_REF] Besançon | On adaptive observers for state affine systems[END_REF][START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF]) while the results about LTI and linear time varying (LTV) systems continue to be completed [START_REF] Zhang | Adaptive Kalman filter for actuator fault diagnosis[END_REF]. In most reported results about adaptive observer design, including those recalled above, in addition to the traditional observability condition for state estimation, a persistent excitation (PE) condition is required to ensure the convergence of the algorithms. Roughly speaking, the PE condition implies sufficiently rich information in sensor signals guaranteeing the well-posedness of the joint stateparameter recursive estimation problem. It is well known in classical adaptive estimation problems that PE is essential for parameter estimation [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF][START_REF] Shimkin | Persistency of excitation in continuous-time systems[END_REF][START_REF] Narendra | Stable adaptive systems[END_REF][START_REF] Astrom | Adaptive control[END_REF]. However, in practice, it may happen that the PE condition associated to an adaptive observer is not satisfied, and this situation may persist for different adaptive observers, because intrinsically the available sensor signals do not contain sufficient information for the considered recursive estimation problem. In other words, the considered recursive estimation problem is ill-posed. What can be done in this case? The easiest answer would be: it is not possible to apply adaptive observers. However, it is well known that the so-called inverse problems in various engineering fields are often ill-posed [START_REF] Chavent | Nonlinear Least Squares for Inverse Problems[END_REF], yet practical solutions are frequently implemented and applied. Is it possible to do something similar for adaptive observers? In this paper, like in typical solutions to ill-posed inverse problems, regularization will be introduced into the adaptive observer initially presented in [START_REF] Zhang | Adaptive observer for multiple-inputmultiple-output (MIMO) linear time varying systems[END_REF] for general multi-input multi-output (MIMO) LTV systems. The regularization technique is also known in robust adaptive control, mainly focused on LTI systems [START_REF] Ioannou | Robust Adaptive Control[END_REF]. Because the considered LTV system adaptive observer is based on a state-parameter decoupling technique, instead of explicitly minimizing some criterion, the proposed regularization is not introduced as a penalty term within a minimized criterion. Moreover, the introduced regularization must preserve the decoupling transformation involved in the design of the adaptive observer. It is widely acknowledged that the introduction of regularization in estimation problems generally leads to biased estimates. This is the price to pay to solve such ill-posed problems. Nevertheless, prior knowledge about model parameters, if available, can be used to reduce the bias. These aspects of regularization will be discussed in this paper after the presentation of the proposed regularized adaptive observer.

PROBLEM FORMULATION

Throughout this paper, for any vector v, its Euclidean norm is denoted by v . For any matrix M , its matrix norm induced by the Euclidean vector norm is denoted by M . For a symmetric matrix M , the inequality M > 0 (or M ≥ 0) means M is (semi)-positive definite. This paper will consider continuous-time MIMO LTV systems in the form of ẋ

(t) = A(t)x(t) + B(t)u(t) + Φ(t)θ + w(t) (1a) y(t) = C(t)x(t) + v(t)
(1b) where x(t) ∈ R n is the state, u(t) ∈ R q the input (control), y(t) ∈ R m the output, ẋ(t) = dx(t)/dt, θ ∈ R p the unknown constant parameter vector, A(t), B(t), C(t), Φ(t) are appropriate size matrix-valued functions of the time t, w(t) ∈ R n and v(t) ∈ R m represent uncertainties in the state and output equations. At the initial instant t 0 , the initial state x(t 0 ) is unknown. In order to keep the problem formulation and analysis within the framework of ordinary differential equations (ODE), without resorting to the continuous time stochastic system theory, it is assumed that the uncertainties w(t) and v(t) are unknown arbitrary bounded functions of t. The system matrices A(t), B(t), C(t) are typically constant (case of LTI systems), periodic, or depending on some known scheduling parameter (LPV systems), whereas the matrix Φ(t) often has a different nature, typically filled with exogenous signals (inputs), sometimes with injected outputs. In existing adaptive observer design methods, it is usually assumed that the signals contained in Φ(t) involve sufficiently rich variations so that the parameter vector θ can be estimated by a recursive algorithm. Such an assumption is usually known as a persistent excitation (PE) condition, as the one recalled in the next section (see Assumption 3). The lack of excitation may imply that the unknown parameter vector θ is not uniquely determined by the available information assumed above. In other words, the PE condition ensures that the recursive estimation of the parameter vector θ is a well-posed problem. In practice, it may happen that the PE condition is not satisfied. The purpose of this paper is to jointly estimate the state x(t) and the parameter θ from the input u(t), the output y(t) and the time varying matrices A(t), B(t), E(t), C(t), Φ(t), despite deficient excitations. Like in the study of typical inverse problems, regularization will be used to address the lack of PE condition.

Assumption 1. (boundedness and continuity). A(t), B(t), C(t), Φ(t) are bounded matrix-valued piecewise continuous functions of t, and u(t), w(t), v(t) are bounded vector-valued piecewise continuous functions of t. While A(t), B(t), C(t), Φ(t), u(t) are known or available from sensor signals, w(t), v(t) are unknown.

2 Assumption 2. (observability). The pair [A(t), C(t)] is uniformly completely observable, in the sense of the uniform positive definiteness of the observability Gramian, as defined in [START_REF] Kalman | New methods in Wiener filtering theory[END_REF]. 2

RECALLING A CONVENTIONAL ADAPTIVE OBSERVER

This section recalls a general adaptive observer for MIMO LTV systems, whose convergence has been established under a PE condition. It will serve as the basis for the results in the next section addressing deficient excitations by means of regularization.

The adaptive observer for the LTV system (1) proposed in [START_REF] Zhang | Adaptive observer for multiple-inputmultiple-output (MIMO) linear time varying systems[END_REF] is in the form of ẋ

(t) = A(t)x(t) + B(t)u(t) + Φ(t) θ(t) + K(t)[y(t) -C(t)x(t)] + Υ(t)ΓΥ T (t)C T (t)[y(t) -C(t)x(t)] (2a) θ(t) = ΓΥ T (t)C T (t)[y(t) -C(t)x(t)] (2b) Υ(t) = [A(t) -K(t)C(t)]Υ(t) + Φ(t), (2c) with the initialization x(t 0 ) = x0 (3a) θ(t 0 ) = θ0 (3b) Υ(t 0 ) = 0 n×p , (3c)
where x(t) ∈ R n is the state estimate, θ(t) ∈ R p the parameter estimate, Υ(t) ∈ R n×p a matrix of auxiliary variables, K(t) ∈ R n×m the state estimation gain matrix, Γ ∈ R p×p the parameter gain matrix, x0 ∈ R n and θ0 ∈ R p are initial values of the state and parameter estimates, and 0 n×p is the n × p zero matrix initializing the auxiliary matrix Υ(t).

Remark 1. The last term in equation ( 2a) is equal to Υ(t) θ(t). It compensates for the error due to the replacement of the true θ by its estimate θ(t) in equation (2a). This extra term plays an important role in the convergence analysis of the adaptive observer. See [START_REF] Zhang | Adaptive observer for multiple-inputmultiple-output (MIMO) linear time varying systems[END_REF] for more details. 2

In general, the state estimation gain K(t) is chosen such that, if the term Φ(t)θ was omitted in the state equation (1a), K(t) would lead to a convergent state observer in the form of ẋ

(t) = A(t)x(t) + B(t)u(t) + K(t)[y(t) -C(t)x(t)]. (4)
In the time invariant (LTI) case, various methods for designing such observer gains are available. General time varying (LTV) systems are less well studied. In this case, observer gain design can be based on the well-known Kalman filter or on the Kalman-like observer [START_REF] Besançon | On adaptive observers for state affine systems[END_REF]. Despite the deterministic framework considered in this paper, the Kalman gain will be used for K(t), as an general LTV observer gain ensuring the convergence of the observer (4) (see Remark 2 below). This gain K(t) is computed as

Ṗ (t) = A(t)P (t) + P (t)A T (t) + Q(t) -P (t)C T (t)R -1 (t)C(t)P (t) (5) K(t) = P (t)C T (t)R -1 (t) (6
) with a positive definite initial matrix P (t 0 ). In the Kalman filter literature, usually P (t) ∈ R n×n is known as the covariance matrix of the state estimate, Q(t) ∈ R n×n and R(t) ∈ R m×m are respectively the state and output noise covariance matrices. In this paper, the considered problem is formulated in a deterministic framework, therefore Q(t), R(t) are treated as tuning parameters, typically chosen as constant matrices, both symmetric positive definite.

Remark 2. It is known [START_REF] Kalman | New methods in Wiener filtering theory[END_REF][START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF] that, under Assumption 2 (uniform complete observability), and under the uniform complete controllability assumption (which is trivially satisfied when Q(t) is chosen to be a positive definite matrix), the Kalman gain K(t) is bounded and ensures the exponential stability of the error dynamics of the state observer (4). More specifically, the Kalman gain K(t) ensures that the homogenous ODE ξ

(t) = [A(t) -K(t)C(t)]ξ(t), (7) with ξ(t) ∈ R n , is exponentially stable. 2
Assumption 3. (PE, assumed in this section only). The matrix Φ(t) contains sufficient variations such that the matrix Υ(t), which is driven by Φ(t) through (2c), satisfies

t+T t Υ T (τ )C T (τ )C(τ )Υ(τ )dτ ≥ αI p , (8)
for some positive constants T and α, and for all t ≥ t 0 . 2 Remark 3. The auxiliary matrix Υ(t) contains signals obtained by linearly filtering Φ(t) through the linear filter (2c). Therefore, Assumption 3 is indeed about the properties of Φ(t), which is usually filled with exogenous signals (inputs), sometimes with injected outputs. For LTI systems (with constant A, C, K), equation ( 2c) becomes an LTI filter of Φ(t). In this case the PE assumption can be expressed in terms of the frequency components of Φ(t).

It is shown in [START_REF] Zhang | Adaptive observer for multiple-inputmultiple-output (MIMO) linear time varying systems[END_REF] that, under Assumptions 1, 2 and 3, if K(t) is chosen such that the homogenous system (7) is exponentially stable, then, with any initial state estimate x0 and any initial parameter estimate θ0 , the state and parameter estimation errors of the adaptive observer (2) both converge exponentially to zero, if the uncertainties w(t), v(t) are omitted (w(t) = 0 and v(t) = 0). Without omitting w(t), v(t), which are assumed bounded (Assumption 1), the state and parameter estimation errors remain bounded.

REGULARIZED ADAPTIVE OBSERVER

In what follows, no PE condition will be assumed. In particular, let us forget Assumption 3. In order to overcome the lack of PE condition, the adaptive observer ( 2) is modified as

ẋ(t) = A(t)x(t) + B(t)u(t) + Φ(t) θ(t) + K(t)[y(t) -C(t)x(t)] + Υ(t)ΓΥ T (t)C T (t)[y(t) -C(t)x(t)] -Υ(t)ΓΛ[ θ(t) -θ] (9a) θ(t) = ΓΥ T (t)C T (t)[y(t) -C(t)x(t)] -ΓΛ[ θ(t) -θ] (9b) Υ(t) = [A(t) -K(t)C(t)]Υ(t) + Φ(t),
(9c) with the same initial conditions as in (3) and the Kalman gain K(t) as in (6). The only modifications w.r.t. the adaptive observer (2) are the extra (blue) terms -Υ(t)ΓΛ[ θ(t)-θ] in (9a) and -ΓΛ[ θ(t) -θ] in (9b), with a symmetric positive definite matrix Λ ∈ R p×p and a vector θ ∈ R p . The positive definite matrix Λ controls the degree of regularization added in the algorithm. The vector θ is a guess of the true value of θ. Naturally, θ can be set to θ0 , the initial parameter estimate used in (3b), but this choice is not necessary. It is possible to take θ = 0 if no prior knowledge about θ is available. Usually, to regularize an estimation algorithm designed by minimizing some criterion, a chosen penalty term is added to the minimized criterion. In contrast, in the proposed algorithm (9), two regularization terms have been added in (9a) and in (9b), in order to preserve the state-parameter decoupling effect of the transformation defined later in (12), as initially introduced in [START_REF] Zhang | Adaptive observer for multiple-inputmultiple-output (MIMO) linear time varying systems[END_REF]. This transformation will be used in Proposition 2. Before analyzing the error dynamics of this regularized adaptive observer, let us first ensure the boundedness of the auxiliary variable Υ(t) ∈ R n×p . Proposition 1. Under Assumptions 1 and 2, the auxiliary variable Υ(t) driven by Φ(t) through (9c) is bounded. 2

Proof. The Kalman gain K(t) is bounded and ensures the exponential stability of the homogenous system (7), according to Remark 2. Then, based on Lemma 1 (in the appendix at the end of this paper), the auxiliary variable Υ(t) driven by bounded Φ(t) through ( 9c) is bounded. 2 The behavior of the state and parameter estimation errors, namely x(t) x(t) -x(t) (10) θ(t) θ -θ(t), (11) will be analyzed in this paper. As an intermediate step, let us define the decoupling transformation, as originally introduced in [START_REF] Zhang | Adaptive observer for multiple-inputmultiple-output (MIMO) linear time varying systems[END_REF], η(t) x(t) -Υ(t) θ(t), ( 12) and analyze the behavior of this transformed error variable η(t).

Proposition 2. The transformed error variable η(t) defined in ( 12) is governed by the ODE η(t) = [A(t) -K(t)C(t)]η(t) + w(t) -K(t)v(t). (13) Moreover, under Assumptions 1 and 2, the homogeneous part of this ODE is exponentially stable, and there exists a positive constant a such that, for sufficiently large t so that the effect of the initial value η(t 0 ) is negligible,

η(t) ≤ a ( w + Ǩ v), ( 14
)
where w, v and Ǩ are respectively upper bounds of w(t) , v(t) and K(t) . 2

Proof. Rewrite (9a) as

ẋ(t) = A(t)x(t) + B(t)u(t) + Φ(t) θ(t) + K(t)[y(t) -C(t)x(t)]
+ Υ(t) θ(t).

(15) Then it is straightforward to derive from ( 1) and ( 15) that

ẋ(t) = [A(t) -K(t)C(t)]x(t) + Φ(t) θ(t) -Υ(t) θ(t) + w(t) -K(t)v(t).
(16) Some simple computations with the variable η(t) defined in ( 12) then lead to

η(t) = [A(t) -K(t)C(t)]η(t) + [A(t) -K(t)C(t)]Υ(t) + Φ(t) -Υ(t) θ(t) -Υ(t) θ(t) -Υ(t) θ(t) + w(t) -K(t)v(t) (17) = [A(t) -K(t)C(t)]η(t) -Υ(t) θ(t) -Υ(t) θ(t) + w(t) -K(t)v(t).
(18) where the important simplification {• • • } = 0 was based on (9c). The true parameter vector θ is assumed constant, therefore θ = 0, and θ(t) = θθ(t) = -θ(t). Then,

-Υ(t) θ(t) -Υ(t) θ(t) = -Υ(t)[ θ(t) -θ(t)] = 0, ( 19
) and equation ( 18) becomes ( 13), which is now proved. It then follows from Remark 2 that, under Assumptions 1 and 2, the homogenous part of ( 13), which is identical to (7), is exponentially stable. Now apply Lemma 1 (in the appendix of this paper) to ( 13), whose homogenous part is already shown to be exponentially stable with some α and β. The terms w(t) -K(t)v(t) correspond to U (t), and therefore w + Ǩ v to γ. For sufficiently large t, the exponentially vanishing last term in ( 42) is negligible, and the remaining term αγ/β corresponds to the right hand side of ( 14), which is then proved with a = α/β. 2

Proposition 3. The parameter estimation error θ(t) satisfies the ODE θ

(t) = -Γ[Υ T (t)C T (t)C(t)Υ(t) + Λ] θ(t) -ΓΥ T (t)C T (t)C(t)η(t) -ΓΥ T (t)C T (t)v(t) + ΓΛ(θ -θ). (20) 2 Proof. θ(t) = θ -θ(t) = 0 -θ(t) (21) = -ΓΥ T (t)C T (t)[y(t) -C(t)x(t)] + ΓΛ[ θ(t) -θ] (22)
Notice that, notably following (12),

y(t) -C(t)x(t) = C(t)x(t) + v(t) (23) = C(t)[Υ(t) θ(t) + η(t)] + v(t), (24) then θ
(t) = -ΓΥ T (t)C T (t)C(t)Υ(t) θ(t) -ΓΥ T (t)C T (t)C(t)η(t) -ΓΥ T (t)C T (t)v(t) + ΓΛ[ θ(t) -θ]. (25)
Rewrite the last term as

ΓΛ[ θ(t) -θ] = -ΓΛ[θ -θ(t) -θ + θ] (26) = -ΓΛ θ(t) + ΓΛ(θ -θ), (27)
and combine -ΓΛ θ(t) with the other homogeneous term, then (20) is proved. 2 Now let us study the stability of the homogeneous part of the ODE (20).

(t) = -Γ[Υ T (t)C T (t)C(t)Υ(t) + Λ]ϑ(t), (28)
with ϑ(t) ∈ R p , is exponentially stable. More specifically, let Ξ(t, t 0 ) ∈ R p×p be the associated state transition matrix such that ϑ(t 2 ) = Ξ(t 2 , t 1 )ϑ(t 1 ) for all time instants t 1 , t 2 ≥ t 0 , then

Ξ(t, t 0 ) ≤ γ max γ min e -λminγmin(t-t0) (29)
for all t ≥ t 0 , with γ max and γ min denoting respectively the largest and the smallest singular values of Γ, and λ min the smallest singular value of Λ. 2

Proof. The matrix Υ T (t)C T (t)C(t)Υ(t) is always positive semidefinite, then Υ T (t)C T (t)C(t)Υ(t) + Λ ≥ Λ ≥ λ min I p , (30)
where λ min > 0 is the smallest singular value of the positive definite matrix Λ. Then, according to Lemma 2 in the appendix of this paper, the homogeneous ODE ( 28) is exponentially stable, and the associated state transition matrix satisfies (29).

2 Proposition 5. Under Assumptions 1 and 2, for sufficiently large t so that the effect of the initial θ(t 0 ) is negligible, there exist positive constants c 1 , c 2 , c 3 such that the parameter estimation error θ(t) is upper bounded as θ(t)

≤ c 1 w + c 2 v + c 3 θ -θ , (31)
with w and v denoting respectively upper bounds of w(t) and v(t) .

2

Proof. The homogeneous part of the ODE (20), namely (28), is exponentially stable, with its state transition matrix satisfying (29).

Apply Lemma 1 to (20) with

α = γ max γ min , β = λ min γ min , (32)
U (t) = -ΓΥ T (t)C T (t)C(t)η(t) -ΓΥ T (t)C T (t)v(t) + ΓΛ(θ -θ), (33)
then, for sufficiently large t, so that the effect of the initial θ(t 0 ) is negligible,

θ(t) ≤ γmax γmin λ min γ min Ǔ , ( 34
)
where Ǔ is an upper bound of U (t) expressed in (33). Basic properties of the Euclidean vector norm and of the induced matrix norm lead to

U (t) ≤ Γ Υ(t) C(t) 2 η(t) + Γ Υ(t) C(t) v(t) + Γ Λ θ -θ ≤ γ max Υ Č2 η + γ max Υ Č v + γ max λ max θ
θ where γ max and λ max are respectively the larges singular values of Γ and Λ, Υ, Č, η, v are respectively upper bounds of Υ(t) , C(t) , η(t) , v(t) . Note that Υ(t) is bounded according to Proposition 1. For large t, η(t) satisfies ( 14). Then

U (t) ≤ γ max Υ Č2 a b ( w + Ǩ v) + γ max Υ Č v + γ max λ max θ -θ . ( 35
)
The right hand side of this inequality is an upper bound of U (t) , which is a linear combination of w, v and θ -θ . This result, combined with (34) then proves (31).

2

Remark 4. Proposition 5 indicates that the regularization term implies a bias of the parameter estimate proportional to (θ-θ) . Therefore, whenever possible, prior knowledge should be used for choosing θ as close as possible to the unknown true parameter value θ, in order to reduce the bias. Proposition 4 shows that the convergence rate of the error dynamics depends on Λ and Γ. In general, a faster convergence is related to a higher sensitivity to uncertainties (or noises). The choice of the regularization parameter Λ could be made following the Bayesian approach, if a stochastic model was established. This approach will be studied in future works. 2

NUMERICAL SIMULATION EXAMPLE

Consider the system ẋ1 ẋ2 ẋ3

=

-1 1 0 -1 0 0 0 -1 -1 x 1 x 2 x 3 + -1 0 0 u+ θ 1 θ 2 θ 3 + w 1 w 2 w 3 y 1 y 2 = 1 0 0 0 0 1 x 1 x 2 x 3 + v 1 v 2 .
Intuitively, the corresponding constant matrix Φ(t) = I 3 may not be sufficiently exciting. This fact is confirmed as follows.

Let σ ∈ R be any constant value. Make a state variable change with z 2 = x 2 + σ replacing x 2 , whereas x 1 and x 2 remain unchanged. After this variable change, the state and output equations are exactly as before, except that θ 1 and θ 3 are replaced respectively by (θ 1 -σ) and (θ 3 +σ). This result implies that all parameter pairs θ 1 , θ 3 corresponding to the same sum value θ 1 +θ 3 lead to the same input-output relationship! It is thus impossible to uniquely determine θ 1 and θ 3 from input-output data.

For the simulation example presented below, the input u(t) u(t) = sin(t) + cos( √ 7t), (36) the parameter vector θ = [1; 0.7; 0.5], and the initial state x(0) = [1; 1; 1]. The uncertainty terms are simulated as

w(t) = 0.1[∆(t); ∆(t/2); ∆(t/3)] (37) v(t) = 0.01[∆(t); ∆(t/2)]
(38) with the triangular wave function ∆(t) 4 |t -t + 0.5 | -1, and the notation x denoting the largest integer less than or equal to x ∈ R. For the regularized adaptive observer, the state estimation gain K(t) is computed with P (0) = I 3 , Q(t) = 0.1I 3 , R(t) = 0.01I 2 , the parameter estimation gain Γ = 20I 3 , and the regularization parameter matrix Λ = 0.0001I 3 . The initial state estimate x0 and parameter estimate θ0 are set to zero values, as well as the prior parameter guess θ.

As shown in Figures 1 and2, despite the deficient excitation and the simulated uncertainties, the state and parameter estimates are close to their simulated (true) values, after a transient period of about 2 seconds. The two parameter estimates θ1 (t) and θ3 (t) have similar bias values, but of opposite signs. This fact is due to the indetermination of the two parameters: all parameter pairs θ 1 , θ 3 corresponding to the same sum value θ 1 + θ 3 lead to the same input-output relationship.

CONCLUSION

The convergence of adaptive observers is usually ensured by some persistent excitation condition. In order to apply such algorithms when this condition is not satisfied, a regularized adaptive observer has been proposed in this paper, inspired by the usual practice for solving ill-posed inverse problems. In future studies, regularization will be studied in a stochastic framework in order to guide the tuning of the regularization parameter. The regularization method proposed in this paper is not the only possible choice. Other possibilities will also be investigated in future studies.

APPENDIX

Lemma 1. Assume that the homogeneous LTV ODE ξ(t) = A(t)ξ(t), (39)

Proposition 4 .

 4 Under Assumptions 1 and 2, The homogeneous part of the ODE (20), namely θ

Fig. 1 .

 1 Fig. 1. Simulated states (in blue) and their estimates (in red).

Fig. 2 .

 2 Fig. 2. Simulated parameter values (in blue) and their estimates (in red).

with ξ(t) ∈ R n and A(t) ∈ R n×n , is exponentially stable, such that its state transition matrix Φ(t, t 0 ) ∈ R n satisfies Φ(t, t 0 ) ≤ αe -β (t-t0) (40) for some positive constants α, β and for all t ≥ t 0 . Then, the solution of the non homogeneous LTV matrix ODE Ẋ

with X(t) ∈ R n×m , which is driven by the bounded input U (t) ∈ R n×m satisfying U (t) ≤ γ for all t ≥ 0, satisfies

2

Remark that this result holds also for m = 1, when X(t) and U (t) are column vectors.

Proof.

then

Inequality ( 42) is then proved.

) where γ min is the smallest singular value of Γ. Then

By Grönwall's inequality, for all t ≥ t 0 ,

Let γ max denote the largest singular value of Γ, then 2βγmin(t-t0)