David Guerrero⁺

Patrick Niérat⁺

Jean-Claude Thill

Emmanuel Cohen⁺

IFSTTAR Gustave Eiffel

Pre-print of a paper published in Journal of Transport Geography, Visualising Transport Geography Series Visualizing the maritime connectivity of African countries

Introduction

Containerization is still on-going in Africa. In the last decades, a number of new port and terminal projects have emerged in parallel with the reorganization of services by shipping lines.

To support the analysis of the maritime connectivity of African countries, this paper proposes a visualization of the Liner Shipping Bilateral Connectivity Indicator (LSBCI).

This research aims to contribute to the current debate on the ways in which containerization is shaping the connectivity of African countries [START_REF] Mboya | The Maritime Silk Road Initiative: Connecting Africa[END_REF] by advancing the use of effective map visualizations of port connectivities. Previous studies have shown the improvement of maritime connections of African countries to Asia. For example, the countries of East Africa are better connected to the Indian Ocean than to Europe [START_REF] Ducruet | The worldwide maritime network of container shipping: spatial structure and regional dynamics[END_REF]Notteboom, 2012, Niérat and[START_REF] Nierat | UNCTAD maritime connectivity indicators: review, critique and proposal[END_REF]. Some have argued that these improvements have mainly affected the connections of Africa with Asia, but without significantly changing the connectivity within Africa or between Africa and other world regions. Others have advanced the counter-argument that the enhancement of maritime connectivity of Africa with Asia also leads to an improvement of connectivity among African countries and also with other regions [START_REF] Kuo | There's One Major Pitfall for African Countries Along China's Next Silk Road[END_REF], cited by Mboya, 2021). Another debate that is germane to our work is that of regional economic integration in Africa versus dependency relationships with outside actors, the processes undergirding transition towards integration, and the role of freight network connectivity [START_REF] Arizala | Africa comes together[END_REF]. As the evidence to inform these debates has been limited, confusing, and sometimes contradictory, this research purports to shed light onto these debates by proposing a visualization of bilateral connectivity of African countries with other countries around the globe. It is hoped this visualization will help clarity the changing dynamics behind commercial relationships of Africa countries and behind the connectivity of Africa ports. The following sections present the data and method (section 2), the visualization and interpretation of the interregional and intra-regional links (section 3). The last section concludes on this work.

Data and method

Maritime connectivity is the access to regular and frequent transport services combined with a diversity of service opportunities brought by competition between multiple service providers [START_REF] Wilmsmeier | The impact of port characteristics on international maritime transport costs[END_REF]. It is often measured through various indicators based on vessel movements. One of them is the LSBCI, which measures the quality of container transport connections between pairs of countries. It is the average of five normalized components [START_REF] Fugazza | Liner shipping connectivity as determinant of trade[END_REF]: (a) the number of transshipments required to get from country A to country B; (b) the number of direct service connections common to both countries A and B; (c) the number of common service connections by country pair with a single transshipment; (d) the level of competition on services that connect country A and country B; (e) the size of the largest ship on the weakest route connecting countries A and B.

In this study we analyze the LSBCI links of 39 African countries and nine geographical groupings of countries outside of Africa. Summarizing the links for these groupings enables us to tease out some of the most salient relationships, leaving aside more localized detail. The LSBCI between each geographical group and each African country is computed on the basis of the original LSBCI statistic as follows. The LSBCI between a country group and any African country is equal to the maximum LSBCI link of all the countries in the group and the country under consideration. For this study we have used the most recent data available, namely 2020. To facilitate the visualization, Africa is partitioned into four sub-regions, based on the UN partition2 (Table 1). The maximum LSBCI is in North Africa while the highest average is in the Southern part of Africa. The values for the West and East regions are very similar.

Visualization of interregional and intra-regional links Figure 1. Strongest LSBCI links of African countries

Figure 1 depicts country or world region associated with the strongest LSBCI link of each African country. In particular, Figure 1a displays the links with regions outside Africa. Amongst the non-African regions, Europe, China and the Middle East and South Asia are the most prominent. While North and West African countries are rather oriented to Europe, a majority of East African countries are oriented towards the Middle East and South Asia. China has very prominent connectivity with some of the countries of West Africa (Benin, Togo, Ghana, Congo), in the Southern tip of Africa, and on a limited basis in East Africa (Djibouti and Kenya), where it exhibits the strongest LSBCI links. In many of the latter countries (marked with an asterisk) the LSBCI with China is even stronger than with their partners within Africa.

In Figure 1b, we depict the strongest connection of each country with the countries of other African sub-regions. It highlights the well-known hub positions of Morocco and South Africa ports competing to attract the feeder flows of West African countries. It is worthwhile noting that the connectivity of Morocco is mostly related to Tanger Med while the connectivity of South Africa stems from several ports (Durban, Cape Town and Ngqura).Conclusions LSBCI appears to be a useful indicator to assess maritime connectivity in Africa, but it can be difficult to interpret because of the complexity stemming from the multiple dimensions it encompasses. To help make sense of it, this paper proposed a set of map visualizations which may be useful for countries aiming to link the commercial success of ports with economic development and disentangle the contested state of logistical connectivity and integration of Africa.

The map visualizations showed that West Africa and East Africa appear as two separated subsystems. Morocco, South Africa and Egypt stand out in our visual analysis as they are by far the best connected countries of Africa These countries have asserted themselves as nation hubs in the continental freight shipping network. However, only the former two countries play mediating roles within the region, while the latter remains largely focused on North Africa and East-West trade routes.

In addition to the former works, the present work revealed that the largest maritime nations of South and West Africa, are now also tightly connected to China. This supports the hypothesis that recent improvements in African connectivity have mainly affected the connections of Africa with Asia and more specifically with China. More research is needed however to assess if these changes have also improved (or not) the connectivity within Africa and with other world regions.

This study shows that the position of individual countries and of large ensembles of countries on the global trade map inform network connectivities. Thus, it is suggested that geography matters in the organization of the African maritime connections, but other factors -such as the strategies of maritime companies-play important roles as well.

Table 1 . Descriptive statistics of the LSBCI in Africa

 1

		Max	Min	Avg Median Std Dev
	North	0.57	0.09	0.18	0.16	0.09
	West	0.36	0.08	0.16	0.16	0.05
	South	0.39	0.11	0.19	0.19	0.06
	East	0.36	0.08	0.16	0.16	0.05
	Africa	0.57	0.08	0.17	0.17	0.06
	Non-Africa	0.97	0.09	0.24	0.20	0.13
	Source: Authors, based on UNCTAD, 2020

Corresponding author (david.guerrero@univ-eiffel.fr)

For the sake of simplicity, the UN subregions of Central Africa and Western Africa were merged.


Rémi Hackert

Nadja Schilling

Martin S Fischer

Mechanical self-stabilization, a working hypothesis for the study of the evolution of body proportions in terrestrial mammals?

Keywords: stability locomotion body proportion running quadruped mammals. Résumé stabilité dynamique, locomotion, proportion, segment, course, mammifères quadrupèdes

Locomotion is a behaviour resulting from the interaction of the nervous and muculo-skeletal systems and the environment. However the musculo skeletal systems of some terrestrial mammals present an intrinsic ability to realize a dynamic stable locomotion. Actual anthropomorphic passive walkers demonstrate that a pure mechanical system with leg and arms is able to walk down an inclined plane. Numerical simulations confirm that self-stabilization of the mechanics is acting in running too. The necessity to ensure the dynamic stability of a cyclic locomotion set physical constraints to the musculo-skeletal system. A description of the locomotor apparatus with neuromechanical variables like the stiffness -accessible to the experimentation-enables for maintaining the number of degrees of freedom of biomechanical models as low as possible. The maximization of the robustness of the mechanical self-stabilization of the models with regard to the body proportions represents for future simulations an optimization criterion which should bring a new light to the comprehension of the body proportions.

Introduction

By the transition from the sprawled reptilian leg configuration to the parasagittal mammalian limb, one additional segment was added in the fore and hind limbs (scapula and elongation of the tarsus). Thus, scapula and femur, humerus and tibia, ulna and tarsus became functional correspondent in mammals [START_REF] Marey | Analyse des mouvements du cheval par la chronophotographie[END_REF][START_REF] Kuznetsov | Energetical profit of the third segment in parasagittal legs[END_REF]. During in-phase gaits (bound, halfbound, gallop), extensive flexions and extensions occur in the posterior thoracic and the lumbar regions of the spine especially in mammals [START_REF] Gasc | Comparative aspects of gait, scaling and mechanics in mammals[END_REF]. These back movements act to some extent as an additional proximal segment modulating the position of the pelvis up to 40 degree [START_REF] Schilling | Intervertebral movements during locomotion in small mammals[END_REF]. Mammalian posture seems to follow the "crouched posture and high fulcrum" [START_REF] Fischer | Crouched posture and high fulcrum, a principle in the locomotion of small mammals. The example of the rock hyrax (Procavia capensis) (Mammalia. Hyracoidea)[END_REF] i.e. scapular pivot in fore and hip joint in hind limbs are held at the same level. But intralimb proportions and relations between the functional corresponding segments within the kinematical chains are still poorly understood if well described by allometry relations within a family [START_REF] Lilje | Scaling of long bones in ruminants with respect to the scapula[END_REF].

The locomotor apparatus evolved under the selective pressure of structural, (eco-) physiological and mechanical constraints. Among them efficiency in energy consumption is certainly one of the most important constraints and is expressed in two dual ways to handle with energy during (loco-) motion.

First: reduction of muscular work as much as possible. Kustnezov [START_REF] Kuznetsov | Energetical profit of the third segment in parasagittal legs[END_REF] interpreted the incorporation of the third segment as an energy saving adaptation which enables the reduction of the mechanical work of leg muscles against each other during the stance phase. The author calculates from his model the kinematics which minimizes muscular work during locomotion over flat ground and found a kinematics comparable to the kinematics observed in small mammals ( [START_REF] Schilling | Kinematic analysis of treadmill locomotion of the tree shrews Tupaia glis (Scandentia: Tupaiidae)[END_REF][START_REF] Schmidt | Cineradiographic study of forelimb movements during quadrupedal walking in the brown lemur (Eulemur fulvus, Primates: Lemuridae)[END_REF] and overview in [START_REF] Fischer | Basic limb kinematics of small therians mammals[END_REF]).

Second: storage or conversion of kinetic energy during one phase to recover it in another phase of the movement [START_REF] Mcn | Elastic Mechanisms in Animal Movement[END_REF][START_REF] Alexander | Elastic structure in the back and their role in galloping insomes mammals[END_REF].

Cavagna et al. [START_REF] Cavagna | Mechanical work in terrestrial locomotion: two basic mechanisms for minimizing energy expenditure[END_REF][START_REF] Fedak | Energetics and mechanics of terrestrial locomotion. II. Kinetic energy changes of the limbs and body as a function of speed and body size in birds and mammals[END_REF] described two mechanisms: The inverted pendulum during slow motions and the elastic mechanism during fast movements. In the first case, kinetic and gravitational potential energy associated with the motion of the centre of body mass differ in their phase and this results in a transfer of kinetic into potential energy and back within the locomotor cycle. In contrast to this, kinetic and gravitational potential energies vary in phase in the elastic mechanism and thus there is no conversion between kinetic and gravitational potential energy during fast locomotion but in place the kinetic energy is provisory stored in elastic element -tendons and muscles [START_REF] Tskhovrebova | Elasticity and unfolding of single molecules of the giant muscle protein titin[END_REF][START_REF] Labeit | Titin: giant proteins in charge of muscle ultrastructure and elasticity[END_REF][START_REF] Linke | Stretching molecular springs: elasticity of titin filaments in vertebrate striated muscles[END_REF] and is restored in a next phase of the movement. Elastic energy storage is assumed for small mammals as well as for larger mammals [START_REF] Cavagna | Mechanical work in terrestrial locomotion: two basic mechanisms for minimizing energy expenditure[END_REF] although there is no need to assume it in order to explain their energy balance [START_REF] Heglund | Energetics and mechanics of terrestrial locomotion. III. Energy changes of the center of mass as a function of speed and body size in birds and mammals[END_REF][START_REF] Heglund | Speed, stride frequency and energy cost per stride: how do they change with body size and gait[END_REF].

The named strategies are a response to the physical constraints applied by interactions of the body with its environment. Mechanical interaction as one of the components of this interaction (beside physico-chemical) is often explained as the necessity to support the body weight i.e. resist gravitational forces. But this is also true during locomotion. Inverse dynamic analysis demonstrates that the net torques created in the joints by muscles work mainly against gravity [START_REF] Witte | Torques patterns of the limbs of small therian mammals during locomotion on flat ground[END_REF][START_REF] Fischer | Kinematics, EMG, and Inverse-dynamics of the therian forelimb -a synthetical approach[END_REF]. However, anti gravitational action is only one vector of the mechanical interaction.

The stability of locomotion particularly at key events as escaping or hunting if stability and manoeuvrability are most important-is a second vector of this interaction. The study of the relationships between stability and morphology has to be embedded into one possible approach described by this contribution.

Dynamic stability during locomotion

A standing quadrupedal animal is in static equilibrium. The feet, as points of ground contact, map a polygon called polygon of support. As long as the vertical projection of its centre of body mass remains inside the polygon the animal is said to be statically stable. This is the case during slow walking which is described as a succession of quasi-static equilibrium. Three legs are always in contact to the ground. The triangle of support changes periodically and the vertical projection of the centre of mass then moves from one triangle to the next. At any time the animal is able to stop its forward motion if the speed is low and therefore the quantity of movement (mass time speed) is small. Cartmill et al. [START_REF] Cartmill | Support polygons and symmetrical gaits in mammals[END_REF] show on the basis of a large set of footfall patterns that quadruped mammals prefer limb coordinations that maximize the time of existence of such a polygon of support during walking. If speed is increased, a preference for limb co ordinations that involve diagonal pairs of limbs is given. During trotting, the polygon is reduced to a diagonal line connecting one hand and the ipsilateral foot. Thus, trotting animals have already leaved the frame of a succession of quasi-static equilibrium state to meet dynamic equilibrium. Idem gallop, bound, half-bound. Stability qualifies equilibrium. Equilibrium is said to be statically stable if a system returns to its initial equilibrium after a perturbation as for instance a pendulum after deflection from its vertical position. In case of a cyclic movement the motion of a system is said stable if it is able to go back to its nominal trajectory after perturbation. This restrictive definition of dynamic stability represents more than simply remaining on the feet. Since motion of a living organism underlies variability, the given definition is rarely realized in nature. In many studies on stability of motion systems, particularly in the robotics, a system is more pragmatically assumed stable if it reaches a neighbour stable state after a perturbation even if it does not return exactly to its nominal state.

Addressing the right degree of complexity

Locomotion is generally described as the behaviour which results from the interaction of the musculo-skeletal system and the neural system and environment components. It is a complex task which exhaustive description would require hundreds of variables. The aim of modelization is to reduce the number of variables -i.e. the number of degrees of freedom (d.o.f.) -to a necessary minimum.

Due to the different physical mechanisms acting in slow and fast locomotion, the number of d.o.f. is reduced by addressing these two types of locomotion separately. Applying Newton's first principle of dynamics, the forces exerted by the limbs on the ground (i.e. ground reaction forces) and measured by the use of ergo meters [START_REF] Cavagna | Force platforms as ergometers[END_REF] can be integrated numerically two fold. The integration leads to the motion of the centre of mass (c.o.m.) of the body.

During trotting and in-phase gaits, motions of the centre of mass of mammals occur mainly in the parasagittal plane. At trot, gait symmetry and resulting torsions of the spine around the longitudinal body axis lead this parasagittal motion of the CoM. At in-phase gaits spine bending movements lead to extensive parasaggital movements of the pelvis which contribute up to half of the stride length in small mammals [START_REF] Fischer | The locomotion of Procavia Capensis (Mammalia, Hyracoidea): Zur Evolution des Bewegunssystems bei Säugetieren[END_REF][START_REF] Fischer | Application of cineradiography for the metric and kinematic study of in-phase gaits during locomotion of the pika (Ochotona rufescens, Mammalia: Lagomorpha)[END_REF][START_REF] Fischer | Basic limb kinematics of small therians mammals[END_REF][START_REF] Schilling | Intervertebral movements during locomotion in small mammals[END_REF]. Therefore the lateral excursion of the centre of mass is smaller than its vertical ones. Thus, a modelization of a quadruped mammal during fast locomotion is reasonable in a two dimensional plane whereas a two dimensional model of slow locomotion will be a partial description of the reality. It is assumed that models deal with straight forward locomotion.

Most simulations studies of the last 15 years can be classified into three types which differ in the description of the interaction between neuronal and mechanical systems and emerged after the publication of three basic articles: Taga [START_REF] Taga | Self-organized control of bipedal locomotion[END_REF] modelized a rough musculoskeletal and a rough nervous system separately which interaction produces a stable cyclic motion. McGeer [START_REF] Geer | Passive walking with knees[END_REF] studies the passive walking of a structure along an inclined plane, and investigates the passive properties of the mechanics disregarding voluntary the role of the nervous system. In direct line to the works of Cavagna and Taylor, Blickhan [START_REF] Blickhan | The spring mass model for running and hopping[END_REF] proposed a synthetic model taking into account elastic energy storage during running: the spring-mass model.

Multibody dynamics

Tagas' purpose [START_REF] Taga | Self-organized control of bipedal locomotion[END_REF] was to test whether a stable walk is able to emerge from the interaction of mechanical and nervous systems defined as follows: The mechanical structure was modelized by a series of rigid segments -three for each limb and a rigid trunk. Taga associates a couple of neuro-oscillators with each limb joint, an assumption founded on its existence in a primitive specie -the lamprey [START_REF] Grillner | Neural Network generating locomotor behaviour in lamprey: circuitry, transmitters, membranes properties and simulation[END_REF] and a presumption of its existence in other species. The localization of the neuro-oscillators was recently identified in a tetrapod [START_REF] Cheng | Identification, Localization, and Modulation of Neural Networks for Walking in the Mudpuppy (Necturus Maculatus) Spinal Cord[END_REF]. Each pair of neuron influences the pair of the adjacent proximal joint. A relatively simple mathematical neuron model (integrator) was assumed. Torques acting at the joints were assumed proportional to the output signal of the neurons. Taga was able to get the model walking stably through the settings of the "right" values of the parameters into the model. This study became the core of others, integrating almost all bones and muscles [START_REF] Hase | Computational Evolution of human bipedal walking by a neuro muscular skeletal model[END_REF], optical sense, and obstacle avoidance into the neuro-controller. With this, the number of degree of freedom of Tagas' basic model was considerably increased. The proposed neurocontrollers and their development largely founded the success of actual Japanese humanoid robots.

Morphologists can learn from complex simulations if they address the right level of complexity. But the quality of the extrapolations made with very complex models [e.g. 36] with hundred of variables is rather poor. Due to the extremely large parameters space and the redundancy in the models (however redundancy is also present in living systems), the search for optimal value of some choosen variable with regard to stability does not lead to significant results since changes in the value of one parameter can easily be compensated by the change of others.

Thus for the study of the evolution of musculo skeletal design it seems to be more promising to comprehend a construction step by step and reconstruct it stepwise. In the field opened by Taga's work -i.e. coupling of one mechanical and one neuronal system -as a morphologist the most interesting simulations are the attempt to discover how few features of the reality is necessary to let a given behaviour emerge. With this aim Ijspeert [START_REF] Ijspeert | A connectionist central pattern generator for the aquatic and terrestrial gaits of a simulated salamander[END_REF] investigates for instance the transition from a standing wave to a travelling wave during the locomotion of a salamander transiting from land to water.

Passive walking dynamics

McGeer [START_REF] Geer | Passive walking with knees[END_REF] designed (originally) a toy that was able to walk down along a slope without additional external control, propelled only by the gravitation. Bipedal passive walker models and robots were then developed with anthropomorphic dimension with or without knees and/ or feets [START_REF] Garcia | the simplest walking model: stability, complexity and scaling[END_REF][START_REF] Collins | Efficient Bipedal Robots Based on Passive Dynamic Walkers[END_REF][START_REF] Goswami | A study of the passive gait of a compass-like biped robot: Symmetry and chaos[END_REF]. A periodic input of energy at the hip even enables a stable walk on a horizontal and even ground. More recently Collins et al. [START_REF] Collins | A three-dimensional passive-dynamic walking robot with two legs and knees[END_REF] developed a 3D simulation model includings arms which movements stabilized the model successfully [START_REF] Kuo | Stabilization of lateral motion in passive dynamic walking[END_REF]. Despite the fact that passive walker actually disregard some features of the walking motion (ankle extension at the end of the stance phase for instance), this main result demonstrates that the body mechanics do not only evolved in order to enable an economic walking but also a stable walking and that this mechanical stability is founded on synergies in the motion of the different part of the body. The results give raise to conceptual development. Following this, is it appropriate to subordinate the musculoskeletal system to the nervous system in the case of a cyclic (automatic) motion? Or should we accept the idea that mechanics control the neuronal system just like the neuronal system control mechanics [START_REF] Brook | Artificial Life and Real Robots, Toward a Practice of Autonomous Systems[END_REF]?

Spring mass models for fast locomotion Blickhan [START_REF] Blickhan | The spring mass model for running and hopping[END_REF] found a roughly linear relationship between the ground reaction force and the distance from the centre of mass to the point of ground contact at the preferred hopping frequency of humans. He introduced the spring mass system in order to model this relationship (figure 1A). The spring mass model has only five degrees of freedom (mass, spring stiffness, spring length, speed vector, angle of attack) and fitted the experimental data with a precision more than 20% in the range of measurement. It comes up to the requirement of Cavagna et al. [START_REF] Cavagna | Mechanical work in terrestrial locomotion: two basic mechanisms for minimizing energy expenditure[END_REF] that the energy balance of middle and large sized mammals during fast locomotion cannot be explained without assuming elastic energy storage.

The described linearity is surprising because force length relationships of an isolated muscle are not linear, the geometry of bones head articulation is not circular and thus the relation of the lever arm of a given muscle versus joint aperture is also not linear. The integration of these components leads first to a linear force-length relationship [START_REF] Wagner | Stabilizing function of skeletals muscles: an analytical investigation[END_REF].

Farley et al. [START_REF] Farley | Running springs: Speed and animal size[END_REF][START_REF] Farley | Leg stiffness and stride frequency in human running[END_REF] extended the validity of the spring mass model to the description of trotting quadrupeds. In this description the spring represents the action of a pair of diagonal limbs. Hackert et al. [START_REF] Hackert | Spinal flexion contributes to self-stabilization of pikas' half-bound[END_REF][START_REF] Hackert | Dynamics of half-bound and spinal flexion[END_REF] confronts the spring mass model to in-phase gaits of small mammals and showed that the force length relationships in forelimbs were also roughly linear in small mammals. But linearity is achieved as a consequence of the intense sagittal spine flexion at inphase gaits that shifts the centre of body mass cranially [START_REF] Hackert | Aspect of spinal bending in small mammals[END_REF]. A roughly linearity of the force length relationships means that the stiffness -that means the ratio of the change in the force (measured in Newton) through the corresponding change in a length (deformation) -is almost constant during the stance phase of forelimbs. The role of limb's stiffness (in Nm -1 ) and thus the role of compliance -was already recognized as one determinant of locomotion [START_REF] Blickhan | The spring mass model for running and hopping[END_REF][8][45] [START_REF] Mc Mahon | The mechanics of running: How does stiffness coupled with speed?[END_REF] but its direct involvement in the achievement of a dynamic stable locomotion was demonstrated clearly only recently.

Stiffness of a structure and dynamic stability

The stiffness of a musculo-skeletal structure can be split into two major components: (a) a passive component linked to the intrinsic properties of the materials and (b) an active component (depending on muscle activation and geometry).

Muscles force does not necessary result in movement. If agonist and antagonist muscles are activated synchronously so that torques of agonist and antagonist muscles remain in equilibrium, no movement will occur in the articulation but its stiffness will be increased. Feldman [START_REF] Feldman | Once more on the equilibrium point hypothesis (Λ model) for motor control[END_REF] demonstrates that the control of only one parameter is sufficient to control joint stiffness and aperture.

Dynamic stability of the spring-mass system -i.e. its ability to bounce stably-was studied first by Schwind and Koditchek [START_REF] Schwind | Spring loaded inverted pendulum running: A plant model[END_REF]. They derived an analytical criterion from the linearized (i.e. approximated) equation of motion.

Seyfarth et al. [START_REF] Seyfarth | A movement criterium for running[END_REF] performed dynamic simulation of a non linearized model and demonstrate its ability to bounce stably for some combinations of the forward speed, the spring-leg stiffness and the angle of attack which is the angle spanned by spring-leg with the ground line at the instant of touch down. That is within a range of values of stiffness and angle of attack and mass position a stable behaviour can be generated (figure 1D). Thus small irregularities of the ground which would result in small changes of the mass position relative to the ground can be overcome without resetting the values of the system, i.e. without control. This concept of a mechanical stabilisation without external control is known in physics as self-stabilization.

Approximations are numerous by spring mass models. During forward movement the spring mass model describes only the stance phase of a spring-leg not the swing phase. The spring-leg is assumed to be massless and from the point of view of the dynamics a massless object cannot realize a motion. Thus, in the simulation the leg is artificially positioned with a given angle of attack during the flight phase of the spring mass system before the next ground contact. This is an important restriction but 85% of the mass is included in the head-trunk structure in small mammals.

Limbs of small mammals are light indeed and therefore assuming massless legs is appropriate. Additionally the model assumes that ground reaction forces keep the direction of the spring -, towards the centre of mass and this is only true as well in humans as in quadruped at constant speed and during the middle stance phase [START_REF] Biewener | Locomotory stresses in the limb bones of two small mammals: the ground squirrel and chipmunk[END_REF].

The comparison between the spring mass system and galloping small mammals show additional descriptive limits. The simple spring mass model trajectory undulates with one maximum and one minimum per cycle whereas the centre of mass of galloping small mammals may have more than two extremum within each locomotor cycle. This gives raise to the development of extended springs-mass models in order to take the back dynamics into account (fig 1C). Hackert and Fischer [START_REF] Hackert | Aspect of spinal bending in small mammals[END_REF] propose to see the flexion of the spine in small mammals and the associated forward displacement of the centre of mass as a way to adjust the angle of attack of fore limbs and thus to use mechanical self-stabilization mechanism. Back flexion co-determines the position of the centre of mass and back stiffness sets the rough instant of touch down of the forelimb by controlling the downwards motion of the centre of mass during hind limbs' stance phase. In respect to the simple spring mass system, Seyfarth et al. [START_REF] Seyfarth | Swing-leg retraction: a simple control model for stable running[END_REF] implement a rotation of the spring leg during its late swing phase when the mass has reached the apex of its trajectory. With this, the authors show that a tuning of the instant and verticality of the leg at touch down may influence positively the dynamic stability of the system. This may explain the retraction of the fore limb observed in galloping mammals before touch down [START_REF] Fischer | Basic limb kinematics of small therians mammals[END_REF].

Models and templates: Learning from neuromechanical models stepwise.

The spring mass model developed from the status of a simple mechanical model for human hopping to a neuromechanical template. Full and Koditschek [START_REF] Full | Templates and anchors: neuromechanical hypotheses of legged locomotion on land[END_REF] proposed to face the spring mass template like an asymptotical limit to which running systems tend to. Moreover, the authors conceptualized a method (rather new in the field of motion studies) to handle with complex system involving a lot of degrees of freedom. First: reduce the number of degrees of freedom by projecting the problem onto one plane, second then model it in this plane, then progressively include new additional degrees of freedom. This method is comparable to the early stage of movement learning in child [START_REF] Bernstein | Some emergent problems of the regulation of motor acts[END_REF] when degrees of freedom are introduced and the stiffness of the limbs reduced stepwise during growth [START_REF] Scheppens | The mechanics of running in children[END_REF].

The introduction of the neuromechanical parameter "stiffness" leads to a level of description that enables provisory to "disregard" how the stiffness is generated i.e. to disregard the muscles, their insertion and activation. The number of degree of freedom of the biomechanical model becomes then considerably lower.

Distribution of the global stiffness onto the joints

The spring mass model is a model at the level of organisms. Limb stiffness can be compared between species after normalization by mass and length. Blickhan and Full [START_REF] Blickhan | Similarity in multilegged locomotion: Bouncing like a monopode[END_REF] introduce the relative stiffness = (F max /mg) / (∆ l/l) and found a values around 8 for animals as different as insect and horses. This conveys that animals segmented limbs have comparable material characteristics.

For comparative anatomists global models as the spring mass model are less informative since morphometric data normally are not involved in the models. It is possible to introduce them into a model, by distributing the elasticity (of the spring) onto the joints and thus introducing a massless polysegmental kinematic connected with rotational springs (Fig 1C). The stiffness is then the ratio variation of the joint torque over variation of the joint aperture i.e. the slope of the tangent at the torque-aperture characteristics.

Seyfarth et al. [START_REF] Seyfarth | Stable operation of an elastic three segmented leg[END_REF] studied the stability of a massless three segmented leg very much in detail ( for equations see [START_REF] Seyfarth | Stable operation of an elastic three segmented leg[END_REF] ) in order to describe its stabilizing behaviour in dependence to its initial configuration (zig-zag or arc like) and stiffness laws (constant or variable). The model assumes that the rotational springs act at the joints in a quasi elastic mode without dissipation i.e. joint torques have the form Ti = k i (φ -φ o ) ν (Eq. 1). That is the torques -angle characteristics do not show hysteresis. Neglecting the torques at the foot pad, the torques equilibrium equation then simplifies into 1B). Seyfarth et al. assumed a symmetric loading of the limb during stance phase: the compression of a zig-zag like kinematical chain depends on the stiffness of its joints. With similar values of joints stiffness, a load results in a flexion of all joint angles at a time to the same extent, else the segmental chain would collapse first at the joint of less stiffness than others. But studies on the intralimb kinematics of small mammals [START_REF] Gasc | Asymmetrical gait of the Saharian rodent Meriones shawi shawi (Duvernoy, 1842) (Rodentia, Mammalia): a high speed cineradiographic analysis[END_REF][START_REF] Fischer | Basic limb kinematics of small therians mammals[END_REF] and birds [START_REF] Abourachid | Bipedal locomotion in ratites (Paleognatiform): examples of cursorial birds[END_REF][START_REF] Verstappen | Terrestrial locomotion in the black-billed magpie: kinematic analysis of walking, running and out-of-phase hopping[END_REF] during running show that changes occur in all joints at the same time and thus point to symmetrical loading and stiffness equilibrium.

1 d 1 +T 2 d 2 =0 (Eq. 2) (figure
Assuming symmetrical loading the torques equilibrium equation (Eq. 2) then simplifies into k / k 23 = l 1 / l 3 (figure 1B) whereas the length of the second segment vanish from the equations [START_REF] Scheppens | The mechanics of running in children[END_REF]. Numerical simulations of in place hopping of this elastic limb model lead to the following result: stability is enhanced if a short first segment correspond a long third segment and reciprocally [START_REF] Scheppens | The mechanics of running in children[END_REF]. The figure 1E presents limb geometry of some terrestrial birds. Birds seem to follow this simple "rule".

Actually no quadruped model has been studied in the same detail. One prerequisite is a better experimental knowledge of the torques-aperture characteristics in quadruped mammals in order to limit the parameter space in the simulations.

The experimental determination of the torque-aperture relationships (and following of the stiffness laws) at the level of each joint is accessible performing inverse dynamic calculations of the limbs. These experimental characteristics will also point out the level (distal versus proximal) at which energy dissipation occurs in the limbs during a cycle of gallop.

Spring-mass models disregard up to now fully energy dissipation up to the theoretical work of Berkemeier [START_REF] Berkemeier | Modelling the dynamics of quadruped running[END_REF]. Inverse dynamic analysis is in small quadruped mammals a rather difficult but realizable task [START_REF] Taga | Self-organized control of bipedal locomotion[END_REF] since it is necessary to record separately the force exerted by each limbs. Outgoing from the torque-aperture relationships numerical simulations of a segmented biomechanical model of a quadruped with few degrees of freedom could be calculated looking for "optimal" limb proportion with regard to the robustness of stability i.e. the degree of disturbance the structure will accept to overcome without falling. The robustness of stability can be considered as an optimization criterion in the frame of the mechanical self-stabilization hypothesis.

Concluding remarks

The influence of morphology onto stability cannot be understood intuitively and necessitates the use of numerical simulations. Stability emerges for some value of the parameters -segment's length, articular stiffness…-and is fundamentally unpredicable. The hypothesis that intrinsic stability of the musculo-skeletal mechanics exerted an evolutionary pressure that lead to the real body proportions was recently sustained by some simulation studies [52] [33] which found a good correspondance between the values of the parameters that stabilize spring mass models humans and small mammals running on one hand and the experimental values of stiffness and angle of attack on the other hand.

If stability generally results from the common action of the nervous and mechanical system, the nervous system can find in a self-stabilized mechanics a way to delegate and thus to improve one part of its work particularly at high speed where the reaction times have to be kept short. This would accord well with the decentralized structure of neural control (see Viala et al. and Jamont et al. in this issue).

Actual small mammals, which size is comparable to the oldest fossils of mammals, are a common point of interest for the palaeontology and functional morphology. They are good candidates for the required experimental determination of the torques-aperture characteristics. Their body size enables for the most precise localization of all joint positions including the most proximal using monoplanar videoradiography in a two dimensional frame. Moreover their size enables for the measurement of the ground reaction forces over many locomotor cycles, a prerequisite for a valuable integration of the ground reaction forces and thus for the determination of the position of the centre of mass. Thus small mammals are appropriate to perform inverse dynamic calculations combining high speed videoradiography (500 fps), force measurement and complementary electromyography. The synchronization of all these signals is 100 years after Marey's effort to capture the instant in the movement a realizable [START_REF] Gasc | Propulsive Action of a snake pushing against a single site: its combined analysis[END_REF][22] but exceptional task in integrative animal motion studies. Dynamic simulations on "hybrid" models of model B and C (fig. 1) should reveal whether the enhancement of the robustness of the mechanical self-stabilisation during locomotion acted as one determinant factor during evolution leading to actual body proportions.

Références

Caption of figure 1 A) The spring mass system for running and hopping in humans (Blickhan [START_REF] Blickhan | The spring mass model for running and hopping[END_REF]): the position vector r can be computed

Figure 1 :

 1 Figure 1: Le système masse ressort pour la course et le saut (Blickhan [7]): le vecteur position r est calculé en


Stanislav Aranovskiy

Rosane Ushirobira

Marina Korotina

Alexey Vedyakov

On preserving-excitation properties of Kreisselmeier's regressor extension scheme

In this work, we consider the excitation preservation problem of Kreisselmeier's regressor extension scheme. We analyze this problem within the context of the Dynamic Regressor Extension and Mixing procedure. The well-known qualitative result is that such a scheme preserves excitation. We perform a quantitative analysis and derive lower bounds on the resulting regressor signal considering both persistent and interval excitation cases. We also show that the resulting signal is excited if and only if the original regressor is. Studying the dynamics of the novel regressor, we provide a lower bound on its derivative. Illustrative simulations support our theoretical results.

I. INTRODUCTION

The linear regression equation (LRE) plays a central role in adaptive parameter estimation and adaptive control. It can be found in system identification [START_REF] Ljung | System Identification: Theory for User[END_REF], in model-reference adaptive control [START_REF] Sastry | Adaptive control: Stability, convergence, and robustness[END_REF]- [START_REF] Lavretsky | Combined/composite model reference adaptive control[END_REF] and adaptive pole-placement [START_REF] Saad | Adaptive controllers for discretetime systems with arbitrary zeros: An overview[END_REF], in filtering and prediction [START_REF] Goodwin | Adaptive filtering prediction and control[END_REF], in reinforcement learning [START_REF] Lewis | Reinforcement learning and feedback control: Using natural decision methods to design optimal adaptive controllers[END_REF], and several other areas. The linear regression model is given by

𝑦(𝑡) = 𝜙 ⊤ (𝑡)𝜃 + 𝑤(𝑡), (1)
where 𝑦(𝑡) ∈ R ℓ is the output signal, 𝜙(𝑡) ∈ R 𝑛×ℓ is the regressor, 𝑤(𝑡) ∈ R ℓ is an additive distortion, e.g., a measurement noise, and 𝜃 ∈ R 𝑛 is the vector of unknown constant parameters. The signals 𝑦 and 𝜙 are known (e.g., they are measured), and the distortion signal 𝑤 is unknown. The purpose is to estimate the vector of parameters 𝜃 using the measurements 𝑦 and 𝜙.

There exist various approaches to tackle the parameter estimation problem, where two traditional strategies are the leastsquares method and the gradient estimator. The convergence of parameter estimation schemes relies on the essential premise of a good richness of the regressor 𝜙, realized by the definition of the persistence of excitation (PE) condition, see Definition 1 in Section II. In recent years, various efforts to ease the PE requirement have been suggested, such as concurrent [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF], [START_REF] Kamalapurkar | Concurrent learning for parameter estimation using dynamic state-derivative estimators[END_REF], or composite learning [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF], [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF]. Within this methodology, a dynamic data stack is built to record online historical data discretely, and the convergence of parameter estimation is managed following the interval excitation (IE) condition, a weaker requirement than the persistence of excitation, see Definition 2 in Section II. In [START_REF] Wang | Fixedtime estimation of parameters for non-persistent excitation[END_REF], several fixed-time convergence algorithms have been proposed under the IE assumption on the *This work was supported by Russian Science Foundation grant (project No. 17-79-20341), and by Rennes Metropole, France (project AIS-19C0326). The corresponding author is S. Aranovskiy.

1 IETR -CentaleSupélec, Avenue de la Boulaie, 35576 Cesson-Sévigné, France.

2 Faculty of Control Systems and Robotics, ITMO University, 197101 Saint Petersburg, Russia.

3 Inria, Univ. Lille, CNRS, UMR 9189 -CRIStAL, F-59000 Lille, France.

regressor, and the PE relaxation has also been examined in [START_REF] Cho | Composite model reference adaptive control with parameter convergence under finite excitation[END_REF] in the context of model reference control. To summarize, the persistence of excitation and interval (or sufficient) excitation are nowadays two crucial components in convergence analysis of parameter estimation algorithms. The Dynamic Regressor Extension and Mixing (DREM) procedure has been recently introduced in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF], [START_REF] Ortega | On dynamic regressor extension and mixing parameter estimators: Two luenberger observers interpretations[END_REF]. This procedure allows for translating the LRE (1) to a set of 𝑛 scalar LREs for each element of the vector 𝜃 separately. The DREM approach has been successfully applied to a variety of adaptive control problems and applications, such as direct model reference control [START_REF] Gerasimov | Relaxing the highfrequency gain sign assumption in direct model reference adaptive control[END_REF], time-varying frequency estimation [START_REF] Vedyakov | A globally convergent frequency estimator of a sinusoidal signal with a time-varying amplitude[END_REF], electrical drives velocity estimation [START_REF] Bobtsov | A robust nonlinear position observer for synchronous motors with relaxed excitation conditions[END_REF], and power system applications [START_REF] Schiffer | Online estimation of power system inertia using dynamic regressor extension and mixing[END_REF], [START_REF] Pyrkin | Identification of photovoltaic arrays' maximum power extraction point via dynamic regressor extension and mixing[END_REF]. The DREM procedure also ensures the element-wise transient monotonicity preventing oscillations and peaking, independently of the excitation conditions. Moreover, each element of the estimate of 𝜃 is tuned with a separate scalar gain, which does not affect transients for other elements making the gain tuning more straightforward and transparent.

Two fundamental steps are involved in the DREM method: the dynamic regressor extension (DRE) and mixing steps. In the first step, the original LRE (1) is linearly extended to obtain an extended LRE with a square matrix regressor. In the second step, a nonlinear transformation is employed on the previously generated data to obtain a set of 𝑛 scalar independent LRE for each component of the vector 𝜃 sharing the same new scalar regressor. Finally, an estimator is applied, e.g., a standard gradient estimator or an estimator with the fixed-time convergence [START_REF] Wang | Fixedtime estimation of parameters for non-persistent excitation[END_REF].

Linearly extending the dynamics in the first step of the DREM procedure is a key point. The question is how to perform such an extension so that the excitation level, either PE or IE, of the regressor 𝜙 is preserved. A poor choice can compromise the convergence even if the original regressor 𝜙 is PE, as it has been shown in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF]. In the discrete-time domain, a summation over a fixed window has been proposed in [START_REF] Ortega | New results on parameter estimation via dynamic regressor extension and mixing: Continuous and discrete-time cases[END_REF], but a priori knowledge about the original regressor 𝜙 must be available for choosing the window size. This choice problem has also been considered in [START_REF] Aranovskiy | Parameter identification of linear time-invariant systems using dynamic regressor extension and mixing[END_REF] for a particular class of LRE, where the regressor consists of a finite sum of sinusoidal signals, and the upper frequency bound is known.

One possible dynamics extension preserving the excitation, widely used in adaptive control, is Kreisselmeier's regressor extension introduced in [START_REF] Kreisselmeier | Adaptive observers with exponential rate of convergence[END_REF]. For the LRE (1), Kreisselmeier's regressor extension scheme generates extended matrices Φ and 𝑌 as solutions of

Φ(𝑡) = -𝑎Φ(𝑡) + 𝜙(𝑡)𝜙 ⊤ (𝑡), ∀𝑡 ∈ R + , (2)
𝑌 (𝑡) = -𝑎𝑌 (𝑡) + 𝜙(𝑡)𝑦(𝑡), (3)
where Φ(0) = Φ 0 ≥ 0, 𝑌 (0) = 𝑌 0 , and 𝑎 > 0 is a scalar tuning parameter. The PE preservation properties of (2) are wellknown and are summarized in the following implication:

𝜙 is PE ⇒ Φ(𝑡) > 0 (4)
for all 𝑡 ≥ 𝑇, where 𝑇 is the excitation interval of 𝜙, see Definition 1 in Section II. This PE preservation property of (2) motivates its use in the DREM scheme; the authors in [START_REF] Gerasimov | On key properties of the lion's and kreisselmeier's adaptation algorithms[END_REF], [START_REF] Ortega | On modified parameter estimators for identification and adaptive control. a unified framework and some new schemes[END_REF] proposed such a choice referring to it as Memory Regressor Extension. Specifically, within the DREM context, we are interested in the scalar signal Δ that is the determinant of the extended matrix Φ, Δ(𝑡)

:= det(Φ(𝑡)) , ∀𝑡 ∈ R + , (5)
and for which (4) implies the positiveness and thus the PE property:

𝜙 is PE ⇒ Δ(𝑡) > 0, ∀𝑡 > 𝑇 ⇒ Δ is PE. ( 6
)
The proof of the implication (4) and thus ( 6) is well-known and can be found, e.g., in [START_REF] Ioannou | Robust Adaptive Control[END_REF]Theorem 4.3.3], where the integral cost gradient adaptation algorithm is considered. However, to the best of our knowledge, only the qualitative results (4), ( 6) are available nowadays, and there is no quantitative result for estimating the lower bound of Δ based on the excitation characteristics of 𝜙 and parameters of (2). This paper aims to close this gap by providing a quantitative analysis of excitation preservation via Kreisselmeier's scheme.

Novelty and Contribution. This paper's contribution is the analysis of Kreisselmeier's scheme's properties in the context of the DREM procedure. Specifically, besides the discussed implication 𝜙 is PE ⇒ Δ is PE, we estimate the lower bound of Δ as a function of the excitation characteristics of 𝜙 and the gain 𝑎 in [START_REF] Sastry | Adaptive control: Stability, convergence, and robustness[END_REF]. Moreover, we show that the inverse implication holds and Δ is PE only if 𝜙 is PE; to the best of our knowledge, this intuitive observation was not rigorously proven before. We also study the Interval Excitation property of 𝜙 and show that it is preserved as well; we also provide the quantitative analysis of the resulting interval excitation of Δ.

A preliminary version of the present article was published in [START_REF] Korotina | On parameter tuning and convergence properties of the drem procedure[END_REF]. The novelties in the paper are its several extensions, such as the output 𝑦 is in R ℓ , the equivalence of the PE condition in Theorem 1, and the proof of Theorem 1. Also, an interval excitation analysis of the dynamic extension is given in Proposition 1. Also, compared to [START_REF] Korotina | On parameter tuning and convergence properties of the drem procedure[END_REF], the proof of Proposition 2 is added, and the simulations are revised.

The organization of this paper is as follows. In Section II, some basic material is provided, together with a brief description of the DREM method. The main results on PE and IE conditions for the extension of the dynamics are provided in Section III, and the regressor dynamics is studied in Section IV. In Section V, the numerical simulations illustrate our results.

Notation. The set of positive integers is denoted by N, and the set of reals is denoted by R. For 𝑚, 𝑛 ∈ N, 𝑚, 𝑛 := {𝑝 ∈ N | 𝑚 ≤ 𝑝 ≤ 𝑛} if 𝑚 ≤ 𝑛 and ∅, otherwise. The matrix 𝐼 𝑛 is the 𝑛 × 𝑛 identity matrix, for all 𝑛 ∈ N. For a vector 𝑥 ∈ R 𝑛 , |𝑥| denotes the Euclidean norm of 𝑥. For a function

𝑓 : R + → R, we say that 𝑓 ∈ L 2 if ∫ 𝑡 0 𝑓 2 (𝑠)
𝑑𝑠 converges to a constant as 𝑡 tends to +∞. If the integral does not converge, we write 𝑥 ∉ L 2 .

II. BACKGROUND MATERIAL

The persistence of excitation (PE): First, we present the definition of the (𝑇, 𝜇)-PE property. Definition 1. A bounded signal 𝜙 : R + → R 𝑛×ℓ is (𝑇, 𝜇)persistently excited if there exist 𝑇 > 0 and 𝜇 > 0 such that for all

𝑡 ∈ R + , ∫ 𝑡+𝑇 𝑡 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠 ≥ 𝜇𝐼 𝑛 .
This property is further denoted as 𝜙 ∈ PE, or 𝜙 is PE.

The persistence of excitation property and its connection with the exponential convergence in various estimation schemes are widely known. One relaxation of this condition is the interval (or sufficient) excitation that is used in, e.g., concurrent and composite learning algorithms [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF], [START_REF] Kamalapurkar | Concurrent learning for parameter estimation using dynamic state-derivative estimators[END_REF].

Definition 2. A bounded signal 𝜙 : R + → R 𝑛×ℓ is (𝑡 1 ,𝑇, 𝜇)- interval excited if there exist 𝑡 1 ≥ 0, 𝑇 > 0, and 𝜇 > 0 such that ∫ 𝑡 1 +𝑇 𝑡 1 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠 ≥ 𝜇𝐼 𝑛 .
The fundamental difference is that the persistence of excitation is uniform in time, whereas the interval excitation holds for the particular time interval starting at 𝑡 1 . If 𝑡 1 = 0, then the interval excitation is also called the initial excitation, see [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF].

The gradient estimators [START_REF] Sastry | Adaptive control: Stability, convergence, and robustness[END_REF]: For all 𝑡 ∈ R + , the gradient estimator for the LRE (1) is given by

θ (𝑡) = Γ𝜙(𝑡) 𝑦(𝑡) -𝜙 ⊤ (𝑡) θ (𝑡) , (7)
where θ denotes the estimate of 𝜃 and Γ > 0 is the gain matrix. Define the estimation error θ (𝑡) := θ (𝑡) -𝜃, ∀𝑡 ∈ R + . Then the error dynamics is given by

θ (𝑡) = -Γ𝜙(𝑡)𝜙 ⊤ (𝑡) θ (𝑡) + Γ𝜙(𝑡)𝑤(𝑡), ∀𝑡 ∈ R + .
In the noise-free scenario, i.e., assuming 𝑤 ≡ 0, the gradient estimator ensures exponential convergence to zero of the error θ if and only if the regressor 𝜙 is PE. Otherwise, the gradient estimator is input-to-state stable with respect to the noise 𝑤. Some sufficient (but not necessary) and necessary (but not sufficient) conditions for asymptotic convergence when 𝜙 is not PE have been discussed in [START_REF] Efimov | Robustness of linear timevarying systems with relaxed excitation[END_REF] for ℓ = 1. However, they are slightly technical and can hardly be applied in practice.

The DREM procedure [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF]: To apply the DREM procedure, we start by performing the dynamic regressor extension step, where the goal is to generate an LRE for the same vector of unknown parameters 𝜃 as in (1) but with a square 𝑛 × 𝑛 matrix regressor. To this end, we apply Kreisselmeier's regressor extension scheme given by ( 2), (3) with zero initial conditions; see [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] for other possible schemes. Due to the linearity, the matrices Φ and 𝑌 generated by ( 2) and (3), respectively, satisfy the extended LRE

𝑌 (𝑡) = Φ(𝑡)𝜃 + 𝑊 (𝑡), ∀𝑡 ∈ R + , (8)
where 𝑊 : R + → R 𝑛 is the result of the distortion 𝑤 propagation in [START_REF] Ioannou | Robust Adaptive Control[END_REF]. Next, the mixing step is applied to obtain a set of 𝑛 scalar equations. Recall that for any square and possibly singular 𝑛 × 𝑛 matrix 𝐴, we have adj( 𝐴) 𝐴 = det( 𝐴)𝐼 𝑛 , where adj(•) is the adjoint (also called adjugate) matrix. Multiplying [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF] by adj(Φ(𝑡)) on the left and setting Y (𝑡) := adj(Φ(𝑡)) 𝑌 (𝑡), W (𝑡) := adj(Φ(𝑡)) 𝑊 (𝑡), ∀𝑡 ∈ R + , we get

Y 𝑖 (𝑡) = Δ(𝑡)𝜃 𝑖 + W 𝑖 (𝑡), (9)
where Y 𝑖 , 𝜃 𝑖 , and W 𝑖 are 𝑖-th elements of the vectors Y, 𝜃, and W, respectively, 𝑖 ∈ 1, 𝑛, and the scalar function Δ : R + → R is defined in [START_REF] Saad | Adaptive controllers for discretetime systems with arbitrary zeros: An overview[END_REF]. It is worth noting that for a bounded regressor 𝜙, the vector W is also bounded, and 𝑤 ≡ 0 implies W ≡ 0.

The set of 𝑛 scalar LRE (9) sharing the same bounded scalar regressor Δ is the main result of the DREM procedure. Applying the gradient estimator to [START_REF] Kamalapurkar | Concurrent learning for parameter estimation using dynamic state-derivative estimators[END_REF] as

θ𝑖 (𝑡) = 𝛾 𝑖 Δ(𝑡) Y 𝑖 (𝑡) -Δ(𝑡) θ𝑖 (𝑡) , (10)
where 𝛾 𝑖 > 0 is a scalar tuning parameter, we obtain

θ𝑖 (𝑡) = -𝛾 𝑖 Δ 2 (𝑡) θ𝑖 (𝑡) + 𝛾 𝑖 Δ(𝑡)W 𝑖 (𝑡),
and thus

θ𝑖 (𝑡) = 𝑒 -𝛾 𝑖 ∫ 𝑡 0 Δ 2 ( 𝜏 ) 𝑑 𝜏 𝜃 𝑖 (0) +𝛾 𝑖 ∫ 𝑡 0 𝑒 -𝛾 𝑖 ∫ 𝑡 𝑠 Δ 2 ( 𝜏 ) 𝑑 𝜏 Δ(𝑠)W 𝑖 (𝑠)𝑑𝑠.
Assuming the noise-free scenario 𝑤 ≡ 0, the following prop- the transients for θ𝑖 only. Concerning the case 𝑤 0, the estimator [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF] is input-tostate stable with respect to W 𝑖 if Δ ∈ PE, which is a similar result as for the standard gradient estimator discussed above. Moreover, as it has been shown in [START_REF] Wang | On robust parameter estimation in finite-time without persistence of excitation[END_REF], if W 𝑖 ∈ L 2 and Δ ∉ L 2 , then θ𝑖 is bounded.

To get the exponential convergence with the DREM procedure, the PE property of Δ must be satisfied. As discussed above, it motivates the use of Kreisselmeier's regressor extension scheme (2), (3) yielding the implication [START_REF] Goodwin | Adaptive filtering prediction and control[END_REF]. However, to evaluate the exponential convergence rate or the input-to-state gain with respect to W 𝑖 , one has to estimate the characteristics of the new regressor Δ. These characteristics depend on the excitation of 𝜙 and the parameter 𝑎 in (2); in the following section, we analyze this dependence.

Remark 1. Let us remark that the computation of the adjoint matrix adj(Φ) can be avoided. So the elements Y 𝑖 in (9) can be computed using Cramer's rule as

Y 𝑖 (𝑡) = det Φ 𝑌 ,𝑖 (𝑡) , ∀𝑡 ∈ R + ,
where Φ 𝑌 ,𝑖 is the matrix Φ where the 𝑖th column is replaced by the vector 𝑌 , and 𝑖 ∈ 1, 𝑛. This can be beneficial when only some elements of the parameter vector 𝜃 are estimated.

III. THE EXCITATION-PRESERVING REGRESSOR EXTENSION

A. Excitation propagation Consider Kreisselmeier's regressor extension (2), (3). The applicability of (2), (3) for the DREM procedure and the implication (6) can be derived from the proof of Theorem 4.3.3 in [START_REF] Ioannou | Robust Adaptive Control[END_REF]. However, in that theorem, only the lower bound's positiveness on the smallest eigenvalue of the matrix Φ is established. Extending that result, we present the following theorem providing precise lower bounds for the determinant of the matrix Φ. We also show that the inverse implication in (6) holds supporting a somewhat intuitive observation that the dynamic extension (2) does not create new excitation.

Theorem 1. Consider the bounded signal 𝜙 : R + → R 𝑛×ℓ and let Φ : R + → R 𝑛×𝑛 be a solution of (2) for some initial value Φ(0) = Φ 0 ≥ 0. Let Δ : R + → R be the determinant of Φ. Then if 𝜙 is (𝑇, 𝜇)-PE, then for any positive integer 𝑞 ≥ 1 and for all 𝑡 ≥ 𝑞𝑇, it holds

Δ(𝑡) ≥ 𝜇 𝑛 𝑞 ∑ 𝑘=1 𝑒 -𝑎𝑘𝑇 𝑛 (11) and lim inf 𝑡→∞ Δ(𝑡) ≥ 𝜇 𝑒 𝑎𝑇 -1 𝑛 . (12)
Moreover, the following implication holds

𝜙 ∈ PE ⇔ Δ ∈ PE. (13)
Proof: The proof consists of two parts. First, we show that if 𝜙 is (𝑇, 𝜇)-PE, then the inequalities [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF] and ( 12) hold proving the direct implication in [START_REF] Cho | Composite model reference adaptive control with parameter convergence under finite excitation[END_REF]. Next, we show that the inverse implication in (13) also holds.

Part 1: 𝜙 ∈ PE implies the inequalities [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF], [START_REF] Wang | Fixedtime estimation of parameters for non-persistent excitation[END_REF] and Δ ∈ PE.

The solution of ( 2) is given by

Φ(𝑡) = 𝑒 -𝑎𝑡 Φ(0) + ∫ 𝑡 0 𝜓(𝑡, 𝑠)𝑑𝑠, ∀𝑡 ∈ R + , (14)
where 𝜓(𝑡, 𝑠) := 𝑒 -𝑎 (𝑡 -𝑠) 𝜙(𝑠)𝜙 ⊤ (𝑠).

Consider 𝑡 ≥ 𝑇 and let 𝑞 ≥ 1 be a positive integer number such that 𝑡 ≥ 𝑞𝑇. The integral term in ( 14) can be rewritten as

∫ 𝑡 0 𝜓(𝑡, 𝑠)𝑑𝑠 = ∫ 𝑡 -𝑞𝑇 0 𝜓(𝑡, 𝑠)𝑑𝑠 + 𝑞 ∑ 𝑘=1 ∫ 𝑡 -𝑘𝑇+𝑇 𝑡 -𝑘𝑇 𝜓(𝑡, 𝑠)𝑑𝑠.
For any positive integer 𝑘 ≤ 𝑞 it holds

∫ 𝑡 -𝑘𝑇+𝑇 𝑡 -𝑘𝑇 𝜓(𝑡, 𝑠)𝑑𝑠 = 𝑒 -𝑎𝑡 ∫ 𝑡 -𝑘𝑇+𝑇 𝑡 -𝑘𝑇 𝑒 𝑎𝑠 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠 ≥ 𝑒 -𝑎𝑡 𝑒 𝑎 (𝑡 -𝑘𝑇 ) 𝜇𝐼 𝑛 = 𝜇𝑒 -𝑎𝑘𝑇 𝐼 𝑛 . Then Φ(𝑡) ≥ 𝜇 𝑞 ∑ 𝑘=1 𝑒 -𝑎𝑘𝑇 𝐼 𝑛 + ∫ 𝑡 -𝑞𝑇 0 𝜓(𝑡, 𝑠)𝑑𝑠 + 𝑒 -𝑎𝑡 Φ(0). ( 15
)
For Φ(0) ≥ 0, the sum of the last two terms in the right-hand side of this inequality is a semi positive-definite matrix,

∫ 𝑡 -𝑞𝑇 0 𝜓(𝑡, 𝑠)𝑑𝑠 + 𝑒 -𝑎𝑡 Φ(0) ≥ 0.
Then from [START_REF] Ortega | On dynamic regressor extension and mixing parameter estimators: Two luenberger observers interpretations[END_REF]

Δ(𝑡) ≥ 𝜇 𝑛 𝑒 -𝑎𝑛𝑇 > 0,
and Δ is strictly separated from zero for all 𝑡 ≥ 𝑇.

To get the inequality ( 12), we choose 𝑞 as the largest integer such that 𝑡 ≥ 𝑞𝑇. Then 𝑞 → ∞ as 𝑡 → ∞. Since

lim 𝑞→∞ 𝑞 ∑ 𝑘=1 𝑒 -𝑎𝑘𝑇 = 1 𝑒 𝑎𝑇 -1
, the asymptotic lower bound [START_REF] Wang | Fixedtime estimation of parameters for non-persistent excitation[END_REF] for Δ(𝑡) follows.

Part 2: Δ ∈ PE ⇒ 𝜙 ∈ PE. Now we will show that if Δ is PE, then 𝜙 is also PE. More precisely, we will show that if 𝜙 is bounded and there exist 𝑇 > 0 and 𝜇 > 0 such that for all

𝑡 ∈ R + ∫ 𝑡+𝑇 𝑡 Δ(𝑠) 2 𝑑𝑠 ≥ 𝜇,
then there exist 𝐿 > 0 and 𝛼 > 0 such that for all

𝑡 ∈ R + ∫ 𝑡+𝐿 𝑡 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠 ≥ 𝛼𝐼 𝑛 .
Since the matrix Φ given by ( 14) is bounded for bounded 𝜙, it follows that all its eigenvalues are non-negative and also bounded, and there exists a constant 𝑐 > 0 such that1

𝑐𝜆 𝑚 (Φ(𝑡)) ≥ Δ 2 (𝑡), ∀𝑡 ∈ R + ,
where 𝜆 𝑚 (Φ) is the smallest eigenvalue of Φ. Then

∫ 𝑡+𝑇 𝑡 𝜆 𝑚 (Φ(𝑠)) 𝑑𝑠 ≥ 𝜇 𝑐 . ( 16
)
From ( 14) it follows that for 𝑠 ≥ 𝑡

Φ(𝑠) = 𝑒 -𝑎 (𝑠-𝑡 ) Φ(𝑡) + ∫ 𝑠 𝑡 𝑒 -𝑎 (𝑠-𝜏 ) 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏.
Then recalling [START_REF] Gerasimov | Relaxing the highfrequency gain sign assumption in direct model reference adaptive control[END_REF], for any positive integer 𝑘 it holds

∫ 𝑡+𝑘𝑇 𝑡 𝜆 𝑚 (Φ(𝑠)) 𝑑𝑠 = ∫ 𝑡+𝑇 𝑡 𝜆 𝑚 𝑒 -𝑎 (𝑠-𝑡 ) Φ(𝑡) + ∫ 𝑠 𝑡 𝑒 -𝑎 (𝑠-𝜏 ) 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 𝑑𝑠 ≥ ∫ 𝑡+𝑘𝑇 𝑡 𝜆 𝑚 𝑒 -𝑎 (𝑠-𝑡 ) Φ(𝑡) 𝑑𝑠 + ∫ 𝑡+𝑘𝑇 𝑡 𝜆 𝑚 ∫ 𝑠 𝑡 𝑒 -𝑎 (𝑠-𝜏 ) 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 𝑑𝑠 ≥ 𝑘 𝜇 𝑐 . Note that ∫ 𝑡+𝑘𝑇 𝑡 𝜆 𝑚 𝑒 -𝑎 (𝑠-𝑡 ) Φ(𝑡) 𝑑𝑠 = 𝜆 𝑚 (Φ(𝑡)) ∫ 𝑡+𝑘𝑇 𝑡 𝑒 -𝑎 (𝑠-𝑡 ) 𝑑𝑠 ≤ 1 𝑎 𝜆 𝑚 (Φ(𝑡))
for 𝑎 > 0. Choose 𝑐 0 as 𝑐 0 := 1 𝑎 sup 𝑡 𝜆 𝑚 (Φ(𝑡)). Then we have that for any 𝑘

∫ 𝑡+𝑘𝑇 𝑡 𝜆 𝑚 ∫ 𝑠 𝑡 𝑒 -𝑎 (𝑠-𝜏 ) 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 𝑑𝑠 ≥ 𝑘 𝜇 𝑐 -𝑐 0 . ( 17
)
Note that

𝜆 𝑚 ∫ 𝑠 𝑡 𝑒 -𝑎 (𝑠-𝜏 ) 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 ≤ 𝜆 𝑚 ∫ 𝑠 𝑡 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 ,
and for all 𝑠 satisfying 𝑡 ≤ 𝑠 ≤ 𝑡 + 𝑘𝑇 it holds

𝜆 𝑚 ∫ 𝑠 𝑡 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 ≤ 𝜆 𝑚 ∫ 𝑡+𝑘𝑇 𝑡 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 . Thus ∫ 𝑡+𝑘𝑇 𝑡 𝜆 𝑚 ∫ 𝑠 𝑡 𝑒 -𝑎 (𝑠-𝜏 ) 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 𝑑𝑠 ≤ ∫ 𝑡+𝑘𝑇 𝑡 𝜆 𝑚 ∫ 𝑠 𝑡 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 𝑑𝑠 ≤ ∫ 𝑡+𝑘𝑇 𝑡 𝜆 𝑚 ∫ 𝑡+𝑘𝑇 𝑡 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 𝑑𝑠 = 𝑘𝑇𝜆 𝑚 ∫ 𝑡+𝑘𝑇 𝑡 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 . (18)
Finally, combining ( 17) and ( 18) yields

𝜆 𝑚 ∫ 𝑡+𝑘𝑇 𝑡 𝜙(𝜏)𝜙 ⊤ (𝜏)𝑑𝜏 ≥ 𝜇 𝑐𝑇 - 𝑐 0 𝑘𝑇 .
Since 𝑐 and 𝑐 0 are constants and do not depend on 𝑘, we can choose the positive integer 𝑘 ≥ 1 such that 𝑘 > 𝑐 0 𝑐 𝜇 . Then 𝜙(𝑡) is PE with 𝐿 = 𝑘𝑇 and 𝛼 = 𝜇 𝑐𝑇 -𝑐 0 𝑘𝑇 > 0.

B. Interval Excitation

In the vein of Theorem 1, it can be also shown that the dynamic extension (2) preserves the interval excitation (in the sense of Definition 2) as well.

Proposition 1. Consider the bounded signal 𝜙 : R + → R 𝑛×ℓ and let Φ : R + → R 𝑛×𝑛 be a solution of (2) for some initial value Φ(0) = Φ 0 ≥ 0. Let the signal Δ : R + → R be the determinant of Φ. If the signal 𝜙 is (𝑡 1 ,𝑇, 𝜇)-interval excited for some 𝑡 1 ≥ 0, 𝑇 > 0, and 𝜇 > 0, then the signal Δ is (𝑡 1 ,𝑇, 𝛼)interval excited for some 𝛼 > 0.

Proof: As it is discussed in the proof of Theorem 1, the solution of ( 2) is given by [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF]. Then for

𝑡 1 ≤ 𝑡 ≤ 𝑡 1 + 𝑇 Φ(𝑡) = 𝑒 -𝑎 (𝑡 -𝑡 1 ) Φ(𝑡 1 ) + ∫ 𝑡 𝑡 1 𝑒 -𝑎 (𝑡 -𝑠) 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠, and ∫ 𝑡 1 +𝑇 𝑡 1 𝜆 𝑚 (Φ(𝑡)) 𝑑𝑡 ≥ ∫ 𝑡 1 +𝑇 𝑡 1 𝜆 𝑚 ∫ 𝑡 𝑡 1 𝑒 -𝑎 (𝑡 -𝑠) 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠 𝑑𝑡 ≥ ∫ 𝑡 1 +𝑇 𝑡 1 𝑒 -𝑎 (𝑡 -𝑡 1 ) 𝑓 (𝑡)𝑑𝑡,
where 𝑓 : R + → R is defined as

𝑓 (𝑡) = 𝜆 𝑚 ∫ 𝑡 𝑡 1 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠 .
The function 𝑓 is continuous and non-decreasing, 𝑓 (𝑡 1 ) = 0 and (due to the interval excitation) 𝑓 (𝑡 1 +𝑇) ≥ 𝜇. The function 𝑓 is not necessarily differentiable; however, since 𝜙 is bounded it admits a Lipschitz constant, i.e., there exists a positive constant 𝜌 > 0 such that for any 𝑠 ≥ 𝑡 1 and ℎ ≥ 0 it holds

0 ≤ 𝑓 (𝑠 + ℎ) -𝑓 (𝑠) ≤ 𝜌ℎ,
and 𝜌𝑇 ≥ 𝜇. Define 𝜏 := 𝜇 𝜌 ≤ 𝑇. Then for all 𝑡 ∈ [𝑡 1 + 𝑇 -𝜏, 𝑡 1 + 𝑇] it holds

𝑓 (𝑡) ≥ 𝜌 (𝑡 -(𝑡 1 + 𝑇 -𝜏)) . Thus ∫ 𝑡 1 +𝑇 𝑡 1 𝑒 -𝑎 (𝑡 -𝑡 1 ) 𝑓 (𝑡)𝑑𝑡 ≥ ∫ 𝑡 1 +𝑇 𝑡 1 +𝑇 -𝜏 𝑒 -𝑎 (𝑡 -𝑡 1 ) 𝜌 (𝑡 -(𝑡 1 + 𝑇 -𝜏)) 𝑑𝑡 = 𝜌 𝑎 2 𝑒 -𝑎𝑇 (𝑒 𝑎𝜏 -1 -𝑎𝜏) .
Finally, recalling that Δ(𝑡) ≥ (𝜆 𝑚 (Φ(𝑡))) 𝑛 , the (𝑡 1 ,𝑇, 𝛼)interval excitation follows:

∫ 𝑡 1 +𝑇 𝑡 1 Δ 2 (𝑡)𝑑𝑡 ≥ 𝛼,
where

𝛼 := 𝜌 𝑎 2 𝑒 -𝑎𝑇 𝑒 𝑎𝜇 𝜌 -1 - 𝑎𝜇 𝜌 2𝑛 > 0. (19)
Theorem 1 and Proposition 1 justify using ( 2), (3) for the dynamic regressor extension step of the DREM procedure. Under this choice, the persistence of excitation and the interval excitation properties of the original regressor are always preserved. The obtained bounds [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF], [START_REF] Wang | Fixedtime estimation of parameters for non-persistent excitation[END_REF], and (19) allow for performance evaluation of the DREM-enhanced estimation algorithms, e.g., the convergence rate estimation.

IV. DYNAMICS OF THE REGRESSOR Δ

In Theorem 1, we provide useful lower bounds [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF], [START_REF] Wang | Fixedtime estimation of parameters for non-persistent excitation[END_REF] for the DREM-generated new regressor Δ when Kreisselmeier's regressor extension scheme (2) is used. However, it is also of interest to study the dynamics of Δ. In this section, we present such a study.

Let us consider a time evaluation of Δ that is the determinant of the time-dependent matrix Φ. The matrix Φ is a solution of (2), so Δ obeys Jacobi's formula (see Theorem 8.1 in [START_REF] Magnus | Matrix Differential Calculus with Applications in Statistics and Econometrics, ser. Probabilistics and Statistics[END_REF]):

Δ(𝑡) = tr adj(Φ(𝑡)) Φ(𝑡) , ∀𝑡 ∈ R + ,
where Δ(0) = det (Φ(0)) and tr denotes the matrix trace.

Recall that the eigenvalues of an adjoint matrix can be estimated as follows. Let 𝜆 1,Φ , . . . , 𝜆 𝑛,Φ denote the eigenvalues of Φ. Applying Schur's Lemma, it is then straightforward to show that the eigenvalues of adj(Φ) are given by 𝜆 𝑖,adj(Φ) = Π 𝑗≠𝑖 𝜆 𝑗,Φ , ∀𝑖 = 1, . . . , 𝑛, and for all 𝑖 = 1, . . . , 𝑛 it holds 𝜆 𝑖,adj(Φ) 𝜆 𝑖,Φ = det (Φ) .

In particular

min 𝑖 𝜆 𝑖,adj(Φ) max 𝑖 𝜆 𝑖,Φ = det (Φ) . (21)
Let 𝜆 𝑀 denote the maximum eigenvalue of Φ. Since for 𝑡 ≥ 0, Φ(𝑡) ≥ 0, then 𝜆 𝑀 (𝑡) = 0 implies that all eigenvalues of Φ(𝑡) are zeros, and so are the eigenvalues of adj(Φ(𝑡)). That implies for all 𝑡 ≥ 0 and all 𝑘 ∈ 1, ℓ, 𝜙 𝑘 (𝑡) ⊤ adj(Φ(𝑡)) 𝜙 𝑘 (𝑡) = 0.

On the other hand, if 𝜆 𝑀 (𝑡) > 0, then due to ( 21)

𝜙 ⊤ 𝑘 (𝑡) adj(Φ(𝑡)) 𝜙 𝑘 (𝑡) ≥ |𝜙 𝑘 (𝑡)| 2 Δ(𝑡) 𝜆 𝑀 (𝑡) , ∀𝑡 ∈ R + .
Recall that for the induced matrix norm ∥𝜙∥ it holds ℓ 𝑘=1 |𝜙 𝑘 | 2 ≥ ∥𝜙∥ 2 . Then the derivation above yields the following proposition.

Proposition 2. Let Φ be a solution of (2) and let 𝜆 𝑀 denote the maximum eigenvalue of Φ and Δ = det (Φ). Then:

• if 𝜆 𝑀 = 0, then Δ = 0; • if 𝜆 𝑀 > 0, then Δ(𝑡) ≥ -𝑎𝑛 + ∥𝜙(𝑡) ∥ 2 𝜆 𝑀 (𝑡) Δ(𝑡), ∀𝑡 ∈ R + .
It is also worth noting that an upper bound of the maximum eigenvalue 𝜆 𝑀 can be estimated given an upper bound of 𝜙.

V. SIMULATIONS

To illustrate the results obtained in Section III, we consider the problem of magnitude and phase estimation for sinusoidal signals with known frequencies. To illustrate that the presented results hold not only for scalar linear regression models, we choose ℓ = 2 and 𝑛 = 3. Consider Obviously, the values 𝐴, 𝐵, and 𝜓 can be reconstructed given 𝜃.

It is straightforward to verify that the regressor 𝜙 is (2𝜋, 2𝜋)-PE, i.e., for all 𝑡 ≥ 0

∫ 𝑡+2 𝜋 𝑡 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠 ≥ 2𝜋𝐼 3 . For simulations, we set 𝐵 = -1, 𝐴 = 2 √ 2, and 𝜓 = 3 4 𝜋 providing 𝜃 = -1 -2 1 ⊤ .
First, we consider the noise-free scenario, i.e., 𝑤 ≡ 0. We apply the standard gradient algorithm [START_REF] Lewis | Reinforcement learning and feedback control: Using natural decision methods to design optimal adaptive controllers[END_REF] with Γ = 𝐼 3 and θ (0) = 0; the simulation results are depicted in Fig. 1. Since 𝜙 is PE, the estimation error θ converges to zero; however, the transients exhibit oscillations. Then we apply the dynamic regressor extension (2), [START_REF] Ioannou | Robust Adaptive Control[END_REF], where the only tuning parameter is chosen as 𝑎 = 0.1. The new regressor Δ computed as [START_REF] Saad | Adaptive controllers for discretetime systems with arbitrary zeros: An overview[END_REF] is depicted in Fig. 2 with the lower bound [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF] and with the asymptotic lower bound [START_REF] Wang | Fixedtime estimation of parameters for non-persistent excitation[END_REF]. Finally, the estimation error transient θ for the DREM procedure with the gradient estimator [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF] with 𝛾 𝑖 = 1, 𝑖 = 1, 2, 3, are depicted in Fig. 3 and illustrate performance improvement for the standard gradient estimator; note the monotonicity of the transients.

Next, to evaluate the measurement noise sensitivity, we let both 𝑤 1 and 𝑤 2 be random uniformly distributed between -1 [START_REF] Lewis | Reinforcement learning and feedback control: Using natural decision methods to design optimal adaptive controllers[END_REF] AND THE DREM (2), ( 3), [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF]. The results are summarized in Table I and illustrate the filtering properties of Kreisselmeier's regressor extension scheme.

VI. CONCLUSION

The recently proposed DREM procedure provides significant performance improvement in linear regression parameter estimation, where the principal degree of freedom is the choice of the dynamic extension. In this paper, we have studied a particular choice of this procedure as Kreisselmeier's regressor extension (2), (3) that ensures the preservation of the persistence of excitation property. We have analyzed the excitation propagation properties and estimated the lower bound of the new regressor Δ. We have also shown that the interval excitation property is also preserved and provided quantitative analysis of the new regressor's interval excitation. The obtained results are valuable for further use of the DREM procedure for linear regression estimation.

  erties hold: P1: Δ ∉ L 2 ⇔ | θ| → 0 asymptotically; P2: Δ is PE ⇔ | θ| → 0 exponentially fast; P3: (the element-wise monotonicity) for all 𝑖 ∈ 1, 𝑛 for 𝑡 𝑏 ≥ 𝑡 𝑎 ≥ 0 it holds | θ𝑖 (𝑡 𝑏 )| ≤ | θ𝑖 (𝑡 𝑎 )|; P4: (the element-wise tuning) variations in the gain 𝛾 𝑖 affect

Substituting ( 2 )

 2 , we obtain for all 𝑡 ∈ R + : Δ(𝑡) = tr -𝑎 adj(Φ(𝑡)) Φ(𝑡) + adj(Φ(𝑡)) 𝜙(𝑡)𝜙(𝑡) ⊤ = -𝑎𝑛Δ(𝑡) + tr adj(Φ(𝑡)) 𝜙(𝑡)𝜙(𝑡) ⊤ , where we recall that the dimension of 𝜙 is 𝑛 × ℓ. Let 𝜙 𝑘 ∈ R 𝑛 be the 𝑘-th column of 𝜙, 𝑘 ∈ 1, ℓ. Due to elementary properties of the matrix trace function, it follows for all 𝑡 ≥ 0, tr adj(Φ(𝑡)) 𝜙(𝑡)𝜙(𝑡) ⊤ = ℓ ∑ 𝑘=1 𝜙 𝑘 (𝑡) ⊤ adj(Φ(𝑡)) 𝜙 𝑘 (𝑡), and we obtain Δ(𝑡) = -𝑎𝑛Δ(𝑡) + ℓ ∑ 𝑘=1 𝜙 𝑘 (𝑡) ⊤ adj(Φ(𝑡)) 𝜙 𝑘 (𝑡).

𝑦 1 (Fig. 1 .Fig. 2 .

 112 Fig.1. The estimation error θ under the standard gradient algorithm[START_REF] Lewis | Reinforcement learning and feedback control: Using natural decision methods to design optimal adaptive controllers[END_REF].

Fig. 3 .

 3 Fig.3. The estimation error θ under the DREM procedure (2), (3) and the gradient estimator[START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF].

and 1 .

 1 5 for (7) 12.6 18.8 8.3 MSE•10 5 for (2), (3),[START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF] 0.85 0.44 0.44 For results comparison, we compute the mean squared error value, 𝑀𝑆𝐸 ( θ𝑖 ) )𝑑𝑠, 𝑖 = 1, 2, 3.

  it follows that for all 𝑡 ≥ 𝑞𝑇 the smallest eigenvalue of Φ(𝑡) is not less than 𝜇 𝑞 𝑘=1 𝑒 -𝑎𝑘𝑇 . Thus

		𝑞	𝑛
	det (Φ(𝑡)) ≥ 𝜇 𝑛	∑	𝑒 -𝑎𝑘𝑇	,
		𝑘=1
	and (11) follows.
	The PE property follows by noting that for 𝑡 ≥ 𝑇 it holds

TABLE I MSE

 I OF ESTIMATION ERROR θ𝑖 , 𝑖 = 1, 2, 3, FOR THE STANDARD GRADIENT ALGORITHM

A conservative estimate of 𝑐 is 𝑐 = sup 𝑡 𝜆 𝑀 (Φ(𝑡 ) )

2𝑛-1 , where 𝜆 𝑀 (Φ)is the largest eigenvalue of Φ.