Guillaume Foret

Pascal Bertolino

David Cibaud

Partition projection in videos by global and local block-matching

come

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Imene Chaieb
email: imene.chaieb@francetelecom.com

Jean-Louis Le Roux
email: jeanlouis.leroux@francetelecom.com

Bernard Cousin
email: bernard.cousin@irisa.fr

MPLS-TE Routing: Adopting a Generic Architecture and Evaluating Various Implementation Approaches

Various Multi-Protocol Label Switching-Traffic Engineering (MPLS-TE) Routing Systems have been proposed in the literature to achieve optimization of resources utilization, Quality-of-Service (QoS) and Fast Recovery. This paper proposes a generic architecture for MPLS-TE Routing Systems, which aims to ease the classification and the analysis of these systems. Then this paper defines a set of MPLS-TE classification criteria. The combination of these criteria leads to the identification of main families of MPLS-TE Routing Systems which are finally compared and qualitatively evaluated according to a set of metrics.

I. INTRODUCTION

The emergence of multi-service IP-centric networks, which transport value added services such as VoIP (Voice/Video Telephony over IP), IP TV, Video on Demand and VPN traffic, leads to the requirements for strict QoS delivery (delay, jitter, packet loss), and high availability. Given the drastic increase of last miles capacities, and the reduction of the gap between core and access bandwidth, the over-provisioning approaches followed for years by operators in core and backhaul networks, so as to ensure QoS, are no longer a panacea today in backhaul networks, and may no longer be a valuable approach in core networks at mid term. Hence Traffic Engineering (TE) mechanisms are required so as to optimize network resource utilization, that is to maximize the amount of traffic that can be transported while ensuring the quality of service, with as main objective to reduce network costs and postpone investments. In order to address the traffic increase and satisfy the QoS requirements of multimedia applications, various TE mechanisms are proposed, among those MPLS-TE, a connection oriented mechanism based on the MPLS forwarding paradigm, well suited to TE thanks to its Explicit Routing capabilities. The MPLS-Traffic Engineering approach [START_REF] Awduche | Requirements for traffic engineering over MPLS, rfc 2702[END_REF] allows setting up explicitly routed Traffic Engineering-Label Switched Path (TE-LSP) whose path satisfy a set of traffic engineering constraints, including bandwidth. MPLS-TE combines explicit routing capabilities of MPLS with a constraint based routing paradigm based on dynamic resources discovery (ISIS-TE [START_REF] Smit | IS-IS extensions for traffic engineering[END_REF], OSPF-TE [START_REF] Katz | Traffic engineering (TE) extensions to ospf version 2[END_REF]), constrained path computation, and distributed LSP signalling and resources reservation (RSVP-TE) [START_REF] Awduche | RSVP-TE: Extensions to rsvp for LSP tunnels, rfc 3209[END_REF]. MPLS-TE ensures Traffic Engineering functions such as network resources optimization, strict QoS guarantees, and fast recovery upon link or node failures. For a load balancing purpose, a set of two or more TE-LSPs may be used to route a given aggregate traffic demand between two end points. The TE-Trunk concept defined in [START_REF] Awduche | Requirements for traffic engineering over MPLS, rfc 2702[END_REF] allows accounting for such load balancing. A TE-Trunk is defined as a set of one or more LSPs used to carry an aggregate traffic demand between two points for a given service class. A TE-Trunk is characterized by its reserved bandwidth and a set of TE parameters (e.g. class of service, delay...). In order to efficiently route flows in TE-Tunks, complementary mechanisms are required on top of the standard MPLS-TE control plane. This includes essentially a TE-Trunk Utilization function, responsible for an efficient routing of a set of N flows in a set of M TE-LSPs of one or more TE-Trunks, along with an Adaptability mechanism responsible for adapting the TE-Trunks (LSPs resizing/creation/suppression) according to traffic matrix changes and/or topology modifications (failures). These Utilization and Adaptability functions are actually intimately linked to the MPLS-TE Path Computation function. The combination of the MPLS-TE control plane building blocks (Routing, Path Computation, Signaling) with these additional functions (Utilization and Adaptability) form together what we call a MPLS-TE Routing System. In the literature, there are papers that focus on MPLS-TE Path Computation. Some of these solutions ( [START_REF] Kodialam | Minimum interference routing with applications to MPLS traffic engineering[END_REF] [6] [START_REF] Suri | Profile-based routing: A new framework for MPLS traffic engineering[END_REF]) propose efficient algorithms to place TE-LSPs in networks and satisfy a pre-defined set of flow requests, others take interest to this functionality but in case of network failure. There are also papers which account for TE-LSPs Utilization [START_REF] Phuvoravan | Fast time scale control for MPLS traffic engineering[END_REF] and for Adaptation mechanisms [START_REF] Battiti | A load balancing scheme for congestion control in MPLS networks[END_REF]. Others, study flow admission control and its application in MPLS-TE networks [START_REF] Bosco | Edge distributed admission control in MPLS networks[END_REF]. However, a global study that covers the overall architecture of an MPLS-TE Routing System is not considered. In the remainder of this paper, we firstly propose, a generic architecture to describe the functions of a MPLS-TE Routing System and their interaction. Then, we rely on this architecture to define a set of MPLS-TE classification criteria and we combine these criteria, so as to identify main MPLS-TE System families. Finally we propose a qualitative evaluation and comparison of these families according to a set of evaluation metrics.

II. FUNCTIONAL ARCHITECTURE OF MPLS-TE ROUTING SYSTEMS

In this section we propose a generic architecture to describe an MPLS-TE Routing System (Fig. 1). This is a functional architecture that helps in covering a large solution spectrum. It is comprised of a set of functions also called building blocks. Some of these building blocks are running on routers, others may be running either on routers or on one or more network servers. We distinguish standard MPLS-TE blocks and implementation specific blocks:

• Standard MPLS-TE functions include the TE Topology Discovery function ensured by an IGP-TE protocol (either OSPF-TE or ISIS-TE) and the LSP Signalling function ensured by the RSVP-TE protocol. These standard functions are located in routers.

• Implementation specific functions include the TE-Trunk Agent, the TE-Trunk Path Computation, the TE-Trunk Adaptation and the TE-Trunk Utilization functions. These functions may be located in routers or externalized in one or more network servers. This also comprises the TE-Manager function which is always located in a network server.

In the below sections we focus on the five building blocks in charge of resource optimization: TE-Manager (TM), TE-Trunk AGent (TAG), TE-Trunk Computation (TC), TE-Trunk ADaptation (TAD), and TE-Trunk Utilization (TU).

A. TE-Manager

The TE-Manager (TM) is a functional entity that takes the decision to setup/release/modify TE-Trunks by relying on the forecast traffic matrix (1), that is the set of aggregate traffic demands between each pair of Edge Routers. It sends TE-Trunk setup/deletion/modification requests to the set of one or more TE-Trunk Agents (2). This function is optional, TE-Trunks may be defined by the operator and may be directly configured on TE-Trunk Agents.

B. TE-Trunk Agent

D. TE-Trunk Adaptation

The TE-Trunk Adaptation block (TAD) is in charge of adapting TE-Trunk size to the actual traffic load. It increases TE Trunk size (i.e. it increases the amount of bandwidth reserved for the TE-Trunk), so as to anticipate congestion issues, when the load between a pair of nodes increases; and decreases TE Trunk size so as not to waste unused bandwidth when the load between a pair of nodes decreases. Verification of the TE-Trunk load can be done in a timer driven manner, in which case the TE-Trunk load in the TE-Trunk Utilization databases is periodically checked by the Adaptation block (13) or it can also be done in an event driven manner, in which case the TE-Trunk Utilization block notifies the Adaptation block that a TE-Trunk is congested or is going to be congested [START_REF] Vasseur | Path computation element (PCE) communication protocol (PCEP)[END_REF]. Note that this block is optional and may not be used in every MPLS-TE Routing Systems.

III. IMPLEMENTATION CONSIDERATION: DISCUSSION AND EVALUATION

In the previous section we proposed a functional architecture for MPLS-TE systems, which includes in addition to standard MPLS-TE blocks, specific blocks such as TE-Trunk Computation, TE-Trunk Adaptation and TE-Trunk Utilization. This architecture may help classifying MPLS-TE mechanisms and improve the design of MPLS-TE systems. An MPLS-TE routing system corresponds actually to a specific implementation of this architecture. The blocks of this generic architecture may be located in different elements (Centralized on Network servers or distributed in Edge routers). The performances of an MPLS-TE routing system, in terms of scalability, reactivity and optimality actually depend on various implementation options, including the repartition of the functions. Before discussing these options, a description of some classification criteria which will help the discussion, is proposed.

A. MPLS-TE classification criteria

Several criteria are identified to arrange the various approaches for implementing an MPLS-TE Routing Systems. We distinguish the following:

1) Time Scale:

• Offline (Of f ): TE-Trunks are computed and established periodically based on forecast traffic matrices. This mode allows more time for path computation. This implies that there is no TE-Trunk Adaptation and there is no LSP rerouting upon network failures.

• Online (On): TE-Trunks are modified (TE-Trunks resizing, LSPs re-routing, LSPs creation/deletion) according to traffic matrix evolution, or network failure. In such mode, path computation time should be minimized so a to ensure good reactivity.

B. TE evaluation metrics

In order to perform a qualitative evaluation of the efficiency and the applicability of MPLS-TE Routing Systems, a set of metrics are specified :

• Optimality (Opt): The ability to maximize the amount of traffic that can transit in a network with guaranteed QoS. Different performance objectives can be considered such as the residual bandwidth on the most loaded link, the cumulative bandwidth consumption, or, under congestion, the number of rejected requests or the amount of rejected bandwidth.

• Scalability (Sca): The ability to scale well with an increase of any of the following parameters: Number of links/nodes, number of TE-Trunks, number of TE-LSPs, and number of external elements (e.g. PCEs), etc.

• Stability (Sta): The ability to avoid route oscillations and to minimize any perturbation on the network resulting from the

C. Function Distribution

We discuss in this section the distribution of each architecture's function and its impact on the MPLS-TE Systems performances. There are some functions of the architecture which should be only Distributed, others only Centralized on a network server and others which can be either Distributed or Centralized. When two functional blocks that need to communicate, are not located in the same element (e.g. One is located in an Edge Router and the other is located in the TE server) a standard communication protocol is required to manage the communication and the cooperation between the two blocks. In contrast, when these two blocks are located in the same element (e.g. an Edge Router) a communication protocol is not required, such communication may rely on a software interface (e.g. Inter-Process Communication API).

• The MPLS-TE protocols (RSVP-TE and IGP-TE) are Distributed on the routers (note that the IGP-TE may passively run on the Path Computation Block when it is Centralized, so as to feed the TED).

• The TE-Trunk Utilization block should be Distributed as it is in charge of the routing of incoming flows within the TE-Trunks and of TE-Trunk load measurement on Edge Routers. The centralization of this block may affect the reactivity of the MPLS-TE System due to the amount of information to be communicated between Edge Routers and the TE server.

• By definition, the TE-manager is always Centralized.

• The TE-Trunk Agent can either be Centralized on a network server or Distributed on TE-Trunk Edge Routers. In a Distributed mode, it maintains only TE-Trunks for which it is the headend. In a Centralized scenario, the TE-Trunk Agent has a global knowledge of all the TE-Trunks. In this case, a communication protocol is required to communicate with Edge Routers (LSP configuration). This may rely for instance on a standard configuration protocol (e.g. XML conf or SNMP). The notification of network failures should be event-driven (e.g. SNMP traps) so as to minimize the amount of information between the Edges Routers and the TE-server.

• The TE-Trunk Adaptation function should always be linked to the TE-Trunk Agent, that is if the TE-Trunk Agent is Centralized (respectively Distributed), the TE-Trunk adaptation is also Centralized (respectively Distributed). When the TE-Trunk Adaptation is Centralized, a communication protocol is required between the TE-Trunk Utilization block located in Edge Routers and the TE-Trunk Adaptation, so as to inform about the LSP load. Such notification should be event-driven so as to minimize the communication between Edge Routers and the TE-server (it is not necessary for the Adaptation block to consult periodically the TE-Trunks Utilization database. The TE-Trunk Utilization block sends a message to the Adaptation block only when a threshold is reached). The separation of these two functions (the TE-Trunk Adaptation and the TE-Trunk Agent) would not bring any value and would require the communication of a lot of information. A PCE is defined as an entity that is capable of computing a network path based on a network graph, and applying computational constraints [START_REF] Farrel | Path computation element (PCE) architecture[END_REF]. A PCE serves path computation requests sent by Path Computation Clients (PCCs). The PCE communication Protocol (PCEP) has been defined to support communication between PCCs and PCEs (see [START_REF] Vasseur | Path computation element (PCE) communication protocol (PCEP)[END_REF]). Here the TE-Trunk Agent acts as a PCC and the TE-Trunk Computation block acts as a PCE. Note that the PCE based architecture may also apply when the TE-Trunk Agent and the TE-Trunk Path Computation functions are centralized but not located in the same TE server.

D. Evaluation

By combining the various criteria defined previously and by considering also the Architecture's function distribution discussed in the previous section, a set of MPLS-TE Systems families or approaches are identified and evaluated (the table I summarizes the evaluation results):

1 Internet Engineering Task Force 1) The On/Dis/Unc MPLS-TE approach: This is an Online Distributed based approach where the requests are handled in a Uncoordinated manner. This approach achieves "bad" performances in terms of optimality because of its Uncoordinated and Online schemes. In return, as each Edge Router handles only its own requests, it offers "good" scalability for the MPLS-TE System. However, as the online mode implies LSP creation/deletion/resizing, and the Uncoordinated mode may imply some TE mechanisms such as preemption and crankback, the system is "poor" in terms of stability. However, according to the reactivity, this approach offers, due to its On/Dis scheme, "good" performances. In fact, all functions of the MPLS-TE System are located in the same element (the Edge Router) and hence this does not require heavy communication between Edge Router and TE-server.

2) The On/Dis/Coo MPLS-TE approach: This approach operates in a Coordinated manner with other network Edge Routers. Thus, network resources usage is optimized because each Edge Router computes paths by taking into account all TE-Trunks demands in the network but not as well as if the Offline mode was used where there is no time constraint. Hence, it can ensure "good" performances in terms of optimality. This option offers "bad" performances in terms of scalability, Edge Routers are likely to be saturated because they have to exchange all the information about their own TE-Trunks/LSPs. This cannot scale because of the number of TE-Trunks in the network and their activity rates (resizing,...). So, it seems not relevant to let all the Edge Routers handle all requests. However, as there is no pre-emption or crankback mechanisms in the Coordinated mode, this approach may result in "good" performances in terms of stability. But, it cannot ensure "high" stability because of the Online mode. The "high" reactivity can be also slightly affected by the Coordinated mode. In fact, in a Coordinated mode, each Edge Router takes into account all TE-Trunk requests to compute TE-Trunks paths which may take potentially long time. Thus the reactivity is affected.

3) The On/Cen/Unc MPLS-TE approach: This approach may achieve "bad" performances in terms of optimality because of its Uncoordinated and Online schemes. Also, it may suffer from "poor" scalability performances because of the Centralized mode as the TEserver may not scale with a network size increase. As the online mode implies LSP creation/deletion/resizing, and the Uncoordinated mode may imply some TE mechanisms such as preemption and crankback, the system also suffer from "poor" stability. This approach also achieves "poor" performances in terms of reactivity because of the Centralized mode because the TE-server and the Edge Routers should always communicate. For instance, in case of a topology change (network failure), the recovery upon network failure would imply the following sequence: (1)-Failure notification on the TEserver (it may rely on an SNMP trap), (2)-paths computation and (3)-communication of the new paths to all Edge Routers, which may take long time.

4) The On/Cen/Coo MPLS-TE approach: Compared to the previous approach (On/Cen/Unc), this approach improves the optimality and so ensure "good" performances as we move to a Coordinated mode but it is not highly improved as we are still in an Online mode where the path computation is time constrained. Like the previous approach (On/Cen/Unc), this approach may suffer from "poor" scalability performances because of the Centralized mode and like the (On/Dis/Coo) approach, it may result in "good" performances in terms of stability because there is no preemption or crankback mechanisms in the Coordinated mode. But, it cannot ensure "high" stability because of the Online mode. In terms of reactivity, this approach offers "bad" performances. For instance, in a network failure case, the recovery would imply the following sequence: (1)-Failure discovery on the TE-server, (2)-Coordinated path computation and (3)-communication of the new paths to all Edge Routers, which may take potentially long time.

5) The Off/Dis/Unc MPLS-TE approach: This approach adopts the Offline mode where TE-Trunks paths, potentially including Backup paths, are pre-computed periodically without real time computation constraints, but it remains "poor" in terms of optimality due to its Dis/Unc scheme. In return, this scheme allows to avoid the message exchange between the TE-server and the Edge Routers and to accomplish "good" scalability performances as each Edge Router handle only its own TE-Trunk requests. According to the stability, this approach can achieve "high" performances as there is no TE-Trunk Adaptation (Trunk suppression/creation/resizing) and LSP re-routing. All Offline approaches are by definition not reactive ("bad" reactivity) and stable ("high" stability).

6) The Off/Dis/Coo MPLS-TE approach: This approach can achieve "high" performances in terms of optimality with its Off/Coo because TE-Trunk paths are computed in Offline mode taking into account all TE-Trunks demands in the network which offers a global network optimization with no time constraint. In contrast, this approach offers "bad" performances in terms of scalability because of its Dis/Coo scheme as each Edge Router has to maintain all the TE-Trunk requests.

7) The Off/Cen/Unc MPLS-TE approach: As the Offline mode is adopted, the performances of the MPLS-TE system in terms of Optimality may be improved. But The Uncoordinated scheme of this approach affects these performances. So, like the Off/Dis/Unc approach, this approach suffers from "poor" optimality performances. This approach may be "poor" in terms of scalability because of the Centralized mode as the TE-server may not scale with network size increase (node number, etc.).

8) The Off/Cen/Coo MPLS-TE approach: This approach can ensure "high" performances in terms of optimality. In fact, TE-Trunks placement can be drastically optimized because the TE-Trunk Path Computation function knows all the requests and can perform a Coordinated path computation, with no time limitation. Like the previous approach (Off/Cen/Unc), this approach may be "poor" in terms of scalability due to its Centralized scheme.

As shown in the previous sections, TE criteria either alone or combined can influence the performances of a MPLS-TE System:

• An approach which adopts Centralized scenario, may suffer from scalability and reactivity issues.

• An approach which adopts Online mode may suffer from network stability problems.

• An approach which adopts Offline mode lacks in reactivity.

• An approach which adopts a Dis/Coo scenario faces scalability problems.

• An approach which adopts a Dis/Unc scenario affects the performances of an MPLS-TE System in terms of optimality. The result of this qualitative evaluation is that special care must be taken when combining different TE approaches to build an MPLS-TE Routing System.

IV. CONCLUSION MPLS-TE is being deployed by network operators to better optimize their network resources. The routing in MPLS-TE networks is a large and open issue. Studies aimed to improve MPLS-TE routing in terms of scalability, stability, robustness, optimality and survivability. In this paper, we have proposed a generic architecture

Fig. 1 .

 1 Fig. 1. Generic architecture for MPLS-TE Routing Systems

  The TE-Trunk Agent (TAG) is the heart of the architecture. It controls the TE-Trunks establishments/modifications/deletions in the network. It coordinates the actions of the TE-Manager, the TE-Trunk Adaptation, the TE-Trunk Path computation, the TE-Trunk Utilization and the LSP Signalling blocks. It handles TE-Trunk setup/deletion/modification requests sent by the TE-Manager (1), and TE-Trunk modification requests sent by the TE-Trunk Adaptation block (3). It sends TE-Trunk Computation requests to the TE-Trunk Path Computation block (4). Once paths are computed the TE-Trunk Agent sends LSP setup requests to the RSVP-TE module (7) so as to signal the TE-LSPs along the computed paths. Once the TE-LSPs are setup, the TE-Trunk Agent feeds the TE-Trunks Database which contains information related to the established TE-Trunks (TE-Trunk constraints, TE Trunk paths, etc.) (8). It also communicates the established TE-Trunks and the corresponding LSPs to the TE-Trunk Utilization block (9). In an "Online mode", it may communicate with the IGP-TE (10) and LSP Signalling blocks so as to be notified of network (link/node) and TE-LSPs failures. This communication allows the TE-Trunk Agent to detect failures and call the TE-Trunk Path Computation block so as to reroute the TE-Trunks on paths avoiding failed elements. C. TE-Trunk Path Computation The TE-Trunk Computation block (TC) is a fundamental building block in MPLS-TE Routing Systems. It has to find TE-Trunks paths by operating on the Traffic Engineering Database (TED) (5) fed up by the IGP (6) and considering the TE-Trunks constraints. It handles Trunk Computation requests sent by the TE-Trunk Agent. A request may correspond to a single TE-Trunk or to a set of TE-Trunks. The request may be a Trunk setup request or a Trunk modification one. The output for a given Trunk is a path or a set of paths whose cumulative bandwidth fits the Trunk(s) request.

2 )Fig. 2 .

 22 Fig. 2. An Example of a Centralized MPLS-TE System based on the architecture • Distributed (Dis): The function is Distributed on multiple computing elements. Figure 3 illustrates an MPLS-TE system based on our architecture where TM is centralized and others blocks: TAG, TC, TAD, TU, IGP-TE and RSVP-TE are distributed.

Fig. 3 .

 3 Fig. 3. An Example of a Distributed MPLS-TE System based on the architecture

•

  The TE-Trunk Path Computation block may be Distributed or Centralized. (1) If the TE-Trunk Agent is Centralized, the TE-Trunk Path Computation block should also be Centralized (the Coordinated mode) because the separation of these two functions would not bring any value and would require the communication of a lot of information. (2) But, if the TE-Trunk Agent is Distributed, the TE-Trunk Path Computation block may either be Distributed or Centralized. When the TE-Trunk Agent is distributed and the TE-Trunk path Computation is Centralized, the TE-Trunk Path Computation remains Uncoordinated because the TE-Trunk Agents send requests independently. This requires a path computation communication protocol between the TAG and the TC. Such a protocol is under definition within the Path Computation Element (PCE) working group in the IETF 1 .

TABLE I EVALUATION

 I OF DIFFERENT MPLS-TE APPROACH COMBINATIONS * * " high, " * * " good, " * " poor, "-" bad) for MPLS-TE Routing Systems, that combines MPLS-TE functional blocks such as TE-Trunk Computation, TE-Trunk utilization and TE-Trunk Adaptation. This generic architecture is proposed to facilitate the classification of MPLS-TE Routing solutions, to improve existing mechanisms and to propose new solutions. By relying on this architecture, we can identify and evaluate a set of MPLS-TE Routing approaches using several evaluation metrics. We have showed that some TE criteria, either alone or combined, can influence the performances of a MPLS-TE Routing System.

		Optimality	Scalability	Stability	Reactivity
	On/Dis/Unc	-	* *	*	* * *
	On/Dis/Coo	* *	-	* *	* *
	On/Cen/Unc	-	*	*	*
	On/Cen/Coo	* *	*	* *	*
	Off/Dis/Unc	*	* *	* * *	-
	Off/Dis/Coo	* * *	-	* * *	-
	Off/Cen/Unc	*	*	* * *	-
	Off/Cen/Coo	* * *	*	* * *	-
	(" *


David Coeurjolly

Pierre Gueth Arskan

Jacques-Olivier Lachaud

Regularization of Voxel Art

Keywords: Computing methodologies → Mesh geometry models, Volumetric models, Shape analysis, Digital Geometry, Meshing, Voxel Art, Variational Approach

Figure 1: First row, from left to right: input voxel art decomposed into two labels (sea, island, 64 × 68 × 40) and three interfaces, piecewise smooth regularization with consistent interfaces between labels. Second row: high resolution regularization and low resolution voxel art reconstruction.

INTRODUCTION

In Computer Graphics, the representation of complex objects as data attached to voxels or elements of a regular grid becomes more and more familiar. A key feature of voxel based modeling is that many geometry and topology processing tasks (collision detection, constructive solid geometry, ray-object intersection, homotopic transforms. . . ) are trivially or efficiently defined within this model. Together with very efficient data structures [START_REF] Kämpe | High Resolution Sparse Voxel DAGs[END_REF], voxel grids are widely used to represent or simulate complex phenomena that are volumetric, multi-material and/or dynamic (e.g. [START_REF] Wojtan | Animating Corrosion and Erosion[END_REF]). Voxel based techniques allow efficient rendering by global illumination [START_REF] Crassin | GigaVoxels: ray-guided streaming for efficient and detailed voxel rendering[END_REF]]. Voxel art and low resolution voxel modeling are also present in the game industry thanks to some dedicated modeling tools [@ephtracy [n. d.]]. In the spirit of [START_REF] Kopf | Depixelizing pixel art[END_REF] which addresses the issue of depixelizing pixel art, our method transforms the crispy representation of voxel surfaces into a smooth reconstruction even for low resolution objects. In this work, we present a new variational formulation to efficiently regularize interfaces of voxel sets and partitions in multi-labeled 3D images. Starting from digital surfaces defined as boundaries of voxel sets, we construct a piecewise smooth regular quadrangulation that captures the singularities of the input voxel geometry. As we maintain a mapping between original digital surface quads and regularized ones, we can transfer information (e.g. color, embossing or bump map) from the multilabeled image to the regularized interfaces. Such smooth surfaces can either be used for visualization purposes, or as an intermediate representation for later geometry processing tasks.

In related works, the popular Marching-Cubes (MC) extract isosurfaces from volumetric data by gluing triangles defined by local configurations [START_REF] Lorensen | Marching cubes: A high resolution 3D surface construction algorithm[END_REF]. When Hermite data are available (points and exact normal vector field), Dual-Contouring (DC) methods solve a quadratic problem in each MC local configuration to construct a smoother dual iso-surface [START_REF] Ju | Dual contouring of hermite data[END_REF]. Despite many extensions (adaptiveness, manifoldness. . . ), the overall reconstruction is highly sensitive to normal and position errors, as illustrated in the supplementary. On binary grids, [START_REF] Chica | Pressing: Smooth isosurfaces with flats from binary grids[END_REF] proposes a complex two-step algorithm combining a decomposition of the surface into maximal flat patches and a specific smoothing step. We propose a GPU friendly simpler yet robust formulation. Related works are also found in volumetric meshing (e.g. [START_REF] Faraj | Multi-Material Adaptive Volume Remesher[END_REF]) but mainly focus on the interior regularity of 3D mesh rather than on the interfaces. We process here only multilabel interfaces and thus obtain better piecewise smooth surfaces even on low resolution data. As we regularize surfaces, variational remeshing approaches can also be considered (e.g. [START_REF] Zhang | Guided mesh normal filtering[END_REF], see the supplementary). We provide here an energy formulation dedicated to digital surfaces which is simple, robust and easy to implement efficiently. Finally, for voxel art reconstruction, the work of Muniz et al. [START_REF] Muniz | Polygonal Mesh Extraction From Digital Voxel Art[END_REF]] is mainly focused on colorimetric transfers between voxel objects and MC-like surfaces. Our approach follows a similar idea for color mapping but uses a robust regularization of surfaces geometry.

METHOD OVERVIEW

Let us consider a single voxel object and its boundary defined as isothetic quads. The core of the method relies on a quadratic energy minimization problem. If P denotes the vertices of the input digital surface, F the set of (quadrilateral) faces and n f an estimated normal vector on the face f , we want the quad surface vertex positions P * that minimizes the following energy function:

E( P ) := α n i =1 ∥p i -pi ∥ 2 + β f ∈F e j ∈∂f (e j • n f ) 2 + γ n i =1 ∥ pi -bi ∥ 2 . (1)
where " • " is the standard R 3 scalar product, e j ∈ ∂ f is an edge of the face f (and is equal to some p kp l ) and bi is the barycenter of the vertices adjacent to pi . The first term is a data attachment term so that the regularized surface stays relatively close to the original one. The second one forces the regularized quad to be perpendicular to the input normal vector field. The last one is a fairness term that enhances the quad aspect ratio and avoids degenerate quads. Each term being quadratic, a unique minimizer P * exists. Furthermore, closed forms of the energy gradients can be expressed as linear operators. The consequence is twofold. First, one can obtain P * by solving a linear system (e.g. using robust Cholesky decomposition). Second, we can approximate solutions converging to P * by a simple iterative gradient descent, which allows interactive regularization with a GPU implementation (less than 3ms per gradient step for the Lake scene in the teaser, visually acceptable solution in less than 10 steps). Note that some terms of the energy function in (1) appeared in various forms in previous works on triangular or quad mesh processing. The originality of the contribution consists in an efficient minimization of (1) and its versatility for voxel art regularization as detailed below.

In a mathematical framework where the digital object is the voxelization of a continuous manifold X , we can show that if the normal vector field {n f } converges to the normal vector field on ∂X as the grid step tends to zero (e.g. using [START_REF] Coeurjolly | Multigrid Convergent Principal Curvature Estimators in Digital Geometry[END_REF]), then the regularized surface P * is close to ∂X in the l 2 sense. The method is stable and robust both to position and normal vector perturbations. Experimentally, we show in Supplementary that our approach outperforms Marching-Cubes or Dual-Contouring approaches when the normal vector field is noisy, and is way more stable to perturbations of positions.

Even if the variational formulation (1) is a global quadratic energy, its evaluation and its gradient are local and can be extended to regularize complex multi-labeled interfaces. On multi-labeled images, we rely on the limited number of topological configurations of digital interfaces: each original quad is shared by two voxels with different labels and each edge is a junction of at most 4 interfaces. In the energy formulation, an edge with more that two adjacent quads contributes more than twice to the energy function (1) (in the second and third term) but the overall formulation is perfectly valid and the same minimization tool is used. Thanks to the robustness of the normal vector field estimated per interface, optimal solution for (1) defines regularized patches that perfectly coincide along topological junctions.

As we maintain a mapping between input and regularized quads, colorimetric information on the input voxel set can be pushed to its smooth interface. In supplementary material, we provide more results of voxel art regularization.

Finally, as we reconstruct piecewise smooth models from voxel shapes, we can build refined voxelizations of initial voxel shapes, just by binarizing our piecewise smooth model. This application is extremely useful for interactive modeling with initial coarse sketches. Please note that the release code of our regularization tool will be publicly available under an open source license.


Romeo Ortega

Anton Pyrkin

Alexey Bobtsov

Denis Efimov

Stanislav Aranovskiy

A GLOBALLY CONVERGENT ADAPTIVE INDIRECT FIELD-ORIENTED TORQUE CONTROLLER FOR INDUCTION MOTORS † June 18, 2018

Keywords: Induction motors, field-oriented control, adaptive control, torque control, speed control

One of the most challenging problems in AC drives applications is the design of a simple plug-in adaptation scheme to estimate the unknown rotor resistance and load torque for the industry-standard indirect field oriented control. In this paper we give the first globally convergent solution to this problem for torque control of current-fed induction motors that does not rely on any excitation assumption. Some results on speed regulation are also presented.

Introduction

The industry standard in induction motor (IM) control is the so-called indirect field-oriented control (IFOC), which provides asymptotic regulation of the generated torque (or the rotor speed) and flux modulus around constant references and does not require rotor flux sensors (or estimators) [START_REF] Krause | Analysis of Electric Machinery[END_REF][START_REF] Leonhard | Control of Electrical Drives[END_REF][START_REF] Nam | AC Motor Control and Electric Vehicle Applications[END_REF][START_REF] Peresada | Theoretical and experimental comparison of indirect field-oriented controllers for induction motors[END_REF]. The popularity of IFOC stems from its extreme simplicity and intuitive operation that permits independent tuning of the flux and torque control loops. IFOC relies on the assumption that the stator currents are available as control inputs-an assumption that is often justified in practice by the use of highgain current control loops. A drawback of the standard IFOC scheme is that it requires accurate knowledge of the rotor resistance, which may vary significantly with temperature, frequency and current amplitude. Even though it has been shown in [START_REF] De Wit | Indirect field oriented control of induction motors is robustly globally stable[END_REF] that stability is preserved for very large errors in rotor resistance estimation, this mismatch seriously affects the performance: it degrades the flux regulation, which may lead to saturation or under-excitation, slows down the torque response and induces a steady-state error. It is fair to say that, from the practitioners' viewpoint, the development of a plug-in scheme to incorporate adaptation to IFOC is one of the most relevant open problems in IM control. For further detail on IFOC the reader is referred to the books [START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF][START_REF] Leonhard | Control of Electrical Drives[END_REF][START_REF] Marino | Induction Motor Control Design[END_REF][START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF].

A globally stable adaptive IFOC for the current-fed IM, with the only assumption of the rotor resistance belonging to a discrete known (but arbitrarily large) set, was reported in [START_REF] Chang | Supervisory field-oriented control of induction motors with uncertain rotor resistance[END_REF]. Unfortunately, as shown in [START_REF] Chang | Sur la commande non-linéaire des machine électriques: synchrones et asynchrones[END_REF], the proposed scheme is extremely complex and difficult to tune hampering its application in most practical scenarios. In [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF] a globally convergent adaptive IFOC with unknown rotor resistance and load torque, but assuming measurement of the motor torque, was reported. A theoretically interesting globally stable output feedback adaptive design was reported in [START_REF] Marino | Adaptive output feedback control of current-fed induction motors with uncertain rotor resistance and load torque[END_REF]. However, there are several drawbacks to the proposed controller. First, it is much more complicated than the basic IFOC and is difficult to implement and tune. Second, the stability analysis relies on a critical persistency of excitation requirement [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF] that may be hard to verify in applications. Finally, the scheme is based on feedback linearization that is in open contradiction with the physical operation of the system and, as it is based on exact cancellation of nonlinearities, is a highly non-robust operation. In [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF][START_REF] Peresada | Theoretical and experimental comparison of indirect field-oriented controllers for induction motors[END_REF] some analytical and experimental evidence of these facts are reported. See also [START_REF] Verrelli | On-line identification of winding resistances and load torque in induction machines[END_REF] for some new results on persistence of excitation-based parameter estimation of IM.

In this paper a plug-in adaptive IFOC for regulation of torque of IM that estimates the rotor resistance is proposed. Global regulation of the torque and the flux amplitude is guaranteed measuring only the rotor speed and assuming that load torque is known. The proposed estimator is extremely simple, does not require any persistence of excitation assumption and achieves asymptotic convergence even in the case of zero rotor speed and/or low torque. To remove the practically unreasonable assumption of known load torque, we then add a load torque estimator and establish global boundedness and convergence to a residual set of the torque regulation error. Finally, it is also shown that a slight modification to the estimator can be used for speed regulation with a PI controller. This leads to error equations that exactly coincide with the globally convergent dynamics of torque regulation when the output of the PI converges to a constant. This suggests that the system is amenable to a singular perturbation-like stability analysis-that is, unfortunately, still conspicuous by its absence.

The remainder of the paper is organized as follows. In Section 2 we present the model of the current-fed IM and in Section 3 we recall the torque regulation IFOC. Section 4 formulates the adaptive IFOC problem while Section 5 contains the new plug-in rotor resistance estimator for the adaptive implementation of the torque regulation IFOC with known load torque. In Section 6 we add a load torque estimator to remove the assumption of known load torque. In Section 7 the speed regulation IFOC is briefly presented and some preliminary results of its adaptive implementation are given. Simulation results that illustrate the transient performance and robustness of the torque and speed adaptive IFOCs are presented in Section 8. Our work is wrapped-up with concluding remarks and future work in Section 9.

Model of the Current-Fed Induction Motor

The induction motor in the fixed stator frame is described by the state equations [START_REF] Krause | Analysis of Electric Machinery[END_REF][START_REF] Leonhard | Control of Electrical Drives[END_REF][START_REF] Nam | AC Motor Control and Electric Vehicle Applications[END_REF]

] ˙ i a = L sr R r L s σL 2 r λ a + n P L sr L s σL r ωλ b -γ 0 i a + 1 L s σ v a ˙ i b = L sr R r L s σL 2 r λ b - n P L sr L s σL r ωλ a -γ 0 i b + 1 L s σ v b λa = - R r L r λ a -n P ωλ b + R r L sr L r i a λb = - R r L r λ b + n P ωλ a + R r L sr L r i b ω = 1 D (τ -τ L ) τ = n P L sr L r (λ a i b -λ b i a ) (1)
where In many practical applications high-gain current loops (sometimes with PI actions) of the form v = 1 (i * ab -i ab ) are used to force i ab to track their corresponding references i * ab , where is a small positive number. It is reasonable then to consider the singularly perturbed reduced model obtained by setting → 0, that is

λ ab = [λ a ,
λab = - R r L r λ ab + n P ωJ λ ab + R r L sr L r i ab D ω = τ -τ L τ = n P L sr L r i ab J λ ab
with the skew-symmetric matrix J := 0 -1

L R λ = -λ + u, (2)
D ω = τ -τ L (3) τ = n P L u Jλ, (4)
where, to simplify the notation, we have defined R := R r , L := L r .

Throughout the paper, we will assume that the behaviour of the (socalled) current-fed induction motor is captured by the dynamical model ( 2)-(4). As discussed in [START_REF] Escobar | Regulation and tracking of the non holonomic double integrator[END_REF][START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF] this apparently innocuous system can exhibit an amazingly complex behaviour and poses a significant challenge for control system design.

Indirect Field Oriented Torque Control

The classical IFOC for torque regulation is given as

u = e Jρ d β d L n P τ d β d ρd = R n P τ d β 2 d , (5)
where β d > 0 and τ d > 0 are references for the flux and the torque amplitude, respectively.

The remarkable stability properties of the torque regulation IFOC, which does not require any measurement of the systems state, are summarized in the proposition below. See [START_REF] De Wit | Indirect field oriented control of induction motors is robustly globally stable[END_REF] for its robustness analysis. Proposition 1. Consider the IM model ( 2)-( 4) in closed-loop with the torque regulation IFOC [START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF]. For all initial conditions (λ(0), ω(0), ρ d (0)) ∈ R 4 we have lim

t→∞ τ (t) = τ d , lim t→∞ |λ(t)| = β d , (6)
where | • | is the Euclidean norm.

Proof. First, notice that

e Jρ d = cos ρ d -sin ρ d sin ρ d cos ρ d ,
and define the desired flux vector as

λ d := e Jρ d β d 0 .
Notice that

|λ d | = β d and λ d satisfies L R λd = αJλ d , λ d (0) = β d 0 , (7)
where, to simplify the notation, we defined the key positive constant

α := L n P τ d β 2 d . (8)
The IFOC (5) can then be written as

u = (I + αJ) λ d . (9)
Defining the flux error e λ := λ -λ d and replacing [START_REF] Escobar | Regulation and tracking of the non holonomic double integrator[END_REF] and ( 9) in (2) we get ėλ = -R L e λ , which implies lim t→∞ e λ (t) = 0 and, consequently, lim t→∞ |λ(t)| = β d . Now, using (9), the torque equation (4) becomes

τ = n P L u J(λ d + e λ ) = n P L λ d (I -αJ)Jλ d + t = n P L α|λ d | 2 + t = τ d + t , (10)
where t := n P L u Je λ is an exponentially decaying term and we have used the facts that J = -J and J 2 = -I. This concludes the proof.

It is clear from the rotor dynamics (3) that, since τ (t) → τ d , the torque reference should be chosen equal to τ L to ensure speed remains bounded. 34 Formulation of the Adaptive IFOC Torque Regulation Problem

Consider the IM model ( 2)-( 4) verifying the following assumptions.

Assumption 1. Rotor speed ω is measurable, the parameters L and D are known and the references are such that the constant defined in (8) verifies

α < 1. ( 11
)
Assumption 2. The rotor resistance R is unknown, but two constants R M and R m that verify R α 2 > R M ≥ R ≥ R m > 0, are known.
Design a rotor resistance estimator

χ = F (χ, u, ω, τ L ) R = H(χ, u, ω, τ L ), (12)
where χ ∈ R q is the estimator state such that the IM model ( 2)-(4) in closedloop with the adaptive torque regulation IFOC

u = e J ρd β d L n P τ d β d ρd = R n P τ d β 2 d , (13)
ensures ( 6) for all (λ(0), ω(0), ρd (0),

χ(0)) ∈ R 4+q .
Although Assumption 2 is quite reasonable in applications, Assumption 1 looks quite technical and deserves some clarification. In Subsection 1.4 of the comprehensive book [START_REF] Marino | Induction Motor Control Design[END_REF] it is first shown that the field-oriented version of the IM model (1), with outputs ω and the d-component of the flux, is exactly left invertible. Then, in Subsection 2.1, the authors prove that, for constant desired outputs, the inverse dynamics controller destabilizes the system for large load torques. Finally, in Subsection 5.1, they prove, that when applied to the current-fed model ( 2)-( 4), the reduced order tracking error dynamics has an additional equilibrium-besides the origin-and this equilibrium is unstable if α > 1. In other words, Assumption 1 pertains to the stability of the open-loop, left-inverting control of the IM. 4

A Globally Convergent Adaptive Torque Regulation IFOC

In this section we provide a solution to the adaptive torque regulation IFOC problem defined in Section 3 adding the following, admittedly unpractical, assumption.

Assumption 3. The load torque τ L is known.

Proposition 2. Consider the IM model ( 2)-( 4) verifying Assumptions 1-3 in closed-loop with the adaptive torque regulation IFOC ( 13) with τ d > 0 and the rotor resistance estimator

L λ = -Rλ + Ru (14) ż = γ D n P Rω λ (J +αI)u+ λ Ju 2 + Lτ L n P λ Ju (15) R =    R M if S(z, ω, λ, u) ≥ R M S(z, ω, λ, u) if R M > S(z, ω, λ, u) > R m R m if R m ≥ S(z, ω, λ, u), (16)
where γ > 0 is a tuning gain and we defined the switching function

S(z, ω, λ, u) := z + γ DL n P ω λ Ju. (17)
For all (λ(0), ω(0), ρ d (0), λ(0), z(0)) ∈ R 7 , (6) holds.

Proof. Defining the flux observation error λ := λ -λ and using ( 2) and ( 14) we get

L λ = -Rλ + Ru + Rλ -Ru = -R( λ -λ) -Rλ + Ru = -R λ + R(u -λ), (18)
where R := R -R is the parameter estimation error. Now, from ( 16) we get

Ṙ =    0 if S(z, ω, λ, u) ≥ R M Ṡ if R M > S(z, ω, λ, u) > R m 0 if R m ≥ S(z, ω, λ, u), (19)
where Ṡ is computed from (17) as

Ṡ = ż + γ DL n P ωλ Ju + γ DL n P ω d dt λ Ju ,
with the first right hand term given by [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF]. The derivative of the speed is defined by ( 3) and ( 4) as

ω = - n P DL λ Ju - τ L D . (20)
The term d dt λ Ju is computed as

d dt λ Ju = λ Ju + λ J u = - 1 L Rλ + 1 L Ru Ju + λ J e J ρd J β d Lτ d n P β d R n P τ d β 2 d = - 1 L Rλ Ju -λ u R n P τ d β 2 d = - 1 L Rλ (J + αI) u, (21)
where α is given by ( 8) Substituting ( 15), (20), and ( 21) into (19)-for the case Notice that the parameter γ plays the role of adaptation gain in the proposed estimator.

R M > S(z, ω, λ, u) > R m -yields Ṙ = γ D n P Rω λ (J + αI)u + γ λ Ju 2 + γ Lτ L n P λ Ju -γ λ Ju + Lτ L n P λ Ju -γ DL n P ω 1 L Rλ (J + αI) u = γ λ Ju 2 -λ Ju λ Ju . (22) B( ρd ) 1 Ls+R - γ s B( ρd ) λ φ - R φ R
Defining the two-dimensional vector φ := u -λ it is possible to write (22) as Ṙ = γφ Juu J λ.

Hence, developing Juu J and defining

B(ρ d ) := Juu J = β 2 d e J ρd α 2 -α -α 1 e -J ρd , (23)
the estimator error equations ( 18) and ( 22) can be written in the more familiar form

L λ = -R λ + φ R Ṙ = -γφ B(ρ d ) λ. ( 24
)
In Fig. 1 a block diagram representation of (24) is given. It clearly underscores the close connection with the classical model reference adaptive control error system [START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF][START_REF] Anderson | Stability of Adaptive Systems: Passivity and Averaging Analysis[END_REF]. As is well-known [START_REF] Anderson | Stability of Adaptive Systems: Passivity and Averaging Analysis[END_REF] the operator in the feedback loop of Fig. 1 is passive. Unfortunately, the matrix B(ρ d )-although positive semidefinite-is not constant. Hence, the operator corresponding to the forward part cannot be shown to be passive without further assumptions on ρd . This hampers the application of the standard passivity argument to prove the stability of the feedback system.

To overcome the aforementioned obstacle and be able to carry out standard Lyapunov-like analysis it is necessary, as done in [START_REF] De Wit | Indirect field oriented control of induction motors is robustly globally stable[END_REF], to define coordinates on which the steady-state behaviour of the system corresponds to constant equilibria and not periodic orbits. Towards this end, we define the five-dimensional vector

x := col(λ u, λ Ju, λ u, λ Ju, R). ( 25
)
Straightforward calculations yield

L ẋ1 = -Rx 1 + αx 5 x 2 + Rβ 2 d (1 + α 2 ) L ẋ2 = -Rx 2 -αx 5 x 1 L ẋ3 = -x 5 x 3 + αx 5 x 4 + x 5 β 2 d (1 + α 2 ) L ẋ4 = -x 5 x 4 -αx 5 x 3 ẋ5 = γx 4 (x 4 -x 2 ). ( 26
)
We will, first, analyze the equilibria of (26). From ( 16) and Assumption 2 we have that

R α 2 > R M ≥ x 5 ≥ R m > 0. (27)
Now, from (26) we have the following equivalences

ẋ1 = ẋ2 = 0 ⇔ x 1 x 2 = Rβ 2 d (1 + α 2 ) R 2 + α 2 x 2 5 R -αx 5 ẋ3 = ẋ4 = 0 ⇔ x 3 x 4 = β 2 d 1 -α ,
where we ruled out the case x 5 = 0 in the second equivalence. Now, since

x 4 = -αβ 2 d = 0, we have that ẋ5 = 0 ⇔ x 2 = x 4 ⇔ (x 5 -R)(x 5 - R α 2 ) = 0,
where the second equivalence is obtained replacing the definitions of x 2 and x 4 above. Invoking (27) we see that the only solution of this equation is x 5 = R. Consequently, the system (26) has a unique equilibrium at

x := col(β 2 d , -αβ 2 d , β 2 d , -αβ 2 d , R).
To relate the problem of stabilization of this equilibrium with the control objective notice first that x 2 = -L n P τ and x2 = -L n P τ d . On the other hand, invoking [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF], it is possible to establish the following chain of implications

x 1 x 2 = x1 x2 ⇔ u -u J λ = x1 x2 ⇔ ⇔ λ = 1 |u| 2 u Ju x1 x2 ⇔ λ = 1 β 2 d (1 + α 2 ) u Ju β 2 d -αβ 2 d ⇒ |λ| = β d .
Consequently, the control objective is achieved if we can prove that lim t→∞ (x 1 (t), x 2 (t)) = (x 1 , x2 ) with all signals remaining bounded.

To proceed with the stability analysis it is convenient to shift the equilibrium of (26) to zero. For this purpose, we define the errors e := x -x and, after some basic computations, get the error dynamics

L ė1 = -Re 1 + α(e 5 + R)e 2 -α 2 β 2 d e 5 L ė2 = -Re 2 -α(e 5 + R)e 1 -αβ 2 d e 5 L ė3 = -(e 5 + R)(e 3 -αe 4 ) L ė4 = -(e 5 + R)(e 4 + αe 3 ) ė5 = γ(e 4 -αβ 2 d )(e 4 -e 2 ). (28)
The first observation is that, under Assumption 2, the origin of the system (e 3 , e 4 ) is globally exponentially stable. Indeed, the derivative of the function

2R m L V 1 ,
where we have used the fact that

e 5 + R = R ≥ R m .
Second, considering the function

V 2 (e 1 , e 2 ) := L 2 (e 2 1 + e 2 2 ), we get V2 = -R(e 2 1 + e 2 2 ) -αβ 2 d e 5 (αe 1 + e 2 ). (29)
Since e 5 ∈ L ∞ by construction we conclude from the equation above that e 1 , e 2 ∈ L ∞ as well. Now, define the signals

w 1 := e 1 -αe 2 w 2 := e 2 -e 4 (30)
Evaluating their time derivative along the error dynamics (28) yields

L ẇ1 := -(R -α 2 R)w 1 + α(1 + α 2 ) R(w 2 + e 4 ) L ẇ2 := -(R + α 2 R)w 2 -α R(w 1 + αe 4 -e 3 ) +(e 4 -αβ 2 d )e 5 . (31)
Consider the function

W (w 1 , w 2 , e 5 ) := 1 2 L 1 + α 2 w 2 1 + Lw 2 2 + 1 γ e 2 5 ,
whose derivative verifies

Ẇ = - R -α 2 R 1 + α 2 w 2 1 -(R + α 2 R)w 2 2 +w 1 1 + w 2 2 ≤ - R -α 2 R 2(1 + α 2 ) w 2 1 - 1 2 (R + α 2 R)w 2 2 + 1 + α 2 2(R -α 2 R) 2 1 + 1 2(R + α 2 R) 2 2 , (32)
where we defined the exponentially decaying signals

1 := α(e 5 + R)e 4 2 := α(e 5 + R)(αe 4 -e 3 ),
in the first identity and used the well-known bound 2ab ≤ ka 2 + 1 k b 2 for the last inequality. Recalling that it has been established that e ∈ L ∞ , e 3 and e 4 are exponentially converge to zero, we conclude that there exists a compact invariant set Ω = {e ∈ R 5 : e ≤ C Ω , e 3 = e 4 = 0}, on which (32) becomes

Ẇ ≤ - R -α 2 R 2(1 + α 2 ) w 2 1 - 1 2 (R + α 2 R)w 2 2 ≤ 0,
hence invoking LaSalle's invariance principle we conclude that all trajectories converge to the limit set where Ẇ = 0, i.e. w 1 = 0 = w 2 = 0. Consequently, from (30) follows that on the limit set e 1 = e 2 = 0 and, using (28), e 5 = 0. Thus, lim t→∞ e(t) = 0.

To complete the proof let us compute the torque provided by the controller [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF] as in Section 3. Define the desired flux vector as λd := e J ρd β d 0 .

Notice that

| λd | = β d and λd satisfies λd = α R L J λd , λd (0) = β d 0 . ( 33
)
The control law (13) can then be written as

u = (I + αJ) λd . ( 34
)
Defining the flux error e λ := λ -λd and replacing (33) and ( 34) in ( 2) we get

ėλ = - R L e λ - α L J λd R.
Since R = R -R = e 5 converges to zero then lim t→∞ e λ (t) = 0 and lim t→∞ |λ(t)| = β d . Following [START_REF] Marino | Adaptive output feedback control of current-fed induction motors with uncertain rotor resistance and load torque[END_REF] and using (34), the torque (4) becomes

τ = n P L u J( λd + e λ ) = n P L α| λd | 2 + t = τ d + t ,
which completes the proof.

A Globally Stable Adaptive Torque Regulation IFOC with Load Torque Estimator

The main limitation of the adaptive IFOC of Proposition 2 is, of course, the assumption that the load torque τ L is known. In this section we propose a load torque estimator that ensures boundedness of all solutions and convergence of the error signals to a residual ball, whose radius is proportional to the rotor resistance estimation error.

Proposition 3. Consider the IM model ( 2)-( 4) verifying Assumptions 1 and 2 in closed-loop with the adaptive torque regulation IFOC [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF] with τ d > 0 and the rotor resistance and load torque estimator

L λ = -Rλ + Ru (35) ż = γ D n P Rω λ (J +αI)u+ λ Ju 2 + Lτ L n P λ Ju (36) χ = -kτ L -k n P L λ Ju (37) R =    R M if S(ẑ, ω, λ, u) ≥ R M S(ẑ, ω, λ, u) if R M > S(ẑ, ω, λ, u) > R m R m if R m ≥ S(ẑ, ω, λ, u), (38) τL
= χ -kDω, (39)
where γ, k > 0 are tuning gains and

S(ẑ, ω, λ, u) := ẑ + γ DL n P ω λ Ju. (40)
The following properties are true or all initial conditions (λ(0), ω(0), ρ d (0), λ(0), z(0), χ(0)) ∈ R 8 . Proof. Similarly to the proof of Proposition 2 consider the model of the observation errors. The flux observation error λ is the same as in ( 18) 5 Where O(•) is the uniform big O symbol.

L λ = -R λ + R(u -λ). Now, from (38) we get Ṙ =    0 if S(ẑ, ω, λ, u) ≥ R M Ṡ if R M > S(ẑ, ω, λ, u) > R m 0 if R m ≥ S(ẑ, ω, λ, u), (41)
where Ṡ is computed from (40) as

Ṡ = ż + γ DL n P ωλ Ju + γ DL n P ω d dt λ Ju ,
with the first right hand term given by (36). Using (20) and (21), from (41)-for the case R M > S(ẑ, ω, λ, u) > R m -we get

Ṙ = γ D n P Rω λ (J + αI)u + γ λ Ju 2 + γ Lτ L n P λ Ju -γ λ Ju + Lτ L n P λ Ju -γ DL n P ω 1 L Rλ (J + αI) u = γ λ Ju 2 -λ Ju λ Ju + γ Lτ L n P λ Ju .
where the load torque estimation error τL :

= τL -τ L satisfies τL = -kτ L -k n P L λ Ju -kD ω = -kτ L + k n P L u J λ -k n P L u Jλ + kτ L = -kτ L -k n P L λ Ju.
Define the six-dimensional vector

x := col(λ u, λ Ju, λ u, λ Ju, R, Lτ L n P ).

Straightforward calculations yield

L ẋ1 = -Rx 1 + αx 5 x 2 + Rβ 2 d (1 + α 2 ) L ẋ2 = -Rx 2 -αx 5 x 1 L ẋ3 = -x 5 x 3 + αx 5 x 4 + x 5 β 2 d (1 + α 2 ) L ẋ4 = -x 5 x 4 -αx 5 x 3 ẋ5 = γx 4 (x 4 -x 2 + x 6 ), ẋ6 = -k(x 4 -x 2 + x 6 ). ( 42
)
Similarly to the case of known τ L , we conclude that the system (42) has a unique equilibrium at

x := col(β 2 d , -αβ 2 d , β 2 d , -αβ 2 d , R, 0).
To proceed with the stability analysis it is convenient to shift the equilibrium of (42) to zero and consider the model of errors e := x -x and, after some basic computations, get the error dynamics (43)

L ė1 = -Re 1 + α(e 5 + R)e 2 -
Comparing ( 43) with (28) we notice that the only modification that the load torque estimator has introduced is the addition in ė5 of the signal e 6 , whose dynamics is given in (43).

As shown in the proof of Proposition 2 the origin of the system (e 3 , e 4 ) is globally exponentially stable and the function e 5 ∈ L ∞ by construction. Now, from (29) we can get the bound

V2 ≤ - R L V 2 + 1 R α 2 (1 + α 2 )β 4 d |e 5 | 2 ,
from which we conclude that e 1 , e 2 ∈ L ∞ . Moreover, the bound of the steady state depends on the L ∞ -norm of e 5 . Finally, from the last equation of (43) we conclude that e 6 ∈ L ∞ which means that all errors are bounded.

Therefore, we conclude that the torque and flux amplitude regulation errors enter asymptotically a neighborhood of zero whose radius decreases proportionally to the L ∞ norm of the rotor resistance estimation error. At first glance this seems to be a rather weak property that is, however, not the case. Indeed, by construction, the resistance estimation error lives in the prior knowledge interval [R m , R M ], see ( 27). This implies that the quality of the regulation is improved with better knowledge of the rotor resistance, and in the ideal case, when this parameter is exactly known, we achieve convergence to zero of the errors.

Preliminary Results on Adaptive Speed Regulation IFOC

As is well-known [START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF][START_REF] Leonhard | Control of Electrical Drives[END_REF][START_REF] Marino | Induction Motor Control Design[END_REF][START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF] IFOC can also be used to control rotor speed, instead of torque. This is achieved simply replacing τ d in ( 5) by the output of a PI control loop around the rotor speed error, that is,

τ d = -k P + k I p ω ( 44
)
where we defined the speed error ω := ω -ω d , with ω d ∈ R the desired speed, p := d dt and k P , k I > 0 are arbitrary tuning gains. Similarly to the torque regulation IFOC its speed regulation version ensures global convergence, as indicated in the proposition below. Proposition 4. Consider the IM model ( 2)-( 4) in closed-loop with the speed regulation IFOC ( 5), (44). For all initial conditions of the PI and all (λ(0), ω(0), ρ d (0)) ∈ R 4 we have that all signals remain bounded and

lim t→∞ ω(t) = ω d , lim t→∞ |λ(t)| = β d . ( 45
)
Proof. First, notice that the only modification to the torque regulation IFOC ( 5) is in the definition of τ d . Therefore, we still have the property [START_REF] Marino | Adaptive output feedback control of current-fed induction motors with uncertain rotor resistance and load torque[END_REF], that is τ = τ d + t , with t converging to zero exponentially fast. Replacing the expression above in ( 3), ( 4) we get

D ω = τ d -τ L + t .
The proof is completed replacing (44) above and invoking well-known properties of linear systems.

It is shown in [START_REF] De Wit | Indirect field oriented control of induction motors is robustly globally stable[END_REF] that stability of the speed regulation IFOC is preserved even for large variations of the rotor resistance. However, as is wellknown [START_REF] Nam | AC Motor Control and Electric Vehicle Applications[END_REF][START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF], performance is degraded when this parameter is unknownmotivating the inclusion of a resistance estimator. In this section we propose a slight modification to the estimator of Proposition 2 and the addition of a (fast) filter to the PI controller (44) of the form

τ d = - 1 p + k F k P + k I p ω, ( 46
)
with k F > 0 a large number, to generate an adaptive speed regulation IFOC.

If the filter 1 p+k F is chosen sufficiently fast the difference between the filtered and the unfiltered signal is negligible, although its practical implementation may generate some numerical problems. As will become clear below, the reason for its inclusion stems from the need to compute-without differentiation nor measurement of acceleration-the signal τd for the adaptive implementation.

We consider the adaptive IFOC [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF] with τ d defined by the (filtered) speed PI (46), the estimator ( 14), ( 16), ( 17) and

ż = γ D n P Rω λ (J +αI)u+ λ Ju 2 + Lτ L n P λ Ju + γ D n P ω λ e J ρd 0 L 2 npβ d τd , ( 47
)
where α is given by ( 8) and ( 46), γ > 0 and the tuning gains k F , k P and k I are selected such that the polynomial

δ(s) := s 3 + k F s 2 + k P D s + k I D ( 48
)
is Hurwitz.

We will now proceed to show that, modulo the definition of α, the resulting closed-loop dynamics exactly coincide with the dynamics of the torque regulation IFOC. More precisely, α given in [START_REF] Krause | Analysis of Electric Machinery[END_REF], is a constant parameter in the latter, while in speed regulation it is a function proportional to the output of the (filtered) PI (46). As discussed below, this difference significantly complicates the stability analysis.

First, notice that we have only added to the estimator of Proposition 2 the last right hand term of (47), which equals zero if τ d is constant. For the speed regulation case of interest, this term can be computed-without differentiation-via

τd = - p p + k F k P + k I p ω.
Repeating verbatim the calculations in the proof of Proposition 2 we see that in the computation of u in (21) a term depending on τd appears. To recover the desired form of the parameter error dynamics (22) the the last right hand term of (47) must be added to ż. Notice that the flux observer, and consequently its error dynamics (18), remains also unchanged. Hence, the estimator error equations are given by (24). Moreover, the dynamics of the (partial) state vector (25) is given by (26). Now, define τ := τ -τ d and write the rotor dynamics (3) as

D ω = τ d + τ -τ L .
Replacing (46) above and grouping terms we have

D ω = p 2 + k F p δ(p) (τ -τ L ). (49)
Recalling that τ L is constant and δ(s) is Hurwitz, from (49) we see that, if torque regulation is achieved, that is, if

lim t→∞ |τ (t) -τ d (t)| = 0, then (45) holds.
As indicated in (ii) of Proposition 4 the vector x defined in (25) is only a partial state vector for the closed-loop dynamics to which we need to add the states coming from the PI (46). These new states are fed-back, through the signal α, to the x dynamics yielding a highly complicated interconnected system. For instance, the definition of its steady-state behavior-that in torque regulation IFOC are easily definable constant equilibria-in this case is hard to elucidate.

Simulation Results

In this section we first show simulations of the IM model ( 2)-( 4) in closedloop with the adaptive torque regulation IFOC of Proposition 2. Then, to relax the assumption of knowledge of the load torque τ L , the controller of Proposition 3 is simulated. Finally, simulations for the adaptive speed regulation IFOC discussed in Section 7 are given.

We have simulated the model of a 0.5[KW ] IM available in the Laboratorie de Genie Electrique de Paris, for which experimental results are reported in [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF]. The IM model parameters are given as R = 2.76 [Ω], L = 0.42 [H], D = 0.06 [kg m 2 ], n P = 2. All simulations start from the initial speed ω(0) = 0 and without load torque τ L = 0 and, at time t = 1[sec], the load torque is stepped-up to τ L = 2 [N m]. Notice that we have chosen a worst-case simulation scenario for the rotor resistance estimator. Indeed, for toque regulation the control u is frozen for 0 ≤ t ≤ 1 providing no excitation in this time interval for the estimator, with a similar situation happening for all other controllers. In Figs. 2 and 3 different initial conditions and a fixed adaptation gain γ = 100 are considered. As expected from the discussion above the rotor resistance estimation goes "in the right direction" only after t = 1[sec]-a pattern that is repeated in all simulations. To show the tracking ability of the estimator, in Figs. 4 and5 we present the case when the resistance is changed from its nominal value R = 2.76 [Ω] for 0 ≤ t < 10 to 0.5R for 10 ≤ t < 20 and 1.5R for t ≥ 20. Fig. 4 shows the behaviour for different adaptation gains γ. As expected, convergence time decreases with increasing adaptation gain, at the price of a larger overshoot. Fig. 5 explores the robustness properties of the scheme when we use wrong estimates of L and D, denoted L and D, respectively. We see from Fig. 5 that the algorithm is particularly sensitive to underestimation of L.

Torque regulation with known load torque

Torque regulation with load torque estimator

To test in simulation the performance of the adaptive IFOC with the load torque estimator of Proposition 3 the same simulation scenario of Fig. 4 was repeated-with different estimator adaptation gains k. The results, presented in Fig. 6, show that transient performance degradation of the adaptive IFOC due to the replacement of the exact knowledge of τ L by its estimate is hardly perceptible.

Speed regulation

In Figs. 7 and8 the simulation results for the adaptive speed regulation IFOC discussed in Section 7, with different desired constant speeds and various PI tuning gains, are given. In Fig. 7 we fix the desired speed to ω d = 0.4[rpm] and select the PI controller gains k P , k I , and k F in (46) to make the polynomial (48) equal to (p + a) 3 for different positive constants a. As expected, as we move the poles further to the left, the speed of response increases. In Fig. 8 we fix a = 50 and show the behaviour for different values of the desired speed ω d . In all cases the rotor resistance is consistently estimated and the desired speed and rotor flux norm regulations are satisfactorily achieved. 9 Discussion and Future Extensions D1 The main difficulty of the task addressed in this paper is the constraint that the IFOC structure should be preserved and it is only allowed to add a rotor resistance estimator to track the variations of this parameter. This constraint is consistent with industrial practice where a long successful experience with a given controller, on one hand, discourages practitioners from major modifications to it while, on the other hand, has led them to identify the major culprit of the scheme-in the case of IFOC its sensitivity to variations of the rotor resistance, which happens in normal operation of the IM. Schemes that involve major departures from IFOC, usually requiring high-order complicated implementations, have proven to be only of academic interest. The same remark applies, of course, to controllers relying on the injection of high-gain, e.g., sliding mode or high-gain observer-based, whose noise amplification characteristics makes them unsuitable for motor applications.

D2

The constraint [START_REF] Marino | Induction Motor Control Design[END_REF] of Assumption 1 expressed in terms of the motor parameters and the controller references is

τ d β 2 d < n P L .
Given the usual order of magnitudes of these quantities this inequality is satisfied for a large class of IM-particularly in the cases where there is some freedom in the choice of β d . In any case, we are investigating modifications of the proposed scheme to remove this constraint and enlarge the realm of application of our scheme. Another issue of concern is the sensitivity of the scheme, observed in simulations, to uncertainty on the rotor inductance-in particular, to its underestimation.

D3 As shown in the proof of Proposition 2 the assumption α < 1 is required, on one hand, to ensure uniqueness of the equilibrium. On the other hand, from (31) we see that the dynamics of the coordinate w 1 becomes unstable at the equilibrium R = R if α > 1. Hence, this is a critical structural condition that is not a consequence of the analytical tools used in the paper. Simulation results have shown that, for some initial conditions, trajectories grow unbounded if α > 1. See also the discussion at the end of Section 4.

D4 Unfortunately, the stability analysis of the adaptive speed regulation IFOC discussed in Section 7 is incomplete. The fact that the dynamics of this scheme exactly coincides with the globally convergent dynamics of torque regulation when the output of the PI, i.e., τ d , converges to a constant, suggests that the system is amenable to a singular perturbation-like stability analysis that we are currently investigating.

D5

The IM model ( 2)-( 4) coincides, up to the presence of the term λ in the first equation and the load torque τ L with the celebrated non-holonomic integrator of Brockett for which a vast amount of research has been devoted. Interesting connections, extensions and simplifications between the time-varying controllers used for this system and the IFOC of Section 2 are explored in [START_REF] Escobar | Regulation and tracking of the non holonomic double integrator[END_REF].

2 srL 2 sr

 22 λ b ] is the rotor flux vector, i ab = [i a , i b ] is the stator current vector, and v ab = [v a , v b ] is the vector of stator voltages. R s , R r [Ω] are stator and rotor resistances, L s , L r [H] are the inductances of the stator and rotor windings and L sr [H] is the mutual inductance. σ = 1 -L LsLr > 0 is the total leakage factor of the motor, γ 0 = Rs Lsσ + LsσLrTr , D [kgm 2 ] is the rotor inertia, ω [rad/s] the rotor speed and n P is the number of pole pairs.

Figure 1 :

 1 Figure 1: Block diagram representation of the estimator error equations (24).

V 1 (e 3 , e 4

 134

  (i) All solutions are bounded. (ii) Define the L ∞ norm of the rotor resistance estimation error δ := R(t) -R ∞ . Then, 5 lim t→∞ |τ (t) -τ d | ≤ O(δ) lim t→∞ ||λ(t)| -β d | ≤ O(δ).

α 2 β 2 d e 5 L

 5 ė2 = -Re 2 -α(e 5 + R)e 1 -αβ 2 d e 5 L ė3 = -(e 5 + R)(e 3 -αe 4 ) L ė4 = -(e 5 + R)(e 4 + αe 3 ) ė5 = γ(e 4 -αβ 2 d )(e 4 -e 2 + e 6 ), ė6 = -k(e 4 -e 2 + e 6 ).

Figures 2 -

 2 Figures 2-5 correspond to the adaptive torque regulation IFOC of Proposition 2, that is, assuming τ L is known. The references are set to β d = 1, τ d = τ L , which correspond to α = 0.42 and the a priori estimated interval for R was taken as R m = 1 and R M = 5 < R α 2 = 13.04.

Figure 2 :Figure 3 :

 23 Figure 2: Simulation results of the adaptive torque regulation IFOC of Proposition 2 with γ = 100, λ(0) = (1, 0), λ(0) = (0, 1) and different initial conditions for z. 1. (blue line) z(0) = 0; 2. (red line) z(0) = 3; 3. (black line) z(0) = 5.

Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 45678 Figure 4: Simulation results of the adaptive torque regulation IFOC of Proposition 2 with z(0) = 2, λ(0) = (0, 0), λ(0) = (1, 0) step changes in R and different adaptation gains. 1. (blue line) γ = 30; 2. (red line) γ = 100; 3. (black line) γ = 300.

This is, actually, only a theoretical requirement, since in all practical applications, the presence of viscous friction ensures speed is bounded even if τ d = τ L .

The authors are deeply grateful to the anonymous reviewer that pointed out this very important fact.

(GOSZADANIE 2.8878Russian Foundation for Basic Research (grant 17-58-53129), and the Ministry of Education and Science of Russian Federation (project 14.Z50.31.0031).