Rafik Makhloufi
email: rafik.makhloufi@utt.fr

Guillaume Doyen
email: guillaume.doyen@utt.fr

Grégory Bonnet
email: gregory.bonnet@unicaen.fr

Dominique Gaïti
email: dominique.gaiti@utt.fr

SAAM: a Self-Adaptive Aggregation Mechanism for Autonomous Management Systems

Keywords: Autonomous Networking, Decentralized Aggregation, Management Information, Fuzzy Logic, MADM

In this paper, we propose a decentralized Self-Adaptive Aggregation Mechanism (SAAM) that adapts itself to the supporting network operational behavior by dynamically selecting the best aggregation approach to use. SAAM is based on (1) a fuzzy-based model that estimates the cost and performance of each aggregation scheme and (2) Multiple Attribute Decision Making (MADM) to make decisions on the best approach to use in this context. We validate SAAM by evaluating its fuzzy model and adaptation cost, and by comparing its utility to the one of existing situated and global schemes.

I. INTRODUCTION

Recently, several approaches have emerged in order to manage networks and services in an autonomous way. In this kind of frameworks, the hosting environment is continuously evolving, for example, in terms of size and dynamics. Consequently, the management approach should adapt itself to such different operational states. However, existing autonomous management systems propose to deploy a unique slightly adaptive approach to manage the network elements whatever the nature of their environment is. The same problematic is addressed in the case of aggregation schemes.

In this paper, we propose a self-adaptive management approach that adapts itself to the supporting network operational behavior by dynamically selecting the best approach to use while optimizing the management cost and performance. We especially investigate the use of this architecture in the case of aggregated information monitoring. We call this framework Self-Adaptive Aggregation Mechanism (SAAM). The latter ensures that, whatever the network operational state, the best monitoring approach is used, providing thus the most accurate aggregated information to the decision making processes. To design SAAM, we rely on fuzzy logic to estimate the cost and the performance of each aggregation scheme and also on MADM to decide what is the best aggregation approach to use. We validate SAAM by evaluating its fuzzy model and adaptation cost, and by comparing its utility to the one of existing situated and global schemes.

This paper is organized as follows. Section II discusses the related work on the existing autonomous management systems. Afterwards, Section III describes the proposed selfadaptive aggregation mechanism. Section IV is dedicated to the validation of the proposed framework. Finally, Section V concludes the paper and presents our futur work.

II. RELATED WORK

In the context of autonomous network management, few works propose a self-adaptive approaches that adapt themselves to the supporting network operational behavior.

For example, in [START_REF] Simmons | Achieving high-level directives using strategy-trees[END_REF], the authors propose to use a strategytree on a policy-based management system of a data center. They defined three strategies to process requests for ressources. The proposed mechanism aims at dynamically alter strategies, by choosing the one maximizing the cumulative weekly net profit while limiting the number of SLA (Service Level Agreement) violations. In [START_REF] Moui | Towards self-adaptive monitoring framework for integrated management[END_REF], the authors present a work in progress to define a framework for self-adaptive monitoring relying on the characterization of governability, adaptability and configurability. They present the global architecture of their framework and some concepts like the capabilities levels to be managed in order to perform on line monitoring adaptation. In [START_REF] Maurel | Towards introspectable, adaptable and extensible autonomic managers[END_REF], the authors present an architecture for building adaptable autonomic managers (AM). According to the current and past behavior of the administration tasks executed by the AMs, the proposed approach dynamically decides to add a new task or, to delete or modify the current tasks.

Globally, for [START_REF] Simmons | Achieving high-level directives using strategy-trees[END_REF], the first validation shows that, compared to the case where strategies are executed individually, the proposed approach did not often improve profit. In [START_REF] Moui | Towards self-adaptive monitoring framework for integrated management[END_REF], no details are provided on the design and validation of their mechanism. Finally, contrary to [START_REF] Maurel | Towards introspectable, adaptable and extensible autonomic managers[END_REF], in our case we are not interested in the adaptation of administration tasks, but in the one of the monitoring framework itself.

III. SELF-ADAPTIVE AGGREGATION MECHANISM

A. Considered aggregation schemes

In order to design efficient management frameworks that minimize the cost and maximize the performance of management, we propose a self-adaptive aggregation mechanism. The latter infers the current network operational state and dynamically decides which aggregation approach should be used and/or what changes must occur in it. SAAM considers typical situated and global aggregation schemes that we evaluated in [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF]. These are: (1) a push-pull gossip scheme inspired from [START_REF] Jelasity | Gossip-based aggregation in large dynamic networks[END_REF], where at each round, each node contacts one neighbor chosen uniformly at random and exchanges information with it ; (2) a push tree-based scheme based on GAP [START_REF] Dam | A generic protocol for network state aggregation[END_REF], where nodes communicate their local aggregates to a single root node that computes an overall aggregate 978-1-4673-0269-2/12/$31.00 c ⃝ 2012 IEEE and spreads it on all the interested nodes through a publishsubscribe mechanism ; and finally (3) a pull-based situated scheme based on the membership protocol HyParView [START_REF] Leitao | Large-Scale Peer-to-Peer Autonomic Monitoring[END_REF] and Propagate2All [START_REF] Bawa | Estimating aggregates on a Peer-to-Peer network[END_REF], where each node maintains a view of its h-hops neighbors. We consider a one-hop situated view (SV1) and a two-hops situated view (SV2).

B. SAAM Architecture

The control loop of autonomic computing on each AM consists in four main operational blocks that are Monitor, Analyze, Plan, Execute (MAPE) [START_REF] Kephart | The Vision of Autonomic Computing[END_REF]. As depicted in Figure 1, this model can be applied to the adaptive configuration of the management plane with four components that enable a dynamically selection of a management strategy according to the current context. Thus, the resulting system operates as a meta-management system that provides an autonomous support for the management plane to select the best approach. The global architecture of the proposed system is composed of the following blocks.

1) Network State Monitor (NSM):

This process is responsible for the monitoring of the managed network operational state. It uses sensors to collect data from local or from other AMs and then it sends them to the next component for analysis. In case of distributed data monitoring, these data can be collected through an aggregation scheme. The NSM collects information about the Network Dynamics (ND), Network Size (NS) and aggregated data dynamics (ID: Information Dynamics). These metrics have a direct impact on the cost and performance of aggregation [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF].

2) Cost/Perf Estimator (CPE):

The CPE is composed of two Fuzzy Logic Controllers (FLCs) that analyze the data provided by the NSM and estimate the cost and the performance of each aggregation scheme. The CPE outputs are Convergence Time (CT), Communication Cost (CC) and Aggregation Accuracy (AA). We rely on fuzzy logic [START_REF] Passino | Fuzzy Control[END_REF] because it provides a formal methodology for representing, manipulating, and implementing a human's heuristic knowledge about how to control a system. It allows a performant decision-making under incomplete and uncertain information, here, about the network operational state. The developed FLCs operates according the following main steps:

• Fuzzification: the CPE inputs/outputs are converted into fuzzy variables. The fuzzy sets of each variable are de-fined according to the performance evaluation presented in [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF] (Figure 2). The set of membership functions we defined for FLC variables are Very Low (VL), Low (L), Medium (M), High (H) and very high (VH). • Inference: according to the CPE inputs, in order to evaluate which control rules have to be activated at a given time and then decide what the output should be, each FLC relies on an inference mechanism. Tables I and II illustrate, respectively, the rules used by the FLCs to estimate both the cost and the performance of each aggregation scheme.

Tree Gossip SV1 SV2 ND ND ND ND L M H L M H L M H L M H ID VL VH VL VL VH H M M M M H M M L VH VL VL VH H M L L L M M M M VH VL VL H H M L L L M M L H H VL VL H M L L L L M M L
• Defuzzification: it converts the conclusions reached by the inference mechanism into quantifiable outputs. We consider the center of gravity method which is the most standard and widely known defuzzification one [START_REF] Leekwijck | Defuzzification: criteria and classification[END_REF].

3) Multiple Attribute Decision Making (MADM):

According to the numerical values of the cost and the performance estimated by the FLCs, the MADM [START_REF] Hwang | Multiple Attribute Decision Making[END_REF] components compute the utility (i.e. score) of each aggregation scheme. This component integrates the user preferences through a set of weights that define a tradeoff between cost and performance. We rely on a MADM system because it is efficient and simple to use in decision support systems. It can give managers many dimensions to consider and evaluate possible alternatives under variable degrees. In our study case, MADM enables Local utility computing component: in order to locally (on each AM) compute the utility 𝑈 𝑖 of each aggregation scheme, we use SAW (Simple Additive Weighting) [START_REF] Zhang | Handover decision using fuzzy madm in heterogeneous networks[END_REF]. The latter is a standard MADM approach. It consists of scaling the values of all attributes and summing up them for each alternative to compute an overall score. The latter is a weighted sum of all the attribute values, 𝑈 𝑖 = 𝑤 1 .𝐶𝑇 𝑖 + 𝑤 2 .𝐶𝐶 𝑖 + 𝑤 3 .𝐴𝐴 𝑖 .

Collaborative utility computing component: in the contexte of decentralized monitoring, where some monitors can provide incorrect information about the network state, the AMs should collaborate in order to make consistent decisions. Each AM collaborates with other AMs and computes the average utility of each approach, by using the aggregation scheme SV1.

4) Change Operation Engine:

This component is in charge of reconfiguring the management plane and applying the given changes. Especially, it compares the utility of the current used approach with the best one, and then decides, according to a gain threshold 𝐺𝑎𝑖𝑛𝑇 , if the approach having the best utility is more relevant to be used than the current. This component can be represented through a finite state machine (FSM) with four states: Gossip (G), Tree (T), one-hop situated view (SV1) and a two-hops situated view (SV2). Since we assume that we have the same transition function between each pair of approaches, this FSM can be factorized into two abstract states 𝑆 𝑖 , 𝑆 𝑗 ∈ {G, T, SV1, SV2}. They represent, respectively, the current aggregation approach and the one having the best utility. The gain threshold 𝐺𝑎𝑖𝑛𝑇 enables SAAM to restrict the adaptation to only major environment changes (large 𝐺𝑎𝑖𝑛𝑇 value) or to accurately follow the environment state (low 𝐺𝑎𝑖𝑛𝑇 value).

IV. VALIDATION

A. Simulation Environment

We develop SAAM's blocs and all the considered aggregation schemes by ourselves. To evaluate the performance of SAAM, we rely on the FreePastry simulator, which enables the creation of the network nodes and the messages exchanges. Concerning the tree-based scheme, we used Scribe [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF] to maintain a multicast tree for the diffusion of the global aggregate to all nodes that monitor it. Simulation parameters are illustrated in Table III. We create a network where each node is able to execute one of the previously described aggregation schemes to compute an average aggregate of the local values on nodes. Concerning the network dynamics, we assume a Poisson process of parameter 1/𝜆 that varies in [2.4,240] seconds for the nodes' inter-arrivals and an exponential distribution of parameter 1/𝜇, ranging in [20,2000] minutes, for the session duration. Initially, we create 𝜆 𝜇 nodes, and then other nodes are created according to the poisson process. User preferences W=(0.25,0.25,0.5) Default approach=SV1 Ordering: churn=0⇒(SV1->SV2->Tree->Gossip) churn=1⇒(SV1->SV2->Gossip->Tree)

We validate SAAM by evaluating its capability to always instantiate the best aggregation process and by measuring its adaptation cost. We evaluate the cost and the performance of aggregation with SAAM, by computing its utility in a static environment with various network sizes and also in a dynamic context by varying the churn level and aggregated data dynamics. We evaluate the adaptation cost by measuring it in the case of a collaborative decision making (CDM) and also in the case of local decision making (LDM).

B. Results

In the following results, a 95% confidence interval is computed for each average value represented in the curves. These intervals do not appear when results are very accurate.

1) Aggregation performance: Given that the aggregation approach having the best utility is the one optimizing the cost and the performance of aggregation, we evaluate the performance of SAAM by periodically measuring its utility under different conditions and we compare it to those obtained with each aggregation approach executed separately. Figures 3.a, 3.b and 3.c illustrate the utility of each aggregation approach according to, respectively, network size, network dynamics and aggregated data dynamics.

We notice here that whatever the level of NS, ND or ID, SAAM outperforms the other aggregation approaches, because it periodically selects the one having the best utility. For the given user preferences, gossip often has a small utility, because its needs more time to converge than the tree or situated view. Figure 3.a shows that, globally, SAAM switches to the tree which offers a global view that is more accurate than the one of SV1. Figure 3.b shows that under a very low dynamic network, SAAM uses the tree for aggregation. But, under a higher dynamics (𝑁𝐷 > 2.5), the performance of the tree is affected by this dynamics. Thus, SAAM choses to use a situated view which better supports churn. Figure 3.c shows that SAAM always uses the tree which presents the best utility under a static network and dynamic aggregated data. In summary, these results show that SAAM optimizes the aggregation cost and performance whatever the current context of management information and its environment.

2) Fuzzy model validation: We check the validity of the fuzzy model by measuring how much the distributed FLCs enable the AMs to consistently estimate by themselves, the performance of each aggregation scheme. Figures 4.a and 4.b display respectively how convergence time and communication cost depend on network size. These results are direct outputs of the cost fuzzy controller. We observe in Figure 4.a a large convergence time for gossip, followed by the tree, and SV1 et SV2. We notice in Figure 4.b that all schemes have the same distribution of the communication cost according to the network size. The communication overhead of SV2 is higher than the one of gossip and SV1, and then the tree.

In summary, these results confirm our expectations and they are consistent with those obtained through simulations in [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF] showing the validity of our fuzzy model.

3) Adaptation cost: We measured it by the average number of messages per adaptation round needed by SAAM to adapt itself. It includes synchronization and collaboration messages. Simulation results are illustrated in Figure 4.c, where we also provide the aggregation overhead involved by SAAM. We observe here a linear distribution of the overhead according to the network size. For example, in the case of LDM and when NS=100, the aggregation overhead is almost 262 times higher than the adaptation overhead. This large difference is due to the fact that only some messages are needed by each node for synchronization. However, in the context of CDM, the aggregation overhead is almost twice higher than the adaptation overhead, because SAAM needs more messages to ensure the collaboration.

V. CONCLUSION AND PERSPECTIVES

In this paper, we propose a new decentralized self-adaptive aggregation mechanism SAAM that infers the current context of the environment and dynamically decides what aggregation strategy should be used according to this context. SAAM mainly rely on fuzzy logic and MADM approaches to estimate the performance of the integrated aggregation schemes and to select the best one to use. The validation results show that SAAM efficiently optimizes the aggregation cost and performance with a relatively low adaptation cost. This work opens numerous working perspectives. We are currently performing extended simulations allowing a full featuring of its operational cost and performance. We also plan to apply our proposal to real use cases like the monitoring of complex networks such as P2P live streaming systems. In this context, we also plan to evaluate our approach in semi-real conditions through experiments over PlanetLab.

Fig. 1 .

 1 Fig. 1. Global Architecture of the Self-Adaptive Aggregation Framework

Fig. 2 .

 2 Fig. 2. Fuzzification of the FLCs inputs and outputs: (a) FLC1, (b) FLC2.

Fig. 3 .

 3 Fig. 3. Maximization of the utility according to (a) NS, (b) ND and (c) ID a consistent ordering of aggregation approaches (alternatives) according to multiple quantitative cost and performance criteria. The MADM component is composed of:Local utility computing component: in order to locally (on each AM) compute the utility 𝑈 𝑖 of each aggregation scheme, we use SAW (Simple Additive Weighting)[START_REF] Zhang | Handover decision using fuzzy madm in heterogeneous networks[END_REF]. The latter is a standard MADM approach. It consists of scaling the values of all attributes and summing up them for each alternative to compute an overall score. The latter is a weighted sum of all the attribute values,𝑈 𝑖 = 𝑤 1 .𝐶𝑇 𝑖 + 𝑤 2 .𝐶𝐶 𝑖 + 𝑤 3 .𝐴𝐴 𝑖 .Collaborative utility computing component: in the contexte of decentralized monitoring, where some monitors can provide incorrect information about the network state, the AMs should collaborate in order to make consistent decisions. Each AM collaborates with other AMs and computes the average utility of each approach, by using the aggregation scheme SV1.4) Change Operation Engine:This component is in charge of reconfiguring the management plane and applying the given changes. Especially, it compares the utility of the current used approach with the best one, and then decides, according to a gain threshold 𝐺𝑎𝑖𝑛𝑇 , if the approach having the best utility is more relevant to be used than the current. This component can be represented through a finite state machine (FSM) with four states: Gossip (G), Tree (T), one-hop situated view (SV1) and a two-hops situated view (SV2). Since we assume that we have the same transition function between each pair of approaches, this FSM can be factorized into two abstract states 𝑆 𝑖 , 𝑆 𝑗 ∈ {G, T, SV1, SV2}. They represent, respectively, the current aggregation approach and the one having the best utility. The gain threshold 𝐺𝑎𝑖𝑛𝑇 enables SAAM to restrict the adaptation to only major environment changes (large 𝐺𝑎𝑖𝑛𝑇 value) or to accurately follow the environment state (low 𝐺𝑎𝑖𝑛𝑇 value).

Fig. 4 .

 4 Fig. 4. Validation results on the FLCs outputs (a) CT and (b) CC, and on (c) the adaptation overhead

TABLE I RULES

 I OF THE COST ESTIMATOR FLC.

			Tree		Gossip	SV1		SV2
			CT	CC	CT	CC	CT	CC	CT	CC
		VL	L	VL	M	VL	VL	VL	VL	VL
	NS	L	M	VL	H	M	L	L	L	M
		M	M	L	VH	M	L	M	L	H
		H	M	M	VH	H	L	H	L	VH

TABLE II RULES

 II OF THE PERFORMANCE ESTIMATOR FLC.


Rafik Makhloufi
email: rafik.makhloufi@utt.fr

Guillaume Doyen
email: guillaume.doyen@utt.fr

Grégory Bonnet
email: gregory.bonnet@unicaen.fr

Dominique Gaïti
email: dominique.gaiti@utt.fr

Towards Self-Adaptive Management Frameworks: the Case of Aggregated Information Monitoring

Keywords: Autonomous Networking, Decentralized Aggregation, Management Information, Fuzzy Logic

In this paper, we propose a decentralized selfadaptive management framework that infers the current context of the hosting environment and dynamically decides what management strategy should be used in order to minimize the cost and maximize the performance of management. We investigate the use of this framework in the case of the aggregated information monitoring, where only one scheme (i.e. situated or global) is generally used for collecting aggregated information whatever the nature of the management information and its environment are. We propose a self-adaptive aggregation mechanism based on fuzzy logic, that estimates the cost and the performance of given aggregation schemes.

I. INTRODUCTION

Current networks and services constantly evolve in terms of complexity, size, distribution and dynamics. During the last decade, numerous approaches have emerged in order to manage them in a decentralized and autonomous way. These autonomous approaches are deployed on heterogenous and dynamic environments where it is crucial for these approaches to adapt themselves to the supporting network operational behavior. For example, in mobile ad-hoc networks (MANETs) and peer-to-peer (P2P) ones, the network size is continuously evolving due to, respectively, nodes mobility and peers churn. Consequently, the management approach should adapt itself to such dynamics and operational states.

In this paper, we propose a decentralized self-adaptive management framework that infers the current operational state of the network and dynamically decides what management strategy should be used according to this context. We investigate the use of this framework in the case of the aggregation of management information, where we propose a self-adaptive aggregation mechanism based on fuzzy logic, that estimates the cost and the performance of given aggregation schemes, computes their utilities and then decides collectively what is the best aggregation scheme to use. We focus on the utility evaluation, which is the core of the architecture dedicated to information analysis for the decision-making, and then we validate the developed fuzzy-based model.

The reminder of this paper is organized as follows. In Section 2, we present the related work. We present the functional elements of the proposed adaptive management framework in Section 3. Then, we present the utilization of this approach for an adaptive aggregation in Section 4. Subsequently, we give early validation results of this approach in Section 5. Finally, we conclude this work and we present our perspectives.

II. RELATED WORK

Autonomous management systems [START_REF] Horn | Autonomic computing: IBM's Perspective on the State of Information Technology[END_REF] need to collect local and global management information in order to analyze them and to make management decisions. Several research efforts propose to collect it through aggregation schemes.

In [START_REF] Bawa | Estimating aggregates on a Peer-to-Peer network[END_REF] and [START_REF] Wuhib | Robust monitoring of network-wide aggregates through gossiping[END_REF], the authors propose and evaluate some gossip-and tree-based aggregation schemes. [START_REF] Bawa | Estimating aggregates on a Peer-to-Peer network[END_REF] shows that each category outperforms the other one for a given performance criteria, while [START_REF] Wuhib | Robust monitoring of network-wide aggregates through gossiping[END_REF] shows that the tree outperforms gossip for all the studied criteria. Unfortunately, these studies do not discuss the situated aggregation schemes.

In [START_REF] Makhloufi | Decentralized Aggregation Protocols in Peer-to-Peer Networks: a Survey[END_REF] and [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF], we gave an overview of a set of decentralized aggregation schemes and a multi-criteria evaluation of them, showing that each aggregation scheme is better than the others for a given evaluation criteria. [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF] shows that global schemes (i.e. gossip and tree) provide a good accuracy at the cost of a larger convergence time and overhead, while situated approaches minimize their cost and provide a high reactivity to the Autonomic Managers (AMs) with a low accuracy.

Globally, these studies address decentralized aggregation schemes and propose to evaluate their cost and performance through simulations or algorithmic complexity. They show that aggregation strategies behave differently depending on the network operational state and the management information behavior. Through these studies, we observe that it is crucial to establish an adaptive approach that adapts itself to the supporting network operational behavior in order to be able to design efficient management systems. But, existing systems propose to use one aggregation scheme in a static and ad-hoc way without taking into account the nature of management information and the one of its environment.

III. SELF-ADAPTIVE MANAGEMENT ARCHITECTURE

In order to be able to design efficient frameworks that minimize the cost and maximize the performance of management, we propose to use a self-adaptive approach that infers the current network operational state and dynamically decides according to this one what management strategy should be used and/or what changes must occur on it. This framework takes into account management policies through user preferences representing, for example, a tradeoff between performance and cost of the management process.

As depicted in Figure 1, an autonomous system is composed of a set of interactive AMs that manage their internal behavior and their relationships with other AMs. The control loop of autonomic computing consists of four main functions which are Monitor, Analyze, Plan, Execute (MAPE) [START_REF] Kephart | The Vision of Autonomic Computing[END_REF]. This model can be applied to the adaptive configuration of the management plane with four components that enable a dynamically selection of a management approach according to the current context. The global architecture of the resulting framework is composed of:

• Network state monitor: this process is responsible for the monitoring and data-gathering about the current operational state of the network (e.g. size, dynamics, neighborhood degree). It uses sensors to collect data from local or from other AMs and then it sends them to the next component for analysis. An aggregation scheme can be used to collect this management information. • Utility evaluator: based on the data provided by the network state monitoring engine, the utility evaluator analyzes them, estimates the values cost and the performance of given management strategies. According to these values, a utility function is then used for the computation of a pertinence metric that shows how much a management strategy is relevant. This utility function allows a tradeoff between performance and cost. • Collaborative decision-making: according to the given utility of each management strategy, this component decides if the approach having the best utility is more relevant to be used than the current and what changes should be applied on the selected approach. To do this, each AM collaborates with other AMs in the network in order to make a collective decision on a possible change so that they minimize the cost and maximize the performance of management. • Change operation engine: once the changes that should have occurred are defined by the collaborative decisionmaking mechanism, the change operation component is in charge of reconfiguring the management plane apply the given decisions. This requires synchronized and rapid actions through all AMs so that the change is seamless.

IV. SAAM: A SELF-ADAPTIVE AGGREGATION MECHANISM

In this section, we present a use case of the previous architecture in the context of aggregated information monitoring. Indeed, an autonomous management system need to collect management information from the network elements. This information can be collected through aggregation schemes. Aggregation [START_REF] Makhloufi | Decentralized Aggregation Protocols in Peer-to-Peer Networks: a Survey[END_REF] is intended as a summarizing mechanism of the overall state within the network. For a best aggregation quality, we design a new decentralized self-adaptive aggregation framework (SAAM), based on the previously described architecture. According to network and information characteristics, each decentralized entity of the SAAM computes the utility of each aggregation scheme and then, with the help of the other elements, it decides what is the best aggregation scheme to use.

A. Considered Aggregation Schemes

We consider typical situated and global aggregation schemes that we benchmarked in [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF]. These are:

• Gossip (G): a push-pull gossip scheme inspired from [START_REF] Jelasity | Gossip-based aggregation in large dynamic networks[END_REF],

where in each round, each node contacts one neighbor chosen uniformly at random and exchanges information with it. Thus, a global aggregate will be computed and made available across all the network nodes. • Tree (T): a push tree-based scheme based on GAP [START_REF] Dam | A generic protocol for network state aggregation[END_REF],

where nodes communicate their local aggregates to a single root node that computes an overall aggregate and spreads it on all the interested nodes through a publishsubscribe mechanism. • Situated view: a pull-based situated scheme based on HyParView [START_REF] Leitao | Large-Scale Peer-to-Peer Autonomic Monitoring[END_REF] and Propagate2All [START_REF] Bawa | Estimating aggregates on a Peer-to-Peer network[END_REF], where each node maintains a view of its h-hops neighbors. Thus, each node collects information from its view and then computes a partial aggregate. We consider a one-hop situated view (SV1) and a two-hops situated view (SV2).

B. Utility Evaluator

In this study, we focus on the utility evaluator. As illustrated in Figure 2, the utility evaluator is composed of Fuzzy Logic Controllers (FLCs) and a utility function that can be computed, for example, through MADM (Multiple Attribute Decision Making) [START_REF] Hwang | Multiple Attribute Decision Making[END_REF]. The latter uses the numerical values of the cost and the performance provided by the FLCs for the computation of a pertinence metric that shows how much an aggregation scheme is relevant to be used.

We rely on fuzzy logic because it provides a formal methodology for representing, manipulating, and implementing a human's heuristic knowledge about how to control a system. It allows a performant decision-making under incomplete and uncertain information, here, about the network operational state. Moreover, computing with fuzzy sets is a necessity when (1) the available information is too imprecise to justify the use of numbers, and (2) there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost, and better rapport with reality [START_REF] Zadeh | Fuzzy logic = computing with words[END_REF], [START_REF] Passino | Fuzzy Control[END_REF]. We propose here to use two fuzzy logic controllers to process the inputs on the network state in order to estimate, respectively, the performance and the cost of each aggregation approach. These FLCs operates according the following main steps: fuzzification, inference and defuzzification. 1) Fuzzification: According our previous evaluation studies [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF], [START_REF] Makhloufi | Situated vs. Global Aggregation Schemes for Autonomous Management Systems[END_REF], we distinguish, on one hand, two main factors that impact the performance of an aggregation scheme: network and information dynamics. On the other hand, the cost of an aggregation scheme depends on the network size. In order to consider them as inputs of the FLC, we need to convert these three parameters into fuzzy inputs presenting different degrees of membership so that they can be interpreted and evaluated in an inference mechanism. These factors are defined as follows:

• Network size (NS) [START_REF] Makhloufi | Situated vs. Global Aggregation Schemes for Autonomous Management Systems[END_REF]: it is the number of nodes in the network. This factor directly affects the cost of aggregation schemes. • Network dynamics (ND) [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF]: it consists in nodes arrivals in the system and their departures that can be planned or due to a sudden failure. We characterize it by the sum of both nodes arrival and departure rates. • Information dynamics (ID) [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF]: it determines the evolution degree of a management information over the time. We characterize it through the changing degree (i.e. distance) between two consecutive values of the monitored variable on the managed element. Since in the literature, the most acknowledged studied cost and performance criteria for aggregation schemes are convergence time, communication cost and accuracy, we choose them as outputs of these FLCs. They are defined as follows:

• Convergence Time (𝐶𝑇 ) [START_REF] Makhloufi | Situated vs. Global Aggregation Schemes for Autonomous Management Systems[END_REF]: it is the necessary time between the initialization of the aggregation process and the time 𝑡 when all nodes hold the aggregation results. • Communication Cost (𝐶𝐶) [START_REF] Makhloufi | Situated vs. Global Aggregation Schemes for Autonomous Management Systems[END_REF]: it is the sum of sizes of messages sent between any node pairs (𝑖, 𝑗) during the aggregation process. Since all messages have approximatively the same size, the communication cost is measured by the number of messages sent by all nodes.

• Aggregation Accuracy (𝐴𝐴) [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF]: it is the precision level of the aggregation scheme when computing aggregates. It is measured by the average deviation (i.e. distance) between the estimated aggregates and the actual global aggregate that should be computed. 2) Inference: In order to evaluate which control rules have to be activated at the current time according to the inputs and then decide what the output should be, each FLC relies on an inference mechanism. The latter uses a set of rules, on the form "If premise Then consequent" to estimate the cost and the performance of each aggregation scheme. We define three membership functions for each FLC variable in the rule-base: low, medium and high. Table I and Table II below illustrate, respectively, the rules used by the FLCs to estimate both the performance and the cost of each aggregation scheme. For example, in the case of gossip on Table I: If ND=Medium and ID=Low Then AA=Medium.

Gossip Tree SV1 SV2 ND ND ND ND L M H L M H L M H L M H ID L H M L H L L L L L L L L M H M L H L L L L L L L L H M L L M L L L L L L L L TABLE II RULES OF THE SECOND FLC. Gossip Tree SV1 SV2 CT CC CT CC CT CC CT CC NS L M L M L L L L L M H L M L L L L M H H M M L L M L H
3) Defuzzification: The defuzzification interface converts the conclusions reached by the inference mechanism (i.e. fuzzy sets) into quantifiable outputs. In the case of SAAM, the defuzzification component will converts the fuzzy measures on the cost and the performance of each aggregation approach into numeric values that will be used by the utility function.

To do this, we consider the center of gravity which is the most standard defuzzification method [START_REF] Leekwijck | Defuzzification: criteria and classification[END_REF]. The latter computes the center of gravity of the area under the membership functions of the inference fuzzy outputs.

V. EARLY VALIDATION RESULTS

In this section, we give early validation results of the proposed adaptive approach, focusing on the FLCs of the utility evaluator. This aims at checking if these FLCs enable the AMs to consistently estimate by themselves, in a decentralized way, the cost and the performance of each aggregation scheme.

A. Evaluation Setup

In order to show the influence of dynamics and network topology on the cost and the performance of an aggregation scheme, we rely on the fuzzy logic toolbox of Matlab [START_REF]Matlab -the language of technical computing[END_REF], in which we implement the two FLCs. We propose to characterize their inputs and outputs according to the performance evaluation results provided in [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF] for each aggregation scheme Figures 3 and 5 display respectively how aggregation performance and cost depend on the FLC inputs, i.e., network churn, information dynamics and network size. Results are direct outputs of the fuzzy controller. They provide an abstract view of the relative performance of the given schemes under different conditions. We observe in Figure 3 that, for both gossip and tree, the aggregation accuracy decreases with an increase of network and information dynamics. We also observe that tree is more affected by the network dynamics than gossip. Concerning SV1 and SV2, they ensure a low accuracy whatever the level of dynamics is. We notice in Figure 5.a a high convergence time that increases when having a large number of nodes in the case gossip. This convergence is medium for tree and low for SV1 and SV2, independently of the network size. In Figure 5.b, we see that the communication cost increases according to the network size. Especially, we observe a high communication cost for both gossip and SV2 followed by SV1 and then a low communication cost for tree. However, for example in Figure 5.a, convergence time on the tree is constant while it should have a logarithmic distribution according to the network size (NS). In our prior simulations [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF], CT is proportional to the network size. Globally these results confirm our expectations and they are coherent with those obtained in [START_REF] Makhloufi | Impact of Dynamics on Situated and Global Aggregation Schemes[END_REF] through simulations. This enables each node to consistently estimate, in a decentralized way, the cost and the performance of each aggregation scheme.

VI. CONCLUSION AND PERSPECTIVES

In order to be able to design efficient management frameworks, we propose in this paper to use a self-adaptive architecture that adapts itself to the supporting network operational behavior. We instantiate this architecture in the case of the aggregation schemes, where it is crucial to efficiently provide the required management information to the decision-making process while minimizing its cost and maximizing its performance. In order to estimate them, we used a fuzzy approach and especially two fuzzy controllers that consider network and information dynamics and network size. Validation results confirm the existing simulation ones, enabling an autonomous manager to get a consistent estimation of the cost and the performance of each aggregation scheme. This work opens numerous perspectives. In order to enhance the characterization of the aggregation schemes and the validation results, we are currently working on the enhancement of the utility evaluator with the definition of a consistent utility function. We are also working on the next functional blocs that present strong challenges: the establishment of a global agreement on the best strategy to use, and the rapid and seamless reconfiguration of the autonomous management plane. It is also important to address the adaptation cost of SAAM after implementing all the blocs. At that moment, realistic situations and use cases (e.g. P2P VoD streaming, MANeTs) can be studied by choosing the suitable parameters (e.g. weights of the utility function), allowing a tradeoff between cost and gain.

Fig. 2 .

 2 Fig. 2. Architecture of the Utility Evaluator.

Fig. 3 .

 3 Fig. 3. Aggregation Accuracy for (a) Gossip, (b) Tree, (c) SV1, and (d) SV2 According to Network and Information Dynamics.

  (i.e. gossip, tree, SV1 and SV2). The fuzzification of these variables is shown in Figure4, where we use normalized values for both dynamics and churn. Thus, ND and AA are ranging in [0;100] while ID∈[0;1].

Fig. 5 .

 5 Fig. 5. Estimated Convergence Time (a) and Communication Cost (b) According to the Network Size.

TABLE I RULES

 I OF THE FIRST FLC.

B. Results


Delphine Bresch-Pietri

Miroslav Krstic

Adaptive Trajectory Tracking Despite Unknown Input Delay and Plant Parameters

In a recent paper we presented the first adaptive control design for an ODE system with a possibly large actuator delay of unknown length. We achieved global stability under full state feedback. In this paper we generalize the design to the situation where, besides the unknown delay value, the ODE also has unknown parameters, and where trajectory tracking (rather than equilibrium regulation) is pursued.

Introduction

Input delay arises in many forms, from an actual physical transport delay in various chemical process systems, hydraulically actuated systems, and combustion systems, to the problems where computational delay manifests itself as being equivalent to input delay. Until recently, the only results on adaptive control of systems with input delays dealt only with uncertain parameters in the ODE part of the system [START_REF] Evesque | Adaptive Control of A Class of Time-delay Systems[END_REF], Niculescu and Annaswamy 2003[START_REF] Ortega | Globally stable adaptive controller for systems with delay[END_REF] but not with uncertainty in the delay value itself. The importance of designing adaptive controllers for unknown delay was recognized in (Kolmanovsky et al. 2001, Krstic and[START_REF] Krstic | Multivariable adaptive control of instabilities arising in jet engines[END_REF], however only approximation-based ideas for limited classes of plants were dealt with. The robustness of adaptive backstepping control for linear ODEs to dynamic perturbations that include input delay was studied in [START_REF] Zhou | Adaptive backstepping control of uncertain systems with unknown input time delay[END_REF], however the nominal system did not involve delay and the delay was not estimated nor compensated by the controller.

In a recent paper (Bresch-Pietri and [START_REF] Bresch-Pietri | Delay-adaptive full-state predictor feedback for systems with unknown long actuator delay[END_REF] we presented the first results on delay-adaptive control for a general class of plants, under full state feedback. This result is global-including being global in the initial delay value estimate (one can arbitrarily underestimate or overestimate the delay value initially, and still achieve stabilization adaptively).

However, the result in (Bresch-Pietri and Krstic 2009) contains two limitations, one made for pedagogical reasons and the other which is fundamental. The limitation for which the Email address: krstic@ucsd.edu (Delphine Bresch-Pietri and Miroslav Krstic).

reason is pedagogical is in assuming that there are no unknown parameters in the ODE. This limitation is being removed by this paper. This limitation was imposed in [START_REF] Bresch-Pietri | Delay-adaptive full-state predictor feedback for systems with unknown long actuator delay[END_REF] to prevent the novel ideas on how to develop global adaptivity in the infinite-dimensional (delay) context from being buried under standard but nevertheless complicated details of ODE adaptive control.

The other limitation in (Bresch-Pietri and [START_REF] Bresch-Pietri | Delay-adaptive full-state predictor feedback for systems with unknown long actuator delay[END_REF], which is fundamental, is in assuming that the full actuator state is measured, though the delay value is completely unknown. The physical meaning of this is that the actuator delay is modeled as a transport process, to which the control designer has physical access for measurement but the speed of propagation of this transport process is completely unknown. As we explain in [START_REF] Bresch-Pietri | Delay-adaptive full-state predictor feedback for systems with unknown long actuator delay[END_REF], the problem where the actuator state is not measurable and the delay value is unknown is not solvable globally, since the problem is not linearly parametrized. We show in [START_REF] Bresch-Pietri | Delay-adaptive full-state predictor feedback for systems with unknown long actuator delay[END_REF] how one can solve it locally, however, this is not a very satisfactory result, since it is local both in the initial state and in the initial parameter error. In other words, the initial delay estimate needs to be sufficiently close to the true delay. (The delay can be long, but it needs to be known quite closely.) Under such an assumption, one might as well use a linear controller and rely on robustness of the feedback law to small errors in the assumed delay value.

So, for this reason, in this paper we continue with a full-state feedback design, specifically assuming the measurement of the actuator state. In [START_REF] Bresch-Pietri | Delay-adaptive full-state predictor feedback for systems with unknown long actuator delay[END_REF] we discussed all the possible problems that one can consider with respect to the availability or unavailability of measurement of the ODE state, the actuator state, the knowledge (or lack of knowledge) of the delay value, and the knowledge (or Preprint submitted to Automatica lack thereof) of the ODE parameters. There is a total number of 14 distinct problem combinations. Here we focus on the most interesting one of them, with both the ODE parameters and the delay value unknown, but with full state measurement. An extension to the case where only an output (and not the complete state) of the ODE is available for measurement is easy (with the method of backstepping and Kreisselmeier observers). We don't pursue it here for consistency of concepts-since we must measure the actuator state, we might as well present a result with full measurement of the ODE state.

As in most of the research on control of unstable plants with a long actuator delay [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF][START_REF] Evesque | Adaptive Control of A Class of Time-delay Systems[END_REF][START_REF] Fiagbedzi | Feedback stabilization of linear autonomous time lag systems[END_REF][START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF], Jankovic 2006a, Jankovic 2006b[START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF][START_REF] Klamka | Observer for linear feedback control of systems with distributed delays in controls and outputs[END_REF], Krstic 2008a, Krstic 2008b, Krstic and Smyshlyaev 2008b[START_REF] Kwon | Feedback stabilization of linear systems with delayed control[END_REF][START_REF] Manitius | Finite spectrum assignment for systems with delays[END_REF][START_REF] Michiels | On the delay sensitivity of Smith predictors[END_REF][START_REF] Michiels | Stability and Stabilization of Time-Delay Systems: An Eignevalue-Based Approach[END_REF][START_REF] Mirkin | On the approximation of distributed-delay control laws[END_REF][START_REF] Mondie | Finite spectrum assignment of unstable time-delay systems with a safe implementation[END_REF], Niculescu and Annaswamy 2003[START_REF] Olbrot | Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays[END_REF][START_REF] Richard | Time-delay systems: An overview of some recent advances and open problems[END_REF], Tadmor 1996, Watanabe and Ito 1996[START_REF] Watanabe | An observer for linear feedback control laws of multivariable systems with multiple delays in controls and outputs[END_REF][START_REF] Zaccarian | A cascade interpretation of the Smith predictor and the arising enhanced scheme[END_REF], Zhong 2006a, Zhong 2006b[START_REF] Zhong | Control of integral processes with dead time-Part 2: Quantitative analysis[END_REF], the essence of our approach is "predictor feedback," which we recently showed in (Krstic 2008a, Krstic andSmyshlyaev 2008b) to be a form of backstepping boundary control for PDEs (Krstic and Smyshlyaev 2008c) and extended to nonlinear plants (Krstic 2008b).

In this paper we generalize the design from (Bresch-Pietri and [START_REF] Bresch-Pietri | Delay-adaptive full-state predictor feedback for systems with unknown long actuator delay[END_REF] in two major ways: we extend it to ODEs with unknown parameters and extend it from equilibrium regulation to trajectory tracking. A significant number of new technical issues arise in this problem. The estimation error of the ODE parameters appears in the error models of both the ODE and of the infinite-dimensional (delay) subsystem, which is reflected also in the update law. The update law has to also deal appropriately with ensuring stabilizability with the parameter estimates, for which projection is employed. Finally, our approach for dealing with delay adaptation involves normalized Lyapunov-based tuning, a rather non-standard approach as compared to finite-dimensional adaptive control. In this framework, we need to bound numerous terms involving parameter adaptation rates (both for the delay and for the ODE parameters) in the Lyapunov analysis. Some of these terms are vanishing (when the tracking error is zero), while the others (which are due to the reference trajectory) are non-vanishing. These terms receive different treatment though both are bounded by normalization and their size is controlled with the adaptation gain.

We begin in Section 2 by defining the problem and present the adaptive control design and the main stability theorem in Section 3. Simulations results are show in Section 4, which is then followed by the stability proof in Section 5.

Problem Formulation

We consider the following system

Ẋ(t) = A(θ )X(t) + B(θ )U(t -D) (1) Y (t) = CX(t) , (2)
where X ∈ R n is the ODE state, U is the scalar input to the entire system, D > 0 is an unknown constant delay, the system matrix A(θ ) and the input vector B(θ ) are linearly parametrized, i.e.,

A(θ ) = A 0 + p ∑ i=1 θ i A i (3) B(θ ) = B 0 + p ∑ i=1 θ i B i , (4)
and θ is an unknown but constant parameter vector that belongs to the convex set

Π = {θ ∈ R p |P(θ ) ≤ 0} , (5)
where, by assuming that the convex function P : R p → R is smooth, we assure that the boundary ∂ Π of Π is smooth.

Assumption 1

The set Π is bounded and known. A constant D is known such that D ∈]0; D].

To make stabilization and tracking possible in the presence of unknown parameters, we make some assumption and illustrate, with the help of an example introduced below, that these assumptions are reasonable. In the sequel, C 0 (Π) and C 1 (Π) is the usual notation for continuous and continuously differentiable functions (on the set Π), respectively.

Assumption 2 We assume that the pair (A(θ ), B(θ )) is completely controllable for each θ . Furthermore, we assume that there exists a triple of vector/matrix-valued functions

(K(θ ), P(θ ), Q(θ )) such that K ∈ C 1 (Π), P ∈ C 1 (Π), Q ∈ C 0 (Π)
, the matrices P(θ ) and Q(θ ) are positive definite and symmetric, and the following Lyapunov equation is satisfied for all θ ∈ Π:

P(θ )(A + BK)(θ ) + (A + BK)(θ ) T P(θ ) = -Q(θ ) . (6)
Example 1 Consider the example of a potentially unstable plant

Ẋ1 (t) = θ X 1 (t) + X 2 (t) (7) Ẋ2 (t) = U(t -D) (8) Y (t) = X 1 (t) , (9)
where we assume Π = [-θ ; θ ] and define

A(θ ) = A 0 + θ A 1 = 0 1 0 0 + θ 1 0 0 0 (10) B = B 0 = 0 1 . ( 11
)
Using the backstepping method we construct the triple (K, P, Q) as 12)

K(θ ) = -1 + (θ + 1) 2 θ + 2 (
P(θ ) = 1 2 Q(θ ) = 1 + (1 + θ ) 2 1 + θ 1 + θ 1 , (13)
which satisfies the Lyapunov equation ( 6).

Assumption 3

The quantities

λ = inf θ ∈Π min {λ min (P(θ )), λ min (Q(θ ))} (14) λ = inf θ ∈Π λ max (P(θ )) . ( 15
)
exist and are known.

Example 2 (Example 1 continued) One can show that the eigenvalues of P(θ ) are

λ max (P(θ )) = 2 + (θ + 1) 2 + |θ + 1| (θ + 1) 2 + 1 2 (16) λ min (P(θ )) = 1 λ max P(θ ) , (17)
from which λ and λ area readily obtained over the set Π = [-θ ; θ ].

Assumption 4 For a given smooth function Y r (t), there exist known functions X r (t, θ ) and U r (t, θ ), which are bounded in t and continuously differentiable in the unknown argument θ on Π, and which satisfy

Ẋr (t, θ ) = A(θ )X r (t, θ ) + B(θ )U r (t, θ ) (18) Y r (t) = CX r (t, θ ) . ( 19
) Example 3 (Example 2 continued) Take Y r (t) = sin(t).
Then, the reference trajectory pair for the state and input is

X r (t, θ ) = sin(t) cos(t) -θ sin(t) (20) U r (t, θ , D) = -sin(t + D) -θ cos(t + D) . ( 21
)
Both functions are bounded in t and continuously differentiable in θ .

Control Design

To prepare for our adaptive control design, we represent the plant as

Ẋ(t) = A(θ )X(t) + B(θ )u(0,t) (22) Y (t) = CX(t) (23) Du t (x,t) = u x (x,t) , x ∈ [0, 1) (24) u(1,t) = U(t) , (25)
where

u(x,t) = U(t + D(x -1)) . ( 26
)
The representation of the pure delay as a transport PDE allows a linear parameterization in the unknown delay D.

We consider reference trajectories X r (t) and U r (t), such as described in Assumption 4. Let us introduce the following error variables

X(t) = X(t) -X r (t, θ ) (27) Ũ(t) = U(t) -U r (t, θ , D) (28) e(x,t) = u(x,t) -u r (x,t, θ ) , (29)
with an estimate θ of the unknown θ . When D and θ are known, one can show that the control law

U(t) = U r (t) -KX r (t + D) +K e AD X(t) + D 1 0 e AD(1-y) Bu(y,t)dy (30)
achieves exponential stability of the equilibrium ( X, e) = 0, compensating the effects of the delay D.

When D and θ are unknown, we employ the control law

U(t) = U r (t, θ , D) -K( θ )X r (t + D, θ ) +K( θ ) e A( θ) D(t) X(t) + D(t) 1 0 e A( θ ) D(t)(1-y) B( θ )u(y,t)dy , (31)
based on the certainty equivalence principle. The update laws for the estimates D and θ are chosen based on the Lyapunov analysis (presented in Section 5) as

Ḋ(t) = γ 1 Proj [0, D] {τ D (t)} (32) θ (t) = γ 2 Proj Π {τ θ (t)} , (33)
with adaptation gains γ 1 and γ 2 chosen as positive and "small enough" and with

τ D (t) = - 1 0 (1 + x)w(x,t)K( θ )e A( θ) D(t)x dx 1 + X(t) T P( θ ) X(t) + b 1 0 (1 + x)w(x,t) 2 dx × (A + BK)( θ ) X(t) + B( θ )w(0,t) (34) τ θ (t) = 2 X(t) T P( θ )/b -1 0 (1 + x)w(x,t)K( θ )e A( θ ) D(t)x dx 1 + X(t) T P( θ ) X(t) + b 1 0 (1 + x)w(x,t) 2 dx × (A i X(t) + B i u(0,t)) 1≤i≤p . (35)
The matrix P is defined in Assumption 2, the standard projector operators are given by

Proj [0, D] {τ D } = τ D        0, D = 0 and τ D < 0 0, D = D and τ D > 0 1, else (36)
Proj Π {τ θ } = τ θ                  1, θ ∈ Π or ∇ θ P T τ ≤ 0 I - ∇ θ P∇ θ P T ∇ θ P T ∇ θ P , θ ∈ ∂ Π and ∇ θ P T τ > 0 . (37)
The transformed state of the actuator is needed in the update law and is defined as w(x,t) = e(x,t) -D(t)

x 0 K( θ )e A( θ) D(t)(x-y) B( θ )e(y,t)dy

-K( θ )e A( θ) D(t)x X(t) (38)
and the constant b is chosen such as

b ≥ 4 sup θ ∈Π |PB| 2 ( θ ) D λ . ( 39
)
Theorem 1 Let Assumptions 1-4 hold and consider the closed-loop system consisting of ( 22)-( 25), the control law (31) and the update laws defined by ( 32)-( 39). There exists γ * > 0 such that for any γ ∈ [0, γ * [, there exist positive constants R and ρ (independent of the initial conditions) such that, for all initial conditions satisfying (X 0 , u 0 , D0 , θ 0 ) ∈ R n × L 2 (0, 1)×]0, D] × Π, the following holds:

ϒ(t) ≤ R e ρϒ(0) -1 , ∀t ≥ 0 , ( 40
)
where Furthermore, asymptotic tracking is achieve, i.e.,

ϒ(t) = | X(t)| 2 + 1 0 e(x,t) 2 dx + D(t) 2 + θ (t) T θ (t) . (
lim t→∞ X(t) = 0 , lim t→∞ Ũ(t) = 0 . ( 42
)

Simulations

We return to the system from Examples 1-3 and present simulation results for the closed-loop system consisting of the plant (1)-( 2), the control law (31) and the update laws defined through (32)-( 39). We focus on the issues arising from the large uncertainties in D and θ and from the tracking problem with the reference trajectory (20)-( 21). We take D = 1, θ = 0.5, D = 2D = 2, θ = 0, θ = 2θ = 1. We pick the adaptations gains as γ 1 = 10, γ 2 = 2.3 and the normaliza- tion coefficient as b = 4 |PB| 2 D λ = 3200. We show simulation results for X 1 (0) = X 2 (0) = 0.5, θ (0) = 0, and two different values of D(0).

Figure 1 shows that, as Theorem 1 predicts, perfect asymptotic tracking is achieved (in a single simulation) for the output, state, and input, despite starting with an initial delay estimate of D(0) = 0, namely, for a large parametric uncertainty and with no predictor feedback initially. Figure 2 shows two distinct simulations, starting from two extreme initial values for the delay estimate, one at zero, and the other at twice the true delay value. The dashed simulation starts with a predictor feedback that initially overcompensates the delay, but it later adjusts the value of the delay estimate to obtain a stabilizing value of this parameter in the predictor feedback.

In Figure 2 we observe that θ (t) converges to the true θ in both simulations, whereas this is not the case with D(t). This is consistent with the theory. By examining the error systems ( 44), ( 45), with the help of a persistency of excitation argument, we could infer the convergence of θ (t) but not of D(t).

Proof of the Main Result

In this section, we prove Theorem 1. We start by considering the transformation (38) along with its inverse e(x,t) = w(x,t) + D(t)

x 0 K( θ )e (A+BK)( θ) D(t)(x-y) B( θ ) ×w(y,t)dy + K( θ)e (A+BK)( θ) D(t)x X(t) . (43)
Using these transformations and the models (1) and ( 18), the transformed system is written as

Ẋ(t) = (A + BK)( θ) X(t) + B( θ)w(0,t) + A( θ )X(t) +B( θ )u(0,t) - ∂ X r ∂ θ (t, θ ) θ (t) (44) Dw t (x,t) = w x (x,t) -D(t)p 0 (x,t) -D Ḋ(t)q 0 (x,t) -D θ (t) T p(x,t) -D θ (t) T q(x,t) (45) w(1,t) = 0 , (46)
where D(t) = D -D(t) is the estimation error of the delay, the quantities

A( θ ) = p ∑ i=1 θi A i = p ∑ i=1 (θ i -θi (t))A i (47) B( θ ) = p ∑ i=1 θi B i (48)
are linear in the parameter estimation error θ (t) = θθ (t), and

p 0 (x,t) = K( θ )e A( θ ) D(t)x ((A + BK)( θ ) X(t) +B( θ )w(0,t)) (49) q 0 (x,t) = x 0 K( θ ) I + A( θ ) D(t)(x -y) e A( θ) D(t)(x-y) ×B( θ )e(y,t)dy + K( θ )A( θ )xe A( θ) D(t)x X(t) (50) = x 0 w(y,t) K( θ ) I + A( θ ) D(t)(x -y) ×e A( θ) D(t)(x-y) B( θ ) + D(t) x y K( θ ) × I + A( θ) D(t)(x -ξ ) e A( θ ) D(t)(x-ξ ) BK( θ ) ×e (A+BK)( θ) D(t)(ξ -y) B( θ )dξ dy + KA( θ )x ×e A( θ) D(t)x + x 0 K( θ ) I + A( θ ) D(t)(x -y)
×e A( θ) D(t)(x-y) BK( θ )e (A+BK)( θ) D(t)y dy X(t) .( 51)

The vector-valued functions q(x,t) and p(x,t) are defined through their coefficients as follows, for 1 ≤ i ≤ p,

p i (x,t) = K( θ )e A( θ) D(t)x (A i X(t) + B i u(0,t)) (52) = K( θ )e A( θ) D(t)x ((A i + B i K( θ )) X(t) + B i w(0,t) +A i X r (t, θ ) + B i u r (0,t, θ )) (53) q i (x,t) = D(t) x 0 ∂ K ∂ θi ( θ ) + K( θ )A i D(t)(x -y) ×e A( θ) D(t)(x-y) B( θ ) + K( θ )e A( θ ) D(t)(x-y) B i ×e(y,t)dy + ∂ K ∂ θi + K( θ )A i D(t)x e A( θ) D(t)x × X(t) -K( θ)e A( θ ) D(t)x ∂ X r ∂ θi (t, θ ) + ∂ u r ∂ θi (x,t, θ ) -D(t) x 0 K( θ )e A( θ) D(t)(x-y) B( θ ) × ∂ u r ∂ θi (y,t, θ )dy (54) = D(t) x 0 w(y,t) ∂ K ∂ θi ( θ ) + K( θ )A i D(t)(x -y) ×e A( θ) D(t)(x-y) B( θ ) + K( θ )e A( θ ) D(t)(x-y) B i + D(t) x y ∂ K ∂ θi ( θ ) + K( θ )A i D(t)(x -ξ ) ×e A( θ) D(t)(x-ξ ) B( θ ) + K( θ )e A( θ) D(t)(x-ξ ) B i ×K( θ )e (A+BK)( θ) D(t)(ξ -y) B( θ )dξ dy + ∂ K ∂ θi ( θ ) + K( θ )A i D(t)x e A( θ) D(t)x + D(t) x 0 ∂ K ∂ θi ( θ ) + K( θ)A i D(t)(x -y) ×e A( θ) D(t)(x-y) B( θ ) + K( θ )e A( θ) D(t)(x-y) B i ×K( θ )e (A+BK)( θ) D(t)y dy X(t) -K( θ )e A( θ ) D(t)x ∂ X r ∂ θi (t, θ ) + ∂ u r ∂ θi (x,t, θ ) -D(t) x 0 K( θ )e A( θ) D(x-y) B( θ ) ∂ u r ∂ θi (y,t, θ )dy . (55)
Now, we define the following Lyapunov function candidate

V (t) = Dlog(N(t)) + b γ 1 D(t) 2 + bD γ 2 θ (t) T θ (t) , (56)
where

N(t) = 1 + X(t) T P( θ ) X(t) + b 1 0 (1 + x)w(x,t) 2 dx . ( 57
)
Taking a time derivative of V (t), we obtain

V (t) = - 2b γ 1 D(t)( Ḋ(t) -γ 1 τ D (t)) - 2bD γ 2 θ (t) T ( θ (t) -γ 2 τ θ (t)) + D N(t) p ∑ i=1 θi (t) X(t) T ∂ P ∂ θi ( θ ) X(t) -X(t) T P( θ ) ∂ X r ∂ θi (t, θ ) -X(t) T Q( θ ) X(t) +2 X(t) T P( θ )B( θ )w(0,t) - b D w 2 - b D w(0,t) 2 -2b Ḋ(t) 1 0 (1 + x)w(x,t)q 0 (x,t)dx -2b θ (t) T 1 0 (1 + x)w(x,t)q(x,t)dx , (58)
where we have used an integration by parts. Using the assumptions that D(0) ∈]0; D] and θ (0) ∈ Π, the update laws (32)-( 33), and the properties of the projection operator, we get

V (t) ≤ D N(t) p ∑ i=1 θi (t) X(t) T ∂ P ∂ θi ( θ ) X(t) -X(t) T P( θ ) ∂ X r ∂ θi (t, θ ) -λ min (Q( θ ))| X(t)| 2 +2 X(t) T P( θ )B( θ )w(0,t) - b D w 2 - b D w(0,t) 2 -2b Ḋ(t) 1 0 (1 + x)w(x,t)q 0 (x,t)dx -2b θ (t) T 1 0 (1 + x)w(x,t)q(x,t)dx , (59)
and, substituting the expressions of ( 32) and (33) and using (39) with the Young inequality, we obtain

V (t) ≤ - D 2N(t) λ min (Q)| X| 2 + b D w 2 + 2 b D w(0,t) 2 +2Dbγ 1 1 0 (1 + x)|w(x,t)||p 0 (x,t)|dx N(t) × 1 0 (1 + x)|w(x)||q 0 (x,t)|dx N(t) +Dγ 2 p ∑ i=1 1 0 (1 + x)|w(x,t)||p i (x,t)|dx N(t) +2| X(t) T P( θ )/b||A i X(t) + B i u(0,t)| N(t) × 1 N(t) | X(t) T ∂ P ∂ θi ( θ ) X(t)| + | X(t) T P( θ ) ∂ X r ∂ θi (t, θ )| + 2b 1 0 (1 + x)|w(x,t)||q i (x,t)|dx .(60)
Fuhermore, each signal depending on θ , namely A, B, K, P, ∂ P/∂ θi , ∂ X r /∂ θi and ∂ u r /∂ θi , is given as continuous in θ . Since θ remains in Π, a closed and bounded subset of R p , each signal is bounded in terms of θ and admits a finite upper bound. We denote M A = sup θ ∈Π |A( θ )| and define M P , M B , M K , M A+BK , M ∂ K/∂ θ similarly. Therefore, using Cauchy-Schwartz and Young inequalities, along with ( 49), ( 50), ( 52) and ( 54), we get 1 0

(1 + x)|w(x,t)||p 0 (x,t)|dx ≤ M 0 (| X(t)| 2 + w(t) 2 + w(0,t) 2 ) (61) 1 0 (1 + x)|w(x,t)||q 0 (x,t)|dx ≤ M 0 (| X(t)| 2 + w(t) 2 ) (62) 1 0 (1 + x)|w(x,t)||p i (x,t)|dx + 2| X(t) T P( θ )/b| × |A i X(t) + B i u(0,t)| ≤ M i (| X(t)| 2 + w(t) 2 + w(0,t) 2 + w(t) ) (63) 1 0 (1 + x)|w(x,t)||q i (x,t)|dx ≤ M i (| X(t)| 2 + w(t) 2 + w(t) ) , (64)
where M 0 and M i (1 ≤ i ≤ p) are sufficiently large constants given by

M 0 = M K max {M A+BK + M A , 2M K ((1 + M A D) ×(M B + M B M K (1 + D)) + M A )} e (M A +M A+BK ) D (65) M i = max {|A i | + |B i |M K + |B i | + 2M P /b, 2 sup (t, θ )∈R×Π (|A i ||X r (t, θ )| + |B i ||u r (0,t, θ )|), 2M K sup (t, θ )∈R×Π ∂ X r ∂ θ (t, θ ) + 2 sup (t, θ)∈R×Π ∂ u r ∂ θ (t, θ ) ×(1 + DM K M B ), M ∂ K/∂ θ + M K |A i | D M B + |A i |M K ×(2 D + 2 DM K M B + M K )} e (M A +M A+BK ) D . (66)
Consequently, if we define

M ′ P = max 1≤i≤p sup θ ∈Π ∂ P( θ ) ∂ θi (67) M r = max 1≤i≤p sup θ ∈Π,t≥0 ∂ X r (t, θ ) ∂ θi (68)
using ( 61)-( 64) in ( 60), we obtain

V (t) ≤ - D 2N(t) λ min (Q)| X| 2 + b D w 2 + 2 b D w(0,t) 2 +2Dbγ 1 M 2 0 | X(t)| 2 + w(t) 2 + w(0,t) 2 N(t) × | X(t)| 2 + w(t) 2 N(t) + Dγ 2 p ∑ i=1 M i N(t) | X(t)| 2 + w(t) 2 +w(0,t) 2 + w(t) 1 N(t) M ′ P | X(t)| 2 + M r | P|| X(t)| +2bM i (| X(t)| 2 + w(t) 2 + w(t) ) . ( 69
)
Bounding the cubic and quadric terms with the help of N(t), we arrive at

V (t) ≤ - D 2N(t) λ | X(t)| 2 + b D w(t) 2 + 2 b D w(0,t) 2 + 2Dbγ 1 M 2 0 min {λ , b} | X(t)| 2 + w(t) 2 + w(0,t) 2 N(t) +Dγ 2 p ∑ i=1 M i M ′ P ( 1 λ + 1 2 min {1, b} ) +M P M r ( 1 2 + 1 2 min {1, λ } ) +2bM i ( 1 min {λ , b} + 1 2 min {1, b} + 1) × | X(t)| 2 + w(t) 2 + w(0,t) 2 N(t) . ( 70
)
Defining the following constant,

m = 2 max bM 2 0 , ∑ p i=1 M i (M ′ P + M P M r + 3bM i ) min {1, λ , b} (71)
we finally obtain

V (t) ≤ - D 2N(t) min λ , b D -2(γ 1 + γ 2 )m ×(| X(t)| 2 + w(t) 2 + w(0,t) 2 ) . (72)
Consequently, by choosing

γ * = min {λ , b/D} 4bm (73)
and (γ 1 , γ 2 ) ∈ [0; γ * [ 2 , we make V (t) negative semidefinite and hence

V (t) ≤ V (0), ∀t ≥ 0 . (74)
Starting from this result, we now prove the results stated in Theorem 1. From the transformation (38) and its inverse (43), we obtain these two inequalities

w(t) 2 ≤ r 1 e(t) 2 + r 2 | X(t)| 2 (75) e(t) 2 ≤ s 1 w(t) 2 + s 2 | X(t)| 2 , ( 76
)
where r 1 , r 2 , s 1 , s 2 are sufficiently large positive constants given by 78)

r 1 = 3 1 + D2 M 2 K e 2M A+BK DM 2 B ( 77
)
r 2 = 3M 2 K e 2M A+BK D (
s 1 = 3 1 + D2 M 2 K e 2M A DM 2 B ( 79
)
s 2 = 3M 2 K e 2M A D . (80)
Furthermore, from ( 56) and ( 76), it follows that

D(t) 2 + θ (t) T θ (t) ≤ γ 1 + γ 2 b V (t) (81)
X(t) 2 ≤ 1 λ (e V (t)/D -1) (82) e(t) ≤ s 1 b (e V (t)/D -1) + s 2 X(t) . (83)
Thus, from the definition of ϒ(t), it is easy to show that

ϒ(t) ≤ 1 + s 2 λ + s 1 b + (γ 1 + γ 2 )D b (e V (t)/D -1) (84)
Besides, using (75), we also obtain

V (0) ≤ D( λ + s 2 b + 2s 1 b) + b 1 γ 1 + 1 γ 2 ϒ(0) . (85)
Finally, if we define

R = 1 + s 2 λ + s 1 b + (γ 1 + γ 2 )D b (86) ρ = λ + s 2 b + 2s 1 b + b D 1 γ 1 + 1 γ 2 , (87)
we obtain the global stability result given in Theorem 1.

We now proove tracking. From (74), we obtain the uniform boudedness of X(t) , w(t) , D(t) and θ (t) . From (43), we obtain that e(t) is also uniformly bounded in time.

From (31), we get the uniformly boudedness of U(t) and consequently of Ũ(t) for t ≥ 0. Thus, we get that u(0,t) and e(0,t) are uniformly bounded for t ≥ D. Besides, from (33) and ( 52), we obtain the uniform boundedness of θ (t) for t ≥ D. Finally, with (44), we obtain that d X(t) 2 /dt is uniformly bounded for t ≥ D. As | X(t)| is square integrable, from (72), we conclude from Barbalat's Lemma that X(t) → 0 when t → ∞.

Besides, from (72), we get the square integrability of w(t) . From (76), we obtain the square integrability of e(t) . Consequently, with (31), we obtain the square integrability of Ũ(t). Furthermore, -y) B i e(y,t)dy (90)

d Ũ(t) 2 dt = 2 Ũ(t) K( θ )e A( θ ) D(t) Ẋ(t) + Ḋ(t)G 0 (t) + p ∑ i=1 θi (t)G i (t) + D D H(t) (88) with G 0 (t) = K( θ ) A( θ )e A( θ ) D(t) X(t) + 1 0 (I + A( θ) D(t) ×(1 -y))e A( θ) D(t)(1-y) B( θ )e(y,t)dy (89) G i (t) = ∂ K ∂ θ ( θ ) e A( θ) D(t) X(t) + D(t) 1 0 e A( θ) D(t)(1-y) ×B( θ )e(y,t)dy + K( θ ) A i D(t)e A( θ ) D(t) X(t) + D(t) 1 0 A i D(t)(1 -y)e A D(t)(1-y) B( θ ) +e A D(t)(1
H(t) = K( θ ) B( θ ) Ũ(t) -e A( θ) D(t) B( θ )e(0,t) + 1 0 A( θ ) D(t)e A( θ ) D(t)(1-y) B( θ )e(y,t)dy . ( 91
)
The signals Ḋ, θ1 , . . . , θp are uniformly bounded over t ≥ 0, according to (32)-(33). By using the uniform boundedness of X(t), Ẋ(t), e(t) , Ũ(t) over t ≥ 0 and of e(0,t) for t ≥ D and the uniform boundedness of all the signals which are functions of θ for t ≥ 0, we obtain the uniform boundedness of d Ũ(t) 2 /dt over t ≥ D. Then, with Barbalat's lemma, we conclude that Ũ(t) → 0 when t → ∞.

Conclusions

In this paper we presented a result on global adaptive tracking for ODEs with unknown parameters and long unknown actuator delays, with full state feedback. The control design employs predictor feedback, modified from stabilization to tracking, whereas adaptation employs our relatively novel Lyapunov update law design with normalization. A simulation example with a second order linear plant illustrates the effectiveness of the design.

Fig. 1 .

 1 Fig.1. The system respons of the system (1)-(2) with the reference trajectory (20)-(21) for D = 1, θ = 0.5, θ (0) = 0 and D(0) = 0.

Fig. 2 .

 2 Fig. 2. The system response of the system (1)-(2) with the reference trajectory (20)-(21) for D = 1, θ = 0.5, θ (0) = 0 and two dramatically different initial conditions for D: D(0) = 0 and D(0) = D = 2D = 2. Note that the solid plots in this figure correspond to the same simulation as in Figure 1.

Future

  work should focus on systems with simultaneous state an input delay. The non-adaptive design (Jankovic 2008) Delphine Bresch-Pietri is a graduate student Ecole des Mines de Paris, also known as Mines ParisTech, France, graduating with a Master degree in automatic control in 2009. She was a visiting student at University of California, San Diego, in summer of 2008. Miroslav Krstic is the Sorenson Distinguished Professor and the founding Director of the Cymer Center for Control Systems and Dynamics (CCSD) at UC San Diego. He received his PhD in 1994 from UC Santa Barbara and was Assistant Professor at University of Maryland until 1997. He is a coauthor of eight books: Nonlinear and Adaptive Control Design (Wiley, 1995), Stabilization of Nonlinear Uncertain Systems (Springer, 1998), Flow Control by Feedback (Springer, 2002), Real-time Optimization by Extremum Seeking Control (Wiley, 2003), Control of Turbulent and Magnetohydrodynamic Channel Flows (Birkhauser, 2007), Boundary Control of PDEs: A Course on Backstepping Designs (SIAM, 2008), Delay Compensation for Nonlinear, Adaptive, and PDE Systems (Birkhauser, 2009), and Adaptive Control of Parabolic PDEs (Princeton, 2009). Krstic is a Fellow of IEEE and IFAC and has received the Axelby and Schuck paper prizes, NSF Career, ONR Young Investigator, and PECASE award. He has held the appointment of Springer Distinguished Visiting Professor of Mechanical Engineering at UC Berkeley.

may be a good starting point for an interesting class of systems, and the clf tools (Karafyllis and Jiang 2008) for systems with state delays may be instrumental.


Miroslav Krstic

Delphine Bresch-Pietri

Delay-Adaptive Full-State Predictor Feedback for Systems With Unknown Long Actuator Delay

Stabilization of unstable systems with actuator delay of substantial length and of completely unknown value is an important problem that has never been attempted. We present a Lyapunov-based adaptive control design that achieves global stability, without a requirement that the delay estimate be near the true delay value. We solve the problem by employing a framework where the actuator delay is represented as a transport PDE, by estimating the delay value as the reciprocal of the convection speed in the transport PDE, and by using full state predictor-based feedback.

I. INTRODUCTION

Adaptive control in the presence of actuator delays is a hard problem. To our knowledge, the only existing results are the 1988 result by Ortega and Lozano [START_REF] Ortega | Globally stable adaptive controller for systems with delay[END_REF] and the 2003 results by Niculescu and Annaswamy [START_REF] Niculescu | An Adaptive Smith-Controller for Time-delay Systems with Relative Degree n * ≥ 2[END_REF] and Evesque et al [START_REF] Evesque | Adaptive Control of A Class of Time-delay Systems[END_REF]. These results deal with the problem where the plant has unknown parameters but the delay value is known.

The remaining theoretical frontier, and a problem of great practical relevance, is the case where the actuator delay value is unknown and highly uncertain. This problem is open in general even in the case where no parametric uncertainty exists in the ODE plant. The importance of problems with unknown delays was highlighted in [START_REF] Diop | Preserving stability/performance when facing an unknown timedelay[END_REF], where a simple scheme for delay estimation and controller gain adjustment to preserve closed-loop stability was also presented. An attempt at adaptive design for unknown delay was also made in [START_REF] Krstic | Multivariable adaptive control of instabilities arising in jet engines[END_REF] by applying the Pade approximation, however, while the design was (predictably) successful for the approximate problem, it was not successful for a model with an actual delay of significant length.

In this paper we present the first systematic adaptive control design for a system with unknown actuator delay by focusing on the case

Ẋ(t) = AX(t) + BU(t -D) , (1)
where the full state-both the ODE plant state X ∈ R n and the infinite-dimensional actuator state U(η), η ∈ [t -D,t]are available for measurement, and where the ODE plant parameters are known, but where the delay length D is unknown (though constant) and can have an arbitrarily large value. This problem can be formulated around an actuator delay model given by a transport equation (convective/first-order hyperbolic PDE), namely,

Ẋ(t) = AX(t) + Bu(0,t) (2) Du t (x,t) = u x (x,t) (3)
u(1,t) = U(t) , (4)
where u(x,t) is the state of the actuator, the domain length is known (unity) but the propagation speed 1/D is unknown. The actuator state is related to the input through the following equation

u(x,t) = U(t + D(x -1)) , (5)
which, in particular, gives u(1,t) = U(t) and u(0,t) = U(t -D). The control law around which we build a delayadaptation mechanism is a predictor-based feedback law,

U(t) = K e AD X(t) + D 1 0 e AD(1-y) Bu(y,t)dy , (6)
which achieves exponential stability at u ≡ 0, X = 0 by performing perfect compensation of the actuator delay, and which has been employed in many control design and analysis studies for systems with actuator delays over the last three decades [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], [START_REF] Evesque | Adaptive Control of A Class of Time-delay Systems[END_REF], [START_REF] Fiagbedzi | Feedback stabilization of linear autonomous time lag systems[END_REF], [START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF], [START_REF] Jankovic | Forwarding, backstepping, and finite spectrum assignment for time delay systems[END_REF], [START_REF] Jankovic | Control of cascade systems with time delay-the integral cross-term approach[END_REF], [START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF], [START_REF] Klamka | Observer for linear feedback control of systems with distributed delays in controls and outputs[END_REF], [START_REF] Krstic | On compensating long actuator delays in nonlinear control[END_REF], [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF], [START_REF] Krstic | Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], [START_REF] Kwon | Feedback stabilization of linear systems with delayed control[END_REF], [START_REF] Manitius | Finite spectrum assignment for systems with delays[END_REF], [START_REF] Michiels | On the delay sensitivity of Smith predictors[END_REF], [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems: An Eignevalue-Based Approach[END_REF], [START_REF] Mirkin | On the approximation of distributed-delay control laws[END_REF], [START_REF] Mondie | Finite spectrum assignment of unstable time-delay systems with a safe implementation[END_REF], [START_REF] Niculescu | An Adaptive Smith-Controller for Time-delay Systems with Relative Degree n * ≥ 2[END_REF], [START_REF] Olbrot | Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays[END_REF], [START_REF] Richard | Time-delay systems: An overview of some recent advances and open problems[END_REF], [START_REF] Tadmor | The standard H ∞ problem in systems with a single input delay[END_REF], [START_REF] Watanabe | Finite spectrum assignment and observer for multivariable systems with commensurate delays[END_REF], [START_REF] Watanabe | An observer for linear feedback control laws of multivariable systems with multiple delays in controls and outputs[END_REF], [START_REF] Zaccarian | A cascade interpretation of the Smith predictor and the arising enhanced scheme[END_REF], [START_REF] Zhong | On distributed delay in linear control laws-Part I: Discrete-delay implementation[END_REF], [START_REF] Zhong | Robust Control of Time-delay Systems[END_REF], [START_REF] Zhong | Control of integral processes with dead time-Part 2: Quantitative analysis[END_REF]. Within this framework we obtain a global adaptive stabilization result, for an arbitrarily large and unknown actuator delay value (Sections III and IV).

Without a question, an even more relevant and challenging problem is the one where the full state is not available for measurement, more specifically, when the state of the transport PDE u(x,t), i.e., the actuator state, is not measured. A yet more challenging problem is when, in addition, only an output of the ODE system

Y (t) = CX(t) (7)
is measured, rather than the full state X(t), and, finally, the most challenging in this string of problems is when the ODE plant has parametric uncertainty, i.e., A(θ ), B(θ ),C(θ ), where θ is unknown. (For an exhaustive categorization of adaptive control problems with actuator delay, please see Section II). However, as restrictive as the requirement for measurement of u(x,t) may seem, we do not believe that any delay-adaptive problem without the measurement of u(x,t) is solvable globally because it cannot be formulated as linearly parametrized in the unknown delay D. As a consequence, when the controller uses an estimate of u(x,t), not only do the initial values of the ODE state and the actuator state have to be small, but the initial value of the delay estimation error also has to be small (the delay value is allowed to be large but the initial value of its estimate has to be close to the true value of the delay). This local result is actually proven in our companion paper [START_REF] Bresch-Pietri | Delay-robustness and adaptivity of predictor feedback without actuator state measurement[END_REF].

In our global full-state feedback design we require only one bit of a priori knowledge about the length of the delay:

Assumption 1: An upper bound D on the unknown D > 0 is known.

This upper bound is used in two ways. An adaptation algorithm employing projection keeps the delay estimate below the a priori bound. In addition, based on the upper bound for the delay length, the adaptation gain is selected to be sufficiently small, and a normalization parameter is selected to be sufficiently large, to ensure that adaptation is sufficiently slow to guarantee closed-loop stability. The approach for update law design (Section III) and for the corresponding stability analysis (Section IV) is based on the ideas that we introduced in [START_REF] Krstic | Adaptive boundary control for unstable parabolic PDEs-Part I: Lyapunov design[END_REF] for Lyapunov-based adaptive control of parabolic PDEs. The adaptation and normalization gain choices are conservative. The relevant part of the design is the structure of the adaptation law, not the exact gain values employed in the analysis.

In this paper the only parametric uncertainty considered is the unknown delay. This is done for clarity of presentation, as the presence of unknown parameters in the plant would obscure presentation of new tools for handling the unknown delay. In another companion paper [START_REF] Bresch-Pietri | Adaptive trajectory tracking despite unknown actuator delay and plant parameters[END_REF] we present an extension with unknown plant parameters and where the control objective is not regulation to zero but trajectory tracking.

We start this paper with Section II in which we categorize all the combinations of delay-adaptive, ODE parameteradaptive, full-state, and output-feedback problems arising in the area of adaptive control in the presence of delay.

II. CATEGORIZATION OF ADAPTIVE CONTROL PROBLEMS WITH ACTUATOR DELAY

A finite-dimensional system with actuator delay may come with

• unknown delay (D)

• unmeasured actuator state (u) • unknown parameters in the finite-dimensional part of the plant (A) • unmeasured state of the finite-dimensional part of the plant (X). Each one of these situations introduces a design difficulty, which needs to be dealt with by using an estimator (a parameter estimator or a state estimator). We point out that a state estimator of the actuator state is trivial when the delay is known (one gets the full state by waiting one delay period), however this estimation problem is far from trivial when the delay is also unknown.

The symbols D, u, A, X will be helpful as we try to categorize all the problems in which one, two, three, or all four of these design difficulties may arise. For example, (D, u, X) denotes the case where only the ODE plant parameters are known, whereas the delay is unknown and the state of the actuator and the ODE are unmeasurable.

There are a total of fourteen combinations arising from the four basic problems, (D), (u), (A), and (X). We focus exclusively on problems where the delay is present and is of significant length to require the use of predictor feedback (rather than being treated as a small perturbation through some form of small gain argument). The following list categorizes the fourteen control problems and gives the status of each them:

1) (X), (u), (u, X)-non-adaptive problems solvable using observer-based predictor feedback [START_REF] Krstic | Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF]; 2) (A, X), (A)-solved in [START_REF] Ortega | Globally stable adaptive controller for systems with delay[END_REF], [START_REF] Niculescu | An Adaptive Smith-Controller for Time-delay Systems with Relative Degree n * ≥ 2[END_REF], [START_REF] Evesque | Adaptive Control of A Class of Time-delay Systems[END_REF] but with relative degree limitations; 3) (u, A), (u, A, X)-tractable using the techniques from [START_REF] Ortega | Globally stable adaptive controller for systems with delay[END_REF], [START_REF] Niculescu | An Adaptive Smith-Controller for Time-delay Systems with Relative Degree n * ≥ 2[END_REF], [START_REF] Evesque | Adaptive Control of A Class of Time-delay Systems[END_REF]; 4) (D)-the main result of the present paper (Sections III and IV); 5) (D, X)-tractable as in Point 4 (by adding a standard ODE observer) but not highly relevant; 6) (D, A)-the subject of our companion paper [START_REF] Bresch-Pietri | Adaptive trajectory tracking despite unknown actuator delay and plant parameters[END_REF]; 7) (D, A, X)-tractable using the techniques in Point 6 combined with adaptive backstepping and Kreisslemeier observers; 8) (D, u), (D, u, A), (D, u, A, X)-not tractable globally because of lack of linear parametrization in any situation involving (D) and (u) simultaneously; the case (D, u) is studied in our other companion paper [START_REF] Bresch-Pietri | Delay-robustness and adaptivity of predictor feedback without actuator state measurement[END_REF].

If this combinatorial complexity hasn't already overwhelmed the reader, we should point out that in each of the cases involving unknown parameters, namely (D) and (A), multiple choices exist in terms of design methodology (Lyapunov-based, estimation/swapping-based, passivity/observer-based, direct, indirect, pole placement, etc.). In addition, in output-feedback adaptive problems, namely problems involving (A) and (X), the relative degree plays a major role in determining the difficulty of a problem. Finally, trajectory tracking requires additional tools, as compared to problems of regulation to zero.

So, the present paper addresses only a subset among important problems in adaptive control with actuator delay, but in our opinion the most relevant among the tractable problems.

III. DELAY-ADAPTIVE PREDICTOR FEEDBACK WITH FULL-STATE MEASUREMENT

We consider the system (2)-( 4) where the pair (A, B) is completely controllable. Before we proceed, for a reader familiar with our prior work we point out that the representation (3), ( 4) is different than the representation ȗt ( x,t) = ȗx ( x,t), ȗ(D,t) = U(t), ȗ(0,t) = U(t -D), ȗ( x,t) = U(t + x -D), which we used in [START_REF] Krstic | Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF], and which would be less convenient for adaptive control as it is not linearly parametrized in D.

When D is unknown, we replace ( 6) by the adaptive controller

U(t) = K e A D(t) X(t) + D(t)
1 0 e A D(t)(1-y) Bu(y,t)dy [START_REF] Jankovic | Forwarding, backstepping, and finite spectrum assignment for time delay systems[END_REF] with an estimate D governed by the update law

Ḋ(t) = γProj [0, D] {τ(t)} (9)
where

τ(t) = - 1 0 (1 + x)w(x,t)Ke A D(t)x dx (AX(t) + Bu(0,t)) 1 + X(t) T PX(t) + b 1 0 (1 + x)w(x,t) 2 dx , (10)
the standard projector operator is given by

Proj [0, D] {τ} = τ    0, D = 0 and τ < 0 0, D = D and τ > 0 1, else (11)
the matrix P is the positive definite and symmetric solution of the Lyapunov equation

P(A + BK) + (A + BK) T P = -Q (12)
for any positive definite and symmetric matrix Q, the constant b is chosen to satisfy the inequality

b ≥ 4|PB| 2 D λ min (Q) , (13)
the transformed state of the actuator is given by w(x,t) = u(x,t) -D(t)

x 0 Ke A D(t)(x-y) Bu(y,t)dy

-Ke A D(t)x X(t) , (14)
and the positive adaptation gain γ is chosen "sufficiently large." For this adaptive controller, the following result holds. Theorem 1: Consider the closed-loop system consisting of the plant (2)-( 4), the control law [START_REF] Jankovic | Forwarding, backstepping, and finite spectrum assignment for time delay systems[END_REF], and the parameter update law defined through ( 9)-( 14). Let Assumption 1 hold. There exists γ * > 0 such that for any γ ∈ (0, γ * ), the zero solution of the system (X, u, D -D) is stable in the sense that there exist positive constants R and ρ (independent of the initial conditions) such that for all initial conditions satisfying (X 0 u 0 , D0 ) ∈ R n × L 2 (0, 1) × [0, D], the following holds:

ϒ(t) ≤ R e ρϒ(0) -1 , ∀t ≥ 0 , (15)
where

ϒ(t) = |X(t)| 2 + 1 0 u(x,t) 2 dx + D(t) 2 . (16) Furthermore, lim t→∞ X(t) = 0 , lim t→∞ U(t) = 0 . ( 17
)
IV. PROOF OF STABILITY FOR FULL-STATE FEEDBACK

In this section we prove Theorem 1. We start by considering the transformation ( 14), along with its inverse u(x,t) = w(x,t) + D(t)

x 0 Ke (A+BK) D(t)(x-y) Bw(y,t)dy +Ke (A+BK) D(t)x X(t) .

(

) 18
After a careful calculation, the transformed system can be written as

Ẋ(t) = (A + BK)X(t) + Bw(0,t) ( 19
)
Dw t (x,t) = w x (x,t) -D(t)p(x,t) -D Ḋ(t)q(x,t) (20) w(1,t) = 0 , (21)
where D(t) = D -D(t) is the parameter estimation error, and

p(x,t) = Ke A D(t)x (AX(t) + Bu(0,t)) = Ke A D(t)x ((A + BK)X(t) + Bw(0,t)) (22) q(x,t) = x 0 K I + A D(t)(x -y e A D(t)(x-y) Bu(y,t)dy +KAxe A D(t)x X(t) = x 0 w(y,t) K I + A D(t)(x -y) e A D(t)(x-y) B + D(t) x y K I + A D(t)(x -ξ ) e A D(t)(x-ξ ) BK
×e (A+BK) D(t)(ξ -y) Bdξ dy

+ KAxe A D(t)x + x 0 K I + A D(t)(x -y)
×e A D(t)(x-y) BKe (A+BK) D(t)y dy X(t) .

Now we consider a Lyapunov-Krasovskii type (nonquadratic) functional

V (t) =D log N(t) + b γ D(t) 2 , (24)
where

N(t) = 1 + X(t) T PX(t) + b 1 0 (1 + x)w(x,t) 2 dx . ( 25
)
Taking a time derivative of V (t), we obtain

V (t) = - 2b γ D(t) Ḋ(t) -γτ(t) + D N(t) -X(t) T QX(t) + 2X(t) T PBw(0,t) - b D w(0,t) 2 - b D w(t) 2 -2b Ḋ(t) 1 0 (1 + x)w(x,t)q(x,t)dx , (26)
where we have used integration by parts and w(t) 2 denotes 1 0 w(x,t) 2 dx. Using the assumption that D(0) ∈ [0, D] and the update law ( 9)-( 11) with the help of [15, Lemma E.1] or [17, Lemma 3], we get

V (t) ≤ D N(t) -X(t) T QX(t) + 2X(t) T PBw(0,t) - b D w(0,t) 2 - b D w(t) 2 -2b Ḋ(t) 1 0 (1 + x)w(x,t)q(x,t)dx , (27)
as well as that D(t) ∈ [0, D], ∀t ≥ 0, and Ḋ2 ≤ γ 2 τ 2 . Then, applying Young's inequality and employing [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF], we obtain

V (t) ≤ - D 2N(t) λ min (Q)|X(t)| 2 + b D w(0,t) 2 + 2 b D w(t) 2 +4b Ḋ(t) 1 0 (1 + x)w(x,t)q(x,t)dx , (28)
and, finally, substituting [START_REF] Jankovic | Control of cascade systems with time delay-the integral cross-term approach[END_REF], we arrive at

V (t) ≤ - D 2N(t) λ min (Q)|X(t)| 2 + b D w(0,t) 2 + 2 b D w(t) 2 + 2Dbγ 1 0 (1 + x)|w(x,t)||p(x,t)|dx N(t) × 1 0 (1 + x)|w(x,t)||q(x,t)|dx N(t) . ( 29
)
Then, a lengthy but straightforward calculation, employing the Cauchy-Schwartz and Young inequalities, along with ( 22) and ( 23), yields

≤ Me m D(t) |X(t)| 2 + w(t) 2 , (31)
where M, m are sufficiently large positive constants given by

M = max 2|K| 2 |A + BK| 2 , 2|K| 2 |B| 2 , 1 + 2|K|(1 + |A| D)|B|(1 + D|BK|), |K| 2 (|A| + |(1 + |A| D)|BK|) 2 (32) m = |A| + |A + BK| . ( 33
)
Introducing these two bounds into (29), we get

V (t) ≤ - D 2N(t) λ min (Q)|X(t) 2 + b D w(0,t) 2 + 2 b D w(t) 2 -γ 4bM 2 e 2m D min {λ min (P), b} |X(t)| 2 + w(t) 2 + w(0,t) 2 , (34)
and, finally,

V (t) ≤ - D 2 min λ min (Q), b D -γ 4bM 2 e 2m D min {λ min (P), b} × |X(t)| 2 + w(t) 2 + w(0,t) 2 N(t) . ( 35
)
By choosing γ * = min λ min (Q), b D min {λ min (P), b} 4bM 2 e 2m D [START_REF] Zhong | Control of integral processes with dead time-Part 2: Quantitative analysis[END_REF] and γ ∈ (0, γ * ) we make V negative semidefinite, and hence

V (t) ≤ V (0) , ∀t ≥ 0 . ( 37
)
From this result we now derive a stability estimate. From ( 14) and ( 18) we show that

u(t) 2 ≤ r 1 w(t) 2 + r 2 |X(t)| 2 (38) w(t) 2 ≤ s 1 u(t) 2 + s 2 |X(t)| 2 , ( 39
)
where r 1 , r 2 , s 1 , s 2 are sufficiently large positive constants given by

r 1 = 3 1 + D2 |K| 2 e 2|A+BK| D|B| 2
(40)

r 2 = 3|K| 2 e 2|A+BK| D ( 41
)
s 1 = 3 1 + D2 |K| 2 e 2|A| D|B| 2 (42)
s 2 = 3|K| 2 e 2|A| D . (43)
From ( 24), ( 25) the following two inequalities readily follow:

D2 ≤ γ b V (44) |X| 2 ≤ 1 λ min (P) e V /D -1 . (45)
Furthermore, from ( 24), ( 25) and (38) it follows that

u 2 ≤ r 1 b e V /D -1 + r 2 |X| 2 . ( 46
)
Combining ( 44)-( 46) we get

ϒ(t) ≤ 1 + r 2 λ min (P) + r 1 b + γ Db e V (t)/D -1 . (47)
So, we have bounded ϒ(t) in terms of V (t), and thus, using (37), in terms of V (0). Now we have to bound V (0) in terms of ϒ(0). First, from [START_REF] Niculescu | An Adaptive Smith-Controller for Time-delay Systems with Relative Degree n * ≥ 2[END_REF], [START_REF] Olbrot | Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays[END_REF] it follows that

V ≤ D λ max (P)|X| 2 + 2b w 2 + b γ D2 . (48)
Using (39) we get

V ≤ (Dλ max (P) + 2bDs 2 ) |X| 2 + 2bDs 1 u 2 + b γ D2 . (49)
and hence

V (0) ≤ Dλ max (P) + 2bDs 2 + 2bDs 1 + b γ ϒ(0) . ( 50
) Denoting R = 1 + r 2 λ min (P) + r 1 b + γD b (51) ρ =λ max (P) + 2bs 2 + 2bs 1 + b γD , (52)
we complete the proof of the stability estimate [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF].

Finally, to prove the regulation result we will use ( 35) and Barbalat's lemma. However, we first discuss the boundedness of the relevant signals. By integrating (37) from t = 0 to t = ∞, and by noting that N(t) is uniformly bounded, it follows that X(t), w(t) , and D(t) are uniformly bounded in time. Using (38) we also get the uniform boundedness of u(t) in time. With the Cauchy-Schwartz inequality, from (8) we get uniform boundedness of U(t) for t ≥ 0. From (5) we get the uniform boundedness of u(0,t) for t ≥ D. Using (2) we get uniform boundedness of d|X(t)| 2 /dt for t ≥ D. From [START_REF] Zhong | Robust Control of Time-delay Systems[END_REF] it follows that X(t) is square integrable in time. From this fact, along with the uniform boundedness of d|X(t)| 2 /dt for t ≥ D, by Barbalat's lemma we get that

X(t) → 0 as t → ∞.
What remains is to prove the regulation of U(t). From (35) it follows that w(t) is square integrable in time. Using (38) we get that u(t) is also square integrable in time. With the Cauchy-Schwartz inequality, from [START_REF] Jankovic | Forwarding, backstepping, and finite spectrum assignment for time delay systems[END_REF] we get that U(t) is also square integrable. To complete the proof of regulation of U(t) by Barbalat's lemma, all that remains to show is that dU(t) 2 /dt is uniformly bounded. Towards this end, we calculate

d dt U(t) 2 =2U(t)K e A D(t) Ẋ(t) + Ḋ(t)G 1 (t) + D(t) D G 2 (t) , (53)
where

G 1 (t) =Ae A D(t) X(t) + 1 0 I + A D(t)(1 -y) g(y,t)dy (54)
G 2 (t) =BU(t) -Be A D(t) u(0,t)

+ 1 0
A D(t)g(y,t)dy (55) and g(y,t) = e A D(t)(1-y) Bu(y,t) .

The signal Ḋ(t) is uniformly bounded over t ≥ 0 according to ( 9)- [START_REF] Klamka | Observer for linear feedback control of systems with distributed delays in controls and outputs[END_REF]. By using also the uniform boundedness of X(t), Ẋ(t), u(t) ,U(t) over t ≥ 0, and of u(0,t) over t ≥ D, we get uniform boundedness of dU(t) 2 /dt over t ≥ D. Then, by Barbalat's lemma, it follows that U(t) → 0 as t → ∞.

V. SIMULATIONS

We present the simulation results for the state-feedback scheme in Section III, namely, for the closed-loop system consisting of the plant (2)-( 4), the control law [START_REF] Jankovic | Forwarding, backstepping, and finite spectrum assignment for time delay systems[END_REF], and the parameter update law defined through ( 9)- [START_REF] Krstic | Multivariable adaptive control of instabilities arising in jet engines[END_REF].

We focus on highlighting the most important aspect of our scheme-the ability to handle long delays, in the presence of a large uncertainty on the delay. For this reason we focus on the case of a scalar but unstable ODE (2), with A = 0.75 and B = 1. We take the delay as D = 1, which is larger than A. So, the system's transfer function is X(s)/U(s) = e -s /(s -0.75). We assume that the known upper bound on the delay is D = 2. We take the nominal control gain as K = -A -1 = -1.75 (which means that P = 1, Q = 2). We take the adaptation gain as γ = 23 and the normalization coefficient as b = 4|PB| 2 D λ min (Q) = 2 D = 4. We take the actuator initial condition as u 0 (x) ≡ 0, i.e., as U(θ ) ≡ 0, ∀θ ∈ [-D, 0], and the plant initial condition as X(0) = 0.5.

Hence, the closed-loop system responds to X(0) and to D(0). We perform our tests for two distinctly different values of D(0)-at one extreme we take D(0) = 0 and at the other extreme we take D(0) = D.

The responses are shown in Figure 1. First, they show that, for both initial estimates, the adaptive controller achieves regulation of the state and input to zero. Second, they show that in both cases the estimate D(t) converges towards the true D and settles in its vicinity. The perfect convergence is not achieved in either of the two cases, since the regulation problem does not provide persistency of excitation for parameter convergence. Third, the dashed plot for D(t) shows that the projection operator is active during the first 0.7 seconds. Fourth, we can observe that by about 3 sec, the evolution of the estimate D(t) has been completed.

Fifth, the plots for X(t) and U(t) are very informative in showing four distinct intervals of behavior of the controller and of the closed-loop system. During the first 1 sec, the delay precludes any influence of the control on the plant, so X(t) shows an exponential open-loop growth. At 1 sec, the plant starts responding to the control and its evolution changes qualitatively, resulting also in a qualitative change of the control signal. When the estimation of D(t) ends at about 3 seconds, the controller structure becomes linear. However, due to the delay, the plant state X(t) continues to evolve based on the inputs from 1 second earlier, so, a nonmonotonic transient continues until about 4 seconds. From about 4 seconds onwards, the (X,U) system is linear and the delay is sufficiently well compensated, so the response of X(t) and U(t) shows a monotonically decaying exponential trend of a first order system.

We want to stress that the plots presented here do not show the best performance achievable with the scheme. Quite on the contrary, the plots have been selected to illustrate the less than perfect behaviors, with non-monotonic evolution of all the states in the closed-loop system, that one would obtain when γ and b are not highly tuned.

VI. CONCLUSIONS

As we have explained in Section II, the problem of full state stabilization with known ODE plant parameters but with unknown delay is the central problem in adaptive control of systems with actuator delays. The other problems in the lengthy catalog of problems are extensions of the this central problem. Some of them are solvable globally and some of them only locally.

We present a globally stabilizing adaptive controller which employs the measurement of actuator state and then prove that, when the actuator state is replaced by its adaptive estimate, local stability and regulation are achieved. A stepping stone towards the latter result is a nonadaptive linear robustness result with respect to the delay value employed in the predictor feedback. We have presented the linear result in more detail, to help the reader's intuition regarding the proof of the local adaptive result, which is considerably more complex and presented with a limited amount of detail due to space limitations.

The simulations show the effectiveness of the Lyapunovbased adaptive controller. Whether the initial estimate of the delay is zero or 100% above the true value, the estimator drives the estimate towards the true value, which in turn results in the stabilization of the closed-loop system by the predictor-based adaptive controller.

When the actuator state is not measured but it is estimated, local stability is achieved, as proved in our companion paper [START_REF] Bresch-Pietri | Delay-robustness and adaptivity of predictor feedback without actuator state measurement[END_REF]. The extension to the problem with unknown ODE plant parameters and to trajectory tracking is presented in our other companion paper [START_REF] Bresch-Pietri | Adaptive trajectory tracking despite unknown actuator delay and plant parameters[END_REF].

Fig. 1 .

 1 Fig.1. The system response of they system (2)-(4), (8)-(14) for D = 1 and for two dramatically different values of initial estimate, D(0) = 0 and D(0) = D = 2D = 2.