An Internal Model Based Semi-global Output Feedback Control for Nonlinear Multi-agent Systems

Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the the 20th9-14, World Toulouse, France, July 2017 Proceedings of 20th World The International Federation of Congress Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International Federation of The International of Automatic Automatic Control Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 8202–8207 An An Internal Internal Model Model Based Based Semi-global Semi-global An Internal Model Based Semi-global An Internal Model Based Semi-global Output Feedback Control for Nonlinear An Internal Model Basedfor Semi-global Output Feedback Control Nonlinear Output Feedback Control for Nonlinear Output Feedback Control for Multi-agent Systems Output Feedback Control for Nonlinear Nonlinear Multi-agent Systems Multi-agent Systems Multi-agent Systems Systems Multi-agent Youfeng Su ∗∗

Youfeng Su ∗ Youfeng Su ∗∗ Youfeng ∗ Youfeng Su Su ∗ ∗ College of Mathematics and Computer Science, Fuzhou University, College of Mathematics and Computer Science, Fuzhou University, ∗ Fuzhou, (e-mail: [email protected]). and Computer Science, ∗ College of Mathematics Fuzhou, China. China. [email protected]). ∗ College of of Mathematics Mathematics and(e-mail: Computer Science, Fuzhou Fuzhou University, University, ∗ College and Computer Science, Fuzhou University, Fuzhou, China. (e-mail: [email protected]). Fuzhou, Fuzhou, China. China. (e-mail: (e-mail: [email protected]). [email protected]). Abstract: Abstract: In In this this paper, paper, we we present present an an internal internal model model based based output output feedback feedback control control for for the the cooperative semi-global output regulation of nonlinear strict-feedback multi-agent systems with Abstract: In this paper, we present an internal model based output feedback control for the cooperative semi-global output regulation of nonlinear strict-feedback multi-agent systems with Abstract: In this paper, we present an internal model based output feedback control for the Abstract: In this paper, we present anThe internal model based outputmulti-agent feedback control for the arbitrary non-identical relative degrees. problem is first converted into a cooperative semicooperative semi-global output regulation of nonlinear strict-feedback systems with arbitrary non-identical relative degrees. The problem isstrict-feedback first convertedmulti-agent into a cooperative semicooperative semi-global output regulation of nonlinear systems with cooperative semi-global output regulation of nonlinear strict-feedback multi-agent systems with global stabilization problem by introducing a distributed internal model. We then propose a arbitrary non-identical relative The problem is firstinternal converted into aa cooperative semiglobal stabilization problem bydegrees. introducing a distributed model. then propose arbitrary non-identical relative degrees. The is converted into cooperative semiarbitrary non-identical relative degrees. Theisproblem problem is first first converted into We ahigh-gain cooperative semi-aa distributed output feedback stabilizer that determined by a distributed observer. global stabilization problem by introducing a distributed internal model. We then propose distributed output feedback stabilizer that isaa determined by a distributed high-gain observer.aa global stabilization problem by distributed internal model. We then propose global stabilization problem by introducing introducing distributed by internal model.nonlinear We thensubsystems propose It is this novel is of the states distributed output stabilizer that is determined aa distributed high-gain observer. It is shown shown that that thisfeedback novel observer observer is capable capable of estimating estimating the states of of nonlinear subsystems distributed output feedback stabilizer that is determined by distributed high-gain observer. distributed output feedback stabilizer that is determined bythe a distributed high-gain observer. with arbitrary non-identical relative degrees. It is shown that this novel observer is capable of estimating states of nonlinear subsystems with arbitrary non-identical relative degrees. It is shown that this novel observer is capable of estimating the states of nonlinear subsystems It is shown that this novel observer is capable of estimating the states of nonlinear subsystems with arbitrary non-identical relative degrees. systems; Nonlinear systems; Internal model; Keywords: Cooperative control; Multi-agent with arbitrary non-identical relative © 2017, IFAC (International Federation ofdegrees. Automatic Control) Hosting by Elsevier Internal Ltd. All rights reserved. Keywords: Cooperative control; Multi-agent systems; Nonlinear systems; model; with arbitrary non-identical relative degrees. High-gain observer. Keywords: Cooperative control; Multi-agent systems; Nonlinear systems; Internal model; High-gain observer. Keywords: Cooperative control; Multi-agent systems; Nonlinear systems; Internal model; Keywords: Cooperative control; Multi-agent systems; Nonlinear systems; Internal model; High-gain observer. High-gain observer. observer. High-gain 1. bilization 1. INTRODUCTION INTRODUCTION bilization problem problem is is another another major major difficulty, difficulty, since since the the augmented system is usually an MIMO system, and 1. INTRODUCTION bilization problem is another major difficulty, the augmented system is usually anmajor MIMO system, since and only only 1. INTRODUCTION bilization problem is another difficulty, since the In cooperative INTRODUCTION problem is another major difficulty, since the the distributed controller is permitted. Notice that, a comaugmented system is usually an MIMO system, and only In the the past past decade, decade,1.the the cooperative control control of of multi-agent multi-agent bilization the distributed controller is permitted. Notice that, a comaugmented system is usually an MIMO system, and only augmented system is usually an MIMO system, and only systems has attracted extensive attention from the control In the past decade, the cooperative control of multi-agent mon assumption in controlling the strict-feedback nonlinis Notice a systems hasdecade, attracted attention from the control the In the the cooperative control of mon assumptioncontroller in controlling the strict-feedback the distributed distributed controller is permitted. permitted. Notice that, that, nonlina comcomIn the past past theextensive cooperative control of multi-agent multi-agent distributed controller is permitted. Notice that, a comcommunity. In nonlinear multi-agent systems hasdecade, attracted extensive attention from thesystems control the ear multi-agent systems (see De Persis and Jayawardhana mon assumption in controlling the strict-feedback nonlincommunity. In particular, particular, nonlinear multi-agent systems systems has attracted extensive attention from the control ear multi-agent systems (see De Persis and Jayawardhana mon assumption in controlling the strict-feedback nonlinmon assumption in controlling the strict-feedback nonlinsystems has attracted extensive attention from the control have been considered increasingly often due to their nucommunity. In particular, nonlinear multi-agent systems [2014], Su [2014]) via the output feedback multi-agent systems (see De Persis Jayawardhana have been considered increasingly often due to their nu- ear community. In nonlinear multi-agent systems [2014], Su and and Huang Huang [2014]) the and output feedback is is ear multi-agent systems (see De Persis and Jayawardhana ear multi-agent systems (see the Devia Persis and Jayawardhana community. In particular, particular, nonlinear multi-agent systems merous applications. Output regulation theory has been have been considered increasingly often due to their nuthat subsystems must have identical relative degree. [2014], Su and Huang [2014]) via the output feedback is merous applications. Output regulation theory has been have been considered increasingly often due to their nuthat subsystems must have the identical relative degree. [2014], Su and Huang [2014]) via the output feedback is have been considered increasingly often due to their nuSu and Huang [2014]) via the output feedback is shown be a useful tool for multi-agent control merousto applications. Output regulation theory hasdesign been [2014], It is worth mentioning that some other attempts for the that subsystems must have the identical relative degree. shown to be a useful tool for multi-agent control design merous applications. Output regulation theory has been It is worth mentioning that some other attempts for the that subsystems must have the identical relative degree. merous applications. Output regulation theory has been that subsystems must have the identical relative degree. and analysis, see Delli Priscoli, et.al. [2015], Su and Huang shown to be a useful tool for multi-agent control design leaderless scenario are reported in Isidori, Marconi, and It is worth mentioning that some other attempts for the and analysis, see Delli Priscoli, et.al. [2015], control Su and Huang shown to be a tool multi-agent design leaderless scenario are reported Isidori, Marconi, and It is mentioning that some other attempts for the It is worth worth mentioning that somein other attempts for the shown to be see a useful useful tool for forand multi-agent design [2015], Wieland, Sepulchre, Allg¨ o Zhu, and analysis, analysis, Delli Priscoli, et.al. [2015], Su[2011], and Huang Huang Casadei [2014], Wieland, Sepulchre, and Allg¨ oower [2011], scenario are reported in Isidori, Marconi, and [2015], Wieland, Sepulchre, and Allg¨ ower wer control [2011], Zhu, leaderless and see Delli Priscoli, et.al. [2015], Su and Casadei [2014], Wieland, Sepulchre, and Allg¨ wer [2011], leaderless scenario are reported in Isidori, Marconi, and leaderless scenario are reported in Isidori, Marconi, and and analysis, see Delli Priscoli, et.al. [2015], Su and Huang Chen, and Middleton [2016], and references therein. The [2015], Wieland, Sepulchre, and Allg¨ o wer [2011], Zhu, Zhu, Chen, and Middleton [2016], in which the treatments [2014], Wieland, Sepulchre, Allg¨ [2011], Chen, and Middleton [2016],and and Allg¨ references The Casadei [2015], Wieland, Sepulchre, oower [2011], Zhu, Zhu, Chen, and Middleton [2016], in and which theooower treatments Casadei [2014], Wieland, Sepulchre, and Allg¨ wer [2011], [2015], Wieland, Sepulchre, Allg¨ wer therein. [2011], Zhu, Casadei [2014], Wieland, Sepulchre, and Allg¨ wer [2011], advantage of output based control in it Chen, and and Middleton [2016],and and references therein. The are quite different from those of the leader-following sceChen, and Middleton [2016], in which the treatments advantage of output regulation regulation based control lies lies in that that it Zhu, Chen, Middleton [2016], and references therein. The are quite different from those of the leader-following sceZhu, Chen, and Middleton [2016], in which the treatments Zhu, Chen, and Middleton [2016], in which the treatments Chen, and Middleton [2016], and references therein. The can handle agents with heterogeneous dynamics, external advantage of output regulation based control lies in that it nario. are quite different from those of the leader-following scecan handleofagents with heterogeneous dynamics, external advantage output regulation based control lies in that it nario. are quite different from those of the leader-following quite different from those of the leader-following scesceadvantage ofagents output regulation based control lies inexternal that it are disturbance, and parametric uncertainty, simultaneously. can handle with heterogeneous dynamics, nario. disturbance, and parametric uncertainty, simultaneously. In this paper, we would present a novel output feedback can handle agents with heterogeneous dynamics, external nario. In this paper, we would present a novel output feedback can handle agents with heterogeneous dynamics, external nario. disturbance, and parametric uncertainty, simultaneously. For the scenario, the concontrol for cooperative semi-global output regulation disturbance, and uncertainty, simultaneously. would aa novel output feedback For the leader-following leader-following scenario, the cooperative cooperative con- In control for the the we cooperative semi-global disturbance, and parametric parametric uncertainty, simultaneously. In this this paper, paper, we would present present noveloutput outputregulation feedback In this paper, we would present a novel output feedback trol via the output regulation theory has been conducted For the leader-following scenario, the cooperative conof strict-feedback nonlinear multi-agent systems. The control for the cooperative semi-global output regulation trol via the output regulation theorythe hascooperative been conducted For the leader-following scenario, conof strict-feedback nonlinear multi-agent systems. The gengencontrol for the cooperative semi-global output regulation control for the cooperative semi-global output regulation For the leader-following scenario, the cooperative conwidely for passive nonlinear multi-agent systems (see De trol via viafor thepassive outputnonlinear regulationmulti-agent theory has has systems been conducted conducted eralization and challenge of our problem comparing with strict-feedback nonlinear multi-agent systems. The genwidely (see De of trol the output regulation theory been eralization and challenge of our problem comparing with of strict-feedback nonlinear multi-agent systems. The genstrict-feedback nonlinear multi-agent systems. Thehand, gentrol viaand thepassive outputnonlinear regulation theoryoutput-feedback has systems been conducted Persis Jayawardhana [2014]), widely for multi-agent (seenonDe of the existing literatures lie in two aspects. On one eralization and challenge of our problem comparing with Persis and Jayawardhana [2014]), output-feedback nonwidely for passive nonlinear multi-agent systems (see De the existing literatures lie in two aspects. On one hand, eralization and challenge of our problem comparing with widely for passive nonlinear multi-agent systems (see De eralization and challenge ofinour problem comparing with linear multi-agent systems (see Ding [2015], Dong and Persis and Jayawardhana [2014]), output-feedback nonin contrast with the identical relative degree case in Delli the existing literatures lie two aspects. On one hand, linear multi-agent systems [2014]), (see Ding [2015], Dong nonand the Persis and Jayawardhana output-feedback in contrast with the identical degreeOn case inhand, Delli existing literatures lie in two one existing literatures lieand in relative two aspects. aspects. On one hand, Persis and Jayawardhana nonHuang [2014], Wang, Xu, and Ji [2016]), strictlinear multi-agent multi-agent systems (see Ding [2015], and Dong and the Priscoli, et.al. [2015], Su Huang [2014], we allow the in contrast with the identical relative degree case in Delli Huang [2014], Wang, Xu,[2014]), and Ding Ji output-feedback [2016]), and strictlinear systems (see [2015], Dong and Priscoli, et.al. [2015], Su and Huang [2014], we allow the in contrast with the identical relative degree case in contrast with the identical relative degree caseallow in Delli Delli linear multi-agent systems (see Ding [2015], Dong and in feedback nonlinear multi-agent systems (see Delli Priscoli, Huang [2014], Wang, Xu, and Ji [2016]), and strictagents to have the arbitrary non-identical relative degrees. Priscoli, et.al. [2015], Su and Huang [2014], we the feedback nonlinear multi-agent systems (see Delli Priscoli, Huang [2014], Wang, Xu, and Ji [2016]), and strictagents to have the arbitrary non-identical relative degrees. Priscoli, et.al. [2015], Su and Huang [2014], we allow the et.al. [2015], Su and Huang [2014], we to allow the Huang [2014], Wang, Xu, [2014, and Ji2015]). [2016]), and strict- Priscoli, et.al. [2015], Su and Huang Such a problem feedback nonlinear multi-agent systems (see Delli Priscoli, As aa consequence, our design applies not only benchagents to have the arbitrary non-identical relative degrees. et.al. [2015], Su and Huang [2014, 2015]). Such a problem feedback nonlinear multi-agent systems (see Delli Priscoli, As consequence, our design applies not only to benchagents to have the arbitrary non-identical relative degrees. feedback nonlinear multi-agent systems (see Delli Priscoli, agents to havesuch the arbitrary non-identical relative degrees. is also called the cooperative output regulation problem. et.al. [2015], Su and Huang [2014, 2015]). Such a problem mark systems as Chua’s circuit, Lorenz system, DuffAs a consequence, our design applies not only to benchis also[2015], called Su theand cooperative output regulation problem. et.al. Huang [2014, 2015]). Such a problem mark systems such as Chua’s circuit, Lorenz system, DuffAs aa consequence, our design applies not only to As consequence, our design applies notany only to benchbenchet.al. [2015], Su Huang [2014, 2015]). aproblem. problem A common for this problem is is also also calledtreatment theand cooperative output regulation ing equation, Van del Pol system, but also combination systems such as Chua’s circuit, Lorenz system, DuffA common treatment for handling handling this Such problem is to to mark is called the cooperative output regulation problem. ing equation, Van del Pol system, but also any combination mark systems such as Chua’s circuit, Lorenz system, Duffmark systems such asPol Chua’s circuit, Lorenz system, Duffis also called the cooperative output regulation problem. develop a distributed version of output regulation tools, A common treatment for handling this problem is to of them. Notice that the implementing of the nonlinear ing equation, Van del system, but also any combination develop a distributed version of output regulation tools, A common treatment for handling this problem is to of them. Notice that the implementing of the nonlinear ing equation, Van del Pol system, but also any combination A common treatment for handling this problem is to more specifically, the distributed internal model principle, ing equation, Van del Pol system, but also any combination develop a distributed version of output regulation tools, observers presented in Delli Priscoli, et.al. [2015], Su of them. Notice that the nonlinear more specifically, the distributed model principle, develop aa distributed version of output regulation tools, observers presented in the Delliimplementing Priscoli, et.al.of Su and and of them. Notice that the implementing of the nonlinear develop distributed version ofIninternal output regulation tools, of them.[2014] Notice that the implementing of [2015], therelative nonlinear instead of the centralized one. particular, as shown in more specifically, the distributed internal model principle, Huang highly depends on the identical deobservers presented in Delli Priscoli, et.al. [2015], Su and instead of the centralized one. Ininternal particular, as principle, shown in observers more specifically, the distributed model Huang [2014] highly depends on the identical relative depresented in Delli Priscoli, et.al. [2015], Su observers presented independs Delli Priscoli, et.al. [2015], Su and and more specifically, the distributed internal model principle, Delli Priscoli, et.al. [2015], De Persis and Jayawardhana instead of the centralized one. In particular, as shown in gree assumption, so their designs are no longer applicable. Huang [2014] highly on the identical relative deDelli Priscoli, et.al. [2015], De In Persis and Jayawardhana instead of the centralized one. particular, as shown in gree assumption, so their designs are no longer applicable. Huang [2014] highly depends on the identical relative de[2014] highly depends on the identical relative deinstead of the centralized one. In particular, asJi shown in Huang [2014], Su and Huang [2014], Wang, Xu, and [2016], Delli Priscoli, et.al. [2015], De Persis and Jayawardhana On the other hand, in order to overcome the difficulty gree assumption, so their designs are no longer applicable. [2014], Su and Huang [2014], Wang, Xu, and Ji [2016], Delli Priscoli, et.al. [2015], De Persis and Jayawardhana On the other hand, in order to are overcome theapplicable. difficulty gree assumption, so their designs no longer Delli Priscoli, et.al. [2015], De Persis and Jayawardhana the distributed internal model approach is the only method gree assumption, so their designs are no longer applicable. [2014], Su and Huang [2014], Wang, Xu, and Ji [2016], caused by systems with non-identical relative degrees, we On theby other hand, in order to overcome difficulty the distributed internal model is the only [2014], Su Huang [2014], Wang, Xu, and Ji [2016], caused systems with non-identical relativethe degrees, we On other hand, in order overcome the difficulty [2014], Su and and Huang nonlinear [2014],approach Wang, Xu, and Ji method [2016], On the theby other hand, in order to to overcome the difficulty for handling uncertain multi-agent systems, and the distributed internal model approach is the only method present a distributed observer consisting of two different caused systems with non-identical relative degrees, we for handling uncertain nonlinear multi-agent systems, and present the distributed internal model approach is the only method a distributed observer consisting of two different caused by systems with non-identical relative degrees, we caused by systems with non-identical relative degrees, we the distributed internal model approach is the only method induces the cooperative stabilization problem of the sofor handling uncertain nonlinear multi-agent systems, and parts, where the first part is a distributed dynamic compresent a distributed observer consisting of two different induces the cooperative stabilization problem of the and so- parts, for handling uncertain nonlinear multi-agent systems, where the first part is a distributed dynamic compresent a distributed observer consisting of two different present a distributed observer consisting of two different for handling uncertain nonlinear multi-agent systems, and call augmented system that composed of the multi-agent induces the cooperative stabilization problem of the sopensator based onfirst the network topology, the parts, where the is aa distributed dynamic comcall augmented system that composedproblem of the multi-agent induces the stabilization of the sobased thepart network topology, and and the second second parts, where the first part is dynamic cominduces the cooperative cooperative stabilization problem ofThe the staso- pensator parts, where theon first part is a distributed distributed dynamic comsystem itself and the distributed internal call augmented augmented system that composed of model. the multi-agent multi-agent part is aa reduced order high gain observer adopted from Su pensator based on the network topology, and the second system itself and the distributed internal model. The stacall system that composed of the part is reduced order high gain observer adopted from Su pensator based based on on the the network network topology, topology, and and the the second second call augmented system that composed of the multi-agent pensator system itself and the distributed internal model. The sta[2016] with necessary modifications. It is this distributed part is a reduced order high gain observer adopted from Su system itself and the distributed internal model. The sta[2016] with necessary modifications. It is this distributed part is a reduced order high gain observer adopted from Su  is with a reduced order high gainwith observer adopted from Su system itself distributed model. The sta- part This work hasand beenthe supported in partinternal by National Natural Science observer that is able to deal the difficulty caused  [2016] necessary modifications. It is this distributed This work has been supported in part by National Natural Science observer that is able to deal with the difficulty caused [2016] with necessary modifications. It is this distributed  Foundation of China under grant No. 61403082, in part by Natural [2016] with necessary modifications. It is this distributed This work has been supported in part by National Natural Science by non-identical relative degrees. Technically, we would observer that is able to deal with the difficulty  This workof Foundation China under grantinNo. 61403082, in part by Natural has been supported part by Natural Science by non-identical relative degrees. we caused would observer that is to deal with the caused  Science Foundation Fujian Province of National China under grant No. This work has beenofunder supported inNo. part61403082, by National Natural Science observer that is able able to step deal withTechnically, the difficulty difficulty caused Foundation of China grant part by Natural follow the standard two treatments of the internal Science Foundation ofunder Fujian Province of Chinain under grant No. by non-identical relative degrees. Technically, would Foundation of China grant No. 61403082, in part by Natural follow the standard two step treatments of the by non-identical non-identical relative relative degrees. degrees. Technically, Technically, we weinternal would 2016J06014. Foundation of Chinaofunder grant No. 61403082, inunder part by Natural Science Foundation Fujian Province of China grant No. by we would 2016J06014. follow the standard two step treatments of the internal Science Foundation of Fujian Province of China under grant No. Science Foundation of Fujian Province of China under grant No. follow the standard two step treatments of the internal 2016J06014. follow the standard two step treatments of the internal 2016J06014. 2016J06014.

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Proceedings of the 20th IFAC World Congress Youfeng Su / IFAC PapersOnLine 50-1 (2017) 8202–8207 Toulouse, France, July 9-14, 2017

model approach. Specifically, we convert our problem into the cooperative semi-global stabilization problem of the augmented system firstly, then stabilize it semi-globally by the distributed output feedback control based on the novel observer we presented. Notation: Given the column vectors ai , i = 1, . . . , s, we denote col(a1 , . . . , as ) = [aT1 , . . . , aTs ]T . The compact set ¯ s  {x = col(x1 , . . . , xs ) ∈ Rs : |xi | ≤ R, i = 1, . . . , s}. Q R Given a positive definite function V : Rs → R, the symbol ¯ c (V (x)) denotes the compact set {x ∈ Rs : V (x) ≤ c}. Ω 2. PROBLEM STATEMENT & PRELIMINARIES We consider multiple nonlinear systems in the following strict feedback form z˙i = f0i (zi , x1i , v, w), x˙ si = x(s+1)i , s = 1, . . . , ri − 1, x˙ ri i = f1i (zi , x1i , . . . , xri i , v, w) + bi (w)ui , (1) yi = x1i , i = 1, . . . , N, where zi ∈ Rnzi , xi  col(x1i , . . . , xri i ) ∈ Rri are the state of the ith subsystem, yi , ui ∈ R are the input and output of the ith subsystem, respectively, w ∈ Rnw represents the parametric uncertainty. v ∈ Rnv represents the reference input as well as the disturbance, and is assumed to be generated by the autonomous system v˙ = Sv, y0 = q0 (v, w), (2) where S ∈ Rnv ×nv , and y0 ∈ R is the output of the exosystem. For i = 1, . . . , N , the regulated errors for the subsystems in (1) are defined as ei = yi − y0 . We assume the functions f0i (·), f1i (·), bi (·), i = 1, . . . , N , and q0 (·) are sufficiently smooth functions with f0i (0, 0, 0, w) = 0, f1i (0, . . . , 0, w) = 0, q0 (0, w) = 0, and bi (w) > 0, for all col(v, w) ∈ Rnv +nw . Here ri is called the relative degree of the ith subsystem. In this paper, we allow ri = rj for any i = j, which is opposite to Delli Priscoli, et.al. [2015], Su and Huang [2014], where ri are required to be identical. Plant (1) and exosystem (2) together constitute a multiagent system of N + 1 agents with exosystem (2) as the leader and all the subsystems of (1) as the followers. With respect to (1) and (2), we can define a digraph G¯ = ¯ E} ¯ where V¯ = {0, 1, . . . , N } with the node 0 associated {V, with the exosystem and the other N nodes associated with ¯ j = 0, 1, . . . , N the N followers, respectively, and (j, i) ∈ E, and i = 1, . . . , N , if and only if the control ui can make use of yi −yj for the feedback control. Let A¯ = [aij ]N i,j=0 be any ¯ i.e., for i, j = 0, 1, . . . , N , weighted adjacency matrix of G, aii = 0, aij > 0 if (j, i) ∈ E¯ and aij = 0 otherwise, and ¯ For i = 1, . . . , N , aij = aji if (j, i) is a bidirected edge of E. we define the virtual regulated error for each subsystem N as evi  j=0 aij (yi − yj ). Based on it, we consider the distributed control law of the form (3) ui = ui (ξi ), ξ˙i = gi (ξi , ξvi , evi ), i = 1, . . . , N, nξ i where ui and gi vanish at the origin, ξi ∈ R with nξi to N be defined later, and ξvi  j=1 aij (C¯i ξi − C¯j ξj ) + ai0 C¯i ξi with C¯i ∈ R1×nξi . Let xc = col(z1 , x1 , ξ1 , . . . , zN , xN , ξN ) N and nc = i=1 (nzi + ri + nξi ). Then we define the cooperative regional output regulation problem of system ¯ a real (1) as: given systems (1) and (2), the digraph G, number R > 0, and compact subset V0 × W ⊆ Rnv +nw

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which contains the origin of the respective Euclidean space, find a control law of the form (3) such that for any ¯ nc , every trajectory col(v(0), w) ∈ V0 × W, and xc (0) ∈ Q R of the closed-loop system composed of (1) and (3) starting from xc (0) and v(0) exists and is bounded for all t ≥ 0, and for i = 1, . . . , N , limt→∞ ei (t) = 0. If for any R > 0, and any compact subset V0 × W ⊆ Rnv +nw which contains the origin, the cooperative regional output regulation problem of system (1) is solvable, then we say that the cooperative semi-global output regulation problem of system (1) is solvable. In the rest of this section, for making our presentation selfcontained, we would first repeat briefly on constructing a distributed internal model for system (1), which has been well defined in Su and Huang [2014]. By doing so, our problem can be converted into the equivalent cooperative semi-global stabilization problem of the corresponding augmented system. The existence of this internal model lies in the following assumptions. Assumption 1. The exosystem is neutrally stable, i.e, all the eigenvalues of S are semi-simple with zero real parts. Assumption 2. There exist sufficiently smooth functions zi (v, w) with zi (0, 0) = 0 such that for any col(v, w) ∈ (v,w) Rnv ×nw , ∂zi∂v Sv = f0i (zi (v, w), q0 (v, w), v, w). For i = 1, . . . , N , let x1i (v, w) = q0 (v, w), xsi (v, w) = xri i (v,w) ∂x(s−1)i (v,w) Sv, s = 2, . . . , ri , ui (v, w) = b−1 i (w)( ∂v ∂v ·Sv−f1i (zi (v, w), x1i (v, w), . . . , xri i (v, w), v, w)). The functions zi (v, w), x1i (v, w), . . . , xri i (v, w), ui (v, w) constitute the solution to the regulator equations associated with (1) and (2), and depict the steady-state of zi , x1i , . . . , xri i , ui at which yi = y0 , see in Isidori and Byrnes [1990]. Assumption 3. The function ui (v, w) is a polynomial in v with coefficients depending on w. Assumptions 1 to 3 have been shown to be quite standard Huang [2004]. Specifically, under Assumption 1, given any compact subset V0 , there exists a compact subset V such that, for any v(0) ∈ V0 , v(t) ∈ V for all t ≥ 0. As shown in Huang [2004], Assumptions 1 and 3 together guarantees that each subsystem of system (1) admits a linear steady-state generator as θ˙i (v, w) = Ti Φi Ti−1 θi (v, w), ui (v, w) = Γi Ti−1 θi (v, w), i = 1, . . . , N, (nη −1) (v,w) i (v,w) where θi (v, w) = Ti col(ui (v, w), duidt , . . . , d i (nηiu−1) dt  T   Inηi −1 0(nηi −1)×1 1 , Γi = 0 ), Φi = , 1i 2i , . . . , nηi i (nηi −1)×1 with all the eigenvalues of Φi being of zero real parts, and Ti ∈ Rnηi ×nηi is the unique nonsingular solution of the Sylvester equation Ti Φi − Mi Ti = Qi Γi , with any controllable pair (Mi , Qi ) satisfying that Mi ∈ Rnηi ×nηi is Hurwitz and Qi ∈ Rnηi Nikiforov [1998]. Then we define the internal model for each subsystem of system (1) as η˙ i = Mi ηi + Qi ui , i = 1, . . . , N,

(4)

and we call the system composed of the plant (1) and the internal model (4) as the augmented system. Performing the coordinate and input transformation z¯i = zi −zi (v, w), x ¯si = xsi − xsi (v, w), s = 1, . . . , ri , η¯i = ηi − θi (v, w), u ¯i = ui − Ψi ηi , where Ψi = Γi Ti−1 , gives that the augmented system takes the following form:

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z¯˙i = f¯0i (¯ zi , x ¯1i , v, w), x ¯˙ si = x ¯(s+1)i , s = 1, . . . , ri − 1, x ¯˙ ri = f¯1i (¯ zi , x ¯1i , . . . , x ¯ri i , v, w) + bi (w)Ψ−1 ¯i + bi (w)¯ ui , i η η¯˙ i = (Mi + Qi Ψi )¯ ηi + Q i u ¯i , ¯1i , i = 1, . . . , N, (5) ei = x where f¯0i (¯ zi , x ¯1i , v, w) = f0i (¯ zi +zi (v, w), x ¯1i +x1i (v, w), v, w) − f0i (zi (v, w), x1i (v, w), v, w), f¯1i (¯ zi , x ¯1i , . . . , x ¯ri i , v, w) = f1i (¯ zi +zi (v, w), x ¯1i +x1i (v, w), . . . , x ¯ri i +xri i (v, w), v, w) −f1i (zi (v, w), x1i (v, w), . . . , xri i (v, w), v, w), The augment -ed system (5) has the property that, for all col(v, w), the origin is an equilibrium point and ei is identically zero at the origin. Now let us consider a distributed control law of the form u ¯i = −φl (G1i ζi ), (6) ζ˙i = G2i ζi + G3i ζvi + G4i evi , i = 1, . . . , N, N nζ i where ζi ∈ R , ζvi  j=1 aij (Ci ζi − Cj ζj ) + ai0 Ci ζi with Ci = [1, 0, . . . , 0] ∈ R1×nζi , and  φl (·) is a saturation t if |t| < l function which is defined as φl (t) = , sgn(t)l if |t| ≥ l with the positive real number l to be designed. Then we are ready to define the cooperative regional stabilization ¯ > problem of system (5) as: given a real number R nv +nw which 0, and some compact subset V × W ⊆ R contains the origin, find controller (6), such that, for all col(v, w) ∈ V × W, the equilibrium point at the origin of the closed-loop system composed of (5) and (6) is asymptotically stable with its domain of attraction ¯ n¯c . Furthermore, if for any R ¯ > 0, and any containing Q R compact subset V × W ⊆ Rnv +nw which contains the origin, the cooperative regional stabilization problem of system (5), then we say that the cooperative semi-global stabilization problem of system (5) is solvable. Lemma 1. Under Assumptions 1 to 3, given any R > 0, and any compact subset V0 × W ⊆ Rnv +nw , there exist a ¯ > 0 and a compact subset V ⊆ Rnv , such that, number R if the cooperative regional stabilization problem of system ¯ n¯c × V × W is solvable by the (5) on the compact subset Q R distributed controller of the form (6), then the cooperative regional output regulation problem of system (1) on the ¯ nc × V0 × W is solvable by a distributed compact subset Q R output feedback controller of the form ui = −φl (G1i ζi ) + Ψi ηi , η˙ i = Mi ηi + Qi ui , (7) ζ˙i = G2i ζi + G3i ζvi + G4i evi , i = 1, . . . , N. The proof of Lemma 1 is almost the same as that of Lemma 1 in Su and Huang [2014]. Controller (7) has the form of (3) by viewing that ξi = col(ηi , ζi ) and nξi = nηi + nζi . Thus, Lemma 1 indicates that, under Assumptions 1 to 3, the cooperative semi-global output regulation problem of system (1) can be converted into the cooperative semiglobal stabilization problem of augmented system (5). Thus, in what follows, we only need to focus on the corresponding cooperative semi-global stabilization problem. 3. MAIN RESULT In this section, we first show that system (5) can be stabilized semi-globally by a centralized state feedback controller. Then we adopt a distributed observer to estimate the states of subsystems, which is modified from Su [2016].

Finally, we prove that system (5) can also be stabilized semi-globally by a distributed output feedback controller based on this novel distributed observer. For convenience, let us introduce two more standard assumptions. Assumption 4. For the ith subsystem of system (5), there exists a C 2 positive definite and proper function V0i : Rnzi → R, such that for all col(v, w) ∈ V × W, V0i (¯ zi ) along the trajectory of its zero dynamics, that is, z¯˙i = f¯0i (¯ zi , 0, v, w), satisfies ∂V0i ¯ zi , 0, v, w) ≤ −α0i ||¯ zi ||2 , (8) f0i (¯ ∂ z¯i where α0i , i = 1, . . . , N , are some known positive real numbers. Assumption 5. The digraph G¯ contains a directed spanning tree with the node 0 as the root. Assumption 5 is the standard one for the fixed topology. Assumption 4 is borrowed from Lin and Qian [2001]. It means that the zero dynamic of each subsystem of system (5) is globally asymptotically stable and locally exponentially stable, uniformly in col(v, w) ∈ V × W. It is less stringent than the assumption of input-to-state stability of the inverse dynamics z¯˙i = f0i (¯ zi , x ¯1i , v, w) by considering z¯i as the state and x ¯1i as the input. In what follows, we first show that system (5) can be stabilized semi-globally by a centralized state feedback controller of the form u ¯i = − K(¯ xri i + gγ(ri −1)i x ¯(ri −1)i + · · · + g ri −2 γ2i x ¯2i

+ g ri −1 γ1i x ¯1i ), (9) where the coefficients γsi , s = 1, . . . , ri − 1, i = 1, . . . , N , are chosen so that the polynomials λri −1 + γ(ri −1)i λri −2 + · · · + γ2i λ + γ1i are all stable, and the positive numbers g and K are to be determined later. In order to make system (5) more trackable, we further define the following standard coordinate transformation for each subsystem si , s = 1, . . . , ri − 1, x ˆai = of system (5), x ˆsi = gx¯s−1 ˆ(ri −1)i ), ϑi = x ¯ri i + gγ(ri −1)i x ¯(ri −1)i + · · · + col(ˆ x1i , . . . , x g ri −2 γ2i x ¯2i + g ri −1 γ1i x ¯1i , η˜i = η¯i − b−1 i (w)Qi ϑi . Then system (5) is equivalent to the following form: z¯˙i = fˆ0i (¯ zi , x ˆai , v, w), ˆai + Bi (g)ϑi , x ˆ˙ ai = gAi x η˜˙ i = Mi η˜i + fˆ1i (¯ zi , x ˆai , ϑi , v, w, g), ˙ ˆ ϑi = f2i (¯ zi , x ˆai , η˜i , ϑi , v, w, g) + bi (w)¯ ui ,

(10)

zi , x ˆai , v, w) = f¯0i (¯ zi , x ¯1i , v, w), fˆ1i (¯ zi , x ˆai , ϑi , v, where fˆ0i (¯ −1 ˜ zi , x ˆai , ϑi , v, w, g) + b−1 w, g) = −bi (w)Qi f1i (¯ i (w)Mi Qi ϑi , fˆ2i (¯ zi , x ˆai , η˜i , ϑi , v, w, g) = f˜1i (¯ zi , x ˆai , ϑi , v, w, g) + bi (w) −1 ˜ Q ϑ , f (¯ z , x ˆ , ϑ , v, w, g) = f¯1i (¯ zi , x ˆ1i , . . . , Ψi η˜i + Ψ−1 i i 1i i ai i i g r−2 x ˆ(ri −1)i , ϑi − g ri −1 (γ(ri −1)i x ˆ(ri −1)i + · · · + γ2i x ˆ2i + ˆ1i ), v, w) + g ri (γ(r−2)i x ˆ(ri −1)i + · · · + γ2i x ˆ3i + γ1i x ˆ2i ) + γ1i x ˆ(ri −1)i +· · ·+γ2i x ˆ2i +γ1i x ˆ1i )), gγ(ri −1)i (ϑi −gri −1 (γ(ri −1)i x Iri −2 0(ri −2)×1 , Bi (g) = and Ai = −γ1i −γ2i , . . . , −γ(ri −1)i   0(ri −2)×1 . Let Xi = col(¯ zi , x ˆai , η˜i , ϑi ) and X = 1 g ri −2

col(X1 , . . . , XN ). Then we have the following lemma. Lemma 2. Under Assumption 4, given any arbitrarily ¯ > 0, there exist sufficiently large positive large number R

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¯ and a real numbers g, K, c, and α, which depend on R, C 2 positive definite function WX (·), such that, N ¯ c (WX (X)). ¯ ¯ i=1 (nzi +ri +nηi ) ⊆ Ω (11) Q R Furthermore, given any  > 0, for all col(v, w) ∈ V × ¯ c+ (WX (X)), the derivative of WX (·) W, and all X ∈ Ω along the closed-loop system composed of system (10) and controller (9) satisfies  ˙ X (X) (9)+(10) ≤ −α||X||2 . W (12)

The proof of Lemma 2 is omitted here due to the space limit. Lemma 2implies that any trajectory of X starting in N

(nzi +ri +nηi )

¯¯ compact set Q remains in the compact set R ¯ Ωc+ (WX (X)), and converges to the origin asymptotically as t → ∞. Thus, the equilibrium at the origin of the closedloop system consisting of (9) and (10), or equivalently, (5) and (9) is asymptotically stable with its domain of  i=1

N

(nz +ri +nη )

i ¯ ¯ i=1 i . That is to say, attraction containing Q R system (5) could be stabilized semi-globally by the centralized controller (9). However, this type of controller is not admissible for the multi-agent system (1) for two reasons. First, since the signals x2i (v, w), . . . , xri i (v, w) rely on the uncertainty w, they are not available for the feedback control, so are the states x ¯2i , . . . , x ¯ri i . Second, due to the communication constraint, only those subsystems which are in the neighbor of the leader system can make use of the output of the leader system y0 , i.e., the regulated output ei for the feedback control, while the others cannot. For estimating these infeasible states, we propose a distributed observer of the form ζ˙1i = ζ2i + hβ(ri −1)i ζ1i + τ (evi − ζvi ), ζ˙2i = ζ3i + h2 β(r −2)i ζ1i − hβ(r −1)i (ζ2i + hβ(r −1)i ζ1i ), i

i

i

ζ˙3i = ζ4i + h3 β(ri −3)i ζ1i − h2 β(ri −2)i (ζ2i + hβ(ri −1)i ζ1i ), .. .

ζ˙(ri −1)i = ζri i + hri −1 β1i ζ1i − hri −2 β2i (ζ2i + hβ(ri −1)i ζ1i ), ζ˙ri i = −hri −1 β1i (ζ2i + hβ(ri −1)i ζ1i ), (13) where the coefficients βsi , s = 1, . . . , ri − 1, i = 1, . . . , N , are chosen so that the polynomials λri −1 + β(ri −1)i λri −2 + · · · + β2i λ + β1i are all stable, and the positive real numbers τ and h are to be determined later. Observer (13) is adopted with some necessary modifications from Su [2016], where the cooperative output regulation for linear multi-agent systems is studied. As has shown in Su [2016], observer (13) is consisted of two different parts. The first part, i.e., the dynamic of ζ1i , is in the distributed version, which is to estimate the regulated error evi , or equivalently, ei , while the second part, i.e., the dynamics of ζ2i , . . . , ζri i , is a modified reduced order high gain observer, which is used to estimate the state col(¯ x2i − hβ(ri −1)i ei , x ¯3i − h2 β(ri −2)i ei , . . . , x ¯ri i − hri −1 β1i ei ). Here we restrict, without loss of generality, the common h and τ for each subsystem instead of different hi and τi in Su [2016], respectively, so as to simply the complex and heavy notations and discussions caused by nonlinear systems. Remark 1. The recent paper Su and Huang [2014] presents a distributed full order high-gain observer, by directly replacing the regulated error ei of the standard observer in Atassi and Khalil [1999] with the virtual

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regulated error evi . The stringent limitation of the one in Su and Huang [2014] is that it can only apply to the subsystem with identical related degree. Alternatively, as will be shown later, our distributed observer (13) is able to handle the more case that the subsystems contain any arbitrary nonidentical relative degrees. Having defined the observer, we now consider a control law of the following form u ¯i = − φl (K((hri −1 β1i ζ1i + ζri i ) + gγ(ri −1)i (hri −2 β2i ζ1i + ζ(ri −1)i ) + · · · + g ri −2 γ2i (hβ(ri −1)i ζ1i + ζ2i )

(14) + g ri −1 γ1i ζ1i )), A controller consisted of (13) and (14) is of the distributed form (6) with ζ = col(ζ1i , . . . , ζri i ), nζi = ri , and   G1i = K Λi g ri −2 γ2i · · · γri i ,   hβ(ri −1)i

G2i

1

2 ) −hβ(ri −1)i  h2 (β(ri −2)i − β(r i −1)i  . .. = ..  .  hri −1 (β − β β r −2 β 1i 2i (ri −1)i ) −h i 2i

G3i =



−hri β1i β(ri −1)i

−τ

0(ri −1)×1



, G4i =



−hri −1 β1i

τ

0(ri −1)×1



0 1 .. . 0 0

,

··· 0 ··· 0   . . .. , . .  ··· 1 ··· 0

(15)

ri −2 ri −1−j j g h γ(j+1)i β(ri −j)i + where Λi = g ri −1 γ1i + j=1 hri −1 β1i . Then we can show that the closed-loop system composed of (10), (13), and (14) has the following stability theorem. Theorem 1. Under Assumptions 4 and 5, given any arbi¯ > 0, there exist sufficiently large trarily large number R positive real numbers g, K, h, τ and l, which depend on ¯ such that the equilibrium at the origin of the closedR, loop system consisting of the augmented system (10) and the distributed output feedback controller (13) and (14) is uniformly locally asymptotically stable with domain of at¯ n¯c , where nc = N (nz +2ri +nη ). traction containing Q i i i=1 R Thus, the cooperative semi-global stabilization problem of system (10), or equivalently, system (5) is solvable. Before going on the proof of Theorem 1, let us further depict the closed-loop system under some proper coordination. Let ν1i = ei − ζ1i , ν2i = hri −2 (¯ x2i − hβ1i ei − ζ2i ), ν3i = hri −3 (¯ x3i −h2 β2i ei −ζ3i ), . . . , ν(ri −1)i = h(¯ x(ri −1)i − ¯ri i − hri −1 β(ri −1)i ei − hri −2 β(ri −2)i ei − ζ(ri −1)i ), νri i = x ζri i . Then u ¯i = −φl (KD1i Xi − KD2i (h)νai − KD3i (h)ν1i ) , (16)   N  ν˙ 1i = −τ  aij (ν1i − ν1j ) + ai0 ν1i  + hβ(ri −1)i ν1i j=1

+ hD4i νai ,

ν˙ ai

(17)

= hD5i νai + D6i X˙ i + hD7i ν1i ,

(18)

g ri −2 ,..., where D1i = [01×(nzi +nηi +ri −1) , 1], D2i (h) = [ γ2i hri −2 γ(ri −1)i g γ(ri −1)i β2i g γ2i β(ri −1)i g ri −2 , 1], D3i (h) = β1i + +· · ·+ h h hri −2 γ1i g ri −1 + hri −1 , D4i = [1, 01×(ri −2) ],

D5i =

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 −β   

1 0 ··· −β(r −2)i 0 1 · · · i . . . . . . . . . . . . 0 0 ··· −β2i −β1i 0 0 ··· (ri −1)i

0 0 . . . 1 0





   , D7i =    

2 β(r −2)i − β(r i i −1)i β(r −3)i − β(r −2)i β(r −1)i i i i . . . β1i − β2i β(r −1)i i −β1i β(r −1)i i



 ,  

Proceedings of the 20th IFAC World Congress 8206 Youfeng Su / IFAC PapersOnLine 50-1 (2017) 8202–8207 Toulouse, France, July 9-14, 2017

D6i =

0

(ri −1)×(nz +nη ) i i

01×(nz +nη ) i i

0(r −1)×(r −1) i i

[−g ri −1 γ1i , . . . , −gγ(r −1)i ] i

0(r −1)×1 i 1



.

As a result, under the coordination of col(Xi , ν1i , νai ), the closed-loop system composed of (5), (13), and (14) is equivalent to system composed of (10), (16), (17), and (18). Let ν1 = col(ν11 , . . . , ν1N ), νa = col(νa1 , . . . , νaN ), and ν = col(ν1 , νa ). Then we have the following lemmas. Lemma 3. Under Assumption 5, there exist a C 2 positive definite function Wν (ν) and a sufficient large positive real constant τ which relies only on the parameter h, such that, along systems (17) and (18), the derivative of Wν (ν) satisfies, for all col(v, w) ∈ V × W,  ˙ 2, ˙ ν (ν) (17)+(18) ≤ − h ||ν||2 + κ ||X|| (19) W 4 h where κ is some positive constant independent of h and τ . Lemma 4. Consider the C 2 positive definite WX (X) given the same as that in Lemma 2. Then its derivative along the closed-loop system consisting of (10) and (16) satisfies, for all col(v, w) ∈ V × W,   ˙ X (X) (9)+(10) ˙ X (X) (10)+(16) ≤ W W N      + ϑi (¯ ui + KD1i Xi ) . (20)   i=1

The proofs of Lemmas 3 & 4 ar e omitted due to the space limit. We are now ready to give the proof of Theorem 1.

Proof of Theorem 1: Given any arbitrarily large number ¯ > 0, we first define the feedback gains g and K the R same as those in Lemma 2. We also define the observer gain τ the same as that in Lemma 3, which relies only on the h. We define the C 2 positive definite Lyapunov function W (X, ν) = WX (X) + Wν (ν), where WX (·) and Wν (·) are determined the same as those in Lemmas 2 and 3, respectively. For any > 0, let the positive real number l = K(maxX∈Ω¯ c+ (WX (X)),i=1,...,N {D1i Xi } +  2 maxi=1,...,N {||D2i (1)|| + |D3i (1)|}). In what follows, by two step analysis, we will show that with the suitable choice of  h, any trajectory of col(X(t), ν(t)) starting in N

(nz +2ri +nη )

i ¯ ¯ i=1 i converges to the origin asympthe set Q R totically as t → ∞. In this case, the Ninitial value of ν(t) ¯ ¯ i=1 (nzi +2ri +nηi ) , but relies on not only the compact set Q R also the designed parameter h. Step-1: we first show that the trajectory of col(X(t), ν(t)) ¯ c+ (W (X, ν)) at a finite time will enter the compact set Ω instant T . Note that the saturation function φl in (16) is bounded with the upper bound l independent of h and τ , so ¯i . As a consequence, for all col(v, w) ∈ V × is u   N  W,  i=1 ϑi (¯ ui + KD1i Xi ) ≤ S1 for some positive real constant S1 which is independent of h and τ . Then by ¯ c+ (WX (X)) and for all (12) and (20), for all X ∈ Ω col(v, w) ∈ V × W,  ˙ X (X) (10)+(16) ≤ −α||X||2 + S1 ≤ S1 . (21) W

¯ c (WX (X)) ⊆ Ω ¯ c+ (WX By (11), we have WX (X(0)) ∈ Ω (X)). Let T = 2 . Then Eq. (21) implies that for all t ∈ [0, T ],

(22) WX (X(t)) ≤ WX (X(0)) + S1 t ≤ c + . 2

On the other hand, by (22), we have for all t ∈ [0, T ], ||X(t)|| is bounded, and its upper bound is independent ˙ 2 ≤ S2 for of h and τ . Therefore, by (10) and (18), κ||X|| some positive real constant C2 which is also independent of h and τ . By (19),  ˙ ν (ν) (17)+(18) ≤ − h Wν + S2 , W (23) ¯ h 4λ ¯  mini=1,...,N {λmin (P3i ), 1}. By (23), we have where λ h ¯ 2 h (1 − e− 4λ¯ t )4λS Wν (ν(t)) ≤ e− 4λ¯ t Wν (ν(0)) + , ∀ t ≥ 0. 2 h  ¯ Let h∗1 = 4 λS 2 . Then we have, for all h ≥ h∗1 , h

Wν (ν(t)) ≤ e− 4λ¯ t Wν (ν(0)) + , ∀ t ≥ 0. (24) 4 N

¯ ¯ i=1 (nzi +2ri +nηi ) , Note that, for all col(X(0), ν(0)) ∈ Q R and for all col(v, w) ∈ V × W, Wν (ν(0)) is bounded by a h polynomial of h. Then we have limh→+∞ e− 4λ¯ T Wν (ν(0)) = 0. Thus, there exists h∗2 > 0 such that, for all h > h h∗2 , e− 4λ¯ T Wν (ν(0)) < 4 , which in turn implies that W (X(T ), ν(T )) < c+ , i.e., the trajectory of col(X(t), ν(t)) ¯ c+ (W (X, ν)) at time T . enters Ω Step-2: we then show that the trajectory of col(X(t), ν(t)) ¯ c+ (WX (X))×Ω ¯  (Wν (ν)) will maintain in the compact set Ω 2 and converge to the origin asymptotically as t → ∞. It is noted that the function in the right hand side of (24) is decreasing with respect to t. Then for any h ≥ max{h∗1 , h∗2 }, we have Wν (ν(t)) < 2 , ∀ t ≥ T. That is ¯  (Wν (ν)) to say, ν(t) will remain the the compact set Ω 2 for all t ≥ T . It is also noted that for any h ≥ 1, ||D2i (h)|| ≤ ||D2i (1)|| and |D3i (h)| ≤ |D3i (1)|. Then for ¯ c+ (WX (X)) × Ω ¯  (Wν (ν)), we always all col(X, ν) ∈ Ω 2 have |ui | ≤ l, i.e., the saturation function does not trigger. ¯ c+ (WX (X)) × Then, for any h ≥ 1, all col(X, ν) ∈ Ω ¯ Ω 2 (Wν (ν)), and all col(v, w) ∈ V × W, ˙ 2 ≤ S3 ||X||2 + S4 ||ν||2 , (25) κ||X||   N   S   5 ||X||2 + ιS6 ||ν||2 , ϑi (¯ ui + Ki D1i Xi ) ≤ (26)    ι i=1 for some positive real numbers S3 , . . . , S6 , which are all independent of h and τ , and some arbitrary ι > 0. Thus, by (12), (19), (20), (25), and (26), the derivative of W (·) along the closed-loop system consisting of system (10) and controller (16) to (18) satisfies, for all col(X, ν) ∈ ¯ c+ (WX (X)) × Ω ¯  (Wν (ν)) and all col(v, w) ∈ V × W, Ω 2    S5 S3 ˙  W (X, ν) (10)+(16)+(17)+(18) ≤ − α − − ||X||2 h ι   h S4 − − ιS6 ||ν||2 . − 4 h

ιS3 Letting, in sequence, ι > Sα5 , h∗3 > max{ αι−S , 4(S4 + 3 ιS6 ), 1}, gives that, for all h ≥ maxi=1,2,3 {h∗i , 1},  ˙ (X, ν) (10)+(16)+(17)+(18) ≤ −α ¯ ||(X, ν)||2 . (27) W

where

  S3 S5 h S4 α ¯  min α − − , − − ιS6 . h ι 4 h

¯ c+ (WX (X)) × Ω ¯  (Wν (ν)), we ¯ c+ (W (X, ν)) ⊆ Ω Since Ω 2 can conclude that, once the trajectory of col(X(t), ν(t))

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¯ c+ (W (X, ν)), it remains in compact set Ω ¯ c+ (WX enters Ω ¯  (Wν (ν)), and by (27), converges to the origin (X)) × Ω 2 asymptotically as t → ∞. Therefore, the equilibrium at the origin of the closed-loop system is uniformly asymptotically stable with its domain of attraction containing N (nzi +2ri +nηi ) ¯ ¯ i=1 . The proof is thus completed.  Q R Combining Lemma 1 and Theorem 1 leads to the following main theorem. Theorem 2. Under Assumptions 1 to 5, given any R > 0, and any compact subset V0 × W ⊆ Rnv +nw , there exist sufficiently large numbers g, K, h, τ and l, which depend on R, such that the cooperative regional output regulation ¯ nc , where problem of system (1) on the compact set Q R N nc = i=1 (nzi +2ri +nηi ), can be solved by the distributed output feedback controller (7) with gain matrices given by (15). Thus, the cooperative semi-global output regulation problem of system (1) is solvable. Remark 2. It is worth mentioning that observer (13) can be replaced by ζ˙1i = ζ¯2i + τ (evi − ζvi ), ζ¯˙1i = −hδri i ζ¯1i + ζ¯2i + hδri i ζ1i , ζ¯˙2i = −h2 δ(r −1)i ζ¯1i + ζ¯3i + h2 δ(r i

.. .

i −1)i

ζ1i ,

ζ¯˙(ri −1)i = −hri −1 δ2i ζ¯1i + ζ¯ri i + hri −1 δ2i ζ1i , ζ¯˙ri i = −hri δ1i ζ¯1i + hri δ1i ζ1i , i = 1, . . . , N,

(28)

where the coefficients δsi , s = 1, . . . , ri , i = 1, . . . , N , are chosen so that the polynomials λri +δri i λri −1 +· · ·+δ2i λ+ δ1i are all stable. Note that observer (28) results in that ei is estimated repeatedly by ζ1i and ζ¯1i , and its dimension will be ri + 1, which is higher than that of (13). Remark 3. It is noted that the distributed observer presented in Su and Huang [2014] is obtained by directly replacing the regulated error ei in the corresponding standard centralized version with the virtual regulated error evi . So evi must be used repeatedly in the coordinate transformation when performing the recursive design process. That is why the relative degrees of the subsystems in Su and Huang [2014] are required to be identical. In contrast, with the first part of our observer presented, the coordinate transformation throughout the paper has nothing to do with evi . It is this difference that we can deal with agents with arbitrary non-identical relative degrees.

4. CONCLUSIONS This paper has presented an internal model based output feedback control for the cooperative semi-global output regulation of nonlinear strict-feedback multi-agent systems with arbitrary non-identical relative degrees. A distributed observer is proposed, which is able to estimate the states of nonlinear subsystems with arbitrary non-identical relative degrees. A semi-global Lyapunov function is then developed for conducting the stability analysis of nonlinear closed-loop system. Our future work will focus on studying the same problem subject to the more general switching network topologies.

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REFERENCES Atassi, A. N. and Khalil, H. K. (1999). A separation principle for the stabilization of a class of nonlinear systems. IEEE Transactions on Automatic Control, 44(9): 1672–1687. De Persis, C. and Jayawardhana, B. (2014) On the internal model principle in the coordination of nonlinear systems. IEEE Transactions on Control of Network Systems, 1(3): 272–282. Delli Priscoli, F., Isidori, A., Marconi, L., and Pietrabissa, A. (2015). Leader-following coordination of nonlinear agents under time-varying communication topologies. IEEE Transactions on Control of Network Systems 2(4): 393–405. Ding, Z. (2015). Adaptive consensus output regulation of a class of nonlinear systems with unknown high-frequency gain. Automatica, 51: 348–355. Dong, Y. and Huang, J. (2014). Cooperative global output regulation for a class of nonlinear multi-agent systems. IEEE Transactions on Automatic Control, 59(5): 1348– 1354. Horn, R. A. and Johnson, C. R. (1991). Topics in Matrix Analysis. New York: Cambridge University Press. Huang, J. (2004). Nonlinear Output Regulation: Theory and Applications, Phildelphia: SIAM. Isidori, A. and Byrnes, C. I. (1990). Output regulation of nonlinear systems. IEEE Transactions on Automatic Control, 35(2): 131–140. Isidori, A., Marconi, L., and Casadei, G. (2014). Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory. IEEE Transactions on Automatic Control, 59(10): 2680–2691. Lin, W. and Qian, C. (2001). Semi-global robust stabilization of MIMO nonlinear systems by partial state and dynamic output feedback. Automatica, 37(7): 1093– 1101. Nikiforov, V. O. (1998). Adaptive nonlinear tracking with complete compensation of unknown disturbances. European Journal of Control, 4: 132–139. Su, Y. (2016). Output feedback cooperative control for linear uncertain multi-agent systems with nonidentical relative degrees. IEEE Transactions on Automatic Control, 61(12): 4027-4033. Su, Y. and Huang, J. (2014). Cooperative semi-global robust output regulation for a class of nonlinear uncertain multi-agent systems. Automatica, 50(4): 1053–1065. Su, Y. and Huang, J.(2015). Cooperative global output regulation for nonlinear uncertain multi-agent systems in lower triangular form. IEEE Transactions on Automatic Control, 60(9): 2378–2389. Wang, X., Xu, D., and Ji, H. (2016). Robust almost output consensus in networks of nonlinear agents with external disturbances, Automatica, 70: 303–311. Wieland, P., Sepulchre, R., and Allg¨ower, F. (2011). An internal model principle is necessary and sufficient for linear output synchronization. Automatica, 47(5): 1068– 1074. Zhu, L., Chen, Z., and Middleton, R. (2016). A general framework for robust output synchronization of heterogeneous nonlinear networked systems IEEE Transactions on Automatic Control. 61(8): 2092–2107.

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