Robust adaptive distributed dynamic surface consensus tracking control for nonlinear multi-agent systems with dynamic uncertainties

Author’s Accepted Manuscript Robust adaptive distributed dynamic surface consensus tracking control for nonlinear multi-agent systems with dynamic uncertainties Xiaocheng Shi, Junwei Lu, Ze Li, Shengyuan Xu www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(16)30274-5 http://dx.doi.org/10.1016/j.jfranklin.2016.08.009 FI2689

To appear in: Journal of the Franklin Institute Received date: 17 December 2015 Revised date: 25 June 2016 Accepted date: 9 August 2016 Cite this article as: Xiaocheng Shi, Junwei Lu, Ze Li and Shengyuan Xu, Robust adaptive distributed dynamic surface consensus tracking control for nonlinear multi-agent systems with dynamic uncertainties, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.08.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust adaptive distributed dynamic surface consensus tracking control for nonlinear multi-agent systems with dynamic uncertainties Xiaocheng Shia , Junwei Lub , Ze Lic , Shengyuan Xua,∗ a

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, P.R. China b School of Electrical and Automation Engineering, Nanjing Normal University, 78 Bancang Street, Nanjing 210042, P.R. China c School of Mechanical and Electrical Engineering, Suzhou University of Science and Technology, Suzhou 215000, P.R. China

Abstract This paper proposes an adaptive distributed consensus tracking control approach for uncertain nonlinear multi-agent systems in pure-feedback form under a directed topology where each follower is dominated by dynamic uncertainties and unmeasured states. Radial basis function neural networks (RBFNNs) are employed to compensate the unknown nonlinear functions obtained by recursive design procedure for followers. The distributed dynamic surface controllers are able to eliminate the condition in which the approximation error of the traditional neural networks is bounded. By introducing an available dynamic signal and two smooth scalar functions, the obstacle caused by unmodeled dynamics is conquered. The main advantage of the proposed method is that for M pure-feedback nonlinear followers, only one learning parameter needs to be updated online. It is also shown that the proposed consensus controller can guarantee cooperatively semi-global uni∗

Corresponding author. Tel.: +86 25 84303027; fax: +86 25 84303027. Email address: [email protected] (Shengyuan Xu )

Preprint submitted to Elsevier

August 12, 2016

form ultimate boundedness (CSUUB) of all the signals, and the consensus errors converge to an adjustable neighborhood of the origin. Keywords: Networked pure-feedback systems, adaptive consensus tracking, unmodeled dynamics, dynamic surface control 1. Introduction Research on distributed control of networked multi-agent systems has attracted tremendous attention due to their applications in formation control [19, 2], event-triggered consensus control [3], smart microgrid energy management [6], containment control [14], energy internet [22], distributed estimation [7, 17] and other areas. The consensus problem means to guarantee the states of a group of networked agents to reach an agreement. An important consensus-like problem is called the distributed consensus tracking control, where all the followers are trying to follow the leader, and only a subset of the networked group can get information from a time-varying leader. Numerous results concerning multi-agent distributed tracking control have been published in recent years [4, 21, 12, 15, 16, 5, 33, 20, 35, 18]. In [4], a consensus tracking protocol was given for multi-agent systems with continuous single-integrator dynamics, which had measurement noises and time-delays. In [21], the authors studied leader-following consensus problem for double-integrator multi-agent systems with velocity measurements based on pining control. The work is extended in [15] using only relative position measurements and in [16] to a coordinated tracking algorithm combined with a distributed filter. For more practical purposes, the consensus tracking problems of multi-agent systems were investigated for high-order nonlinear systems [5, 33, 20] and multiple rigid bodies [35, 18]. It should be pointed out that most of these aforementioned studies [4, 21, 15, 16, 5, 33, 20, 35, 18] 2

only deal with nonlinear multi-agent systems with matched uncertainties. However, various physical systems have more complicated dynamics without satisfying the matching conditions. On the other hand, adaptive backstepping technique has become a powerful tool to handle strict-feedback nonlinear systems without matching condition. In [11], adaptive backstepping controller was first designed for parametric strict-feedback systems with over-parametrization to obtain asymptotic tracking performance. By introducing universal function approximators, the systematic and recursive control method was further extended and applied to a large class of SISO nonlinear systems with unstructured uncertainties [1, 26, 30]. Dynamic surface control method [23, 28] has been proven to be useful for solving the problem of explosion of complexity in the traditional adaptive neural or fuzzy network based on backstepping design. Recently, dynamic surface control approach has been improved to control nonlinear multi-agent systems [32, 31]. In [32], a neural network-based distributed consensus tracking problem for uncertain multi-agent strictfeedback systems was studied by using the dynamic surface control technique, in the case where only a small fraction of follows has access to a time-varying leader , while in [31], based on the Lyapunov’s stability theory, a distributed adaptive containment control approach was proposed for uncertain nonlinear multi-agent systems in strict-feedback form by employing neural networks. It was pointed out that the agents in [32, 31] were modeled by strict-feedback systems, and they had affine appearance of the state variables. Unmodeled dynamics and dynamic disturbances usually co-exist in various practical systems. Their existence often have serious effects on the performance and the stability of the control systems. In past decades, a 3

great number of works have been done for several types of nonlinear systems to address this problem based on the adaptive backstepping technique. In [8, 9], two adaptive backstepping control methods were proposed for a class of parametric strict-feedback nonlinear systems with unmeasured input-tostate stable dynamics. The robustness of the unmodeled dynamics and dynamic disturbances is ensured by introducing a Lyapunov function. In [24], a stable adaptive fuzzy robust approach was proposed for strict-feedback canonical nonlinear systems with unmodeled dynamics by combining fuzzy logic systems with the small-gain approach; while in [25], a direct fuzzy adaptive control method was developed for a class of SISO nonlinear systems with completely unknown virtual control directions. By integrating changing supply function with dynamic surface control technique, a new robust adaptive output feedback control scheme was investigated based on a fuzzy state observe in [34]. In [27], an adaptive observe-based neural control problem was discussed for a class of stochastic nonlinear strict-feedback systems with unmodeled dynamics and dead-zone input by using a stochastic small-gain theorem. The authors in [24, 25, 34, 27] use a dynamic signal to compensate for the unmodeled dynamics. However, it is assumed that the upper bound functions with respect to the dynamical disturbance in [8, 9, 24, 25, 34, 27] are known, which is very crucial in the controller design. Motivated by the above observations, we consider the distributed tracking control problem for uncertain pure-feedback multi-agent systems with unmodeled dynamics and dynamic disturbance. The main contributions of the proposed control approach in this paper lie in: (i) Different form the tractional adaptive fuzzy or NN backstepping controllers designed for individual agent system [11, 1, 26, 30, 23, 28], a novel 4

distributed tracking control scheme for nonlinear multi-agent systems needs to consider the coupled terms in dynamics, the communication among the agents, and the ability on information transmission of a time-varying leader and so on. By utilizing dynamic surface control technique, the possible circular argument is avoided in Lyapunov stability analysis. (ii) Each follower node is a pure-feedback integrator incorporated with dynamic uncertainties and unmodeled dynamics. Most of the existing results described the dynamics of nonlinear multi-agent systems with unmatched uncertainties as strict-feedback form[32, 31]. Recently, distributed consensus tracking control problem was investigated for nonlinear pure-feedback multiagent systems in [29], in which the partial derivatives

∂fi,k (¯ xi,k ,xi,k+1 ) ∂xi,k+1

are

strictly assumed to be either positive or negative. However, our approach no longer needs such a restrictive assumption. Besides, unlike in [29], we consider unmodeled dynamics which makes the system model much more complex and practical. (iii) In contrast to the distributed consensus tracking problem for multiagent systems with uncertain nonlinearities based on adaptive neural networks or fuzzy logic systems [5, 33, 20, 32, 31, 29], only one parameter is tuned on line regardless of the weights of neural networks. The rest of the paper is organized as follows. Section 2 formulates the problem. In Section 3, distributed dynamic surface control is proposed for a class of nonlinear multi-agent systems with M followers and a leader under a directed network, then the stability of the closed-loop system is given. Simulation results are performed to demonstrate the effectiveness of the proposed control scheme in Section 4, and Section 5 concludes the paper.

5

2. Preliminaries and Problem Formulation 2.1. Graph theory The communication topology is denoted by a directed graph G = {V, E, Λ} with a set of nodes V = {1, . . . , M }, a set of edges E ⊆ V × V, and an adjacency matrix Λ = [aij ] ∈ RM ×M . (j, i) ∈ E means that agent i can directly receive information from agent j. The neighbor set of vertex i can be described by Ni = {j|(j, i) ∈ E}. aij ∈ Λ is defined as aij > 0 if (j, i) ∈ E and aij = 0 otherwise. Throughout this paper, there is no self-connectivity element, i.e., aii = 0. The graph Laplacian matrix L = D − Λ ∈ RM ×M , where D = diag[d1 , . . . , dM ] is the diagonal in-degree matrix with di = M j=1,j=i aij . A directed graph has a directed spanning tree, if root node exists, there is a directed path from it to every other node in the graph. 2.2. Problem statement Consider the following nonlinear multi-agent systems with M followers and a leader under a directed network. Follower i is described as: ⎧ ⎪ z˙ = q (z, xi , t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙ = fi,1 (xi,1 , xi,2 ) + gi,1 xi,2 ⎪ ⎨ i,1 x˙ i,l = fi,l (¯ xi,l , xi,l+1 ) + gi,l xi,l+1 + Δi,l (xi , z, t) ⎪ ⎪ ⎪ ⎪ x˙ ⎪ i,ni = fi,ni (xi ) + gi,ni ui + Δi,ni (xi , z, t) ⎪ ⎪ ⎪ ⎪ ⎩ yi = xi,1

(1)

¯i,l = [xi,1 , . . . , xi,l ]T ∈ Rl , xi = where i = 1, . . . , M , l = 2, . . . , ni − 1, x [xi,1 , . . . , xi,ni ]T ∈ Rni are the state vectors; ui ∈ R, yi ∈ R are the inxi,q , xi,q+1 ), q = puts and outputs of the control system respectively; fi,q (¯ 1, . . . , ni − 1 and fi,ni (xi ) are the unknown smooth functions; gi,q , q = 1, . . . , ni are constants; z ∈ R0n is the unmodeled dynamics and Δi,l (xi , z, t) 6

are the unknown Lipschitz continuous functions. The control objective of this paper is, for a dynamic leader output signal r(t), which is a smooth function of t, r(t), r(t), ˙ r¨(t) are bounded and available, to find neural networks based adaptive distributed consensus laws ui for M followers such that the follower outputs yi asymptotically synchronize to the time-varying leader r(t) while all of the signals in the closed-loop systems are bounded. To describe the communication among the M followers and a leader, we define a directed graph for the M + 1 agents G = {V, E, Λ} with V = {0, . . . , M }. The communications between followers are described by a sub¯ E, ¯ Λ} ¯ with V¯ = {1, . . . , M }. Then, the Laplacian matrix L graph G¯ = {V, can be Partitioned as

⎞

⎛ L=⎝

0

01×M

−b L¯ + B

⎠

where b = [b1 , . . . , bM ]T represents the communication weight from the leader to followers. bi > 0 if the leader is a neighbor of node i and bi = 0 ¯−Λ ¯ with Λ ¯ = [aij ] ∈ RM ×M otherwise. B = diag[b1 , . . . , bM ], and L¯ = D is the adjacency matrix of the subgraph; aij > 0 if (j, i) ∈ E¯ and aij = 0 ¯ = diag[d1 , . . . , dM ] is the subgraph Laplacian matrix. otherwise, D Remark 1. Each follower described by (1) possesses three types of uncertainties: dynamic disturbance Δ(·), unknown function f (·) and unmodeled dynamics z. Note that the existing nonlinear multi-agent systems did not contain the unmodeled dynamics and dynamic disturbances. Therefore, the system considered in this paper is more general. Assumption 1. The states xj,1 and xj,2 of the jth follower are only known and available for the ith follower satisfying j ∈ Ni , i = 1, . . . , M, j = 1, . . . , M , and , i = j. 7

Assumption 2. There exist unknown nonnegative continuous functions xi,k ) and nonnegative monotone increasing functions ϕi,k,2(z), ϕi,k,1 (¯ xi,k ) + with ϕi,k,1 (0) = ϕi,k,2 (0) = 0, such that Δi,k (xi , z, t) ≤ ϕi,k,1 (¯ ϕi,k,2 (z), ∀i ∈ 1, . . . , M , ∀k ∈ 2, . . . , ni . Remark 2. The functions ϕi,k,1(·) and ϕi,k,2(·) are unknown in this paper while they are known in [8, 9, 24, 25, 34, 27]. Therefore, the assumption with respect to dynamic disturbances and unmodeled dynamics is relaxed. Assumption 3. The unmodeled dynamics z is said to be exponentially inputstate-practically stable(exp-ISpS), that is, for system z˙ = q(z, xi , t), if there exists a Lyapunov function V (z) such that ¯ 2 (z) α ¯ 1 (z) ≤ V (z) ≤ α

(2)

∂V (z) q(z, xi , t) ≤ −cV (z) + γ¯ (|xi,1 |) + d ∂z

(3)

¯ 2 (·), γ¯ (·) are the class k∞ functions, c and d are known positive where α ¯ 1 (·), α constants. Lemma 1. [10] If V is an exp-ISpS Lyapunov function for a system z˙ = q(z, xi , t), that is, (2)and (3) hold, then, for any constant c¯ ∈ (0, c), any initial instant t0 > 0, any initial condition z0 = z(t0 ), r0 > 0, for any continuous function γ¯ such that γ¯ (|xi,1 |) ≥ γ(|xi,1 |), there exists a finite T0 = max{0, ln[V (z0 )/r0 ]/(c − c¯)} ≥ 0, a nonnegative function D(t0 , t); defined for all t > t0 and a signal described by r˙ = −¯ cr + γ¯ (xi,1 ) + d, r(t0 ) = r0

(4)

such that D(t0 , t) = 0 for t ≥ t0 + T0 , and V (z) ≤ r(t) + D(t0 , t) with D(t0 , t) = max{0, e−c(t−t0 ) × V (z0 ) − e−¯c(t−t+0) r0 }. Without loss of generality, we assume γ¯ (|xi,1 |) = γ(|xi,1 |). 8

Lemma 2. [13] For any real continuous function f (x, y), there exist postive smooth scalar functions φ(x) ≥ 0 and υ(y) ≥ 0, such that the following inequality holds | f (x, y) |≤ φ(x) + υ(y)

(5)

where x ∈ Rm , y ∈ Rn . In this paper, we will employ RBFNNs to approximate the unknown smooth functions Hi,l (Zi,l ) ∗T ξi,l (Zi,l ) + δi,l (Zi,l ), ∀Zi,l ∈ ΩZi,l Hi,l (Zi,l ) = θi,l

(6)

∗ is an where l = 1, . . . , ni , i = 1, . . . , M , ΩZi,l is a given compact set, θi,l

unknown ideal constant weight vector defined as ∗ : arg min θi,l

sup

θˆi,l ∈Rli Zi,l ∈ΩZi,l

ˆ i,l (Zi,l ) | | Hi,l (Zi,l ) − H

where δi,l (Zi,l ) denotes the approximation error. the basis function vector ξi,l (Zi,l ) = [si,l1 (Zi,l ), . . . , si,lpi (Zi,l )]T ∈ Rpi with si,lj (zi,l ) being chosen as the commonly used Gaussian functions, which have the form  (Zi,l − vi,lj )T (Zi,l − vi,lj ) si,lj (zi,l ) = exp − 2 ki,lj

(7)

j = 1, . . . , pi , i = 1, . . . , M , vi,lj = [vi,lj1 , . . . , vi,ljqij ]T is the center of the receptive filed and ki,lj is the width of the Gaussian function. 3. Main results For the system (1), dynamic surface control-based distributed consensus tracking design procedure for multiple pure-feedback systems with unmodeled dynamics contains ni steps, where the graph-based error surfaces 9

are employed and RBFNNs are used to compensate the unknown nonlinear terms induced from the the procedure of the controller design. We first define error surfaces si,k and the boundary layer errors yi,k as follows si,1 =

M

aij (yi − yj ) + bi (yi − r(t))

(8)

j=1

si,k = xi,k − zi,k , k = 2, . . . , ni yi,k = zi,k − αi,k

(9) (10)

Before constructing the real control law, we first derive the virtual control signals at each step as follows αi,2 = +

1 1 ˆ T [−ki,1 si,1 − 2 λs i,1 ξi,1 (Zi,1 ) ξi,1 (Zi,1 )] (di + bi )gi,1 2bi,1 M

1 aij gj,1 xj,2 (di + bi )gi,1

(11)

j=1

αi,l+1

1 1 ˆ T = [−ki,l si,l − 2 λs i,l ξi,l (Zi,l ) ξi,l (Zi,l )], gi,l 2bi,l

l = 2, . . . , ni − 1

(12)

ˆ is the estimate of the unknown constant λ which is specified as where λ  2  ∗ ˜ =λ ˆ − λ. The input vectors λ = max θi,l  , i = 1, . . . , M, l = 1, . . . , ni . λ Zi,l ∈ Ωzi,l , l = 1, . . . , ni of RBFNNs will be specified later at each step. Remark 3. In contrast to the previous dynamic surface design [23, 28, 34], M  aij (yi − yj ) + bi (yi − r(t)) are prothe distributed error surfaces si,1 = j=1

posed to treat the networked pure-feedback systems with unmodeled dynamics. Theorem 1. Consider the MASs consisting of (1) under Aussumptions 1-3 and assume that the dynamic leader has directed paths to M followers, the

10

virtual control laws defined by (11) and (12), if the distributed consensus control law is finally chosen as ui =

1 gi,ni

[−ki,ni si,ni −

1 ˆ T (Zi,ni ) ξi,ni (Zi,ni )] λsi,ni ξi,n i 2b2i,ni

(13)

and the adaptive law

γξi,l (Zi,l ) ξi,l (Zi,l ) si,l ˆ ˆ˙ = − γσ λ λ 2 2b i,l i=1 M

ni

T

2

(14)

l=1

For bounded initial conditions, satisfying V (0) ≤ p, there exist constants ki,m > 0, ki,ni > 0, τi,k > 0, γ > 0, σ > 0 such that the consensus tracking errors in the overall closed-loop neural control system are CSUUB and can be smaller than a prescribed error bounded, ki,m , ki,ni and τi,k satisfy ⎧ ⎪ ki,m ≥ 1 + α0 /2, m = 1, ..., ni − 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ki,n ≥ 1/2 + α0 /2  i 2 ⎪ ⎪ ≥ 1 + g 1 τ i,k ⎪ i,k−1 2 + α0 /2, k = 2, ..., ni ⎪ ⎪ ⎪ ⎩ α0 ≤ γσ

(15)

Remark 4. Compared with the existing results of the neural or fuzzy -based adaptive distributed tracking problems in [5, 33, 20, 32, 31, 29], this paper only employs one tuning law in (14) to design all the consensus controllers regardless of the increasing number of the followers or the increasing order of each follower. As a result, the complexity of the design procedure and the computation loads drastically reduce. 3.1. Distributed dynamic surface control design In this section, we propose a neural-based adaptive distributed consensus tracking control scheme.

11

Step 1: Consider the first subsystem of ith follower in(1). Use (3). Its time derivative along (9) and (10) is s˙ i,1 = (di +bi ) x˙ i,1 −

M

aij x˙ j,1 − bi r˙ (t)

j=1 M 1 aij fj,1 (xj,1, xj,2 ) = (di +bi ) [fi,1 (xi,1 , xi,2 ) − di + bi j=1

+ gi,1 (si,2 + yi,2 + αi,2 )] −

M

aij gj,1 xj,2 − bi r˙ (t)

(16)

j=1

Using Young’s inequality, we obtain 1 1 2 2 si,1 si,1 (di + bi )gi,1 si,2 ≤ s2i,2 + (di + bi )2 gi,1 2 2 1 2 1 2 2 + (di + bi )2 gi,1 si,1 si,1 (di + bi )gi,1 yi,2 ≤ yi,2 2 2

(17) (18)

In order to design adaptive control, choose a Lyapunov function candidate as follows 1 1 Vsi,1 = s2i,1 + v 2 λ0

(19)

where λ0 is a positive constant. From Lemma 1, and substituting (17) and (18) into the derivative of Vsi,1 , we have 1 2 1 + s2i,2 + si,1 (di +bi ) gi,1 αi,2 V˙ si,1 ≤ si,1 H(Zi,1 ) + yi,2 2 2 s2i,1 γ¯ (|xi,1 |) −¯ c d + (1 − 2 ) + v+ λ0 λ0 λ0 εγ¯

(20)

where εγ¯ is a design constant. Hi,1 (Zi,1 ) = (di + bi )fi,1 (xi,1 , xi,2 ) −

M

aij fj,1 (xj,1 , xj,2 )

j=1

− bi r˙ (t) +

si,1 γ¯(|xi,1 |) 2 + (di + bi )2 gi,1 si,1 λ0 ε2γ¯ 12

(21)

with ˙ T ∈ Ωzi,1 , j ∈ Ni Zi,1 = [xi,1 , xi,2 , xj,1 , xj,2 , si,1 , r(t)] To overcome the ”explosion of complexity”, the following first-order filter is designed to replace the differentiation of the virtual control signal αi,2 by simpler algebraic operation. τi,2 z˙i,2 + zi,2 = αi,2 , zi,2 (0) = αi,2 (0)

(22)

where τi,2 > 0 is a small time constant that we will choose later. From (10), we known that y˙ i,2 = −yi,2 /τi,2 − α˙ i,2 , as si,1 Hi,1 (Zi,1 ) = ∗T ξ (Z ) + s δ (Z ) ≤ si,1 θi,1 i,1 i,1 i,1 i,1 i,1

1 T (Z )ξ (Z ) λs2i,1 ξi,1 i,1 i,1 i,1 2b2i,1

+ 12 b2i,1 + | si,1 ||

δi,1 (Zi,1 ) |, and noting the distributed first virtual control law in (11), it yields 1 1 2 1 1 ˜ 2 T + s2i,2 − 2 λs ξ (Zi,1 )ξi,1 (Zi,1 ) V˙ si,1 ≤ (−ki,1 + )s2i,1 + yi,2 2 2 2 2bi,1 i,1 i,1 s2 γ¯ (|xi,1 |) −¯ c d 1 2 (Zi,1 ) + (1 − 21 ) + v+ + Ci,1 + ψi,1 2 λ0 λ0 λ0 εγ¯

(23)

where Ci,1 = 12 b2i,1 , continuous function ψi,1 satisfies ˆ v, r(t), r(t)) ˙ |δi,1 (Zi,1 )| ≤ ψi,1 (si,1 , ..., si,ni , yi,2 , ..., yi,ni , λ,

(24)

Noting (11), we have y˙ i,2 = − +

yi,2 1 1 ˆ˙ T + [ki,1 s˙ i,1 + 2 λs i,1 ξi,1 (Zi,1 ) ξi,1 (Zi,1 ) τi,2 (di + bi )gi,1 2bi,1

M ˆ i,1 dξi,1 (Zi,1 )2 1 λs − aij gj,1 x˙ j,2] dt 2b2i,1 j=1

(25)

In view of (25) and by induction for some continuous function ηi,2 , we obtain     ˆ v, r(t), r(t), y˙ i,2 + yi,2  ≤ ηi,2 (si,1 , ..., si,n , yi,2 , ..., yi,n , λ, ˙ r¨(t)) (26) i i  τi,2  13

From (25) and (26), we have 2 yi,2 ˆ v, r(t), r(t), + |yi,2 | ηi,2 (si,1 , ..., si,ni , yi,2 , ..., yi,ni , λ, ˙ r¨(t)) τi,2 2 yi,2 1 2 2 + yi,2 + ηi,2 (27) ≤− τi,2 4

yi,2 y˙ i,2 ≤ −

Step l (2 ≤ l ≤ ni − 1): After differentiating si,l along (1), we obtain s˙ i,l = fi,l (¯ xi,l , xi,l+1 ) + gi,l xi,l+1 + Δi,l (xi , z, t) − z˙i,l

(28)

To stabilize (28), Lyapunov candidate function for ith follower is chosen as 1 Vsi,l = s2i,l 2

(29)

According to Assumption 2 and using Young’s inequality, the time derivative of Vsi,l follows from (28) that xi,l , xi,l+1 ) + gi,l xi,l+1 − z˙i,l ] + |si,l | ϕi,l,1 (¯ xi,l ) V˙ si,l ≤ si,l [fi,l (¯ + |si,l | ϕi,l,2 (z)

(30)

α−1 1 (·) is an increasing function simply because α1 is a class k∞ function. Noting (2) and Lemma 1, we get z ≤ α−1 1 (v(t) + D(t0, t))

(31)

ϕi,l,2 (z) ≤ ϕi,l,2 (α−1 1 (v(t) + D(t0, t)))

(32)

Because ϕi,l,2 (α−1 1 (·)) is a nonnegative smooth function, using Lemma 2, (32) can be rewritten as |si,l | ϕi,l,2 (z) ≤ |si,l | φi,l (v(t)) + |si,l | vi,l (D(t0 , t)) ≤ s2i,l φ2i,l (v(t)) + |si,l | vi,l (D(t0 , t))+ 14

1 4

(33)

where φi,l (·) and vi,l (·) are both unknown nonnegative smooth functions. From Lemma 1, we conclude that if t ≥ t0 +T0 , then D(t0 , t) = 0. Therefore, ∗ such that v 2 (D(t , t)) ≤ v ∗ . Apply there exists an unknown constant vi,l 0 i,l i,l

Young’s inequality. then, it can be easily verified that 1 xi,l , xi,l+1 ) − z˙i,l + 2 si,l ϕi,l,1 (¯ xi,1 ) + si,l φ2i,l (v(t)) + si,l V˙ si,l ≤ si,l [fi,l (¯ εi,l 2

εi,l 1 1 ∗ 1 2 + + vi,l + + yi,l+1 + si,l gi,l αi,l+1 + 2 2 4 4 4 1 2 1 2 ≤ si,l Hi,l (Zi,l ) + si,l+1 + yi,l+1 + si,l gi,l αi,l+1 2 2 ε2i,l 1 1 ∗ + + vi,l + 4 4 4 1 2 si,l gi,l ]+ s2i,l+1

(34)

where εi,l > 0 is a design constant. xi,l , xi,l+1 ) − z˙i,l + Hi,l (Zi,l ) = fi,l (¯

1 si,l ϕi,l,1 (¯ xi,1 ) ε2i,l

2 + si,l φ2i,l (v(t)) + si,l + si,l gi,l

xi,l+1 , si,l , z˙i,l , v]T ∈ Ωzi,l ⊂ Rl+4 Zi,l = [¯

(35) (36)

Similar to step 1, we employ RBFNNs in (6) to approximate unknown continuous function Hi,l (Zi,l ). Considering the lth distributed virtual control law for ith follower in (12) with k = l, we have 1 1 2 1 ˜ 2 T − 2 λs ξ (Zi,l ) ξi,l (Zi,l ) V˙ si,l ≤ −ki,l s2i,l + s2i,l+1 + yi,l+1 2 2 2bi,l i,l i,l + | si,l || δi,l (Zi,l ) | +Ci,l where Ci,l =

ε2i,l 4

+

1 4

+

∗ vi,l 4

+

(37)

b2i,l 2 .

Proceeding Similarly, introduce the lth first-order filter to obtain filtered virtual control law, i.e., τi,l+1z˙i,l+1 + zi,l+1 = αi,l+1 , zi,l+1 (0) = αi,l+1 (0) 15

(38)

There exist some continuous functions ψi,l such that ˆ v, r(t), r(t)) ˙ |δi,l (Zi,l )| ≤ ψi,l (si,1 , ..., si,ni , yi,2 , ..., yi,ni , λ, From Young’s inequality, (37) becomes 1 1 1 2 1 ˜ 2 T − 2 λs ξ (Zi,l ) ξi,l (Zi,l ) V˙ si,l ≤ (−ki,l + )s2i,l + s2i,l+1 + yi,l+1 2 2 2 2bi,l i,l i,l 1 2 + ψi,l (Zi,l ) + Ci,l (39) 2 y

i,l+1 − α˙ i,l+1 , it yields As y˙ i,l+1 = − τi,l+1

    ˆ˙ i,l ξ T (Zi,l ) ξi,l (Zi,l ) y˙ i,l+1 + yi,l+1  ≤ 1 [−ki,l s˙ i,l − 1 λs i,l   2 τi,l+1 gi,l 2bi,l +

ˆ i,l dξi,l (Zi,l )2 1 λs 1 ˆ T ] (40) ξ (Z ) ξ (Z ) + λ s ˙ i,l i,l i,l i,l i,l dt 2b2i,l 2b2i,l

    ˆ v, r(t), r(t), y˙ i,l+1 + yi,l+1  ≤ ηi,l+1 (si,1 , ..., si,n , yi,2 , ..., yi,n , λ, ˙ r¨(t)) (41) i i  τi,l+1  where ηi,l+1 donate some continuous functions. In view of (40) and (41), we have yi,l+1 y˙ i,l+1 ≤ −

2 yi,l+1

τi,l+1

2 + yi,l+1 +

2 ηi,l+1

4

(42)

Step ni : The time derivative of the ni th error surface is s˙ i,ni = fi,ni (xi ) + gi,ni ui + Δi,ni (xi , z, t) − z˙i,ni

(43)

By defining the ni th quadratic function as Vsi,ni = 12 s2i,ni and differentiating it with respect to time t, we obtain V˙ si,ni = si,ni [fi,ni (xi ) + gi,ni ui + Δi,ni (xi , z, t) − z˙i,ni ]

16

(44)

From Assumption 2 and 3, and employing Young’s inequality, we have |si,ni Δi,ni (xi , z, t)| ≤ |si,ni | ϕi,ni ,1 (xi ) + |si,ni | ϕi,ni ,2 (z) ≤ |si,ni | ϕi,ni ,1 (xi ) + |si,ni | ϕi,ni ,2 (α−1 1 (v(t) + D(t0, t))) ≤ |si,ni | ϕi,ni ,1 (xi ) + |si,ni | φi,ni (v(t)) + |si,ni | vi,ni (D(t0 , t)) ∗ 1 vi,n i + + s2i,ni 4 4 ∗ ε2i,ni + 1 + vi,n 1 i ≤ 2 s2i,ni ϕi,ni ,1 (xi ) + s2i,ni φ2i,ni (v(t)) + s2i,ni + 4 εi,ni

≤ |si,ni | ϕi,ni ,1 (xi ) + s2i,ni φ2i,ni (v(t)) +

(45)

Substituting (45) into (44), we obtain V˙ si,ni ≤ si,ni gi,ni ui + si,ni Hi,ni (Zi,ni ) +

∗ ε2i,ni + 1 + vi,n i 4

(46)

where Hi,ni (Zi,ni ) = fi,ni (xi ) +

1

s ϕi,ni ,1 (¯ xi,n ) ε2i,ni i,n

+ si,ni φ2i,ni (v(t)) + si,ni − z˙i,ni Zi,ni = [xi , si,ni , z˙i,ni , v]T ∈ Ωzi,ni ⊂ Rni +3

(47) (48)

Constructing the distributed adaptive actual controllers in (13), and similar to the procedure of the previous steps, we have 2 + 1)s2i,ni − V˙ si,ni ≤ (−ki,n i

1 ˜ 2 T λs ξ (Zi,ni ) ξi,ni (Zi,ni ) 2b2i,ni i,ni i,ni

1 2 (Zi,ni ) + Ci,ni + ψi,n i 2 where Ci,ni =

∗ ε2i,n +1+vi,n i i 4

(49)

are positive constants. Continuous function sat-

ˆ v, r(t), r(t)). ˙ isfies δi,ni (Zi,ni ) ≤ ψi,ni (si,1 , ..., si,ni , yi,2 , ..., yi,ni , λ, 3.2. Stability analysis In this section, we will devote to the stability analysis of the proposed distributed adaptive control system under a directed topology. Let us define 17

some compact sets as follows ˆ v]T : V1 ≤ p} ⊂ RpΩ Ω ={[si,1 , ..., si,ni , yi,2 , ..., yi,ni , λ, Ωd ={[r(t), r(t), ˙ r¨(t)]T : r 2 (t) + r˙ 2 (t) + r¨2 (t) ≤ R0 } ⊂ Rpd Where pΩ = 2M ni + 2 − M . p > 0 is a design constant, and ni ni M

˜2 1 1 λ 2 2 ( si,l + yi,k )+ v+ V1 = 2 λ0 2γ i=1

l=1

k=2

We can easily see that Ω×Ωd is compact set in RpΩ +pd . There exists constant Mi,l such that | ψi,l |≤ Mi,l on Ω × Ωd , and ηi,k has a maximum Ni,k on Ω × Ωd . l = 1, · · · , ni , k = 2, · · · , ni , i = 1, · · · , M . Choose the Lyapunov function candidate as V =

ni ni M

˜2 1 λ 2 [ Vsi,l + yi,k ]+ 2 2γ i=1 l=1

(50)

k=2

Differentiating V with respect to time t leads to V˙ =

ni ni M

1 ˜ ˆ˙ λ [ yi,k y˙ i,k ] + λ V˙ si,l + γ i=1 l=1

(51)

k=2

Substituting (23), (39), (49), (27) and (42) into (51), and applying (14), it yields V˙ ≤

M

{[−ki,1 +

i=1

n i −1 1 2 + (di + bi )2 gi,1 ]s2i,1 + (−ki,l + 1+g2i,l )s2i,l 2 l=2

2 + (−ki,n i

ni M 2 2

yi,k ηi,k 3 2 2 ) + 1)si,ni } + [ (− + yi,k + τi,k 2 4 i=1 k=2 ni

1 2 (Zi,1 ) + + (di +bi )2 ψi,1 2 + (1 −

k=2

s2i,1 γ¯ (|xi,1 |) ˜λ ˆ ) −σ λ λ0 ε2γ¯

c¯ 1 2 ψi,k (Zi,k )] − v 2 λ0

+

ni M

i=1 l=1

18

Ci,l

(52)

According to Perfect square trinomial, the following inequality holds ˜2) ˜λ ˆ = −σ λ( ˜ λ ˜ + λ) ≤ σ( 1 λ2 − 1 λ −σ λ 2 2

(53)

Let i i

1 1 2 1 2 2 Mi,k (Zi,k ) + N + μ1 = (di +bi )2 Mi,1 2 2 4 i,k

+

ni M

i=1 l=1

n

n

k=2

k=2

1 Ci,l + σλ2 2

(54)

Substituting (15), (53) and (54) into (52), we obtain V˙ ≤ −α0 V + μ1 +

M

(1 −

i=1

s2i,1 γ¯ (|xi,1 |) ) λ0 ε2γ¯

(55)

Let us define Ωsi,1 = {si,1 ||si,1 | ≤ εγ¯ }, i = 1, . . . , M , where εγ¯ > 0 is a constant. Hence, the following analysis is developed in three cases. Case 1: |si,1 | > εγ¯ , ∀i = 1, ..., M , obviously, function 1 − s2i,1 /ε2γ¯ < 0, (55) becomes V˙ ≤ −α0 V + μ1

(56)

Case 2: |si,1 | ≤ εγ¯ , ∀i = 1, ..., M , from (3), we known x1,1 , . . . , xM,1 are bounded. Since γ(·) is a nonnegative class k∞ function, there exists a positive constant μ1 such that M

s2i,1 γ¯ (|xi,1 |) (1 − 2 ) ≤ μ0 λ0 εγ¯ i=1

(57)

V˙ ≤ −α0 V + μ1 + μ0

(58)

And then, we have

Case 3: Let Ωj = {j : |sj,1| > εγ¯ , j = 1, ..., M }, Ωi = {i : |si,1 | ≤ εγ¯ , j = M  s2 γ ¯ (|xi,1 |) (1 − εi,1 ≤ 1, ..., M }, there exists a constant μ2 > 0 such that 2 ) λ0 i=1

19

γ ¯

 i∈Ωi

(1 −

s2i,1 γ ¯ (|x |) ) λi,1 ε2γ¯ 0

≤ μ2 . So we obtain V˙ ≤ −α0 V + μ1 + μ2

(59)

Integrating the three kinds of cases, we have V˙ ≤ −α0 V + μ

(60)

Where μ = μ1 + max{μ0 , μ2 }. If V = p and α0 > μ/p, then V˙ ≤ p. It implies that V (t) ≤ p, ∀t ≥ 0 for V (0) ≤ p.Furthermore   μ −α0 t μ + V (0) − 0 ≤ V (t) ≤ e α0 α0

(61)

Therefore, all signals in the closed-loop systems, i.e. , si,1 , ..., si,ni , yi,2 , ..., yi,ni , i = ˆ are CSUUB. This completes the whole proof. 1, ..., M and λ 4. Simulation Examples

Figure 1. Digraph for a group of three followers F 1 − F 3 and one leader L

In this section, we present simulation results to prove the feasibility of the proposed distributed adaptive control approach, the following multi-agent systems consist of one leader and three followers. 20

1.2 y1

1

y2

0.8

y3 yd

0.6

r(t),yi

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

0

5

10 Time (sec)

15

20

Figure 2. Output of three followers yi and one leader r(t) (yellow line)

80 u1 60

u2 u3

40 20

ui

0 −20 −40 −60 −80 −100

0

5

10 Time (sec)

15

20

Figure 3. Distributed adaptive controllers ui

Each follower is denoted by the following nonlinear systems with unmodeled dynamics in pure-feedback form:

21

0.1 0.095 0.09 0.085 0.08 ˆ λ 0.075 0.07 0.065 0.06 0.055 0.05

0

5

10 Time (sec)

15

20

ˆ Figure 4. One tuning parameter λ

1.4

z v

1.2

1

z,v

0.8

0.6

0.4

0.2

0

0

5

10 Time (sec)

15

20

Figure 5. Unmodeled dynamics z and a dynamic signal v

Follower 1: ⎧ ⎪ ⎪ x˙ = x21,1 + x1,2 + 0.1 sin(t) ⎪ ⎨ 1,1 x˙ 1,2 = 2x1,1 + x1,1 x1,2 + u1 + 0.1 sin(t) + x1,1 z ⎪ ⎪ ⎪ ⎩ y =x 1

11

22

(62)

3 s1 s2

2.5

s3

si,1

2

1.5

1

0.5

0

0

5

10 Time (sec)

15

20

Figure 6. Consensus errors si,1

Follower 2: ⎧ ⎪ ⎪ x˙ = x2,1 e−0.5x2,1 + x2,2 ⎪ ⎨ 2,1

x˙ 2,2 = x2,1 x22,2 + u2 + 0.1z 2 cos(0.2x2,2 t) ⎪ ⎪ ⎪ ⎩ y =x 2 2,1

Follower 3: ⎧ ⎪ x˙ 3,1 = 2x3,1 sin(x3,1 t)+x3,1 x3,2 + x3,2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 3,2 = x2 + x3,1 x3,2 + x3,2 cos( x3,1 +x2 3,2 ) 3,1 1+x ⎪ ⎪ + u3 + 2z 2 + 0.2 cos(0.5x3,1 t) ⎪ ⎪ ⎪ ⎪ ⎩ y3 = x3,1

1,1

(63)

(64)

where z˙ = −z + x21,1 . The directed communication topology for the simulation is described by Fig.1. According to the graph theory, we choose aij = 1 if j ∈ Ni , aij = 0 otherwise, b = [1, 0, 0]T . The leader single r(t) is r(t) = 0.5sin(t).

23

The distributed adaptive control laws are chosen for i = 1, 2, 3 as follows: αi,2 = +

1 1 ˆ T [−ki,1 si,1 − 2 λs i,1 ξi,1 (Zi,1 ) ξi,1 (Zi,1 )] (di + bi )gi,1 2bi,1 M

1 aij gj,1 xj,2 (di + bi )gi,1

(65)

j=1

ui =

1 gi,ni

[−ki,ni si,ni −

1 ˆ T (Zi,ni ) ξi,ni (Zi,ni )] λsi,ni ξi,n i 2b2i,ni

γξi,l (Zi,l ) ξi,l (Zi,l ) si,l ˆ ˆ˙ = − γσ λ λ 2 2b i,l i=1 M

ni

T

(66)

2

(67)

l=1

with ˙ T ∈ ΩZ1,1 Z1,1 = [x1,1 , x1,2 , s1,1 , r(t)] Z1,2 = [x2,1 , x2,2 , x1,1 , x1,2 , s2,1 , r(t)] ˙ T ∈ ΩZ2,1 Z1,3 = [x3,1 , x3,2 , x1,1 , x1,2 , s3,1 , r(t)] ˙ T ∈ ΩZ3,1 Zi,ni = [xi , si,ni , z˙i,ni , v]T ∈ ΩZi,ni , i = 2, 3 The design parameters of the distributed adaptive controller are taken as γ = 100; σ = 0.05; k1,1 = 10, k2,1 = 10, k3,1 = 150, k1,2 = 50, k2,2 = 5, k3,2 = 1.2; b1,1 = 1, b2,1 = 15, b3,1 = 10, b1,2 = 1, b2,2 = 22.5, b3,2 = 10; gi,l = 1, i = 1, 2, 3, l = 1, 2; The time constants of the first order filter are τ1,2 = τ2,2 = 0.01, τ3,2 = 0.001; The parameters of the RBFNNs are chosen as v1,1j = 0.15j[1, 1, 1]T , j = 1, ..., p1,1 , v1,2j = 0.15j[1, 1, 1, 1, 1]T , j = 1, ..., p1,2 , v2,1j = 0.1 × (j − 10)[1, 1, 1, 1, 1]T , j = 1, ..., p2,1 , v2,2j = 0.5 × (j − 10)[1, 1, 1, 1, 1]T , j = 1, ..., p2,2 , v3,1j = (j − 5)[1, 1, 1, 1, 1]T , j = 1, ..., p3,1 , v3,2j = (j − 5) × 0.5[1, 1, 1, 1, 1]T , j = 1, ..., p3,2 , p1,1 = p1,2 = p2,1 = p2,2 = 20, p3,1 = p3,2 = 9, ki,lj = 1, j = 1, . . . , pi,l , i = 1, 2, 3, l = 1, 2; The possible dynamic single is defined as v˙ = −0.6v+2.5x41,1 +0.625; The initial conditions are set to x1 (0) = [0.1, −0.1]T , x2 (0) = [−0.1, 0.5]T , x3 (0) = [0.5, 0]T , z(0) = 24

ˆ 0.1, v(0) = 0.1, z1,2 (0) = z2,2 (0) = 0.1, z3,2 (0) = 0.5, λ(0) = 0.1. Simulation results are shown in Figures.2-6. Figure 1 shows that only Follower 1 can receive the leader’s output. Figures 2-6 show simulation results by employing virtual control laws (65), actual control laws (66) and only one adaptive law (67). It can be seen that all the followers asymptotically synchronize to the leader, and the tuning law is independent of the number of followers and the order of each follower. 5. Conclusion In this paper, a robust adaptive distributed control approach has been proposed for pure-feedback nonlinear multi-agent systems where each follower is with inherent dynamic disturbances and unmodeled dynamics under a directed network topology. An available dynamic signal is employed to compensate the unmodeled dynamics based on the separation technique and Young’s inequality. RBFNNs are utilized to approximate the unknown continuous functions derived from the distributed tracking controller design procedure. By applying dynamic surface control technique to each adaptive distributed controller, the computational complexity is greatly reduced. Compared with the related results in the literature, only one tuning parameter is adjusted online throughout the design process. Finally, simulation results have been demonstrated the effectiveness of the proposed approach. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant 61374087, the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT13072, and a 25

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