Distributed Output-feedback Tracking for Stochastic Nonlinear Multi-agent Systems with Time-varying Delays

In this paper, we investigate the distributed output-feedback tracking control for stochastic nonlinear multi-agent systems (MASs) with time-varying delays. We propose a new distributed stochastic homogeneous domination method. Specifically, we first design distributed output-feedback controllers for the corresponding nominal systems. Then, by selecting the gains of controllers and observers, we solve the distributed tracking problem for stochastic MASs. After that, based on the coordinates transformation, with a proper Lyapunov-Krasoviskii (L-K) functional, it can be shown that the tracking error can be adjusted to arbitrarily small and all the states of the closed-loop system are bounded in probability. Finally, we give a simulation example to demonstrate the effectiveness of the control scheme.


I. INTRODUCTION
D UE to the ubiquity of stochastic noise in applications, the study on stochastic systems has attracted attention in the fields of biology and environmental science. Since stochastic stabilization theory was proposed by [1], the research on its design and stabilization problem has been made great progress [2]- [6]. [7] studied the finite-time stabilization of the feedforward systems by adding a power integrator and sign function. In addition, stability of timedelay systems is also an important topic [8]- [10]. Based on matrix inequality, the H-index with Markov jump systems and its application in H-fault detection filter (FDF) were studied in [11].
Multi-agent systems (MASs) exist widely in control engineering, such as power and traffic systems [12]- [13]. It has received widespread concern. For nonlinear MASs, by inequality technique, a leader-following consensus problem was considered in [14]. Draw support from neural network (NN) approximation, [15] solved the unknown dynamics problem. [16] adopted a two-layer distributed hierarchical control strategy to deal with systems with unknown and inconsistent control direction. For stochastic nonlinear MASs, [17] investigated consensus of partially mixed impulse time-delay systems by comparison principle. In particular, distributed output tracking is becoming more and more popular. [18]- [19] solved the stochastic distributed output tracking problems by developing a new distributed integrator backstepping method. [20] proposed a distributed hybrid eventtriggered mechanism to save communication resources. But the schemes in [18]- [20] required the all the states of agents are available.
From practical perspectives, the agents' states may not always be known or measurable. For this reason, it is necessary to study output-feedback tracking schemes. For single-agent systems, [21]- [22] first designed homogeneous observers to estimate unmeasurable states, and then the output-feedback controller was designed to solve the tracking problem. For multi-agent systems, [23]- [24] solved distributed outputfeedback tracking problems for nonlinear systems. However, there are rare results on distributed output-feedback tracking control for stochastic systems with time-varying delay.
Inspired by the previous discussion, this paper studies the distributed tracking problem of stochastic nonlinear timedelay MASs by output-feedback. The main contributions include: 1) The systems under investigation is more general than that in [18] and [19]. [18] and [19] did not consider timedelay, and required that all the states are measurable. The existence of time-varying delay makes it difficult to select a proper Lyapunov-Krasoviskii (L-K) functional. The unmeasurable states makes the distributed controllers design more challenging.
2) Due to the influence of the Hessian term and timevarying delay, the distributed homogeneous domination approach developed in [23] is invalid. A new design scheme is developed in this paper.
The remainder of this paper is organized as follows. Section II is for preliminaries and problem formulation. In Section III states the main results. Section IV gives a simulation example. Section V is the conclusion.

A. PRELIMINARIES
Consider the stochastic time-delay system Definition 1 [10]: For any given V (x(t), t) ∈ C 2,1 associated with system (1), the differential operator L is defined as Lemma 1 [22]: For (x, y) ∈ R 2 the inequality holds: where v > 0, the constants p > 1 and q > 1 satisfy (p − 1)(q − 1) = 1. Lemma 2 [10]: Let x 1 , x 2 , · · · , x n , p be positive real numbers, then In this paper, we consider a networkḠ = (V,Ē) including N followers and one leader (labeled by 0). De- if the leader can directly send information to the ith follower, and b i = 0, otherwise. Let the followers' digraph be G = (V, E, A). The Laplacian of G is set as L = diag( j∈N1 a 1j , j∈N2 a 2j , · · · , j∈N N a Nj )−(a ij ) N×N . Define H = B + L. More notations about graph theory can be found in [28].

B. PROBLEM FORMULATION
The followers' dynamics are described by and y i ∈ R are the state, input, output of the ith follower, respectively. d ij (t) : R + → [0, d], j = 1, 2, are time-varying delay.
Assumption 4: Remark 1: When τ i = 0, Assumption 1 reduces to that in [25]. In fact, τ i admits constant disturbances, while µ i andμ i allow diminishing disturbances. τ i is independent of µ i and µ i .

A. NOMINAL MASS ANALYSIS
Consider the following nominal MASs: From Lemma 1 in [18] and Assumption 3, we can conclude H is invertible. Define According to [23], we construct distributed reduced-order observers for (3) aṡ and the distributed output-feedback controller as whereẑ i2 = z i1 + w i2 , α i and c j1 > 0 are constants. For system (3), construct the Lyapunov function as where from (3), (5), and (6), we geṫ Lemma 3 : 1) E(Z) and V (Z) are homogeneous of degree 1 and 3 respectively, with the dilation weight: 2) the derivative of V (Z) satisfieṡ where constant c 0 > 0 and Z = (

Remark 2:
In the corresponding results on deterministic systems [23], it uses the Lyapunov function of the form Due to the existence of 1 2 Tr{g ∂ 2 V ∂x 2 g T } in stochastic differential, the Lyapunov function (11) is invalid for stochastic system.
Let's consider a simple example where y r (t) = sin t.
Obviously, by adjusting the gain L, 1 2 Tr g ∂ 2 V ∂z 2 g T cannot be made arbitrarily small. In this paper, we employ quartic Lyapunov functions.
Proof . From (13) and Lemma 1 in [18], we have From (4) and (18), we then have According to Assumption 2, we get where M > 0 is a constant.

VOLUME 4, 2016
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3187104
2) All the states of the closed-loop system are bounded in probability.
Step 4. From (41) we obtain Let ξ = Z(t) and together with (41) we get By (49) and Definition 1 in [19] , ξ is bounded in probability. Using (43) and Assumption 2, we get y i (t) = z i1 (t) is bounded in probability, and Notice that ξ i1 (t), ξ i2 (t) and z i1 (t) are bounded in probability, by (50), we get z i2 (t) is bounded in probability, i = 1, 2, · · · , N . By (13), we can conclude x i1 (t) and x i2 (t) are bounded in probability, i = 1, 2, · · · , N . Thence, we are able to conclude that all the states of the closed-loop system are bounded in probability. The proof is completed. Remark 3: The parameter ε is an arbitrary positive constant, which is pre-given according to the control objective. In the control scheme, We first set ε, then we design the control (depends on ε). Usually, the smaller requires larger control.
Remark 4: The nonlinear drift terms and nonlinear diffusions terms in (2) make all the existing distributed control methods invalid. To overcome this difficulty, we develop a new distributed stochastic homogeneous domination method.
Specifically, we first focus on the nominal MASs analysis to produce negative terms, then we use these negative terms to dominate nonlinear terms appeared in the distributed outputfeedback control design. The novelty of this approach is that no precise information of the nonlinearities is needed. Thus, this approach provided a new perspective to deal with the distributed output-feedback control problem.

IV. A SIMULATION EXAMPLE
Consider the following MASs. Fig. 1 shows the topology.

V. CONCLUSION
For stochastic nonlinear MASs, a new distributed outputfeedback tracking scheme is proposed in this paper. The MASs simultaneously consider time-varying delays, unmeasurable states, and Hessian terms. We construct a L-K functional to deal with the time-delays terms. Distributed observers and distributed output-feedback controllers are designed to solve the output tracking problem.
In the future work, we will consider generalizing the results in this paper to more general systems such as those in [28]- [30].