Practical output consensus control of uncertain nonlinear multi-agent systems without using the higher-order states of neighbours

Currently, most existing consensus methods require each agent to access the higher-order states of its neighbouring agents, thereby increasing communication and computation costs in a multi-agent system. This work considers the practical consensus problem without such requirements for multi-agent systems in a strict-feedback form. Speciﬁcally, this study addresses the distributed control problem with asymmetric time-varying output constraints under a general communication topology that contains a directed spanning tree. One salient feature of this work is to present a new dynamic reference output for dealing with both the directed topology and the satisfaction of the output constraints. Based on this reference output, a distributed consensus scheme is developed such that each agent utilizes only its states and the output information of its neighbours to update the control input. The proposed method is straightforward to implement because the complex analytic computation of the derivatives of the virtual controllers is not needed. Theoretical and simulation veriﬁcations of the proposed scheme, ensuring that output consensus can be achieved without violation of the constraints and that all closed-loop signals remain bounded, are rigorously studied.


FIGURE 1
Schematic of the single-link direct-drive manipulator management and safety regulations, and their output should be bounded within predetermined conditions. Although there have been various control design methods proposed for individual output-constrained nonlinear systems with several control techniques, e.g. barrier functions [19][20][21], funnel control [22], and prescribed performance control [23][24][25], the development of distributed consensus strategies for output-constrained multi-agents remains quite a challenge because of the increasing design complexity of these techniques, arising from the interacting system dynamics. Recently, the problem of consensus reaching for multi-agents subject to state constraints was considered in [26]. Nevertheless, this seminal result is restricted to simple first-integrator agents. Subsequently, by requiring that the interaction topologies be undirected and connected, consensus solutions were provided for output-constrained nonlinear multi-agent systems [27,28]. By introducing the reference output to address the symmetric output constraint, a distributed consensus controller was developed for the output-constrained multi-agent system with unknown control directions in [29]. Notice that apart from [11,15], all the aforementioned studies related to higher-order multi-agent systems require that the neighbours' higher-order states of each agent be available to it. Such a requirement poses a tremendous challenge in cases where only the outputs of neighbouring agents can be measured. On the other hand, the complex analytic calculation of the derivatives of the virtual control laws was imposed in [16-18, 27, 29], leading to the welldocumented "explosion of complexity" problem and therefore making these proposed methods difficult to implement in practice.
In this work, a higher-order strict-feedback multi-agent subject to asymmetric time-varying output constraints is investigated. The communication topology among agents is a directed graph with a spanning tree. A new dynamic reference output integrated with the output constraints is presented to handle the directed communication topology. With the aid of this reference output, a distributed strategy is designed so that each agent's control input can be derived, invoking only its states and the output information of its neighbours without calculating the derivatives of the virtual controllers. Based on the nature of the directed graph, we show all closed-loop signals are bounded and that the requirements of output constraints are always met. The limit superior of functions and the mean-value theorem are technically introduced to demonstrate that the consensus errors can be adjusted arbitrarily small. Compared with the literature, the main contribution of this work lies in three aspects: (i) Different from the current consensus results [16,18,[26][27][28] that rely on the symmetric positive semidefiniteness of the Laplacian matrix and thereby are limited to the cases of undirected graphs, the designed distributed strategy is able to solve the output consensus problem and the output-constrained control problem under a general directed graph. (ii) Contrary to the investigations presented in [16][17][18], where the high-frequency gains of the agents are assumed to be constant and the uncertain dynamics satisfies the "linearityin-parameters" assumption, in this paper, we deal with a more general system in which both the high-frequency gains and the dynamics are time-varying functions with unknown analytical expressions. More importantly, the time-varying output constraints are not considered in [16][17][18]. (iii) In contrast to the output-constrained consensus control method necessitating the availability of the higherorder states of neighbouring agents [27,29], the solution described herein requires that only the output information Load positions Tracking errors Trajectories of the tracking errors z i,1 for 1 ≤ i ≤ 4 of neighbouring agents be accessible, thereby greatly easing the burden of communication. In addition, the structure of the proposed control solution is more straightforward than that of [27,29] since the explicit calculation of the partial derivatives of the virtual controllers is no longer needed.
The remainder of the article is arranged as follows. Section 2 presents the theoretical background and formulates the issue to be solved. Section 3 presents our new reference output and controller framework. Section 4 provides the simulation results on a set of electromechanical systems, while Section 5 summarizes the results and discusses future developments. Tracking errors Tracking errors Trajectories of the tracking errors z i,3 for 1 ≤ i ≤ 4 Notation: We denote with 1 m and 0 m , respectively, the m−vector of all ones and all zeros, and let I m represent the m−dimensional identity matrix. We use diag{k i } to represent the diagonal matrix with diagonal entries k 1 to k N . For a vector function u(t ), it is said that u ∈  Load velocities Motor current Input voltages Trajectories of the input control voltages u i for 1 ≤ i ≤ 4 with  is defined by a i j > 0 if ( j, i ) ∈  and a i j = 0 otherwise. We suppose that a ii = 0 and the topology is time-invariant. The

Technical lemmas
In the following, we give two technical lemmas that are of use in proving our main results. Consider the initial-value probleṁ

Problem statement
In this work, a multi-agent system consisting of N (N ≥ 2) agents is considered. The dynamics of the ith, i = 1, … , N , agent is described bẏ where n represents the order of the agent system, and m = 1, … , n − 1 is the integer index. u i ∈ R denotes the control input of the ith agent and y i ∈ R is the output.
are uncertain nonlinear functions that include unmodelled system dynamics and disturbances. Furthermore, it is assumed that unknown nonlinearities, i.e. f i, j (x i, j , t ) and g i, j (x i, j , t ), are locally Lipschitz inx i, j and piecewise continuous in t . The problem to be solved in this work is to design a consensus algorithm u i for the ith, i = 1, … , N , agent using only its own states and the neighbouring agent output information so that (i) in the closed-loop system, all signals remain bounded, (ii) all agents can realize practical output consensus, i.e. lim sup t →∞ |y i (t ) − y j (t )| ≤ , ∀1 ≤ i ≠ j ≤ N , and (iii) the output constraints of each agent are never violated, i.e.
Here, is a positive constant that can be adjusted arbitrarily small, and k l (t ) and k u (t ) (satisfying k l (t ) < k u (t )) are known bounded continuous functions with bounded first derivatives and represent the user-defined time-varying output constraints.
To solve the above-stated issue, we make the following hypothesis on the dynamics of the agents.

Assumption 3. The initial outputs of all agents satisfy the constraints
The communication topology of the N agents modelled by  is considered to satisfy the following assumption.

Assumption 4.  has a directed spanning tree.
Remark 1. Assumption 1 indicates that functions |f i,m | and |ḡ i,m | may not increase arbitrarily large due to changes in t . Assumption 1 is reasonable because the time-dependent component of uncertainty can be largely attributed to the external influence of the environment, which has limited energy and therefore is bounded. Assumption 2 suggests that the signs of g i,m , called the control directions of the agent, are known. By simply analysing or identifying the system structure, Assumption 2 can be satisfied in practice. Assumptions 1 and 2 impose a global controllability condition on (2) and are common in the context of strictfeedback systems [32,33]. System (2) satisfying Assumptions 1 and 2 constitutes a significant class of higher-order nonlinear multi-agents that can represent various mechanical systems such as pendulum systems [34], single-link manipulators [35], and jet engine compressors [32]. Hence, the multi-agent systems consisting of multiple practical applications can be described by the agent (2). From now onwards, without losing generality, we

CONSENSUS CONTROL DESIGN AND MAIN RESULT
This section exhibits the control scheme design and stability analysis for the multi-agent system (2) subject to time-varying output constraints. A nonlinear transformed function integrated with the asymmetric output constraints is introduced as where i = (y i − k l (t ))∕(k u (t ) − k l (t )) is designed to convert the original agent system with time-varying output constraints into an equivalent unconstrained system. Also, to realize the purpose of reaching a consensus under the condition of directed communication, we lay out the novel dynamic feedback to yield a reference output for the ith agent aṡ in which i > 0 is a constant and i,0 (0) can be arbitrarily selected.
Remark 2. The key idea behind the design of the reference output (5) is as follows. First, we select s i to overcome the difficulties caused by the time-varying asymmetric output constraint. In other words, it can be seen from the proof of Theorem 1 that maintaining the boundedness of s i (t ), i = 1, … , N is equivalent to assuring k l (t ) < y i (t ) < k u (t ) for all t ≥ 0. Second, we design the reference output dynamics to deal with directed communication topology conditions.
We propose the distributed control input u i of the ith agent inspired by the backstepping technique and the prescribed performance design technique [23,24], which enables the transformed output s i to converge towards the reference output i,0 . To start the development, the tracking error variables are introduced for the ith agent as follows where i,m−1 can be considered as the intermediate control signals to be designed below. The intermediate control signals i,m , m = 1, … , n − 1 and the actual controller u i for the ith agent are designed as Remark 3. Within the framework of the prescribed performance technique, some research efforts have emerged to approach distributed leader-following issues of multi-agent systems; see, for instance, [3, 8 24, 36]. Nevertheless, these results are built on the basis of the attributes that the related matrices (Laplacian matrices plus diagonal matrices) for the leader-following issues are positive stable and fail to solve the consensus problem considered herein, as the related matrices (Laplacian matrices) are merely semi-positive stable. This is the first time to address the (leaderless) consensus issue of the multi-agent systems subject to output constraints relying on the prescribed performance technique under directed graphs, up to the best of the authors' knowledge.
The subsequent theorem proves the use of our proposed controller to solve the problem described in Subsection 2.3.

Theorem 1.
Consider the nonlinear multi-agent system of N agents (2) and suppose that Assumptions 1-4 hold. Then the application of the proposed distributed control (7) ensures that (i) in the closed-loop system, each signal remains bounded, (ii) the output constraints of each agent are never violated, i.e. k l (t ) < y i (t ) < k u (t ), ∀t ≥ 0, ∀i = 1, … , N , and (iii) all agents can achieve practical output consensus.
Proof. The state variables of the ith, i = 1, … , N , agent in (2) can be expressed as functions of i,m , i , and t as where k e (t ) = k u (t ) − k l (t ) and m = 2, … , n. Notice that the nonlinearities g i, and f i, are the functions ofx i, and t , thus they can be presented as , t ), where = 1, … , n. For ease of notation, the arguments of the g i, , f i, −functions will be dropped whenever no confusion would arise. The time derivatives of i, and i are given bẏ −k e (t ) ) where Owing to the introduction of time-varying constraints and the time-varying system nonlinearities, the dynamics in (10) is non-autonomous, which complicates the stability analysis because the solution of the non-autonomous system (10) is dependent on both t and 0. To cope with this situation, we prove Theorem 1 according to the following two phases. We define the nonempty and open set In Phase A, the existence and uniqueness of a maximal solution of (10) over the set Ω for a time interval [0, t f ) is ensured, i.e. (t ) ∈ Ω, ∀t ∈ [0, t f ). Subsequently, in Phase B, we will demonstrate that our algorithm assures for all t ∈ [0, t f ) that (i) in the closed-loop system, every signal is bounded and (ii) (t ) stays within a compact subset of Ω.  (10), over the set Ω the map f of the resulting dynamical systeṁ= f ( , t ) is locally Lipschitz in , and locally integrable and continuous on t for each fixed . In the light of Lemma 2, the existence and uniqueness of a maximal solution of (10) over the set Ω on [0, t f ) is obtained.
Phase B: From Phase A, it is ensured that (t ) ∈ Ω for all t ∈ [0, t f ). Next, we will show that the reference outputs i,0 (t ) are bounded on [0, t f ). Integrating both sides of (10) with respect to t gives the solution where (t ) = ∫ For the exponential matrix e −Lt , we can obtain that is the Jordan block associated with the nonzero eigenvalues ofL, and It is noted that J N −1 is a Hurwitz matrix. Then, for some positive k 0 and 0 , we are able to show that By (12) and the fact P 1 1 ∈  ∞ [0, t f ), we obtain that ∈  ∞ [0, t f ). As a result, we get that 0 ∈  ∞ [0, t f ). Next, we use a recursive stepwise process to validate each closed-loop signal's boundedness and the fact that (t ) evolves in a compact subset of Ω, ∀t ∈ [0, t f ).
Step 1: Now, consider the following Lyapunov function V i,1 = 2 i,1 ∕2, where i = 1, … , N . Its time derivative along (10) yields for all t ∈ [0, t f ) where l i ∈ R 1×N is the ith row of the matrixL. , andḡ i,1 , there exist positive con- . For all t ∈ [0, t f ), the use of the aforementioned analysis and (7) giveṡ . By the Cauchy-Schwarz inequality and i (t ) < 1 on [0, t f ), it can be achieved that for all t ∈ [0, t f ),

Consequently, the intermediate control signal i,1 ( i,1 (t )) is bounded on [0, t f ). Taking the inverse logarithmic function in
Utilizing (13), the boundedness of i,1 (t ), the fact 0 ∈  ∞ [0, t f ), and s i,1 = i,1 (t ) i,1 + i,0 , we have that |s i,1 (t )| ≤ s * i,1 for a positive constant s * i,1 , ∀t ∈ [0, t f ). Hence, we conclude from (4) that where (10) and the bound- Step where i, j = i, j +1 (t ) i, j +1 with j = 2, … , n − 1 and i,n = 0. Note that i (t ), k e (t ), k l (t ), i,1 (t ), … , i,m (t ), i,1 (t ), … , i,m (t ), and i,1 ( i,1 (t )), … , i,m−1 ( i,m−1 (t )) remain bounded on [0, t f ). Thus, employing Assumption 1 and the continuity off i,m , g i,m , andḡ i,m , there exist pos- for all t ∈ [0, t f ). By virtue of the aforementioned analysis and recalling (7), we obtain that for all t ∈ [0, t f ) Accordingly, the intermediate control signal i,m ( i,m (t )) and u i ( i,n (t )) are bounded on [0, t f ). By taking the inverse logarithmic function, we obtain that Next, the fact that t f can be extended to ∞ will be demonstrated. Notice by (13), (14), (16) where Ω * is a nonempty and compact subset of Ω. Based on Lemma 3, the solution is global, i.e. t f = ∞. Based on (9), it is obvious that x i,1 (t ), … , x i,n (t ) are bounded on [0, ∞). Furthermore, by employing the definition of i and 0 < i (t ) < 1, we have which guarantees the satisfaction of the output constraints during the operation. It remains to show that output consensus can be achieved. Taking into account (12) and (13), we get wherek Noting that e −Lt Introducing the following vector e = [ 1, for all 1 ≤ i ≠ j ≤ N . By (13) Notice from (6) that Therefore, we can get from (21), (22), and (23) that lim sup t →∞ |s i (t ) − s j (t )| ≤ i, j , where i, j = 2(1 + √ N )k 0̄∞ ∕ 0 +̄i i,1,∞ +̄j j,1,∞ . According to the mean-value theorem, we can conclude that for all t ≥ 0 with h i, j ∈ (min{y i , y j }, max{y i , y j }). Since k l , k u ∈  ∞ [0, ∞) and k l (t ) < k u (t ), there exists a positive constant k * e satisfying 0 < k e (t ) ≤ k * e for all t ≥ 0. Finally, we arrive at , which implies for arbitrarily small > 0, there exists T i, j > 0 such that |y i (t ) − y j (t )| < (k * e i, j ∕4) + , ∀t > T i, j . Note that̄∞ = max{ m,1,∞ } and m,1,∞ , m = 1, … , N are positive design parameters that are independent ofk 0 , 0 and̄m. Thus, the residual errors i, j can be reduced arbitrarily small by adjusting m,1,∞ . This together with the arbitrariness of shows that the consensus error |y i − y j | can also be made arbitrarily small by choosing m,1,∞ . The proof is complete. □ Remark 4. Theorem 1 shows that our proposed method can achieve output consensus in the presence of system uncertainties. In fact, system uncertainties are dealt with by the prescribed performance technique. It can be observed from the proof of Theorem 1 that system uncertainties affect only the magnitude of * i,m via d * i,m . However, inequalities (13) and (16) hold no matter how large the finite bound * i,m is, leaving unaltered the established stability properties.
Remark 5. In this paper, a new dynamic reference output is proposed for dealing with both the directed topology and the satisfaction of the output constraints. Similar efforts can be founded in the significant work [38], in which an optimal signal generator is presented to achieve the optimal output consensus. The main difference between our reference output and the optimal signal generator is that the key elements and the construction of their dynamic equations are different due to the different requirements on the agent output. That is, in [38], the gradient on the optimal signal generator is used, while the nonlinear transformed function integrated with the asymmetric output constraints (4) is employed in our algorithm.
Remark 6. This paper discusses the distributed control problem with asymmetric time-varying output constraints. It should be noted that due to uncertain technical difficulties, the output bound of each agent in this paper is required to be the same at each moment. How to relax this requirement is an interesting open question.

SIMULATION STUDY
To exhibit our control method's application, we consider a set of four single-link manipulators driven by permanent magnet brush direct current motors, as shown in Figure 1. The ith, i = 1, 2, 3, 4, manipulator is governed by the following dynam-ics [35] where , J i represents the rotor inertia, M i,o denotes the load mass, m i is the link mass, L i,o denotes the length of the link, g is the standard gravitational acceleration, R i,o is the load radius, B i,o is the coefficient of viscous friction at the joint, q i denotes the position of the load, I i demotes the motor armature current, K i, demotes the coefficient characterizing the electromechanical conversion of armature current to torque, L i is the armature inductance, R i is the armature resistance, K i,b is the counter-electromotive force coefficient, V i,e is the input control voltage, d i,1 and d i,2 denote time-varying disturbance terms. The simulation parameters are given in Table 1, and all of them are not available for designing the controller. The control goal is to design the distributed control voltage V i,e to enable all the manipulators' load positions q i to realize consensus subject to asymmetric position constraints  Figure 3, from which the consensus has been satisfactorily achieved and the requirement on the position constraints of the manipulators is satisfied all of the time, despite the presence of uncertain nonlinearities f i,m and g i,m , m = 2, 3. Figures 4-6 present the evolution of the tracking errors z i,1 , z i,2 , and z i,3 defined in (6) along with their corresponding performance bounds. Moreover, the boundedness of the load velocities x i,2 and the motor current x i, 3 can be observed in Figures 7  and 8, respectively. Figure 9 shows the input voltages u i of the four manipulators. From these results, we can see that the presented distributed method is able to achieve the consensus control of the manipulators without the violation of the position constraints, as stated in Theorem 1.

CONCLUSION
In the present paper, a general framework has been established to control nonlinear multi-agent systems with asymmetric time-varying output constraints under general communication topologies that have a directed spanning tree. Compared to the relevant results in the literature, our proposed control algorithm requires minimum information from neighbouring agents, namely, their output information, so that practical output consensus can be realized at remarkably low communication costs. Furthermore, this solution is more straightforward to perform because there is no need to explicitly calculate the partial derivatives of the intermediate control signals. Simulation results on electromechanical systems have been provided to demonstrate the feasibility of implementing the algorithm. Further research is currently underway to extend the approach to the situation of switching communication topologies.