Distributed Control of Nonlinear Systems With Unknown Time-Varying Control Coefficients: A Novel Nussbaum Function Approach

In this article, distributed control of uncertain multiagent systems (MAS) with completely unknown nonlinearities, unknown time-varying control coefficients, and multiple unknown control directions is investigated. First, a new theorem with a series of novel Nussbaum functions are presented, which solve the outstanding problem in control of nonlinear systems with unknown time-varying control coefficients and unknown signs. Second, global consensus of MAS with unknown time-varying control coefficients and unknown system nonlinearities under directed graph is first achieved, and the transient tracking performance is also guaranteed. Third, a novel filter is proposed for each agent which does not require a prior knowledge of time derivative of leader. Finally, simulation results show the effectiveness of the proposed control schemes.


I. INTRODUCTION
I N CONTROL systems, when the sign of the control coefficients is unknown, the Nussbaum function N (ξ) is commonly applied as a solid control tool since it can alternatively change its sign to find the right control direction as ξ varies [1], [2], [3], [4], [5], [6], [7], [8], [9]. A fundamental lemma given in [10] which guarantees the boundedness of a Lyapunov-like energy function with a Nussbaum function is used in many papers to deal with control systems with unknown control directions, where the control coefficients are constants; see [1], [2], and many other similar works. In addition, different Nussbaum functions are also proposed in [1], [6], and [7] to handle more complicated cases. For example, when multiple control inputs with different unknown control coefficients are considered, such as multiagent systems or interconnected large-scale systems, new Nussbaum functions are designed in [7] to make the summation of multiple Nussbaum functions applicable to guarantee boundedness of the closed-loop system. However, when the control coefficients are totally unknown and time-varying, the control problem will be totally different and much more difficult. The stability analysis procedure in [10], which is for the case of unknown constant control coefficients with unknown signs, could not be used for the time-varying case. It could be observed that many existing results apply the common Nussbaum functions and stability analysis designed for constant control coefficients directly to the cases of unknown and time varying coefficients. A counter-example will be given later to show that the signals in closed-loop system may be unbounded with a special type of coefficient. Although [12] tries to solve this problem, but it needs the assumption that ξ > 0 andξ ≥ 0. This assumption makes the control scheme only applicable for some special types of systems, such as linear systems. Therefore, this is still an outstanding problem which deserves further investigation.
For the past decades, the consensus problems of MAS have attracted more and more attention in the control community. As a promising tool for uncertain nonlinear control problem [13], [14], [15], adaptive backstepping method is a typical approach for solving the consensus problems of uncertain high-order MAS, to name a few, [15], [16], [17], [18]. Though many remarkable results are obtained for linear MAS [15], [16], [16], [17], [18], [19], [20], practical systems are always inherently nonlinear and are also more difficult than the linear cases. For nonlinear MAS with matched conditions that nonlinear dynamics or uncertainties are only in the presence of the last state subsystems where control inputs exist, Wang et al. [21] designed an asymptotic consensus method with newly constructed Nussbaum functions. Bechlioulis  a model-free prescribed performance controller using barrier functions. Su [20] solved the problem that uncertain system parameters belong to an unknown and noncompact set. Zou et al. [23] investigated the case that system nonlinearities are switched and Liu and Huang [24] investigated the case that the communication topology is switched. For nonlinear MAS with unmatched uncertainties or coupled variable terms, fruitful results have also been obtained as shown in [25], [26], [27], [28]. To deal with unmatched linearly parameterized nonlinear terms, the authors in [26] and [27] introduced local compensators to estimate the unknown information of the leader. Using a high-gain observer, Zhang et al. [28] dealt with the unmatched uncertainties which satisfy Lipschitz conditions. Under the condition that the growths of system functions are bounded by some known functions, Hua et al. [25] developed a finite-time consensus control method. Nevertheless, these methods require the uncertain nonlinearities of systems to be bounded by some known functions or to be linearly parameterized, which means partial knowledge of these uncertain terms are required.
For MAS with completely unknown nonlinearities and constant control gains, semiglobal consensus control results are obtained in [29] and [30] by using neural networks (NN) as universal approximators. An adaptive fuzzy approximators-based control method is proposed in [31] such that system nonlinearities and control gains are allowed to be unknown functions, while only semiglobal consensus control can be achieved owing to fuzzy approximators. Similar results are also obtained in [32] and [33] by using fuzzy or NN approximators. It is worth pointing out that, for MAS under directed graph, some filters are constructed in [34] and [35], in which global consensus are achieved with unknown or uncertain system nonlinearities, but their control gains are constants in [34] and [35]. To guarantee the transient performance of consensus errors, a prespecified performance control method is proposed in [29]. So far, though remarkable progress on MAS has been made in literature [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] and so on, it is worth mentioning that, let alone the unknown control directions problem discussed previously, there are still some outstanding problems left for the consensus of MAS. For example, how to achieve global consensus control of MAS with unknown system nonlinearities, time-varying control gains, and directed graph? How to achieve a global control scheme without using the initial conditions of the agents? How to design filters of MAS without the prior knowledge of the time derivatives of the leader? In this article, we aim to investigate the global leader-following consensus of MAS with completely unknown and unmatched uncertainties, in which the signs of the time-varying unknown control coefficients are also unknown. The control objective is to achieve a global consensus, meanwhile guaranteeing the transient performance of the consensus errors, without setting initial conditions for barrier functions or using the bounds of time derivatives of leader's information. It is worth emphasizing that all the existing control techniques cannot achieve such an objective.
In this article, new type of Nussbaum functions are designed and a new paradigm of stability analysis is given to show the boundedness of the Lyapunov-like energy function with multiple Nussbaum functions. In addition, since it is proposed for the MAS, the summation of multiple Nussbaum functions is also applicable. The main contributions are summarized as follows.
1) A series of novel Nussbaum functions are proposed, which solves the outstanding problem in control of nonlinear systems with multiunknown time-varying control coefficients. The signs of these coefficients are allowed to be unknown, and their bounds are assumed to be unknown time-varying functions rather than constants. Stability analysis of the closed-loop system with a Lyapunov-like energy function is established. 2) Global consensus for MAS with unknown time-varying control coefficients and unknown system nonlinearities under directed graph is first achieved in this article, by combining the output of the distributed filters for each agent. 3) Different from existing filter-based consensus control techniques, i.e., [34], [35] and so on, in which the bounds of the time derivatives of leader's output are assumed to be known and used in the filter design, in this article a novel type of distributed filters is proposed to estimate the desired output signals for MAS. The novel filters do not require a prior knowledge of time derivative of the leader.

II. PROBLEM STATEMENT AND PRELIMINARIES
Consider a class of uncertain multiagent systems as follows: are the states, the control input and the output of the ith subsystem, respectively. The system nonlinearities f i,m (·), g i,m (·) : R + × R m → R are continuous functions for all t andx i,m .
The desired trajectory y d for the outputs of the subsystems is bounded and only known by part of the N subsystems, as well asẏ d , whlieÿ d is bounded and unknown to all subsystems.
Suppose that the information transmission condition among the group of N subsystems can be represented by a directed graph G . ., N} denotes the set of indexes corresponding to each subsystem. The edge (i, j) ∈ E indicates that subsystem j could obtain information from subsystem i, but not necessarily vice versa. In this case, subsystem j is called neighbor of subsystem i, and vice versa. Denote the set of neighbors for subsystem i as N i Introduce an in-degree matrix Δ such that Δ = diag(Δ i ) ∈ R N ×N with Δ i = j∈N i a ij being the ith row sum of A. Then, the Laplacian matrix of L is defined as L = Δ − A. Define B = diag{μ 1 , μ 2 , . . ., μ N }, where μ i = 1 means the y d is accessible directly by subsystem i, and otherwise, we have μ i = 0.

Assumption 1:
The directed graph G contains a spanning tree, and the desired trajectory y d is accessible to at least one subsystem, i.e., N i=1 μ i > 0. Assumption 2: There exist continuous positive un- where (4) implies the signs of g i,n (t,x i,n ) are unknown. Lemma 1 [35]: Based on Assumption 1, the matrix (L + B) is nonsingular, and there exists an diagonal matrix P = diag{P 1 , . . ., P N } with constants P i > 0, i = 1, . . ., N, such that Q = P (L + B) + (L + B) T P is positive definite.
The objective of this article is to develop a fully distributed control method for system (1) under Assumptions 1-2, such that transient tracking performance can be guaranteed, while the consensus error could be made arbitrarily small. At the same time, the initial conditions of the states are not required in control design. First we will show that the traditional Nussbuam functions method could not solve the problem we considered in this article, although they have been used extensively in literatures. Then a series of new Nussbaum functions will be proposed with their variables being designed to be monotonic, which can not only be applied for a single control system but also be used for the multiagent systems.
Remark 1: In existing methods, the bounds for g i,m (t,x i,m ) are constants and the bounds for f i,m (t,x i,m ) are always known functions multiplying unknown constants in [5], [27], [35], and so on. However, in this article, the bounds of g i,m (t,x i,m ) and f i,m (t,x i,m ) are all unknown functions. The unknown functions will make the control design much more difficult. For example, as shown in (47), we have to design a compact set to enclose all consensus errors, and finally to prove that the compact set is an invariant set. If the bounds are constants, there is no need to do so.
Remark 2: For Assumption 2, similar assumptions could be found in [36] and [38]. In the model of the control system (1), model uncertainties, external disturbances, unknown system parameters, etc., may appear in unknown system functions f i,m (t,x i,m ) and unknown control gains g i,m (t,x i,m ), which may be rewritten as f i,m (d(t),x i,m ) and g i,m (d(t),x i,m ). Unbounded d(t) may cause that (1) could not be controlled. To rule out this case, we invoke Assumption 2.
As for being any unknown function of the system states.
is, and therefore, Assumption 2 is satisfied. In this case, it can be seen that the unknown bounds of f i,m (t,x i,m ) exist and we do not require to know what its bound is to verify Assumption 2.
Similar analysis can be made for f i,m (t,x i,m ) containing other bounded time-varying variables.
Remark 3: The control directions, namely, the signs of g i,n (t,x i,n ), are unknown, which must be treated with little knowledge of nonlinearities. The control gain functions g i,n (t,x i,n ) are not constraint by any constant, which makes the methods in [1], [6], [7], and [21] inapplicable. New Nussbaum functions should be proposed and additional technique should be incorporated to deal with this problem.

III. PROBLEMS IN EXISTING NUSSBAUM FUNCTIONS-BASED METHODS
We now show that when dealing with unknown time-varying coefficients with unknown signs, the main conclusion related to the existing Nussbaum functions does not hold, motivating us to propose a novel Nussbaum function together with a skillful design of its variable to address this issue in the next section.
The following lemma has been commonly used in dealing with control problems with unknown control directions, where N (·) is a Nussbaum function and g(·) is an unknown time-varying coefficient. Normally N (·) is chosen as N (x) = x 2 sin(x) or N (x) = e x cos(x) and so on.
Remark 5: For traditional Nussbaum functions-based stability analysis, normally a contradiction is invoked to show the boundedness of the closed-loop system by assuming ζ approaches infinity, which will lead V (t) approaching −∞. Therefore, a contradiction is invoked in this case. However, when the control coefficients are unknown and time-varying, as shown in our Justification, V g could still approach +∞ even though ζ is bounded. Thus Lemma 2, which has been extensively used for a similar problem formulation, is actually not correct. This problem will be solved in the next section, where a new type of Nussbaum function and a skillful stability analysis paradigm will be proposed.

IV. NOVEL RESULTS WITH MULTI-NUSSBAUM FUNCTIONS
In this section, to deal with the problem raised in the previous section, we construct a series of novel Nussbaum functions, which can not only deal with unknown time-varying control coefficients but also can deal with multiple unknown timevarying control coefficients. Therefore, the proposed Nussbaum functions could be applied to distributed control case, such as multiagent systems. The new Nussbaum functions are designed as follows: with ω being any positive constant, and i = 1, 2, . . ., N. For N i (ζ), first, we have the following Lemma, which is critical in establishing the main conclusion of the Nussbaum functions. and where k = 1, 2, . . ., . ., N represent the signs of unknown control gains or system functions, i.e., b i = 1 or b i = −1, and δ is a positive constant defined in (21).
Proof: From the definition of ε 1 , it is easy to verify that 1 ≤ ε 1 < 3. Define where where is a positive integer. Moreover, it can be easily verified that k i is an odd integer when b i = 1, and k i is an even integer when b i = −1, which implies b i sin(k i π + π/2) = −1. Using the similar analysis as (16), we also have Furthermore, the length of interval of Ω i is 2 1−i πω −1 , and (16), (17), and i sin(k i π + π/2) = −e ζ 2 i < 0 by noting k i is an odd integer when b i = 1 and k i is an even integer when b i = −1, based on the previous analysis, we Noting (12) and the property of function sin(·), we can obtain that It follows from (18) and (19) that for ∀ζ ∈ [ζ m,N ,ζ M,N ], with δ > 0 defined as Therefore, from (20) and (21), we know (13) holds. This completes the proof.
With Lemma 3, we have the following main results.
are unbounded. To show the boundedness of ζ i (t), we only need to prove the boundedness ofζ Max (t), which will be achieved by seeking a contradiction. Suppose thatζ Max (t) is unbounded. Then, there must exist a monotonously increasing sequence {t l }, l = l 0 , l 0 + 1, l 0 + 2, . . ., such thatζ Max (t l ) = πω −1 (2l + ε 1 ) + ε 3 , where l 0 is an arbitrary positive integer satisfying πω −1 (2l Using (22) and (23), we have , and define compact sets It is easily verified that M j (t l ) is an even function with respect toζ i (t l ). Therefore, we only consider the case whereζ i (t l ) > 0.
For |ζ i (t l )| < K l , we have with positive constants G M and C i,j , j = 1, 2, 3 defined as It follows from (12) where C i,2 are defined as (33). It follows from (13) with positive constants C i,3 defined in (33), and G m = min i∈J {|G − i |, |G + i |} > 0. Substituting (29) into (25) and noting (31), (34), and (35), we have Noting that where It is obviously that e 2ε 2 π ω (2l+ε 1 )+ε 2 2 grows much faster than K l since K l is a linear function with respect to l. Therefore, from (38), we know that V (t l ) → −∞ as l → +∞, which leads to a contradiction since V (t l ) is a positive Lyapunov function. Thus, we know ζ Max (t) is bounded, which impliesζ i (t) and hence ζ i (t) are bounded. Furthermore, bounded. This completes the proof.
Remark 6: Compared with [1], [6], [21], Theorem 1, as a new result, is presented for control design with multiple unknown time-varying control gains. N i (ζ i ) has an arbitrary positive frequency ω, which means the selection of parameters of N i (ζ i ) is more free than others. Moreover, compared with [6], choosing parameters of N i (ζ i ) do not require any knowledge of system parameters. Compared with [21], N i (ζ i ) can deal with the case that control gains are time-varying.

Remark 7:
The key for solving the problem raised in Section III is that the variables of Nussbaum functions, i.e., ζ i (·) in Theorem 1, should be monotonically increasing or decreasing, so that the value changes for ζ, as shown in (8) can be avoided. Therefore, novel Nussbaum functions are constructed as (11), with ζ i being designed as monotonic functions in this article. The monotonicity of ζ for the novel Nussbaum functions is critical, which avoids repeatedly jumping of ζ similar with (8), and therefore, M in (10) will be eliminated and the existing problems will not happen.
Define J 0 as an arbitrary subset of J, then, similarly, we have the following results.
Theorem 2 suggests the results still hold in case of single Nussbaum function when J 0 is a set including a single element, which is commonly for single-input-single-output systems.

V. DESIGN OF DISTRIBUTED ADAPTIVE CONTROLLER
In this section, the distributed tracking control design for multiagent systems (1) will be designed based on Nussbaum functions proposed previously. To facilitate the control design distributedly, first, design a filter (q i,1 , q i,2 ) for each agent i, i = 1, . . ., N.

A. Filter Design
Design a filter for ith agent as with where and c 0 , c 1 , c 2 , c 3 , and T s are positive design parameters satisfying 0 < c 2 < 1. Then, we have the following lemma.
Lemma 4: Consider the closed-loop system consisting of N filters (40) satisfying Assumption 1 with local controller (41). Then, q i,1 and q i,2 are bounded for ∀t ≥ 0, and the outputs of the filters satisfy |q i,

and Y (t) is defined as (100).
Proof: See the Appendix. Remark 8: It can be observed from the filters that the knowledge on bounds ofẏ d and y (2) d are no longer required, which is different from all filter-based consensus control schemes in literature. Therefore, the control scheme is more robust than the existing ones since network transmission of the bounds of time derivatives of y d (t) is no longer required and the calculation on the derivatives may amplify the noise in practice.

B. Design of Adaptive Distributed Controller
Introduce the following error variables and change of coordinates: with the adaptive parameters ξ i updated bẏ where λ i,m , γ i , and i are the positive design parameters for i = 1, . . ., N, and N i (·), i = 1, 2, . . ., N are Nussbaum function which are continuous differentiable defined in (11). Remark 9: It can be observed from (43)-(47) that, the process of setting initial conditions, which is required in all the existing barrier functions-based controller [29], [36], and [38], is no longer needed in this article, since we have introduced a arctan(·) to the barrier functions-based controller first in this article.

VI. STABILITY ANALYSIS
We are at the position of giving the main results of leaderfollowing consensus control with guaranteed transient performance.
Theorem 3: Consider the closed-loop system consisting of N uncertain nonlinear subsystems (1) satisfying Assumption 1-2, the intermediate control signals (45), the distributed controller (46), and the adaptive parameters update function (47). Then, we have the following properties.
1) All the signals in the closed-loop system are globally uniformly bounded.
It can be easily shown that e i (0) ∈ Ω 0 ∀i ∈ J. Define e max (t) = max i∈J,m∈P {|e i,m (t)|}. There are two cases for e max (t) to be considered. Case 1) e max (t) < M 0 holds for ∀t ≥ 0, then e i (t) ∈ Ω 0 for ∀t ≥ 0. Case 2) There exists a t M such that t M = inf{t|e max (t) = M 0 , t ≥ 0}, where t M > 0 represents the first time that e max (t) reaches at M 0 . In this case, the proposition e i (t) ∈ Ω 0 naturally holds for 0 ≤ t ≤ t M . Therefore, it is only required to prove that e i (t) ∈ Ω 0 for ∀t ∈ [t M , +∞).
It should be noted that, for ∀t ∈ [t M , +∞), we have which will be useful in the following analysis.
Define the open set for i ∈ J and m ∈ P . In the following part, we will prove that e i (t) ∈ Ω 0 for ∀t ∈ [t M , +∞) based on (49) and (51).
From (43), we have Noting (44) and (45), we have It can be observed from (52) that x i,1 is a continuous function of e i,1 , q i,1 , and k i,1 , where q i,1 and k i,1 are time-varying bounded functions. Similar analysis can be made for x i,2 and x i,m .
Step j (2 ≤ j ≤ n − 1): Consider the following positive definite functions Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Similar as the previous step, it follows from (1), (43), (44), (52), and (53) that the time derivative of V i,j iṡ with Noting that x i,m , m = 1, 2, . . ., j are bounded on Ω e in view of (49), (51) and the fact that α i,m−1 , e i,j , and e i,j+1 are bounded on Ω e . Since k i,j+1 , k i,j ,k i,j , q i,1 , q i,2 are bounded, employing the extreme value theorem owing to the continuity of f * i,j (·), g * i,j (·) and g i,j (·), we arrive at where c j,1 , c j,2 , and c j,3 are some positive constants. Then, substituting (68) and (69) into (65) yieldṡ It follows from (70) thatV i,j ≤ 0 when |tan e i,j | ≥ c j,3 /λ i,j c j,1 , which implies From (71) we have that As a result, the control signal α i,j is bounded. Moreover, invoking (53) for m = j + 1, we also can obtain the boundedness of x i,j+1 . Furthermore, take time-derivative of α i,j yieldṡ Therefore, it also can be easy to conclude the boundedness oḟ α i,j .
Step n: Consider the following Lyapunov functions: Similar as the previous steps, we havė Similarly, employing the extreme value theorem owing to the continuity of f * i,n (·), g * i,n (·), and g i,n (·), we arrive at with c n,1 , c n,2 , and c n,3 being some positive constants. Then, substituting (78) into (75) yieldṡ where G i,n (t) = is strictly positive or negative and is bounded by noticing (77).
Divide the set J into two parts, i.e., Then, for ∀i ∈ J 1 , by noting (47), it can be obtained thaṫ ξ i = 0, and thus, On the other hand, for ∀i ∈ J 2 , it follows from (46) and (47) thatV where Substituting (83) into (82) yieldṡ Integrating (84) over [0, t], we have (85) Notice that J 2 is a subset of J and G i,n (t) is strictly positive or negative and is bounded, then the conditions of Theorem 2 are satisfied, and therefore, V N 2 (t), ξ i (t) are bounded by using (85) and Theorem 2. Thus, we have V N 2 (t) ≤ C M with C M being a positive constant.
Noting that (81) holds for ∀i ∈ J 1 and V N 2 (t) ≤ C M for ∀i ∈ J 2 , thus we have and therefore, for ∀i ∈ J. Then, the boundedness of u i can be obtained by noting (46), (87) and the boundedness of ξ i (t). Furthermore, (62), (72), and (87) imply that Invoking that e i (t) ∈ Ω 0 holds for 0 ≤ t ≤ t M , it follows from (88) that which means Ω 0 is an invariant set for e i (t). Hence, all closed loop signals are global bounded. Moreover, it can be concluded from (62) that for ∀t ≥ 0. Noting (90) and 0 < k i,1 < 1, we have Using Lemma 4 and (91), one obtains which implies Property 2) holds. This completes the proof. Remark 10: It can be observed that, combining with the novel filters, the global consensus for MAS with f i,m (t,x i,m ) and g i,m (t,x i,m ) being unknown functions under directed graph is first achieved in this article, which suggests that our control scheme is more robust. It is worth mentioning that, although g i,1 (·) are allowed to be time-varying functions for the MAS with directed graph by using filters designed in [34], [35], and so on, all filters designed in these works require the bounds oḟ y d , y (2) d and so on, and none of these works consider the global consensus control under unknown time-varying control gains.
To further show the control performance under unknown different control directions, we change the control gain functions to be g 3,2 = −1 and g 4,2 = −1, while all the other parameters, system functions and conditions are not changed. Then, the simulation results are reported as in Figs. 6 and 7.
It can be seen in Figs. 6 and 7 that, though the control gain functions are changed without modifying the controllers, the tracking performance is still very well, and fairly good control performance is achieved.

VIII. CONCLUSION
This article designs a novel distributed adaptive consensus method for nonlinear MAS with unknown control directions and dynamic system nonlinearities by constructing new Nussbaum functions and combining them with barrier functions. A novel stability result is present for the control problem with multiunknown control directions based on the constructed Nussbaum functions. With this stability result, the control design is achieved for nonlinear multiagent system with unknown nonidentical control directions, and the stability of whole controlled system is established. By virtue of using barrier functions, little knowledge of system nonlinearities is required for control design in the sense that the bound functions for system dynamic are completely unknown. Simulation results illustrate the effectiveness of the proposed scheme.

ACKNOWLEDGMENT
The author Zongcheng Liu would like to thank Prof. Xinmin Dong, who was a professor with the department of flight control and electronic engineering in Air Force Engineering University. Prof. Dong was the supervisor of Liu and gave great help and valuable advice on Liu's researches. The authors would also like to thank Charalampos P. Bechlioulis, who is an Associate Professor with the department of electrical and computer engineering at the University of Patras, for his valuable suggestions, and thank the Editor, Associate Editor, and anonymous Reviewers for their constructive comments, which helped to improve this article considerably.