Logic-Based Switching Mechanism for Multiple Parametric-Strict-Feedback Systems With Unknown Control Directions

In this paper, we consider the cooperative control of multiple parametric-strict-feedback (PSF) systems with unknown control directions and uncertain parameters. A logic-based switching mechanism which depends on the incremental errors of the analysis function between two consecutive switching moments is proposed such that the unknown control direction can be matched. Then, asymptotic output synchronization is realized by recursively designed controllers in which the states of agent $i$ and its neighbors are incorporated, meanwhile, parametric uncertainties are compensated via parameter estimators, which are all tuned online in a distributed manner. Compared with the previous works, the algorithm proposed in this paper has fewer potential switches and better transient performance. Simulation results are presented to verify the effectiveness of the proposed algorithm.


I. INTRODUCTION
A great deal of work has been done for the class of systems in parameter-strict-feedback(PSF) form ever since backstepping design has been proposed in [1], [2]. Early works have mainly focused on the stabilization or regulation of such system, see [3]- [6] and many references therein. It is worth noting that the control direction or the sign of the high-frequency gain determines the characteristics of feedback, negative feedback, or positive feedback, which is important in controller design. Usually, it is assumed to be known and positive. However, this assumption becomes unrealistic in some cases, such as uncalibrated visual servo control in [7] and autopilot design of surface vehicles [8]. In these cases, the Nussbaum function has been proposed to solve the problem of stabilization or regulation for a single system with an unknown control direction [9]- [13]. However, rare work has been done for multiple systems with unknown control directions, due to the difficulty in analysing the cooperative behavior in the case of multiple Nussbaum functions interacting simultaneously.
The associate editor coordinating the review of this manuscript and approving it for publication was Haibin Sun . Since the rise of cooperative control research, parameterstrict-feedback system has attracted the researchers' attention in this area. Early works have involved the output synchronization of multiple PSF systems, see [14]- [18] and so on. In these works, uncertainties and nonlinearities include unknown parameter, state constraint and input saturation and so on. Thence, the agents in the networked system have nonidentical dynamics. Recently, agent with unknown control direction has been taken into consideration. Our previous work [19] has studied the consensus of multiple first-order integrators. After that, researchers have extended the agent's dynamic to PSF systems [20]- [25], etc. When the signs of the control coefficients are unknown and the systems are in PSF forms, the backstepping control is quite involved and Nussbaum-type functions are normally adopted. But due to the critical properties of Nussbaum-function, i.e., lim k→∞ sup 1 k k 0 N (δ)dδ = +∞ and lim k→∞ inf 1 k k 0 N (δ)dδ = −∞, the control signals may be extremely large and may damage the actuator or equipment in real system.
As another effective method to tackle the unknown control direction, logic-based switching has been proposed in VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ [26]- [28], in which the unknown sign of the input coefficient is matched online by a parameter switched between positive and negative values. After that, it has been applied to various kinds of dynamics, e.g., nonlinear systems with the powers of positive odd rational numbers [29], stochastic feedforward systems [30], stochastic systems with state constraint [31], systems with actuator faults [32] and so on. [33] has applied the method to multiple first-order integrator systems with unknown control directions, but the amount of agents in the network has been incorporated in the switching rule for each agent. Therefore, the switching rule has not been worked in a fully distributed manner and may have numerous potential switches. In [34], the agent's dynamic has been extended to high-order strict-feedback systems. But the control objective has been achieved by driving all agents' outputs to 0. Motivated by the works mentioned above, in this brief, we aim to solve the output synchronization problem of multiple PSF systems with unknown control directions via logicbased switching mechanism and backstepping design. The main contributions of this brief are summarized as follows: i. The mechanism is a combination of backstepping design, tunning function design and logic-based switching, which are all worked in distributed manners. The sign of the control coefficient b i is matched by the switching parameter K i which is dependent on the incremental errors between consecutive switching moments. Output synchronization controller is designed recursively and the parametric uncertainties in agents' dynamics are compensated by parameter estimators. ii. Compared with [33], [34], the advantages of the algorithm in this brief can be seen from two aspects. Firstly, the switching mechanism is dependent on the incremental errors rather than the total errors as many previous related works did. It is more sensitive to mismatch between true control direction and the estimated ones, so the authentic control direction can be identified quickly. The last but not least is that the potential switches are much fewer. That is, for each agent, it has only 2 potential switch values, i.e., 1 or −1. It reduces the calculative burden and greatly facilitates the formula derivation. The rest of the paper is organized as follows. Section II presents some preliminaries. Then, the problem formulation and control objective are proposed in Section III. Section IV presents the main result with the theoretical proof. A numerical example is proposed in Section V to illustrate the effectiveness of proposed method. Finally, Section VI concludes the paper.

II. PRELIMINARIES
In what follows, some notations and lemma are given for the later expression and performance analysis.
Let R m×n denotes the family of m × n real matrices. For matrix M , M ≥ (≤) 0 means that M is a positive (negative) semi-definite matrix and M > (<) 0 means that M is a positive (negative) definite matrix. Null(M ) denotes its null space. For a continuous differentiable function f : R n → R, the row vector of ∂f /∂x is [∂f /∂x 1 , · · · , ∂f /∂x n ].
The communication topology among N agents is characterized by G = (V, E), where V = {1, 2, . . . , N } is a nonempty finite set of N nodes, an edge set E ⊆ V × V is used to model the communications among agents. The neighbor set of node i is denoted by N i = {j|j ∈ V, (i, j) ∈ E}. j / ∈ N i means that there is no information flow from node j to node i. A sequence of successive edges in the form {(i, k), (k, l), . . . , (m, j)} is defined as a path from node i to node j. For a digraph, it has a spanning tree if there exists at least one node that can reach all the other nodes in the network.
A weighted adjacency matrix A = [a ij ] ∈ R N ×N of graph G satisfies that a ii = 0 (∀i) and a ij > 0 (i = j) if (i, j) ∈ E and 0 otherwise. The Laplacian matrix L ∈ R N ×N of graph G is defined to satisfy L ij = −a ij , ∀i = j and L ii = j =i a ij . It is well-known that c1 N is the null space of Laplacian matrix L when the communication graph has a spanning tree, with c is some constant and 1 N = [1, 1, . . . , 1] T ∈ R N . Barbalat's Lemma: Consider the function φ : R + → R. If φ is uniformly continuous and lim t→∞ t 0 φ(τ )dτ exists and is finite, then, lim t→∞ φ(t) = 0.

III. PROBLEM FORMULATION
Consider a network of N agents with the dynamic of agent i being described by the following PSF system: x in ] T ∈ R n , u i ∈ R, y i ∈ R are the state, input and output of agent i, respectively. b i = 0 is the unknown control coefficient, particularly, its sign that represents the control direction is unknown.

A. CONTROL OBJECTIVE
Our goal is to design u i for all agents in graph G, such that their outputs are asymptotically synchronized, i.e., lim t→∞ (y i (t) − y j (t)) = 0, ∀i, j ∈ V, while the closed-loop signals are guaranteed to be bounded.
Remark 1: We revisit the problem of output synchronization of multiple nonlinear systems with unknown control directions [21], [23], [24], [25], [35], etc, the sign of the unknown coefficients b i is matched by a Nussbaum-type gain such that the control direction can be tuned online for the synchronization controller to play the right role. Due to the properties of the Nussbaum function, large control overshoot appears frequently. Then, poor transient performance follows.
As is known, switching mechanism is an effective method to tackle the unknown control direction. Recently, researchers use switching mechanism to match the unknown control directions in multi-agent system [33] where agents have firstorder integrator dynamics. The sign of b i is matched by a switching parameter which is updated online. But the switching rule is complicated in a sense, the potential switching values are too many, i.e., 2 (N −i) with N is the amount of the agents. On the other hand, it is not totally distributed, since agent i needs to the global information N . In [34], the agent's dynamic has been extended to high-order nonlinear system. But synchronization is realized by driving all agents' outputs to 0. Motivated by the above mentioned works and our previous work on the asymptotic tracking for a single PSF system [36], in this paper, we focus on solving the output synchronization of multiple PSF systems with unknown control directions by a logic-based switching mechanism.

IV. MAIN RESULT
Next, we have the following main result of this paper. For the sake of simplicity, in what follows, Theorem 1: For agent i described by (1), there is a distributed controller and K i (t) = 1 or K i (t) = −1 is the switching signal which serves as the estimation of the sign of b i . For agent i, when the kth switching occurs at t = T with c > 0 and where a 2 , · · · , a n > 0 and φ im , w im , ψ jm are calculated by with γ 1 , γ 2 > 0 and tuning functions τ i , τ 10,2022 such that asymptotic output synchronization of the network can be achieved, i.e., lim t→∞ (y i (t) − y j (t)) = 0 and the other signals in the closed-loop system are bounded, provided that the digraph has a spanning tree.
Proof: Now, we start the step-by-step design procedure and the proof is carried out at the same time.

Final step n: Similarly, by
Define thence, the time derivative of V in for all t except the switching moments can be described by: takingρ i , u i as (3) and (2) Noting that Then, (23) is reduced tȯ The remaining part of the proof is divided into two parts.
In the first part, we show that if there is only a finite number of switching times, y i (t) → y j (t), ∀i, j as t → ∞ and the closedloop system is guaranteed to be bounded.
On the other hand, from the structure of M j and ρ i are all bounded from above. Since the closed-loop system is smooth after the final switching occurs, the boundedness of ξ i2 , · · · , ξ in deduce the boundedness fo x i2 , · · · , x in . Then the closed-loop signals are bounded on [T i.e., x 2 i1 ξ 2 i1 and ξ 2 i2 , · · · , ξ 2 in are integrable on [0, ∞). By Barbalat's Lemma, one has lim t→∞ x 2 i1 (t)ξ 2 i1 (t) = 0, i.e., lim t→∞ x i1 (t) = 0 or lim t→∞ ξ i1 (t) = 0. No matter which case, when the directed communication graph has a spanning tree, one has lim Part II. Seeking a contradiction, suppose on the contrary, that there are infinite numbers of switching moments. Then, there exists an integer σ i such that similarly, it can be obtained Thus T (i) σ i = ∞ because after rearranging the terms for (29), it is plain to see that (4) can never be satisfied for all t > T (i) So there is only a finite number of switching moments. The proof is completed.
Remark 2: In [33] and [34], the similar problem has been investigated. But there are fundamental differences between this work and them. Compared to [33], in this brief, K i has fewer potential choices, that is, one agent has only two potential choices. Moreover, the switching mechanism (4) is dependent on the incremental errors rather than the total errors, i.e., the integration is proceeded in [t k , t k+1 ) rather than [t 0 , t k+1 ). So, the computational burden is reduced naturally. In [34], the controller drives all x i1 to 0 such that consensus is realized, while in our work, x i1 may be converged to other trajectories theoretically, since we drive x i1 ξ i1 to 0. The reason lies in the structure of the analysis function. In this brief, it is derived from x i1 . So, the proof is simplified since there is no need to construct an ISS system for proof. Notice that M (4) can be unified by some constantM large enough, such that the switching mechanism is simplified. But as a result, the rate of convergence would be slowed down. Since the switching becomes less frequently, some 'wrong matches' of K i and b i may last longer.
Remark 3: It can be seen that x j1 , · · · , x jn need to be transferred through communication network. In order to reduce the communication burden, [38] and [39] have designed a reference output to agent i to track, such that only partial state information is necessary for communication. As a tradeoff, the controller's dimension is enlarged since it has to construct a reference trajectory for each agent, where the dynamic of the reference trajectory is of the same order as agent i.

V. SIMULATION RESULTS
In this section, an example is presented to verify the effectiveness of the proposed controller in Theorem 1. Comparison is made between this mechanism and the Nussbaum gain method. To this end, we consider the output synchronization problem of a group of three agents, denoted by '1'-'3' in Fig 1. For the sake of simplicity, a ij are set to be 1 when a ij > 0. Be the same problem formulation as [25], the dynamic of agent i is ] T , y i and u i are the state, output and input of agent i, respectively. θ i is the unknown parameter.
To be more specific, the initial values of agents are chosen as the same as [25], i.e., and all the initial values of estimators are set to be 0. Then, we set control gains a 2 = 2, γ 1 = 0.2, γ 2 = 0.     asymptotically synchronized, i.e.,lim t→∞ (y i (t) − y j (t)) = 0 (∀i, j ∈ V).   is bounded. Thus, the simulation results well confirm the theoretical issues in Theorem 1.
With the same simulation example, comparison between this paper and the previous work [25] has been made in Fig 7.  Fig 7.a and Fig 7.c are the x i1 and x i2 by the controller in Theorem 1 respectively. Fig 7.c and Fig 7.d are the x i1 and x i2 by the Nussbaum method in [25]. Firstly, it can be seen their effectiveness, i.e., output synchronization and bounded maintenance are achieved by these the different methods. Secondly, it can be seen that the transient performance of Fig 7.a is better than that of Fig 7.b Concretely, it can be seen that output synchronization is achieved at 2s in Fig 7.a. But in Fig 7.b, output synchronization is achieved after 3s. That is to say, logic-based switching mechanism possess faster convergence. But there are imperfections in this method, i.e., switching leads to tiny chattering which can be seen in Fig 7.c.
Our future research will focus on the agents in non-strict feedback forms [40], [41]. Moreover, to further reduce the calculation burden caused by integration in the switching mechanism (4), we will consider time-triggered scheme in [42] such that data is periodic sampled.

VI. CONCLUSION
This literature investigates the output synchronization problem of multiple PSF systems with unknown control directions and parameter uncertainties under a directed graph. Based on VOLUME 10, 2022 backstepping design and logic-based switching mechanism, a distributed controller is designed recursively for each agent such that asymptotic output synchronization can be achieved. Meanwhile, the closed-loop system maintains bounded. The simulation example shows the efficiency of the presented scheme. Comparison is made between the literature and previous work.