Decentralized Control of Multiple Strict-Feedback Systems With Unknown Control Directions

This paper is concerned with the decentralized control problem of networked high-order nonlinear systems that can be transformed into strict feedback forms with parameter uncertainties and unknown control directions under a directed communication graph. A decentralized controller is designed recursively for each agent to realize output synchronization and guarantee the overall system to be bounded. Instead of a Lyapunov-based argument which is commonly used in the literature, an inductive contradiction argument is employed. Moreover, not like other existing works, we use the Nussbaum function in the analysis function form rather than its derivative, which not only facilitates the proof but also enlarges the schedule’s application scope. Simulation results are presented to verify the effectiveness of the proposed scheme.

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 The associate editor coordinating the review of this manuscript and approving it for publication was Zheng Chen . positive and known. However, this assumption becomes unrealistic in some cases, such as uncalibrated visual servo control in [24] and autopilot design of surface vehicles [25]. 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 [26]- [30]. However, rare work has been done for multiple systems with unknown control directions, because it is difficult to show cooperative behavior without sacrificing distributivity. Another basic requirement is that the networked system should remain bounded. Since the entire system may have multiple Nussbaum-type functions interacting simultaneously, analysis becomes extremely difficult. Our previous work [31] has studied the consensus of multiple first-order integrators, that is, the agent has a model ofẋ i = b i u i with unknown high-frequency gain b i . After that, we have extended the agent's dynamic to a high-order lowertriangular system [32], but the signs of all b i are assumed to be the same. Recently, [33] has applied the same hypothesis about the unknown control directions, but with prior knowledge of their bounds. Motivated by the above discussion, this paper aims to solve the output synchronization problem in more complex situations, i.e., networked agents with unknown control directions and uncertainty, but no additional assumptions about their control directions are required. The objective of this paper is to design a decentralized controller for each agent whose dynamic is in the form of high-order strict feedback, with parameter uncertainty and unknown control coefficient (sign and amplitude), to achieve output synchronization, and the entire system remains bounded.
The main contributions of this brief are summarized as follows: i. Existing works have assumed that agents have known control directions [12] or unknown but identical control directions [32], [34], [35] or unknown and nonidentical control directions but with a restriction on its bounds [33]. In this paper, all these restrictions are removed, that is, these agents can have nonidentical unknown control directions, and the controller is designed without prior knowledge of the b i . ii. The analysis function is crucial for recursively designing the decentralized controller. Usually, the Nussbaum function appears in the derivative of the Lyapunov function [26]- [28]. In this paper, the Nussbaum function appears in the analysis function itself rather than its derivative, which not only facilitates proofs based on contradictions but also expands the scope of application of the schedule. The rest of the paper is organized as follows. In Section II, we present some basic notions and preliminaries of the graph theory. Then, the problem formulation and control objective are proposed in Section III. In Section IV, the decentralized controller is proposed. A numerical example is proposed in Section V to illustrate the effectiveness of proposed decentralized controllers. Finally, the conclusion is drawn in Section VI.

A. NOTIONS
Throughout this paper, R m×n denotes the family of m × n real matrices. M ≥ (≤) 0 means that M is a positive (negative) semi-definite matrix, M > (<) 0 means that M is a positive (negative) definite matrix. Null(M ) denotes the null space of matrix M , sup(·), inf (·) denote the least upper bound and the greatest lower bound, respectively, and sign(·) is the classical signum function. For a continuous differentiable function f : R n → R, the row vector of ∂f /∂x is [∂f /∂x 1 , · · · , ∂f /∂x n ].

B. GRAPH THEORY
Here, we introduce some graph terminologies that can also be found in [1]- [3]. A weighted graph is denoted 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 an undirected graph, it is said to be connected if there is a path from node i to node j, for all the distinct nodes i, j ∈ V.
A weighted adjacency matrix A = [a ij ] ∈ R N ×N where a ii = 0 (∀i) and a ij > 0 (i = j) if (i, j) ∈ E and 0 otherwise. In an undirected graph, a ij = a ji , where the information exchange is uniformly balanced. In what follows, we set a ij = 1 when a ij > 0, without loss of any generality. In addition, we define the in-degree of node i as d i = j a ij and D = diag{d i } ∈ R N ×N is thus the in-degree matrix. Then, the Laplacian matrix of graph is L = D − A. 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 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.

C. NUSSBAUM-TYPE FUNCTION
A Nussbaum-type function N(·) is the one with the following properties [26] Commonly used Nussbaum-type functions include e k 2 cos(k), k 2 sin(k) and k 2 cos(k).

III. PROBLEM FORMULATION
Consider a network of N agents with the dynamic of agent i described by 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 control direction, i.e., high-frequency gain, whose sign and amplitude are both unknown.
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 overall system is guaranteed to be bounded.
Remark 1: In existing works [12], [33]- [38], the coefficients b i (sign and amplitude) are assumed to be known or unknown but identical or some assumptions have been made on b i . In this paper, the signs of b i can be nonidentical, which means that some agents may have positive high-frequency gains while others may have negative high-frequency gains. In the circumstance, the voltage equalization control of power source which is composed by multiple ultra-capacitors can be formulated [39], where the reversal of positive and negative VOLUME 10, 2022 poles may happen occasionally, either by misoperation in assembling or inappropriate use.

IV. MAIN RESULT A. DECENTRALIZED CONTROLLER DESIGN
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 (2), there is a decentralized controller with the Nussbaum function N 0 (s i ) = s 2 i cos(s i ) and s i is updated byṡ with 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 such that output synchronization of the network can be achieved, i.e., lim t→∞ ||y i (t) − y j (t)|| = 0 and the overall system is guaranteed to be bounded provided that the digraph has a spanning tree.
Proof: Now, we start the step-by-step design procedure. The proof is carried out at the same time.
Step m: jm are chosen as (6) and (8), We can see thaṫ Final step n: Similarly, by (22) Noting that jn are chosen as (6) and (8), j are updated by (4) and (7). Then, (24) is reduced tȯ Next, we will prove the existence of the closed-loop system solution on the time interval [0, +∞). Let us denote by [0, t f ) that the maximum interval of existence of the closed-loop system solution. First, assume on the contrary that, at t f , the state s i escapes to the positive infinity, no matter what the sign of b i is, there is a strictly increasing, infinite sequence s 1 i , s 2 i , · · · , with the property lim k→+∞ s k i = t f , such that On the other hand, (25) shows thatV in (t) ≤ 0, which contradicts lim k→+∞ V in (s k i ) = +∞. Therefore, s i is within a compact set on [0, t f ), i.e., s i is bounded on [0, t f ). Similarly, if we assume that, at t f , the state s i escapes to the negative infinity, the same conclusion can be given due to the property of the Nussbaum function N 0 (·), see (1).
Remark 2: Compared with the existing works [32], [33], the main contribution of this paper is that, the unknown b i in agent's model can be with different signs without other additional assumptions. It is a significant progress for many practical problems. For example, in the circumstance, the voltage equalization control of ultracapacitor-type power source can go on wheels even in fault mode.
Usually, when combined with backstepping design, the Nussbaum item appears in the derivative of the Lyapunov function, see [26], [28], [31], [33]. But in this paper, the Nussbaum item has been added in the analysis function itself, see (22). There are two reasons for this. Firstly, it facilitates the contradiction argument when the Nussbaum item appears in V in rather thanV in , since we can get the conclusion that V in (t) → +∞ if s i is unbounded, which contradicts (24). Secondly, it is essentially an extension of the known control direction case. A stabilizer for a single system is firstly proposed in our previous work [40] in this way. To the best of our knowledge, it is the first time that this method is applied in a multi-agent system. Remark 3: By applying the above scheme, the outputs y i (t) are driven to the equilibrium eventually. If we alter the controller slightly, outputs of agents can be synchronized to other desired trajectories as well. For example, by setting , where (·) is the desired trajectory. Remark 4: Distributed control and decentralized control are applicable to large scale systems composed by multiple subsystems. Each subsystem can only acquire partial information. The main difference lies in their control objectives. In distributed control, researchers mainly focus on subsystems' behaviors, such as consensus or synchronization of the multi-agent system. But in decentralized control, not only subsystem's behavior but also the overall system's behavior are considered, just like the control objective of this paper, i.e., output synchronization as well as the boundedness of the overall system are the points of focus. Typically, in traditional decentralized control, the partial information needed for subsystem i is not formulated by a communication graph, i.e., subsystem i may receive redundant information which does not appear in its controller. Motivated by the cooperative control, in this paper, we formulate the information exchange by the graph theory, such that not only a graph condition has been added in the result but also the information subsystem i received is used in its controller.

V. SIMULATION RESULTS
In this section, an example is presented to verify the effectiveness of the proposed controller in Theorem 1. 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. The dynamic of agent i is the same as that in [28]         By Theorem 1, a decentralized controller (3) together with parameter updaters (7) are designed for each agent. The simulation results are shown in Fig. 2 to Fig. 6. It can be seen in Fig 2 that the outputs of all agents in the network are asymptotically synchronized, i.e., lim t→∞ ||y i (t) − y j (t)|| = 0 (∀i, j ∈ V). Fig. 3 shows that x i2 (∀i ∈ V) are bounded and Fig. 4 shows that all s i are bounded. It can be seen in Fig. 5 and Fig. 6 that both parameter updaters, i.e.,θ i for agent i itself andθ (i) j for its neighbors, are also guaranteed to be bounded. So, the overall system is bounded. Thus, the simulation results well confirm the theoretical issues in Theorem 1. Compared with the related references [32], [33], [35], the proposed controller results in faster converge of the parameter estimators.
Our future research will focus on the convergence speed of such a networked system with time varying delays [41] and agents in non-strict feedback forms [42].

VI. CONCLUSION
This paper investigates the output synchronization problem of multiple strict feedback systems with parameter uncertainties and unknown control directions, in the circumstance of directed communication graph and agents may have different control direcions, which is the first highlight of this paper. A decentralized controller is designed recursively for each agent such that output synchronization can be achieved. Meanwhile, the overall system maintains bounded. A new analysis method is proposed, in which the Nussbaum item appears in the analysis function itself directly rather than its derivative, such that the proof can be simplified significantly, which is the second highlight of the paper. The simulation example shows the efficiency of the presented scheme.