Data-Driven Containment Control for a Class of Nonlinear Multi-Agent Systems: A Model Free Adaptive Control Approach

: This paper studies the containment control problem of heterogeneous multi-agent systems (MASs) with multiple leaders. The follower agent dynamics are assumed to be unknown and nonlinear. First, each follower is transformed into an incremental data description based on the dynamic linearization technique. Then, a distributed model-free adaptive containment control law is proposed such that all followers will be driven into the convex hull of the leaders. Furthermore, the algorithm is extended to the time-switching and dynamic leaders case. As a data-driven approach, the proposed controller design uses only the received input and output (I/O) data of these agents rather than agent mathematical models. Finally, to test the potential in real applications, three representative examples considering various environment factors, including external disturbances, are simulated to show the effectiveness and resilience of this method.


Introduction
Inspired by the coordinated behavior in nature, such as the flocking of birds or migrating of herds, the distributed control of multi-agent systems has been studied with increasing attention [1].Due to the distributed and flexible structure, the MASs based methods have great potential in solving the coordination problem for complex systems.The applications have been reported in various fields, including vehicle cooperation [2], multi-robot systems [3], and traffic signal control [4].Until now, numerous algorithms have been designed concerning three types of coordinated tasks, i.e., consensus control, formation control, and containment control [5].Unlike the single leader setting in consensus or formation control, containment control is regarded as more challenging and requires all the followers to converge to a convex hull spanned by multiple leaders.Containment control also has many potential applications.For instance, a group of robots try to travel across an unsafe tunnel, but only a portion of them are equipped with sensors [6].Then, the ones with sensors can act as leaders and the others will ensure safety by staying in the area constructed by leaders.
According to the dynamics of agents, the results of containment control can be roughly classified into four categories: (1) the first-order integral system [7,8]; (2) the secondorder or high-order integral system [6,[9][10][11][12] ; (3) the general linear system [13][14][15]; (4) the nonlinear system [16][17][18][19][20] .The pioneering work for a collection of single-integrator agents is investigated in [7], where a "stop-go" rule based hybrid control scheme is designed for leaders, whereas consensus-like local interaction rules are applied for followers.However, the result of [7] is only discussed under a fixed undirected topology.To explore the effect of dynamic communication, the containment problem of single-integrator agents under switching directed network topologies is considered in [8].In [9], a type of feedback control law with velocity measurements is proposed for double-integrator MASs and an experimental platform consisting of five wheeled mobile robots is used to validate the result.To reduce the communication burden, [10] developed a finite-time containment control algorithm for double-integrator MASs using only position measurements.In [11], the effects of time-varying delays are taken into consideration, where the linear matrix inequality method and the Lyapunov functional method are jointly used to ensure the containment.In [12], the noise in agent information transmission is formulated and corresponding timevarying state feedback containment control laws are proposed for both first-order and second-order integral MASs.As a extension of [12], ref. [6] considers both measurement noise and polynomial disturbance in high-order integral MASs, where a containment controller is designed with a proportional term and the nth-order integral terms.
Recent work begins to focus on the general linear or nonlinear MASs.In [13], based on the relative outputs of neighboring agents, dynamic observer-type containment controllers are built for both continuous-time and discrete-time general linear MASs.The results of time-varying uncertainties and constant time delays for general linear MASs are shown in [14,15], respectively.As a special kind of nonlinear agents, the rigid body MASs formed by Euler-Lagrange expression is studied in [16], in which a sliding-mode estimator based containment controller is given.Considering the parametric uncertainties of Euler-Lagrange MASs, [17] develops a modified sliding mode containment controller based on the one in [16] without using velocity information.Some results of more general nonlinear agents have also been reported.In [18], the fuzzy logic is utilized to approximate the dynamics of Lipschitz nonlinear MASs and an observer-based containment controller is designed.In [19], sliding mode controllers are proposed to address the finite-time containment problem of MASs with Lipschiz-like nonlinearity and external disturbances.Based on neural networks (NNs) and backstepping techniques, a finite-time containment controller is applied to the high-order nonlinear MASs with dynamic uncertainties in [20].
Model-free adaptive control (MFAC), initially proposed in [21], is an effective approach to deal with unknown nonlinear systems.With 20 years of development, MFAC has now provided a systematic control framework with a wide range of engineering applications [22][23][24][25][26][27][28][29][30][31][32][33].The partial-form dynamic linearization based MFAC controller is designed for a class of single-input and single-output unknown nonlinear system in [22], and it is extended to the multi-input and multi-output system in [23].Based on the view of an ideal controller, the controller dynamic linearization based MFAC is proposed in [24] and then enhanced by radial basis function NNs in [25].To address the repetitive tasks over a finite time interval, model-free adaptive iterative learning control (MFAILC) is presented in [26].It is reported in [27] that MFAC has been applied to over 150 yields.The latest applications and advances can be found in [28][29][30][31][32], including weld penetration [28], steamwater heat exchanger [29], traffic systems [30,31], and the stability analysis of full-form dynamic linearization based MFAC [32].For the brief review of MFAC, readers are referred to [27,33].
MFAC has also been employed in the coordination of MASs [34][35][36][37][38][39][40][41][42].MFAC is first used to address the consensus problem for a group of unknown heterogenous nonlinear MASs in [34].Based on the controller dynamic linearization technique and Newton-type optimization method, a data-driven ideal leader-follower consensus tracking algorithm is designed in [35].Then, the consensus tracking issue under repetitive environment is solved by using MFAILC in [36].In [37], a MFAC based predictive consensus scheme is presented for MASs with communication constraints.In [38], the consensus problem for the MASs under time delays is solved by the Smith estimator enhanced MFAC scheme.For the formation control of MASs, [39] gives a data-driven formation control scheme based on MFAILC.Considering the iterative-varying disturbance, the algorithm in [39] is extended to a robust form formation control law in [40].In [41], an event-triggered MFAC method is used to perform the containment task where the MASs are suffering from denial-ofservice attacks.A model-free adaptive containment control scheme is investigated for a class of multi-input multi-output nonlinear MASs in [42].However, the MASs in [34][35][36][37][38][39][40] only contain one single leader, while no exploration has been conducted for multileaders case.The work in [41,42] only found the result that the defined local containment error is bounded or converged, which is not exactly equivalent to the containment goal.Inspired by above observations, a novel distributed model-free adaptive containment control scheme is proposed in this paper.Compared with existing work, the main contributions of this work are threefold: (1) A novel distributed model-free adaptive containment controller is developed.Compared with previous work [34][35][36][37][38][39][40], the MASs extend to the multiple-leader case.By virtue of the MFAC framework, this controller is endowed with the data-driven feature, i.e., the design process only utilizes I/O data of the agents rather than their mathematical models.Moreover, this method does not need the extra training process that is required in the NNs based algorithm [20].(2) The global containment error is proposed to overcome the analysis nonequivalent problem in [41,42].Then, the Lyapunov function based stability analysis is conducted to guarantee that the containment goal is achieved, which is different from the conventional contraction mapping principle used in MFAC based protocols (e.g., ref. [35]).
The remaining part of the paper is organized as follows.Section 2 presents some preliminary knowledge and gives problem formulation.Section 3 shows the main results, including the controller design and stability analysis.Section 4 extends the existing result to a time-switching topology and dynamic leaders case.In Section 5, three numerical examples are provided.Finally, the conclusion is discussed in Section 6.

Preliminaries and Problem Formulation 2.1. Symbol Notations and Graph Theory
Let R, R n and R n×m denote the set of real numbers, n × 1 real vectors, and n × m real matrices, respectively.The diagonal matrix and identity matrix are denoted by diag(•) and I; 1 n = [1, 1, ..., 1] ⊤ ∈ R n is a vector with all elements are 1; ∥a∥ denotes the Euclidean norm for a vector a ∈ R n ; card(S) stands for the number of elements in the set S.
The follower and leader agents are presented by nodes in the set F = {1, ..., N} and W = {N + 1, ..., N + M}, respectively.The communication between these agents is described by a directed graph G = {V , E , A } whose node set is V = F ∪ W. The edges in the set E ⊂ V × V represent the communication links between agents.The expression A = [a ij ] ∈ R (N+M)×(N+M) (i, j ∈ V ) is defined as an adjacency matrix with elements a ij = 1 if and only if the directed edge (j, i) ∈ E , otherwise a ij = 0.The in-degree matrix of a graph G is defined by a diagonal matrix D = diag(d 1 , ..., d N+M ) with diagonal entries The neighbors of the ith agent are denoted by the set N i = {j|(j, i) ∈ E }.A directed path from node i to node j is defined as a sequence of consecutive edges which begin from node i and end in node j.A directed graph is strongly connected if a path exists between any pair of nodes.The Laplacian matrix is defined as L = D − A .Due to the fact that leaders do not receive any information from other agents, the Laplacian matrix L can be partitioned as the following block matrix [8]: where In order to give the readers a better understanding of this paper, the major notations are listed in Notations part.

Problem Formulation
The MASs consist of N follower agents and the dynamics of the ith agent is described as: where y i (k) ∈ R, u i (k) ∈ R are the agent output and control input of the ith agent at time instant k, respectively, and f i (•) is an unknown nonlinear function.
Denote the output of the leader as w l (k) ∈ R, l = N + 1, ..., N + M, which can only be received by a subset of follower agents.

Definition 1 ( Convex Set and Convex Hull
Definition 2 (Containment Control [10]).The MASs achieve containment if all followers are driven by a distributed control law and move into the convex hull spanned by the leader set W, that is, ∀i ∈ 1, ..., N, there exist an element wi ∈ Co(W ), such that where wi is the convergence value for the ith follower agent.
The following assumptions are given before further discussions.Assumption 1.The partial derivative of f i (•) with respect to u i (k) is continuous.Assumption 2. For all k with finite exceptions, if ∥∆u i (k)∥ ̸ = 0, the follower agent dynamics (2) satisfies the following generalized Lipschitz condition, that is and b is a positive constant.Remark 1. Assumption 1 is a general condition for controller design.Assumption 2 gives a bound of the change rate of the agent's output, i.e., the agent's output change should be finite if the control input change is bounded.A wide range of real agent systems satisfy the assumptions.For instance, the velocity change of an autonomous car must be finite with a bounded change of the accelerator.Similar assumptions can also be found in [22][23][24][25][26][27][28][29]32,33].
Under Assumptions 1 and 2, the unknown agent dynamics (2) can be transformed into the following dynamic linearization model and then the distributed control law will be designed based on it.

Lemma 1 ([44]
).Consider follower agents with dynamics (2) satisfying Assumptions 1 and 2. If ∥∆u i (k)∥ ̸ = 0 holds, then agent (2) can be transformed into a compact form dynamic linearization (CFDL) data model: where the time-varying variable ϕ i (k) ∈ R is named as the pseudo partial derivative (PPD), which is bounded for any time instant k.
Remark 2. By virtual of CFDL technique, the unknown nonlinear agent dynamics (2) are now transformed into an incremental form data description (3) in every operation point with a bounded PPD.In implementation, with proper selected algorithms, PPD can be estimated by using the I/O data of the agents and the estimated value can also be proved to be bounded, which is shown in Theorem 1.Here, the CFDL data model is derived under the condition ∥∆u i (k)∥ ̸ = 0.In fact, if the case ∥∆u i (k)∥ = 0 happens at some sampling time, a new CFDL data model can also be established after shifting In the next section, the data model (3) will be used to design the distributed control law and derive the stability analysis result.

Main Results
In this section, the data driven containment control law is first designed under the stationary leaders and fixed topology case.
Let ξ i (k) denote the local containment error: where a ij (i, j ∈ V ) is the entry in the adjacency matrix, and y i (k) and w l (k) denote the outputs of the follower agent i and leader agent l, respectively.From this definition, ξ j (k) can be regarded as the tracking error of the ith agent with all its neighboring agents at time instant k.
The distributed model free adaptive containment control (MFACC) is designed as follows: Distributed PPD updating algorithm: PPD reset mechanism: Distributed containment input law: where ρ i ∈ (0, 1], η ∈ (0, 2], µ > 0 and λ > 0 are adjustable parameters, ϕ i (k) is the estimated value of ϕ i (k), ϕ i (1) is the initial value of ϕ i (k), and ε > 0 is a preset small constant.Remark 3. The PPD is estimated by the distributed updating algorithm ( 5) and (6), which is the well-known projection algorithm [22][23][24][25][26][27][28][29][30][31][32][33] and derived by minimizing the following performance index J( 2 .With this distributed algorithm, the PPDs of follower agents are estimated by using their own I/O data.Theoretical results show these estimated values will be bounded if some parameters are appropriately chosen, which is presented in the next section.Furthermore, due to the heterogeneity of the agent dynamics, the trajectories of the estimated PPD ϕ i (k) may vary from different follower agents.Remark 4. The distributed containment control law (7) is derived by minimizing the following in- As we can see, this performance index tries to decrease the errors between the ith agent and its neighbors.It will be proved in the following part that the MASs will finally achieve the containment control goal.Observing the control scheme ( 5)-( 7), one will find it only uses the local I/O data to estimate the PPD and calculate the input for each follower agent, i.e., no model information of f i (•) is involved; that is why it is called a data-driven method.The control gain in ( 7) is also adaptively tuned by ϕ i (k), which is continuously updated by the new received I/O data.In addition, the protocols ( 5)- (7) will regress to the consensus control law in [34] when there is only one leader.
The following assumptions and lemma are prepared before the main result.Assumption 3.For all follower agents, the sign of the PPD ϕ i (k , where ε is an arbitrarily small positive constant. Without loss of generality, it is assumed that ϕ i (k) > ε > 0 in the discussion.Remark 5. Assumption 3 indicates that the follower agent's output incremental direction caused by its input do not change during the containment control process.Similar assumptions can also be found in model-based control theory.Assumption 4. The communication graph among followers is strongly connected, and each follower can receive the information through a direct path from at least one leader.Assumption 4 guarantees that there are no isolated followers and the containment control problem is well-posed.
In the stationary leader case, the trajectory of the lth leader is written as w l (k) = w l (l ∈ W ), where w l is a constant.The main result is summarized as Theorem 1.
Theorem 1. Assume that the MASs dynamics satisfies Assumptions 1-3 and the communication topology satisfies Assumption 4. In the stationary leaders case, that is, w l (k) = w l (l ∈ W ), let the distributed MFACC ( 5)-( 7) be applied.If the parameter ρ i (i = 1, ..., N) is selected to satisfy the following convergence condition then there exists a λ min > b 2 /4, and the MASs will achieve the containment control objective defined in Definition 2 when λ > λ min and k → ∞.
Proof.The proof consists of three steps.
Step one is to prove that the PPD estimation is bounded.
Step two is to derive the convergence condition.
Step three is to prove the convergence of the global containment error.
Step 1 The boundedness of In the other case, the PPD estimation error is defined as φi (k) = ϕ i (k) − ϕ i (k).Sub- tracting ϕ i (k) from both sides of the PPD estimation algorithm (5) yields Define . Then, substituting CFDL data description (3) into (8) gives From Lemma 1, it can be known that ϕ i (k) is bounded, and assume the bound of it as b.Taking the norm on both sides of (9) leads to Note the fact that the term ) is monotonically increasing with respect to |∆u(k − 1)|.Thus, there must exist a constant δ 1 to satisfy the following inequality when 0 < η < 2 and µ > 0: Substituting ( 11) into (10) gives which means φi (k) is bounded.Since the boundedness of ϕ i (k) is guaranteed by Lemma 1, ϕ i (k) is also bounded.
Step 3 Prove the convergence of the global containment error.
Define the upper bounds of the followers, leaders, and the MASs as follows: Note that one can then define y − (k) = min i y i (k), w − = min l w l , and Ω − (k) = min{y − (k), w − } for the lower bound.
For the upper bound, Ω + (k) = w + means all the followers goes into the convex hull of the leaders.Therefore, define the upper global containment error as follows: for the lower global containment error).Then, one considers the following Lyapunov function V 1 (k) = e + (k) 2 for further analysis.The time difference can be written as Rewrite y i (k + 1) as where γ ij (k) is the entries of Γ(k).
(2) e + (k) ̸ = 0, i.e., y For the ith follower, s i (k) = 1 means the outputs of the ith follower and its neighbors are all equal to y + (k), which also means only the first term in the right side of ( 22) exists.Substituting y j (k) = y + (k) into (22) gives Otherwise, if s i (k) ̸ = 1, the following results can be concluded from Equation ( 22): Combining ( 24) and (25) gives It is clear that Therefore, ∆V 1 (k + 1) ≤ 0 holds.Now, one only needs to prove that ∆V 1 (k + 1) ̸ ≡ 0.
Assume that there exists a time instant ) for all h > 0, i.e., card(S 1 (h)) > 0 must hold for all h > 0.
According to Assumption 4, at least one agent in S 1 (h) whose neighbor's output is less than y + (K 1 ), i.e., there exist a certain follower Repeating the process in ( 27), one gets which contracts to card(S 1 (h)) > 0. Therefore, the assumption is violated.According to Lemma 2, one concludes that e + (k) will converge to 0. This completes the proof.Remark 6.According to the analysis in Theorem 1, if there is only one leader, all followers will converge to the leader's output.In other words, consensus is a special case of the containment control discussed in this paper.

Extension to Switching Topologies and Dynamic Leaders
Due to transmission failure and other factors, the communication topology between agents may change in the containment process.On the other hand, the trajectories of leaders usually change over time in practical agent coordination tasks.Therefore, the proposed containment design is extended to the switching topologies and dynamic leaders situation.
A switching graph G (k) associated with time instant k is now used to indicate the communication topology among agents.The graph switches from the set G = {G 1 , G 2 , ..., G ψ }, where ψ ∈ Z + is the total number of possible graphs.The time-varying adjacency matrix and the Laplacian matrix are denoted by A (k) = [a ij (k)] ∈ R (N+M)×(N+M) (i, j ∈ V ) and L (k), respectively.The two blocks in (1) are also time-varying and written as L 1 (k) and L 2 (k).The in-degree matrix in time instant k is denoted by For a specific interaction graph G l in the set G, the corresponding in-degree matrix is defined as D l = diag d l 1 , ..., d l N+M .The neighbors of the ith agent in time instant k are denoted by the set N i (k).Assumption 5.For every graph in G, the communication graph among followers is strongly connected and each follower can receive the information via a directed path from at least one leader.
Next, we consider the time varying leaders.Assume that the trajectories of leaders have lower and upper bounds, that is w− ≤ w l (k) ≤ w+ , l = N + 1, ..., N + M where w− and w+ are two constants.

Proof. Define the time-varying leaders' output collective stack vectors as w
In this case, Equation (17) becomes where The γii (k) is now written as If the condition 0 < ρ i < 1 max l=1,2,..,ψ d l i holds, one has 0 < d i (k)ρ i < 1.From the analysis in ( 18)- (20), it is easy to verify that Γ(k) also is a stochastic matrix when λ > λ min (λ min > b 2 /4).
Rewrite y i (k + 1) as Following the same steps from ( 23) to (26), one concludes One only needs to prove that there does not exist a time instant K 2 such that According to Assumption 5, at least one agent in S 2 (h) whose neighbor's output is less than y + (K 2 ), i.e., there exists a certain follower which contracts to card(S 2 (h)) > 0. Therefore, the assumption is violated.In summary, one concludes that ẽ+ (k) converges to 0. This completes the proof.

Remark 7.
When the leaders become time-invariant after time instant, one has w+ = max l w l and w− = min l w l , and the result in Theorem 2 will regress to the result in Theorem 1.

Simulation
This section gives three numerical examples to verify the theoretical results.The example settings and parameter selections are shown in Table 1.It is noteworthy that the agent dynamics presented in the simulation are only used to generate the I/O data and the controller design process contains no model information.
All tests in this paper are conducted in MATLAB R2018a with the Intel CPU Core (TM) i7-11800H @ 2.30 GHz (Intel, Santa Clara, CA, USA).The sample time is chosen as 0.1 s.The time step is set to be 500 for Example 1 and 1000 for Examples 2-3.Four follower agents are labeled as agents 1-4 and two leaders are labeled as agents 5-6, i.e., F = {1, 2, 3, 4} and W = {5, 6}.The communication network is fixed in this case as shown in Figure 1, in which the topology among followers is strongly connected.Moreover, the information from agents 5-6 can only be received by agent 1 and agent 4, respectively.The dynamics of four followers is described as It is worth noting that the followers are heterogeneous and include linear and nonlinear dynamics.Agent 1 is a linear agent, which is widely used in autonomous vehicle formation control [2].The cosine nonlinear dynamics of agent 2 is taken from [46].The nonaffine nonlinear descriptions of agents 3-4 are picked from [36].The heterogeneity and nonlinearity of followers become a great challenge to the controller design.
The outputs and inputs of agents are shown in Figures 2 and 3 (with λ = 2), respectively.They show that the follower agents are driven into the convex hull of leaders gradually.At the time instant k = 165 and k = 330, the convex hull moves to another location due to the change of the leaders' outputs.The followers can still converge to the new convex hull, which verifies the result in Theorem 1.To further study the influence of the key parameter, the tracking performance with different λ (i.e., λ = 5, 2, 1, and 0.5) is presented in the sub-figures in Figure 2. The results show that, within a certain range, the smaller the λ value, the faster the tracking speed of follower agents.However, one should also note that the overshoots and oscillations may be intensified if the λ value is too small.The PPD estimation (with λ = 2) is shown in Figure 4. Due to the heterogeneous dynamics of followers, the PPD estimations are quite different from each other even though they have the same initial value.

Example 2
The time-switching topology and dynamic leaders are simulated in this case.The followers dynamics are assumed to be the same with Case 1.The extended algorithm Equations ( 5), (6), and (30) will be implemented in the MASs.The communication topology is assumed to switch over three graphs, i.e., S G = {G 1 , G 2 , G 3 }, which are presented in Figure 5.An extra time-varying signal σ(k) is introduced to indicate the switching process.The definition of σ(k) is given as follows: The dynamic trajectories of the leaders are selected as Agent 5 : w 5 (k) = (0.5 + 0.001k)sin(πk/150) Agent 6 : w 6 (k) = (0.5 + 0.001k)sin(πk/150) − 1 Agent 7 : w 7 (k) = 0.3sin(πk/100) − 0.5 From the description, one can see the amplitude of leader 5 and 6 increases gradually over time.
All the initial values and the parameters are the same with Case 1 while λ = 0.5.From Figure 5, the in-degree matrices are D 1 = diag(2, 1, 3, 2, 0, 0), D 2 = diag(3, 1, 2, 2, 0, 0), and D 3 = diag(2, 1, 2, 3, 0, 0) for graphs G 1 ,G 2 , and G 3 , respectively.Thus, it is easy to check ρ i = 0.3 < 1/ max l=1,2,..,ψ d l i = 1 3 also satisfies the convergence condition in Theorem 2. Figure 6 depicts the changes of indicative signal σ(k) from k = 1 to k = 50, where one can observe that the communication network randomly switches among three graphs in different time instants.Figures 7-9 show the outputs, inputs and estimated PPD of the follower agents, respectively.In Figure 7, the green curves present the envelopes of the leaders' trajectory, i.e., these two curves are the upper and lower boundaries of the convex hull.It can be found all followers are driven into the space within these two curves, which verifies the result in Theorem 2. Furthermore, the trajectories in Figures 7-9 are not smooth, which is caused by the quick switch of the communication topologies and the dynamic changes of the leaders.

Example 3
A more practical simulation with external disturbance is conducted in this case.The followers' dynamics and topologies are the same as Example 2. The leaders' outputs are modified as piece-wise functions to simulate the transition of different types of tasks.where ŵl (k) and ŷi (k) are the real information that the follower receives from leaders and its neighboring followers, respectively, and n i (k) is the channel disturbance following the normal distribution with intensity β.

Notations
The following abbreviations are used in this manuscript: the vector collecting all follower agents' outputs u(k) the vector collecting all follower agents' inputs ξ(k) the vector collecting all local containment errors w s \w(k) the vector collecting all stationary\dynamic leaders' outputs w + \w − the upper\lower bound of stationary leaders' outputs w+ \ w− the upper\lower bound of dynamic leaders' outputs y + (k)\y − (k) the upper\lower bound of followers' outputs Ω + (k)\Ω − (k) the upper\lower bound of the MASs with stationary leaders Ω+ (k)\ Ω− (k) the upper\lower bound of the MASs with dynamic leaders e + (k)\ ẽ+ (k) the upper global containment error with stationary\dynamic leaders

Table 1 .
Example settings and parameter selections.
\D (k)the in-degree matrix with fixed\switching topologies L \L (k)the Laplacian matrix with fixed\switching topologies Co(W )the convex hull of the set Wy i (k)\ ŷi (k)the output without\with disturbances of the ith follower agent u i (k)the input of the ith follower agent f i (.)the unknown follower dynamics wi the convergence value for the ith follower agent w l (k)\ ŵl (k)the output without\with disturbances of the lth leader agentϕ i (k) the pseudo partial derivative ϕ i (k)the estimated pseudo partial derivative φi(k)the pseudo partial derivative estimation errorξ i (k)\ ξi (k)the local containment error with fixed\switching topologies