Observer-based bipartite containment control of multi-agent systems with input and output quantization

Authors: yanqing  hou
yan  yan
shuanghe  yu
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-01 09:45:20

Abstract: This paper presents a sliding mode control (SMC) method for the bipartite containment control of multi-agent systems (MASs) under input and output quantization. First, an observer based on the super-twisting algorithm (STA) is used to estimate the unmeasured velocities of followers in finite time, using only quantized position information. Second, an SMC controller is developed to achieve bipartite containment control for followers under input and output quantization. The stability analysis ensures that sliding variables and bipartite containment position errors converge within specified ranges determined by the quantization parameters and STA-based observer parameters. Finally, simulation results validate the effectiveness of the proposed SMC method.

Main paper: 
 i. introduction: 
 In recent years, the cooperative control of multi-agent systems (MASs) has gained significant attention for tasks such as exploration of resources, marine geodesy and cartography, and unmanned aerial vehicle swarm show [1] . Studies have focused on fundamental MAS control mechanisms, including consensus, formation, and flocking strategies [2] . Among these, consensus has been a primary focus in cooperative control research [3] . Consensus problems are typically divided into three categories: leader-free consensus, single-leader trajectory tracking, and multi-leader containment control [4] . Containment control, where followers move into the convex hull formed by leaders, is particularly important for addressing practical issues [5] .Quantization refers to reducing data size by representing it with fewer bits or levels of precision, thereby conserving network bandwidth and minimizing communication overhead [17] . In practical applications, quantization control is widely employed in network information processing [7] . However, the discontinuity inherent in output quantization complicates controller design and stability analysis, receiving considerable attention [8] .The super-twisting algorithm (STA) is extensively utilized in controllers and observers due to its continuous control signals and finite-time convergence properties [9] [10] . In [11] , an STA-based observer is developed to estimate velocity and utilize actuator fault information to achieve multi-agent consensus despite actuator faults. Additionally, [12] introduces a third-order super-twisted extended state observer to estimate the position and velocity of a sensorless drive system in interior permanent magnet synchronous motors. Furthermore, [13] presents an adaptive STA-based observer with a predetermined convergence time in discrete time.SMC is extensively applied across various applications due to its robust performance in the presence of external disturbances and uncertainties [14] . A periodic event-triggered SMC law is designed in [15] to solve the path-following problem of underactuated surface vehicles. The SMC is received even more attention under the influence of quantization [16] . Therefore, designing a quantized sliding mode controller in a MAS can enhance the robustness of the system while saving communication resources. This approach reduces the need for frequent updates of actuators, promoting energy efficiency and minimizing wear. Since quantized SMC of MASs is less researched, this paper investigates this direction.The main contributions of this paper are summarized below. Notations: R, R n , and R n×n denote the sets of all real numbers, n × 1 column vectors, and n × n real matrices, respectively. (•) ij represents the element of a matrix in row i and column j. For a matrix A, A T and A -1 denote its transpose and inverse, respectively. For x ∈ R n , ∥x∥ and ∥x∥ ∞ denote the Euclidean norm and the infinity norm of x, respectively.This paper discusses MASs consisting of M leaders and N followers. The communication network topology is represented by a directed graph G = {V, E}, where V = {ν 1 , ν 2 , . . . , ν N +M } denotes the set of nodes, and E ⊆ {V × V} denotes the set of edges. (ν i , ν j ) ∈ E indicates that the information of the jth agent can be used for the ith agent and the opposite is false. Self-edges are not allowed in the graph, i.e., (ν i , ν i ) / ∈ E. In a signed graph, the nodes set consists of two subgroups V 1 and V 2 , whereV 1 ∪ V 2 = V and V 1 ∩ V 2 = ∅. Consider the adjacency matrix A = [a ij ] ∈ R (N +M )×(N +M ) of a directed graph G = {V,E}, if there exists an edge (ν i , ν j ) ∈ E, then a ij = 1 (ν i , ν j ∈ V q , q = 1, 2), which denotes the cooperative interaction, and a ij = -1 (ν i ∈ V p , ν j ∈ V q , p, q = 1, 2, p ̸ = q), which denotes the antagonistic interaction, and a ij = 0 otherwise. Then the in-degree matrix D is rewritten asD = diag {d 1 , d 2 , . . . , d N +M }, where d i = N +M j=1 |a ij |,and the Laplace matrix L is rewritten as L = D -A with l ii = d i and l ij = -a ij for i ̸ = j. By using (1), the Laplace matrix is normalized to obtain ℓ.By normalizing the Laplace matrix L of the directed graph G = {V, E}, we getEQUATIONThen the normalized directed Laplacian matrix ℓ can be rewritten asEQUATIONwhereℓ 1 ∈ R N ×N , ℓ 2 ∈ R N ×M , and satisfied -ℓ -1 1 ℓ 2 1 M ×1 = 1 N ×1 .For saving communication resources, we consider the communication signals among leader and followers are quantized. We use the uniform quantizer which is defined asEQUATIONwhere ζ ∈ R is the input of the quantizer, q(ζ) is the quantized value and µ > 0 is the quantization parameter. Define the quantization error ase ζ = q(ζ)-ζ. It follows that |e ζ | ≤ 0.5µ.The dynamics of the ith follower are described by (i = 1, . . . , N ):EQUATIONwherex i ∈ R n , v i ∈ R n represent position and velocity of the ith follower, respectively, u i ∈ R n is control input of the ith follower. And f (x i , v i , t) is a nonlinear function, d i is an external disturbance. Definition 1. Let there exist a set C h ⊆ R n . For any x, ȳ ∈ C h , and θ ∈ [0, 1], C h is said to be a convex set if the point θx + (1 -θȳ) is in C h .Given the convex hull Co(χ) which is the minimal convex set containing all points in χ using a set of points χ = {χ 1 , χ 2 , . . . , χ N }, and is defined asCo(χ) = { N i=1 θ i χ i |χ i ∈ χ, θ i > 0, N i=1 θ i = 1}. Assumption 1. Given a sequence of positive constants κ 1 , κ 2 , . . . , κ M satisfying M i=1 κ i = 1, for any y, z, y i , z i ∈ R n , i = 1, 2, . . . M there exist two constants γ 1 , γ 2 > 0 satisfying ∥f (y, z, t) - M i=1 κ i f (y i , z i , t) ∥ ∞ ≤γ 1 ∥y - M i=1 κ i y i ∥ ∞ + γ 2 ∥z - M i=1 κ i z i ∥ ∞ (5)The control objective of this paper is to design STAbased state observers and distributed bipartite containment controllers for MASs in the presence of unmeasured velocities and input and output quantization to make the position of each follower satisfyEQUATIONwhere h(t) ∈ Co{η 1(N +1) , . . . , η 1(N +M ) } denotes the convex hull formed by leaders, and δ > 0 is a constant.Define the error functions asEQUATIONWe can get ėxi = e vi , and according to (7) , we obtainėvi = N +M j=1 |a ij | (f (x i , v i , t) + u i (t)) + g i (t) - N j=1 a ij (f (x j , v j , t) + u j (t)) - N +M j=N +1 a ij ẍj (8) where g i (t) = N +M j=1 |a ij |d i (t)- N j=1 a ij d j (t) with g i (t) ≤ 2d max .The sliding mode variable s i is designed asEQUATIONFrom ( 8)-( 9), the derivative of s i becomesEQUATIONBy using the SMC technology and from (10) , the control input of the ith follower can be taken asu i (t) = -ce vi -f (x i , v i , t) -Ksign(s i ) + N j=1 a ij (f (x j , v j , t) + u j (t)) + N +M j=N +1a ij ẍj (11) where K > 2d max .Substituting the quantized control input into (3), we obtainEQUATIONwhere ûi (t) = q (u i (t)). Let the quantization error of control inputs be e ui (t) = ûi (t) -u i (t), which satisfies ∥e ui ∥ ∞ ≤ 0.5µ.In practice, the agent's actual position can be accurately obtained by GPS, but the velocity cannot [17] . Therefore, this paper considers a situation where the velocity is not measurable. Then, the STA-based observer is used to observe the velocity. The input signal of the STA-based observer is xik , i = 1, . . . , N , k = 1, . . . , n. Let xik and vik be the estimates of signals x ik and v ik , respectively. The STA-based observer equation is described as followsEQUATIONwhere f 1ik is the kth element of f 1i , k 1 and k 2 are the observer parameters, and f 1i (x i , vi , t) = f (x i , vi , t) + ûi . Define the observation error variables xik = xik -x ik and ṽik = vik -v ik . According to ( 12)-( 13), the observation error equation is described as followsEQUATIONwhere F 1ik (x i , xi , v i , vi ) is the kth element of F 1i (x i , xi , v i , vi ) and F ik (x i , xi , v i , vi ) = f 1i (x i , vi , t) - f 1i (x i , v i , t) -ûi -d i (t)> 3 L + 2 L2 k 2 1, the observation error converges in finite time toEQUATION)whereζ i = [ζ 1i , ζ 2i ] T = |x ik | 1/2 sign (x ik ) , ṽik T , χ i = λmax{P } λmin{P } Ξ, Ξ = (k1θ1+2k2θ2) √ 2µ (1-κ)λmin{Q} , θ 1 = p 2 11 + p 2 12 , θ 2 = p 212 + 1, and κ ∈ (0, 1) is a constant. Matrices Q = q 11 q 12 q 12 q 22 and P = p 11 p 12 p 12 p 22 are symmetric matrices withq 11 = k 1 2k 2 + k 2 1 -2 L /2, q 12 = -k 1 k 1 + 2 L/k 1 /2, q 22 = k 1 /2, p 11 = 2k 2 + k 21 /2, p 12 = -k 1 /2, and p 22 = 1. And settling time of convergence T si isEQUATIONDetailed proof can be found in reference [10] .With only the quantized information being available, the tracking error e xi and e vi as defined in (7) becomeEQUATIONDue to quantization, the control input of the ith follower as defined in (11) becomeu i (t) = -cē vi -f (x i , vi , t) -K i sign(š i ) + N j=1 a ij (f (x j , vj , t) + ûj (t)) + N +M j=N +1 a ij xj (18)whereEQUATION)We present the analysis to show the sliding variable s i can converge to a bounded region in finite time.Theorem 2. Consider the MAS (12) with the sliding variable (19) and the SMC strategy (18) by using the quantized outputs and inputs information. If control parameters satisfyEQUATIONwhere ε > 0 is a small constant, then the sliding variable s i can converge to the region ∥s i ∥ ∞ ≤ cµ + 2χ i and the overall bipartite containment position errors of each follower can converge to the regionEQUATIONwhere i = 1, 2, . . . , N .Proof: Define the Lyapunov function asV i = 1 2 s T i s i . Considering R (ψ i ) ≈ R( ψi ) and substituting (18) into (10), we have Vi =s T i (t) ṡi (t) =s T i (t) f (x i , v i , t) -f (x i , vi , t) + e ui + ce vi -cē vi + N j=1 a ij (f (x j , vj , t) -f (x j , v j , t)) + N +M j=N +1 a ij xj -ẍj + N j=1 a ij (û j (t) -ûj (t))-K i sign(š i ) + g i (t) .(22) From (3), we haveEQUATIONAfter applying ( 23)-( 24) to (22) yieldsEQUATIONNoting the term sgn (š i (t)) ∈ R 3 in (25), we now establish the set Υ ik where it is possible sgn (š ik (t)) ̸ = sgn (s ik (t)), k = 1, . . . , n.The difference between the sliding variable ŝi , which is affected by quantization, and the sliding variable s i staisfy∥š i -s i ∥ ∞ = ∥c (ê xi -e xi )+(ē vi -e vi ) ∥ ∞ ≤ cµ+2χ i (26)For (e xi , e vi ) / ∈ Υ ik , k = 1, 2, 3, we have sgn(š i ) = sgn(s i ). According to the condition (20), (25) becomesEQUATIONEQUATION)The settling time can be calculated asEQUATIONwhich means that the sliding variable s i of the ith follower must enter the quasi-sliding mode bound when t ≥ T i . Next, we show the actual position of followers and the farthest distance at corresponding the convex hull. According to ( 9) and (26), we obtain∥e xi (t)∥ = ∥e xi (0)e -ct + t 0 e -c(t-τ ) s(τ )dτ ∥ < ∥e xi (0)∥e -ct + √ n (cµ + 2χ i ) t 0 e -c(t-τ ) dτ = ∥e xi (0)∥e -ct + √ n (cµ + 2χ i ) c (1 -e -ct ). (29) Since c > 0, we have ∥e xi (t)∥ < √ n(cµ+2χi) cas t → ∞. Therefore, the distance between the position of the ith follower and ideal position should satisfylim t→∞ ∥η 1i -h(t)∥ ≤ √ n (cµ + 2χ i ) cThis ends the proof.In this section, we give a simulation to demonstrate the bipartite containment control for MASs. Consider the MAS consisting of three leader agents and four followers. The trajectories of each leaders are x 5 = [0.1t, 2.5 sin( t 30 ) + 10, arctan( In this paper, we studied sliding mode bipartite containment control for MASs under input and output quantization. We addressed the challenge of unmeasurable follower velocities by designing an STA-based observer to estimate these velocities within a finite time. Using quantized position and observed velocity information, we developed a distributed bipartite containment control law that enables the followers to converge near the convex hull formed by the leaders and the symmetric convex hull. Our analysis shows that the sliding variable and bipartite containment position errors converge to a bounded region determined by the quantization parameters and the upper bound of observation errors. Simulation results support our theoretical findings. 
 
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Reviews: review: title: title: This paper can be accepted.
main_review: main_review: 1.	Innovation: The paper suggests a new control method using sliding mode control (SMC) and super-twisting algorithm (STA) for managing multi-agent systems (MAS) with quantized inputs and outputs. It’s an interesting approach, but we need more experiments and real-world tests to see if it really makes a difference compared to existing methods.

2.	Data Sources and Methods: The study relies on simulated data to test the proposed method. While the simulations support the theory, it’s missing comparisons with real-world data or more complex scenarios. It’d be good to discuss how this method might work in real life and clarify how the simulation data was created and why certain model parameters were chosen.

3.	Theory: The theoretical part is solid with clear stability analysis and control design. However, some explanations and formulas are a bit brief. Adding more details or an appendix for key steps would help readers follow the derivation better.

4.	Experiments: The experiments are well done and show the method’s effectiveness. However, they only cover a few scenarios. Testing in more complex situations, like with different disturbances and quantization levels, would strengthen the results.

5.	Figures and Tables: The figures and tables are clear, but some could use better color or line styles to improve readability. More detailed annotations could also help explain the results better.

6.	Comparisons: The paper compares the new method with existing sliding mode and quantization control methods, which is good. But it could also benefit from comparing with other advanced control techniques, like robust or fuzzy control, to provide a fuller picture of its strengths and weaknesses.

7.	Spelling and Grammar: The paper is mostly well-written with minor grammar issues. Some sentences are a bit wordy and could be simplified. A quick language check could improve clarity, especially for longer sentences.

8.	Conclusion: The conclusion wraps up the research well, but adding a look at future research directions or potential applications would make it even stronger and more forward-thinking.
title: title: Accept after modification
main_review: main_review: This paper primarily proposes a sliding mode control approach for bipartite containment control in multi-agent systems with input and output quantization. Overall, the article presents a comprehensive narrative. However, several points merit attention:

1.The manuscript contains minor grammatical and formatting issues, such as the incorrect ordering of references (e.g., reference 17 appears before reference 7 in the introduction) and inaccuracies in the phrasing, such as the term ‘an STA-based observer’ following a reference. It is advisable to review the entire article to enhance its rigor and readability.
2.In the introduction, the acronym SMC should be introduced with its full form upon its first appearance. 
3.The conclusion stating ‘It follows that |eζ| ≤ 0.5µ.’ lacks adequate justification within the text. Please provide a reference or additional explanation to substantiate this conclusion.
4.Figure 4 displays incorrect units for the variable x. The units for position are incorrectly listed as m/s, whereas they should be in meters (m).
title: title: Accept
main_review: main_review: This paper presents a sliding mode control (SMC) method for the bipartite containment control of multi-agent systems (MASs) under input and output quantization. Overall, the language and organization are satisfactory. It can be accepted as a conference paper.