A practical tracking control method for event-triggered cooperative index of multi-manipulator system

Abstract

The present invention discloses an event-triggered cooperative index practical tracking control method for a multi-manipulator system. The method comprises the following steps: constructing a multi-agent system consisting of a follower subsystem described by the Euler-Lagrange equation and a leader system described by a linear system, wherein the Euler-Lagrange equation represents the dynamic model of the manipulator system and the linear system generates reference trajectory information for the multi-manipulator system; designing a distributed event-triggered observer to observe the state information of the leader system for the follower subsystem; and designing a distributed event-triggered feedback control law and an event-triggered mechanism based on the state information to solve the event-triggered cooperative index practical tracking problem for the multi-Euler-Lagrangian system, ensuring that the trajectory tracking error can converge exponentially to any small neighborhood of the origin. This method ensures that the distributed event-triggered observer only relies on the sampling information of the neighboring subsystems and that the trajectory tracking error can converge exponentially to any specified neighborhood of the origin.

Description

Technical Field

[0001] The present invention relates to the field of automatic control technology, and in particular to a practical tracking control method for event-triggered collaborative index of a multi-manipulator system. Background Art

[0002] The robotic arm system is a complex system with high nonlinearity and strong coupling, which can be described by the Euler-Lagrangian system. Due to its operational flexibility, the robotic arm has been applied in many engineering fields such as industrial assembly and aerospace, and the control problem of the robotic arm system has also attracted widespread attention in the control community. Therefore, how to design a control strategy to ensure high-precision and effective control of the robotic arm system is an important and challenging research topic. In recent years, the use of adaptive control, H ∞ Great progress has been made in the research on trajectory tracking control problems of manipulator systems by combining methods such as control and sliding mode control, and combining new sampling control methods such as event triggering and time triggering. Combined with the sampling control method based on event triggering, it is not only possible to design control strategies that are directly executed on the digital platform, but also possible to utilize the event triggering mechanism to reduce the number of sampling times and the use of communication resources. With the development of robotics technology, higher requirements are also placed on the speed and accuracy of trajectory tracking for manipulators. Based on distributed observers, adaptive control technology and event-triggered control strategy, the present invention proposes a practical tracking control method of event-triggered collaborative index for manipulator systems, and conducts simulation verification on it.

[0003] Based on event-triggered control methods, neural network control methods, and backstepping design methods, fixed-time consistent tracking problems (CN202210536923, CN202311111574), finite-time consistent tracking problems (CN202210991150), given performance control problems (CN202310496159), and preset time control problems (CN202211690101) of multi-manipulator systems have been studied. However, the design of their event-triggered mechanisms still relies on continuous information from neighboring subsystems. Based on event-triggered control methods, the consistency control problem of multi-Eulerian-Lagrangian systems under DoS attacks has been studied (CN202210854102), but the controller design is based on continuous-time signals. Summary of the Invention

[0004] To address the challenges of the prior art, the present invention aims to provide a practical tracking control method for the event-triggered collaborative index of a multi-manipulator system. This method utilizes a distributed observer, adaptive control techniques, and an event-triggered control strategy to investigate the practical tracking of the collaborative index of a multi-manipulator system. Under the event-triggered mechanism, a distributed event-triggered observer and a distributed event-triggered state feedback control law are designed to implement trajectory tracking control of the multi-manipulator system, ensuring that the trajectory tracking error converges exponentially to an arbitrary small neighborhood of the origin.

[0005] The present invention is achieved through at least one of the following technical solutions.

[0006] A practical tracking control method for event-triggered collaborative index of a multi-manipulator system, comprising the following steps:

[0007] Step 1: Construct a multi-agent system consisting of N follower subsystems described by Euler-Lagrange equations and a leader system described by a linear system. The Euler-Lagrange equations represent the dynamic model of the robotic arm system, and the linear system generates the reference trajectory information of the multi-robotic arm system.

[0008] Step 2: Design a distributed event-triggered observer to observe the status information of the leader system for each follower subsystem;

[0009] Step 3: Design a distributed event-triggered feedback control law and event-triggered mechanism to solve the practical tracking problem of event-triggered cooperative index for multi-Euler-Lagrangian systems.

[0010] Step 4: Use the Lyapunov method to prove the stability of the closed-loop system and ensure that the trajectory tracking error can converge exponentially to any small neighborhood of the origin;

[0011] Step 5: Verify the effectiveness of the proposed control strategy through MATLAB simulation.

[0012] Furthermore, the multi-manipulator system can be described by the following N Euler-Lagrange equations:

[0013]

[0014] in represents the generalized position vector, represents the generalized velocity vector, represents the generalized inertia matrix, denotes the Coriolis and centripetal force vectors, represents the gravity vector, represents the generalized force vector. The Euler-Lagrange system satisfies the following three standard properties:

[0015] Property 1: Matrix Mi (q i ) is a symmetric positive definite matrix, the matrix is an antisymmetric matrix.

[0016] Property 2: There exist some positive constants μ1, μ2, μ3, μ4 and n-dimensional identity matrix I such that for any q i , μ1I≤M i (q i )≤μ2I, and ‖G i (q i )‖≤μ4.

[0017] Property 3: For any in is a known regression matrix, is an uncertain parameter vector.

[0018] Assuming a reference trajectory Generated by a leader system as follows:

[0019]

[0020] in is the state of the leader system, and are the system matrix and output matrix respectively, and both are composed of constants.

[0021] A multi-agent system with N+1 agents can be composed of a follower subsystem and a leader system, where the follower subsystem and the leader system are represented by formula (1) and formula (2) respectively. We can use To define a multi-agent system, where the node set Edge Set The i-th follower subsystem in the follower subsystem can be represented by node i=1,…,N, and the leader system can be represented by node 0. The neighbor set of the i-th subsystem can be defined as The weighted adjacency matrix can be defined as in And for i≠j, when hour otherwise

[0022] For each follower system i=1,…,N and the leader system, the position tracking error of the trajectory is defined as e i =q i -q0 and velocity tracking error are

[0023] To solve the event-triggered collaborative tracking control problem of a multi-manipulator system, we need the following two assumptions:

[0024] Assumption 1: All eigenvalues ​​of the matrix S0 are semisimple and have zero real part.

[0025] Assumption 2: In In the network, every node i=1,…,N is reachable from node 0.

[0026] Define symbols λ min (X) and λ max (X) represents the smallest eigenvalue and the largest eigenvalue of the matrix X, respectively, and ‖x0‖ or ‖X0‖ represents the Euclidean norm of the vector x0 or the matrix X0.

[0027] Furthermore, we introduce a distributed event-triggered observer, which can be viewed as a dynamic compensator and can estimate the state of the leader system for each follower system i=1,…,N. The distributed event-triggered observer can be described as follows:

[0028]

[0029] in is the state of the distributed event-triggered observer, μ0 is an arbitrary positive number, yes The estimated state and only the discrete state Right now Decide, represents the set of triggering times of the i-th subsystem, represents the set of non-negative integers, represents the event trigger sampling time of the i-th subsystem and It is determined by the event triggering mechanism.

[0030] For i = 0, 1, ..., N, define the observation error and estimation error So, we get

[0031]

[0032] set up and We get

[0033]

[0034] in For i≠j, and IN and are N-dimensional and n0-dimensional unit vectors respectively.

[0035] Since N1 is a Hurwitz matrix, for any given symmetric positive definite matrix Q1, there is always another symmetric positive definite matrix P1 such that

[0036] Furthermore, for i=1,…,N, let the position estimation error between the follower subsystem and the event-triggered observer be and velocity estimation error set up

[0037]

[0038] Where α0 is a positive real number. We further get

[0039]

[0040] According to Property 3, there exists a known matrix and an uncertain parameter vector Θ i , making

[0041]

[0042] set up

[0043]

[0044] From (3), (6) and (9), we have

[0045]

[0046] in

[0047] Then, for i=1,…,N, we consider the distributed event-triggered control law as follows:

[0048]

[0049] in

[0050]

[0051] is a symmetric positive definite matrix, γ i 、k i1 and k i2 are some positive real numbers.

[0052] Furthermore, define

[0053]

[0054] Then, substituting the control law (11) into formula (1) yields

[0055]

[0056] Combining (8) and (10) we can further obtain

[0057]

[0058] in

[0059] Furthermore, the Lyapunov function is selected as

[0060]

[0061] Based on property 2, we have

[0062]

[0063] set up

[0064]

[0065] in k1>1 and 0<λ1<1.

[0066] For i=1,…,N, we consider the event triggering mechanism as follows:

[0067]

[0068] Where β1, β2 and β3 are positive numbers and satisfy β1>0, and β4>0.

[0069] Furthermore,

[0070]

[0071] c7=c2+α0c6+c5‖C0‖, b8=b6+b1‖C0‖, c8=c6+c1‖C0‖#

[0072] b9=b7+b4‖C0‖,c9=c7+c4‖C0‖#(20)

[0073] in It is a compact set, α0>a0.

[0074] According to (15), (17), (18) and Property 1, we get that the Lyapunov function (15) satisfies

[0075]

[0076]

[0077] Combining (21), (22) and (23), we get

[0078]

[0079] Furthermore, based on the event triggering mechanism (19), we get

[0080]

[0081] According to (25), we get

[0082]

[0083] According to the definition of V0(t), we get satisfy

[0084]

[0085] and s i (t) Satisfaction

[0086]

[0087] According to the event triggering mechanism (19), we get

[0088]

[0089] According to (29), we can get

[0090]

[0091] because Therefore According to (5), we can get

[0092]

[0093] According to (4), and χ i The definition of Therefore

[0094]

[0095] According to (10), Therefore, we get

[0096]

[0097] From (28), (32) and (33) we can get

[0098]

[0099] Therefore, according to the definition of trajectory tracking error, we get the position tracking error e i (t) Satisfaction

[0100]

[0101] and velocity tracking error satisfy

[0102]

[0103] Furthermore, from (35) and (36), it can be seen that the trajectory tracking error of the multi-manipulator system can converge to any small neighborhood of the origin in an exponential form. The steady-state tracking error is adjusted by γ, and the convergence speed is adjusted by a0.

[0104] Furthermore, from (11), we can see that the distributed event-triggered control law is discrete and can be directly executed on a digital platform. In addition, from (3), (11), and (19), we can see that the designed distributed event-triggered observer, distributed event-triggered control law, and event-triggered mechanism only need to rely on the discrete information of neighboring subsystems, thus greatly reducing the use of communication resources.

[0105] Compared with existing technologies, the present invention offers the following advantages: Based on an event-triggered control strategy, a distributed event-triggered observer, and adaptive control technology, it designs a distributed event-triggered control law that can be executed directly on a digital platform. This solves the problem of practical tracking of event-triggered collaborative exponentials in multi-manipulator systems, ensuring that the trajectory tracking error converges exponentially to any small neighborhood of the origin, with both the steady-state error and the convergence rate adjustable. Furthermore, the designed distributed event-triggered observer, distributed event-triggered control law, and event-triggered mechanism rely solely on discrete information from neighboring subsystems, significantly reducing the use of communication resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0106] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0107] Figure 1 is a communication network topology diagram in this embodiment;

[0108] Figure 2 is a schematic diagram of the reference trajectory generated by the leader system in this embodiment;

[0109] Figure 3(a) and Figure 3(b) are the parameters γ in this embodiment. i =400 and γ i Schematic diagram of the event trigger function curve under the condition of =800;

[0110] Figure 4(a) and Figure 4(b) are the parameters γ in this embodiment. i Schematic diagram of the tracking error curves of the two position state components under the condition of =400;

[0111] Figure 5(a) and Figure 5(b) are the parameters γ in this embodiment. i Schematic diagram of the tracking error curves of the two position state components under the condition of =800;

[0112] Figure 6(a) and Figure 6(b) are the parameters γ in this embodiment. i Schematic diagram of the tracking error curves of the two velocity state components under the condition of =400;

[0113] Figure 7(a) and Figure 7(b) are the parameters γ in this embodiment. i Schematic diagram of the tracking error curves of the two velocity state components under the condition of =800;

[0114] Figure 8 This is a flow chart of a practical tracking control method for an event-triggered collaborative index of a multi-manipulator system according to this embodiment. DETAILED DESCRIPTION

[0115] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0116] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings.

[0117] This embodiment provides a practical tracking control method for event-triggered collaborative index of a multi-manipulator system, which solves the collaborative trajectory tracking problem of the multi-manipulator system, including the following steps:

[0118] Step 1: Construct a multi-agent system consisting of N follower subsystems described by Euler-Lagrange equations and a leader system described by a linear system. The Euler-Lagrange equations represent the dynamic model of the robotic arm system, and the linear system generates the reference trajectory information of the multi-robotic arm system.

[0119] Step 2: Design a distributed event-triggered observer to observe the status information of the leader system for each follower subsystem;

[0120] Step 3: Design a distributed event-triggered feedback control law and event-triggered mechanism to solve the practical tracking problem of event-triggered cooperative index for multi-Euler-Lagrangian systems.

[0121] Step 4: Use the Lyapunov method to prove the stability of the closed-loop system and ensure that the trajectory tracking error can converge exponentially to any small neighborhood of the origin;

[0122] Step 5: Verify the effectiveness of the proposed control strategy through MATLAB simulation.

[0123] As an embodiment, in step 1, further, the multi-manipulator system can be described by the following N Euler-Lagrange equations:

[0124]

[0125] in represents the generalized position vector, represents the generalized velocity vector, represents the generalized inertia matrix, denotes the Coriolis and centripetal force vectors, represents the gravity vector, represents the generalized force vector. The Euler-Lagrange system satisfies the following three standard properties:

[0126] Property 1: Matrix M i (q i ) is a symmetric positive definite matrix, the matrix is an antisymmetric matrix.

[0127] Property 2: There exist some positive constants μ1, μ2, μ3, μ4 and n-dimensional identity matrix I such that for any q i , μ1I≤M i (q i )≤μ2I, and ‖G i (q u )‖≤μ4.

[0128] Property 3: For any in is a known regression matrix, is an uncertain parameter vector.

[0129] Assuming a reference trajectory Generated by a leader system as follows:

[0130]

[0131] in is the state of the leader system, and are the system matrix and output matrix respectively, and both are composed of constants.

[0132] A multi-agent system with N+1 agents can be composed of a follower subsystem and a leader system, where the follower subsystem and the leader system are represented by formula (1) and formula (2) respectively. We can use To define a multi-agent system, where the node set Edge Set The i-th follower subsystem in the follower subsystem can be represented by node i=1,…,N, and the leader system can be represented by node 0. The neighbor set of the i-th subsystem can be defined as The weighted adjacency matrix can be defined as in And for i≠j, when hour otherwise

[0133] For each follower system i=1,…,N and the leader system, the position tracking error of the trajectory is defined as e i =q i -q0 and velocity tracking error are

[0134] To solve the event-triggered collaborative tracking control problem of a multi-manipulator system, we need the following two assumptions:

[0135] Assumption 1: All eigenvalues ​​of the matrix S0 are semisimple and have zero real part.

[0136] Assumption 2: In In the network, every node i=1,…,N is reachable from node 0.

[0137] Define symbols λ min (X) and λ max (X) represents the smallest eigenvalue and the largest eigenvalue of the matrix X, respectively, and ‖x0‖ or ‖X0‖ represents the Euclidean norm of the vector x0 or the matrix X0.

[0138] Step 2: We further introduce a distributed event-triggered observer, which can be viewed as a dynamic compensator and can estimate the state of the leader system for each follower system i=1,…,N. The distributed event-triggered observer can be described as follows:

[0139]

[0140] in is the state of the distributed event-triggered observer, μ0 is an arbitrary positive number, yes The estimated state and only the discrete state Right now Decide, represents the set of triggering times of the i-th subsystem, represents the set of non-negative integers, represents the event trigger sampling time of the i-th subsystem and It is determined by the event triggering mechanism.

[0141] For i = 0, 1, ..., N, define the observation error and estimation error So, we get

[0142]

[0143] set up and We get

[0144]

[0145] in For i≠j, and I N and are N-dimensional and n0-dimensional unit vectors respectively.

[0146] Since N1 is a Hurwitz matrix, for any given symmetric positive definite matrix Q1, there is always another symmetric positive definite matrix P1 such that

[0147] Step 3: Further, for i=1,…,N, let the position estimation error between the follower subsystem and the event-triggered observer be and velocity estimation error set up

[0148]

[0149] Where α0 is a positive real number. We further get

[0150]

[0151] According to Property 3, there exists a known matrix and an uncertain parameter vector Θ i , making

[0152]

[0153] set up

[0154]

[0155] From (3), (6) and (9), we have

[0156]

[0157] in

[0158] Then, for i=1,…,N, we consider the distributed event-triggered control law as follows:

[0159]

[0160] in

[0161]

[0162] is a symmetric positive definite matrix, γ i 、k i1 and k i2 are some positive real numbers.

[0163] Furthermore, define

[0164]

[0165] Then, substituting the control law (11) into formula (1) yields

[0166]

[0167] Combining (8) and (10) we can further obtain

[0168]

[0169] in

[0170] Furthermore, the Lyapunov function is selected as

[0171]

[0172] Based on property 2, we have

[0173]

[0174] set up

[0175]

[0176] in k1>1 and 0<λ1<1.

[0177] Furthermore, for i=1,…,N, we consider the event triggering mechanism as follows:

[0178]

[0179] Where β1, β2 and β3 are positive numbers and satisfy β1>0, and β4>0.

[0180] Step 4: Further, set

[0181]

[0182] c7=c2+α0c6+c5‖C0‖, b8=b6+b1‖C0‖, c8=c6+c1‖C0‖#

[0183] b9=b7+b4‖C0‖,c9=c7+c4‖C0‖#(20)

[0184] in It is a compact set, α0>a0.

[0185] According to (15), (17), (18) and Property 1, we get that the Lyapunov function (15) satisfies

[0186]

[0187]

[0188] Combining (21), (22) and (23), we get

[0189]

[0190] Furthermore, based on the event triggering mechanism (19), we get

[0191]

[0192] According to (25), we get

[0193]

[0194] According to the definition of V0(t), we get satisfy

[0195]

[0196] and s i (t) Satisfaction

[0197]

[0198] According to the event triggering mechanism (19), we get

[0199]

[0200] According to (29), we can get

[0201]

[0202] because Therefore According to (5), we can get

[0203]

[0204] According to (4), and χ i The definition of Therefore

[0205]

[0206] According to (10), Therefore, we get

[0207]

[0208] From (28), (32) and (33) we can get

[0209]

[0210] Therefore, according to the definition of trajectory tracking error, we get the position tracking error e i (t) Satisfaction

[0211]

[0212] and velocity tracking error satisfy

[0213]

[0214] Furthermore, from (35) and (36), it can be seen that the trajectory tracking error of the multi-manipulator system can converge to any small neighborhood of the origin in an exponential form. The steady-state tracking error is adjusted by γ, and the convergence speed is adjusted by a0.

[0215] Furthermore, from (11), we can see that the distributed event-triggered control law is discrete and can be directly executed on a digital platform. In addition, from (3), (11), and (19), we can see that the designed distributed event-triggered observer, distributed event-triggered control law, and event-triggered mechanism only need to rely on the discrete information of neighboring subsystems, thus greatly reducing the use of communication resources.

[0216] Step 5. In this example, we consider a multi-Eulerian-Lagrangian system consisting of three two-degree-of-freedom manipulators, whose dynamic equations are as follows:

[0217]

[0218] where q i =col(q i1 ,q i2 ), τ i =col(τ i1 ,τ i2 ),

[0219]

[0220] where g is the gravity vector, Θ i =col(Θ i1 ,Θ i2 ,Θ i3 ,Θ i4 ,Θ i5 ). Let the reference trajectory q0 = col(q 01 ,q 02 ), where q 01 (t) = sin 0.5t and q 02 (t) = cos 0.5t, which can be generated by (2),

[0221]

[0222] And v0(0)=col(0,1). Obviously, for i=1,2,3, there is e i =col(e i1 ,e i2 )and It is represented as a communication network topology diagram, such as Figure 1 shown. From (38) and Figure 1 It is easy to see that assumptions 1 and 2 are satisfied. For i=1, 2, 3, the controller parameters are selected as follows:

[0223] g=9.8,μ0=1,α0=1,γ i =400,γ i =800,k i1 =96,k i2 =92#

[0224] Λ i =100I, λ=0.2, Q1=2I, β1=50, β2=2, β4=200#(40)

[0225] Where I is an identity matrix of appropriate dimensions. The initial values ​​are chosen as follows:

[0226] Θ1=col(0.6,1.1,0.1,0.5,0.3),Θ2=col(0.8,1.2,0.1,0.8,0.4)#

[0227]

[0228] According to the above parameters, we perform simulation based on the distributed event-triggered control law (11) under the event-triggered mechanism (19) and obtain the simulation results as follows: Figure 2 -As shown in Figure 7. Figure 2 represents the reference trajectory generated by the leader system. Figure 3(a) and Figure 3(b) respectively represent γ i =400 and γ i =800 two sets of event triggering functions under different parameter conditions. Figure 4 (a) and Figure 4 (b) show the parameter γ i =400, the tracking error curves of the two position state components, Figure 5 (a) and Figure 5 (b) show the parameter γ i = 800, the tracking error curves of the two position state components, Figure 6 (a) and Figure 6 (b) show the parameter γ i =400, the tracking error curves of the two velocity state components, Figure 7 (a) and Figure 7 (b) show the parameter γ i = 800 under the condition of the tracking error curve of the two speed state components. As shown in Figure 4-Figure 7, under the condition of γ i =400 and γ i =800 Under two different sets of parameter conditions, the position and velocity tracking errors of the system can converge in an exponential form, and by selecting different parameters, the trajectory tracking error can be guaranteed to converge to an arbitrarily small neighborhood of the origin.

[0229] The above is the content of the present invention. Any changes made according to the technical solution of the present invention, as long as the resulting functions and effects do not exceed the scope of the technical solution of the present invention, shall fall within the scope of protection of the present invention.

Claims

1. A practical tracking control method for event-triggered synergy index of a multi-manipulator system, characterized in that: The following steps are involved: Step 1: Construct a multi-agent system consisting of N follower subsystems described by Euler-Lagrange equations and a leader system described by a linear system. The Euler-Lagrange equations represent the dynamic model of the robotic arm system, and the linear system generates the reference trajectory information of the multi-robotic arm system. Step 2: Design a distributed event-triggered observer to observe the status information of the leader system for each follower subsystem; Step 3: Based on the state information, a distributed event-triggered feedback control law and an event-triggered mechanism are designed to solve the event-triggered cooperative exponential tracking problem of the multi-agent system, ensuring that the trajectory tracking error can converge exponentially to any small neighborhood of the origin; The multi-follower subsystem can be described by the following N Euler-Lagrange equations: in represents the generalized position vector, represents the generalized velocity vector, represents the generalized inertia matrix, denotes the Coriolis and centripetal force vectors, represents the gravity vector, represents the generalized force vector; the follower subsystem satisfies the following three standard properties: Property 1: Matrix M i (q i ) is a symmetric positive definite matrix, the matrix is an antisymmetric matrix; Property 2: There exist some positive constants μ1, μ2, μ3, μ4 and n-dimensional identity matrix I such that for any q i , μ1I≤M i (q i )≤μ2I, and ‖G i (q i )‖≤μ4; Property 3: For any vector in is a known regression matrix, is an uncertain parameter vector; Reference trajectory Generated by a leader system as follows: in is the state of the leader system, and They are the system matrix and the output matrix, and both are composed of constants; A multi-agent system with N+1 agents consists of a follower subsystem and a leader system, where the follower subsystem and the leader system are represented by formula (1) and formula (2) respectively; To define a multi-agent system, where the node set Edge Set The i-th follower subsystem in the follower subsystem is represented by node i=1,…,N, and the leader system is represented by node 0; The neighbor set of the i-th follower subsystem is defined as The weighted adjacency matrix is ​​defined as in And for i≠j, when hour otherwise For each follower subsystem i=1,…,N and the leader system, the position tracking error of the trajectory is defined as e i =q i -q0 and velocity tracking error are In order to solve the event-triggered cooperative tracking control problem of multi-agent systems, the following two assumptions are proposed: Assumption 1: All eigenvalues ​​of the matrix S0 are semisimple and have zero real part; Assumption 2: In In the example, every node i=1,…,N is reachable from node 0; Define symbols λ min (X) and λ max (X) represents the smallest eigenvalue and the largest eigenvalue of the matrix X, respectively, and ‖x0‖ or ‖X0‖ represents the Euclidean norm of the vector x0 or the matrix X0.

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