Time-triggered intermittent control protocol method for leadership following consistency of multi-agent system

Abstract

The invention discloses a time-triggered intermittent control protocol method for leadership following consistency of a multi-agent system, and the method comprises the steps: 1, constructing a multi-agent system which comprises a leadership and # imgabs0 followers; 2, constructing a time triggering mechanism; 3, constructing a mathematical representation form of the time-triggered intermittent control protocol; 4, constructing an error system; 5, establishing a sufficiency criterion of error system index stability; and step 6, proposing a time-triggered intermittent control protocol design method. The constructed mathematical expression of the time-triggered intermittent control protocol further comprises variables m and m1 to be designed in addition to a control gain matrix to be designed; when the exponential stability of the error system is established, the constructed mixed Lyapunov functional is irrelevant to the number of agents, and is only required to be positive definite at the sampling moment; the time-triggered intermittent control protocol method is provided by taking an intermittent ratio as a performance index, and the information transmission quantity between intelligent agents can be effectively reduced.

Description

Technical Field

[0001] The present invention belongs to the field of automatic control technology, and in particular to a time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system. Background Art

[0002] A multi-agent system is a complex system composed of multiple agents that can exchange information. Each agent has independent dynamics and certain sensing, computing and execution capabilities. Compared with a single agent, a multi-agent system completes complex tasks through communication between agents. The consistency of a multi-agent system is achieved by designing a distributed control protocol to make the state of each agent consistent. It is widely used in engineering fields such as smart grid frequency synchronization, satellite attitude adjustment and drone combat.

[0003] Intermittent control protocol is an effective method to achieve consistency in multi-agent systems. The characteristic of intermittent control protocol is that the control signal of each agent exists in the control time interval and disappears in the free time interval. Compared with continuous control protocol, intermittent control protocol only relies on the information of the agent and its neighboring agents in the control time interval and acts on the agent through actuators; compared with pulse control protocol, the control signal of intermittent control protocol is continuous in the control time interval, which can achieve more ideal performance. Intermittent control protocol is between continuous control protocol and pulse control protocol, and has their comprehensive advantages, achieving satisfactory performance while saving resources.

[0004] The existing intermittent control protocol methods mainly include periodic intermittent control protocol, non-periodic intermittent control protocol, intermittent sampling data control protocol and intermittent control protocol based on sampling data. Periodic intermittent control protocol and non-periodic intermittent control protocol rely on the real-time status or output information of the agent itself and its neighboring agents. Intermittent sampling data control protocol relies on the status or output information of all sampling moments, and divides each sampling interval into a control time interval and a free time interval, which will cause the actuator to frequently "turn on" and "turn off". Although the intermittent control protocol based on sampling data only relies on the status or output information of the sampling moment in the control time interval, it requires the number of sampling intervals in the control time interval and the free time interval to be given in advance, and their values ​​cannot be dynamically designed according to the intermittent ratio, and thus, the amount of transmitted data cannot be effectively reduced. Therefore, the existing intermittent control protocol methods have not yet solved the problem of reducing the amount of information transmission between agents. Summary of the invention

[0005] In view of the shortcomings of the prior art, the technical problem to be solved by the present invention is to provide a time-triggered intermittent control protocol method that can effectively reduce the amount of information transmission between intelligent agents, thereby achieving leadership and following consistency in a multi-agent system.

[0006] In order to solve the above technical problems, the technical solution adopted by the present invention is: a time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system, comprising the following steps:

[0007] Step 1: Construct a multi-agent system, which includes a leader and Followers;

[0008] Step 2: Build a time trigger mechanism;

[0009] Step 3: Construct a mathematical representation of the time-triggered intermittent control protocol;

[0010] Step 4: Construct the error system;

[0011] Step 5: Establish the sufficiency criterion for exponential stability of the error system;

[0012] Step 6: Propose a design method for a time-triggered intermittent control protocol.

[0013] In the above-mentioned time-triggered intermittent control method for leader-follower consistency of a multi-agent system, in step 1, the specific process of constructing a multi-agent system is as follows:

[0014]

[0015] in, are the state and control input of the ith agent, The state of being a leader, and is a known constant matrix.

[0016] In the above-mentioned multi-agent system leader-follower consistency time-triggered intermittent control protocol method, in step 2, the specific process of constructing the time trigger mechanism is as follows: Assuming that all agents are synchronously and non-periodically sampled, the sampling time sequence is recorded as: when When and m≥2, The quotient and remainder after division by m are denoted k and l respectively; therefore, and l∈{0,1,2,...,m-1}; for and m1<m, the time trigger mechanism is described as follows: x i (t km+l ) for transmission; otherwise, no transmission is performed; wherein m represents the total number of sampling intervals in each operation cycle, and m1 represents the number of sampling intervals in the control time interval.

[0017] In the above-mentioned time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system, in step 3, the specific process of constructing the mathematical representation of the time-triggered intermittent control protocol is as follows:

[0018] The time-triggered control protocol for the i-th agent is designed as follows

[0019]

[0020] is the control gain matrix to be designed, x0(t km+l ) and x i (t km+l ) represent the leader and the i-th agent at sampling time t km+l The state of. Define the time interval T km+l =[t km+l ,t km+l+1 )and They are called the control time interval and the free time interval respectively; the difference between any two consecutive sampling moments is the sampling period, denoted as t km+l+1 -t km+l =h km+l ∈[h n ,h N ], l∈{0,1,2,...,m}, where h n and h N are the lower and upper bounds of the sampling period respectively.

[0021] In the above-mentioned time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system, in step 4, the process of obtaining the error system through the mathematical representation of the time-triggered intermittent control protocol and the state equation of the agent system is as follows:

[0022] Definition i (t) = x i (t)-x0(t), the following error system is obtained through the control protocol and the system state equation

[0023]

[0024] in, express dimensional identity matrix, represents the Kronecker product,

[0025] In the above-mentioned time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system, in step 5, a sufficient criterion for exponential stability of the error system is established, and the specific process is as follows:

[0026] By analyzing the different dynamic characteristics of the error system in the control time interval and the free time interval and constructing a mixed Lyapunov functional, the following sufficient criterion for the exponential stability of the error system is established;

[0027] Theorem 1: For α>0, β>0, d1>0, d2>0, h N ≥h n >0 and the control gain matrix If there exists a positive definite diagonal matrix Positive definite matrix and the symmetric matrix Diagonal Matrix g=1,2,3, any matrix Existence, and So that the following inequality holds for Establishment

[0028]

[0029]

[0030] m<d2,

[0031] (α(h n -h N )-βh N )m+(α+β)h N m1>d1,

[0032] m-m1>0,

[0033] in,

[0034]

[0035] Π1=[r1 r2 r4], Π2=[r3 r0 r1], Π3=[r2 r3],

[0036] Y1=-Sym{[E2 E20]}, Y2=Sym{[E1-E12E2]},

[0037] Then the error system is exponentially stable and its decay rate is

[0038] The above-mentioned time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system, in step 6, the specific process of the time-triggered intermittent control protocol design method is as follows:

[0039] Using the linear matrix inequality technique and the sufficient criterion for the exponential stability of the error system established in step 5, the following theorem is given to ensure that the multi-agent system achieves exponential consistency under the time-triggered intermittent control protocol;

[0040] Theorem 2: For α>0, β>0, d1>0, d2>0, h N ≥h n >0,θ g (g=1,2,3), if there exists a positive definite diagonal matrix Positive definite matrix and the symmetric matrix Diagonal Matrix Any matrix as well as So that the following inequality holds for Establishment

[0041]

[0042] m<d2,

[0043] (α(h n -h N )-βh N )m+(α+β)h N m1>d1,

[0044] m-m1>0, where

[0045]

[0046] Π1=[r1 r2 r4], Π2=[r3 r0 r1], Π3=[r2 r3],

[0047]

[0048] Then the multi-agent system achieves exponential consistency under the time-triggered intermittent control protocol, and its control gain matrix is The attenuation rate is

[0049] On this basis, the following time-triggered intermittent control protocol design method is proposed;

[0050] Step 1: Choose α, β, θ1, θ2, θ3, and h n The value of

[0051] Step 2: Use the Yamlmi Optimization Toolbox to solve the first three inequalities in Theorem 2. If h can be solved, N , go to step 3, otherwise, go back to step 1;

[0052] Step 3: Using α, β and h from step 1 n , h in step 2 N The value of d1 and d2 is selected to minimize the last three inequality constraints in Theorem 2. (equivalent to maximizing the interval ratio Calculate m and m1; if there is a solution, go to step 4; otherwise, go to step 1;

[0053] Step 4: Use the above α, β, θ1, θ2, θ3, h n and h N , according to the first three inequalities in Theorem 2, find The advantages of the time-triggered intermittent control protocol method for the leader-follower consistency of the multi-agent system of the present invention are:

[0054] 1. The mathematical expression of the constructed time-triggered intermittent control protocol includes the variables m and m1 to be designed in addition to the control gain matrix to be designed;

[0055] 2. When establishing the exponential stability of the error system, the constructed mixed Lyapunov functional is independent of the number of agents and is only required to be positive definite at the sampling time;

[0056] 3. The proposed time-triggered intermittent control protocol design method uses the intermittent ratio as the performance indicator, which can effectively reduce the amount of information transmission between intelligent agents.

[0057] The time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system of the present invention can be widely applied to actual physical systems, especially robot systems, drone formations, vehicle systems, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 A schematic diagram of information transmission between intelligent agents of the present invention;

[0059] Figure 2 is the topological diagram of the multi-agent system;

[0060] Figure 3 is a graph showing the variation of sampling period over time;

[0061] Figure 4 is the trajectory curve of the first component of the agent state;

[0062] Figure 5 is the trajectory curve of the second component of the agent state;

[0063] Figure 6 This is the output curve of the time-triggered intermittent control protocol. DETAILED DESCRIPTION

[0064] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0065] In the present invention, unless otherwise stated, directional words such as "upper" and "lower" generally refer to the upper and lower parts of the device in actual use or working state, specifically the drawing direction in the attached drawings; while "inside" and "outside" refer to the outline of the device. In addition, in the description of this application, the term "including" means "including but not limited to". The terms first, second, third, etc. are used only as labels and do not impose numerical requirements or establish an order. The term "plurality" means "two or more".

[0066] A time-triggered intermittent control protocol method for leader-follower consistency in a multi-agent system comprises the following steps:

[0067] Step 1: Construct a multi-agent system, which includes a leader and Followers. The specific process is as follows:

[0068] The relationships between agents are shown in the topological graph. Indicates. Among them, is a non-empty set of nodes, represents the edge set, is a weighted adjacency matrix. Means node From the node Get status information. Each node has no influence on itself, that is, if Weighted edge ω ij > 0, otherwise, ω ij = 0. If Topology is an undirected graph. The Laplace matrix of When i=j, When i≠j, l ij =-ω ij .

[0069] When a leader is added to graph G, a new topology is formed in, is the set of all nodes, is an edge set, is a weighted adjacency matrix. The relationship between agents can be represented by the matrix It indicates that, is the Laplace matrix when there is no leader, Represents the relationship between the leader and the follower. If Available Information 0i >0, otherwise, l 0i =0.

[0070] Information transmission between agents is as follows Figure 1 The specific process of building a multi-agent system is as follows:

[0071]

[0072] in, are the state and control input of the ith agent, The state of being a leader, and is a known constant matrix.

[0073] Step 2: Construct a time trigger mechanism. The specific process is as follows: Assume that all agents are synchronously and non-periodically sampled, and the sampling time sequence is recorded as: when When and m≥2, The quotient and remainder after division by m are denoted k and l respectively; therefore, and l∈{0,1,2,...,m-1}; for and m1<m, the time trigger mechanism is described as follows: x i (t km+l ) for transmission; otherwise, no transmission is performed; wherein m represents the total number of sampling intervals in each operation cycle, and m1 represents the number of sampling intervals in the control time interval.

[0074] Step 3: Build a time-triggered intermittent control protocol. The specific process is as follows:

[0075] The time-triggered control protocol of the i-th agent is designed as follows:

[0076]

[0077] is the control gain matrix to be designed, x0(t km+l ) and x i (t km+l ) represent the leader and the i-th agent at sampling time t km+l The state of. Define the time interval T km+l =[t km+l ,t km+l+1 )and They are called the control time interval and the free time interval respectively. The difference between any two consecutive sampling moments is the sampling period, denoted by t km+l+1 -t km+l =h km+l ∈[h n ,h N ], l∈{0,1,2,...,m}, where h n and h Nare the lower and upper bounds of the sampling period respectively.

[0078] Step 4: Construct the error system. The specific process is as follows:

[0079] Definition i (t) = x i (t)-x0(t), the following error system is obtained through the control protocol (2) and the system state equation (1):

[0080]

[0081] in, express dimensional identity matrix, represents the Kronecker product,

[0082] Step 5: Establish sufficient criteria for exponential stability of error system. The specific process is as follows:

[0083] The following Lemma 1 plays a key role in proving the exponential stability of the error system.

[0084] Lemma 1: The state vector of system (3) satisfies the following condition

[0085]

[0086] in, Proof: When t∈T km+l , When , we can know from (3) and the Grown-Bellman lemma that

[0087]

[0088] when From (3) and the Grown-Bellman lemma, we know that

[0089]

[0090] The proof is complete.

[0091] Sym{X} means X+X T , * is the symmetric term in the matrix. The vector is defined as follows:

[0092]

[0093] r0=[0 n 0 n 0 n 0 n ] T , r1=[I n 0n 0 n 0 n ] T , r2=[0 n I n 0 n 0 n ] T ,

[0094] r3=[0 n 0 n I n 0 n ] T , r3=[0 n 0 n 0 n I n ] T .

[0095] The sufficient criterion for the exponential stability of the error system is given by Theorem 1.

[0096] Theorem 1: For α>0, β>0, d1>0, d2>0, h N ≥h n >0 and the control gain matrix If there exists a positive definite diagonal matrix Positive definite matrix and the symmetric matrix Diagonal Matrix g=1,2,3, any matrix So that the following inequality holds for Established,

[0097]

[0098] m<d2, (9)

[0099] (α(h n -h N )-βh N )m+(α+β)h N m1>d1, (10)

[0100] m-m1>0, (11)

[0101] in,

[0102]

[0103]

[0104] Π1=[r1 r2 r4], Π2=[r3 r0 r1], Π3=[r2 r3],

[0105] Y1=-Sym{[E2 E20]}, Y2=Sym{[E1-E1 2E2]},

[0106] Then the error system (3) is exponentially stable, and its decay rate is described as Proof: For system (3), the mixed Lyapunov functional is constructed as follows

[0107]

[0108] in,

[0109]

[0110] because V(t) is continuous and positive definite at the sampling time.

[0111] When t∈T km+l , When V1(t) is differentiated with respect to t along the first equation of system (3), and 2αV1(t) is added, we get

[0112]

[0113] According to the integral inequality of free matrices, we have

[0114]

[0115] For any G1, G2, G3, according to the error system (3), the following equation holds:

[0116]

[0117] According to equations (15) to (19), we know that when t∈T km+l , Sometimes, there is

[0118]

[0119] in, Obviously, the following equation holds

[0120]

[0121] From equations (6) and (21), we can know that Ψ1+h km+l Ψ2<0. From equations (7) and (22) and the Schur complement lemma, we can get Ψ1+h km+l Ψ3<0. According to formula (20), when t∈T km+l , Sometimes, there are

[0122]

[0123] when When , we calculate V2(t) along the second equation of system (3) and take the derivative with respect to time t, and add -2βV2(t), we get

[0124]

[0125] Furthermore, there are

[0126]

[0127] When t∈T km+l , When , according to equations (23) and (24), we have

[0128]

[0129] According to Lemma 1, (10) and (25), we can get

[0130]

[0131]

[0132] in,

[0133] when From equations (10), (23) and (24), we can derive

[0134]

[0135] According to Lemma 1, equations (10) and (27), we have

[0136]

[0137]

[0138] From equations (26) and (28), we can see that for t∈[t0,+∞), the following inequalities hold:

[0139]

[0140] Among them, σ max =max{σ1,σ2}. Therefore, the error system (3) is exponentially stable. The proof is complete.

[0141] Step 6: Propose a design method for a time-triggered intermittent control protocol. The specific process is as follows:

[0142] Based on Theorem 1, using linear matrix inequality technology, we obtain the following Theorem 2.

[0143] Theorem 2: For α>0, β>0, d1>0, d2>0, h N ≥h n >0,θ g (g=1,2,3), if there exists a positive definite diagonal matrix Positive definite matrix and the symmetric matrix Diagonal Matrix Any matrix as well as So that the following inequality holds for Establishment

[0144]

[0145] m<d2, (33)

[0146] (α(h n -h N )-βh N )m+(α+β)h N m1>d1,(34)

[0147] m-m1>0,(35)

[0148] in,

[0149]

[0150] Π1=[r1 r2 r4], Π2=[r3 r0 r1], Π3=[r2 r3],

[0151]

[0152] Then the multi-agent system (1) achieves exponential consensus under the time-triggered intermittent control protocol (2), and its control gain matrix is The attenuation rate is Proof: Let G θ =θ g G(θ=1,2,3),

[0153] By multiplying (30) before and after and Z4, (31) multiplied by and Z6, (32) multiplied by and Z2, it is not difficult to see that (6), (7) and (8) are equivalent to (30), (31) and (32) respectively. By Theorem 1, the error system is exponentially stable. Furthermore, the multi-agent system (1) achieves exponential consistency under the time-triggered intermittent control protocol (2). The proof is complete.

[0154] Based on Theorem 2, the following time-triggered intermittent control protocol design method is proposed.

[0155] The design method of time-triggered intermittent control protocol is:

[0156] Step 1: Choose α, β, θ1, θ2, θ3, and h n The value of .

[0157] Step 2: Use the YamlmiP optimization toolbox to solve conditions (30) to (32). If h can be solved N , go to step 3. Otherwise, go back to step 1.

[0158] Step 3: Using α, β and h from step 1 n , h in step 2 N The value of d1 and d2 is taken to minimize under conditions (33) to (35). (equivalent to maximizing the interval ratio Calculate m and m1. If a solution exists, go to step 4; otherwise, go to step 1.

[0159] Step 4: Use the above α, β, θ1, θ2, θ3, h n and h N , according to conditions (30) to (32), find

[0160] In the simulation, consider Figure 2 The linear multi-agent system shown in Figure 1 consists of a leader and three followers, where the parameters in the linear multi-agent system are Laplacian Matrix and

[0161] According to the time-triggered intermittent control protocol design method, let α = 0.15, β = 0.58, θ1 = θ2 = θ3 = 1, h n = 0.1, we can get h N =0.11. Take d1=0.01, d2=60, the calculation result is m=49, m1=40. Then, the control gain matrix is ​​obtained Set the initial state x0 =

[85] T , x1=

[64] T , x2=[2-2] T , x3=

[13] T , draw Figure 3-Figure 6 .Depend on Figure 3 It can be seen that the sampling period varies with time in the range of 0.1 to 0.11. Figure 4 and Figure 5It can be seen that the multi-agent system (1) achieves exponential consistency under the designed time-triggered intermittent control protocol (2). Figure 6 It can be seen that the output signal of the designed time-triggered intermittent control protocol is updated at the non-periodic sampling time in each working interval [0,4.102), [5.161,9.251)..., and disappears in the rest interval [4.102,5.161), [9.251,10.29)..., which is in line with the characteristics of the time-triggered intermittent control protocol.

[0162] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Any changes, modifications, additions or substitutions made by ordinary technicians in this technical field within the essential scope of the present invention should fall within the protection scope of the present invention.

Claims

1. A time-triggered intermittent control protocol method for leader-follower consistency in a multi-agent system, characterized in that: The steps include: Step 1: Construct a multi-agent system, which includes a leader and Followers; Step 2: Build a time trigger mechanism; Step 3: Construct a mathematical representation of the time-triggered intermittent control protocol; Step 4: Construct the error system; Step 5: Establish the sufficiency criterion for exponential stability of the error system; Step 6: Propose a design method for a time-triggered intermittent control protocol.

2. The time-triggered intermittent control method for leader-follower consistency of a multi-agent system according to claim 1 is characterized by: In step 1, the specific process of building a multi-agent system is as follows: in, are the state and control input of the ith agent, The state of being a leader, and is a known constant matrix.

3. The time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system according to claim 1 is characterized by: In step 2, the specific process of constructing the time trigger mechanism is as follows: Assuming that all agents are synchronously and non-periodically sampled, the sampling time sequence is recorded as: when When and m≥2, The quotient and remainder after division by m are denoted k and l respectively; therefore, and l∈{0,1,2,...,m-1}; for and m1<m, the time trigger mechanism is described as follows: x i (t km+l ) for transmission; otherwise, no transmission is performed; wherein m represents the total number of sampling intervals in each operation cycle, and m1 represents the number of sampling intervals in the control time interval.

4. The time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system according to claim 1 is characterized by: In step 3, the specific process of constructing the mathematical representation of the time-triggered intermittent control protocol is as follows: The time-triggered control protocol for the i-th agent is designed as follows is the control gain matrix to be designed, x0(t km+l ) and x i (t km+l ) represent the leader and the i-th agent at sampling time t km+l state; define the time interval T km+l =[t km+l ,t km+l+1 )and They are called the control time interval and the free time interval respectively; the difference between any two consecutive sampling moments is the sampling period, denoted as t km+l+1 -t km+l =h km+l ∈[h n ,h N ], l∈{0,1,2,...,m}, where h n and h N are the lower and upper bounds of the sampling period respectively.

5. The time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system according to claim 1 is characterized in that: In step 4, the process of obtaining the error system through the mathematical representation of the time-triggered intermittent control protocol and the state equation of the intelligent system is as follows: Definition i (t) = x i (t)-x0(t), the following error system is obtained through the control protocol and the system state equation in, express dimensional identity matrix, represents the Kronecker product, 6. The time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system according to claim 1, characterized in that: In step 5, a sufficient criterion for the stability of the error system exponential is established, and the specific process is as follows: By analyzing the different dynamic characteristics of the error system in the control time interval and the free time interval and constructing a mixed Lyapunov functional, the following sufficient criterion for the exponential stability of the error system is established: Theorem 1: For α>0, β>0, d1>0, d2>0, h N ≥h n >0 and the control gain matrix If there exists a positive definite diagonal matrix Positive definite matrix and the symmetric matrix Diagonal Matrix g=1,2,3, any matrix Existence, and So that the following inequality holds for Establishment m<d2, (a(h n -h N )-βh N )m+(a+b)h N m1>d1, m-m1>0, in, Π1=[r1 r2 r4], Π2=[r3 r0 r1], Π3=[r2 r3], Y1=-Sym{[E2 E20]}, Y2=Sym{[E1-E12E2]}, Then the error system is exponentially stable and its decay rate is 7. The time-triggered intermittent control protocol method for leader-follower consistency of a multi-agent system according to claim 1 is characterized by: In step 6, the specific process of the time-triggered intermittent control protocol design method is as follows: Using the linear matrix inequality technique and the sufficient criterion for the exponential stability of the error system established in step 5, the following theorem is given to ensure that the multi-agent system achieves exponential consistency under the time-triggered intermittent control protocol; Theorem 2: For α>0, β>0, d1>0, d2>0, h N ≥h n >0,θ g (g=1,2,3), if there exists a positive definite diagonal matrix Positive definite matrix and the symmetric matrix Diagonal Matrix Any matrix as well as So that the following inequality holds for Establishment m<d2, (a(h n -h N )-βh N )m+(a+b)h N m1>d1, m-m1>0, in, Π1=[r1 r2 r4], Π2=[r3 r0 r1], Π3=[r2 r3], Then the multi-agent system achieves exponential consistency under the time-triggered intermittent control protocol, and its control gain matrix is The attenuation rate is On this basis, the following time-triggered intermittent control protocol design method is proposed; Step 1: Choose α, β, θ1, θ2, θ3, and h n The value of Step 2: Use the Yamlmi Optimization Toolbox to solve the first three inequalities in Theorem 2. If h can be solved, N , go to step 3, otherwise, go back to step 1; Step 3: Using α, β and h from step 1 n , h in step 2 N The value of d1 and d2 is selected to minimize the last three inequality constraints in Theorem 2. (equivalent to maximizing the interval ratio ), calculate m and m1; if there is a solution, go to step 4; otherwise, go to step 1; Step 4: Use the above α, β, θ1, θ2, θ3, h n and h N , according to the first three inequalities in Theorem 2, find

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