Control method for switching positive system in communication network based on event trigger mechanism

Abstract

The invention discloses a control method for switching a positive system in a communication network based on an event trigger mechanism. The method comprises the following steps: firstly, establishing a switching positive system state space model of a network control positive system, then constructing a controller of the network control switching positive system under an event trigger mechanism, giving MDMDT residence time constraint, and introducing a mode-dependent segmented Lyapunov function. A condition that a network control switching system is positive is designed and verified, a condition that finite time L1 of the network switching positive system is bounded under constraint of different residence time conditions is finally designed and verified, a state feedback controller is given, the control method can ensure that subsystems stably operate within the finite time, the system performance is optimized, and the system performance is improved. And meanwhile, the control benefit of the network control system is ensured.

Description

Technical Field

[0001] The invention belongs to the field of automation technology and modern control technology, and relates to a control method of a switching positive system in a communication network based on an event trigger mechanism. Background Art

[0002] The network control system allows sensors, controllers and actuators distributed in different geographical locations to access the communication network through a reasonable layout, and finally forms a fully distributed closed-loop control system with real-time feedback, breaking the limitation of the past that a single node must be controlled by a single node, and truly achieving full system decentralization, system digitization, multi-node interconnection, communication and control. In addition, the network control system can deconstruct the traditional control system with high coupling and complex internal interactions into a modular distributed system; through reasonable design and architecture in the early stage, different modules undertake different tasks to improve the work efficiency and response speed within the system; the distributed design makes the maintenance of the system more convenient after it goes online, and it is easy to expand and upgrade in different directions in the future. The network control system established through the 5G communication network has been widely used in aerospace, power systems, traffic control, economic management, telemedicine, and dangerous special environments.

[0003] Traditional control is executed in a periodic manner, that is, the time-triggered mechanism, which has the advantages of predictability and easy implementation. However, from the perspective of resource utilization, the time-triggered mechanism will waste communication resources and battery equipment energy. In addition, if the sampling period is small, a large number of redundant signals will be released to the communication network with limited bandwidth, causing network congestion. In order to reduce the load of the network, an event-triggered mechanism can be introduced to avoid the transmission of unnecessary data packets. The event-triggered mechanism refers to the decision of whether to transmit sampled data based on a given event trigger condition. Compared with the time-triggered mechanism, the event-triggered mechanism only transmits signals when the trigger condition is violated, rather than according to the time period, which can save a lot of computing and communication resources. The control system based on the event-triggered mechanism is mainly composed of two key elements: event-triggered control law and feedback controller. The physical device sends the necessary sampling signal to the communication network through the event-triggered mechanism, so that it reaches the controller to generate a control signal, and then transmits the control signal to the actuator through the communication network, and finally transmits the control signal to the physical device to realize control. Therefore, the design of the event-triggered control law plays a key role in the stable operation of the system.

[0004] The switching signal is one of the important components of the switching system. The average dwell time (ADT) is a commonly used switching signal in the prior art. It requires that the average dwell time of the system in each subsystem meets certain constraints. As long as the switching signal meets the constrained dwell time, the system is stable or bounded. However, the actual situation is that due to the limitations of various physical factors, the mode of a system cannot be arbitrarily and quickly switched to another system mode. For example, systems such as traffic lights and the operation of robotic arms must stay for a period of time after being activated. Therefore, a new dwell time constraint needs to be proposed to meet this application scenario. Summary of the invention

[0005] In view of the shortcomings of the prior art, the present invention proposes a control method for a switching positive system in a communication network based on an event-triggered mechanism, uses modern control theory and technology to establish a state space model of the switching control positive system, designs a switching controller for the network control system, ensures the positivity and L1 boundedness of the closed-loop system, and proposes a switching signal based on MDDT constraints to provide an independent upper and lower bound of the residence time for each switching mode, thereby solving the disturbance of the network control system and making up for the shortcomings of the existing residence time constraints.

[0006] An efficient control method for switching positive systems in communication networks based on event triggering mechanism, the specific steps are as follows:

[0007] Step 1: Establish the state space model of the switching positive system of the network control positive system

[0008] Build a state-space model of a continuous-time system for control systems

[0009]

[0010] Where x(t)∈R n , y(t)∈R m ,u(t)∈R u , w(t)∈R r They are the system state, system output, controller input, and disturbance input of the network control system, and the disturbance input satisfies:

[0011]

[0012] Where T f is a given time constant, t0=0 is the initial time, x0 is the initial state of the system; σ(t): [t0, ∞) → N = {1, 2, ... N} is the switching signal of the switching system, N represents the number of subsystems, when σ(t) = i∈ N , indicating that the i-th subsystem is activated. is a constant matrix of suitable dimension.

[0013] Step 2: Design the system’s event triggering mechanism and MDMDT residence time constraints

[0014] s2.1、Construct the controller of the network control switching positive system under the event trigger mechanism:

[0015]

[0016] Among them, F σ(t) ∈R u×n is the event-triggered controller gain matrix and satisfies: is the system at the event triggering time t q Based on the sampling state, the following event trigger mechanism is established:

[0017] ‖x e (t)‖1>α‖x(t)‖1,0<α<1

[0018] in, Represents sampling error.

[0019] Under this event triggering mechanism, the following closed-loop switching positive system is obtained according to the state space model:

[0020]

[0021] s2.2. Set the MDMDT residence time constraint:

[0022] T ki ≥T i * , T ki =t k+1 -t k , t∈[t k , t k+1 ), k∈Z

[0023] The time interval [t k , t k+1 ) is divided into S parts, each part has a length of The interval of the sth part

[0024] Select the Lyapunov function: V(x(t))=x T (t)v σ(t) (t), and but:

[0025]

[0026] in, Taking the derivative with respect to t, we get:

[0027]

[0028] S2.3. Based on the event triggering mechanism and residence time constraints in S2.1 and S2.2, the controller u(t) of the switched positive system is redesigned:

[0029]

[0030] Then we can get a new closed-loop system:

[0031]

[0032] Among them, F i，s ∈R u×n , is modally dependent and piecewise, satisfying

[0033] Step 3: Design the network control switching positive system to be positive:

[0034] s3.1、State positivity of the design system:

[0035] For any initial condition and Next, if Matrix and When the system status

[0036] s3.2、Positivity of design system output:

[0037] For any and Both

[0038] s3.3, Conditions for the system to be positive:

[0039] Design Matrix is a Metzler matrix, and ensure that the matrix where I is the identity matrix of appropriate dimensions, 1 n×n is an n×n matrix with all elements equal to 1.

[0040] Step 4: Solve the event-triggered controller gain:

[0041] when vector and When θ>ε>0, if there exists a constant: vector satisfy:

[0042]

[0043] v i,0 -v j，S <0

[0044]

[0045] Then the closed-loop system in s2.3 is constrained by MDMDT with respect to (ε, θ, T f , d, σ(t)) is bounded in finite time L1, so the event-triggered controller gain satisfies:

[0046] When t∈S is hour:

[0047] when hour,

[0048] The present invention has the following beneficial effects:

[0049] 1. A unique 1-norm-based event triggering mechanism is designed.

[0050] 2. In order to meet the residence time constraint of MDMDT, a new discrete linear copositive Lyapunov function (DLCLF) is proposed for the design of the controller. This function is a mode-dependent piecewise function and can be successfully used in finite-time control based on event-triggered mechanism. The self-affine condition of its underlying control scheme depends on the system matrix Ai and has compatibility with systems with time-varying, uncertain and nonlinear characteristics.

[0051] 3. By performing matrix decomposition on the event-triggered controller gain matrix, the design conditions are expressed in LP format, which provides a more concise form and reduces the computational cost compared to the widely used LMI format. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 Schematic diagram of the networked control system framework.

[0053] Figure 2 Schematic diagram of networked control system model. DETAILED DESCRIPTION

[0054] The present invention will be further explained below with reference to the accompanying drawings;

[0055] The control method of the switching positive system in the communication network based on the event trigger mechanism has the following specific steps:

[0056] Step 1: Establish the state space model of the switching positive system of the network control positive system

[0057] Establish as Figure 1 The networked control system shown in the figure samples the system state by sensors and sends the relevant data of the system state to the event trigger mechanism. The system state after being screened by the event trigger mechanism is sent to the state feedback controller under the MDMDT constraint, and then the control quantity obtained by the state feedback controller is input into the actuator. Finally, the data is fed back to the original system and the above process is repeated. For this networked control system, the state space model of its continuous time system is established:

[0058]

[0059] in, w(t)∈R r They are the system state, system output, controller input, and disturbance input of the network control system, and the disturbance input satisfies:

[0060]

[0061] Where T f is a given time constant, t0=0 is the initial time, x0 is the initial state of the system; σ(t): [t0, ∞) → N =ξ1, 2, ... N} is the switching signal of the switching system, N represents the number of subsystems, when σ(t) = i∈ N , indicating that the i-th subsystem is activated. is a constant matrix of suitable dimension.

[0062] Step 2: Design the system’s event triggering mechanism and MDMDT residence time constraints

[0063] like Figure 2 As shown in Figure 1, the communication network system is divided into two modes: busy and idle, and switches between the two modes according to the number of packets transmitted in the network. In actual communication networks, when there is frequent use or too many users, actuator failures and saturation are inevitable in the system. In order to avoid the collapse of the communication network, an event trigger mechanism needs to be used to limit the network speed of certain communication channels at a certain time.

[0064] s2.1、Construct the controller of the network control switching positive system under the event trigger mechanism:

[0065]

[0066] Among them, F σ(t) ∈R u×n is the event-triggered controller gain matrix and satisfies: is the system at the event triggering time t q Based on the sampling state, the following event trigger mechanism is established:

[0067] ‖x e (t)‖1>α‖x(t)‖1,0<α<1

[0068] in, Represents sampling error.

[0069] Under this event triggering mechanism, the following closed-loop switching positive system is obtained according to the state space model:

[0070]

[0071] s2.2. Set the MDMDT residence time constraint:

[0072] T ki ≥T i * , T ki =t k+1 -t k , t∈[t k , t k+1 ), k∈Z

[0073] The time interval [t k , t k+1 ) is divided into S parts, each part has a length of The interval of the sth part The time interval for each sub-part.

[0074] Select the Lyapunov function: V(x(t))=x T (t)v σ(t) (t), and but:

[0075]

[0076] in, Taking the derivative with respect to t, we get:

[0077]

[0078] S2.3. Based on the event triggering mechanism and residence time constraints in S2.1 and S2.2, the controller u(t) of the switched positive system is redesigned:

[0079]

[0080] Then we can get a new closed-loop system:

[0081]

[0082] Among them, F i，s ∈R u×n is modality-dependent and piecewise, satisfying

[0083] Step 3: Design the network control switching positive system to be positive:

[0084] s3.1、State positivity of the design system:

[0085] For t∈[t0,t1), under any initial conditions Next, according to the event triggering conditions in step 2, we can get:

[0086]

[0087] After simplification, we get: Bring it into the closed-loop system established in step 1:

[0088]

[0089] in, Define a specified set: Ξ={a:x a (t)=0}, for have:

[0090]

[0091] where a∈Ξ, [Φ i ] ab , [D i ] ab They are matrices Φ i , D i The element in the bth row and ath column of i is a Metzler matrix, it can satisfy And in [D i ] ab ≥0 is certain to hold; therefore, for any and Under the initial conditions, we can get thereby The implicit condition As the initial condition for the next time interval [t1, t2).

[0092] Repeat the above steps for have:

[0093]

[0094] Similarly, for any and Under the initial conditions, we can get thereby The implicit condition As the next time interval [t k+1 , t k+2 ) initial conditions.

[0095] Therefore, for any initial condition and Next, if Matrix and hour, That is, the system state is positive.

[0096] s 3.2、Design system output positivity:

[0097] for The closed-loop system in step 2 exists:

[0098]

[0099] Among them [C i ] ab , [E i ] ab Represents the matrix C i 、E i The element in row b and column a, [C i ] ab ,[E i ] ab >0 holds, and for any and Both Thus there is

[0100] s 3.3、Conditions for the system to be positive:

[0101] Design Matrix is a Metzler matrix, and ensure that the matrix where I is the identity matrix of appropriate dimensions, 1 n×n is an n×n matrix with all elements equal to 1.

[0102] Step 4: Solve for the controller gain:

[0103] s4.1、Design the closed-loop system in s2.3 under the MDMDT constraint about (ε, θ, T f , d, σ(t)) is finite-time L1 boundedness condition: vector and Satisfies θ>ε>0, if there exists a constant: vector satisfy:

[0104]

[0105] The event-triggered controller gain satisfies:

[0106] When t∈S is hour:

[0107] When t∈[t k +T i * , t k+1 )hour,

[0108] s4.2. For the closed-loop system under the MDMDT constraint, (ε, θ, T f The verification of the finite-time L1 boundedness condition of , d, σ(t)) is as follows:

[0109] First, we verify the positivity of the closed-loop system based on the existing positivity conditions. It is worth noting that Can guarantee Therefore, from the conditions in step 4.1, we can get:

[0110]

[0111] Substitute the event-triggered controller gain matrix into the matrix:

[0112]

[0113] Its implicit condition is a Metzler matrix, and the closed-loop system is a switched positive system, and at the same time:

[0114] Next, we verify the boundedness of finite time L1 and define the time interval [t k , t k+1 ) is the switching interval, and the switching points are t k With t k+1 ; Time interval [t q , t q+1 ) is the event trigger interval, and the event trigger points are tq and t q+1 , select the linear copositive Lyapunov function designed in step 2.3, and then consider two time intervals:

[0115] s4.2.1. For t∈[t k , t k +T i * ), consider two possibilities for the finite-time stability of the closed-loop system:

[0116] s4.2.1.1、Case 1: Assuming the switching interval [t k , t k +T i * ) does not contain any time trigger point, that is, t q <t k , t q+1 >t k +T i * , taking the derivative of the Lyapunov function in step 2.3 with respect to time t, we get:

[0117]

[0118] Combining the conditions in step 4.2, we can get:

[0119]

[0120] According to the control gain matrix under the event triggering condition and the preset conditions in step 4.1, there is an inequality group:

[0121]

[0122]

[0123]

[0124] Then the inequality about the Lyapunov function can be written as follows:

[0125]

[0126] And then:

[0127]

[0128] Combining the conditions in step 4.1, we can get:

[0129]

[0130] And because ξ1ε <V i，s<ξ2θ, then: Right now:

[0131] From t k Integrating up to t gives:

[0132]

[0133] s4.2.1.2, Case 2: Assuming the switching interval [t k , t k +T i * ) contains several event trigger points, namely Then for t∈[t q+l , t k +T i * ), repeat the steps of case 1, then we have:

[0134]

[0135] Similarly, for t∈[t q+l-1 , t q+l ),exist:

[0136]

[0137] Similarly, for t∈[t q+l-2 , t q+l-1 ),have:

[0138]

[0139] For each event trigger interval [t k , t q+1 ), [t q+1 , t q+2 ),…,[t q+l-1 , t q+l ), [t q+l , t k +T i * ) Repeat the above steps, and according to the conclusions drawn for each event trigger interval, k , t k +T i * ), we can get:

[0140]

[0141] Therefore, regardless of the interval [t k , t k+T i * ) Whether there is an event trigger point, inequality Always holds true.

[0142] s4.2.2 For t∈[t k +T i * , t k+1 ), we still discuss two cases of finite-time stability of closed-loop systems:

[0143] s4.2.2.1, Case 1: Assuming the switching interval [t k +T i * , t k+1 ) does not contain any time trigger point, that is, t q <t k +T i * , t q+1 >t k+1 , taking the derivative of the Lyapunov function in step 2.3 with respect to time t, we get:

[0144]

[0145] Combining the above proofs, we can get:

[0146]

[0147] Therefore, according to the conditions in step 4.1, we can get: And then: Then for this formula from t k +T i * to k+1 Points can be obtained:

[0148]

[0149] s4.2.2.2, Case 2: Assuming the switching interval Contains several event trigger points, namely Then for t∈[t q+l , t k+1 ), then:

[0150]

[0151] Similarly, for t∈[tq+l-1, tq+l), there exists:

[0152]

[0153] For t∈[t q+l-2 , t q+l-1 ),have:

[0154]

[0155] For each event trigger interval [t k +T i * , t q+1 ), [t q+1 , t q+2 ),…,[t q+l-1 , t q+l ), [t q+l , t k+1 ) Repeat the above steps, and according to the conclusions drawn for each event trigger interval, k +T i * , t k+1 ), we can get:

[0156]

[0157] Then combined with the time interval [t k , t k +T i * ) and [t k +T i * , t k+1 ) can be discussed in the time domain [t k , t k+1 )have:

[0158]

[0159] The switching moment t k , there exists σ(t)=i,t∈[t k , t k+1 ),as well as

[0160] Then for the closed-loop system, the Lyapunov function and the conditions in step 4.1 can be obtained:

[0161]

[0162] And then:

[0163]

[0164] For t∈[t0, T f ), from the above formula we can get:

[0165]

[0166] From the conditions in step 4.1, we can get:

[0167]

[0168] V σ(t) (x(t)) = x T (t)v(t)>ξ1x T (t)ε

[0169] Then we have: There are So x T (t)ε<1, so the closed-loop system is about (θ, ε, T f , d, σ(t)) are finite-time stable.

[0170] According to the closed-loop system about (θ, ε, T f , d, σ(t)) finite time stability, thus further proving its finite time L1 boundedness, still choose the Lyapunov function in step 2.3, and also consider two time intervals:

[0171] s4.2.3. For t∈[t k , t k +T i * ), consider two possibilities for the finite-time L1 boundedness of the closed-loop system:

[0172] s4.2.3.1、Case 1: Assuming the switching interval [t k , t k +T i * ) does not contain any time trigger point, that is, t q <t k , t q+1 >t k +T i * , combined with the steps above and the Lyapunov function in step 2.3, we can get:

[0173]

[0174] From t k Integrating to t, we can get:

[0175]

[0176] s4.2.3.2, Case 2: Assuming the switching interval Contains several event trigger points, namely for We can get:

[0177]

[0178] Similarly, for t∈[tq+l-1, tq+l), we have:

[0179]

[0180] For t∈[tq+f-2, tq+l-1), we have:

[0181]

[0182] By analogy, we can get:

[0183]

[0184] In summary, regardless of the switching interval Is there an event trigger point? Inequality V σ(t) (x(t))≤ s4.2.4 is always true We still discuss two cases of finite-time L1 boundedness of closed-loop systems:

[0185] s4.2.4.1, Case 1: Assuming the switching interval Does not contain any time trigger point, that is t q+1 >t k+1 , we can get:

[0186]

[0187] From t k Integrate to t:

[0188]

[0189] s4.2.4.2, Case 2: Assuming the switching interval Contains several event trigger points, namely Then for t∈[t q+l , t k+1 ):

[0190]

[0191] Similarly, for t∈[t q+l-1 , t q+l ),have:

[0192]

[0193] For t∈[t q+l-2 , t q+l-1 )have:

[0194]

[0195] By analogy, we can get:

[0196]

[0197] Then combined with the time interval as well as From the discussion on time domain [t k , t k+1 )have:

[0198]

[0199] Also because is established, then the above equation has the following inequality relationship:

[0200]

[0201] Combined with the zero initial condition, the above formula can be further obtained: Let t = T f , multiply both sides simultaneously Then we have:

[0202]

[0203] In summary, the closed-loop system is about (θ, ε, T f , d,σ(t)) is finite time L1 bounded.

[0204] s4.3. Finally, discuss Zeno's behavior.

[0205] Assume ‖w(t)‖2<η1,η1∈R + , then there will be a sufficiently small positive constant η2 that satisfies ‖x(t)‖2>η2 when the closed-loop system is positive and bounded in finite time L1. According to the designed event triggering conditions, in order to avoid continuous sampling and updating of the control law, a lower bound T * ∈R + Given between any two consecutive samples to analyze Zeno behavior.

[0206] definition: as well as Then according to the relationship between 1-norm and 2-norm: We have: Then for t∈[t k , t k+1 )and Can be obtained from From the definition of , we get:

[0207]

[0208] According to the Cauchy-Schwarz inequality, we get:

[0209]

[0210] Combined with The discussion of the relationship yields:

[0211]

[0212] in: Perform definite integral on both sides of the above equation:

[0213]

[0214] Also because therefore:

[0215]

[0216] In summary, we can get a closed-loop system, in the time domain of the event trigger law [t k , t k+1 ) * for:

[0217]

Claims

1. A control method for a switching system in a communication network based on an event trigger mechanism, characterized in that: The specific steps are as follows: Step 1: Establish a state space model of the network control switching positive system; Step 2: Design the event trigger mechanism of the system, construct the controller of the network control switching positive system under the event trigger mechanism, and transform the event trigger controller gain matrix F σ(t) ∈R u×n Divided into Two parts, where u and n represent the controller input dimension and system state dimension respectively; design the average residence time constraint based on modal dependence; Step 3, design the conditions for the network control switching positive system to be positive; Step 4: Design the finite time L1 boundedness condition of the networked switching positive system under the MDMDT residence time condition constraint, and solve the event-triggered controller gain matrix.

2. The control method of the switching system in the communication network based on the event trigger mechanism as claimed in claim 1, characterized in that: In step 1, the following state space model is established for the control system continuous time system: Where x(t)∈R n , y(t)∈R m ,u(t)∈R u , w(t)∈R r They are the system state, system output, controller input, and disturbance input of the network control system, and the disturbance input satisfies: Where T f is a given time constant, t0=0 is the initial time, x0 is the initial state of the system; σ(t):[t0,∞)→N={1,2,…N} is the switching signal of the switching system, N represents the number of subsystems, when σ(t)=i∈N, it means that the i-th subsystem is excited, is a constant matrix of suitable dimension.

3. The control method of switching positive system in communication network based on event trigger mechanism as claimed in claim 2, characterized in that: Establish the following event trigger mechanism: ||xe(t)||1>α||x(t)||1,0<α<1, in, Represents the sampling error; Under this event triggering mechanism, the following closed-loop switching positive system is obtained according to the state space model: The controller of the network control switching positive system is: Among them, Fσ (t) ∈Ru ×n is the event-triggered controller gain matrix and satisfies: It is the sampling state of the system at the event triggering time tq.

4. The control method of the switching system in the communication network based on the event trigger mechanism as claimed in claim 3, characterized in that: The average dwell time constraint based on modal dependence is: The time interval [tk,tk +1 ) is divided into S parts, each part has a length of The interval of the sth part Select the Lyapunov function: V(x(t))=xT(t)vσ (t) (t), and but: in, Taking the derivative with respect to t, we get:

5. The control method of switching positive system in communication network based on event trigger mechanism as claimed in claim 4, characterized in that: Based on the event triggering mechanism and the dwell time constraint, the controller u(t) of the switched positive system is redesigned: Then the following closed-loop system is obtained:

6. The control method of switching positive system in communication network based on event trigger mechanism as claimed in claim 5, characterized in that: When the matrix is a Metzler matrix, and the matrix The system is positive when , where I is the identity matrix of appropriate dimension, 1 n×n is an n×n matrix with all elements equal to 1; when vector and satisfy When there are constants: ζ>0,ξ1>0,ξ2>0, vector satisfy: ξ1ε<νι ,s <ξ2θ we ,0 -νj ,S <0 Then the closed-loop system is L1 bounded in finite time with respect to (ε,θ,Tf,d,σ(t)) under the MDMDT constraint.

7. The control method of switching positive system in communication network based on event trigger mechanism as claimed in claim 5, characterized in that: The event triggers the controller gain to satisfy: When t∈Sis: when hour,

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