Control methods for switching positive systems based on event triggering mechanisms in communication networks

Abstract

This invention discloses a control method for a switching positive system based on an event-triggered mechanism in a communication network. The method first establishes a state-space model of the switching positive system of the network control system, then constructs a controller for the network control switching positive system under the event-triggered mechanism, and provides MDMDT residence time constraints, introducing a modally dependent piecewise Lyapunov function. The conditions for the network control switching system to be positive are designed and verified. Finally, the finite-time L1 bounded conditions for the networked switching positive system under different residence time constraints are designed and verified, and a state feedback controller is provided. This control method can ensure the smooth operation of the subsystem within a finite time, optimize system performance, and simultaneously guarantee the control benefits of the network control system.

Description

Technical Field

[0001] This invention belongs to the fields of automation technology and modern control technology, and relates to a control method for a switching positive system based on an event-triggered mechanism in a communication network. Background Technology

[0002] Networked control systems allow sensors, controllers, and actuators distributed across different geographical locations to connect to a communication network through a rational layout, ultimately forming a fully distributed, real-time feedback closed-loop control system. This breaks the limitations of past systems that required single-node-to-single-node control, truly achieving fully decentralized, digitalized, and multi-node interconnected communication and control. Furthermore, networked control systems can decompose traditionally highly coupled and complex internally interactive control systems into modular distributed systems. Through rational design and architecture in the early stages, different modules can undertake different tasks, improving the system's internal efficiency and response speed. The distributed design makes post-launch maintenance more convenient and facilitates future expansion and upgrades in different directions. Networked control systems built using 5G communication networks have been widely applied in aerospace, power systems, traffic control, economic management, telemedicine, and hazardous and special environments.

[0003] Traditional control is executed periodically, i.e., time-triggered mechanisms, which have the advantages of predictability and ease of implementation. However, from a resource utilization perspective, time-triggered mechanisms waste communication resources and battery power. Furthermore, if the sampling period is short, a large number of redundant signals will be released into the bandwidth-limited communication network, causing network congestion. To alleviate network load, event-triggered mechanisms can be introduced to avoid sending unnecessary data packets. Event-triggered mechanisms determine whether to transmit sampled data based on pre-defined event trigger conditions. Compared to time-triggered mechanisms, event-triggered mechanisms only transmit signals when the trigger condition is violated, rather than based on a time period, thus saving significant computational and communication resources. Control systems based on event-triggered mechanisms mainly consist of two key elements: event-triggered control laws and feedback controllers. Physical devices send necessary sampled signals to the communication network via the event-triggered mechanism, which then reaches the controller to generate control signals. These control signals are then transmitted to the actuators via the communication network, and finally back to the physical devices to achieve control. Therefore, the design of the event-triggered control law plays a crucial role in the stable operation of the system.

[0004] The switching signal is a crucial component of a switching system. Average Dwell Time (ADT) is a commonly used switching signal in existing technologies. It requires the average dwell time of the system in each subsystem to meet certain constraints. If the switching signal satisfies this dwell time constraint, the system is considered stable or bounded. However, in reality, due to various physical limitations, a system mode cannot arbitrarily and rapidly switch to another system mode. For example, traffic lights and robotic arms require a dwell time after activation. Therefore, a new dwell time constraint is needed to meet the requirements of such applications. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention proposes a control method for a switching positive system based on an event-triggered mechanism in a communication network. It utilizes modern control theory 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. This provides an independent upper and lower bound for the dwell time of each switching mode, addresses disturbances in the network control system, and compensates for the deficiencies of existing dwell time constraints.

[0006] An efficient control method for a handover system based on an event-triggered mechanism in a communication network is proposed, with the following specific steps:

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

[0008] Establish a state-space model of the continuous-time system of the control system.

[0009]

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

[0011]

[0012] Where T f Given a time constant, t0 = 0 is the initial time, and x0 is the initial state of the system; σ(t): [t0, ∞) → N ={1, 2, ..., N} is the switching signal of the switching system, and N represents the number of subsystems. When σ(t) = i∈ N When, it indicates that the i-th subsystem is activated. It is a constant matrix with appropriate dimensions.

[0013] Step 2: Design the system's event triggering mechanism and MMDDT dwell time constraints.

[0014] s2.1 Constructing a controller for the network control switching positive system under an event-triggered mechanism:

[0015]

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

[0017] ||x e (t)‖1>α‖x(t)‖1,0<α<1

[0018] in, This represents the sampling error.

[0019] Under this event-triggered mechanism, the following closed-loop switching positive system is obtained based on the state-space model:

[0020]

[0021] s2.2, Set MDMDT residence time constraints:

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

[0023] Time interval [t] k , t k+1 It is divided into S parts, and the length of each part is... The interval of the s-th part

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

[0025]

[0026] in, Differentiating it with respect to t, we get:

[0027]

[0028] Based on the event triggering mechanism and dwell time constraints in s2.1 and s2.2, the controller u(t) of the switching positive system is redesigned:

[0029]

[0030] This leads to a new closed-loop system:

[0031]

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

[0033] Step 3: Design the conditions for the network control to switch the positive system to positive:

[0034] s3.1, Design the state positivity of the system:

[0035] For any initial conditions and If Matrix and At that time, the system state

[0036] s3.2, Design the positiveness of the system output:

[0037] For any and All

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

[0039] Design Matrix It is a Metzler matrix, and ensures that the matrix... Where I is an identity matrix with appropriate dimensions, 1 n×n It is an n×n matrix in which all elements are 1.

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

[0041] when vector and If a constant exists when θ>ε>0: vector satisfy:

[0042]

[0043] v i,0 -v j，S <0

[0044]

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

[0046] When t∈S is hour:

[0047] when hour,

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

[0049] 1. A unique event triggering mechanism based on the 1-norm was designed.

[0050] 2. To meet the dwell time constraint of MMDDT, a novel discrete linear common positive Lyapunov function (DLCLF) is proposed for the design of the controller. This function is a modally dependent piecewise function that can be successfully applied to finite-time control based on event triggering mechanism. Its underlying control scheme is self-affine conditionally dependent 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 gain matrix of the event-triggered controller, the design conditions are represented in LP format, which provides a more concise form and reduces computational costs compared to the widely used LMI format. Attached Figure Description

[0052] Figure 1 This is a schematic diagram of a networked control system framework.

[0053] Figure 2 This is a schematic diagram of a networked control system model. Detailed Implementation

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

[0055] The control method for a switching positive system based on an event-triggered mechanism in a communication network includes the following steps:

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

[0057] Establish such as Figure 1 The networked control system shown in the diagram samples the system state from sensors and sends the obtained system state data to an event-triggered mechanism. The system state, after being filtered by the event-triggered mechanism, is then sent to a state feedback controller constrained by MDMDT (Multi-Mode Development and Control Theory). The control input obtained from the state feedback controller is then input to the actuator, and finally, the data is fed back to the original system, repeating the above process. For this networked control system, a state-space model of its continuous-time system is established:

[0058]

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

[0060]

[0061] Where T f Given a time constant, t0 = 0 is the initial time, and x0 is the initial state of the system; σ(t): [t0, ∞) → N =ξ1,2,...N} is the switching signal of the switching system, and N represents the number of subsystems. When σ(t) = i∈ N When, it indicates that the i-th subsystem is activated. It is a constant matrix with appropriate dimensions.

[0062] Step 2: Design the system's event triggering mechanism and MMDDT dwell time constraints.

[0063] like Figure 2 As shown, the communication network system operates in two modes: busy and idle, switching between them based on the number of data packets transmitted. In real-world communication networks, actuator failures and saturation inevitably occur when usage is frequent or there are too many users. To prevent network crashes, an event-triggered mechanism is needed to limit the network speed of certain communication channels at specific times.

[0064] s2.1 Constructing a controller for the network control switching positive system under an event-triggered mechanism:

[0065]

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

[0067] ||x e (t)‖1>α‖x(t)‖1,0<α<1

[0068] in, This represents the sampling error.

[0069] Under this event-triggered mechanism, the following closed-loop switching positive system is obtained based on the state-space model:

[0070]

[0071] s2.2, Set MDMDT residence time constraints:

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

[0073] Time interval [t] k , t k+1 It is divided into S parts, and the length of each part is... The interval of the s-th part For each sub-part, there is a time interval.

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

[0075]

[0076] in, Differentiating it with respect to t, we get:

[0077]

[0078] Based on the event triggering mechanism and dwell time constraints in s2.1 and s2.2, the controller u(t) of the switching positive system is redesigned:

[0079]

[0080] This leads to a new closed-loop system:

[0081]

[0082] Among them, F i，s ∈R u×n It is modally dependent and segmented, satisfying

[0083] Step 3: Design the conditions for the network control to switch the positive system to positive:

[0084] s3.1, Design the state positivity of the system:

[0085] For t∈[t0, t1), under any initial conditions Based on the event triggering conditions in step 2, we can obtain the following:

[0086]

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

[0088]

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

[0090]

[0091] Where a∈Ξ, [Φ i ] ab [D] i ] ab Φ i D i The element in the b-th row and a-th column, when Φ i When it is a Metzler matrix, it can satisfy... And in [D i ] ab ≥0 is always true; therefore, for any and Under the initial conditions, we can obtain thereby Implicit conditions 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 obtain thereby Implicit conditions As the next time interval [t] k+1 , t k+2 The initial conditions of ).

[0095] Therefore, for any initial conditions and If Matrix and hour, That is, the system state is positive.

[0096] s 3.2 Design the positive output of the system:

[0097] for The closed-loop system in step 2 has the following characteristics:

[0098]

[0099] Among them [C] i ] ab [E] i ] ab Represent matrix C respectively i E i The element in row b and column a, [C i ] ab [E] i ] ab >0 holds true, and therefore for any and All Therefore,

[0100] s 3.3 Conditions for a positive system:

[0101] Design Matrix It is a Metzler matrix, and ensures that the matrix... Where I is an identity matrix with appropriate dimensions, 1 n×n It is an n×n matrix in which all elements are 1.

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

[0103] s4.1, Design the closed-loop system in s2.3 under MDMDT constraints with respect to (ε, θ, T) f Finite-time L1 boundedness conditions for d, σ(t)): vector and If there exists a constant that satisfies θ > ε > 0: 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, Regarding the closed-loop system under MDMDT constraints with respect to (ε, θ, T) f The verification of the finite-time L1 boundedness condition for d, σ(t) is as follows:

[0109] First, based on the existing positive conditions, we verify the positiveness of the closed-loop system. It is worth noting that... It can be guaranteed Therefore, from the conditions in step 4.1, we can obtain:

[0110]

[0111] Substituting the gain matrix of the event-triggered controller, we have:

[0112]

[0113] Its implicit conditions It is a Metzler matrix, therefore the closed-loop system is a switching positive system, and also has:

[0114] Next, we verify the finite-time L1 boundedness by defining the time interval [t]. k , t k+1 ) represents the switching interval, and the switching points are t and t respectively. k With t k+1 Time interval [t] q , t q+1 ) represents the event triggering interval, and the event triggering points are t and t, respectively.q and t q+1 We 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 the two possibilities for the finite-time stability of the closed-loop system:

[0116] s4.2.1.1, Case 1: Assume the switching interval [t] k , t k +T i * It does not contain any time trigger points, i.e., 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 obtain:

[0119]

[0120] Based on the control gain matrix under the event triggering conditions and the preset conditions in step 4.1, we have a set of inequalities:

[0121]

[0122]

[0123]

[0124] The inequalities concerning Lyapunov functions can then be written in the following form:

[0125]

[0126] Furthermore:

[0127]

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

[0129]

[0130] And because ξ1ε <V i，sIf ξ2θ < 2θ, then: Right now:

[0131] From t in the above equation k Integrating up to t, we get:

[0132]

[0133] s4.2.1.2, Case 2: Assume the switching interval [t] k , t k +T i * It contains several event trigger points, namely Then for t∈[t q+l , t k +T i * If we repeat the steps in 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 based on the conclusions drawn for each event trigger interval, for t∈[t k , t k +T i * From this, we can obtain:

[0140]

[0141] Therefore, regardless of the interval [t] k , t k+T i * Does an event trigger point exist? (Inequality) It always holds true.

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

[0143] s4.2.2.1, Case 1: Assume the switching interval [t] k +T i * , t k+1 It does not contain any time trigger points, i.e., 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] Based on the proof above, we can conclude that:

[0146]

[0147] Therefore, based on the conditions in step 4.1, we can obtain: Furthermore: Then, regarding this expression from t... k +T i * to t k+1 The integral yields:

[0148]

[0149] s4.2.2.2, Case 2: Assuming a switching interval It contains several event trigger points, namely Then for t∈[t q+l , t k+1 Then we have:

[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 based on the conclusions drawn for each event trigger interval, for t∈[t k +T i * , t k+1 From this, we can obtain:

[0156]

[0157] Then, combining the time interval [t] k , t k +T i * ) and [t k +T i * , t k+1 From the discussion of ), we can obtain the following for the time domain [t] k , t k+1 )have:

[0158]

[0159] For the instant of switching t k There exists σ(t) = i, t ∈ [t k , t k+1 ),as well as

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

[0161]

[0162] Furthermore:

[0163]

[0164] For t∈[t0, T] f From the above formula, we can obtain:

[0165]

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

[0167]

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

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

[0170] According to the closed-loop system with respect to (θ, ε, T) f To further prove its finite-time L1 boundedness, we still choose the Lyapunov function from step 2.3, and consider two time intervals:

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

[0172] s4.2.3.1, Case 1: Assume the switching interval [t] k , t k +T i * It does not contain any time trigger points, i.e., t q <t k , t q+1 >t k +T i * Combining the steps above and the Lyapunov function in step 2.3, we can obtain:

[0173]

[0174] From t in the above equation k Integrating up to t, we get:

[0175]

[0176] s4.2.3.2, Case 2: Assuming a switching interval It contains several event trigger points, namely for We can obtain:

[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] Following this logic, we can obtain:

[0183]

[0184] In summary, regardless of the switching range Does an event trigger point exist, and what is the inequality V? σ(t) (x(t))≤ s4.2.4 is always true. We will continue to discuss the two cases of finite-time L1 boundedness of closed-loop systems:

[0185] s4.2.4.1, Case 1: Assuming a switching interval It does not contain any time trigger points, i.e. t q+1 >t k+1 We can obtain:

[0186]

[0187] From t in the above equation k Integrate up to t:

[0188]

[0189] s4.2.4.2, Case 2: Assuming a switching interval It 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] Following this logic, we can obtain:

[0196]

[0197] Then, combining the time interval as well as From the discussion, we can obtain the following for the time domain [t] k , t k+1 )have:

[0198]

[0199] And because If this is true, then the above equation holds the following inequality:

[0200]

[0201] Combining the zero initial condition, the above equation further becomes: Let t = T f Both sides ride at the same time Then we have:

[0202]

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

[0204] s4.3 Finally, Zeno's behavior will be discussed.

[0205] Assume that ||w(t)||2 < η1, η1 ∈ R + Then there exists a sufficiently small positive constant η2 such that, given a positive closed-loop system and a finite time L1 bounded condition, ||x(t)|2>η2. Based on the designed event triggering conditions, to avoid continuous sampling and updating of the control law, a lower bound T is established. * ∈R + Given between any two consecutive samples, to analyze Zeno's behavior.

[0206] definition: as well as Then, based on the relationship between the 1-norm and the 2-norm: We have: Then for t∈[t k , t k+1 )and From From the definition we get:

[0207]

[0208] According to the Cauchy-Schwarz inequality:

[0209]

[0210] Combined with The discussion of the relationship yielded the following:

[0211]

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

[0213]

[0214] And because therefore:

[0215]

[0216] In summary, we can obtain the closed-loop system in the time domain [t] of the event triggering law. k , t k+1 The lower bound T of ) * for:

[0217]

Claims

1. A control method for a switching positive system based on an event-triggered mechanism in a communication network, characterized in that: The specific steps are as follows: Step 1: Establish the following state-space model of the network control switching positive system: in, These are the system state, system output, controller input, and disturbance input of the network control system, respectively, and the disturbance input satisfies: in It is a given time constant. It is the initial time. This is the initial state of the system; This is the switching signal for the switching system, where N represents the number of subsystems. When, it indicates that the i-th subsystem is activated. It is a constant matrix with appropriate dimensions; Step 2: Design the system's event triggering mechanism, construct the controller for the network control switching positive system under the event triggering mechanism, and set the gain matrix of the event triggering controller. Divided into Two parts, where u and n represent the controller input dimension and the system state dimension, respectively; design an average residence time constraint based on modal dependence; Step 3: Design the conditions for the network control to switch the positive system to positive; Step 4: Design the finite-time L1 bounded condition of the networked switching positive system under the MMDDT dwell time constraint, and solve for the event-triggered controller gain matrix.

2. The control method for a switching positive system based on an event-triggered mechanism in a communication network as described in claim 1, characterized in that: Establish the following event triggering mechanism: in, This represents the sampling error; under this event-triggered mechanism, the following closed-loop switching positive system is obtained based on the state-space model: The controller of the network control switching positive system is: in, It is the gain matrix of the event-triggered controller, and satisfies: , ; It is the system at the moment the event is triggered. The sampling state under the following conditions.

3. The control method for a switching positive system based on an event-triggered mechanism in a communication network as described in claim 2, characterized in that: The average residence time constraint based on modality dependency is: Time interval It is divided into S parts, and the length of each part is The interval of the s-th part ; Selecting Lyapunov functions: ,and ,but: in, , Taking its derivative with respect to t, we get: 。 4. The control method for a switching positive system based on an event-triggered mechanism in a communication network as described in claim 3, characterized in that: Based on the aforementioned event triggering mechanism and dwell time constraints, the controller for switching to the positive system... Redesign: This leads to the following closed-loop system: 。 5. The control method for a switching positive system based on an event-triggered mechanism in a communication network as described in claim 4, characterized in that: When matrix It is a Metzler matrix, and the matrix is... When the system is positive, I is an identity matrix of suitable dimension. For a given set where all elements are 1 1-order matrix; when ,vector satisfy If a constant exists: ,vector , ,satisfy: Then the closed-loop system under MMDDT constraints regarding The time L1 is bounded.

6. The control method for a switching positive system based on an event-triggered mechanism in a communication network as described in claim 4, characterized in that: The gain of the event trigger controller satisfies: when hour: ; when hour, .

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