A distributed dynamic event triggering control method for urban smart water system

Abstract

The present invention discloses a distributed dynamic event triggering control method under a switching signal failure, comprising the following steps: S1, establishing a positive switching system state space model of an urban smart water control system; S2, setting and optimizing the conditions of the positive switching system state space model; S3, verifying the urban smart water control system. A state space model of the urban smart water control system is established using a positive switching system. The present invention designs dynamic feedback controllers for normal and faulty switching signals respectively to ensure that the system can still operate safely and smoothly when the system switching signal fails. The distributed output feedback control method improves the fault tolerance of the system operation and optimizes the system communication efficiency. The dynamic event triggering control strategy can dynamically adapt to changes in system performance, reduce system resource loss, and ensure the control efficiency of the urban smart water control system.

Description

Technical Field

[0001] The present invention belongs to the field of automation technology and intelligent control, and specifically refers to a distributed dynamic event triggering control method for an urban smart water system. Background Art

[0002] Cities are the main body of national economic and social development. With the continuous progress of science and technology, it has become an inevitable trend to promote the transformation and upgrading of cities from digital cities to smart cities. The rapid development of cities has put forward higher requirements for the utilization of urban environmental resources, smart services and innovative development, and also put forward new requirements for the rapid acquisition and analysis of urban information, the solution and processing of urban problems, and the guarantee of efficient, low-carbon and scientific operation of cities. With the people's growing demand for high-quality life, the standards for urban water resource utilization are becoming increasingly stringent. Designing a more efficient and environmentally friendly urban smart water control system has become a research hotspot.

[0003] At present, the construction of smart city water systems based on the Internet of Things, artificial intelligence, and automation has improved the control methods of urban water control systems by using distributed information technology and intelligent means. Although it can meet the water treatment needs brought about by the rapid development of cities, different water treatment control schemes are selected according to the water use time nodes and the amount of water use, and based on the characteristic that water use is always non-negative,

[0004] In actual systems, due to system structure or external factors, the state of the system cannot be measured. The output feedback control method uses sensors to detect the output value of the system and then control the system. However, there may still be situations where sensor failures affect the normal operation of the water system.

[0005] In addition, the traditional centralized feedback control and distributed output feedback control improve the system's fault tolerance and data communication efficiency. Among the commonly used trigger strategies, the time trigger mechanism controls the transmission of system signals through periodic intermittent sampling. Due to its periodic sampling characteristics, it often causes waste of system resources and increases design costs. In addition, the event trigger mechanism is a triggering method based on setting event trigger thresholds, which avoids unnecessary periodic sampling in the time trigger mechanism. However, the traditional event trigger strategy sets a fixed trigger threshold, so it is impossible to adjust the trigger conditions in time according to changes in system performance, which still leads to conservative system performance. Summary of the invention

[0006] The purpose of the present invention is to address the problems raised by modeling an urban smart water control system using a positive switching system, provide a distributed dynamic event triggering control method for an urban smart water system, establish a state space model of the urban smart water control system using modern control theory and technology, combine the dynamic event triggering mechanism and the distributed control method, and design a distributed dynamic event triggering controller for the urban smart water control system. The system can still ensure positivity and stability in the event of a switching signal failure.

[0007] The present invention provides a distributed dynamic event triggering control method for an urban smart water system, comprising the following steps:

[0008] S1. Establish the state space model of the positive switching system of the urban smart water control system.

[0009] The specific form is as follows:

[0010]

[0011] in, Z∈N + are the system state, control input, disturbance input and measurable output of the sth sensor node of the urban smart water system respectively. The piecewise constant function σ(t) represents the switching law of the system, and its value is in the finite set J∈N + In, assuming A σ(t) is the Metzler matrix, B σ(t) ≥0, D σ(t) ≥0, when When the system matrix is ​​A i , B i , D i ,

[0012] S2. Condition setting and optimization of the state space model of the positive switching system;

[0013] S3. Verification of Urban Smart Water Control System

[0014] S3-1. Positive verification of the urban smart water control system under normal switching conditions;

[0015] S3-2, Positive verification of the urban smart water control system in the case of switching signal failure;

[0016] S3-3. Stability verification of the urban smart water control system under normal switching conditions;

[0017] S3-4. Verification of the stability of the urban smart water control system under switching signal failure.

[0018] Preferably, step S2 includes the following sub-steps:

[0019] S2-1. Establish a distributed dynamic trigger mechanism for the urban smart water control system;

[0020] S2-2, construct a dynamic output feedback controller based on the state observer system;

[0021] S2-3, build a closed-loop system for urban smart water control systems;

[0022] S2-4. Design the conditions for the smooth operation of the urban smart water control system.

[0023] Preferably, in step S2-1, a distributed dynamic trigger mechanism of the urban smart water control system is established, and its construction form is as follows:

[0024] Define the sampling error of the event generator, the expression is as follows:

[0025]

[0026] in, y s (t l ) is the number of events generated by the sth event generator at the event triggering time t l , the output signal when l∈N;

[0027] The output will be released according to the following dynamic event trigger conditions:

[0028]

[0029] Where β>0, θ>0, and when the initial conditions η(t0)=η0 and ρ>0, η(t) is an internal dynamic variable that satisfies

[0030]

[0031] Preferably, in step S2-2, a dynamic output feedback controller based on a state observer system is constructed, and its construction form is as follows:

[0032]

[0033] in, is the observer state, G i and L i is the observer gain matrix, K i represents the controller gain matrix, is the set of sensor subscripts s′ that satisfy the distribution condition, The value is 0 or 1, and the error is defined as

[0034]

[0035] Preferably, in step S2-3, a closed-loop system of the urban smart water control system is constructed, and the construction method thereof is as follows:

[0036] definition Then, under the controller in step S2-2, the closed-loop system of the urban smart water control system in step S1 can be expressed as:

[0037]

[0038] in,

[0039]

[0040] Preferably, in step S2-4, the conditions for the smooth operation of the urban smart water control system are designed, and the construction form thereof is as follows:

[0041] Design constant ε ι >0, ι=1,2,...,6, ε5>ε6, θ>0, γ>0, λ>1, 0≤β<1, ρ>μ and vector δ it ≥0,

[0042] The conditional expressions set according to the above constants are as follows:

[0043]

[0044]

[0045] 1 m B i (p i +q i )1 n ≤ε2q i ,

[0046]

[0047] p i ≤λp j ,q i ≤λq j ,

[0048] For any ι=1,2,...,n and Established, where q iι ,ι=1,2,...,n,q i is the ιth element, Then, in the case of normal switching, the switching condition at the average dwell time

[0049]

[0050] The observer and the urban smart water control system in step S2-2 are positive and stable, where the control gain matrix is

[0051]

[0052] When the switching signal fails, the switching condition is

[0053]

[0054] The observer and the urban smart water control system in step S2-2 are positive and stable, where the fault controller gain matrix is

[0055]

[0056] The following description is given for step S2-4:

[0057] Explanation of setting conditions: The conditions for the smooth operation of the designed urban smart water control system refer to all the inequalities in steps S2-4;

[0058] Instructions for setting the formula:

[0059] Set formula and That is, the gain matrix is ​​decomposed into the form of the sum of vector products through the matrix decomposition technology, so that the set conditions are converted into a linear form.

[0060] Symbolic explanation of the set constants: ε1 is the constant that ensures that the matrix is ​​Metzler; ε2, ε3 and ε4 are the coefficients of the conditional linear transformation; ε5 is the matrix coefficient that ensures that the system is positive in the case of switching faults; ε6 is the design fault controller gain matrix K fi β is the trigger coefficient of the dynamic event trigger condition; θ is the coefficient of the internal dynamic variable in the dynamic event trigger condition; ρ is the coefficient of the internal dynamic variable in the internal dynamic variable formula; γ is the L1 gain performance indicator; λ is the coefficient of the Lyapunov function switching; μ and are the coefficients of the Lyapunov function state recursion under normal switching and switching failure conditions, respectively;

[0061] Set the symbol of the vector: p i and q i is the Lyapunov function vector; ξ ij , δ iι and They are the designed controller gain matrix K i , the state observer gain matrix G i and L i A vector of ξ i and The vector to be linearly transformed.

[0062] Preferably, in step S3-1, the positive verification method of the urban smart water control system under normal switching conditions is as follows:

[0063] Given any initial state x(t0)≥0, and the disturbance signal ω(t0)≥0, then the output y s (t0)≥0, Combined with the dynamic event triggering conditions in step 2-1, we have

[0064]

[0065] Then, we can deduce

[0066]

[0067] Therefore, the dynamic output feedback controller in step S2-1 and the closed-loop system in step S2-2 satisfy and in,

[0068]

[0069] because and Using the conditions in step S2-4, we can deduce G i is the Metzler matrix, and Therefore, -B i K i ≥0, G i -B i K i is also a Metzler matrix,

[0070] Combine the conditions in steps S2-4: Conclusion

[0071]

[0072] Using the conditions in step S2-4: 1 m B i (p i +q i )1 n ≤ε2q i ,have Combine the conditions in steps S2-4:

[0073]

[0074] Conclusion

[0075]

[0076] According to the conditions in step S2-4: Conclusion

[0077]

[0078] therefore,

[0079]

[0080] Use the controller gain matrix in step S2-4 to derive A i +B i K i is the Metzler matrix, so we have is the Metzler matrix and For x(t0) and The i-th element x i (t0)≥0, And ω(t0)≥0, we have and Therefore, it is possible to obtain and That is to say, the closed-loop system of the dynamic output feedback controller and the urban smart water system in step S2-2 is positive.

[0081] Preferably, in step S3-2, the positive verification process of the urban smart water control system in the case of a switching signal failure is as follows:

[0082] Average dwell time switching condition This means that the switching signal fails. At this time, the fault gain controller matrix is ​​proposed in step S2-4. Note that ε5>ε6, so it is substituted into the condition in step S2-4: Conclusion

[0083]

[0084] This means

[0085]

[0086] Using the fault controller gain matrix A is given i +B i K fi is the Metzler matrix,

[0087] Substitute the fault controller gain matrix into the conditions in step S2-4:

[0088]

[0089] Conclusion

[0090]

[0091] therefore, So we get is a Metzler matrix. Using the method in step S3-1, it can be obtained that the closed-loop system of the dynamic output feedback controller and the urban smart water control system in step S2-2 is positive.

[0092] Preferably, in step S3-3, the method for verifying the stability of the urban smart water control system under normal switching conditions is as follows:

[0093] Choose the linear cosine Lyapunov function as

[0094]

[0095] Among them, v i =(p i q i ),

[0096] Consider ω(k) = 0, then

[0097]

[0098] in,

[0099] Using the gain matrix in step S2-4, we have

[0100]

[0101] Combine the conditions in steps S2-4:

[0102]

[0103]

[0104] Derived Therefore, for the switching moment Can get in According to the conditions in step S2-4: p i ≤λp j ,q i ≤λq j ,

[0105] Conclusion

[0106]

[0107] Thus, the above inequality can be further transformed into

[0108]

[0109] This means in and They are Minimum and maximum elements, using average dwell time to switch conditions, Therefore, the urban smart water control system is exponentially stable.

[0110] Consider ω(t)≠0,

[0111]

[0112] in,

[0113] Define Γ(t)=γ‖ω(t)‖1-‖e(t)‖1. According to the linear copositive Lyapunov function, the Lyapunov function inequality can be obtained:

[0114]

[0115] Using the conditions in step S2-4:

[0116]

[0117] Conclusion

[0118]

[0119] G i q i -K i B i (p i +q i )+1 n +μq i ≤0,

[0120]

[0121] Substituting the above formula into the Lyapunov function inequality, we have Given Furthermore, the urban smart water control system in step 1 has a performance of γ Gain stability.

[0122] Preferably, in step S3-4, the method for verifying the stability of the urban smart water control system in the case of a switching signal failure is as follows:

[0123] Substitute the fault controller gain matrix in step S2-4 into the condition: In

[0124]

[0125] Therefore, based on the conditions in step S2-4:

[0126]

[0127] There is a constant Make

[0128]

[0129] Furthermore, we can deduce Using the Lyapunov function, we get

[0130]

[0131] in and They are The minimum and maximum elements of the switch condition are determined by the average residence time, This means that the city's smart water control system still meets the requirements in the event of a switching signal failure. Gain stable.

[0132] The beneficial effects of the present invention are as follows:

[0133] The method of the present invention establishes a state space model of the urban smart water control system based on the positive switching system, and designs a distributed dynamic event trigger controller. With the help of linear cosine Lyapunov function and dynamic event trigger law, the distributed control method can ensure the stable operation of the system in the case of abnormal switching signal, thereby improving the safety of the system. The dynamic trigger control strategy can effectively adapt to the changes in system performance, reduce the conservatism of system performance utilization, and ensure the efficiency of resource utilization. BRIEF DESCRIPTION OF THE DRAWINGS

[0134] Urban smart water control system water supply control flow chart, Figure 2It is a framework diagram of a positive switching system based on a distributed dynamic event triggering mechanism. DETAILED DESCRIPTION

[0135] The present invention is further described below in conjunction with specific embodiments, but the present invention is not limited to these specific embodiments. Those skilled in the art should recognize that the present invention covers all possible alternatives, improvements and equivalents within the scope of the claims.

[0136] This embodiment provides a distributed dynamic event triggering control method for an urban smart water system.

[0137] The specific steps of the method of the present invention include the following:

[0138] S1. Establish the state space model of the positive switching system of the urban smart water control system.

[0139] Specifically, step S1 includes

[0140] S1-1 Describe the control process of the urban water control system

[0141] Urban production and domestic water is produced by professional equipment through multiple complex process flows, and then delivered to various users. The urban water supply process is complex, and the main processes include water source water intake, water distribution in distribution wells, water delivery through pipe networks, secondary treatment at water plants, comprehensive dispatching by water control centers, and the use of water pumps to purify and disinfect water plants to produce water resources that meet the corresponding standards for production and domestic use (see Figure 1 Water supply control flow chart of urban smart water control system). Among them, the main function of the sensor is to monitor the data information such as the liquid level, pressure and flow of each water supply pipeline, and transmit the data information back to the water control center through the event trigger mechanism for comprehensive water supply scheduling. Considering the characteristic that the water volume is always non-negative, the present invention intends to use a positive switching system to model the urban smart water control system. In addition, a distributed dynamic event triggering strategy and a switching signal fault control method are also introduced (see Figure 2 Framework diagram of the positive switching system based on the distributed dynamic event trigger mechanism), which ensures that the urban smart water control system can still operate stably in the event of a switching signal failure.

[0142] S1-2 Establish the state space model of the urban smart water control system based on the above description

[0143]

[0144] in, Z∈N +are the system state, control input, disturbance input and measurable output of the sth sensor node of the urban smart water system. The piecewise constant function σ(t) represents the switching signal of the system, and its value is in the finite set J∈N + Assume A σ(t) is the Metzler matrix, B σ(t) ≥0, D σ(t) ≥0, when When the system matrix is ​​A i , B i , D i ,

[0145] S2. Condition setting and optimization of the state space model of the positive switching system;

[0146] Furthermore, step S2 includes the following sub-steps:

[0147] S2-1. Establish a distributed dynamic trigger mechanism for urban smart water control systems;

[0148] Its construction form is as follows:

[0149] Define the sampling error of the event generator, the expression is as follows:

[0150]

[0151] in, y s (t l ) is the number of events generated by the sth event generator at the event triggering time t l , the output signal when l∈N;

[0152] The output will be released according to the following dynamic event trigger conditions:

[0153]

[0154] Where β>0, θ>0, and when the initial conditions η(t0)=η0 and ρ>0, η(t) is an internal dynamic variable that satisfies

[0155]

[0156] S2-2, construct a dynamic output feedback controller based on the state observer system;

[0157] Its construction form is as follows:

[0158]

[0159] in, is the observer state, G i and L i is the observer gain matrix, K i represents the controller gain matrix, is the set of sensor subscripts s′ that satisfy the distribution condition, The value is 0 or 1, and the error is defined as

[0160]

[0161] In the above technical solution, a dynamic feedback controller is designed for normal and faulty switching signals to ensure that the system can still operate safely and smoothly when the system switching signal fails. The distributed output feedback control method improves the fault tolerance of the system operation and optimizes the system communication efficiency. The dynamic event trigger control strategy can dynamically adapt to changes in system performance, reduce system resource loss, and ensure the control efficiency of the urban smart water control system.

[0162] S2-3, build a closed-loop system for urban smart water control systems;

[0163] The construction method is as follows:

[0164] definition Then, under the controller in step S2-2, the closed-loop system of the urban smart water control system in step S1 can be expressed as:

[0165]

[0166] in,

[0167]

[0168] S2-4. Design the conditions for the smooth operation of the urban smart water control system.

[0169] Its construction form is as follows:

[0170] Design constant ε ι >0, ι=1,2,...,6, ε5>ε6, θ>0, γ>0, λ>1, 0≤β<1, ρ>μ and vector δ iι ≥0,

[0171] The conditional expressions set according to the above constants are as follows:

[0172]

[0173] 1 m B i (p i +q i )1 n ≤ε2q i ,

[0174]

[0175]

[0176] For any ι=1,2,...,n and Established, where q iι ,ι=1,2,...,n,q i is the ιth element, Then, in the case of normal switching, the switching condition at the average dwell time

[0177]

[0178] The observer and the urban smart water control system in step S2-2 are positive and stable, where the control gain matrix is

[0179]

[0180] It can be understood that the control gain matrix described above is decomposed into the form of the sum of vector products by using the matrix decomposition technology, so that the set conditions are converted into a linear form.

[0181] When the switching signal fails, the switching condition is

[0182]

[0183] The observer and the urban smart water control system in step S2-2 are positive and stable, where the fault controller gain matrix is

[0184]

[0185] The following description is given for step S2-4:

[0186] It should be noted that the conditions for the smooth operation of the designed urban smart water control system refer to all the inequalities in steps S2-4;

[0187] Among them, ε1 is a constant to ensure that the matrix is ​​Metzler; ε2, ε3 and ε4 are the coefficients of conditional linear transformation; ε5 is the matrix coefficient to ensure that the system is positive in the case of switching faults; ε6 is the designed fault controller gain matrix K fi β is the trigger coefficient of the dynamic event trigger condition; θ is the coefficient of the internal dynamic variable in the dynamic event trigger condition; ρ is the coefficient of the internal dynamic variable in the internal dynamic variable formula; γ is the L1 gain performance indicator; λ is the coefficient of the Lyapunov function switching; μ and are the coefficients of the Lyapunov function state recursion under normal switching and switching failure conditions, respectively;

[0188] Among them, p i and q i is the Lyapunov function vector; ξ ij , δ iι and They are the designed controller gain matrix K i , the state observer gain matrix G i and L i A vector of ξ i and The vector to be linearly transformed.

[0189] S3. Verification of Urban Smart Water Control System

[0190] S3-1. Positive verification of the urban smart water control system under normal switching conditions;

[0191] Specifically, the positive verification method of the urban smart water control system under normal switching conditions is as follows:

[0192] Given any initial state x(t0)≥0, and the disturbance signal ω(t0)≥0, then the output y s (t0)≥0, Combined with the dynamic event triggering conditions in step 2-1, we have

[0193]

[0194] Then, we can deduce

[0195]

[0196] Therefore, the dynamic output feedback controller in step S2-1 and the closed-loop system in step S2-2 satisfy and in,

[0197]

[0198] because and Using the conditions in step S2-4, we can deduce G i is the Metzler matrix, and Therefore, -B i K i ≥0, G i -B i K i is also a Metzler matrix,

[0199] Combine the conditions in steps S2-4: Conclusion

[0200]

[0201] Using the conditions in step S2-4: 1 m B i (p i +q i )1 n ≤ε2q i ,have Combine the conditions in steps S2-4:

[0202]

[0203] Conclusion

[0204]

[0205] According to the conditions in step S2-4: Conclusion

[0206]

[0207] therefore,

[0208]

[0209] Use the controller gain matrix in step S2-4 to derive A i +B i K i is the Metzler matrix, so we have is the Metzler matrix and For x(t0) and The i-th element x i (t0)≥0, And ω(t0)≥0, we have and Therefore, it is possible to obtain and That is to say, the closed-loop system of the dynamic output feedback controller and the urban smart water system in step S2-2 is positive.

[0210] S3-2, Positive verification of the urban smart water control system in the case of switching signal failure;

[0211] Specifically, the positive verification process of the urban smart water control system under the condition of switching signal failure is as follows:

[0212] Average dwell time switching condition This means that the switching signal fails. At this time, the fault gain controller matrix is ​​proposed in step S2-4. Note that ε5>ε6, so it is substituted into the condition in step S2-4: Conclusion

[0213]

[0214] This means

[0215]

[0216] Using the fault controller gain matrix A is given i +B i K fi is the Metzler matrix,

[0217] Substitute the fault controller gain matrix into the conditions in step S2-4:

[0218]

[0219] Conclusion

[0220]

[0221] therefore, So we get is a Metzler matrix. Using the method in step S3-1, it can be obtained that the closed-loop system of the dynamic output feedback controller and the urban smart water control system in step S2-2 is positive.

[0222] It can be understood that according to step S3-1 and step S3-2, the urban smart water control system is positive both when the switching signal is normal and when the switching signal fails, so the positivity of the urban smart water control system is proven.

[0223] S3-3. Stability verification of the urban smart water control system under normal switching conditions;

[0224] Specifically, the verification method for the stability of the urban smart water control system under normal switching conditions is as follows:

[0225] Choose the linear cosine Lyapunov function as

[0226]

[0227] Among them, v i =(p i q i ),

[0228] Consider ω(k) = 0, then

[0229]

[0230] in,

[0231] Using the gain matrix in step S2-4, we have

[0232]

[0233] Combine the conditions in steps S2-4:

[0234]

[0235] Derived Therefore, for the switching moment Can get in According to the conditions in step S2-4: p i ≤λp j ,q i ≤λq j , it is concluded

[0236]

[0237] Thus, the above inequality can be further transformed into

[0238]

[0239] This means in and They are Minimum and maximum elements, using average dwell time to switch conditions, Therefore, the urban smart water control system is exponentially stable.

[0240] Consider ω(t)≠0,

[0241]

[0242] in,

[0243] Define Γ(t)=γ‖ω(t)‖1-‖e(t)‖1. According to the linear copositive Lyapunov function, the Lyapunov function inequality can be obtained:

[0244]

[0245] Using the conditions in step S2-4:

[0246]

[0247] Conclusion

[0248]

[0249] G i q i -K i B i (p i +q i )+1 n +μq i ≤0,

[0250]

[0251] Substituting the above formula into the Lyapunov function inequality, we have Given Furthermore, the urban smart water control system in step 1 has a performance of γ Gain stability.

[0252] S3-4. Verification of the stability of the urban smart water control system under switching signal failure.

[0253] Specifically, the method for verifying the stability of the urban smart water control system under the condition of switching signal failure is as follows:

[0254] Substitute the fault controller gain matrix in step S2-4 into the condition: In

[0255]

[0256] Therefore, based on the conditions in step S2-4:

[0257]

[0258] There is a constant Make

[0259]

[0260] Furthermore, we can deduce Using the Lyapunov function, we get

[0261]

[0262] in and They are The minimum and maximum elements of the switch condition are determined by the average residence time, This means that the city's smart water control system still meets the requirements in the event of a switching signal failure. Gain stable.

[0263] It can be understood that, in combination with step S3-3 and step S3-4, it can be concluded that the urban smart water control system is stable.

[0264] It can be seen that this embodiment provides a distributed dynamic event trigger control strategy to improve the system signal transmission efficiency. In addition, the core of switching system modeling lies in coordinating the switching signals of subsystem operations. The present invention fully considers the problem of premature system switching in the case of switching signal failure, which causes system instability. Therefore, based on the distributed dynamic event trigger controller, the present invention proposes a new switching signal fault controller, which can ensure the smooth and reliable operation of the smart water system.

[0265] The embodiments of the present invention are described in detail above with reference to the accompanying drawings, but the present invention is not limited to the described embodiments. For those skilled in the art, various changes, modifications, substitutions and variations of these embodiments including components are made without departing from the principles and spirit of the present invention, and still fall within the scope of protection of the present invention.

Claims

1. A distributed dynamic event triggering control method for an urban smart water system, characterized in that: The following steps are included: S1. Establish the state space model of the positive switching system of the urban smart water control system. The specific form is as follows: in, Z∈N + are the system state, control input, disturbance input and measurable output of the sth sensor node of the urban smart water system respectively. The piecewise constant function σ(t) represents the switching law of the system, and its value is in the finite set J∈N + In, assuming A σ(t) is the Metzler matrix, B σ(t) ≥0, D σ(t) ≥0, When σ(t)=i∈S, let the system matrix be A i , B i , D i , S2, Condition setting and optimization of the state space model of the positive switching system, The following sub-steps are included: S2-1. Establish a distributed dynamic trigger mechanism for the urban smart water control system. Its construction form is as follows: Define the sampling error of the event generator, the expression is as follows: in, y s (t l ) is the number of events generated by the sth event generator at the event triggering time t l , the output signal when l∈N; The output will be released according to the following dynamic event trigger conditions: Where β>0, θ>0, and when the initial conditions η(t0)=η0 and ρ>0, η(t) is an internal dynamic variable that satisfies S2-2, construct a dynamic output feedback controller based on the state observer system, Its construction form is as follows: in, is the observer state, G i and L i is the observer gain matrix, K i represents the controller gain matrix, is the set of sensor subscripts s′ that satisfy the distribution condition, The value is 0 or 1, and the error is defined as S2-3. Build a closed-loop system for urban smart water control systems. The construction method is as follows: definition Then, under the controller in step S2-2, the closed-loop system of the urban smart water control system in step S1 can be expressed as: in, S2-4. Design the conditions for the smooth operation of the urban smart water control system. Its construction form is as follows: Design constant ε ι > 0, ι = 1,2,...,6, ε5 > ε6, θ > 0, γ > 0, λ > 1, 0 ≤ β < 1, ρ > μ and vector p i > 0, q i > 0, ξ i < 0, δ il ≥ 0, The conditional expressions set according to the above constants are as follows: For any l=1,2,…,n and Established, where q il ,l=1,2,...,n,q i is the ιth element, Then, in the case of normal switching, the switching condition at the average dwell time The observer and the urban smart water control system in step S2-2 are positive and stable, where the control gain matrix is When the switching signal fails, the switching condition is The observer and the urban smart water control system in step S2-2 are positive and stable, where the fault controller gain matrix is The following description is given for step S2-4: Among them, ε1 is the constant that ensures that the matrix is ​​Metzler; ε2, ε3 and ε4 are the coefficients of conditional linear transformation; ε5 is the matrix coefficient that ensures that the system is positive in the case of switching faults; ε6 is the designed fault controller gain matrix K fi β is the trigger coefficient of the dynamic event trigger condition; θ is the coefficient of the internal dynamic variable in the dynamic event trigger condition; ρ is the coefficient of the internal dynamic variable in the internal dynamic variable formula; γ is the L1 gain performance indicator; λ is the coefficient of the Lyapunov function switching; μ and are the coefficients of the Lyapunov function state recursion under normal switching and switching failure conditions, respectively; p i and q i is the Lyapunov function vector; and They are the designed controller gain matrix K i , the state observer gain matrix G i and L i A vector of ξ i and Linear transformation of vectors; S3. Verification of Urban Smart Water Control System S3-1. Positive verification of the urban smart water control system under normal switching conditions; S3-2, Positive verification of the urban smart water control system in the case of switching signal failure; S3-3. Stability verification of the urban smart water control system under normal switching conditions; S3-4. Verification of the stability of the urban smart water control system under switching signal failure.

2. The distributed dynamic event triggering control method of the urban smart water system according to claim 1 is characterized in that: In step S3-1, the positive verification method of the urban smart water control system under normal switching conditions is as follows: Given any initial state x(t0)≥0, and the disturbance signal ω(t0)≥0, then the output y s (t0)≥0, Combined with the dynamic event triggering conditions in step 2-1, we have Then, we can deduce Therefore, the dynamic output feedback controller in step S2-1 and the closed-loop system in step S2-2 satisfy and in, Due to q i >0,δ iι >0, and Using the conditions in step S2-4, we can deduce G i is the Metzler matrix, L i M>0 and K i <0, therefore, -B i K i ≥0, G i -B i K i is also a Metzler matrix, Combine the conditions in steps S2-4: Conclusion Using the conditions in step S2-4: 1 m B i (p i +q i )1 n ≤ε2q i ,have Combine the conditions in steps S2-4: Conclusion According to the conditions in step S2-4: Conclusion therefore, Use the controller gain matrix in step S2-4 to derive A i +B i K i is the Metzler matrix, so we have is the Metzler matrix For x(t0) and The i-th element x i (t0)≥0, And ω(t0)≥0, we have and Therefore, it is possible to obtain and That is to say, the closed-loop system of the dynamic output feedback controller and the urban smart water system in step S2-2 is positive.

3. The distributed dynamic event triggering control method of the urban smart water system according to claim 2 is characterized in that: In step S3-2, the positive verification process of the urban smart water control system in the case of a switching signal failure is as follows: Average dwell time switching condition This means that the switching signal fails. At this time, the fault gain controller matrix is ​​proposed in step S2-4. Note that ε5>ε6, so it is substituted into the condition in step S2-4: Conclusion This means Using the fault controller gain matrix A is given i +B i K fi is the Metzler matrix, Substitute the fault controller gain matrix into the conditions in step S2-4: Conclusion therefore, So we get is a Metzler matrix. Using the method in step S3-1, it can be obtained that the closed-loop system of the dynamic output feedback controller and the urban smart water control system in step S2-2 is positive.

4. The distributed dynamic event triggering control method for urban smart water system according to claim 3 is characterized in that: In step S3-3, the method for verifying the stability of the urban smart water control system under normal switching conditions is as follows: Choose the linear cosine Lyapunov function as Among them, v i =(p i q i ), Consider ω(k) = 0, then in, Using the gain matrix in step S2-4, we have Combine the conditions in steps S2-4: Derived Therefore, for the switching moment Can get in According to the conditions in step S2-4: p i ≤λp j ,q i ≤λq j , it is concluded Thus, the above inequality can be further transformed into This means in and They are Minimum and maximum elements, using average dwell time to switch conditions, Therefore, the urban smart water control system is exponentially stable. Consider ω(t)≠0, in, Define Γ(t)=γ‖ω(t)‖1-‖e(t)‖1. According to the linear copositive Lyapunov function, the Lyapunov function inequality can be obtained: Using the conditions in step S2-4: Conclusion Substituting the above formula into the Lyapunov function inequality, we have Given Furthermore, the urban smart water control system in step 1 has a performance of γ Gain stability.

5. The distributed dynamic event triggering control method for urban smart water system according to claim 4 is characterized in that: In step S3-4, the method for verifying the stability of the urban smart water control system in the case of a switching signal failure is as follows: Substitute the fault controller gain matrix in step S2-4 into the condition: In Therefore, based on the conditions in step S2-4: There is a constant Make Furthermore, we can deduce Using the Lyapunov function, we get in and They are The minimum and maximum elements of the switch condition are determined by the average residence time, This means that the city's smart water control system still meets the requirements in the event of a switching signal failure. Gain stable.

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