Background
With the acceleration of the urbanization process, people have higher and higher pursuits for the quality of life. Water resources are an indispensable part of urban life and production and are life resources on which human beings rely for survival. In the 21 st century, the phenomena of excessive consumption, pollution and the like of water resources occur, and the situations of water resource shortage and water supply difficulty occur in the construction process of cities. Therefore, the environmental protection and reasonable distribution of water resources are important for the development of cities. The water supply and drainage system and the water service pipe network of the city form a key component part for stable operation and development of the city. In cities, the demands for water consumption in various time periods and different areas are different, and reasonable allocation of water resources is required under limited water resources. In the process, the opening problem of the valve threshold value, the water storage capacity of the pipe network and the like all have certain influence on stable water supply. Therefore, the stable and effective water supply scheme is adopted, and the key to solve the problems of the valve threshold opening degree and the water storage capacity of the water service pipe network in the urban water service water supply system is solved.
In real life, there is a class of variables that always remain non-negative, such as the number of biological populations, the concentration of substances in a chemical reaction, the number of waiting people in a queue, etc. Such systems consisting of non-negative quantities can be described precisely by positive system modeling. Positive systems require that their input and output states remain non-negative. Because the volumes of water in the pumping station and the pipe network of the urban water service system are non-negative, the positive system modeling method is adopted in the research of the application. Meanwhile, the water storage capacity of a pipe network of the water service system is limited, and the opening degree of a pump station and a valve is also limited, which are the saturation problems of the system. Therefore, the water service system is more reasonable by utilizing the state saturation positive system to model, and the controller based on the event trigger mechanism is adopted to appropriately limit the flow in the period of insufficient water quantity of a pipe network or peak water consumption of residents, so that the stable operation of the urban water supply system can be ensured.
Disclosure of Invention
The invention aims to solve the problems of instability of water storage capacity in a pump station, pipe network threshold values and water flow distribution process in an urban water supply system, and provides a control method of a state saturation positive system based on an event trigger mechanism. By constructing event trigger conditions and designing an event trigger controller, the problem of unreasonable distribution of a water supply system of water services is solved, real-time supply of water is realized, the problems of water supply shortage, difficulty and the like are avoided, and the life quality of urban residents is improved. The specific technical scheme is as follows:
the method comprises the following specific steps:
step 1, establishing a state space model of the urban water service water supply system by acquiring water flow data of a pipe network of the urban water supply system:
x(t+1)=sat(Ax(t))+Bu(t),
wherein
Showing the water flow in the water storage tanks of the urban water service system at the time t, n showing the number of the water storage tanks,
the opening degree of the water pipe network or the water pipe valve at t sampling moments is shown, m represents the number of the water valves,
a real column vector expressed in m dimensions;
and
a matrix representing the appropriate dimensions, A, B is a matrix of weighting constants designed by the modeling system based on the acquired data; considering the fact that the water system requires water flow to be non-negative, i.e., ensuring that x (t) and u (t) are always non-negative, the present invention models the water system according to a positive system model, satisfying A ≧ 0, B ≧ 0, for each element in matrices A and B, i.e., requiring all elements in matrix A, B to be non-negative.
sat:
Is a compound defined as sat (u) ═ sat (u)
1),...,sat(u
m))
TSaturation vector function of, wherein sat (u)
l)=sgn(u
l)min{|u
l|,1},l∈{1,...,m}。
Step 2, establishing an event trigger control mechanism of the water supply system
Where theta is a constant and has theta > 0,
is the sampling error. II-
1Represents the 1 norm of the vector, i.e., the sum of the absolute values of all elements within the vector.
Step 3, designing an event trigger controller of the water supply system, comprising the following steps:
step 3.1 design event trigger controller
Wherein the content of the first and second substances,
f is a controller gain matrix of the designed water affair system;
step 3.2 following step 1, step 2 and step 3.1:
step 3.3 following step 1, step 3.1 and step 3.2:
Step 3.4, a linear residual positive Lyapunov function is constructed for the water service network management system:
V(t)=xT(t)v,
wherein the value of the vector satisfies that v is more than 0, namely each element in the vector is a positive number, and
is a real column vector of dimension n. In order to ensure the stable operation of the water affair system, the difference equation for calculating the Lyapunov function is as follows:
ΔV=V(t+1)-V(t)=xT(t+1)v-xT(t)v,
where the symbol T represents the transpose of a matrix or vector.
Step 3.5 design constants λ > 0, θ > 0, ρ > 0, and β > 0, if there is an n-dimensional variable vector v > 0, ξs>0,ξ>0,zl< 0, z < 0 such that the following inequality
zl<z,ξs<ξ,
βv-ξs≥0,
1.. 2. for each lnN is true, i 1, …, m, and s 1,
Step 3.6, designing a water supply pipe network system x (t +1) ═ sat (ax (t)) + Bu (t)) for water service, and triggering a controller at event
Next, the modeling system is positive and stable; first, according to step 3.3, the positivity of the system is ensured by the lower bound of the modeling system function, i.e.
Step 3.7, calculating a difference equation of the Lyapunov function of the modeling system to satisfy the following conditions:
ΔV<0.
step 3.8 obtained according to step 3.3 and step 3.4:
step 3.9 according to the conditions set forth in step 3.5, one can obtain
Further, it can be deduced
Step 3.10 from step 3.8 and step 3.9 can be obtained
Thus combining the positive of the system in step 3.6: x (k) ≧ 0, we can get
ΔV<0;
Step 3.11 in summary, combine step 3.1 to stepStep 3.10 may obtain the event-triggered controller gain matrix as:
and the controller auxiliary gain matrix is:
the invention provides a control method for improving stable operation of an urban water supply system, which provides a state saturation control technology based on event triggering according to the problems of water supply shortage or shortage and the like in the operation process of the current water supply system.
Detailed Description
The present invention is further explained below.
The saturation control method for the water supply of the urban water service system comprises the following steps:
step 1, firstly, collecting data of water storage capacity change in a pump station and a water service pipe network in an urban water service water supply system, and establishing a following space state model by using the data:
x(t+1)=sat(Ax(t)+Bu(t)),
wherein
Representing the water flow in the water storage tanks of the urban water service system at time t, n representing the number of water service pipe networks,
representing the opening of the valves on the pump station or water service pipe network at the moment t, and m representing the number of the valves, wherein
And
representing the real column vectors in the n and m dimensions, respectively.
And
a matrix representing the appropriate dimensions, A, B is a matrix of weighting constants designed by the modeling system based on the acquired data; considering the actual situation that the water affair system requires that the water flow is all non-negative, namely, ensuring that x (t) and u (t) are always non-negative, the invention models the water affair system according to a positive system model, and satisfies that A is more than or equal to 0, B is more than or equal to 0, and more than or equal to each element in the matrix AB, namely all elements in the matrix AB are required to be non-negative.
sat:
Is a compound defined as sat (u) ═ sat (u)
1),...,sat(u
m))
TSaturation vector function of, wherein sat (u)
l)=sgn(u
l)min{|u
ι|,1},ι∈{1,...,m}。
Step 2, establishing an event trigger mechanism of the urban water supply system
Wherein the constant theta is greater than 0 and,
indicating the sampling error, i.e. the difference between the value of the sampling instant and the current instant,
‖·‖
1represents the 1 norm of the vector, i.e., the sum of the absolute values of all elements within the vector.
Step 3, designing an event trigger controller of the water supply system, which is characterized by comprising the following steps:
step 3.1 design event trigger controller
Wherein the content of the first and second substances,
f is a controller gain matrix of the designed water affair system;
combining with the step 2 to obtain
Further, it can be deduced
Wherein
And is
Is a set of n x n dimensional diagonal matrices with diagonal elements of 0 or 1.
And is provided with
Can represent
Step 3.2 following step 1, step 2 and step 3.1:
step 3.3 following step 1, step 3.1 and step 3.2:
Step 3.4, a linear residual positive Lyapunov function is constructed for the water service network management system:
V(t)=xT(t)v,
wherein the value of the vector satisfies that v is more than 0, namely each element in the vector is a positive number, and
is a real column vector of dimension n. In order to ensure the stable operation of the water affair system, the difference equation for calculating the Lyapunov function is as follows:
ΔV=V(t+1)-V(t)=xT(t+1)v-xT(t)v,
where the symbol T represents the transpose of a matrix or vector.
Step 3.5 design constants λ > 0, κ > 0, θ > 0, ρ > 0 and β > 0, if there is an n-dimensional variable vector v > 0, ξs>0,ξ>0,zl< 0, z < 0, such that the following inequality
zl<z,ξs<ξ,
βv-ξs≥0,
1.. 2. for each lnWhere l is 1, …, m, and s is 1, …, n are all true,
Step 3.6, designing a water supply pipe network system x (t +1) ═ sat (ax (t)) + Bu (t)) for water service, and triggering a controller at event
Next, the modeling system is positive and stable; first, according to step 3.3, the positivity of the system is ensured by the lower bound of the modeling system function, i.e.
Step 3.7, calculating a difference equation of the Lyapunov function of the modeling system to satisfy the following conditions:
ΔV<0;
step 3.8 obtained according to step 3.3 and step 3.4:
step 3.9 according to the conditions set forth in step 3.5, one can obtain
Further, it can be obtained
Step 3.10 from step 3.8 and step 3.9 can be obtained
Positive working of the system according to step 3.6, i.e.: x (k) is not less than 0, and we can deduce
ΔV<0;
Step 3.11 in summary, combining step 3.1 to step 3.10, the gain matrix of the event-triggered controller can be obtained as follows:
and the controller auxiliary gain matrix is: