Background
With the modernization development of cities, the urban water management problem is increasingly concerned by people. The phenomena of water resource shortage, water supply difficulty and the like begin to appear in the urban construction process. The urban water service pipe network system is an important infrastructure of a city and is a material foundation on which the city lives and develops. The water supply system for city water service pipe network consists of water source, pump station, water supply tank, water supply pipe network, water pipe valve, water using department, etc. Wherein the proper operation of the pumping station and water shut-off valves is very important for the daily supply of water. At present, due to the shortage of water resources caused by the rapid development of cities, a water supply system of an urban water service pipe network must be operated at full load with long time and high precision in the peak period of water consumption. This can lead to pump station and water pipe valve to break down and consequently cause the unstability of water supply system and appear phenomenons such as lack of water, cutting off water supply. Therefore, it is very necessary to adopt a reasonable water supply scheme to solve the problem of water supply caused by pump station and valve failure in the water supply system.
Furthermore, the peak water usage periods in various water usage departments (e.g., schools, factories, residential buildings, etc.) in a city are uncertain. For example, in a plant, the peak water usage for each department of the plant is random, the current time period may be department 1, and the next time period may be department 1 or department 2. Fig. 1 (see attached drawings) shows a water supply network of a designated physical area, and water sources cannot supply water to a plurality of water using departments simultaneously during peak water use due to water resource shortage. Therefore, if the school is in the peak water consumption period in the current time period, the valves of the pump B and the pump C are closed, and the valve of the pump A is opened to supply water to the school; the next time water department (maybe school, maybe factory, maybe district) if it is factory, then close the valves of pumps A and C, open the valve of pump B to supply water to factory. It can be seen that this random behavior is more suitably represented by a markov jump process. Because the volume of water in the water supply tank of the water supply system, the opening degree of a pump station and a water pipe valve and the like are all non-negative, the change of the volume of water in the water supply tank of the water supply system can be more effectively analyzed by adopting an analysis method of a positive system. In order to ensure the safe and reliable water use and the stable operation of a water supply system, a positive Markov jump system with an actuator fault is used as a model, a corresponding analysis method is adopted, and a control method based on state feedback can realize the safe and reliable control of the water supply fault of an urban pipe network system.
Disclosure of Invention
The invention aims to provide a method for controlling water supply faults of an urban water service pipe network system, aiming at the problems of pump station and water pipe valve faults in the water supply process of the urban water service pipe network system.
The invention adopts a state feedback control method of a positive Markov jump system to control the water volume of a water supply tank and the opening values of a pump station and a water pipe valve in the urban water service pipe network water supply system, and carries out safe and effective control on the problem of water supply failure of the urban water service pipe network system by designing a reliable controller of the positive Markov jump system with actuator failure.
The method comprises the following specific steps:
step 1, establishing a state space model of the water volume of a water supply tank of an urban water service pipe network system, wherein the specific method comprises the following steps:
1.1 consider the water supply network of the urban water service pipe network system. A water supply network typically comprises a set of water supply pipes, water tanks of different volumes, and a plurality of pumping stations and valves for managing the water flow to supply water to users. Fig. 1 (see the drawings of the specification) shows the relationship between these elements. The water tank of fig. 1 provides water storage capacity for the entire water supply network to ensure that sufficient water service is provided for the user. The mass balance expression relating to the volume v stored in the nth supply tank can be written as a discrete time difference equation
Wherein
Representing the amount of water flowing from the jth element into the nth tank,
representing the amount of water flowing out of the nth tank to the h element (referring to the user). Further, Δ t is a sampling time, and k is a discrete time. For all k, the constraint on the tank volume is expressed as upsilon
n(k) For convenience, ≧ 0, the dynamic behavior of these elements is described as a function of volume. The model does not take into account errors arising from the geometry of the water tank.
The water supply network takes into account two control actuators: valves and pumps. Variable q representing valve opening degree when valve is assumed to be openuNegative, full valve closure represents a valve opening value of 0.
Nodes represent network points where water flow converges or diverges in the network and must adhere to mass conservation relationships.
Considering the elements described above, a control-oriented model can be obtained by adding these elements and the corresponding dynamic descriptions. In a general form, a water volume dynamic expression that considers all of these elements can be written as
x(k+1)=h(x(k),u(k)),
Wherein x (k) e RnRepresents the system state, u (k) e RrRecording as system input (i.e. opening values of valves and pump stations), h: Rn×Rm→RnIs an arbitrary system state function, k is equal to N+. In the present inventionIn a water supply network, a discrete-time state space model can be written
Wherein x (k) ═ x
1(k),x
2(k),...,x
n(k)]
T∈R
nIndicating the volume of water supplied to the tank and n representing the number of tanks. u. of
f(k)∈R
rIndicating opening q of water pipe valve and pump with failure
uAnd r represents the total number of water pipe valves and pumps. R
nRepresenting a real column vector of dimension n.
And
representing a system matrix of appropriate dimensions. For each g
kAll are belonged to S
(
Is directed to the matrix
I.e., all elements within the matrix are non-negative). g
kRepresenting a markov jump process that takes values in a finite set S ═ 1,2,.., M },
and (4) the following steps.
For convenience, let's note g
kI, i ∈ S, then there is
1.2 design Markov jump signal gkThe transfer rate satisfies the following conditions:
Pr(gk+1=j|gk=i)=πij,
where for each i, j ∈ S there is a
ij≥0,
Step 2, designing a state feedback controller of water quantity change in a water supply tank of the urban water service pipe network system, which comprises the following specific steps:
2.1 design the control input model with faults as
uf(k)=Hiu(k).
Matrix HiIs an unknown fault matrix and satisfies
Wherein
h ijAnd
is a diagonal matrix
H iAnd
and is a given constant, and ρ > 1 is a given constant value.
2.2 design State feedback controller as
u(k)=(Ki+ΔKik)x(k)
And
ΔKik=FiGik
wherein K
i∈R
r×nIs the nominal controller gain, Δ K, to be designed
ikIs the fluctuation of the controller gain, G
ik∈R
r×nIs the decision matrix, F, to be designed
i∈R
r×rIs a known non-negative matrix and for 0 < sigma
1<σ
2Satisfy the requirement of
2.3 design
Its associated definitions are indicated at step 1.1, step 2.1 and step 2.2. Simultaneously constructing a random complementary Li ya Ponuff function
V(x(k),gk=i)=xT(k)vi,
Wherein
v
i∈R
nIs an n-dimensional real column vector and each element in the column is a positive number. Calculating the difference of the Lyapunov function:
where T represents the transpose of the matrix,
K
iand G
ikThe definition of (c) is identical to that in step 2.2.
2.4 design constants
Sum vector v
i∈R
n,
And
such that the following inequality
For each i e S, where
ρ > 1 is a given constant value, σ
2>σ
1> 0 are respectively a matrix F
iThe upper and lower bound parameters of (a),
2.5 design water service pipe network water supply system
Feedback controller u in a reliable state
f(k)=H
i(K
i+F
iG
ik) x (k) the system is randomly stable. For any initial condition x
0∈R
nMode r
0The state of the system belonging to the epsilon S meets the following conditions:
wherein E {. represents the mathematical expectation, | | | · | | | not calculation1Represents the standard 1 norm, i.e., the sum of the absolute values of the vector elements. Firstly, calculating the infinitesimal operator satisfaction of the Lyapunov function
ΔV(x(k),gk=i)<0.
2.6 the following inequality relationships can be obtained according to step 2.5:
2.7 from step 2.5, step 2.6:
ΔV(x(k),gk=i)≤-γ||x(k)||1,
combining with step 2.3, further converting into:
from step 2.4, the following relationship holds:
2.8 according to the conditions set forth in step 2.4, the nominal controller gain K is taken into account
iIs divided into non-negative components
And a non-positive component
Gain perturbation matrix G
ikIs divided into
And
the following forms are available:
2.9 from steps 2.7 and 2.8 the following inequality relationship is obtained:
combining with the step 2.3, the following can be obtained:
ΔV(x(k),gk=i)<0.
2.10 the reliable state feedback controller of the water supply fault process of the urban water service pipe network system can be obtained by integrating the steps from 2.3 to 2.9, and the form is as follows:
The invention has the following beneficial effects:
the method of the invention aims at the problem of water supply faults of the urban water service pipe network system, establishes a state space model of the water quantity in the water supply tank, designs a reliable state feedback controller by constructing a random complementary Li ya Punuo function and considering an actuator fault model, and can effectively solve the problems of water supply difficulty, water supply faults and the like caused by aging, faults and the like of a pump station and a water-closing valve in the urban water supply process. The designed controller can also better deal with the uncertain problems caused by external factors in the water supply system. The invention adopts the feedback control of the positive Markov jump system, designs a more reliable state feedback controller under the condition of considering the system with the actuator fault, makes up the defects of the general system and the control method, and increases the applicability of the controller and the capability of processing a more complex system.
The specific implementation mode is as follows:
the method is characterized in that a model of the water quantity in a water supply tank of the urban water service pipe network system is established by taking the urban water service pipe network water supply system as an actual object, taking the opening degrees of a pump station and a water shut valve in the system as input and taking the volume of water in the water supply tank as output.
Step 1, establishing a state space model of the water volume of a water supply tank of an urban water service pipe network system, wherein the specific method comprises the following steps:
1.1 firstly, collecting water volume data in a water supply tank of the urban water service pipe network system, and establishing a state space model of the water volume of the water supply tank of the urban water service pipe network system by using the data, wherein the form is as follows:
wherein x (k) ═ x
1(k),x
2(k),...,x
n(k)]
T∈R
nIndicating the volume of water supplied to the tank and n representing the number of tanks. u. of
f(k)∈R
rIndicating opening q of water pipe valve and pump with failure
uAnd r represents the total number of water pipe valves and pumps. R
nRepresenting a real column vector of dimension n.
And
representing a system matrix of appropriate dimensions. For each g
kAll are belonged to S
(
Is directed to the matrix
I.e., all elements within the matrix are non-negative). g
kRepresenting a markov jump process that takes values in a finite set S ═ 1,2,.., M },
and (4) the following steps. For convenience, let's note g
kI, i ∈ S, then there is
1.2 design Markov jump signal gkThe transfer rate satisfies the following conditions:
Pr(gk+1=j|gk=i)=πij,
where for each i, j ∈ S there is a
ij≥0,
Step 2, designing a state feedback controller of water quantity change in a water supply tank of the urban water service pipe network system, which comprises the following specific steps:
2.1 design the control input model with faults as
uf(k)=Hiu(k).
Matrix HiIs an unknown fault matrix and satisfies
Wherein
h ijAnd
is a diagonal matrix
H iAnd
and is a given constant, and ρ > 1 is a given constant value.
2.2 design State feedback controller as
u(k)=(Ki+ΔKik)x(k)
And
ΔKik=FiGik
wherein K
i∈R
r×nIs the nominal controller gain, Δ K, to be designed
ikIs the fluctuation of the controller gain, G
ik∈R
r×nIs the decision matrix, F, to be designed
i∈R
r×rIs a known non-negative matrix and for 0 < sigma
1<σ
2Satisfy the requirement of
2.3 design
Its associated definitions are indicated at step 1.1, step 2.1 and step 2.2. Simultaneously constructing a random complementary Li ya Ponuff function
V(x(k),gk=i)=xT(k)vi,
Wherein
v
i∈R
nIs an n-dimensional real column vector and each element in the column is a positive number. Calculating the difference of the Lyapunov function:
where T represents the transpose of the matrix,
K
iand G
ikThe definition of (c) is identical to that in step 2.2.
2.4 design constants
Sum vector v
i∈R
n,
And
such that the following inequality
For each i e S, where
ρ > 1 is a given constant value, σ
2>σ
1> 0 is a matrix F
iThe upper and lower bound parameters of (a),
2.5 design water service pipe network water supply system
Feedback controller u in a reliable state
f(k)=H
i(K
i+F
iG
ik) x (k) the system is randomly stable. For any initial condition x
0∈R
nMode r
0The state of the system belonging to the epsilon S meets the following conditions:
wherein E {. represents the mathematical expectation, | | | · | | | not calculation1Represents the standard 1 norm, i.e., the sum of the absolute values of the vector elements. Firstly, calculating the infinitesimal operator satisfaction of the Lyapunov function
ΔV(x(k),gk=i)<0.
2.6 the following inequality relationships can be obtained according to step 2.5:
2.7 from step 2.5, step 2.6:
ΔV(x(k),gk=i)≤-γ||x(k)||1,
combining with step 2.3, further converting into:
from step 2.4, the following relationship holds:
2.8 according to the conditions set forth in step 2.4, the nominal controller gain K is taken into account
iIs divided into non-negative components
And a non-positive component
Gain perturbation matrix G
ikIs divided into
And
the following forms are available:
2.9 from steps 2.7 and 2.8 the following inequality relationship is obtained:
combining with the step 2.3, the following can be obtained:
ΔV(x(k),gk=i)<0.
2.10 the reliable state feedback controller of the water supply fault process of the urban water service pipe network system can be obtained by integrating the steps from 2.3 to 2.9, and the form is as follows: