CN114675531A - PI control method of interval uncertain water tank liquid level control system - Google Patents

PI control method of interval uncertain water tank liquid level control system Download PDF

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CN114675531A
CN114675531A CN202210323866.0A CN202210323866A CN114675531A CN 114675531 A CN114675531 A CN 114675531A CN 202210323866 A CN202210323866 A CN 202210323866A CN 114675531 A CN114675531 A CN 114675531A
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control system
liquid level
water tank
level control
tank liquid
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周晓月
曹宇
张俊锋
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Hangzhou Dianzi University
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    • G05B11/36—Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential
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Abstract

The invention discloses a PI control method of an interval uncertain water tank liquid level control system with external disturbance input. The method firstly utilizes a positive switching system with external disturbance input to establish a state space model of the water tank liquid level control system. The PI controller with the interval uncertain positive switching system with the external disturbance input is designed by means of a multiple linear residual positive Lyapunov function and a matrix decomposition technology, so that the liquid level of the multi-capacity water tank system is reasonably controlled, and normal and smooth operation of life and production of people is guaranteed. Almost all systems contain uncertainties which can compromise the performance of the system and even lead to instability of the system. Therefore, it is necessary to design the PI controller based on the interval uncertainty system, and the production benefit of the actual control system is guaranteed.

Description

PI control method of interval uncertain water tank liquid level control system
Technical Field
The invention belongs to the field of automation technology and modern control, and relates to a modeling of a multi-tank liquid level control system based on an interval uncertain positive switching system and a PI control method with external disturbance input.
Background
With the development of society in China, automatic control is implemented very early. The liquid level control system is widely applied in China, is common process control in industry, and has no influence on production. Liquid level control is commonly used in daily production and life, for example, water supply of a water tower cannot meet the requirement of continuous water supply when the water level in the water tower is too low, and normal production and life of people are affected; in order to ensure the normal operation of the boiler, the liquid level of the boiler needs to be maintained to be a normal standard value, the liquid level of the boiler is too low, the boiler is easy to be burnt to dry, serious accidents occur, and the liquid level of the boiler is too high, so that steam is easy to carry water and overflow danger exists. The water level control of the water tank is also applied to the water control of the yellow river, and the water level of the yellow river is detected by the liquid level control system, so that the condition that the water level of the yellow river is too high and the life risk and property loss are brought to people without understanding is avoided. In addition, liquid level control is also very popular in other industries such as petrochemical industry, steel smelting, food and pharmacy. Therefore, the liquid level control technology has become an important part of industrial automation, and has an irreplaceable position in the field of automatic control.
The multi-container water tank liquid level control system is a platform for observing, measuring and monitoring the change of the flow and liquid level parameters of a water tank in the simulation industrial production, has the advantages of strong function, simple and convenient application, small volume and the like, and is beneficial to solving a plurality of problems of the actual industry. In a tank level control system, water flow dynamics in the tank control system are considered. The amount of water in the water reservoir is mainly related to the amount of water flowing into the water reservoir and the amount of water flowing out of the water reservoir. A complete water tank liquid level control system mainly comprises accessories such as a water tank, a valve, a liquid level transmitter and the like. When the liquid level transmitter detects that the water level in the water tank changes, the controller drives the actuating mechanism (namely the valve) to perform corresponding action (namely water inlet or water discharge) so as to keep the water level in the water tank stable. Considering the nonnegative characteristic of water quantity and the mutual influence among a plurality of water tanks, the water tank liquid level control system is difficult to accurately determine, and the water tank liquid level control system is easy to be influenced by sudden external interference such as damage of components and parts, change of external environment and the like, the water quantity dynamic system can be characterized by an interval uncertain positive switching system with external interference input. In the industrial production process, the liquid level of the production device is often required to be maintained at a certain value or changed according to a certain rule so as to meet the requirements of the production process. Conventional control systems mostly employ a proportional controller to achieve the desired results, and the use of a proportional controller alone in the industry may not eliminate the deviation well to bring the actual values of the controlled variables into agreement with the predetermined values of the process requirements. In order to solve the above mentioned problems, the present invention adopts a PI control method, i.e. a PI controller is designed in the present invention. The PI (proportional-integral) controller is composed of a proportional unit P and an integral unit I, is a linear controller in practice, forms a control deviation according to a given value and an actual output value, and linearly combines the proportion and the integral of the deviation to form a control quantity to realize the control of a controlled object so as to lead the actual value of the controlled variable to be consistent with a preset value required by the process. The PI controller combines proportional (P) control and integral (I) control, and has the functions of proportional control and integral control for eliminating deviation in time. The PI controller is widely applied to liquid level, temperature, flow and other control systems in industrial production at present. In industrial control, sudden changes in the control system always occur due to component failure, environmental sudden changes, and other unexpected factors. Mutations often affect the performance of the system and even destroy the stability of the system. In addition, almost all systems contain uncertainties, which can also destroy the performance of the system and even cause system instability. In order to solve the problems, the invention aims to design a PI controller aiming at a water tank liquid level control system which has an uncertain interval of external disturbance input and can switch a plurality of water tanks back and forth under the condition of meeting a certain switching law, so that the performance requirement of the multi-tank liquid level control system is further improved.
Aiming at the problems, the invention utilizes the modern control theory technology to establish a state space model of the multi-tank liquid level control system, designs a PI controller and a proportional and integral gain matrix for the tank system, analyzes the positivity and the stability of the PI controller and ensures that the liquid level in the multi-tank liquid level control system is maintained at a certain value. In conclusion, the design of the water tank liquid level control system and the PI control method based on the interval uncertain positive switching system modeling has important scientific research significance and practical application significance.
Disclosure of Invention
The invention aims to solve the problem that liquid level needs to be controlled in life production, a multi-tank control system device is used for researching a water tank liquid level control system, and a PI control method of an interval uncertain water tank liquid level control system with external disturbance input is provided.
A PI control method of an interval uncertain water tank liquid level control system with external disturbance input comprises the following steps:
step 1, establishing a positive switching system state space model of a multi-tank liquid level control system with external disturbance input;
step 2, establishing a PI control law of a water tank liquid level control system;
step 3, designing an integral part of the PI controller;
step 4, establishing a switching condition met by a switching signal sigma (t);
step 5, designing the stable operation condition of the water tank liquid level control system;
step 6, a positive verification process of the water tank liquid level control system;
step 7, the stability of the water tank liquid level control system is improved, and the water tank liquid level control system has l1Verification process of gain performance gamma.
The specific method of the step 1 is as follows:
establishing a positive switching system state space model of the multi-tank liquid level control system with external disturbance input:
Figure BDA0003572738630000021
y(t)＝Cσ(t)x(t)+Fσ(t)ω(t)
wherein the content of the first and second substances,
Figure BDA0003572738630000022
representing the amount of water in the tank at time t,
Figure BDA0003572738630000023
representing an operation on the derivative of the vector x (t),
Figure BDA0003572738630000024
is the water flow of r controllable valves at time t,
Figure BDA0003572738630000025
the quantity of water flowing out of the water tank is shown by s sensors at the time t,
Figure BDA0003572738630000026
and the external interference on the water tank liquid level control system caused by sudden external changes such as damage of elements of the water tank liquid level control system or environmental changes at the moment t is shown. The function σ (t) represents the switching law and is from a finite set
Figure BDA0003572738630000031
Taking the value in the step (1). When σ (t) ═ p, the pth subsystem is activated, where,
Figure BDA0003572738630000032
Figure BDA0003572738630000033
is a system matrix of the multi-container water tank liquid level control system and meets the uncertainty of the interval
Figure BDA0003572738630000034
Figure BDA0003572738630000035
Wherein the content of the first and second substances,
Figure BDA0003572738630000036
is based on the upper bound, A, of the system matrix obtained from the actual measurements made on the systemσ(t),Bσ(t),Cσ(t),Eσ(t),Fσ(t)Is the lower bound of the system matrix.
Figure BDA0003572738630000037
The real matrix space, the positive integer set and the non-negative integer set respectively represent real vectors with dimensions of n, r and s, n multiplied by n, n multiplied by r, s multiplied by n and s multiplied by r. [ x ] of1(t),x2(t),...,xn(t)]TRepresents a vector [ x ]1(t),x2(t),...,xn(t)]The transposing of (1).
The specific method of the step 2 is as follows:
establishing a PI control law of a water tank liquid level control system, wherein the PI control law is constructed in the following form:
up(t)＝KPpCpx(t)+KPpFpω(t)+KIpe(t)
wherein the content of the first and second substances,
Figure BDA0003572738630000038
and
Figure BDA0003572738630000039
respectively, a proportional gain matrix and an integral gain matrix of the p-th subsystem to be designed, and e (t) is an integral part of the PI controller.
The specific method in step 3 is as follows:
designing an integral part of the PI controller, wherein the integral part is constructed in the following form:
Figure BDA00035727386300000310
wherein alpha is a tuning parameter and alpha is more than 0.
The specific method of the step 4 is as follows:
the switching condition satisfied by the switching signal sigma (t) is established and constructed as follows
Figure BDA00035727386300000311
Wherein N isσ(t0And t) denotes the time t0The number of handovers from time t, τ representing the average residence time, N0Representing the jitter bound.
The specific method of the step 5 is as follows:
the conditions for designing the stable operation of the water tank liquid level control system are as follows:
5.1 design constants ζ > 0, α > 0, β > 1, μ > 0, λ >1,γ＞0，
Figure BDA0003572738630000041
(Vector)
Figure BDA00035727386300000417
And
Figure BDA0003572738630000042
(Vector)
Figure BDA0003572738630000043
such that:
Figure BDA00035727386300000418
Figure BDA0003572738630000044
Figure BDA0003572738630000045
Figure BDA0003572738630000046
Figure BDA0003572738630000047
Figure BDA0003572738630000048
Figure BDA0003572738630000049
Figure BDA00035727386300000410
Figure BDA00035727386300000420
for any purpose
Figure BDA00035727386300000411
If p ≠ q and j ≠ 1, 2.. r, then, under the PI control law in step 2 and the average dwell time switching condition:
Figure BDA00035727386300000412
the tank level control system is positive, stable and has1The gain performance γ. Wherein I is an identity matrix having compatible dimensions; Σ is one summation symbol; 1sAn s-dimensional column vector representing all elements as 1, 1rAn r-dimensional column vector representing all elements as 1,
Figure BDA00035727386300000419
r-dimensional column vectors representing that the jth element is 1 and other elements are all 0; vector superscript(p)And subscript p each represents the vector for the p-th subsystem, superscript(q)Represents the vector for the qth sub-system, and p, q both belong to
Figure BDA00035727386300000413
p is not equal to q; vector quantity
Figure BDA00035727386300000414
The superscript + in (b) indicates that all elements of the vector are positive, the vector
Figure BDA00035727386300000415
Upper mark in-All elements representing the vector are negative.
5.2 designing a proportional gain matrix and an integral gain matrix of the water tank liquid level control system as follows respectively:
Figure BDA00035727386300000416
and satisfies the following conditions:
Figure BDA0003572738630000051
Figure BDA0003572738630000052
wherein the content of the first and second substances,
Figure BDA0003572738630000053
upper mark in+Indicating that all elements of the gain matrix are positive,
Figure BDA0003572738630000054
upper mark in-Indicating that all elements of the gain matrix are negative.
Step 6, the positive verification process of the water tank liquid level control system is as follows:
6.1 obtaining the following data according to the state space model of the water tank liquid level system in the step 1, the integral part of the PI controller in the step 3 and the PI control law designed in the step 2:
Figure BDA0003572738630000055
Figure BDA0003572738630000056
6.2 given
Figure BDA0003572738630000057
Then
Figure BDA0003572738630000058
Thus, step 6.1 can be converted into:
Figure BDA0003572738630000059
wherein- α IsIs a diagonal matrix with s rows and s columns having diagonal elements of-alpha,
Figure BDA00035727386300000510
represents the relative quantity
Figure BDA00035727386300000511
And (6) derivation.
6.4 due to
Figure BDA00035727386300000516
Bp≥B p≥0，
Figure BDA00035727386300000517
And beta > 0 to obtain
Figure BDA00035727386300000512
Using condition (1) in step 5.1 yields:
Figure BDA00035727386300000513
therefore, it is not only easy to use
Figure BDA00035727386300000514
Is a Metzler matrix which has the characteristic that off-diagonal elements are not negative. From the interval uncertainty:
Figure BDA00035727386300000515
therefore, the temperature of the molten metal is controlled,
Figure BDA0003572738630000061
is a Metzler matrix。
6.5 Using condition (2) in step 5.1:
Figure BDA0003572738630000062
from the uncertainty of the interval:
Figure BDA0003572738630000063
6.6 combining condition (3) in step 5.1 gives:
Figure BDA0003572738630000064
from the interval uncertainty:
Figure BDA0003572738630000065
6.7 knowing C from interval uncertaintyp≥C p≥0,Fp≥F pNot less than 0, and combining the steps of 6.4-6.6 to obtain the following product:
Figure BDA0003572738630000066
is a Metzler matrix, and
Figure BDA0003572738630000067
thus, the positivity of the tank level control system was demonstrated.
Step 7, the stability of the water tank liquid level control system is improved, and the water tank liquid level control system has l1The verification process of the gain performance γ is as follows:
7.1 for the p subsystem, design multiple linear complementary positive Lyapunov function
Figure BDA0003572738630000068
Wherein upsilon is(p)＝(v'(p)T v”(p)T)T，
Figure BDA0003572738630000069
Assume that σ (t) is in the interval (t)0The switching sequence in t) is
Figure BDA00035727386300000610
Wherein, Nσ(t0T) is in the interval (t)0T) satisfies the switching law established in step 4 with respect to the switching signal σ (t). And (3) carrying out derivation on the multiple linear complementary positive Lyapunov function to obtain:
Figure BDA00035727386300000611
binding interval uncertainty, the above equation is converted to:
Figure BDA0003572738630000071
wherein the content of the first and second substances,
Figure BDA0003572738630000072
is a multiple linear complementary positive Lyapunov function
Figure BDA0003572738630000073
The derivative function of (a).
7.2 Using condition (4) in step 5.1, one can obtain:
Figure BDA0003572738630000074
Figure BDA0003572738630000075
7.3 combining condition (8) in step 5.1 and step 7.2 gives:
Figure BDA0003572738630000076
Figure BDA0003572738630000077
thus, it can be derived:
Figure BDA0003572738630000078
7.4 combining conditions (5) - (7) in step 5.1 gives:
Figure BDA0003572738630000079
7.5 integrating the two sides of the equation unequal sign of step 7.4 simultaneously and recycling the condition (8) in step 5.1 to obtain:
Figure BDA00035727386300000710
that is to say
Figure BDA00035727386300000711
This is true. Wherein the content of the first and second substances,
Figure BDA00035727386300000712
‖·‖1represents the 1 norm of the vector, i.e., the sum of the absolute values of all the elements in the vector,
Figure BDA00035727386300000713
the maximum amount of water flowing out of the water tank collected by the sensor at time t is shown.
Therefore, the interval uncertain tank liquid level control system with external disturbance input is stable and has I1The gain performance γ.
The invention has the following beneficial effects:
the method firstly utilizes an interval uncertain positive switching system with external disturbance input to establish a state space model of the water tank liquid level control system. The PI controller is designed by means of a multiple linear residual-positive Lyapunov function and a matrix decomposition technology, so that the system is reasonably controlled by liquid level, and normal production and life of people are guaranteed. The PI controller is designed based on the interval uncertain system with external disturbance input, so that the stability of the system is enhanced, the performance of the system is improved, and the production benefit is ensured.
Drawings
FIG. 1 is a schematic view of a multi-tank level control system according to the present invention.
FIG. 2 is a schematic diagram of a PI control framework based on a multiple tank level control system with external disturbance input.
Detailed Description
The present invention will be further described with reference to specific examples.
As shown in fig. 1 and 2, the PI control method of the present invention is based on a tank level control system modeled by a positive switching system with an external disturbance input. The method comprises the following specific steps:
step 1, establishing a positive switching system state space model of a multi-tank liquid level control system with external disturbance input;
Figure BDA0003572738630000081
y(t)＝Cσ(t)x(t)+Fσ(t)ω(t)
wherein the content of the first and second substances,
Figure BDA0003572738630000082
representing the amount of water in the tank at time t,
Figure BDA0003572738630000083
representing an operation on the derivative of the vector x (t),
Figure BDA0003572738630000084
is the water flow of r controllable valves at time t,
Figure BDA0003572738630000085
indicates the time t passes through sThe sensor collects the obtained water quantity flowing out of the water tank,
Figure BDA0003572738630000086
and the external interference on the water tank liquid level control system caused by sudden external changes such as damage of elements of the water tank liquid level control system or environmental changes at the moment t is shown. The function σ (t) represents the switching law and is from a finite set
Figure BDA0003572738630000087
A medium value. When σ (t) ═ p, the pth subsystem is activated, where,
Figure BDA0003572738630000088
Figure BDA0003572738630000089
is a system matrix of the multi-container water tank liquid level control system and meets the uncertainty of the interval
Figure BDA00035727386300000810
Figure BDA00035727386300000811
Wherein the content of the first and second substances,
Figure BDA00035727386300000812
is based on the upper bound of the system matrix from actual measurements made on the system,A σ(t),B σ(t),C σ(t),E σ(t),F σ(t)is the lower bound of the system matrix.
Figure BDA0003572738630000091
The real matrix space, the positive integer set and the non-negative integer set respectively represent real vectors with dimensions of n, r and s, n x n, n x r, s x n and s x r. [ x ]1(t),x2(t),...,xn(t)]TRepresents a vector [ x1(t),x2(t),...,xn(t)]The transposing of (1).
Step 2, establishing a PI control law of a water tank liquid level control system;
establishing a PI control law of a water tank liquid level control system, wherein the PI control law is constructed in the following form:
up(t)＝KPpCpx(t)+KPpFpω(t)+KIpe(t)
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003572738630000092
and
Figure BDA0003572738630000093
respectively, a proportional gain matrix and an integral gain matrix of the p-th subsystem to be designed, and e (t) is an integral part of the PI controller.
Step 3, designing an integral part of the PI controller;
designing an integral part of the PI controller, wherein the integral part is constructed as follows:
Figure BDA0003572738630000094
wherein alpha is a tuning parameter and alpha is more than 0.
Step 4, establishing a switching condition met by a switching signal sigma (t);
the switching condition satisfied by the switching signal sigma (t) is established and constructed as follows
Figure BDA0003572738630000095
Wherein N isσ(t0T) denotes the time t0The number of handovers from time t, τ representing the average residence time, N0Indicating a jitter bound.
Step 5, designing the stable operation condition of the water tank liquid level control system;
the conditions for designing the stable operation of the water tank liquid level control system are as follows:
5.1 design constants ζ > 0, α > 0, β > 1, μ > 0, λ > 1, γ > 0,
Figure BDA0003572738630000096
(Vector)
Figure BDA0003572738630000097
and
Figure BDA0003572738630000098
(Vector)
Figure BDA0003572738630000099
such that:
Figure BDA00035727386300000910
Figure BDA00035727386300000911
Figure BDA00035727386300000912
Figure BDA0003572738630000101
Figure BDA0003572738630000102
Figure BDA0003572738630000103
Figure BDA0003572738630000104
Figure BDA0003572738630000105
Figure BDA0003572738630000106
for any purpose
Figure BDA0003572738630000107
If p ≠ q and j ≠ 1, 2.. r, then, under the PI control law in step 2 and the average dwell time switching condition:
Figure BDA0003572738630000108
the tank level control system is positive, stable and has1The gain performance γ. Wherein I is an identity matrix having compatible dimensions; Σ is one summation symbol; 1sAn s-dimensional column vector representing all elements 1, 1rAn r-dimensional column vector representing all elements as 1,
Figure BDA0003572738630000109
r-dimensional column vectors which represent that the jth element is 1 and other elements are 0; vector superscript(p)And the subscript p each represents the vector for the p-th subsystem, the superscript(q)Represents the vector for the qth sub-system, and p, q both belong to
Figure BDA00035727386300001010
p is not equal to q; vector quantity
Figure BDA00035727386300001011
Upper mark in+All elements representing the vector are positive, the vector
Figure BDA00035727386300001012
Upper mark in-All elements representing the vector are negative.
5.2 designing a proportional gain matrix and an integral gain matrix of the water tank liquid level control system as follows respectively:
Figure BDA00035727386300001013
and satisfies the following conditions:
Figure BDA00035727386300001014
Figure BDA00035727386300001015
wherein the content of the first and second substances,
Figure BDA00035727386300001016
upper mark in+Indicating that all elements of the gain matrix are positive,
Figure BDA00035727386300001017
upper mark of (1)-Indicating that all elements of the gain matrix are negative.
Step 6, the positive verification process of the water tank liquid level control system is as follows:
6.1 according to the state space model of the water tank liquid level system in the step 1, the integral part of the PI controller in the step 3 and the PI control law designed in the step 2, obtaining:
Figure BDA0003572738630000111
Figure BDA0003572738630000112
6.2 given
Figure BDA0003572738630000113
Then
Figure BDA0003572738630000114
Thus, step 6.1 can be converted into:
Figure BDA0003572738630000115
wherein- α IsIs a diagonal matrix with s rows and s columns having diagonal elements of-alpha,
Figure BDA0003572738630000116
represents the relative quantity
Figure BDA0003572738630000117
And (6) derivation.
6.4 due to
Figure BDA0003572738630000118
Bp≥Bp≥0，
Figure BDA0003572738630000119
And beta > 0 to obtain
Figure BDA00035727386300001110
Using condition (1) in step 5.1 yields:
Figure BDA00035727386300001111
therefore, it is not only easy to use
Figure BDA00035727386300001112
Is a Metzler matrix which has the characteristic that off-diagonal elements are not negative. From the interval uncertainty:
Figure BDA00035727386300001113
therefore, the temperature of the molten metal is controlled,
Figure BDA00035727386300001114
is a Metzler matrix.
6.5 Using condition (2) in step 5.1:
Figure BDA00035727386300001115
from the uncertainty of the interval:
Figure BDA00035727386300001116
6.6 combining condition (3) in step 5.1 gives:
Figure BDA0003572738630000121
from the interval uncertainty:
Figure BDA0003572738630000122
6.7 knowing from the interval uncertainty
Figure BDA0003572738630000123
Combining the steps 6.4-6.6 to obtain the following products:
Figure BDA0003572738630000124
is a Metzler matrix, and
Figure BDA0003572738630000125
thus, the positivity of the tank level control system was demonstrated.
Step 7, the stability of the water tank liquid level control system is improved, and the water tank liquid level control system has l1The verification process of the gain performance γ is as follows:
7.1 for the p subsystem, design multiple linear complementary positive Lyapunov function
Figure BDA0003572738630000126
Wherein upsilon is(p)＝(v'(p)T v”(p)T)T，
Figure BDA0003572738630000127
Assume that σ (t) is in the interval (t)0The switching sequence in t) is
Figure BDA0003572738630000128
Wherein N isσ(t0T) is in the interval (t)0T) satisfies the switching law established in step 4 with respect to the switching signal σ (t). And (3) carrying out derivation on the multiple linear complementary positive Lyapunov function to obtain:
Figure BDA0003572738630000129
binding interval uncertainty, the above equation is converted to:
Figure BDA00035727386300001210
wherein the content of the first and second substances,
Figure BDA00035727386300001211
is a multiple linear complementary Lyapunov function
Figure BDA00035727386300001212
The derivative function of (a).
7.2 Using condition (4) in step 5.1, one can obtain:
Figure BDA0003572738630000131
Figure BDA0003572738630000132
7.3 combining condition (8) in step 5.1 and step 7.2 gives:
Figure BDA0003572738630000133
Figure BDA0003572738630000134
thus, it can be derived:
Figure BDA0003572738630000135
7.4 combining conditions (5) - (7) in step 5.1 gives:
Figure BDA0003572738630000136
7.5 integrating the two sides of the equation unequal sign of step 7.4 simultaneously and recycling the condition (8) in step 5.1 to obtain:
Figure BDA0003572738630000137
that is to say
Figure BDA0003572738630000138
This is true. Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003572738630000139
‖·‖1represents the 1 norm of the vector, i.e., the sum of the absolute values of all the elements in the vector,
Figure BDA00035727386300001310
the maximum amount of water flowing out of the water tank collected by the sensor at time t is shown.
Therefore, the interval uncertain tank liquid level control system with external disturbance input is stable and has I1The gain performance gamma.

Claims (8)

1. A PI control method of an interval uncertain water tank liquid level control system with external disturbance input is characterized by comprising the following steps:
step 1, establishing a positive switching system state space model of a multi-tank liquid level control system with external disturbance input;
step 2, establishing a PI control law of a water tank liquid level control system;
step 3, designing an integral part of the PI controller;
step 4, establishing a switching condition met by a switching signal sigma (t);
step 5, designing the stable operation condition of the water tank liquid level control system;
step 6, a positive verification process of the water tank liquid level control system;
step 7, the stability of the water tank liquid level control system is improved, and the water tank liquid level control system has l1Verification process of gain performance gamma.
2. The PI control method for the interval uncertain water tank liquid level control system with the external disturbance input as claimed in claim 1, wherein the specific method in step 1 is as follows:
establishing a positive switching system state space model of the multi-tank liquid level control system with external disturbance input:
Figure FDA0003572738620000011
y(t)＝Cσ(t)x(t)+Fσ(t)ω(t)
wherein the content of the first and second substances,
Figure FDA0003572738620000012
representing the amount of water in the tank at time t,
Figure FDA0003572738620000013
representing the operation of taking the derivative of the vector x (t),
Figure FDA0003572738620000014
is the water flow of r controllable valves at time t,
Figure FDA0003572738620000015
the quantity of water flowing out of the water tank is shown by s sensors at the time t,
Figure FDA0003572738620000016
the external interference on the water tank liquid level control system caused by sudden external changes such as component damage or environmental change of the water tank liquid level control system at the moment t is shown; the function σ (t) represents the switching law and is from a finite set
Figure FDA0003572738620000017
Taking a middle value; when σ (t) ═ p, the pth subsystem is activated, where,
Figure FDA0003572738620000018
Figure FDA0003572738620000019
is a system matrix of the multi-container water tank liquid level control system and meets the uncertainty of the interval
Figure FDA00035727386200000110
Figure FDA0003572738620000021
Wherein the content of the first and second substances,
Figure FDA0003572738620000022
is based on the upper bound of the system matrix from actual measurements made on the system,A σ(t),B σ(t),C σ(t),E σ(t),F σ(t)is the lower bound of the system matrix;
Figure FDA0003572738620000023
respectively representing real matrix spaces, positive integer sets and non-negative integer sets of n-dimensional, r-dimensional and s-dimensional real vectors, n x n-dimensional, n x r-dimensional, s x n-dimensional and s x r-dimensional; [ x ]1(t),x2(t),...,xn(t)]TRepresents a vector [ x1(t),x2(t),...,xn(t)]The transposing of (1).
3. The PI control method for the interval uncertain water tank liquid level control system with external disturbance input as claimed in claim 2, wherein the specific method in step 2 is as follows:
establishing a PI control law of a water tank liquid level control system, wherein the PI control law is constructed in the following form:
up(t)＝KPpCpx(t)+KPpFpω(t)+KIpe(t)
wherein the content of the first and second substances,
Figure FDA0003572738620000024
and
Figure FDA0003572738620000025
respectively, a proportional gain matrix and an integral gain matrix of the p-th subsystem to be designed, and e (t) is an integral part of the PI controller.
4. The PI control method for the interval uncertain water tank liquid level control system with external disturbance input as claimed in claim 3, wherein the specific method in step 3 is as follows:
designing an integral part of the PI controller, wherein the integral part is constructed as follows:
Figure FDA0003572738620000026
wherein alpha is a tuning parameter and alpha is more than 0.
5. The PI control method for the interval uncertain water tank liquid level control system with external disturbance input as claimed in claim 4, wherein the specific method of step 4 is as follows:
establishing a switching condition satisfied by a switching signal sigma (t), wherein the construction form is as follows:
Figure FDA0003572738620000027
wherein N isσ(t0T) denotes the time t0The number of handovers from time t, τ representing the average residence time, N0Representing the jitter bound.
6. The PI control method for the interval uncertain tank level control system with external disturbance input as claimed in claim 5, wherein the specific method in step 5 is as follows:
the conditions for designing the stable operation of the water tank liquid level control system are as follows:
5.1 design constants ζ > 0, α > 0, β > 1, μ > 0, λ > 1, γ > 0,
Figure FDA0003572738620000031
(Vector)
Figure FDA0003572738620000032
and
Figure FDA0003572738620000033
(Vector)
Figure FDA0003572738620000034
Figure FDA0003572738620000035
such that:
Figure FDA0003572738620000036
Figure FDA0003572738620000037
Figure FDA0003572738620000038
Figure FDA0003572738620000039
Figure FDA00035727386200000310
Figure FDA00035727386200000311
Figure FDA00035727386200000312
Figure FDA00035727386200000313
Figure FDA00035727386200000314
for any purpose
Figure FDA00035727386200000315
If p ≠ q and j ≠ 1, 2.. r, then, under the PI control law in step 2 and the average dwell time switching condition:
Figure FDA00035727386200000316
the tank level control system is positive, stable and has1Gain performance γ; wherein I is an identity matrix having compatible dimensions; Σ is a summationA symbol; 1sAn s-dimensional column vector representing all elements as 1, 1rAn r-dimensional column vector representing all elements as 1,
Figure FDA00035727386200000317
r-dimensional column vectors which represent that the jth element is 1 and other elements are 0; vector superscript(p)And subscriptspAll represent vectors, superscripts, for the p-th subsystem(q)Represents the vector for the qth sub-system, and p, q both belong to
Figure FDA00035727386200000318
p is not equal to q; vector quantity
Figure FDA00035727386200000319
Upper mark in+All elements representing the vector are positive, the vector
Figure FDA00035727386200000320
Upper mark in-All elements representing the vector are negative;
5.2 designing a proportional gain matrix and an integral gain matrix of the water tank liquid level control system as follows respectively:
Figure FDA0003572738620000041
and satisfies the following conditions:
Figure FDA0003572738620000042
Figure FDA0003572738620000043
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003572738620000044
upper mark in+Indicating that all elements of the gain matrix are positive,
Figure FDA0003572738620000045
superscript in (b) -indicates that all elements of the gain matrix are negative.
7. The PI control method for the interval uncertain tank level control system with external disturbance input as claimed in claim 6, wherein the specific method of step 6 is as follows:
the positive verification process of the water tank liquid level control system is as follows:
6.1 according to the state space model of the water tank liquid level system in the step 1, the integral part of the PI controller in the step 3 and the PI control law designed in the step 2, obtaining:
Figure FDA0003572738620000046
Figure FDA0003572738620000047
6.2 given
Figure FDA0003572738620000048
Then
Figure FDA0003572738620000049
Thus, step 6.1 can be converted into:
Figure FDA00035727386200000410
wherein- α IsIs a diagonal matrix with s rows and s columns having diagonal elements of-alpha,
Figure FDA00035727386200000411
represents the relative quantity
Figure FDA00035727386200000414
Derivation is carried out;
6.4 due to
Figure FDA00035727386200000412
And beta > 0 to obtain
Figure FDA00035727386200000413
Using condition (1) in step 5.1 yields:
Figure FDA0003572738620000051
therefore, it is possible to
Figure FDA0003572738620000052
The matrix is a Metzler matrix which has the characteristic that non-diagonal elements are not negative; from the interval uncertainty:
Figure FDA0003572738620000053
therefore, the temperature of the molten metal is controlled,
Figure FDA0003572738620000054
is a Metzler matrix;
6.5 Using condition (2) in step 5.1:
Figure FDA0003572738620000055
from the uncertainty of the interval:
Figure FDA0003572738620000056
6.6 combining condition (3) in step 5.1 gives:
Figure FDA0003572738620000057
from the interval uncertainty:
Figure FDA0003572738620000058
6.7 knowing from the interval uncertainty
Figure FDA0003572738620000059
Combining the steps 6.4-6.6 to obtain the following products:
Figure FDA00035727386200000510
is a Metzler matrix, and
Figure FDA00035727386200000511
8. the PI control method for the interval uncertain tank level control system with external disturbance input as claimed in claim 7, wherein the specific method of step 7 is as follows:
stability of water tank level control system and having l1The verification process of the gain performance γ is as follows:
7.1 for the p subsystem, designing a multiple linear complementary positive Lyapunov function
Figure FDA0003572738620000061
Wherein for any
Figure FDA0003572738620000062
Can all obtain
Figure FDA00035727386200000610
Assume that σ (t) is in the interval (t)0The switching sequence in t) is
Figure FDA0003572738620000063
Wherein N isσ(t0T) is in the interval (t)0T) the number of switching times, which satisfies the switching law about the switching signal σ (t) established in step 4; and (3) carrying out derivation on the multiple linear complementary positive Lyapunov function to obtain:
Figure FDA0003572738620000064
binding interval uncertainty, the above equation is converted to:
Figure FDA0003572738620000065
wherein the content of the first and second substances,
Figure FDA0003572738620000066
is a multiple linear complementary Lyapunov function
Figure FDA0003572738620000067
A derivative function of;
7.2 Using condition (4) in step 5.1, one can obtain:
Figure FDA0003572738620000068
Figure FDA0003572738620000069
7.3 combining condition (8) in step 5.1 and step 7.2 gives:
Figure FDA0003572738620000071
Figure FDA0003572738620000072
Figure FDA0003572738620000073
Figure FDA0003572738620000074
thus, it can be derived:
Figure FDA0003572738620000075
7.4 combining conditions (5) - (7) in step 5.1 gives:
Figure FDA0003572738620000076
7.5 integrating the two sides of the equation unequal sign of step 7.4 simultaneously and recycling the condition (8) in step 5.1 to obtain:
Figure FDA0003572738620000077
namely:
Figure FDA0003572738620000078
if true; wherein the content of the first and second substances,
Figure FDA0003572738620000079
‖·‖1represents the 1 norm of the vector, i.e., the sum of the absolute values of all the elements in the vector,
Figure FDA00035727386200000710
the maximum amount of water flowing out of the water tank collected by the sensor at time t is shown.
CN202210323866.0A 2022-03-30 2022-03-30 PI control method of interval uncertain water tank liquid level control system Pending CN114675531A (en)

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115259362A (en) * 2022-08-26 2022-11-01 海南大学 Non-fragile PI control method for urban sewage treatment system

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115259362A (en) * 2022-08-26 2022-11-01 海南大学 Non-fragile PI control method for urban sewage treatment system
CN115259362B (en) * 2022-08-26 2023-05-16 海南大学 Non-fragile PI control method of urban sewage treatment system

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