CN116400581A - Undisturbed switching PID control method for stable operation of sewage treatment system - Google Patents

Undisturbed switching PID control method for stable operation of sewage treatment system Download PDF

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CN116400581A
CN116400581A CN202310238904.7A CN202310238904A CN116400581A CN 116400581 A CN116400581 A CN 116400581A CN 202310238904 A CN202310238904 A CN 202310238904A CN 116400581 A CN116400581 A CN 116400581A
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sewage treatment
treatment system
switching
controller
undisturbed
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黄梦醒
张沛
张俊锋
吴迪
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Hainan University
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Hainan University
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    • G05—CONTROLLING; REGULATING
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    • G05B11/00—Automatic controllers
    • G05B11/01—Automatic controllers electric
    • G05B11/36—Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential
    • G05B11/42—Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential for obtaining a characteristic which is both proportional and time-dependent, e.g. P. I., P. I. D.

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Abstract

The invention discloses a non-disturbance switching PID control method for a multi-mode system, which is based on a positive switching system modeling method, a non-disturbance switching technology and a PID control method and is used for collecting data of sewage discharge and water yield in a sewage treatment process. The method is beneficial to improving the control precision and can also improve the running stability of the system. The method fully considers the characteristics of non-negativity, multi-modal property and the like of the control variable and the switching oscillation problem of the controller during modeling, and designs a disturbance-free switching PID controller based on the characteristics. The undisturbed switching strategy is provided to realize smooth switching of the controller and improve the stability of the system. Meanwhile, the application of the linear programming method greatly improves the calculation efficiency, and has greater advantages for solving the calculation of a complex system.

Description

Undisturbed switching PID control method for stable operation of sewage treatment system
Technical Field
The invention belongs to the field of engineering technology and automatic control, and particularly relates to a non-disturbance switching PID control method for smooth running of a sewage treatment system.
Background
Currently, with the development of economy and the increase of population, the demand for water resources is increasing. However, the problems of limited water resources, high difficulty in developing water resources, serious water pollution and the like prevent the development and the utilization of water resources, and many countries face the water shortage crisis, and meanwhile, the further development of economy is restricted. To alleviate the water supply shortage, many countries are beginning to build sewage treatment systems to realize secondary use of water resources. At present, the sewage treatment efficiency of most developed countries is higher than that of China, and the main reasons for slow development of the sewage treatment industry in China are as follows: the sewage treatment technology is late in starting, and the problems of low treatment efficiency, low automation degree, high energy consumption and the like exist; the management capability is not strong; insufficient funds or lack of funds. Therefore, highly automated sewage treatment techniques are proposed to have a critical role in improving sewage treatment efficiency. The existing urban sewage treatment is roughly divided into primary treatment, secondary treatment and tertiary treatment. The secondary treatment is also called as biological treatment system, mainly comprises an aeration tank and a secondary sedimentation tank, and is the most important part in the sewage treatment process. The microorganism concentration, the dissolved oxygen concentration and the like in the sewage biochemical treatment process have important influence on the water quality after sewage treatment, so that the flow of discharged sewage is required to meet the sewage quantity required by an aeration tank and a sedimentation tank. Therefore, the control of the discharge amount of sewage by the automatic control technology is of great significance for improving the quality of the effluent. Based on the analysis, the invention provides a novel automatic control technology of the sewage treatment system, thereby improving the quality of effluent water of sewage treatment and ensuring safe and stable operation of the system.
Considering that the mode of an actual sewage treatment system is not constant, taking urban sewage discharge as an example, the sewage discharge amount is 6 in the morning: 30-10.30 and night 19:00-22:30 will have two high peaks and the other periods will not have high displacement. Therefore, describing an actual sewage treatment system by a single dynamic model is not accurate, and problems such as increased difficulty in controller design, untimely control and the like can be caused. The invention is intended to model a sewage treatment system using a switching system with multiple modes. Meanwhile, the state of the sewage treatment system, namely the sewage discharge amount, is non-negative, and the existing sewage treatment model, such as a general switching system model, a general state space model and the like, is adopted to cause modeling redundancy, so that resource conservation is not facilitated, and modeling accuracy is improved. Therefore, the invention provides a direct-conversion system which takes both non-negativity and multi-modal into consideration to model the sewage treatment system, thereby improving the modeling precision and being more in line with the actual system. When the sewage treatment system is switched from one mode to another mode, the corresponding controller can also generate switching to achieve the effect of one-to-one control. However, when the controller performs switching, a switching buffeting phenomenon often occurs. When the vibration of the controller is serious, the quality of the discharged water is not up to standard, and the system is unstable to operate. Aiming at the problem of controller switching vibration, the invention designs the undisturbed switching controller by introducing certain limiting conditions, achieves the aim of stable switching of the controller among a plurality of modes, realizes smooth switching and improves the system stability. The traditional control system mainly controls proportion, but due to the large scale and the complex whole of the sewage treatment system and the influence of factors such as the performance limitation of devices such as a sensor, an actuator and the like, the real-time performance and the accuracy of the sewage discharge amount of the actual sewage treatment system are not very high, so that the system error is increased. Therefore, accurate control of the amount of sewage discharge cannot be achieved by only proportional control. In order to reduce the system error as much as possible, the invention proposes a control strategy based on a PID controller. The PID controller (proportional-integral-derivative controller) performs deviation adjustment on the whole control system by setting a proportional unit P, an integral unit I and a derivative unit D, so that the actual value of the controlled variable is consistent with the given value. The PID controller combines proportional (P) control, integral (I) control and differential (D) control, integrates the advantages of timely proportional control, static error elimination of integral control, oscillation reduction of differential control and the like, and ensures that the control process is timely, accurate and stable. Compared with a general quadratic matrix inequality method, the linear programming method adopted by the invention has greater advantages in terms of calculation method for processing large-scale calculation.
Based on the analysis, the invention establishes a state space model of the direct-switching system of the sewage treatment system by utilizing the modern control theory technology, designs a non-disturbance switching PID controller for the sewage treatment system, realizes smooth switching of the controller, ensures high-efficiency, safe and stable operation of the system, and improves the precision and stability of the system. Meanwhile, the solution is carried out by adopting a linear programming method, so that the calculation load is reduced, the calculation resources are saved, and the method has important reference value and practical significance for treating large-scale systems like sewage systems and the like.
Disclosure of Invention
Aiming at the sewage treatment process, the invention establishes a positive and multi-mode characteristic direct-switching system model, and provides a non-disturbance switching PID control method for the stable operation of the sewage treatment system.
A non-disturbing switching PID control method for smooth operation of a sewage treatment system comprises the following steps:
step 1, establishing a state space model of a sewage treatment process;
step 2, constructing a non-disturbing switching PID controller of the sewage treatment system;
step 3, introducing a reference controller;
step 4, introducing a controller undisturbed switching mechanism;
step 5, constructing a non-disturbing switching PID control closed loop system of the sewage treatment system;
step 6, designing a linear programming form for ensuring the stable operation of the sewage treatment system;
step 7, positive verification of a closed-loop control system of the sewage treatment system;
step 8, verifying the stability of a closed-loop control system of the sewage treatment system;
and 9, verifying the undisturbed property of the PID controller of the sewage treatment system.
Further, the step 1 specifically comprises the following steps:
1.1, collecting input and output data of a sewage treatment system to describe an actual system;
1.2, establishing a state space model of the sewage treatment process.
Further, the step 2 specifically comprises the following steps:
u(k)＝K σ(k) y(k)+L σ(k) e(k)+M σ(k) (Δy(k)),
wherein e (k) =y (k-1) +e (k-1) - δe (k-1) represents the integral term of the controller, the derivative term of the controller satisfying Δy (k) =y (k) -y (k-1) =c σ(k) x(k)-C σ(k) x(k-1)，K σ(k) 、L σ(k) 、M σ(k) Proportional, integral and differential gain matrices, denoted sigma (k) m-1 )＝q,σ(k m )＝p,σ(k m+1 ) =r; the method comprises the steps of designing the gain of a non-disturbing switching PID controller in the sewage treatment process, wherein the specific form is as follows:
Figure BDA0004123455370000031
wherein the symbols are
Figure BDA00041234553700000313
Representing a transpose of a vector or matrix, 1 r R-dimensional column vector representing all elements 1,/->
Figure BDA0004123455370000032
R-dimensional column vector, ζ, representing that the i-th element is 1 and the remaining elements are 0 p 、η p And theta p Is an s-dimensional column vector, ">
Figure BDA0004123455370000033
Is an n-dimensional column vector.
Further, the form of the reference controller in step 3 is as follows:
u * (k)＝K * y(k)+L * e(k)+M * (Δy(k))，
wherein u is * (k) Representing the desired control input, K * ,L * ,M * Is a given matrix and respectively satisfies
Figure BDA0004123455370000034
Further, specific constraint conditions of the controller undisturbed switching mechanism in step 4 are as follows:
Figure BDA0004123455370000035
wherein alpha is more than or equal to 0, beta is more than or equal to 0, gamma is more than or equal to 0 and is an undisturbed performance index,
Figure BDA00041234553700000312
and->
Figure BDA0004123455370000036
Control input u (k) and desired control input u, respectively * (k) Is>
Figure BDA0004123455370000037
A row element.
Further, the step 5 specifically comprises the following steps:
combining the state space model of the step 1 and the undisturbed PID control model of the step 2, the PID closed-loop control system of the sewage treatment control system is as follows:
Figure BDA0004123455370000038
wherein,,
Figure BDA0004123455370000039
Figure BDA00041234553700000310
Figure BDA00041234553700000311
I s representing an identity matrix in s x s dimensions.
Further, in step 6, the following conditions are satisfied for designing a linear programming form that ensures smooth operation of the sewage treatment system:
design constants 0 < delta < 1,0 < rho < 1, lambda > 1, iota=1, …, r,
Figure BDA0004123455370000041
vector->
Figure BDA0004123455370000042
And->
Figure BDA0004123455370000043
(Vector)
Figure BDA0004123455370000044
So that
Figure BDA0004123455370000045
Figure BDA0004123455370000046
Figure BDA0004123455370000047
Figure BDA0004123455370000048
Figure BDA0004123455370000049
Figure BDA00041234553700000410
Figure BDA00041234553700000411
Figure BDA00041234553700000412
Figure BDA00041234553700000413
Figure BDA00041234553700000414
Figure BDA00041234553700000415
Figure BDA00041234553700000416
For any p, q e S, then, under the undisturbed switching PID controller designed in step 2, the closed loop system of the sewage treatment system is positive and exponentially stable, and the average residence time condition is designed as:
Figure BDA00041234553700000417
further, the positive verification process of the closed loop control system of the sewage treatment system in step 7 is as follows:
from the specific form of the undisturbed switching PID controller designed in step 2 and condition (a) in step 6, we get
Figure BDA00041234553700000418
I.e. indicate
Figure BDA00041234553700000419
Combining the conditions in step 2 and step 6
Figure BDA00041234553700000420
Obtaining
Figure BDA0004123455370000051
I.e. indicate
Figure BDA0004123455370000052
Further, it is known that->
Figure BDA0004123455370000053
Thus, the positive performance of the closed loop control system of the sewage treatment system is ensured.
Further, the stability verification process of the closed loop control system of the sewage treatment system in step 8 is as follows:
construction of Lyapunov function
Figure BDA0004123455370000054
Wherein,,
Figure BDA0004123455370000055
for V p Difference is obtained to obtain
Figure BDA0004123455370000056
Deducing from the conditions (b) - (e) in step 6
Figure BDA0004123455370000057
It is known that each subsystem is stable. Further, combining the condition (f) in step 6 with the designed average residence time, we obtain
Figure BDA0004123455370000058
Finally, after finishing, obtain
Figure BDA0004123455370000059
Wherein,,
Figure BDA00041234553700000510
thus, the stability of the closed loop control system of the sewage treatment system is ensured.
Further, the procedure of verifying the non-interference of the PID controller of the sewage treatment system in step 9 is as follows:
first, the following equation is obtained according to steps 3 and 4:
Figure BDA00041234553700000511
wherein,,
Figure BDA00041234553700000512
and->
Figure BDA00041234553700000513
Respectively represent K p And K * Is>
Figure BDA00041234553700000514
A row element.
Then, according to the gain matrix form in step 2, obtain
Figure BDA0004123455370000061
Combining conditions (g) and (h) in step 6
Figure BDA0004123455370000067
The following inequality is obtained
Figure BDA0004123455370000062
According to conditions (i) and (j) in step 6
Figure BDA0004123455370000068
Obtaining
Figure BDA0004123455370000063
Similarly, according to conditions (k) and (l) in step 6 and
Figure BDA0004123455370000069
and 1-delta > 0, can be obtained
Figure BDA0004123455370000064
The following inequality is derived by combining the three inequalities
Figure BDA0004123455370000065
And then can obtain
Figure BDA0004123455370000066
Therefore, an undisturbed switching PID controller of the wastewater treatment system is effective.
The invention provides a design method of a PID controller capable of performing undisturbed switching, which is based on a direct-switching system model, an undisturbed switching technology and a PID control method and aims at data acquisition of sewage flow in the sewage treatment process, so that the control system can be ensured to operate efficiently, safely and stably. Meanwhile, the linear programming method is adopted to reduce the calculation load and save the calculation resources. The data privacy is secure. The model built by the invention fully considers the characteristics of the positive, multi-modal, switching oscillation and the like of an actual system, and has higher application value.
Drawings
FIG. 1 is a flow chart of a sewage treatment process;
FIG. 2 is a schematic diagram of a smooth running undisturbed switching PID control architecture for a wastewater treatment system of the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
As shown in FIG. 1, a dynamic model of the sewage treatment process is established by taking the sewage treatment process as a research object, urban flow flowing into the sewage treatment system as a control input, and civil water yield after three-stage treatment as an output.
The invention relates to a non-disturbing switching PID control method for stably operating a sewage treatment system, which comprises the following steps:
step 1, combining data transmission dynamic process data packet quantity data to establish a state space model of the sewage treatment process.
1.1, collecting input and output data of a sewage treatment system to describe an actual system:
considering the sewage treatment process as shown in fig. 1, sewage to be treated first enters a grating and a primary sedimentation tank, and solid pollutants in a suspended state in the sewage are mainly removed, and the process is also called a pretreatment process. The pretreated sewage enters an aeration tank and a secondary sedimentation tank containing microorganisms for biochemical treatment, and organic pollutants in colloid and dissolved state in the sewage are mainly removed. And then, continuing to perform tertiary treatment on the sewage after secondary treatment to obtain the quoted domestic water, and allowing part of untreated sewage to flow into the river channel. The whole sewage treatment process is finished. Wherein, in the biochemical treatment process, the concentration of the reflux microorganism, the concentration of the dissolved oxygen and the like have great influence on the quality of the effluent water, so that the amount of sewage flowing into the aeration tank and the secondary sedimentation tank should be strictly controlled. In view of the large and complex sewage treatment system and the limitations of equipment, errors are inevitably generated in the process of controlling the system, and therefore, a controller with superior performance is applied to the sewage treatment system to improve the control accuracy. The actual sewage treatment system is modeled as a direct-swing system model in consideration of non-negativity of sewage discharge amount and multi-modal characteristics. Because the corresponding controller can also switch when the switching system is switched, the switching oscillation phenomenon is very easy to occur, and therefore, a design method of a disturbance-free switching PID controller capable of realizing smooth switching is provided. In addition, in order to reduce the system calculation load and improve the calculation efficiency, a linear programming method is adopted. Fig. 2 is a diagram of the undisturbed switching PID control structure.
1.2, collecting sewage discharge data in the sewage treatment process, and establishing a state space model of the sewage treatment process, wherein the form is as follows:
x(k+1)＝A σ(k) x(k)+B σ(k) u(k),k∈N,
y(k)＝C σ(k) x(k),k∈N,
wherein,,
Figure BDA0004123455370000071
representing the sewage flow in an aeration tank or a secondary sedimentation tank in the sewage treatment system, < >>
Figure BDA0004123455370000074
Indicating the control input, i.e. the flow of sewage into the aeration tank or secondary sedimentation tank,/>
Figure BDA0004123455370000072
Represents the flow of sewage after treatment by an aeration tank or a secondary sedimentation tank. Sigma (k) represents the law obeyed by the modal changes of the sewage treatment system and is defined as S= {1,2, …, N }, N ε N + And (3) taking the value. A is that σ(k) ,B σ(k) ,C σ(k) Represents a system matrix and presumes +.>
Figure BDA0004123455370000075
Figure BDA0004123455370000076
Respectively representing n-dimension, r-dimension, s-dimension Euclidean matrix space and positive integer set.
Step 2, constructing a non-disturbing switching PID controller of a sewage treatment system, wherein the specific method is as follows:
u(k)＝K σ(k) y(k)+L σ(k) e(k)+M σ(k) (Δy(k)),
wherein e (k) =y (k-1) +e (k)-1) - δe (k-1 represents the integral term of the controller, the derivative term of the controller satisfying Δy (k) =y (k) -y (k-1) =c σ(k) x(k)-C σ(k) x(k-1)，K σ(k) 、L σ(k) 、M σ(k) Proportional, integral and differential gain matrices, denoted sigma (k) m-1 )＝q,σ(k m )＝p,σ(k m+1 ) =r. The method comprises the steps of designing the gain of a non-disturbing switching PID controller in the sewage treatment process, wherein the specific form is as follows:
Figure BDA0004123455370000081
wherein the symbols are
Figure BDA00041234553700000816
Representing a transpose of a vector or matrix, 1 r R-dimensional column vector representing all elements 1,/->
Figure BDA0004123455370000082
R-dimensional column vector, ζ, representing that the i-th element is 1 and the remaining elements are 0 p 、η p And theta p Is an s-dimensional column vector, ">
Figure BDA0004123455370000083
Is an n-dimensional column vector.
Step 3, introducing a reference controller, wherein the reference controller is in the following form:
u * (k)＝K * y(k)+L * e(k)+M * (Δy(k))，
wherein u is * (k) Representing the desired control input, K * ,L * ,M * Is a given matrix and respectively satisfies
Figure BDA0004123455370000084
And 4, introducing a controller undisturbed switching mechanism, wherein the specific constraint conditions are as follows:
Figure BDA0004123455370000085
wherein alpha is more than or equal to 0, beta is more than or equal to 0, gamma is more than or equal to 0 and is an undisturbed performance index, u j (k) And
Figure BDA0004123455370000086
control input u (k) and desired control input u, respectively * (k) Is>
Figure BDA0004123455370000087
A row element.
Step 5, constructing a non-disturbing switching PID control closed loop system of the sewage treatment system:
combining the state space model of the step 1 and the undisturbed PID control model of the step 2, the closed-loop control system of the sewage treatment control system is as follows:
Figure BDA0004123455370000088
wherein,,
Figure BDA0004123455370000089
Figure BDA00041234553700000810
Figure BDA00041234553700000811
I s representing an identity matrix in s x s dimensions.
And 6, designing the following linear programming form for ensuring the stable operation of the sewage treatment system:
the design constant is more than 0 and less than or equal to 10 and less than or equal to rho 1 lambda > 1,
Figure BDA00041234553700000817
Figure BDA00041234553700000812
vector->
Figure BDA00041234553700000813
And->
Figure BDA00041234553700000814
(Vector)
Figure BDA00041234553700000815
So that
Figure BDA0004123455370000091
Figure BDA0004123455370000092
Figure BDA0004123455370000093
Figure BDA0004123455370000094
Figure BDA0004123455370000095
Figure BDA0004123455370000096
Figure BDA0004123455370000097
Figure BDA0004123455370000098
Figure BDA0004123455370000099
Figure BDA00041234553700000910
Figure BDA00041234553700000911
Figure BDA00041234553700000912
For any p, q e S, then, under the undisturbed switching PID controller designed in step 2, the closed loop system of the sewage treatment system is positive and exponentially stable, and the average residence time condition is designed as:
Figure BDA00041234553700000913
and 7, positive verification of a closed-loop control system of the sewage treatment system comprises the following steps:
from the specific form of the undisturbed switching PID controller designed in step 2 and condition (a) in step 6, we get
Figure BDA00041234553700000914
I.e. indicate
Figure BDA00041234553700000915
Combining the conditions in step 2 and step 6
Figure BDA00041234553700000916
Obtaining
Figure BDA00041234553700000917
I.e. indicate
Figure BDA00041234553700000918
Further, it is known that->
Figure BDA00041234553700000919
Thus, the positive performance of the closed loop control system of the sewage treatment system is ensured.
Step 8, the stability verification process of a closed-loop control system of the sewage treatment system is as follows:
construction of Lyapunov function
Figure BDA0004123455370000101
Wherein,,
Figure BDA0004123455370000102
for V p Difference is obtained to obtain
Figure BDA0004123455370000103
Deducing from the conditions (b) - (e) in step 6
Figure BDA0004123455370000104
It is known that each subsystem is stable. Further, combining the condition (f) in step 6 with the designed average residence time, we obtain
Figure BDA0004123455370000105
Finally, after finishing, obtain
Figure BDA0004123455370000106
Wherein,,
Figure BDA0004123455370000107
thus, the stability of the closed loop control system of the sewage treatment system is ensured.
Step 9, the undisturbed verification process of the PID controller of the sewage treatment system is as follows:
first, the following equation is obtained according to steps 3 and 4:
Figure BDA0004123455370000108
wherein,,
Figure BDA0004123455370000109
and->
Figure BDA00041234553700001010
Respectively represent K p And K * Is>
Figure BDA00041234553700001011
A row element.
Then, according to the gain matrix form in step 2, obtain
Figure BDA00041234553700001012
Combining conditions (g) and (h) in step 6
Figure BDA00041234553700001013
The following inequality is obtained
Figure BDA00041234553700001014
According to conditions (i) and (ii) in step 6(j) And +.>
Figure BDA0004123455370000111
Obtaining
Figure BDA0004123455370000112
Similarly, according to conditions (k) and (l) in step 6 and
Figure BDA0004123455370000113
and->
Figure BDA0004123455370000114
Can obtain
Figure BDA0004123455370000115
The following inequality is derived by combining the three inequalities
Figure BDA0004123455370000116
And then can obtain
Figure BDA0004123455370000117
Therefore, an undisturbed switching PID controller of the wastewater treatment system is effective.

Claims (10)

1. A non-disturbing switching PID control method for smooth operation of a sewage treatment system is characterized by comprising the following steps:
step 1, establishing a state space model of a sewage treatment process;
step 2, constructing a non-disturbing switching PID controller of the sewage treatment system;
step 3, introducing a reference controller;
step 4, introducing a controller undisturbed switching mechanism;
step 5, constructing a non-disturbing switching PID control closed loop system of the sewage treatment system;
step 6, designing a linear programming form for ensuring the stable operation of the sewage treatment system;
step 7, positive verification of a closed-loop control system of the sewage treatment system;
step 8, verifying the stability of a closed-loop control system of the sewage treatment system;
and 9, verifying the undisturbed property of the PID controller of the sewage treatment system.
2. The undisturbed switching PID control method for smooth operation of a sewage treatment system according to claim 1, wherein step 1 is specifically as follows:
1.1, collecting input and output data of a sewage treatment system to describe an actual system;
1.2, establishing a state space model of the sewage treatment process.
3. The undisturbed switching PID control method for smooth operation of a sewage treatment system according to claim 2, wherein step 2 is specifically as follows:
u(k)＝K σ(k) y(k)+L σ(k) e(k)+M σ(k) (Δy(k)),
wherein e (k) =y (k-1) +e (k-1) - δe (k-1) represents the integral term of the controller, the derivative term of the controller satisfying Δy (k) =y (k) -y (k-1) =c σ(k) x(k)-C σ(k) x(k-1)，K σ(k) 、L σ(k) 、M σ(k) Proportional, integral and differential gain matrices, denoted sigma (k) m-1 )＝q,σ(k m )＝p,σ(k m+1 ) =r; the method comprises the steps of designing the gain of a non-disturbing switching PID controller in the sewage treatment process, wherein the specific form is as follows:
Figure FDA0004123455360000011
wherein the symbols are
Figure FDA0004123455360000012
Representing a transpose of a vector or matrix, 1 r R-dimensional column vector representing all elements 1,/->
Figure FDA0004123455360000013
R-dimensional column vector, ζ, representing that the i-th element is 1 and the remaining elements are 0 p 、η p And theta p Is an s-dimensional column vector, ">
Figure FDA0004123455360000014
Is an n-dimensional column vector.
4. A smooth-running non-intrusive switching PID control method of a sewage treatment system according to claim 3, wherein the reference controller in step 3 is in the form of:
u * (k)＝K * y(k)+L * e(k)+M * (Δy(k))，
wherein u is * (k) Representing the desired control input, K * ,L * ,M * Is a given matrix and respectively satisfies
Figure FDA0004123455360000021
5. The undisturbed switching PID control method for smooth operation of a sewage treatment system according to claim 4, wherein specific constraint conditions of the undisturbed switching mechanism of the controller in step 4 are:
Figure FDA0004123455360000022
wherein alpha is more than or equal to 0, beta is more than or equal to 0, gamma is more than or equal to 0 and is an undisturbed performance index,
Figure FDA0004123455360000023
and->
Figure FDA0004123455360000024
Control input u (k) and desired control input u, respectively * (k) Is>
Figure FDA0004123455360000025
A row element.
6. The undisturbed switching PID control method for smooth operation of a sewage treatment system according to claim 1, wherein step 5 is specifically as follows:
combining the state space model of the step 1 and the undisturbed PID control model of the step 2, the PID closed-loop control system of the sewage treatment control system is as follows:
Figure FDA0004123455360000026
wherein,,
Figure FDA0004123455360000027
Figure FDA0004123455360000028
Figure FDA0004123455360000029
I s representing an identity matrix in s x s dimensions.
7. The undisturbed switching PID control method for smooth operation of a wastewater treatment system according to claim 6, wherein in step 6, the following conditions are satisfied for designing a linear programming form for ensuring smooth operation of the wastewater treatment system:
design constant
Figure FDA00041234553600000210
Vector->
Figure FDA00041234553600000211
And->
Figure FDA00041234553600000212
(Vector)
Figure FDA00041234553600000213
So that
Figure FDA0004123455360000031
Figure FDA0004123455360000032
Figure FDA0004123455360000033
Figure FDA0004123455360000034
Figure FDA0004123455360000035
Figure FDA0004123455360000036
Figure FDA0004123455360000037
Figure FDA0004123455360000038
Figure FDA0004123455360000039
Figure FDA00041234553600000310
Figure FDA00041234553600000311
Figure FDA00041234553600000312
For any p, q e S, then, under the undisturbed switching PID controller designed in step 2, the closed loop system of the sewage treatment system is positive and exponentially stable, and the average residence time condition is designed as:
Figure FDA00041234553600000313
8. the non-disturbing switching PID control method for smooth operation of sewage treatment system according to claim 7, wherein the positive verification process of the closed loop control system of the sewage treatment system in step 7 is as follows:
from the specific form of the undisturbed switching PID controller designed in step 2 and condition (a) in step 6, we get
Figure FDA00041234553600000314
I.e. indicate
Figure FDA00041234553600000315
Combining the conditions in step 2 and step 6
Figure FDA00041234553600000316
Obtaining
Figure FDA0004123455360000041
I.e. indicate
Figure FDA0004123455360000042
Further, it is known that->
Figure FDA0004123455360000043
Thus, the positive performance of the closed loop control system of the sewage treatment system is ensured.
9. The undisturbed switching PID control method for smooth operation of a wastewater treatment system according to claim 8, wherein the stability verification process of the closed loop control system of the wastewater treatment system in step 8 is as follows:
construction of Lyapunov function
Figure FDA0004123455360000044
Wherein,,
Figure FDA0004123455360000045
for V p Difference is obtained to obtain
Figure FDA0004123455360000046
Deducing from the conditions (b) - (e) in step 6
Figure FDA0004123455360000047
It is known that each subsystem is stable. Further, combining the condition (f) in step 6 with the designed average residence time, we obtain
Figure FDA0004123455360000048
Finally, after finishing, obtain
Figure FDA0004123455360000049
Wherein,,
Figure FDA00041234553600000410
thus, the stability of the closed loop control system of the sewage treatment system is ensured.
10. The non-intrusive switching PID control method for smooth operation of a sewage treatment system according to claim 9, wherein the non-intrusive verification process of the PID controller of the sewage treatment system in step 9 is as follows:
first, the following equation is obtained according to steps 3 and 4:
Figure FDA0004123455360000051
wherein,,
Figure FDA0004123455360000052
and->
Figure FDA0004123455360000053
Respectively represent K p And K * Is>
Figure FDA0004123455360000054
A row element.
Then, according to the gain matrix form in step 2, obtain
Figure FDA0004123455360000055
Combining conditions (g) and (h) in step 6
Figure FDA0004123455360000056
The following inequality is obtained
Figure FDA0004123455360000057
According to conditions (i) and (j) in step 6
Figure FDA0004123455360000058
Obtaining
Figure FDA0004123455360000059
Similarly, according to conditions (k) and (l) in step 6 and
Figure FDA00041234553600000510
and 1-delta > 0, can be obtained
Figure FDA00041234553600000511
The following inequality is derived by combining the three inequalities
Figure FDA00041234553600000512
And then can obtain
Figure FDA00041234553600000513
Therefore, an undisturbed switching PID controller of the wastewater treatment system is effective.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117430237A (en) * 2023-10-30 2024-01-23 海南大学 PID control method of sewage treatment system based on PI observer

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117430237A (en) * 2023-10-30 2024-01-23 海南大学 PID control method of sewage treatment system based on PI observer

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