CN115259362B - Non-fragile PI control method of urban sewage treatment system - Google Patents
Non-fragile PI control method of urban sewage treatment system Download PDFInfo
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- 239000010865 sewage Substances 0.000 title claims abstract description 152
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- 238000005273 aeration Methods 0.000 claims description 7
- 238000010992 reflux Methods 0.000 claims description 7
- 238000004062 sedimentation Methods 0.000 claims description 7
- 238000004064 recycling Methods 0.000 claims description 6
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- 239000010841 municipal wastewater Substances 0.000 description 1
- 239000002957 persistent organic pollutant Substances 0.000 description 1
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- C02F—TREATMENT OF WATER, WASTE WATER, SEWAGE, OR SLUDGE
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Abstract
The invention discloses a non-fragile PI control method of an urban sewage treatment system. The method comprises the following steps: 1. establishing a state space model of the urban sewage treatment system; 2. establishing an actuator fault model of the urban sewage treatment system; 3. establishing a non-fragile PI control law of the urban sewage treatment system; 4. designing an integrating part of the PI controller; 5. establishing a switching condition satisfied by a switching signal; 6. designing the stable running condition of the urban sewage treatment system; 7. a positive verification process of the urban sewage treatment system; 8. and (3) verifying the stability of the urban sewage treatment system. The invention establishes a state space model of the urban sewage treatment system by utilizing the forward switching system, and proposes and designs a non-fragile PI controller design method with actuator faults by means of a multi-linear residual forward Lyapunov function and matrix decomposition technology, so that urban sewage is reasonably treated, smooth running of life production of people is ensured, and effective operation of the sewage treatment system is ensured.
Description
Technical Field
The invention belongs to the technical field of engineering, and relates to a non-fragile PI control method of an urban sewage treatment system, which is used for controlling the urban sewage treatment system by utilizing a direct conversion technology, a matrix decomposition technology and a PI controller design method and considering the problems of actuator faults, non-fragile property and the like.
Background
With the progress of society, the sewage problem is a serious problem in the urban development process, and the sewage problem is a problem which is difficult to avoid in the life at present. The sewage treatment is a process for purifying sewage to meet the water quality requirement of being discharged into a certain water body or reused. The increasing shortage of water resources and the increasing pollution of water become an important mode of water resource protection at present, and the sewage treatment has no neglect to influence on production. At present, sewage treatment is also widely used, such as: is used in various fields of construction, agriculture, transportation, energy, petrifaction, environmental protection, urban landscapes, medical treatment, catering and the like. In addition, sewage treatment is increasingly going into the daily life of common people. Thus, sewage treatment technology has become an important part of industrial automation, and there is an irreplaceable place in the field of automatic control.
The sewage treatment system consists of three parts: a drainage pipeline system for collecting and conveying sewage and rainwater in a barrage, a sewage treatment system and a sewage advanced treatment and recycling system. The sewage treatment aims at economically and reasonably solving the problems of management, treatment and utilization of the sewage in the barrage, and the treatment degree to be achieved is determined according to the total pollutant discharge amount control target, the sewage quality, the discharge water body function and flow, the sewage outlet, the water quantity and other factors. The main purpose of sewage treatment is to make the treated effluent reach a certain discharge requirement, so that the environment is not polluted, and the self-cleaning capability of the water body is fully utilized, so as to save the cost. When considering the sewage treatment scheme, the degree to which sewage should be treated is first determined, and is generally classified into three stages: the first-stage treatment is also called pretreatment, and mainly removes solid pollutants in a suspended state in sewage; the secondary treatment can greatly remove organic pollutants in colloid and dissolved state in the sewage; the third stage treatment further removes contaminants that the biological treatment failed to remove on the basis of the second stage treatment. All parts of the sewage treatment system are coordinated and matched with each other, so that sewage can be treated in time, and the sewage treatment efficiency is greatly improved.
Considering that the sewage flow rate in the urban sewage treatment system has non-negative characteristics, and the sewage treatment process needs to coordinate and match each part of the sewage treatment system to treat the sewage. If the non-negative characteristic of sewage flow in the urban sewage treatment system is ignored, the modeling analysis by using the general system theory (non-positive system theory) can cause resource redundancy and waste on the model, and the representation meaning of the redundancy in the actual process is relatively weak. Therefore, it is necessary to model the municipal sewage treatment system using a direct-swing system. In addition, the urban sewage treatment system is inevitably subjected to actuator fault phenomena caused by factors such as device aging, abrasion, external environment and the like in the actual operation process, and the problem of non-vulnerability of performance deterioration of the feedback control system caused by disturbance of the controller, so that the system cannot safely and stably operate, and even paralysis is caused. In order to solve the problems, the invention aims to consider modeling the urban sewage treatment system based on the direct conversion system, and designs a non-fragile PI controller by taking the actuator failure phenomenon which is inevitably caused by factors such as device aging, abrasion, external environment and the like in the operation process of an actual system into consideration, so that the performance requirement of the urban sewage treatment system is further improved. The PI controller combines proportional (P) control and integral (I) control, so that the proportional control function is timely controlled, and the integral control function has the capability of eliminating deviation. In addition, the PI controller not only can improve the stability performance of the system, but also can ensure the dynamic performance of the system. Therefore, the PI controller is of great significance in controlling the urban sewage treatment system.
In summary, the non-fragile PI control method of the urban sewage treatment system has important practical significance and important engineering value.
Disclosure of Invention
The invention provides a non-fragile PI control method of a municipal sewage treatment system, which is based on a forward switching system model, an actuator fault and a non-fragile PI control method, collects input and output data of the municipal sewage treatment system, establishes a state space model of the municipal sewage treatment system and aims at solving the problem that the input and output data of the municipal sewage treatment system are not fragile. The non-fragile PI control method of the urban sewage treatment system considering the actuator faults designs the non-fragile PI controller with the actuator faults by means of multiple linear residual Lyapunov functions and matrix decomposition technology.
The invention aims at overcoming the defects of the prior art and provides a non-fragile PI control method of an urban sewage treatment system.
The technical scheme adopted by the invention for solving the technical problems comprises the following steps:
step 1, establishing a state space model of an urban sewage treatment system, wherein the specific method comprises the following steps:
1.1, input and output data of the municipal sewage treatment system are collected to describe an actual system:
considering the municipal sewage treatment system, the municipal sewage treatment system is a drainage pipeline system for collecting and conveying sewage and rainwater in a barrage. The urban sewage treatment system comprises a sewage treatment system and a sewage advanced treatment and recycling system. The activated sludge process is an aerobic biological treatment of sewage, and is also the most widely used method for treating municipal sewage. Wherein the concentration of microorganisms in the reflux is an important controlled parameter, such a method requires precise control of the concentration of microorganisms in the reflux. The flow chart of the activated sludge process for treating the municipal sewage is shown in fig. 1, the municipal sewage is firstly discharged into an aeration tank for treatment, and enters a sedimentation tank from the aeration tank, and the reflux concentration of microorganisms discharged from the sedimentation tank is detected. Considering that the sewage flow and the microorganism concentration in the sewage treatment system are non-negative, and the sewage treatment process requires that all parts of the sewage treatment system are mutually coordinated and matched. Therefore, the urban sewage treatment system can adopt the modeling of the direct conversion system, and take the problems of actuator fault phenomena possibly occurring in the running process of the system due to factors such as device aging, abrasion, external environment and the like and non-fragility of the feedback control system performance deterioration caused by the disturbance of the controller into consideration, so as to prevent the urban sewage treatment system from being paralyzed caused by the actuator fault and the non-fragility. A block diagram of a non-fragile PI control process designed to take into account actuator faults is shown in fig. 2.
1.2 constructing a state space model of the urban sewage treatment system:
y(t)＝C σ(t) x(t), (1)
wherein ,
the microbial concentration in the sewage at time t. />
Actuator fault input signal for sigma (t) th subsystem, < >>
Is sewage inflow amount flowing into an aeration tank in the urban sewage treatment system. />
And s represents the dimension of y (t) for the microorganism reflux concentration discharged from the sedimentation tank of the treated municipal sewage treatment system. The function σ (t) represents the switching law and is derived from the finite set q= {1, 2..j }, J e N + And (3) taking the value. When σ (t) =p, the p-th subsystem is activated, where p e Q. Here->
The system matrix of the urban sewage treatment system can be obtained by data collected in the actual process. For convenience, when σ (t) =p, the system matrix may be denoted as a p ,B p ,C p . Assuming matrix A p The Metzler characteristic is satisfied (non-principal diagonal non-negative),
respectively representing n-dimensional, r-dimensional and s-dimensional vector spaces; />
Respectively representing n×n, n×r, s×n real matrix spaces; n (N) + N represents a positive integer set and an integer set, respectively. />
Representing vector x 1 (t),x 2 (t),...,x n (t)]Is a transpose of (a).
Step 2, building an actuator fault model of the urban sewage treatment system, wherein the construction form is as follows:
wherein ,Lp For an unknown fault diagonal matrix with upper and lower bounds, the following is satisfied:
wherein ,/>
L p Upper and lower bound matrix for a given fault diagonal matrix,/->
And ρ > 1.
Step 3, establishing a non-fragile PI control law of the urban sewage treatment system, wherein the construction form is as follows:
u p (t)＝(K p +ΔK p )C p x(t)+(F p +ΔF p )e(t),
wherein ,Kp ,F p E (t) is the integral part of the PI controller for the proportional and integral gain matrix of the p-th subsystem to be designed. ΔK p ,ΔF p Is a possible controller gain disturbance and satisfies ΔK p ＝E p H p ，ΔF p ＝P p Q p ，H p ,Q p Decision matrix for the p-th subsystem to be designed, E p ,H p Is a known non-negative matrix and satisfies
wherein ,0＜δ1 ＜δ 2 ,0＜δ 3 ＜δ 4 。
And 4, designing an integral part of the PI controller, wherein the integral part has the following construction form:
wherein alpha is a tuning parameter and satisfies alpha > 0.
Step 5, establishing a switching condition satisfied by a switching signal sigma (t), wherein the construction form is as follows:
wherein ,Nσ (t 0 T) represents time t 0 The number of switches between time t and τ represents the average residence time, N 0 Representing the jitter bound and being a given non-negative constant.
And 6, designing the conditions for smooth running of the urban sewage treatment system as follows:
6.1 if constants ρ > 1, λ > 1, α > 0, μ > 0, ζ > 0, δ are present 2 ＞δ 1 ＞0、δ 4 ＞δ 3 > 0, n-dimensional vector
And s-dimensional vector
The following inequality is caused:
for any (p, Q) ∈q, p+.q and j=1, 2, once again, r holds, where,
Θ ps2 ＝ρB p L p ，/>
Θ ps4 ＝ρB p L p E p then, under PI control law and average residence time in step 3, the following are satisfied:
the urban sewage treatment system is positive and stable, wherein I is an identity matrix with compatible dimensions; sigma is a sum symbol; 1 s An s-dimensional column vector representing all elements as 1,
an r-dimensional column vector representing the j-th element as 1 and the other elements as 0; vector superscript (p) And subscripts p All representing vectors for the p-th subsystem, superscript (q) Representing vectors for the qth subsystem, and p, Q all belong to Q, p+.q; vector->
Superscript in (3) + All elements representing the vector are positive, vector +.>
Superscript in (3) - All elements representing the vector are negative.
6.2, the proportional gain matrix, the integral gain matrix and the decision matrix of the p subsystem of the designed sewage treatment system are as follows:
and satisfy the following
wherein ,
superscript in (3) + All elements representing the gain matrix are positive, and (2)>
Superscript in (3) - All elements representing the gain matrix are negative.
And 7, the positive verification process of the urban sewage treatment system is as follows:
7.1, according to the state space model of the urban sewage treatment system established in the step 1.2, the actuator fault model established in the step 2, the non-fragile PI control law established in the step 3 and the integral part of the PI controller designed in the step 4, the integral part is obtained:
wherein ,
representing vectors +.>
Derivative, -alpha I s A diagonal matrix with-alpha diagonal elements for an s row and an s column.
7.2 due to
Easy to get->
and />
From the condition (1) in step 6.1:
7.3 from step 3
Step 6.2 can obtain:
so that the number of the parts to be processed,
is a Metzler matrix, which is a non-negative matrix of off-diagonal elements.
7.4 due to
Thereby obtaining the product,
thus, A is obtained p +B p L p K p C p +B p L p E p H p C p Is a Metzler matrix.
7.5 binding
And step 6.2 available->
And because of
Therefore, get +.>
Obviously, -alpha I s Is a diagonal matrix with a diagonal element-alpha. Combining the steps 7.4 to obtain: />
Is a Metzler matrix. Thus, the positivity of the municipal sewage treatment system was demonstrated.
Step 8, the stability verification process of the urban sewage treatment system is as follows:
8.1 for the p-th subsystem, a multiple linear residual Lyapunov function is designed
wherein ,/>
Assume that the switching signal sigma (t) is in the interval (t 0 The switching sequence in t) is +.>
wherein ,Nσ (t 0 T is in the interval (t) 0 T), and satisfying the switching law related to the switching signal sigma (t) established in the step 5, and deriving the multiple linear residual positive Lyapunov function to obtain:
8.2 is obtainable from condition (2) in step 6.1:
thus, it can be derived that:
8.3 from conditions (3) - (4) in step 6.1:
8.4 integrating both sides of the inequality of the step 8.3 and recycling the condition (5) in the step 6.1 to obtain the following components:
further, the stability conditions of the obtained municipal sewage treatment system are as follows:
wherein ,ρ1 ,ρ 2 Respectively is vector v (p) Minimum element and maximum element of (a) in the above list. I.I. | 1 Representing the 1-norm of the vector, i.e. the sum of the absolute values of all elements in the vector.
The beneficial effects of the invention are as follows:
the invention provides a non-fragile PI control method for an urban sewage treatment system, and a corresponding structure diagram is shown in figure 2. The invention provides a non-fragile PI control method of an urban sewage treatment system considering an actuator fault based on a positive switching system model, an actuator fault and a non-fragile PI control method, aiming at input and output data of the urban sewage treatment system, and designs the non-fragile PI controller with the actuator fault by means of a multiple linear residual Lyapunov function and a matrix decomposition technology.
Drawings
FIG. 1 is a flow chart of a municipal sewage treatment system.
Fig. 2 is a schematic diagram of a non-fragile PI control architecture for a municipal wastewater treatment system.
Detailed Description
The invention will be further illustrated with reference to specific examples.
The invention provides a non-fragile PI control method with an actuator fault for an urban sewage treatment system based on positive switching system modeling.
The method of the invention is specifically implemented as follows:
step 1, establishing a state space model of an urban sewage treatment system, wherein the specific method comprises the following steps:
1.1, input and output data of the municipal sewage treatment system are collected to describe an actual system:
considering an urban sewage treatment system, the urban sewage treatment system is a system consisting of a drainage pipeline system for collecting and conveying sewage and rainwater in a barrage, a sewage treatment system, a sewage advanced treatment and recycling system and the like. The activated sludge process is an aerobic biological treatment of sewage, which is also the most widely used process for treating municipal sewage, wherein the concentration of microorganisms in the return stream is an important controlled parameter, and this process requires precise control of the concentration of microorganisms in the return stream. The flow chart of the activated sludge process in treating urban sewage is shown in figure 1 (see the attached drawing of the specification), the urban sewage is firstly discharged into an aeration tank for treatment, and enters a sedimentation tank from the aeration tank, and the reflux concentration of microorganisms discharged from the sedimentation tank is detected. Considering that the sewage flow and the microorganism concentration in the sewage treatment system are non-negative, and the sewage treatment process requires that all parts of the sewage treatment system are mutually coordinated and matched. Therefore, the urban sewage treatment system can adopt the modeling of the direct conversion system, and take the problems of actuator fault phenomena possibly occurring in the running process of the system due to factors such as device aging, abrasion, external environment and the like and non-fragility of the feedback control system performance deterioration caused by the disturbance of the controller into consideration, so as to prevent the urban sewage treatment system from being paralyzed caused by the actuator fault and the non-fragility. A block diagram of a non-fragile PI control process designed to take actuator failure into account is shown in fig. 2 (see the drawing of the specification).
1.2 constructing a state space model of the urban sewage treatment system:
y(t)＝C σ(t) x(t),
wherein ,
the microbial concentration in the sewage at time t. />
Actuator fault input signal for sigma (t) th subsystem, < >>
Is sewage inflow amount flowing into an aeration tank in the urban sewage treatment system. />
Is treated urban sewageThe microorganism reflux concentration of the treatment system discharged from the sedimentation tank, s represents the dimension of y (t). The function σ (t) represents the switching law and is derived from the finite set q= {1, 2..j }, J e N + And (3) taking the value. When σ (t) =p, the p-th subsystem is activated, where p e Q. Here->
The system matrix of the urban sewage treatment system can be obtained by data collected in the actual process. For convenience, when σ (t) =p, the system matrix may be denoted as a p ,B p ,C p . Assuming matrix A p The Metzler characteristic is satisfied (non-principal diagonal non-negative),
respectively representing n-dimensional, r-dimensional and s-dimensional vector spaces; />
Respectively representing n×n, n×r, s×n real matrix spaces; n (N) + N represents a positive integer set and an integer set, respectively. />
Representing vector x 1 (t),x 2 (t),...,x n (t)]Is a transpose of (a).
Step 2, building an actuator fault model of the urban sewage treatment system, wherein the construction form is as follows:
wherein ,Lp For an unknown fault diagonal matrix with upper and lower bounds, the following is satisfied:
wherein ,/>
L p Upper and lower bound matrix for a given fault diagonal matrix,/->
And ρ > 1.
Step 3, establishing a non-fragile PI control law of the urban sewage treatment system, wherein the construction form is as follows:
u p (t)＝(K p +ΔK p )C p x(t)+(F p +ΔF p )e(t),
wherein ,Kp ,F p E (t) is the integral part of the PI controller for the proportional and integral gain matrix of the p-th subsystem to be designed. ΔK p ,ΔF p Is a possible controller gain disturbance and satisfies ΔK p ＝E p H p ，ΔF p ＝P p Q p ，H p ,Q p Decision matrix for the p-th subsystem to be designed, E p ,H p Is a known non-negative matrix and satisfies
wherein ,0＜δ1 ＜δ 2 ,0＜δ 3 ＜δ 4 。
And 4, designing an integral part of the PI controller, wherein the integral part has the following construction form:
wherein alpha is a tuning parameter and satisfies alpha > 0.
Step 5, establishing a switching condition satisfied by a switching signal sigma (t), wherein the construction form is as follows:
wherein ,Nσ (t 0 T) represents time t 0 Time of arrivalThe number of switches between the instants t, τ representing the average dwell time, N 0 Representing the jitter bound and being a given non-negative constant.
And 6, designing the conditions for smooth running of the urban sewage treatment system as follows:
6.1 if constants ρ > 1, λ > 1, α > 0, μ > 0, ζ > 0, δ are present 2 ＞δ 1 ＞0、δ 4 ＞δ 3 > 0, n-dimensional vector
And s-dimensional vector
The following inequality is caused:
for any (p, Q) ∈q, p+.q and j=1, 2, once again, r holds, where,
Θ ps2 ＝ρB p L p ，/>
Θ ps4 ＝ρB p L p E p . Then, under PI control law and average residence time in step 3, the following are satisfied:
the urban sewage treatment system is positive and stable, wherein I is an identity matrix with compatible dimensions; sigma is a sum symbol; 1 r An r-dimensional column vector representing all elements as 1,
an r-dimensional column vector representing the j-th element as 1 and the other elements as 0; vector superscript (p) And subscripts p All representing vectors for the p-th subsystem, superscript (q) Representing vectors for the qth subsystem, and p, Q all belong to Q, p+.q; vector->
Superscript in (3) + All elements representing the vector are positive, vector +.>
Superscript in (3) - All elements representing the vector are negative.
6.2, the proportional gain matrix, the integral gain matrix and the decision matrix of the p subsystem of the designed sewage treatment system are as follows:
and satisfy the following
wherein ,
superscript in (3) + All elements representing the gain matrix are positive, and (2)>
Superscript in (3) - All elements representing the gain matrix are negative.
And 7, the positive verification process of the urban sewage treatment system is as follows:
7.1, according to the state space model of the urban sewage treatment system established in the step 1.2, the actuator fault model established in the step 2, the non-fragile PI control law established in the step 3 and the integral part of the PI controller designed in the step 4, the integral part is obtained:
wherein ,
representing vectors +.>
Derivative, -alpha I s A diagonal matrix with-alpha diagonal elements for an s row and an s column.
7.2 due to
Easy to get->
And
from the condition (1) in step 6.1: />
7.3 from step 3
Step 6.2 can obtain:
so that the number of the parts to be processed,
is a Metzler matrix, which is a non-negative matrix of off-diagonal elements.
7.4 due to
Thereby obtaining the product,
thus, A is obtained p +B p L p K p C p +B p L p E p H p C p Is a Metzler matrix.
7.5 binding
And step 6.2 available->
And because of
Therefore, get +.>
Obviously, -alpha I s Is a diagonal matrix with a diagonal element of-alpha for s rows and s columns. Combining the steps 7.4 to obtain:
is a Metzler matrix. Thus, the positivity of the municipal sewage treatment system was demonstrated.
Step 8, the stability verification process of the urban sewage treatment system is as follows:
8.1 for the p-th subsystem, a multiple linear residual Lyapunov function is designed
wherein ,/>
Assume that the switching signal sigma (t) is in the interval (t 0 The switching sequence in t) is +.>
wherein ,Nσ (t 0 T is in the interval (t) 0 T), and satisfying the switching law related to the switching signal sigma (t) established in the step 5, and deriving the multiple linear residual positive Lyapunov function to obtain:
8.2 is obtainable from condition (2) in step 6.1:
thus, it can be derived that:
8.3 from conditions (3) - (4) in step 6.1:
8.4 integrating both sides of the inequality of the step 8.3 and recycling the condition (5) in the step 6.1 to obtain the following components:
further, the stability conditions of the obtained municipal sewage treatment system are as follows:
wherein ,ρ1 ,ρ 2 Respectively is vector v (p) Minimum element and maximum element of (a) in the above list. I.I. | 1 Representing the 1-norm of the vector, i.e. the sum of the absolute values of all elements in the vector.
Claims (1)
1. A non-fragile PI control method of an urban sewage treatment system is characterized by comprising the following steps:
step 1, establishing a state space model of an urban sewage treatment system;
step 2, establishing an actuator fault model of the urban sewage treatment system;
step 3, establishing a non-fragile PI control law of the urban sewage treatment system;
step 4, designing an integrating part of the PI controller;
step 5, establishing a switching condition met by a switching signal sigma (t);
step 6, designing the stable running condition of the urban sewage treatment system;
step 7, a positive verification process of the urban sewage treatment system;
step 8, verifying the stability of the urban sewage treatment system;
the specific method of the step 1 is as follows:
1.1, collecting and sorting input and output data of an urban sewage treatment system;
1.2, constructing a state space model of the urban sewage treatment system;
y(t)＝C σ(t) x(t),
wherein ,
the microorganism concentration in the sewage at the time t; />
Actuator fault input signal for sigma (t) th subsystem, < >>
Sewage inflow amount flowing into an aeration tank in the urban sewage treatment system; />
The concentration of the microorganism reflux discharged from the sedimentation tank of the treated municipal sewage treatment system is s which represents the dimension of y (t); the function σ (t) represents the switching law and is derived from the finite set q= {1, 2..j }, J e N + Taking a value of the middle value; when σ (t) =p, the p-th subsystem is activated, where p ε Q, ++>
The system matrix is a system matrix of the urban sewage treatment system, and can be obtained by data arrangement collected in the actual process; for convenience, when σ (t) =p, the system matrix may be denoted as a p ,B p ,C p The method comprises the steps of carrying out a first treatment on the surface of the Assuming matrix A p Meet Metzler characteristics, < >>
Respectively representing n-dimensional, r-dimensional and s-dimensional vector spaces; />
Respectively representing n×n, n×r, s×n real matrix spaces; n (N) + N represents a positive integer set and an integer set respectively; />
Representing vector x 1 (t),x 2 (t),...,x n (t)]Is a transpose of (2);
the specific method of the step 2 is as follows:
an actuator fault model of the urban sewage treatment system is established, and the construction form is as follows:
wherein ,Lp For an unknown fault diagonal matrix with upper and lower bounds, the following is satisfied:
wherein ,/>
L p For the upper and lower bound matrix of a given fault diagonal matrix, L p ≥0，/>
And ρ > 1;
the specific method of the step 3 is as follows:
the method establishes a non-fragile PI control law of the urban sewage treatment system, and has the following structural form:
u p (t)＝(K p +ΔK p )C p x(t)+(F p +ΔF p )e(t),
wherein ,Kp 、F p Respectively representing the proportional and integral gain matrixes of the p subsystem to be designed, wherein e (t) is an integral part of the PI controller; ΔK p 、ΔF p Is a possible controller gain disturbance and satisfies ΔK p ＝E p H p ，ΔF p ＝P p Q p ，H p 、Q p Decision matrix for the p-th subsystem to be designed, E p 、H p Is a known non-negative matrix and satisfies delta 1 I≤E p ≤δ 2 I,δ 3 I≤P p ≤δ 4 I, wherein 0 < delta 1 ＜δ 2 ,0＜δ 3 ＜δ 4 ；
The specific method of the step 4 is as follows:
the integral part of the PI controller is designed, and the construction form is as follows:
wherein alpha is a tuning parameter and satisfies alpha > 0;
the specific method in step 5 is as follows:
the switching conditions satisfied by the switching signal sigma (t) are established in the following construction form:
wherein ,Nσ (t 0 T) represents time t 0 The number of switches between time t and τ represents the average residence time, N 0 Represents the jitter bound and is a given non-negative constant;
the specific method of the step 6 is as follows:
the conditions for designing the smooth running of the urban sewage treatment system are as follows:
6.1 if constants ρ > 1, λ > 1, α > 0, μ > 0, ζ > 0, δ are present 2 ＞δ 1 ＞0、δ 4 ＞δ 3 > 0, n-dimensional vector
And s-dimensional vector
So that the following inequality holds:
for any (p, Q) ∈q, p+.q and j=1, 2,..r holds; wherein,
Θ ps2 ＝ρB p L p ，/>
Θ ps4 ＝ρB p L p E p then the PI control law and average residence time in step 3 satisfy:
the urban sewage treatment system is positive and stable, wherein I is an identity matrix with compatible dimensions; sigma is a sum symbol; 1 r An r-dimensional column vector representing all elements as 1,
an r-dimensional column vector representing the j-th element as 1 and the other elements as 0; the superscript (p) and the subscript p of the vector each represent the vector for the p-th subsystem, the superscript (Q) represents the vector for the Q-th subsystem, and p, Q each belong to Q, p+.q; vector->
The superscript + in (a) indicates that all elements of the vector are positive, vector +.>
Superscript in (a) -meaning that all elements of the vector are negative; />
6.2, the proportional gain matrix, the integral gain matrix and the decision matrix of the p subsystem of the designed sewage treatment system are as follows:
and satisfy the following
wherein ,
superscript in (3) + All elements representing the gain matrix are positive, and (2)>
Superscript in (3) - All elements representing the gain matrix are negative;
the positive verification process of the urban sewage treatment system in the step 7 is as follows:
7.1, according to the state space model of the urban sewage treatment system established in the step 1.2, the actuator fault model established in the step 2, the non-fragile PI control law established in the step 3 and the integral part of the PI controller designed in the step 4, the integral part is obtained:
wherein ,
representing the derivation of the vector x (t), - αI s A diagonal matrix with diagonal elements of-alpha for an s row and an s column;
7.2 due to
Easy to get->
and />
From the condition (1) in step 6.1:
7.3 from step 3
Step 6.2 can obtain: />
wherein ,
is a Metzler matrix, which is a non-negative matrix of off-diagonal elements;
7.4 due to
Thereby obtaining the product,
thus obtaining A p +B p L p K p C p +B p L p E p H p C p Is a Metzler matrix;
7.5 binding
And step 6.2 available->
And because of
Therefore, get +.>
Obviously, -alpha I s Is a diagonal matrix with diagonal elements of s rows and s columns as-alpha; combining the steps 7.4 to obtain: />
Is a Metzler matrix; thus, the positivity of the municipal sewage treatment system was demonstrated;
the stability verification process of the urban sewage treatment system in the step 8 is as follows:
8.1 for the p-th subsystem, a multiple linear residual Lyapunov function is designed
wherein ,/>
Assume that the switching signal sigma (t) is in the interval (t 0 The switching sequence in t) is +.>
wherein ,Nσ (t 0 T is in the interval (t) 0 T), and satisfying the switching law related to the switching signal sigma (t) established in the step 5, and deriving the multiple linear residual positive Lyapunov function to obtain: />
8.2 is obtainable from condition (2) in step 6.1:
thus, it can be derived that:
8.3 from conditions (3) - (4) in step 6.1:
8.4 integrating both sides of the inequality of the step 8.3 and recycling the condition (5) in the step 6.1 to obtain the following components:
the stability conditions of the urban sewage treatment system are further as follows:
wherein ,ρ1 ,ρ 2 Respectively is vector v (p) Minimum element and maximum element of (a); I.I. | 1 Representing the 1-norm of the vector, i.e. the sum of the absolute values of all elements in the vector.
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| KR100796456B1 (en) * | 2007-06-22 | 2008-01-21 | 태화강재산업 주식회사 | Waste water treatment process control apparatus and method |
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| CN110110496A (en) * | 2019-06-06 | 2019-08-09 | 杭州电子科技大学 | A kind of reliable control method of the urban water supply based on nonlinear system modeling |
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