A PI control method for a water tank level control system with interval uncertainty

Abstract

The present invention discloses a PI control method for an interval uncertain water tank level control system with external disturbance input. The method of the present invention first establishes a state space model of the water tank level control system using a positive switching system with external disturbance input. With the help of multiple linear copositive Lyapunov functions and matrix decomposition technology, a PI controller of the interval uncertain positive switching system with external disturbance input is designed, so that the liquid level of the multi-capacity water tank system is reasonably controlled, thereby ensuring the normal and smooth progress of people's life and production. Almost all systems contain uncertainty, and the uncertainty of the system will destroy the performance of the system and even cause the instability of the system. Therefore, it is very necessary to design a PI controller based on the interval uncertain system to ensure the production efficiency of the actual control system.

Description

Technical Field

[0001] The present invention belongs to the field of automation technology and modern control, and relates to the modeling of a multi-capacity water tank level control system based on an interval uncertain positive switching system and a PI control method with external disturbance input. Background Art

[0002] With the development of society in my country, automated control has been implemented very early. Liquid level control systems have also been widely used in my country. Liquid level control systems are a common process control system in industry, and their impact on production cannot be ignored. Liquid level control is ubiquitous in our daily lives. For example, when the water level in a water tower is too low, it cannot meet the continuous water supply demand, disrupting people's normal work and life. To ensure normal operation of a boiler, the liquid level must be maintained at a normal standard. A low level can easily dry out the boiler, causing serious accidents, while an excessively high level can easily cause steam to carry water and create a risk of overflow. Water tank level control is also used in Yellow River flood control. Liquid level control systems monitor the water level of the Yellow River to prevent uninformed high water levels, which could result in life-threatening and property damage. Liquid level control is also widely used in other industries, such as petrochemicals, steelmaking, food, and pharmaceuticals. Therefore, liquid level control technology has become a vital component of industrial automation and holds an irreplaceable position in the field of automatic control.

[0003] A multi-tank level control system is a platform for observing, measuring, and monitoring changes in flow and level parameters within a water tank, simulating industrial production. It offers powerful functionality, ease of use, and a compact size, helping to address many practical industrial challenges. In a water tank level control system, the dynamics of water volume within the control system are considered. The water volume in a water tank primarily refers to the amount of water flowing into and out of the tank. A complete water tank level control system primarily consists of components such as the water tank, valves, and a level transmitter. When the level transmitter detects a change in the water level within the tank, the controller drives the actuator (i.e., valve) to perform the corresponding action (i.e., fill or drain) to maintain a stable water level. Given the non-negative nature of water volume, the mutual influence between multiple tanks, the difficulty of accurately determining the water tank level control system, and its susceptibility to sudden external disturbances such as component damage and environmental changes, the water volume dynamics described above can be characterized as an interval-uncertain positive switching system with external disturbance inputs. In industrial production processes, the liquid level of production equipment is often required to be maintained at a certain value or to change according to a specific pattern to meet production process requirements. Traditional control systems often use proportional controllers to achieve the desired results. However, using proportional controllers alone in industry may not effectively eliminate deviations, ensuring that the actual value of the controlled variable is consistent with the predetermined value required by the process. To address the aforementioned issues, the present invention employs PI control, specifically designing a PI controller. A PI (proportional-integral) controller, consisting of a proportional unit (P) and an integral unit (I), is essentially a linear controller. A control deviation is formed by combining the set value and the actual output value. The proportional and integral of the deviation are linearly combined to form the control variable, thereby controlling the controlled object to ensure that the actual value of the controlled variable is consistent with the predetermined value required by the process. The PI controller combines proportional (P) and integral (I) control, providing both the timely control of proportional control and the deviation elimination capability of integral control. PI controllers are currently widely used in industrial control systems for liquid level, temperature, flow, and other applications. In industrial control, control system fluctuations are often caused by component failures, environmental changes, and other unexpected factors. These fluctuations often affect system performance and even undermine system stability. In addition, almost all systems contain uncertainty, which can also disrupt system performance and even lead to system instability. To address these issues, the present invention proposes a PI controller for a water tank level control system with an uncertain external input range and multiple water tanks switching back and forth under certain switching laws, further improving the performance requirements of the multi-tank level control system.

[0004] To address the aforementioned issues, this paper utilizes modern control theory to establish a state-space model for a multi-tank level control system. A PI controller, along with proportional and integral gain matrices, is designed for the system. The positivity and stability of the controllers are analyzed to maintain the liquid level within the multi-tank control system at a constant value. In summary, the design of a water tank level control system and PI control method based on interval-uncertain positive switching system modeling has significant scientific and practical application implications. Summary of the Invention

[0005] The purpose of the present invention is to address the problem of liquid level control in daily life and production, use a multi-capacity water tank control system device to study the water tank level control system, and provide a PI control method for an interval uncertain water tank level control system with external disturbance input.

[0006] A PI control method for an interval uncertain water tank level control system with external disturbance input comprises the following steps:

[0007] Step 1: Establish a state space model of a positive switching system of a multi-tank level control system with external disturbance input;

[0008] Step 2: Establish the PI control law of the water tank level control system;

[0009] Step 3: Design the integral part of the PI controller;

[0010] Step 4: Establish the switching conditions satisfied by the switching signal σ(t);

[0011] Step 5: Design the conditions for the stable operation of the water tank level control system;

[0012] Step 6: Positive verification process of the water tank level control system;

[0013] Step 7: Verify the stability of the water tank level control system and its l1 gain performance γ.

[0014] Step 1:

[0015] Establish the state space model of the positive switching system of the multi-tank level control system with external disturbance input:

[0016]

[0017] y(t)=C σ(t) x(t)+F σ(t) ω(t)

[0018] in, represents the amount of water in the tank at time t, represents the operation of finding the derivative of the vector x(t), is the water flow of r controllable valves at time t, It represents the amount of water flowing out of the water tank collected by s sensors at time t, It represents the external interference to the water tank level control system caused by sudden external changes such as damage to the components of the water tank level control system or environmental changes at time t. The function σ(t) represents the switching law and is derived from the finite set When σ(t)=p, the pth subsystem is activated, where They are all system matrices of the multi-tank level control system and meet the interval uncertainty in, is the upper bound of the system matrix obtained by actual measurement of 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. Respectively represent n-dimensional, r-dimensional, s-dimensional real vectors, n×n-dimensional, n×r-dimensional, s×n-dimensional, s×r-dimensional real matrix spaces, positive integer sets and non-negative integer sets. [x1(t),x2(t),...,x n (t)] T represents a vector [x1(t),x2(t),...,x n (t)].

[0019] Step 2:

[0020] The PI control law of the water tank level control system is established, and its construction form is as follows:

[0021] u p (t) = K Pp C p x(t)+K Pp F p ω(t)+K Ip e(t)

[0022] in, and are the proportional gain matrix and integral gain matrix of the pth subsystem to be designed, and e(t) is the integral part of the PI controller.

[0023] Step 3:

[0024] Design the integral part of the PI controller, which is constructed as follows:

[0025]

[0026] Wherein, α is a tuning parameter and α>0.

[0027] Step 4:

[0028] Establish the switching condition satisfied by the switching signal σ(t), which is constructed as follows

[0029]

[0030] Among them, N σ (t0, t) represents the number of switches between time t0 and time t, τ represents the average dwell time, and N0 represents the jitter bound.

[0031] Step 5:

[0032] The conditions for designing a stable operation of the water tank level control system are as follows:

[0033] 5.1 Design constants ζ>0, α>0, β>1, μ>0, λ>1, γ>0, vector and vector So that:

[0034]

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043] For any p≠q and j=1,2,...,r holds true, then, under the PI control law in step 2 and the average residence time switching condition:

[0044]

[0045] Under this condition, the water tank level control system is positive, stable and has l1 gain performance γ. Where I is an identity matrix with compatible dimensions; ∑ is a summation symbol; 1s represents an s-dimensional column vector whose elements are all 1, 1 r represents an r-dimensional column vector whose elements are all 1, Represents an r-dimensional column vector whose jth element is 1 and other elements are 0; the superscript of the vector (p) and subscript p both represent the vector of the p-th subsystem, and the superscript (q) Represents the vector for the qth subsystem, and both p and q belong to p≠q; vector The superscript + in the vector indicates that all elements of the vector are positive. Superscript in - Indicates that all elements of this vector are negative.

[0046] 5.2 Design the proportional gain matrix and integral gain matrix of the water tank level control system:

[0047]

[0048] And satisfy:

[0049]

[0050]

[0051] in, Superscript in + Indicates that all elements of the gain matrix are positive, Superscript in - Indicates that all elements of the gain matrix are negative.

[0052] Step 6: The positive verification process of the water tank level control system is as follows:

[0053] 6.1 Based on the state space model of the water tank level system in step 1, the integral part of the PI controller in step 3, and the PI control law designed in step 2, we can obtain:

[0054]

[0055]

[0056] 6.2 Given but Therefore, step 6.1 can be transformed into:

[0057]

[0058] Among them, -αI s is a diagonal matrix with s rows and s columns and a diagonal element of -α, Represents a vector Find the derivative.

[0059] 6.4 Due to B p ≥ B p ≥0, and β>0, we get Using condition (1) in step 5.1, we can obtain:

[0060]

[0061] so is a Metzler matrix, which has the characteristic that the non-diagonal elements are non-negative. From the interval uncertainty, we can see that:

[0062]

[0063] therefore, is a Metzler matrix.

[0064] 6.5 Using condition (2) in step 5.1, we obtain:

[0065]

[0066] From the uncertainty of the interval we can see that:

[0067] 6.6 Combined with condition (3) in step 5.1, we obtain:

[0068]

[0069] From the interval uncertainty, we can know that:

[0070] 6.7 From the interval uncertainty, we know that C p ≥ C p ≥0,F p ≥ F p ≥0, combined with steps 6.4-6.6, we can get: is a Metzler matrix, and Therefore, the positivity of the water tank level control system is proven.

[0071] Step 7: The verification process of the stability of the water tank level control system and its l1 gain performance γ is as follows:

[0072] 7.1 For the pth subsystem, design multiple linear copositive Lyapunov functions Among them, (p) =(v' (p)T v” (p)T ) T , Assume that the switching sequence of σ(t) in the interval (t0, t) is Among them, N σ (t0, t) is the number of switching times in the interval (t0, t), which satisfies the switching law for the switching signal σ(t) established in step 4. Derivatives of the above multilinear copositive Lyapunov function yield:

[0073]

[0074] Combined with interval uncertainty, the above formula is transformed into:

[0075]

[0076] in, is a multilinear copositive Lyapunov function The derivative function of .

[0077] 7.2 Using condition (4) in step 5.1, we can obtain:

[0078]

[0079]

[0080] 7.3 Combining condition (8) in step 5.1 with step 7.2 gives:

[0081]

[0082]

[0083] Therefore, it can be concluded that:

[0084]

[0085] 7.4 Combining conditions (5)-(7) in step 5.1, we obtain:

[0086]

[0087] 7.5 Integrate both sides of the inequality sign in step 7.4 and recycle condition (8) in step 5.1 to obtain:

[0088]

[0089] That is to say Established. Among them, ‖·‖1 represents the 1-norm of the vector, which is the sum of the absolute values ​​of all elements in the vector. It represents the maximum amount of water flowing out of the water tank collected by the sensor at time t.

[0090] Therefore, the interval uncertain water tank level control system with external disturbance input is stable and has l1 gain performance γ.

[0091] The beneficial effects of the present invention are as follows:

[0092] The method of the present invention first establishes a state-space model of a water tank level control system using an interval-uncertain positive switching system with external disturbance input. Using multilinear copositive Lyapunov functions and matrix decomposition techniques, a PI controller is designed, enabling reasonable level control and ensuring normal production and daily life. Designing a PI controller for this interval-uncertain system with external disturbance input enhances system stability, improves system performance, and ensures production efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0093] Figure 1 It is a schematic diagram of the multi-capacity water tank liquid level control system of the present invention.

[0094] Figure 2 It is a schematic diagram of the PI control framework of a multi-tank level control system with external disturbance input. DETAILED DESCRIPTION

[0095] The present invention will be further described below with reference to specific examples.

[0096] like Figure 1 and 2 As shown, the PI control method of the water tank level control system based on the positive switching system modeling with external disturbance input of the present invention. The specific steps of the method of the present invention include the following:

[0097] Step 1: Establish a state space model of a positive switching system of a multi-tank level control system with external disturbance input;

[0098]

[0099] y(t)=C σ(t) x(t)+F σ(t) ω(t)

[0100] in, represents the amount of water in the tank at time t, represents the operation of finding the derivative of the vector x(t), is the water flow of r controllable valves at time t, It represents the amount of water flowing out of the water tank collected by s sensors at time t, It represents the external interference to the water tank level control system caused by sudden external changes such as damage to the components of the water tank level control system or environmental changes at time t. The function σ(t) represents the switching law and is derived from the finite set When σ(t)=p, the pth subsystem is activated, where They are all system matrices of the multi-tank level control system and meet the interval uncertainty in, is the upper bound of the system matrix obtained by actual measurement of 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. Respectively represent n-dimensional, r-dimensional, s-dimensional real vectors, n×n-dimensional, n×r-dimensional, s×n-dimensional, s×r-dimensional real matrix spaces, positive integer sets and non-negative integer sets. [x1(t),x2(t),...,x n (t)] T represents a vector [x1(t),x2(t),...,x n (t)].

[0101] Step 2: Establish the PI control law of the water tank level control system;

[0102] The PI control law of the water tank level control system is established, and its construction form is as follows:

[0103] u p (t) = K Pp C p x(t)+K Pp F p ω(t)+K Ip e(t)

[0104] in, and are the proportional gain matrix and integral gain matrix of the pth subsystem to be designed, and e(t) is the integral part of the PI controller.

[0105] Step 3: Design the integral part of the PI controller;

[0106] Design the integral part of the PI controller, which is constructed as follows:

[0107]

[0108] Wherein, α is a tuning parameter and α>0.

[0109] Step 4: Establish the switching conditions satisfied by the switching signal σ(t);

[0110] Establish the switching condition satisfied by the switching signal σ(t), which is constructed as follows

[0111]

[0112] Among them, N σ (t0, t) represents the number of switches between time t0 and time t, τ represents the average dwell time, and N0 represents the jitter bound.

[0113] Step 5: Design the conditions for the stable operation of the water tank level control system;

[0114] The conditions for designing a stable operation of the water tank level control system are as follows:

[0115] 5.1 Design constants ζ>0, α>0, β>1, μ>0, λ>1, γ>0, vector and vector So that:

[0116]

[0117]

[0118]

[0119]

[0120]

[0121]

[0122]

[0123]

[0124]

[0125] For any p≠q and j=1,2,...,r holds true, then, under the PI control law in step 2 and the average residence time switching condition:

[0126]

[0127] Under this condition, the water tank level control system is positive, stable and has l1 gain performance γ. Where I is an identity matrix with compatible dimensions; ∑ is a summation symbol; 1 s represents an s-dimensional column vector whose elements are all 1, 1 r represents an r-dimensional column vector whose elements are all 1, Represents an r-dimensional column vector whose jth element is 1 and other elements are 0; the superscript of the vector (p) and subscript p both represent the vector of the p-th subsystem, and the superscript (q) Represents the vector for the qth subsystem, and both p and q belong to p≠q; vector Superscript in + Indicates that all elements of the vector are positive, and the vector Superscript in - Indicates that all elements of this vector are negative.

[0128] 5.2 Design the proportional gain matrix and integral gain matrix of the water tank level control system:

[0129]

[0130] And satisfy:

[0131]

[0132]

[0133] in, Superscript in + Indicates that all elements of the gain matrix are positive, Superscript in - Indicates that all elements of the gain matrix are negative.

[0134] Step 6: The positive verification process of the water tank level control system is as follows:

[0135] 6.1 Based on the state space model of the water tank level system in step 1, the integral part of the PI controller in step 3, and the PI control law designed in step 2, we can obtain:

[0136]

[0137]

[0138] 6.2 Given but Therefore, step 6.1 can be transformed into:

[0139]

[0140] Among them, -αI s is a diagonal matrix with s rows and s columns and a diagonal element of -α, Represents a vector Find the derivative.

[0141] 6.4 Due to B p ≥B p ≥0, and β>0, we get Using condition (1) in step 5.1, we can obtain:

[0142]

[0143] so is a Metzler matrix, which has the characteristic that the non-diagonal elements are non-negative. From the interval uncertainty, we can see that:

[0144]

[0145] therefore, is a Metzler matrix.

[0146] 6.5 Using condition (2) in step 5.1, we obtain:

[0147]

[0148] From the uncertainty of the interval we can see that:

[0149] 6.6 Combined with condition (3) in step 5.1, we obtain:

[0150]

[0151] From the interval uncertainty, we can know that:

[0152] 6.7 From the interval uncertainty, we know that Combined with steps 6.4-6.6, we can get: is a Metzler matrix, and Therefore, the positivity of the water tank level control system is proven.

[0153] Step 7: The verification process of the stability of the water tank level control system and its l1 gain performance γ is as follows:

[0154] 7.1 For the pth subsystem, design multiple linear copositive Lyapunov functions Among them, (p) =(v' (p)Tv” (p)T ) T , Assume that the switching sequence of σ(t) in the interval (t0, t) is Among them, N σ (t0, t) is the number of switching times in the interval (t0, t), which satisfies the switching law for the switching signal σ(t) established in step 4. Derivatives of the above multilinear copositive Lyapunov function yield:

[0155]

[0156] Combined with interval uncertainty, the above formula is transformed into:

[0157]

[0158] in, is a multilinear copositive Lyapunov function The derivative function of .

[0159] 7.2 Using condition (4) in step 5.1, we can obtain:

[0160]

[0161]

[0162] 7.3 Combining condition (8) in step 5.1 with step 7.2 gives:

[0163]

[0164]

[0165] Therefore, it can be concluded that:

[0166]

[0167] 7.4 Combining conditions (5)-(7) in step 5.1, we obtain:

[0168]

[0169] 7.5 Integrate both sides of the inequality sign in step 7.4 and recycle condition (8) in step 5.1 to obtain:

[0170]

[0171] That is to say Established. Among them, ‖·‖1 represents the 1-norm of the vector, which is the sum of the absolute values ​​of all elements in the vector. It represents the maximum amount of water flowing out of the water tank collected by the sensor at time t.

[0172] Therefore, the interval uncertain water tank level control system with external disturbance input is stable and has l1 gain performance γ.

Claims

1. A PI control method for an interval uncertain water tank level control system with external disturbance input, characterized in that: The steps include: Step 1: Establish a state space model of a positive switching system of a multi-tank level control system with external disturbance input; Step 2: Establish the PI control law of the water tank level control system; Step 3: Design the integral part of the PI controller; Step 4: Establish the switching conditions satisfied by the switching signal σ(t); Step 5: Design the conditions for the stable operation of the water tank level control system; Step 6: Positive verification process of the water tank level control system; Step 7: Verification process of the stability of the water tank level control system and its l1 gain performance γ; Step 1: Establish the state space model of the positive switching system of the multi-tank level control system with external disturbance input: y(t)=C σ(t) x(t)+F σ(t) ω(t) in, represents the amount of water in the tank at time t, represents the operation of finding the derivative of the vector x(t), is the water flow rate of r controllable valves at time t, It represents the amount of water flowing out of the water tank collected by s sensors at time t, It represents the external interference to the water tank level control system caused by sudden external changes such as damage to the components of the water tank level control system or changes in the environment at time t; the function σ(t) represents the switching law and is derived from the finite set When σ(t)=p, the pth subsystem is activated, where They are all system matrices of the multi-tank level control system and meet the interval uncertainty in, is the upper bound of the system matrix obtained by actual measurement of 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; Respectively represent n-dimensional, r-dimensional, s-dimensional real vectors, n×n-dimensional, n×r-dimensional, s×n-dimensional, s×r-dimensional real matrix spaces, the set of positive integers and the set of non-negative integers; represents a vector [x1(t),x2(t),…,x n (t)]; Step 2: The PI control law of the water tank level control system is established, and its construction form is as follows: u p (t)=K Pp C p x(t)+K Pp F p ω(t)+K Ip e(t) in, and are the proportional gain matrix and integral gain matrix of the p-th subsystem to be designed, and e(t) is the integral part of the PI controller; Step 3: Design the integral part of the PI controller, which is constructed as follows: Where α is a tuning parameter and α>0; Step 4: The switching condition satisfied by the switching signal σ(t) is established, and its construction form is as follows: Among them, N σ (t0, t) represents the number of switches between time t0 and time t, τ represents the average dwell time, and N0 represents the jitter bound.

2. The PI control method for an interval uncertain water tank level control system with external disturbance input according to claim 1 is characterized in that: Step 5: The conditions for designing a stable operation of the water tank level control system are as follows: 5.1 Design constants ζ>0, α>0, β>1, μ>0, λ>1, γ>0, vector and vector So that: For any and j=1,2,...,r holds true, then, under the PI control law in step 2 and the average residence time switching condition: Under this condition, the water tank level control system is positive, stable and has l1 gain performance γ; where I is an identity matrix with compatible dimensions; ∑ is a summation symbol; 1 s represents an s-dimensional column vector whose elements are all 1, 1 r represents an r-dimensional column vector whose elements are all 1, represents an r-dimensional column vector whose jth element is 1 and the other elements are 0; the superscript (p) and subscript p of the vector both represent the vector for the pth subsystem, and the superscript (q) represents the vector for the qth subsystem, and both p and q belong to vector The superscript + in the vector indicates that all elements of the vector are positive. The superscript - in indicates that all elements of the vector are negative; 5.2 Design the proportional gain matrix and integral gain matrix of the water tank level control system: And satisfy: in, The superscript + in indicates that all elements of the gain matrix are positive. The superscript - in indicates that all elements of the gain matrix are negative.

3. The PI control method for an interval uncertain water tank level control system with external disturbance input according to claim 2 is characterized in that: Step 6: The positive verification process of the water tank level control system is as follows: 6.1 Based on the state space model of the water tank level system in step 1, the integral part of the PI controller in step 3, and the PI control law designed in step 2, we can obtain: 6.2 Given but Therefore, step 6.1 is transformed into: in, -αI s is a diagonal matrix with s rows and s columns and a diagonal element of -α, Represents a vector Derivative; 6.4 Due to and β>0, we get Using condition (1) in step 5.1, we can obtain: so is a Metzler matrix, which has the characteristic that the off-diagonal elements are non-negative; from the interval uncertainty, we can see that: therefore, is a Metzler matrix; 6.5 Using condition (2) in step 5.1, we obtain: From the uncertainty of the interval we can see that: 6.6 Combined with condition (3) in step 5.1, we obtain: From the interval uncertainty, we can know that: 6.7 From the interval uncertainty, we know that Combined with steps 6.4-6.6, we can get: is a Metzler matrix, and 4. The PI control method for an interval uncertain water tank level control system with external disturbance input according to claim 3 is characterized in that: Step 7: The verification process of the stability of the water tank level control system with l1 gain performance γ is as follows: 7.1 For the pth subsystem, design multiple linear copositive Lyapunov functions Among them, for any All can be concluded Assume that the switching sequence of σ(t) in the interval (t0, t) is Among them, N σ (t0, t) is the number of switching times in the interval (t0, t), which satisfies the switching law for the switching signal σ(t) established in step 4. Derivative the above multilinear copositive Lyapunov function to obtain: Combined with interval uncertainty, the above formula is transformed into: in, is a multilinear copositive Lyapunov function The derivative of 7.2 Using condition (4) in step 5.1, we obtain: 7.3 Combining condition (8) in step 5.1 with step 7.2 gives: Therefore, it follows that: 7.4 Combining conditions (5)-(7) in step 5.1, we obtain: 7.5 Integrate both sides of the inequality sign in step 7.4 and recycle condition (8) in step 5.1 to obtain: That is: Established; among them, ‖·‖1 represents the 1-norm of the vector, which is the sum of the absolute values ​​of all elements in the vector. It represents the maximum amount of water flowing out of the water tank collected by the sensor at time t.

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