A dual-capacity water tank level control method based on SVDF
By performing SVD decomposition and linear transformation on the Hessian matrix and utilizing the unconstrained solution information to optimize the suboptimal solution in the conjugate space, the difficulty of the MPC algorithm in quickly calculating the constrained solution is solved, and efficient water tank level control is achieved.
Patent Information
- Application Number
- CN202211244953.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-10-12
- Publication Date
- 2025-09-26
- Estimated Expiration
- 2042-10-12
AI Technical Summary
Existing model predictive control (MPC) algorithms have difficulty in quickly computing solutions that satisfy constraints, especially in multi-input and multi-output control systems. Existing technologies often rely on controllers with large storage capacity, overly simplified models, and cannot fully utilize unconstrained solution information.
The Hessian matrix is decomposed by a method based on singular value decomposition (SVD) to obtain the conjugate matrix. The suboptimal solution of the conjugate space is obtained through linear transformation and system constraint optimization. The unconstrained solution information is used to meet the system constraints and improve the calculation speed.
At each sampling moment, the unconstrained solution is selectively retained to obtain a suboptimal solution that satisfies the constraints, ensuring the safe operation of the system and improving the control effect. At the same time, the calculation speed is significantly improved and the occurrence of infeasible solutions is avoided.
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Abstract
Description
Technical Field
[0001] The present invention relates to water tank liquid level control, and in particular to a double-capacity water tank liquid level control method based on SVDF. Background Art
[0002] In the 1970s, a new type of computer control algorithm emerged in the European and American industrial sectors: predictive control. This advanced model-based control technology, also known as model predictive control (MPC), offers advantages over traditional proportional-integral-derivative (PID) control in that it can handle constraints and exhibits excellent control effectiveness for multi-input, multi-output (MIMO) control systems. MPC offers a new approach to controlling complex industrial processes characterized by widespread uncertainty, constraints, and nonlinearities. After more than 30 years of development, MPC has gained widespread use and recognition in industrial process control, and has been gradually applied to fields such as energy, aerospace engineering, and the automotive industry.
[0003] With the increase in the number of controlled variables in industrial processes and the development of embedded devices in industrial processes, some factories have put forward more stringent requirements on the timeliness of control in actual applications. Therefore, in recent years, there has been increasing attention on the problem of online MPC calculation. How to quickly calculate the MPC solution that meets the constraints while ensuring control performance within the system sampling interval has become a difficult problem that the current industry urgently needs to solve.
[0004] Currently, there are many existing technologies for improving the calculation speed of MPC, but most of them have disadvantages such as dependence on controller storage capacity, over-simplification of models, and inability to fully utilize unconstrained solution information. Summary of the Invention
[0005] The purpose of the present invention is to overcome the defects of the above-mentioned prior art and provide a dual-capacity water tank level control method based on SVDF that fully utilizes unconstrained solution information and improves calculation speed.
[0006] The purpose of the present invention can be achieved by the following technical solutions:
[0007] A dual-capacity water tank level control method based on SVDF includes the following steps:
[0008] Establishing a state space model of the double-capacity water tank to obtain a first objective function of the model predictive control, converting the first objective function of the model predictive control into a second objective function related to the prediction input based on the state space model, wherein the parameters of the second objective function include a Hessian matrix, and performing SVD decomposition on the Hessian matrix to obtain a conjugate matrix;
[0009] If this is the sampling moment, a linear transformation is performed on the control input based on the conjugate matrix, and a third objective function of the conjugate space is obtained based on the linear transformation and the second objective function. Based on the third objective function of the conjugate space, an unconstrained optimal solution of the conjugate space is obtained;
[0010] Obtaining the system constraints of the conjugate space based on the linear transformation and the actual system constraints, and transforming the third objective function of the conjugate space into a function related to the suboptimal solution of the conjugate space and the unconstrained optimal solution of the conjugate space;
[0011] Based on a third objective function related to the conjugate space suboptimal solution and the unconstrained optimal solution of the conjugate space, a relationship function among the conjugate space suboptimal solution, a solution that does not satisfy the system constraints of the conjugate space, and the first solution that satisfies the system constraints of the conjugate space is obtained; and with the system constraints of the conjugate space as a constraint condition, a linear search is performed on the relationship function to obtain the conjugate space suboptimal solution;
[0012] Based on the conjugate space suboptimal solution and the linear transformation, a resulting suboptimal solution is obtained, and the control variable of the first moment of the resulting suboptimal solution is input into the double-capacity water tank to control the liquid level.
[0013] Furthermore, the specific expression of the second objective function related to the prediction input is:
[0014]
[0015] Where k is the kth moment, is the prediction input, H is the Hessian matrix, M is the first coefficient, and c is a constant.
[0016] Furthermore, the specific expression of the Hessian matrix and the specific expression of the first coefficient M are:
[0017] H=F T QF+R
[0018] M=x(k) T E T Q T F
[0019] Among them, F is the second coefficient, Q is the first weight coefficient matrix, R is the second weight coefficient matrix, x(k) is the state of the system at time k, and E is the third coefficient.
[0020] Furthermore, the specific expression for performing linear transformation on the control input based on the conjugate matrix is:
[0021]
[0022] in, is the input in the conjugate space, and P is the conjugate matrix.
[0023] Furthermore, the specific expression of the unconstrained optimal solution of the conjugate space is:
[0024]
[0025] in, is the unconstrained optimal solution in the conjugate space, G is the eigenvalue matrix, P is the conjugate matrix, and M is the first coefficient.
[0026] Furthermore, the specific expression for converting the third objective function of the conjugate space into a function related to the suboptimal solution of the conjugate space and the unconstrained optimal solution of the conjugate space is:
[0027]
[0028] in, is the suboptimal solution in conjugate space, is the unconstrained optimal solution in conjugate space, and G is the eigenvalue matrix.
[0029] Furthermore, the specific expression of the relationship function between the suboptimal solution in the conjugate space, the solution that does not satisfy the system constraints in the conjugate space, and the first solution that satisfies the system constraints in the conjugate space is:
[0030]
[0031] in, is the suboptimal solution in conjugate space, U r is the solution that does not satisfy the system constraints of the conjugate space, U r+1 is the first solution that satisfies the system constraints in conjugate space, and λ is the weight coefficient.
[0032] Furthermore, the weight coefficient λ satisfies 0≤λ≤1.
[0033] Furthermore, the specific expression of the suboptimal solution is:
[0034]
[0035] in, is the suboptimal solution in conjugate space, P is the conjugate matrix, The result is a suboptimal solution.
[0036] Furthermore, the method also includes: if the present moment is not a sampling moment, directly using the control quantity of the previous moment to input into the double-capacity water tank.
[0037] Compared with the prior art, the present invention has the following beneficial effects:
[0038] (1) Make full use of the unconstrained solution information and selectively retain the unconstrained solution at each sampling moment to obtain a suboptimal solution that meets the constraints. While ensuring that the constraints are met and the system operates safely, a satisfactory control effect is achieved.
[0039] (2) Since the solution at each sampling moment is obtained by only using unconstrained solutions instead of calculating the optimization problem, the calculation speed is greatly improved, the lack of feasible solutions at the sampling moment is avoided, and the safe and stable operation of the control algorithm is maintained. BRIEF DESCRIPTION OF THE DRAWINGS
[0040] Figure 1 is a flow chart of the present invention;
[0041] Figure 2 This is a structural diagram of a double-capacity water tank of the present invention;
[0042] Figure 3 This is a graph showing the change in the liquid level of the first water tank controlled by the SVDF algorithm according to an embodiment of the present invention;
[0043] Figure 4 This is a graph showing the liquid level change of the No. 2 water tank controlled by the SVDF algorithm of an embodiment of the present invention. DETAILED DESCRIPTION
[0044] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.
[0045] This embodiment provides a dual-capacity water tank level control method based on SVDF. The flow chart of the method is as follows: Figure 1 As shown, the method comprises the following steps:
[0046] (1) Establish a state space model of the double-capacity water tank, obtain the first objective function of the model predictive control, and transform the first objective function of the model predictive control into the predicted input based on the state space model. The related second objective function, the parameters of the second objective function include the Hessian matrix, and the information features are extracted by performing SVD decomposition on the Hessian matrix to obtain a conjugate matrix;
[0047] (2) If this is the sampling moment, the unconstrained optimal solution information is selectively retained to obtain the suboptimal solution that meets the constraints. The specific calculation process of the suboptimal solution is:
[0048] If this is the sampling moment, the control input is linearly transformed based on the conjugate matrix. Based on the linear transformation and the second objective function, the third objective function of the conjugate space is obtained. Based on the third objective function of the conjugate space, the unconstrained optimal solution of the conjugate space is obtained.
[0049] Based on linear transformation and actual system constraints, the system constraints of the conjugate space are obtained, and the third objective function of the conjugate space is transformed into a function related to the suboptimal solution of the conjugate space and the unconstrained optimal solution of the conjugate space;
[0050] Based on the third objective function related to the conjugate space suboptimal solution and the unconstrained optimal solution of the conjugate space, a relationship function among the conjugate space suboptimal solution, the solution that does not satisfy the conjugate space system constraints, and the first solution that satisfies the conjugate space system constraints is obtained. With the conjugate space system constraints as the constraint conditions, a linear search is performed on the relationship function to obtain the conjugate space suboptimal solution.
[0051] Based on the conjugate space suboptimal solution and linear transformation, the resulting suboptimal solution is obtained;
[0052] (3) Use the control quantity of the suboptimal solution at the first moment to input the controlled double-capacity water tank system to control the liquid level of the double-capacity water tank.
[0053] The specific process in step (1) is as follows:
[0054] S1.1 obtains the state space model of the double-capacity water tank through the system identification method. The model is specifically described as:
[0055]
[0056]
[0057]
[0058] Where u1 is the main circuit regulating valve opening, and u2 is the auxiliary circuit inverter signal. h1 is the liquid level in water tank 1, and h2 is the liquid level in water tank 2. To ensure the safety of the dual-capacity water tank equipment and maximize its service life, the following constraints apply during equipment operation:
[0059] 0≤u1≤17
[0060] 0≤u2≤17
[0061] -1≤Δu1≤1
[0062] -1≤Δu2≤1
[0063] S1.2 Based on the state space model of the double-capacity water tank, the first objective function of the model predictive control is written as The relevant second objective function. The specific process is as follows:
[0064] Discretize the transfer function model in the state space model of the double-capacity water tank obtained by identification to obtain the discrete linear model of the system:
[0065] x(k+1)=Ax(k)+Bu(k)
[0066] in and are the state and input variables of the system at time k respectively. The first objective function of model predictive control is:
[0067]
[0068] Among them, Q and R are weight coefficient matrices, and they satisfy
[0069]
[0070]
[0071] and They are respectively the predicted state set and input set in a finite time domain, and
[0072]
[0073]
[0074] where N p Represents the prediction time domain and the control time domain.
[0075] The discrete linear model of the system can be recursively deduced to obtain the following relationship between the predicted state and input:
[0076]
[0077] in,
[0078]
[0079] Since x(k) is known at every moment, the first objective function can then be replaced by this relationship to eliminate the predicted state Transformed into predicted input The related second objective function:
[0080]
[0081] Where k is the kth moment, is the prediction input, H is the Hessian matrix, M is the first coefficient, c is a constant, Q and R are weight coefficient matrices, and
[0082] H=F T QF+R
[0083] M=x(k) T E T Q T F
[0084] Among them, F is the second coefficient, Q is the first weight coefficient matrix, R is the second weight coefficient matrix, x(k) is the state of the system at time k, and E is the third coefficient.
[0085] S1.3 Get and predict input After the relevant second objective function is obtained, the information features are extracted by performing SVD decomposition on the Hessian matrix to obtain the conjugate matrix P. The expression of the conjugate matrix P is:
[0086]
[0087] where n σ =N p n u , G is the eigenvalue matrix, σ i is the eigenvalue of the Hessian matrix and satisfies
[0088]
[0089] The specific process of step (2) is as follows:
[0090] S2.1 If this is the sampling moment, the control input is first linearly transformed based on the conjugate matrix. The specific process of linear transformation is:
[0091] From the state-space model of the double-capacity water tank, we can see that since the model does not change, the second coefficient matrix F remains unchanged during the prediction process. If the weight coefficient matrices Q and R do not change during the calculation process, the Hessian matrix H of the objective function will also remain unchanged. Therefore, the control input can be linearly transformed as follows based on the conjugate matrix P to project it into a new space:
[0092]
[0093] in, is the input in the conjugate space, P is the conjugate matrix, is the prediction input.
[0094] After completing the linear transformation in S2.2, the third objective function of the conjugate space is obtained based on the linear transformation and the second objective function. Based on the third objective function of the conjugate space, the unconstrained optimal solution of the conjugate space is obtained. The specific process is as follows:
[0095] Based on the linear transformation and the second objective function, the third objective function of the conjugate space is obtained. The expression of the third objective function is:
[0096]
[0097] in, is the input in the conjugate space, P is the conjugate matrix, M is the first coefficient, G is the eigenvalue matrix, and c is a constant.
[0098] At this point, the unconstrained optimal solution of the conjugate space can be obtained:
[0099]
[0100] Based on linear transformation, the unconstrained optimal solution in conjugate space can be transformed into the unconstrained optimal solution in real space:
[0101]
[0102] After obtaining the unconstrained optimal solution of the conjugate space in S2.3, the system constraints of the conjugate space are obtained based on the linear transformation and the actual system constraints. The specific process is as follows:
[0103] To conveniently express the MPC quadratic programming problem, we first use the following formula to express the actual system constraints that can ensure the safe operation of the equipment:
[0104]
[0105] This then forms the constrained optimization problem that MPC solves at this time:
[0106]
[0107]
[0108] According to the linear transformation, the system constraint form of the conjugate space is:
[0109]
[0110] S2.4 After obtaining the system constraints of the conjugate space, the third objective function of the conjugate space is converted into a function related to the suboptimal solution of the conjugate space and the unconstrained optimal solution of the conjugate space. Based on the third objective function related to the suboptimal solution of the conjugate space and the unconstrained optimal solution of the conjugate space, the relationship function between the suboptimal solution of the conjugate space, the solution that does not satisfy the system constraints of the conjugate space, and the first solution that satisfies the system constraints of the conjugate space is obtained. The specific process is as follows:
[0111] Assume that the suboptimal solution in conjugate space is First, the third objective function of the conjugate space is transformed into a function related to the suboptimal solution of the conjugate space and the unconstrained optimal solution of the conjugate space. Specifically, it is in the following form:
[0112]
[0113] in, is the suboptimal solution in conjugate space, is the unconstrained optimal solution in conjugate space, and G is the eigenvalue matrix.
[0114] Due to the unconstrained optimal solution in conjugate space So the value of the objective function is determined by the second term of the above formula. The larger the absolute value, the larger the target value, and the corresponding larger eigenvalue σ i The larger the target value, the greater the impact.
[0115]
[0116] Among them U 0 ,…,U r They all do not satisfy the conjugate space system constraints, U r+1 It is the first solution that satisfies the system constraints in conjugate space, the suboptimal solution in conjugate space Written as U r with U r+1 The form between them, that is, the relationship function between the suboptimal solution in conjugate space, the solution that does not satisfy the system constraints in conjugate space, and the first solution that satisfies the system constraints in conjugate space:
[0117]
[0118] in, is the suboptimal solution in conjugate space, U r is the solution that does not satisfy the system constraints of the conjugate space, U r+1 is the first solution that satisfies the system constraints in conjugate space, and λ is the weight coefficient.
[0119] S2.5 uses the system constraints in conjugate space as constraints, performs a linear search on the relationship function, and obtains the suboptimal solution in conjugate space.
[0120] In the relational function, 0≤λ≤1, linear search λ makes Satisfy the system constraints of conjugate space, that is,
[0121]
[0122] Get the suboptimal solution in conjugate space
[0123] S2.6 is based on the conjugate space suboptimal solution and linear transformation to obtain the suboptimal solution
[0124]
[0125] The suboptimal solution not only satisfies the constraints but also retains the information of the unconstrained solution as much as possible, and has a better control effect.
[0126] In step (2), if the current moment is not the sampling moment, the control quantity of the previous moment is directly used to input the double-capacity water tank.
[0127] In step (3), the control variable of the suboptimal solution at the first moment is input into the controlled double-capacity water tank system.
[0128] In order to verify the effectiveness and speed of the proposed SVDF algorithm, the SVDF algorithm is applied to the CS4000 process control experimental device. The structure of the double-capacity water tank is as follows: Figure 2 As shown, the liquid level setting value of the double-capacity water tank is changed and tracked by the SVDF algorithm, and the obtained Figure 3 and Figure 4 The experimental results show that the liquid levels of both tanks can be quickly and accurately tracked to the given set points. In addition, during operation, the system inputs can be kept within the constraints.
[0129] The above describes in detail the preferred embodiments of the present invention. It should be understood that those skilled in the art can make numerous modifications and variations based on the concepts of the present invention without inventive effort. Therefore, any technical solutions that can be derived by those skilled in the art through logical analysis, reasoning, or limited experimentation based on the concepts of the present invention and the prior art should be within the scope of protection defined by the claims.
Claims
1. A dual-capacity water tank level control method based on SVDF, characterized in that: The following steps are involved: Establishing a state space model of the double-capacity water tank to obtain a first objective function of the model predictive control, converting the first objective function of the model predictive control into a second objective function related to the prediction input based on the state space model, wherein the parameters of the second objective function include a Hessian matrix, and performing SVD decomposition on the Hessian matrix to obtain a conjugate matrix; If this is the sampling moment, a linear transformation is performed on the control input based on the conjugate matrix, and a third objective function of the conjugate space is obtained based on the linear transformation and the second objective function. Based on the third objective function of the conjugate space, an unconstrained optimal solution of the conjugate space is obtained; Obtaining the system constraints of the conjugate space based on the linear transformation and the actual system constraints, and transforming the third objective function of the conjugate space into a function related to the suboptimal solution of the conjugate space and the unconstrained optimal solution of the conjugate space; Based on a third objective function related to the conjugate space suboptimal solution and the unconstrained optimal solution of the conjugate space, a relationship function among the conjugate space suboptimal solution, a solution that does not satisfy the system constraints of the conjugate space, and the first solution that satisfies the system constraints of the conjugate space is obtained; and with the system constraints of the conjugate space as a constraint condition, a linear search is performed on the relationship function to obtain the conjugate space suboptimal solution; Based on the conjugate space suboptimal solution and the linear transformation, a result suboptimal solution is obtained, and the control variable of the first moment of the result suboptimal solution is input into the double-capacity water tank to control the liquid level; The specific expression for converting the third objective function of the conjugate space into a function related to the suboptimal solution of the conjugate space and the unconstrained optimal solution of the conjugate space is: in, is the suboptimal solution in conjugate space, is the unconstrained optimal solution in conjugate space, G is the eigenvalue matrix; The specific expression of the relationship function between the suboptimal solution in the conjugate space, the solution that does not satisfy the system constraints in the conjugate space, and the first solution that satisfies the system constraints in the conjugate space is: in, is the suboptimal solution in conjugate space, U r is the solution that does not satisfy the system constraints of the conjugate space, U r+1 is the first solution that satisfies the system constraints in conjugate space, and λ is the weight coefficient.
2. The method for controlling the liquid level of a dual-capacity water tank based on SVDF according to claim 1, characterized in that: The specific expression of the second objective function related to the prediction input is: Where k is the kth moment, is the prediction input, H is the Hessian matrix, M is the first coefficient, and c is a constant.
3. The method for controlling the liquid level of a dual-capacity water tank based on SVDF according to claim 2 is characterized in that: The specific expressions of the Hessian matrix and the first coefficient M are: H=F T QF+R M=x(k) T E T Q T F Among them, F is the second coefficient, Q is the first weight coefficient matrix, R is the second weight coefficient matrix, x(k) is the state of the system at time k, and E is the third coefficient.
4. The method for controlling the liquid level of a dual-capacity water tank based on SVDF according to claim 1, characterized in that: The specific expression for performing linear transformation on the control input based on the conjugate matrix is: in, is the input in the conjugate space, and P is the conjugate matrix.
5. The method for controlling the liquid level of a dual-capacity water tank based on SVDF according to claim 1, characterized in that: The specific expression of the unconstrained optimal solution of the conjugate space is: in, is the unconstrained optimal solution in the conjugate space, G is the eigenvalue matrix, P is the conjugate matrix, and M is the first coefficient.
6. The method for controlling the liquid level of a dual-capacity water tank based on SVDF according to claim 1, characterized in that: The weight coefficient λ satisfies 0≤λ≤1.
7. The method for controlling the liquid level of a dual-capacity water tank based on SVDF according to claim 1, characterized in that: The specific expression of the suboptimal solution is: in, is the suboptimal solution in conjugate space, P is the conjugate matrix, The result is a suboptimal solution.
8. The method for controlling the liquid level of a dual-capacity water tank based on SVDF according to claim 1, characterized in that: The method also includes: if the current moment is not a sampling moment, directly using the control quantity of the previous moment to input into the double-capacity water tank.