CN113568309B - On-line space-time control method for temperature field - Google Patents
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Abstract
The invention provides an online space-time control method for a temperature field, which comprises the following steps: a nonlinear mapping function is introduced to realize the mapping of nonlinear data from a low-dimensional space to a high-dimensional space, and a space kernel function is established to represent the space nonlinear dynamics of a temperature field; projecting the actual output onto the updated space kernel function, establishing a Lagrange time multiplier model, and reconstructing the time dynamics of the temperature field; designing an online updating strategy, and updating and adjusting the spatial dynamic distribution and time-varying dynamic output of the model by using a spatial kernel function and a time Lagrange multiplier model; based on the constructed space-time dynamic model, an online space-time inverse control method is established by utilizing the Taylor expansion theory, and online updating of the controller parameters is realized according to real-time tracking errors of the system and in combination with an online updating strategy.
Description
Technical Field
The invention relates to the field of control of time-varying nonlinear distribution parameter systems, in particular to an online space-time control method for a temperature field.
Background
For the known DPSs, the usual control method first converts the partial differential equation to a finite ordinary differential equation using a first main modeling method (ODEs). Then DPSs is converted into a typical multiple-input multiple-output (MIMO) system, and many classical controllers such as PID controllers, fuzzy control, neural network control, model Predictive Control (MPC) and adaptive control are proposed. While such control methods have achieved many successful applications, they require knowledge of both partial differential equations and boundary conditions, which are often difficult to meet in practical industrial processes due to their time-varying nature and lack of sufficient physical knowledge. For unknown DPSs, data-driven control strategies are of great interest and are designed based on input and output data. Most of these strategies use learning methods to identify spatial basis functions from a set of snapshots. Then, H-infinity control, model Predictive Control (MPC) and Neural Network (NN) are employed. Obviously, it is desirable that the data-driven controller can be designed directly based on spatio-temporal data, but little work has been reported in this regard. Recently, a three-dimensional fuzzy control method has been proposed to enhance the capability of conventional two-dimensional fuzzy control in DPSs, which integrates the additional dimension of spatial information on the basis of conventional two-dimensional fuzzy sets. However, the derivation process is complex and the calculation amount is large. In addition, most of the existing control methods are designed according to a steady process, and cannot adapt to the time-varying characteristics of a dynamic process.
Disclosure of Invention
The invention provides an online space-time control method for a temperature field, which aims to solve the problem of the existing control method in the aspect of time-varying nonlinear distributed parameter system control.
To achieve the above object, an embodiment of the present invention provides an online space-time control method for a temperature field, including:
Step 1, a nonlinear mapping function is introduced to construct a space kernel function so as to represent the space nonlinear dynamics of a temperature field; dynamically updating the space kernel function according to the modeling error, and reconstructing a new Lagrangian multiplier model by projecting the actual output onto the updated space kernel function;
Step 2, designing an online updating strategy, and updating and adjusting the spatial dynamic distribution and time-varying dynamic output of the model by using a spatial kernel function and a time Lagrange multiplier model;
And step 3, establishing an online space-time inverse control method by utilizing a Taylor expansion theory according to the constructed space-time dynamic model, and realizing online updating of the controller parameters according to real-time tracking errors of the system and by combining an online updating strategy.
Wherein, the step 1 specifically includes:
Mapping an original low-dimensional space to a high-dimensional space by using a space kernel function, and constructing a space-time LS-SVM model as follows:
The following objective function is constructed:
Wherein, Is a modeling error representing a regularization factor that trades off between approximation accuracy and generalization;
The solution of the equation is obtained by solving the Lagrangian multiplier method as follows:
further converting it into a matrix form:
by introducing a kernel function, the solutions for parameters a and b are obtained as follows:
LS-SVM is used to construct models α i(tk) and b (t k) to predict its value at any time, resulting in a spatiotemporal LS-SVM model as follows:
wherein, the step 2 specifically includes:
The following gaussian radial basis functions are typically used as the spatial kernel function:
where σ is the width of the gaussian radial basis function;
The following objective function is used to update the parameters to minimize modeling errors:
the updated value of σ at time t k+1 is calculated by the following formula:
wherein, σ (t k+1)＝σ(tk)+γΔσ(tk), γ is the adjustment factor, and the time-varying spatial kernel function is updated as:
In the online modeling process, new data are collected, the Lagrangian multiplier coefficient sum is updated according to a formula (5) by utilizing the data and a space kernel function, and the regression time sequence parameter is updated as follows:
zi(tk+1)＝[αi(u(tk));…;αi(u(tk-p+1));u(tk+1);…;u(tk-d+1)] (11)
Wherein p and d represent regression steps;
The core matrix in formula (7) is
The time coefficient model may be updated as:
The lagrangian multiplier η τ and the deviation θ s are updated as:
wherein the method comprises the steps of ,αi(u(tL+1))＝[αi(u(t1)),αi(u(t2)),…,αi(u(tL+1))]T,ητ＝[η1,···,ηL+1]T,
The lagrangian time coefficient multiplier in equation (13) is re-expressed as:
The online model is obtained as follows: wherein b (t k+1)＝g(u(tk+1))/(v) Is a bias term whose update procedure is identical to/>Wherein, the step 3 specifically includes:
from the spatiotemporal model in equation (16), taking into account the first-order taylor expansion, it can be derived:
Wherein Δu (t k)＝u(tk)-u(tk-1),Rk[Δu(tk) ] is a higher-order infinite term, satisfying the following relationship:
Further, a first derivative of equation (17) may be obtained:
wherein the method comprises the steps of ,qi(tk-1)＝[αi(u(tk-2));…;αi(u(tk-p-1));u(tk-2);…;u(tk-d)];
Selecting the following gaussian function as k α(z(ti),z(tj)
First order derivative in formula (19)Can be converted into:
substituting equations (19) and (21) into equation (17), the system output may be converted to:
model uncertainty and external disturbance of a space-time LS-SVM model are defined as Formula (22) may be converted to:
Wherein, v k satisfies And/>Is a sufficiently small positive number;
A target temperature r (x, t k) is set for the control system, delta input deltau (t k) is input according to the Taylor expansion formula when When there is a sufficiently small positive number, the higher order infinitesimal terms satisfy the following relationship:
Wherein, Ρ 0 is a positive number;
Formula (23) may be:
The incremental control rate is thus obtained as:
where ε is a small positive number to prevent the denominator of the above formula from being 0.
The scheme of the invention has the following beneficial effects:
the on-line space-time control method for the temperature field realizes the mapping of nonlinear data from low-dimensional space to high-dimensional space by introducing a nonlinear mapping function, and establishes a space kernel function to represent the space nonlinear dynamics of the temperature field; projecting the actual output onto the updated space kernel function, establishing a Lagrange time multiplier model, and reconstructing the time dynamics of the temperature field; designing an online updating strategy, and updating and adjusting the spatial dynamic distribution and time-varying dynamic output of the model by using a spatial kernel function and a time Lagrange multiplier model; based on the constructed space-time dynamic model, an online space-time inverse control method is established by utilizing the Taylor expansion theory, and online updating of the controller parameters is realized according to real-time tracking errors of the system and in combination with an online updating strategy. According to the invention, the space kernel function is dynamically updated according to the actual modeling error, and the actual output is projected onto the updated space kernel function to reconstruct a new Lagrangian multiplier model, so that an online space-time model of the temperature field is finally formed. The on-line space-time inverse control method established based on the space-time data model not only can effectively reconstruct DPSs space-time dynamics, but also ensures stable control of temperature distribution in a temperature field. The design of the strategy considers the actual spatial distribution and time dynamics of the system and accords with the mechanism characteristics of the distributed parameter system. Based on the online space-time LS-SVM model, the online updating of the controller parameters is realized by utilizing the Taylor expansion theory according to the real-time tracking error of the system and combining with an online updating strategy, so that the tracking stability of the model to time-varying dynamics is effectively improved.
Drawings
FIG. 1 is a schematic diagram of an online space-time control method of a distributed parameter system based on data driving;
FIG. 2 is a schematic diagram of a space-time modeling method of a distribution parameter system based on LS-SVM;
FIG. 3 is a schematic diagram of an online space-time update strategy based on LS-SVM;
FIG. 4 is a schematic diagram of the structure of the heating furnace and a schematic diagram of the sensor space layout;
FIG. 5 is a schematic diagram of input signals and system outputs;
FIG. 6 is a schematic diagram of a time coefficient update process;
FIG. 7 is a schematic diagram of modeling performance of the proposed method;
FIG. 8 is a schematic representation of the actual control inputs under the influence of the inventive method;
FIG. 9 is a schematic diagram of system output and control tracking error;
FIG. 10 is a schematic diagram of tracking performance using sensors s3, s5, s8, s10 as an example;
FIG. 11 is a schematic diagram of the update process of the time coefficient in the presence of disturbances;
FIG. 12 is a schematic diagram of modeling performance of the proposed method;
FIG. 13 is a schematic diagram of actual control inputs under the influence of the inventive method;
FIG. 14 is a schematic diagram of system output and control tracking error;
FIG. 15 is a schematic diagram of the tracking performance of the proposed method in the presence of disturbances.
Detailed Description
In order to make the technical problems, technical solutions and advantages to be solved more apparent, the following detailed description will be given with reference to the accompanying drawings and specific embodiments.
As shown in fig. 1, an embodiment of the present invention provides an online space-time control method for a temperature field, including: a nonlinear mapping function is introduced to realize the mapping of nonlinear data from a low-dimensional space to a high-dimensional space, and a space kernel function is established to represent the space nonlinear dynamics of a temperature field; projecting the actual output onto the updated space kernel function, establishing a Lagrange time multiplier model, and reconstructing the time dynamics of the temperature field; designing an online updating strategy, and updating and adjusting the spatial dynamic distribution and time-varying dynamic output of the model by using a spatial kernel function and a time Lagrange multiplier model; based on the constructed space-time dynamic model, an online space-time inverse control method is established by utilizing the Taylor expansion theory, and online updating of the controller parameters is realized according to real-time tracking errors of the system and in combination with an online updating strategy. According to the invention, the space kernel function is dynamically updated according to the actual modeling error, and the actual output is projected onto the updated space kernel function to reconstruct a new Lagrangian multiplier model, so that an online space-time model of the temperature field is finally formed. On the basis, the Taylor expansion theory is utilized, a space-time inverse control method based on a data model is provided, tracking errors are utilized and an online updating strategy is combined, online updating of controller parameters is achieved, and tracking stability of the model to time-varying dynamics is improved.
The on-line space-time control method fully considers the dynamic characteristics of the distributed parameter system in time and space, considers the processing of space-time dynamics and the on-line updating strategy of the model in the design of the controller, effectively solves the influence of interference on the actual control process, and ensures the stability of the whole control system.
Wherein, the step 1 specifically includes: the space-time modeling method based on the LS-SVM has higher fitting precision when the traditional LS-SVM method processes a nonlinear model, and some scholars expand the nonlinear model to space and time dimensions and apply the nonlinear model to modeling of a distributed parameter system, as shown in figure 2. Firstly, mapping an original low-dimensional space into a high-dimensional space by using a space kernel function to realize linearization of a nonlinear relation; then, the original space-time modeling problem can be converted into a time sequence modeling problem, and the time sequence modeling problem is effectively verified in a complex industrial system.
The constructed space-time LS-SVM model is as follows:
to solve the modeling problem, the following objective function is constructed:
Wherein, Is a modeling error representing a regularization factor that trades off between approximation accuracy and generalization;
The solution of the equation is obtained by solving the Lagrangian multiplier method as follows:
further converting it into a matrix form:
by introducing a kernel function, such as a Radial Basis Function (RBF), the solutions for parameters a and b are obtained as follows:
LS-SVM is used to construct models α i(tk) and b (t k) to predict its value at any time, resulting in a spatiotemporal LS-SVM model as follows:
Wherein, the step 2 specifically includes: on-line updating strategy based on space-time LS-SVM, the space-time LS-SVM modeling method focuses on an off-line modeling process, and time-varying DPS cannot be modeled. The present invention therefore proposes an online spatiotemporal update strategy to describe the time-varying dynamics of DPS. As shown in fig. 3:
the following gaussian radial basis functions are generally chosen as spatial kernel functions, which exhibit good performance:
where σ is the width of the gaussian radial basis function;
The following objective function is used to update the parameters to minimize modeling errors:
the updated value of σ at time t k+1 is calculated by the following formula:
wherein, σ (t k+1)＝σ(tk)+γΔσ(tk), γ is the adjustment factor, and the time-varying spatial kernel function is updated as:
and (3) an online updating strategy of the time coefficient multiplier, wherein in the online modeling process, new data are collected, the Lagrange multiplier coefficient sum is updated according to a formula (5) by utilizing the data and a space kernel function, and the regression time sequence parameter is updated as follows:
zi(tk+1)＝[αi(u(tk));…;αi(u(tk-p+1));u(tk+1);…;u(tk-d+1)] (11)
Wherein p and d represent regression steps;
The core matrix in formula (7) is
The time coefficient model may be updated as:
The lagrangian multiplier η τ and the deviation θ s are updated as:
Wherein the method comprises the steps of ,αi(u(tL+1))＝[αi(u(t1)),αi(u(t2)),…,αi(u(tL+1))]T,ητ＝[η1,···,ηL+1]T,
The lagrangian time coefficient multiplier in equation (13) is re-expressed as:
The online model is obtained as follows:
Wherein b (t k+1)＝g(u(tk+1)), Is a bias term whose update procedure is identical to/>
Wherein, the step 3 specifically includes: an on-line space-time inverse control method based on a data model, and an inverse control method based on Tylor expansion theory are generally used for neural network control of a centralized parameter system (LPSs), but are not applicable to DPSs. On the basis, the method is popularized to dynamic process control, and an online space-time inverse control method suitable for time-varying nonlinear dynamic process control is provided.
From the spatiotemporal model in equation (16), taking into account the first-order taylor expansion, it can be derived:
Wherein Δu (t k)＝u(tk)-u(tk-1),Rk[Δu(tk) ] is a higher-order infinite term, satisfying the following relationship:
Further, a first derivative of equation (17) may be obtained:
wherein the method comprises the steps of ,qi(tk-1)＝[αi(u(tk-2));…;αi(u(tk-p-1));u(tk-2);…;u(tk-d)];
Selecting the following gaussian function as k α(z(ti),z(tj)
First order derivative in formula (19)Can be converted into:
substituting equations (19) and (21) into equation (17), the system output may be converted to:
model uncertainty and external disturbance of a space-time LS-SVM model are defined as Formula (22) may be converted to:
Wherein, v k satisfies And/>Is a sufficiently small positive number;
In the actual control process, most of chemical reaction heating rods, casting and forging heating furnaces and the like are typical slow-changing systems. Thus, once the target temperature r (x, t k) is set for the control system. Delta input Deltau (t k), when according to the Taylor expansion formula When there is a sufficiently small positive number, the higher order infinitesimal terms satisfy the following relationship:
Wherein, Ρ 0 is a positive number;
Therefore, the higher order infinitesimal terms are negligible, which is reasonable because in many practical processes, such as the inertia of thermal processes, these processes cannot change too fast in a small time interval. Formula (23) may be:
The incremental control rate is thus obtained as:
Where ε is a small positive number to prevent the denominator of the above formula from being 0. The on-line control model dynamically models the system, and simultaneously, the design part of the controller considers the spatial distribution and the time dynamic of the system and accords with the mechanism characteristics of the distributed parameter system.
According to the on-line space-time control method for the temperature field, a space kernel function is dynamically updated according to modeling errors, and a new Lagrangian multiplier model is reconstructed by projecting actual output onto the updated space kernel function; and the updated space kernel function is combined with the updated Lagrangian multiplier, so that the reconstruction of DPSs space-time dynamics is effectively realized. An online updating strategy is designed to adjust the space-time dynamics of the model, and the space dynamic distribution and time-varying dynamic output of the model are updated and adjusted by using a space kernel function and a time Lagrange multiplier model respectively. The design of the strategy considers the actual spatial distribution and time dynamics of the system and accords with the mechanism characteristics of the distributed parameter system. Based on the online space-time LS-SVM model, an online space-time inverse control method is established by utilizing the Taylor expansion theory, and online updating of the controller parameters is realized according to real-time tracking errors of the system and in combination with an online updating strategy, so that the tracking stability of the model to time-varying dynamics is effectively improved.
The course of the use of the proposed method is shown here on the basis of two experiments carried out on a heating furnace and its effectiveness is evaluated. In an industrial process, the heating process of the forging or casting is generally performed in a heating furnace, as shown in fig. 4. In the heating furnace, four heaters (H1 to H4) are placed and driven by the corresponding power amplifiers. During the heating process, 12 sensors (S1 to S12) are uniformly distributed on the workpiece to collect temperature data.
The effectiveness of the controller was verified with two indicators of Relative Error (RE) and Mean Absolute Percent Error (MAPE) as follows:
(1) Temperature control without interference
In the experiment, the target temperature was set at 85 deg.c, in the initial experiment, output data was generated using a random input signal excitation system to estimate the initial values of the model parameters, as shown in fig. 5. This will yield 200 sets of experimental data from 12 sensors with a sampling interval of 10s. These input/output data are then used to build a model.
On this basis, the model is updated online according to real-time data, wherein the updating process of the time coefficients alpha 3,α8 and alpha 10 is shown in fig. 6. As can be seen from fig. 6, the lagrangian time multiplier also changes dynamically as the temperature increases, presenting an increasing trend, consistent with an actual industrial process. The approximate performance of this model is shown in figure 7. From both figures, it can be seen that the model approximates the actual process well. When the temperature rises and tends to stabilize during training and testing, the relative model error is small and is bounded by [0,1.5% ]. Based on this on-line model, controllers were designed, and control inputs for heaters u3 and u4 are shown in fig. 8.
The control results of the heating temperature are shown in fig. 9 and 10, and it is thus known that the heating process can efficiently track the target temperature due to small tracking errors. According to the trend of fig. 8 and 9, when the temperature gradually approaches the target, the control process may be divided into three stages: 1) The control input is rapidly increased to 50V to increase the convergence speed to the target; 2) When the temperature component reaches the target value, the control input is reduced to slow down the heating process until the control input is equal to zero. In order to more clearly illustrate the tracking process, the tracking details of the four sensors (S3, S5, S8, S10) are shown here in fig. 10.
As can be seen from fig. 10, the temperature profile of each spatial location shows a consistent steady trend, with a final control error in a small range. And compared with the conventional fuzzy-PID control method, the control performance of the method is verified. The RE and RSME comparison results for all sensors are shown in tables 1 and 2.
TABLE 1
TABLE 2
From these tables, it can be seen that the proposed method has better control performance than the fuzzy PID, since the MAPE and RE of the proposed controller for all sensors are smaller than the fuzzy PID, which is also demonstrated at most spatial location points. For example, when using a conventional fuzzy PID controller, RE and MAPE on sensor 3 and sensor 5 are twice as high as the proposed inverse controller.
(2) Temperature control without interference
Then, the robustness of the control method in the presence of external disturbances was verified by another experiment. In the experiment, at the 50 th and 90 th sampling, external disturbance was generated by opening and closing the oven door, and the duration of each disturbance was 2s. In the experiment, the time coefficient and the space kernel function are updated online, and the updating process of the time coefficient (alpha 3,α8,α10) is shown in fig. 11. The convergence performance of the model is shown in fig. 12. It can be seen that the relative model error is small and the model is very close to the actual process. Further, control experiments were performed, and control inputs (for example, u3 and u 4) of the heater are shown in fig. 13. The control performance using this control law is shown in fig. 14-15. As can be seen from these figures, when the temperature component reaches the target value, the control input is lowered to slow down the heating process until zero. As shown in the tracking details (S3, S5, S8, S10) of fig. 15, the actual heating process can efficiently track the target temperature c with a small tracking error even if there is a disturbance. Thus, the proposed control method is very robust to disturbed time-varying non-linear processes.
While the foregoing is directed to the preferred embodiments of the present invention, it will be appreciated by those skilled in the art that various modifications and adaptations can be made without departing from the principles of the present invention, and such modifications and adaptations are intended to be comprehended within the scope of the present invention.
Claims (3)
1. An on-line space-time control method for a temperature field, comprising:
Step 1, a nonlinear mapping function is introduced to construct a space kernel function so as to represent the space nonlinear dynamics of a temperature field; dynamically updating the space kernel function according to the modeling error, and reconstructing a new Lagrangian multiplier model by projecting the actual output onto the updated space kernel function;
Step 2, designing an online updating strategy, and updating and adjusting the spatial dynamic distribution and time-varying dynamic output of the model by using a spatial kernel function and a time Lagrange multiplier model;
step 3, according to the constructed space-time dynamic model, utilizing a Taylor expansion theory, and according to the real-time tracking error of the system and combining an online updating strategy, realizing online updating of the controller parameters; wherein, the step 1 specifically includes:
Mapping an original low-dimensional space to a high-dimensional space by using a space kernel function, and constructing a space-time LS-SVM model as follows:
The following objective function is constructed:
Wherein, Is a modeling error representing a regularization factor that trades off between approximation accuracy and generalization;
The solution of the equation is obtained by solving the Lagrangian multiplier method as follows:
further converting it into a matrix form:
by introducing a kernel function, the solutions for parameters a and b are obtained as follows:
LS-SVM is used to construct models α i(tk) and b (t k) to predict its value at any time, resulting in a spatiotemporal LS-SVM model as follows:
2. the on-line space-time control method for temperature field according to claim 1, wherein the step 2 specifically comprises:
The following gaussian radial basis functions are typically used as the spatial kernel function:
where σ is the width of the gaussian radial basis function;
The following objective function is used to update the parameters to minimize modeling errors:
the updated value of σ at time t k+1 is calculated by the following formula:
wherein, σ (t k+1)＝σ(tk)+γΔσ(tk), γ is the adjustment factor, and the time-varying spatial kernel function is updated as:
In the online modeling process, new data are collected, the Lagrangian multiplier coefficient sum is updated according to a formula (5) by utilizing the data and a space kernel function, and the regression time sequence parameter is updated as follows:
zi(tk+1)＝[αi(u(tk));…;αi(u(tk-p+1));u(tk+1);…;u(tk-d+1)] (11)
Wherein p and d represent regression steps;
The core matrix in formula (7) is
The time coefficient model may be updated as:
The lagrangian multiplier η τ and the deviation θ s are updated as:
Wherein the method comprises the steps of ,αi(u(tL+1))＝[αi(u(t1)),αi(u(t2)),…,αi(u(tL+1))]T,ητ＝[η1,···,ηL+1]T,
The lagrangian time coefficient multiplier in equation (13) is re-expressed as:
The online model is obtained as follows:
Wherein b (t k+1)＝g(u(tk+1)), Is a bias term whose update procedure is identical to/>
3. The on-line space-time control method for temperature field according to claim 2, wherein the step 3 specifically comprises:
from the spatiotemporal model in equation (16), taking into account the first-order taylor expansion, it can be derived:
Wherein Δu (t k)＝u(tk)-u(tk-1),Rk[Δu(tk) ] is a higher-order infinite term, satisfying the following relationship:
Further, a first derivative of equation (17) may be obtained:
wherein the method comprises the steps of ,qi(tk-1)＝[αi(u(tk-2));…;αi(u(tk-p-1));u(tk-2);…;u(tk-d)];
Selecting the following gaussian function as k α(z(ti),z(tj)
First order derivative in formula (19)Can be converted into:
substituting equations (19) and (21) into equation (17), the system output may be converted to:
model uncertainty and external disturbance of a space-time LS-SVM model are defined as Formula (22) may be converted to:
Wherein, v k satisfies And/>Is a sufficiently small positive number;
A target temperature r (x, t k) is set for the control system, delta input deltau (t k) is input according to the Taylor expansion formula when When there is a sufficiently small positive number, the higher order infinitesimal terms satisfy the following relationship:
Wherein, Ρ 0 is a positive number;
Formula (23) may be:
The incremental control rate is thus obtained as:
where ε is a small positive number to prevent the denominator of the above formula from being 0.
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