CN110609476B - Multivariate nonlinear dynamic system model prediction control method based on Gaussian process model - Google Patents

Multivariate nonlinear dynamic system model prediction control method based on Gaussian process model Download PDF

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CN110609476B
CN110609476B CN201910889495.0A CN201910889495A CN110609476B CN 110609476 B CN110609476 B CN 110609476B CN 201910889495 A CN201910889495 A CN 201910889495A CN 110609476 B CN110609476 B CN 110609476B
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任密蜂
张旭霞
程兰
续欣莹
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Abstract

The invention discloses a multivariate nonlinear dynamic system model predictive control method based on a Gaussian process model, belonging to the technical field of predictive control of multivariate nonlinear dynamic system models; the technical problem to be solved is as follows: the improvement of a multivariate nonlinear dynamic system model predictive control method based on a Gaussian process model is provided; the technical scheme for solving the technical problem is as follows: the method comprises the following steps: the method comprises the following steps: establishing an external dynamic PLS framework: step two: predicting output data, and decoupling a dynamic GP-PLS model to obtain a plurality of single-input single-output systems in an implicit space; step three: controlling by using a dynamic GP-PLS model, and designing a model prediction controller in each single-input single-output subsystem; step four: obtaining an optimal control action by minimizing an objective function; step five: reconstructing a model prediction control result in the hidden space back to the original space, and controlling the original space; the method is applied to a multivariate nonlinear dynamic system model.

Description

Multivariate nonlinear dynamic system model prediction control method based on Gaussian process model
Technical Field
The invention discloses a multivariate nonlinear dynamic system model predictive control method based on a Gaussian process model, and belongs to the technical field of predictive control of multivariate nonlinear dynamic system models.
Background
With the rapid development of the industry and the information science technology, the industrial production scale is larger and larger, and the production process and the production flow become more and more complex, which provides a significant challenge for the traditional mechanism modeling and control strategy, and is especially applied to the industries of petroleum, chemical industry, metallurgy, machinery and the like. Model Predictive Control (MPC) is an advanced computer control algorithm that computes the first input value of a future state optimization sequence of a system to be applied to the system based on current and past operating states of the system, and the computation is repeated for a subsequent control time. The MPC does not need to obtain an accurate mathematical model of the controlled system accurately, but obtains a performance index function through rolling optimization to obtain a control signal of the current moment in an optimized mode, and finally corrects the accuracy of the prediction model through feedback correction, but the predicted value of the model cannot deviate greatly, and the accuracy of the model determines whether the prediction control can obtain a good effect to a great extent. In model predictive control, rolling optimization is the biggest difference from traditional optimal control, the traditional optimal control judges global optimization by using a performance function, and the optimization of the model predictive control is not completed off-line at one time, but is repeated and is a stable process. Therefore, the optimization method is suitable for complex industrial processes with dynamic characteristic change and uncertain factors.
In the industrial control process, most of the systems are nonlinear systems with noise and delay, and the conventional linear control algorithm neglects the nonlinear links or uses linear approximation, so that the linear control algorithm is still widely applied to the nonlinear systems, particularly the linear feedback algorithm which is greatly developed in recent years; however, when the system parameters are uncertain or contain disturbance and noise signals, the algorithm cannot ensure the robustness of the system, so that the control effect is obviously reduced; today, with the rapid development of scientific technology, the system is more and more complex and has higher uncertainty, and meanwhile, the system has higher requirements on the control effect of the system, so that the whole control system becomes very complex, and the complex system generally has the characteristics of nonlinearity, multivariable, noise, large hysteresis and the like.
The nonlinear predictive control has been a hot research problem in industrial production control, but with the current research results, the following problems still exist:
at present, in the control field, an RBF neural network is mainly used as a nonlinear prediction model in prediction control, but the structural design of the neural network still mainly depends on experience, the number of optimized parameters is large, and the training process is complex, so that the neural network has many limitations;
secondly, before the MPC is implemented, a well-described model must be developed for the process, but an accurate first-principle model is not only difficult to obtain, but also because complete process knowledge is often scarce, a data-driven model identified from input/output data is usually preferred for MPC design, whereas when the MPC is implemented using the data-driven model, especially when a well-described model is constructed from highly correlated data, the process is complicated;
for a large-scale multiple-input multiple-output (MIMO) system, in MIMO processing, solving of control actions becomes expensive and time-consuming, and cross-coupling of process variables causes difficulty in designing a controller, which has a great limitation on real-time application of MPC in industry;
fourth, for a system with a higher dimension, the traditional MPC algorithm not only leads to a complex design of the controller, but also increases the calculation amount of the controller, so that the control algorithm needs to be improved to overcome the defects in the traditional model predictive control.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention aims to solve the technical problems that: an improvement of a multivariate nonlinear dynamic system model predictive control method based on a Gaussian process model is provided.
In order to solve the technical problems, the invention adopts the technical scheme that: a multivariate nonlinear dynamic system model predictive control method based on a Gaussian process model comprises the following steps:
the method comprises the following steps: establishing an external dynamic PLS framework:
step 1.1: let two scaled datasets U and Y, where: u is an input data matrix with the size of nxm, and Y is an output data matrix with the size of nxl;
the data sets U and Y are normalized (mean 0 and variance 1) to obtain a scaling matrix WxAnd Wy；
Step 1.2: obtaining a score matrix T and a load matrix P of a data set U, and a score matrix V and a load matrix Q of a data set Y by the PLS external model;
step 1.3: let tiAnd viThe ith orthogonal column vector of the scoring matrix T and V respectively, and the T is obtainediAnd viBy v internal relations ofi＝fGP(ti) Establishing a nonlinear internal model of the GP process, representing an algebraic relation between an input latent variable and an output latent variable, wherein f (t)i) Index score vector tiAnd viAn algebraic relationship of (c);
step 1.4: load vectors for data sets U and Y are calculated, respectively:
Figure BDA0002208163270000021
and
Figure BDA0002208163270000022
the formula is derived from a nonlinear iterative partial least squares algorithm, wherein:
Figure BDA0002208163270000023
and
Figure BDA0002208163270000024
transposes of the ith vectors of the loading matrices P and Q, respectively, i representing the number of loops;
tiand viThe ith orthogonal column vector of the scoring matrices T and V, respectively;
Figure BDA0002208163270000025
and
Figure BDA0002208163270000026
transpose of the ith orthogonal column vector of the scoring matrices T and V, respectively;
step 1.5: computing residual matrix Ea+1And Fa+1If the residual error meets the convergence condition or the principal element number reaches a set value, ending the algorithm to obtain a GP-PLS prediction model, and if the residual error does not meet the convergence condition or the principal element number reaches the set value, turning to the step 1.1;
step two: after a GP-PLS prediction model is obtained according to the algorithm, output data are predicted, and a multivariable system in an original space is decoupled through a dynamic GP-PLS model to obtain a plurality of single-input single-output systems in a hidden space;
step three: controlling by using a dynamic GP-PLS model, and designing a model prediction controller in each single-input single-output subsystem;
step four: by minimizing the objective function JikObtaining the optimal control action;
step five: and reconstructing the model prediction control result in the hidden space back to the original space through the correlation between the original space and the hidden space, and controlling the original space.
Compared with the prior art, the invention has the following beneficial effects:
the method for using the rolling optimization is the biggest difference from the traditional optimal control, the traditional optimal control judges the global optimization by using a performance function, the optimization of the model predictive control is not completed off line at a time, but is repeated and is a stable process, so the method is more suitable for the complex industrial process with dynamic characteristic change and uncertain factors;
secondly, the invention establishes a dynamic GP-PLS model, extracts the characteristic information of the dynamic process through a PLS framework, eliminates the co-linearity of data and reduces the dimension of input variables. An input score vector t is then built by GPiAnd output score vector viThe dynamic GP-PLS combines the advantages of PLS and GP, namely feature extraction of GP, the robustness of other nonlinear processing capacity and PLS method, and the GP nonlinear internal model has fewer parameters and excellent parametersThe method is easier to realize, and the variance information can represent the confidence degree of prediction and is a characteristic which is not possessed by other methods;
thirdly, model prediction control of an original space is relatively difficult, optimization parameters are more, decoupling is performed in a hidden space, the MIMO problem is converted into a plurality of SISO problems, and the control problem is time-saving and simple.
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The invention is further described below with reference to the accompanying drawings:
FIG. 1 is a schematic illustration of predictive control of the present invention;
FIG. 2 is a block flow diagram of a model predictive control method of the present invention applied to a multivariable nonlinear dynamical system;
FIG. 3 is a schematic view of the structure of a continuous stirring reaction tank to which the present invention is applied;
FIG. 4 is a diagram illustrating the predicted results of the first latent variable dynamic GP-PLS model according to the present invention;
FIG. 5 is a diagram showing the predicted results of a second latent variable dynamic GP-PLS model according to the present invention;
FIG. 6 is a graph showing the change in the concentration of the product A in the continuous stirring reaction vessel according to the present invention with time;
FIG. 7 is a graph of temperature T versus time for a continuous stirred tank reactor according to the present invention.
Detailed Description
As shown in fig. 1 to 7, the present invention proposes a Model Predictive Control (MPC) scheme based on a Gaussian Process (GP) model in a dynamic Partial Least Squares (PLS) framework, which is applied to a multivariable nonlinear dynamical system; the method takes the Gaussian process as a nonlinear prediction model, not only is simple in modeling and few in parameters, but also is easier to super-parameter train, and the variance information can reflect the prediction accuracy while the predicted output value can be obtained. Moreover, the multi-step prediction based on the single-step prediction is simple, and the time complexity of the prediction model training can be reduced while a better prediction effect can be obtained; for a multiple-input multiple-output (MIMO) system, in order to eliminate cross coupling of process variables, avoid decoupling control and loop pairing and reduce complexity of calculated amount, the multiple-input multiple-output (MIMO) system is decoupled into a single loop through a dynamic PLS structure, model prediction control is respectively carried out on SISO in a hidden space, and a control target is realized by adopting simple controller design and less calculation time.
The method mainly comprises a dynamic GP-PLS modeling stage and a control stage.
The method comprises the following steps: firstly, establishing a dynamic GP-PLS prediction model:
(1) external dynamic PLS framework:
first consider two scaled data sets, U (an input (predictor) data matrix, size nxm) and Y (an output (predictor) data matrix, size nxl), where n, m, l represent the number of observations, manipulated variables and process variables, respectively.
Data sets U and Y are represented as:
Figure BDA0002208163270000041
Figure BDA0002208163270000042
the specific relationship between the data sets U and Y is:
Figure BDA0002208163270000043
ti＝[ti,k,ti,k-1,…,ti,k-n+1]T，pi＝[pi,1,pi,2,…,pi,m]T (2)
Figure BDA0002208163270000044
vi＝[vi,k,vi,k-1,…,vi,k-n+1]T，qi＝[qi,1,qi,2,…,qi,l]T (4)
wherein the number of principal components a may be determined by statistical techniques, such as cross-validation or heuristics, as
Figure BDA0002208163270000045
(predicted value of Y) percentage of variance retained. T and V both represent scoring matrices, and P and Q are loading matrices for T and V, respectively. Furthermore, tiAnd viThe ith orthogonal column vector of the scoring matrices T and V, respectively, and piAnd q isiThe ith vector of load matrices P and Q, respectively. Ea+1And Fa+1Are the residual matrices of X and Y, respectively.
Considering the nonlinear iterative partial least squares (NIPALS) algorithm, the following relationship can be obtained:
Figure BDA0002208163270000046
ti＝Xwi (6)
wherein, wiIs a weight vector.
In the internal model, an algebraic relation between input and output latent variables is obtained by a least square method.
vi＝f(ti)+ei (7)
Wherein f (t)i) Refers to the score vector tiAnd viAlgebraic relation of eiRefers to the residual vector.
The uncertainty of the model is then refined based on the internal GP model.
For each loop, the GP model is used to characterize the input ti,k(i ═ 1,2, …, a) and an output vi,k(i ═ 1,2, …, a) dynamic relationships of relationships between the underlying variables.
One GP model is a set of random variables with a joint distribution:
Figure BDA0002208163270000051
for any input Zi＝{zi,1,…,zi,kV and output vi＝{vi,k,…,vi,k-n+1Set of (d) }, average vector μiAnd ZiRespectively, refer to the input vectors of the gaussian process model, not the controlled variables.
For dynamic modeling, zi,kPast manipulated latent variables and past output latent variables may be included, respectively, as follows:
zi，k＝[ti，k-1，…，ti，k-b+1，vi，k-1，…，vi，k-d+1]T (9)
where subscripts b and d refer to the past b manipulated latent variables and the past d output latent variables, respectively. L isiIs a suitable normalization constant. CiIs made of a parameterized covariance function
Figure BDA0002208163270000052
A covariance matrix of the defined data.
Giving a data set Di＝(Zi,vi) Due to the combined density P (v)i,k+1,vi) Is also Gaussian, so v can be easily inferredi,k+1(ii) a The conditional distribution obtained by Bayesian theorem is:
Figure BDA0002208163270000053
it is also gaussian, and then gives the posterior distribution:
Figure BDA0002208163270000054
finally, the prediction mean and the prediction variance of the GP model are solved, which are respectively:
Figure BDA0002208163270000055
Figure BDA0002208163270000061
wherein, gT(zi,k+1)＝[C(zi,k+1,zi,1)…C(zi,k+1,zi,k)]，μi,k+1Is vi,k+1The predicted average value of (a) is,
Figure BDA0002208163270000062
is the standard deviation of the prediction.
Vector gT(zi,k+1)C-1Can be viewed as a smoothing term that weights the training output to weight the new input vector zi,k+1And (6) performing prediction. The standard deviation of the prediction provides a confidence level for the model prediction, since higher variances indicate that the region of the input vector contains little noise corrupted data.
And finally, selecting a covariance function and optimizing parameters of the covariance function.
The covariance function is very important and common choices are:
Figure BDA0002208163270000063
wherein a isi0、ai1、wi,s、qiV and vi0Is a defined hyper-parameter which is,
hyperparameter vi0The overall scale of the local correlation is controlled,
ai1allowing different distance measurements in each input dimension,
s and qiIs an estimate of the variance of the noise.
δiIs a Kronecker delta parameter if r1＝r2，δiIs 1, otherwise δiThe value of (d) is 0.
Under a gaussian process model, where the priors are gaussian distributed, the hyper-parameters can be estimated by maximizing the log-likelihood function:
Figure BDA0002208163270000064
wherein theta isi＝[ai0,ai1,wis,qi,νi0]。
The maximum log-likelihood function can be solved using the derivative of the log-likelihood function to the hyper-parameter:
Figure BDA0002208163270000065
can obtain vi,k+1，
Figure BDA0002208163270000066
The covariance function of (a).
And (3) establishing a dynamic GP-PLS model.
The GP-DPLS method is fundamentally different from the conventional PLS method in that an internal model of a dynamic PLS is built using a non-linear GP model instead of linear regression, the dynamic GP-PLS method extracts feature information of a dynamic process through PLS, eliminates co-linearity of data, and reduces the dimension of an input variable, then obtains a score vector for each step decomposition, and builds an input score vector t through GPiAnd output score vector viThe non-linear internal model improves the non-linear processing capability of the internal model.
Obviously, the GP-DPLS method inherits the advantages of PLS and GP, namely the robustness of the PLS method, the feature extraction and other non-linear processing capability of GP. Compared with other non-linear PLS methods using non-linear functions (such as neural networks) to build the PLS internal model, the GP-PLS model has significantly fewer non-linear internal model parameters and easier parameter optimization.
Step two, model prediction control:
considering a single input single output system, the past output trajectory of the system is given by ytIndicating that the predicted value of the system in the future is ypAnd (4) showing. At discrete samplingAt time k, the input value y of the system at that timekAs shown, the system prediction value at this time for the future time k + j is yp(k+j|k)。
In FIG. 1, L denotes a prediction time domain, C denotes a control time domain, and C is general<L. In addition, there are two traces in the figure: y iswAnd
Figure BDA0002208163270000071
wherein
Figure BDA0002208163270000072
Set value, y, representing system outputwThe trace is indicated in the sense that when a disturbance occurs, the system should follow trace ywFinally returning to the set track
Figure BDA0002208163270000073
As can be seen from the effect of these two trajectories, the ultimate goal of the control system is to cause the output of the system to track the specified trajectory
Figure BDA0002208163270000074
Thus, it is possible to provide
Figure BDA0002208163270000075
Must exist otherwise the control system loses the meaning of control optimization.
And step three, in a dynamic PLS framework, implementing a multivariable MPC rolling optimization strategy by using a GP model.
In the traditional controller design, the output of the controller is usually finished off-line once, namely once calculation is carried out according to a model and a control target of the system, the control method has the advantages that the control quantity of the system at each moment can be obtained by carrying out once calculation, and the model predictive control is carried out by adopting a rolling type optimization method, namely on-line repeated control. During a certain control period, the controller calculates the optimal control quantity in the period according to a control target, wherein the most common control target is determined by the minimum value of the two-norm measure of the predictive performance. If the MPC controller is designed in raw space, a typical objective function is:
Figure BDA0002208163270000076
wherein
Figure BDA0002208163270000077
yk+jAnd Δ uk+j-1Respectively, set point output, process output and input delta, P is the length of the prediction horizon and M is the length of the control horizon, by minimizing the objective function, the optimal control action can be obtained.
If the MPC is designed in the original space, the objective function is described by the above equation, and using the PLS model, the correlation between the original objective function and the potential variable space objective function is:
Figure BDA0002208163270000081
the objective function in the original space is slightly smaller than the sum of the objective functions in the potential variable space, which means that control performance in the potential variable space can be sacrificed, in effect, because PLS reduces and decouples the system, potentially affecting the optimization of the MPC technique.
The identified GP model can be directly applied to MPC controller design, the prediction variance of the GP model can provide uncertainty information, and for the ith single loop, the objective function used in the simulation is as follows:
Figure BDA0002208163270000082
where Δ tiThe method is solved by minimizing a performance index function at each time point, wherein a gradient descent method is applied to minimize the performance index function, the predicted variance takes model uncertainty into account, and the variance information is taken into account in the optimization process, so that the method has better robustnessThe control system comprises the following specific steps:
Figure BDA0002208163270000083
where η is the step size of the gradient descent method, for ease of understanding, the expression of the difference between the revised desired output and the predicted output is described here:
Figure BDA0002208163270000084
the partial derivative term in the above equation can be extended as follows:
Figure BDA0002208163270000085
from the GP model, the following partial derivatives can be obtained:
Figure BDA0002208163270000086
Figure BDA0002208163270000087
further, the derivative of the covariance function is:
Figure BDA0002208163270000091
if it is not
Figure BDA0002208163270000092
Then
Figure BDA0002208163270000093
Obtaining:
Figure BDA0002208163270000094
solving for Δ t with a gradient descent can be obtainediExpression (c):
Figure BDA0002208163270000095
finally, for a separate loops of the MPC controller, the overall objective function is:
Figure BDA0002208163270000096
multivariable MPC control strategies can be implemented by estimating the optimal control action that minimizes the objective function to cause the output to track the set point, which can be directly applied to the control process to achieve the desired process variable.
The control method is adopted to control the specific industrial production, and the experimental contents and data are as follows:
a Continuous Stirred Tank Reactor (CSTR) is a widely used chemical reactor for realizing a polymerization reaction, plays a very important role in industrial production processes of petroleum, medicine, reagents, food, synthetic materials and the like, and is a typical highly nonlinear strongly coupled chemical reaction system in process industry; in order to verify the feasibility of the method of the present invention, a simulation study of the CSTR system using the above method will be performed.
The CSTR characteristic can be represented by the following system of continuous-time nonlinear differential equations:
Figure BDA0002208163270000097
Figure BDA0002208163270000098
in the formula, CaIs the equilibrium concentration of the product;
t is the reaction temperature;
Cafis the feed concentration;
q is the material flow;
Tfand TcfRespectively the material temperature and the coolant temperature;
Figure BDA0002208163270000101
V,p,pcand hA is a coefficient holding constant of the chemical reaction.
The above-mentioned one two-input two-output, strongly correlated and strongly nonlinear multivariable system.
In the experimental process, the number of samples of the system is 100, the sampling interval is 0.1s, the system is used as an industrial control system, Gaussian noise is added to the output of the system to simulate the actual situation, a dynamic GP-PLS prediction model is established offline and trained, the number of the principal elements is 2 (obtained by cross validation), the first 80 sample points are used as a training set, the last 20 sample points are used as a test set, and the predicted value generated by the GP-PLS prediction model is compared with the actual output value. After a GP-PLS prediction model is built in a hidden space, model prediction control is performed on a first pair of latent variables and a second pair of latent variables, respectively, with a prediction time domain P of 10 and a control time domain M of 2, and the experimental results are shown in fig. 3, 4, 5 and 6.
Experimental results show that for a two-input two-output nonlinear system with severe correlation, the GP-PLS prediction model has a good training effect, and the smaller the variance is, the better the fitting effect of the model is, namely the GP model can express the confidence coefficient of the model, which is a characteristic that other nonlinear models do not have.
Then MPC control is carried out on the two loops under the hidden space, and the original space is reconstructed, so that when given change of a certain output is carried out, the corresponding controlled quantity can well follow the given value, and for the other controlled quantity, only an interference effect is added, so that the two outputs of Ca and T of CSTR can be overcome and stabilized on the given value quickly, and the two outputs can be well controlled. The output value of the prediction model can well track the set value, and compared with the original space, the controller is simpler in design and shorter in calculation time, and the complexity of the prediction model training is reduced while a better prediction effect is obtained.
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present invention.

Claims (1)

1. A multivariate nonlinear dynamic system model predictive control method based on a Gaussian process model is characterized in that: the method comprises the following steps:
the method comprises the following steps: establishing an external dynamic partial least squares framework:
step 1.1: let two scaled datasets U and Y, where: u is an input data matrix with the size of nxm, and Y is an output data matrix with the size of nxl;
carrying out standardization on the data sets U and Y, wherein the mean value is 0, the variance is 1, and a scaling matrix W is obtainedxAnd Wy；
Step 1.2: obtaining a score matrix T and a load matrix P of a data set U, and a score matrix V and a load matrix Q of a data set Y by a partial least square external model;
step 1.3: let tiAnd viThe ith orthogonal column vector of the scoring matrix T and V respectively, and the T is obtainediAnd viBy v internal relations ofi＝fGP(ti) Establishing a non-linear internal model of the Gaussian process, and representing an algebraic relation between an input latent variable and an output latent variable, wherein f (t)i) Index score vector tiAnd viAn algebraic relationship of (c);
step 1.4: load vectors for data sets U and Y are calculated, respectively:
Figure FDA0003475601140000011
and
Figure FDA0003475601140000012
the formula is derived from a nonlinear iterative partial least squares algorithm, wherein:
Figure FDA0003475601140000013
and
Figure FDA0003475601140000014
transposes of the ith vectors of the loading matrices P and Q, respectively, i representing the number of loops;
tiand viThe ith orthogonal column vector of the scoring matrices T and V, respectively;
Figure FDA0003475601140000015
and
Figure FDA0003475601140000016
transpose of the ith orthogonal column vector of the scoring matrices T and V, respectively;
step 1.5: computing residual matrix Ea+1And Fa+1If the residual error meets the convergence condition or the principal component number reaches a set value, ending the algorithm to obtain a prediction model of the nonlinear system based on the Gaussian process and the partial least square method, and if the residual error does not meet the convergence condition or the principal component number reaches the set value, turning to the step 1.1;
step two: after a prediction model of a nonlinear system based on a Gaussian process and a partial least square method is obtained according to the algorithm, output data are predicted, and a multivariable system in an original space is decoupled through a nonlinear system dynamic model based on the Gaussian process and the partial least square method to obtain a plurality of single-input single-output systems in a hidden space;
step three: controlling by using a nonlinear system dynamic model based on a Gaussian process and a partial least square method, and designing a model prediction controller in each single-input single-output subsystem;
step four: by minimizing the objective function JikObtaining the optimal control action;
step five: and reconstructing the model prediction control result in the hidden space back to the original space through the correlation between the original space and the hidden space, and controlling the original space.
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