CN103176406B - A kind of continuous time filter converts the method for discrete time filter to - Google Patents

Abstract

The invention provides the method that continuous time filter converts discrete time filter to, sampling controlled system input quantity and output quantity, utilize identification algorithm to obtain the continuous time model of system; Continuous time filter is built according to continuous time model; According to the employing time of the coefficient in continuous time filter equation and discrete system, the method for partial differential equation is adopted to calculate discrete time filter matrix of coefficients; Build discrete time filter structure.This method carries out discrete transform to designing the continuous time filter obtained, obtain discrete time filter, compared with current discrete filter method for designing, decrease and the process of discretize is carried out to controlled device and carries out the process of design of filter based on discrete system, make use of the advantage based on continuous time system design of filter theory, design can not only design the wave filter obtaining excellent performance, and the systems design and development time can also be saved, be applicable to very much the practical implementation of digital control system.

Description

A kind of continuous time filter converts the method for discrete time filter to

Technical field

The present invention relates to industrial automation control system design field in iron and steel metallurgical industry, be specifically related to a kind of method that continuous time filter converts discrete time filter to.

Technical background

Classical control theory and modern control theory are all take continuous time model as research object, the design of filter theory of a lot of maturation also designs based on continuous time filter structure, and currently having entered the digital control epoch, the wave filter of form needed to convert discrete-time version to and just can be applied to reality continuous time.Current discrete time filter is designed with special method for designing, but the basis of those methods for designing is plant models of discrete time, and the form that filter characteristic design also must convert in discrete system, often make troubles for deviser.

Summary of the invention

The technical problem to be solved in the present invention is: provide a kind of continuous time filter to convert the method for discrete time filter to, neither affect the analysis to controlled device continuous time, can be applied in digital control system again by designing the wave filter obtained.

The present invention for solving the problems of the technologies described above taked technical scheme is: a kind of continuous time filter converts the method for discrete time filter to, it is characterized in that: it comprises the following steps:

S1, sampling controlled system input quantity and output quantity, utilize identification algorithm to obtain the continuous time model of system;

S2, according to continuous time model build continuous time filter;

S3, employing time according to the coefficient in continuous time filter equation and discrete system, the method for partial differential equation is adopted to calculate discrete time filter matrix of coefficients;

S4, structure discrete time filter structure.

By such scheme, in described step S1, continuous time model is $\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\ω}}w\left(t\right)\\ y\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ Wherein x (t) is the quantity of state of t system, represent the differential of x (t), u (t) is input quantity, y (t) is output quantity, and w (t) is 1 × 1 process random noise, and v (t) is for measuring random noise, A, B, C, B _ωfor State Equation Coefficients matrix, and A is n × n matrix, and B is n × 1 matrix, and C is 1 × n matrix, B _ωfor n × 1 matrix.

Further, in described step S2, the continuous time filter formula of structure is: wherein be the estimated value of x (t), K is filter gain matrix;

In described step S3, discrete time filter matrix of coefficients comprises AD, BD, KD,

Computing formula is respectively:

$\mathrm{AD}=e^{(A-K\×C)T_S},$ $\mathrm{BD}={\∫}_0^{T_S}{e}^{(A-\mathrm{KC})t}\mathrm{dt}\×B,$ $\mathrm{KD}={\∫}_0^{T_S}{e}^{(A-\mathrm{KC})t}\mathrm{dt}\×K,$

Wherein T _sfor the sampling time of discrete system, t is variable integral time;

The discrete time filter structure built in described step S4 is:

$\hat{x}(k+1)=\mathrm{AD}\×\hat{x}\left(k\right)+\mathrm{BD}\×u\left(k\right)+\mathrm{KD}\×y\left(k\right),$

Wherein for T _s× k moment system estimation state, k be greater than zero arithmetic number.

Beneficial effect of the present invention is: this method carries out discrete transform to designing the continuous time filter obtained, thus obtain discrete time filter, compared with current discrete filter method for designing, decrease and the process of discretize is carried out to controlled device and carries out the process of design of filter based on discrete system, make use of the advantage based on continuous time system design of filter theory, design can not only design the wave filter obtaining excellent performance, and the systems design and development time can also be saved, be applicable to very much the practical implementation of digital control system.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of one embodiment of the invention.

Fig. 2 is the first state estimation curve of reactive power compensator.

Fig. 3 is reactive power compensator the second state estimation curve.

Fig. 4 is reactive power compensator output reactive power curve and estimation curve of output.

Embodiment

Fig. 1 is the process flow diagram of one embodiment of the invention, and it comprises the following steps: S1, sampling controlled system input quantity and output quantity, utilizes identification algorithm to obtain the continuous time model of system; S2, according to continuous time model build continuous time filter; S3, employing time according to the coefficient in continuous time filter equation and discrete system, the method for partial differential equation is adopted to calculate discrete time filter matrix of coefficients; S4, structure discrete time filter structure.

In step S1, continuous time model is $\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\ω}}w\left(t\right)\\ y\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ Wherein x (t) is the quantity of state of t system, represent the differential of x (t), u (t) is input quantity, y (t) is output quantity, and w (t) is 1 × 1 process random noise, and v (t) is for measuring random noise, A, B, C, B _ωfor State Equation Coefficients matrix, and A is n × n matrix, and B is n × 1 matrix, and C is 1 × n matrix, B _ωfor n × 1 matrix.

In step S2, the continuous time filter formula of structure is: wherein be the estimated value of x (t), K is filter gain matrix.

In step S3, discrete time filter matrix of coefficients comprises AD, BD, KD,

Computing formula is respectively:

$\mathrm{AD}=e^{(A-K\×C)T_S},$ $\mathrm{BD}={\∫}_0^{T_S}{e}^{(A-\mathrm{KC})t}\mathrm{dt}\×B,$ $\mathrm{KD}={\∫}_0^{T_S}{e}^{(A-\mathrm{KC})t}\mathrm{dt}\×K,$

Wherein T _sfor the sampling time of discrete system, t is variable integral time.

The discrete time filter structure built in described step S4 is:

$\hat{x}(k+1)=\mathrm{AD}\×\hat{x}\left(k\right)+\mathrm{BD}\×u\left(k\right)+\mathrm{KD}\×y\left(k\right),$

Wherein for T _s× k moment system estimation state, k be greater than zero arithmetic number.

Below in conjunction with instantiation, the present invention will be further described.

Certain steel mill 6.5kV bus is connected to a TCR type reactive power compensator, every phase reactive inductor amount L=18.7mH, opened loop control is utilized to carry out Model Distinguish experiment to compensation system, sample the reactive power y (t) that thyristor control angle u (t) and system export in 1 second, and the sampling period is T _s=0.0001 second, continuous time state equation discrimination method is utilized to obtain reactive power compensator system model and be:

$\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=\left[\begin{array}{cc}-10.9025531534668& -188.7790221402286\\ 148.7801599579912& 2.4037216595325\end{array}\right]x\left(t\right)+\left[\begin{array}{c}-1856.51\\ -115.25\end{array}\right]u\left(t\right)+\left[\begin{array}{c}-0.0017876\\ -0.0117144\end{array}\right]w\left(t\right)\\ y\left(t\right)=\begin{array}{}\end{array}\left[\begin{array}{cc}21765000& -4458700\end{array}\right]x\left(t\right)+v\left(t\right)\end{array}\right.$

Can obtain Kalman Bush filter gain matrix K according to Design on Kalman Filter method is:

$K=\left[\begin{array}{c}-0.001638768022482\\ -0.011733290026127\end{array}\right]$

Then construct Kalman Bush wave filter according to step S2, its expression formula is:

$\stackrel{\·}{\hat{x}}\left(t\right)=\left[\begin{array}{cc}-10.9025531534668& -188.7790221402286\\ 148.7801599579912& 2.4037216595325\end{array}\right]\hat{x}\left(t\right)+\left[\begin{array}{c}-1856.51\\ -115.25\end{array}\right]u\left(t\right)$ $+\left[\begin{array}{c}-0.001638768022482\\ -0.011733290026127\end{array}\right](y\left(t\right)-\left[\begin{array}{cc}21765000& -4458700\end{array}\right]\hat{x}\left(t\right)).$

Above-mentioned Kalman Bush wave filter abbreviation can be obtained:

$\stackrel{\·}{\hat{x}}\left(t\right)=\left[\begin{array}{cc}35656.9& -7495.6\\ 255524.1& -52312.9\end{array}\right]\hat{x}\left(t\right)+\left[\begin{array}{c}-1856.51\\ -115.25\end{array}\right]u\left(t\right)+\left[\begin{array}{c}-0.001638768022482\\ -0.011733290026127\end{array}\right]y\left(t\right).$

Above-mentioned matrix $A-\mathrm{KC}=\left[\begin{array}{cc}35656.9& -7495.6\\ 255524.1& -52312.9\end{array}\right],$ Then utilize step S3 design factor matrix $\mathrm{AD}=e^{{\mathrm{AT}}_S}=\left[\begin{array}{cc}2.452598527193898& -0.336557186227178\\ 11.473252509925388& -1.497320942505076\end{array}\right];$

Matrix of coefficients BD:

$\mathrm{BD}={\∫}_0^{T_S}{e}^{(A-\mathrm{KC})t}\mathrm{dt}\×B=\left[\begin{array}{c}-0.36920844550756081862953751393513\\ -1.4017445244134212603146452327079\end{array}\right];$

Matrix of coefficients KD:

$\mathrm{KD}={\∫}_0^{T_S}{e}^{(A-\mathrm{KC})t}\mathrm{dt}\×K={10}^{-6}\×\left[\begin{array}{c}-0.066992584219293\\ -0.527939339286366\end{array}\right].$

On the basis obtaining above-mentioned coefficient of dispersion matrix A D, BD, KD, just can construct the Kalman Bush wave filter of discrete-time version, its structure is as follows:

$\hat{x}(k+1)=\left[\begin{array}{cc}2.452598527193898& -0.336557186227178\\ 11.473252509925388& -1.497320942505076\end{array}\right]\hat{x}\left(k\right)$ $+\left[\begin{array}{c}-0.36920844550756081862953751393513\\ -1.4017445244134212603146452327079\end{array}\right]u\left(k\right)+{10}^{-6}\×\left[\begin{array}{c}-0.066992584219293\\ -0.527939339286366\end{array}\right]y\left(k\right).$

In order to verify the correctness of the discrete time filter conversion method that the present invention proposes, the above-mentioned discrete form wave filter be converted to is substituted into real system herein to test, obtain 2 kinds of state estimation curves as shown in accompanying drawing 2 and accompanying drawing 3, due to estimate system state cannot and the state of reality between contrast, in order to verify the error between observed reading and actual value, the estimation calculated according to estimated state is exported export with reality and contrast, its curve as shown in Figure 4.

Can find from accompanying drawing 4, almost overlap between the output valve that estimation obtains and the output valve of reality, be enough to show that the Kalman Bush wave filter of discretize has good estimation effect, have the state observation performance obtained based on continuous time system design completely, the continuous filter demonstrating the present invention's proposition converts correctness and the science of the method for discrete time filter to.

Claims (1)

1. continuous time filter converts a method for discrete time filter to, it is characterized in that: it comprises the following steps:

S1, sampling controlled system input quantity and output quantity, utilize identification algorithm to obtain the continuous time model of system;

S2, according to continuous time model build continuous time filter;

S3, sampling time according to the coefficient in continuous time filter equation and discrete system, the method for partial differential equation is adopted to calculate discrete time filter matrix of coefficients;

S4, structure discrete time filter structure;

In described step S1, continuous time model is $\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\ω}}w\left(t\right)\\ y\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ Wherein x (t) is the quantity of state of t system, represent the differential of x (t), u (t) is input quantity, y (t) is output quantity, and w (t) is 1 × 1 process random noise, and v (t) is for measuring random noise, A, B, C, B _ωfor State Equation Coefficients matrix, and A is n × n matrix, and B is n × 1 matrix, and C is 1 × n matrix, B _ωfor n × 1 matrix;

In described step S2, the continuous time filter formula of structure is: $\stackrel{\stackrel{\·}{^}}{x}\left(t\right)=A\hat{x}\left(t\right)+\mathrm{Bu}\left(t\right)+K(y\left(t\right)-C\hat{x}\left(t\right)),$ Wherein be the estimated value of x (t), K is filter gain matrix;

In described step S3, discrete time filter matrix of coefficients comprises AD, BD, KD,

Computing formula is respectively:

$\mathrm{AD}=e^{(A-K\×C)T_s},\mathrm{BD}={\∫}_0^{T_s}{e}^{(A-\mathrm{KC})t}\mathrm{dt}\×B,\mathrm{KD}={\∫}_0^{T_s}{e}^{(A-\mathrm{KC})t}\mathrm{dt}\×K,$

Wherein T _sfor the sampling time of discrete system, t is variable integral time;

The discrete time filter structure built in described step S4 is:

$\hat{x}(k+1)=\mathrm{AD}\×\hat{x}\left(k\right)+\mathrm{BD}\×u\left(k\right)+\mathrm{KD}\×y\left(k\right),$

Wherein for T _s× k moment system estimation state, k be greater than zero arithmetic number.