CN103176406A - Method for transforming continuous-time filter to discrete-time filter - Google Patents

Abstract

The invention provides a method for transforming a continuous-time filter to a discrete-time filter. The method includes: sampling input quantity and output quantity of a controlled system, and utilizing an identification algorithm for obtaining a continuous-time model of the system; constructing the continuous-time filter according to the continuous-time model; adopting a partial differential equation method to compute a coefficient matrix of the discrete-time filter according to coefficients in a continuous-time filter equation and adopted time of a discrete system; and constructing a discrete-time filter structure. The discrete-time filter is obtained by performing discrete transformation to the designed continuous-time filter. Compared with an existing discrete-time filter design method, the method has the advantages that discretization processes of a controlled object and filter designing processes based on the discrete system are decreased, and by utilizing advantages of a filter design theory based on a continuous-time system, the filter excellent in performances can be designed, system design and development time can be saved, and the method is particularly suitable for practical engineering application of digital control systems.

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, a lot of ripe design of filter theories also are based on the continuous time filter structure and design, and currently having entered the digital control epoch, the wave filter of form need to convert the discrete time form to and just can be applied to reality continuous time.Discrete time filter is designed with special method for designing at present, but the basis of those methods for designing is plant models of discrete time, and filter characteristic design also must convert the form in discrete system to, often makes troubles for the deviser.

Summary of the invention

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

The present invention solves the problems of the technologies described above the technical scheme of taking to be: 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, build continuous time filter according to continuous time model;

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

S4, structure discrete time filter structure.

Press such scheme, in described step S1, continuous time model is $\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\ω}}w\left(t\right)\\ y\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ The quantity of state of x (t) etching system when being t wherein, The differential of expression x (t), u (t) are that input quantity, y (t) are output quantity, and w (t) is 1 * 1 process random noise, and v (t) is for measuring random noise, A, B, C, B _ωBe the State Equation Coefficients matrix, and A is n * n matrix, B is n * 1 matrix, and C is 1 * n matrix, B _ωBe 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 the filter gain matrix;

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

Computing formula is respectively:

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

T wherein _SBe the sampling time of discrete system, t is variable integral time;

The discrete time filter structure that builds 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

Be T _S* k system estimation constantly state, k are the arithmetic number greater than zero.

Beneficial effect of the present invention is: this method is carried out discrete transform to designing the continuous time filter that obtains, thereby obtain discrete time filter, compare with current discrete filter method for designing, reduced controlled device is carried out the process of discretize and carried out the process of design of filter based on discrete system, utilized the advantage based on continuous time system design of filter theory, design not only can design the wave filter that obtains excellent performance, and can also save the systems design and development time, be fit to very much the practical implementation of digital control system.

Description of drawings

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

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

Fig. 3 is reactive power compensator the second state estimation value 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, and utilize identification algorithm to obtain the continuous time model of system; S2, build continuous time filter according to continuous time model; S3, according to the coefficient in the continuous time filter equation, and employing time of discrete system, adopt the method for partial differential equation to calculate the discrete time filter matrix of coefficients; S4, structure discrete time filter structure.

In step S1, continuous time model is $\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\ω}}w\left(t\right)\\ y\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ The quantity of state of x (t) etching system when being t wherein,

The differential of expression x (t), u (t) are that input quantity, y (t) are output quantity, and w (t) is 1 * 1 process random noise, and v (t) is for measuring random noise, A, B, C, B _ωBe the State Equation Coefficients matrix, and A is n * n matrix, B is n * 1 matrix, and C is 1 * n matrix, B _ωBe n * 1 matrix.

In step S2, the continuous time filter formula of structure is:

Wherein

Be the estimated value of x (t), K is the filter gain matrix.

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

Computing formula is respectively:

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

T wherein _SBe the sampling time of discrete system, t is variable integral time.

The discrete time filter structure that builds 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

Be T _S* k system estimation constantly state, k are the arithmetic number greater than zero.

The present invention will be further described below in conjunction with instantiation.

Be connected to a TCR type reactive power compensator on certain 6.5kV of steel mill bus, every phase reactance inductance value L=18.7mH, utilize open loop to control compensation system is carried out the Model Distinguish experiment, the reactive power y (t) of sampling interior thyristor control angle u (t) in 1 second and system's output, the sampling period is T _S=0.0001 second, utilize state equation discrimination method acquisition continuous time reactive power compensator system model to 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 get Kalman Bush filter gain matrix K according to the Design on Kalman Filter method is:

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

According to step S2 structure Kalman Bush wave filter, 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 got:

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

Above-mentioned matrix $A-\mathrm{KC}=\left[\begin{array}{cc}35656.9& -7495.6\\ 255524.1& -52312.9\end{array}\right],$ 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 that obtains above-mentioned coefficient of dispersion matrix A D, BD, KD, just can construct the Kalman Bush wave filter of discrete time form, 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, this paper tests the above-mentioned discrete form wave filter substitution real system that is converted to, 2 kinds of state estimation curves as shown in accompanying drawing 2 and accompanying drawing 3 have been obtained, due to the system state of estimating can't and the state of reality between compare, in order to verify the error between observed reading and actual value, will export according to the estimation that estimated state calculates

Contrast with reality output, its curve as shown in Figure 4.

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

Claims (3)

1. a continuous time filter converts the method for discrete time filter to, and 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, build continuous time filter according to continuous time model;

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

S4, structure discrete time filter structure.

2. continuous time filter according to claim 1 converts the method for discrete time filter to, it is characterized in that: in described step S1, continuous time model is $\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\ω}}w\left(t\right)\\ y\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ The quantity of state of x (t) etching system when being t wherein,

The differential of expression x (t), u (t) are that input quantity, y (t) are output quantity, and w (t) is 1 * 1 process random noise, and v (t) is for measuring random noise, A, B, C, B _ωBe the State Equation Coefficients matrix, and A is n * n matrix, B is n * 1 matrix, and C is 1 * n matrix, B _ωBe n * 1 matrix.

3. continuous time filter according to claim 2 converts the method for discrete time filter to, it is characterized in that: in described step S2, the continuous time filter formula of structure is: Wherein

Be the estimated value of x (t), K is the filter gain matrix;

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

Computing formula is respectively:

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

T wherein _SBe the sampling time of discrete system, t is variable integral time;

The discrete time filter structure that builds 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

Be T _S* k system estimation constantly state, k are the arithmetic number greater than zero.

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