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

Abstract

The invention provides a method for converting a continuous time filter into a discrete time filter, sampling the input and output of the controlled system, and using the identification algorithm to obtain the continuous time model of the system; constructing a continuous time filter according to the continuous time model; The coefficients in the filter equation and the adoption time of the discrete system are calculated by the partial differential equation method to calculate the discrete-time filter coefficient matrix; the discrete-time filter structure is constructed. This method performs discrete transformation on the designed continuous-time filter to obtain a discrete-time filter. Compared with the current discrete filter design method, it reduces the process of discretization of the controlled object and the filter based on the discrete system. The design process takes advantage of the continuous-time system filter design theory. The design can not only design a filter with excellent performance, but also save system design and development time, which is very suitable for the practical application of digital control systems.

Description

A Method of Converting Continuous Time Filters to Discrete Time Filters

technical field

The invention relates to the design field of industrial automatic control systems in the iron and steel metallurgy industry, in particular to a method for converting a continuous time filter into a discrete time filter.

technical background

Both classical control theory and modern control theory take the continuous-time model as the research object. Many mature filter design theories are also designed based on the continuous-time filter structure. Now that we have entered the era of digital control, continuous-time filters require Converting to discrete time form can be applied in practice. At present, there are special design methods for discrete-time filter design, but the basis of those design methods is the discrete-time controlled object model, and the filter characteristic design must also be converted into the form in the discrete system, which often brings inconvenient.

Contents of the invention

The technical problem to be solved by the present invention is to provide a method for converting a continuous-time filter into a discrete-time filter, which does not affect the analysis of the controlled object in continuous time, and can apply the designed filter to a digital control system middle.

The technical scheme that the present invention takes for solving the above-mentioned technical problem is: a kind of continuous time filter is converted into the method for discrete time filter, it is characterized in that: it comprises the following steps:

S1. Sampling the input and output of the controlled system, and using the identification algorithm to obtain the continuous time model of the system;

S2. Constructing a continuous-time filter according to the continuous-time model;

S3. According to the coefficients in the continuous-time filter equation and the adoption time of the discrete system, the discrete-time filter coefficient matrix is calculated by the method of partial differential equations;

S4. Construct a discrete-time filter structure.

According to the above scheme, in the step S1, the continuous time model is $\left\{\begin{array}{c}\stackrel{\&Center\; Dot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\ω}}w\left(t\right)\\ \mathrm{the\; y}\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ where x(t) is the state quantity of the system at time t, Indicates the differential of x(t), u(t) is the input quantity, y(t) is the output quantity, w(t) is 1×1 process random noise, v(t) is measurement random noise, A, B, C , B _ω is the state equation coefficient matrix, and A is an n×n matrix, B is an n×1 matrix, C is a 1×n matrix, and B _ω is an n×1 matrix.

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

In the step S3, the discrete-time filter coefficient matrix includes AD, BD, KD,

The calculation formulas are:

$\mathrm{<span\; class="notranslate">\; AD</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; KC</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; KD</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; KC</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; ,</span>$

Where T _S is the sampling time of the discrete system, and t is the integration time variable;

The discrete-time filter structure constructed in the described step S4 is:

$\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; AD</span>}<span\; class="notranslate">\; \&times;</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; KD</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

in Estimated state of the system at time T _S ×k, k is a positive real number greater than zero.

The beneficial effects of the present invention are: the method performs discrete transformation on the designed continuous-time filter to obtain a discrete-time filter. Process and process of filter design based on discrete system, taking advantage of the advantages of filter design theory based on continuous time system, the design can not only design a filter with excellent performance, but also save system design and development time, very suitable for digital control System engineering practical application.

Description of drawings

Fig. 1 is a flowchart of an embodiment of the present invention.

Fig. 2 is the first state estimation value curve of the reactive power compensation device.

Fig. 3 is the second state estimation value curve of the reactive power compensation device.

Fig. 4 is the output reactive power curve and the estimated output curve of the reactive power compensation device.

Detailed ways

Fig. 1 is the flow chart of an embodiment of the present invention, and it comprises the following steps: S1, sample controlled system input quantity and output quantity, utilize the identification algorithm to obtain the continuous time model of system; S2, construct continuous time filter according to continuous time model ; S3. According to the coefficients in the continuous-time filter equation and the adoption time of the discrete system, the discrete-time filter coefficient matrix is calculated by the method of partial differential equations; S4. The discrete-time filter structure is constructed.

In step S1, the continuous time model is $\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\&omega;}}w\left(t\right)\\ \mathrm{the\; y}\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ where x(t) is the state quantity of the system at time t, Indicates the differential of x(t), u(t) is the input quantity, y(t) is the output quantity, w(t) is 1×1 process random noise, v(t) is measurement random noise, A, B, C , B _ω is the state equation coefficient matrix, and A is an n×n matrix, B is an n×1 matrix, C is a 1×n matrix, and B _ω is an n×1 matrix.

In step S2, the continuous time filter formula constructed is: in is the estimated value of x(t), and K is the filter gain matrix.

In step S3, the discrete-time filter coefficient matrix includes AD, BD, KD,

The calculation formulas are:

$\mathrm{<span\; class="notranslate">\; AD</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; KC</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; KD</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; KC</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; ,</span>$

Where T _S is the sampling time of the discrete system, and t is the integration time variable.

The discrete-time filter structure constructed in the described step S4 is:

$\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; AD</span>}<span\; class="notranslate">\; \&times;</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; KD</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

in Estimated state of the system at time T _S ×k, k is a positive real number greater than zero.

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

A TCR-type reactive power compensation device is connected to a 6.5kV bus in a steel factory. The reactance and inductance of each phase is L=18.7mH. The open-loop control is used to conduct a model identification experiment on the compensation device, and the thyristor control angle u(t ) and the reactive power y(t) output by the system, the sampling period is T _S =0.0001 second, and the system model of the reactive power compensation device obtained by using the continuous time state equation identification method is:

$\left\{\begin{array}{c}\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{cc}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10.9025531534668</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 188.7790221402286</span>}\\ \mathrm{<span\; class="notranslate">\; 148.7801599579912</span>}& \mathrm{<span\; class="notranslate">\; 2.4037216595325</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1856.51</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 115.25</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.0017876</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.0117144</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\begin{array}{}\end{array}\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 21765000</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4458700</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\end{array}\right.$

According to the Kalman filter design method, the Kalman-Busch filter gain matrix K can be obtained as:

$\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; 001638768022482</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.011733290026127</span>}\end{array}\right]$

Then construct the Kalman-Busch filter according to step S2, and its expression is:

$\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{cc}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 10.9025531534668</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 188.7790221402286</span>}\\ \mathrm{<span\; class="notranslate">\; 148.7801599579912</span>}& \mathrm{<span\; class="notranslate">\; 2.4037216595325</span>}\end{array}\right]\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1856.51</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 115.25</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>$ $<span\; class="notranslate">\; +</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.001638768022482</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.011733290026127</span>}\end{array}\right]<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 21765000</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4458700</span>}\end{array}\right]\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$

Simplify the Kalman-Busch filter above to get:

$\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 35656.9</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 7495.6</span>}\\ \mathrm{<span\; class="notranslate">\; 255524.1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 52312.9</span>}\end{array}\right]\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1856.51</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 115.25</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.001638768022482</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.011733290026127</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$

the above matrix $A-\mathrm{KC}=\left[\begin{array}{cc}35656.9& -7495.6\\ 255524.1& -52312.9\end{array}\right],$ Then use step S3 to calculate the coefficient matrix $\mathrm{AD}=e^{{\mathrm{AT}}_S}=\left[\begin{array}{cc}2.452598527193898& -0.336557186227178\\ 11.473252509925388& -1.497320942505076\end{array}\right];$

Coefficient matrix BD:

$\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; KC</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.36920844550756081862953751393513</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1.4017445244134212603146452327079</span>}\end{array}\right]<span\; class="notranslate">\; ;</span>$

Coefficient matrix KD:

$\mathrm{<span\; class="notranslate">\; KD</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; KC</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; 10</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; \&times;</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.066992584219293</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.527939339286366</span>}\end{array}\right]<span\; class="notranslate">\; .</span>$

On the basis of obtaining the above discrete coefficient matrices AD, BD, KD, the Kalman-Busch filter in discrete time form can be constructed, and its structure is as follows:

$\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 2.452598527193898</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.336557186227178</span>}\\ \mathrm{<span\; class="notranslate">\; 11.473252509925388</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1.497320942505076</span>}\end{array}\right]\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>$ $<span\; class="notranslate">\; +</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.36920844550756081862953751393513</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1.4017445244134212603146452327079</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; 10</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; \&times;</span>\left[\begin{array}{c}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.066992584219293</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.527939339286366</span>}\end{array}\right]\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$

In order to verify the correctness of the discrete-time filter conversion method proposed by the present invention, this paper substitutes the discrete-time filter obtained by the above conversion into the actual system for testing, and obtains two state estimation curves as shown in accompanying drawings 2 and 3 , since the estimated system state cannot be compared with the actual state, in order to verify the error between the observed value and the actual value, the estimated output calculated according to the estimated state Compared with the actual output, the curve is shown in Figure 4.

From Figure 4, it can be found that the estimated output value almost coincides with the actual output value, which is enough to show that the discretized Kalman-Busch filter has a good estimation effect, which is completely consistent with that obtained based on the continuous-time system design. The state observation performance verifies the correctness and scientificity of the method for converting the continuous filter into the discrete time filter proposed by the present invention.

Claims (1)

1. A method for converting a continuous-time filter into a discrete-time filter, characterized in that: it may further comprise the steps: S1. Sampling the input and output of the controlled system, and using the identification algorithm to obtain the continuous time model of the system; S2. Construct a continuous-time filter according to the continuous-time model; S3. According to the coefficients in the continuous-time filter equation and the sampling time of the discrete system, the discrete-time filter coefficient matrix is calculated by the method of partial differential equations; S4, constructing a discrete-time filter structure; In the step S1, the continuous time model is $\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{\mathrm{\&omega;}}w\left(t\right)\\ \mathrm{the\; y}\left(t\right)=\mathrm{Cx}\left(t\right)+v\left(t\right)\end{array}\right.,$ where x(t) is the state quantity of the system at time t, Indicates the differential of x(t), u(t) is the input quantity, y(t) is the output quantity, w(t) is 1×1 process random noise, v(t) is measurement random noise, A, B, C , B _ω is the state equation coefficient matrix, and A is an n×n matrix, B is an n×1 matrix, C is a 1×n matrix, and B _ω is an n×1 matrix; In the described step S2, the continuous time filter formula of construction is: $\stackrel{\stackrel{\&Center\; Dot;}{^}}{x}\left(t\right)=A\hat{x}\left(t\right)+\mathrm{Bu}\left(t\right)+K(\mathrm{the\; y}\left(t\right)-C\hat{x}\left(t\right)),$ in is the estimated value of x(t), K is the filter gain matrix; In the step S3, the discrete-time filter coefficient matrix includes AD, BD, KD, The calculation formulas are: $\mathrm{<span\; class="notranslate">\; AD</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; KC</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; KD</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; KC</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; ,</span>$ where T _s is the sampling time of the discrete system, and t is the integration time variable; The discrete-time filter structure constructed in the described step S4 is: $\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; AD</span>}<span\; class="notranslate">\; \&times;</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; KD</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$ in Estimated state of the system at time T _s ×k, k is a positive real number greater than zero.