CN106844828A - Two dimension H is realized based on digraph opinion∞The method of wave filter FM II state-space models - Google Patents

Abstract

The invention discloses one kind based on the theoretical realization two dimension H of two-dimentional digraph_∞The method of wave filter FM models, the method using two-dimentional digraph construction transmission function proper polynomial, then obtains state matrix A using the arc collection and vertex set of two-dimentional digraph first₁,A₂, and then determine ψ matrixes, determine to realize matrix B by ψ matrixes₁,B₂.FM model methods are constructed compared to simple digraph, it is first local in calculating process to realize denominator, then entirety obtains FM realizations, solution matrix B₁,B₂, substantial amounts of matrix computations and complex array can be produced during C, so as to substantially increase difficulty in computation and complexity.The present invention is implemented in combination with FM models with ψ matrix methods by digraph is theoretical, and this method possesses that calculating is simple, and obtain realize that order of matrix number is relatively low.

Description

Two dimension H is realized based on digraph opinion∞The method of wave filter FM-II state-space models

Technical field

Two dimension H is realized the present invention relates to wave filter technology, more particularly to a kind of being discussed based on digraph_∞Wave filter FM-II shapes The method of state space model.

Background technology

Filtering plays very important effect in control theory research, and it is one of important data processing method.This The method of kind is to recover status signal from observation noise output signal is carried.Most widely used at present is H_∞Filtering, H_∞Filtering It is by H_∞Norm is applied in performance indications, and predicted state vector is carried out from observable signal.The H of uncertain system_∞Filtering Analysis Need to construct a multinomial or uncertain parameters in LFR.And passing through FM models, the indefinite model problems of LFR are in algebra On be equivalent to a realization for nD systems.Therefore, an efficient nD is realized to H_∞Filtering control theory suffers from major contribution.

Have the FM models of accomplished in many ways multidimensional Rational Transfer both at home and abroad at present.These methods can substantially divide It is three classes：One class first obtains local FM and realizes that obtaining overall FM again realizes.Given transfer function matrix is decomposed into and is comprised only One-dimensional monomial and with product, to each factor construction Linear Fractional LFR (Linear Fractional Represention), overall LFR is obtained using LFR coupling calculating volumes.Equations of The Second Kind is that direct entirety FM is realized.

The content of the invention

The technical problem to be solved in the present invention is for defect of the prior art, there is provided one kind is real based on digraph opinion Existing two dimension H_∞The method of wave filter FM-II state-space models.

The technical solution adopted for the present invention to solve the technical problems is：Two dimension H is realized based on digraph opinion_∞Wave filter The method of FM-II state-space models, comprises the following steps：

1) to the H of uncertain system_∞Filtering is analyzed, and builds two dimension H_∞The FM models of filtering, the two-dimentional H_∞Filtering FM models are as follows：

X (i, j)=A₁x(i-1,j)+A₂x(i,j-1)+B₁u(i-1,j)+B₂u(i,j-1) (1)

Y (i, j)=Cx (i, j)+Du (i, j) (2)

Wherein x (i, j) represents state vector, and u (i, j) represents external disturbance input, and y (i, j) represents controlled output；A₁, A₂,B₁,B₂, C, D are real number matrix, and the transmission function of system (1) and (2) is

Wherein d (z₁,z₂) represent proper polynomial, z₁,z₂Represent and postpone operation；I_rIt is r rank unit matrixs；

2) the Two-Dimensional Discrete Systems G (z for giving₁,z₂), by proper polynomial d (z₁,z₂) resolve into a series of list Item formula, then one digraph circulation of each monomial correspondence；

3) the digraph circulation for combining all monomials obtains proper polynomial d (z₁,z₂) circulation, by two-dimentional digraph Arc collection and vertex set obtain state matrix A₁,A₂；

4) using in ψ matrix methods：A₁=A₁₀+D_HT1C,A₂=A₂₀+D_HT2C, determines matrix A₁₀,A₂₀, by ψ=C (I- A₁₀z₁-A₂₀z₂)^-1, obtain ψ matrixes；

5) for transmission function molecule n (z₁,z₂)=ψ (B₁z₁+B₂z₂), accomplished matrix B₁,B₂。

By such scheme, step 2) in represent proper polynomial decompose monomial digraph circulation specific practice be, by d (z₁,z₂) decompose as follows：

To a monomial d_i(z₁,z₂) the digraph circulation of construction monomial, vertex set number M=degd_i(z₁,z₂), i= 1…p；Wherein degd_i(z₁,z₂) it is 2-d polynomial d (z₁,z₂) in monomial number of times maximum.

By such scheme, step 3) middle combination monomial digraph circulation d_i(z₁,z₂) obtain proper polynomial d (z₁,z₂) The detailed process of circular treatment is as follows：

One digraph D is made up of nonempty finite set V and S, and wherein V and S is respectively the vertex set and arc collection of digraph D, Each element of vertex set V is referred to as the summit of digraph D, and each element of S is referred to as the arc of digraph D；One two dimension has To figure D⁽²⁾=(V, S), wherein V={ v₁,v₂,...,v_nAnd S={ ξ₁,ξ₂Respectively represent digraph D vertex set and arc collection； Digraph D⁽²⁾Rank be summit in D number, digraph D⁽²⁾Scale be D⁽²⁾The number of middle arc；If from vertex v_iTo v_jIn the presence of (z₁,z₂) arc, then corresponding matrix (A₁,A₂) jth row, the i-th row (being its coefficient represented by correspondence arc) for 0, wherein i, J=1 ..., n.

By such scheme, step 3) in combination monomial digraph circulation follow following principle：It is multinomial that combination obtains feature The digraph of formula will not produce new circulation；All of monomial digraph meets at last summit；Proper polynomial is oriented The coefficient of the arc that last summit is gone out is characterized polynomial coefficient in figure.

The beneficial effect comprise that：One avoids complicated solving ψ matrix processes using two-dimentional digraph theory, and And n-D discrete system FM models can be generalized to by n dimension digraph theories.Secondly it is whole to make use of ψ matrixes to realize digraph Body realizes transmission function, it is to avoid complicated solving realizes the process of matrix.

Brief description of the drawings

Below in conjunction with drawings and Examples, the invention will be further described, in accompanying drawing：

Fig. 1 is the method flow diagram of the embodiment of the present invention；

Fig. 2 is monomial d₁(z₁,z₂),d₂(z₁,z₂),d₃(z₁,z₂),d₄(z₁,z₂),d₅(z₁,z₂) two-dimentional digraph；

Fig. 3 is D⁽²⁾Realize proper polynomial d (z₁,z₂) it is unsatisfactory for the schematic diagram of condition；

Fig. 4 is D⁽²⁾Realize proper polynomial d (z1, z₂) meet the schematic diagram of condition.

Specific embodiment

In order to make the purpose , technical scheme and advantage of the present invention be clearer, with reference to embodiments, to the present invention It is further elaborated.It should be appreciated that specific embodiment described herein is only used to explain the present invention, limit is not used to The fixed present invention.

According to reality digraph of the invention it is theoretical with_ψMatrix method method overall procedure such as Fig. 1.Comprise the following steps：

Step S1：Two-dimentional H_∞The transmission function of wave filter FM models

Step S2：By proper polynomial d (z₁,z₂) a series of monomial is resolved into, for each monomial correspondence one Individual digraph circulation.

The value of monomial：d₁(z₁,z₂)=a₁z₂, such as Fig. 2 (a)；d₂(z₁,z₂)=a₂z₂, such as Fig. 2 (b)；Such as Fig. 2 (c)；Such as Fig. 2 (d)；Such as Fig. 2 (e)；Correspondence 5 Digraph is circulated.Vertex set number M=degd_i(z₁,z₂), i=1 ... 5, then M_i=3.

Step S3：The digraph circulation of combination monomial is according to following condition：

1. combination obtains the digraph of proper polynomial and will not produce new circulation.

2. all of monomial digraph meets at last summit.

3. the coefficient of the arc that last summit is gone out is characterized polynomial coefficient in proper polynomial digraph.

According to vertex set number 3 in monomial, construction such as Fig. 3 proper polynomial d (z₁,z₂) circulation, from the figure 3, it may be seen that all lists The digraph circulation of item formula meets at last vertex v of summit₃Then meet condition 2.Digraph circulation number remains as 5 and meets bar Part 1, without the new circulation of generation (not having new monomial to produce).Each term coefficient a in proper polynomial_iNot all by most Latter summit to other summits arc coefficient determine (Coefficient by vertex v₁To v₂Arc coefficient determine), no Meet condition 3.Then by vertex set number M+1, construction such as Fig. 4 proper polynomial d (z₁,z₂) circulation.As shown in Figure 4, all monomials Digraph circulation meets at last vertex v of summit₄Then meet condition 2.Digraph circulation number remains as 5, new without producing Circulation then meet condition 1.Each term coefficient a in proper polynomial_iThe arc coefficient all gone out by last summit determines, then Meet condition 3.

It is hereby achieved that state matrix A in ψ matrix methods₁,A₂。

Step S4：Using in ψ matrix methods:A₁=A₁₀+D_HT1C,A₂=A₂₀+D_HT2C, determines matrix A₁₀,A₂₀, by ψ=C (I-A₁₀z₁-A₂₀z₂)-¹Obtain ψ matrixes.From formula (12)

Can be obtained by formula (9) and C=[0 00 1]

ψ=[z₁z₂ z₂ z₁1] and ψ z₁∪ψz₂Comprising all items of transmission function, then the ψ meets condition, otherwise returns to step Rapid S3 reconfigures monomial digraph.

Step S5：Using n (z₁,z₂)=ψ (B₁z₁+B₂z₂), accomplished matrix B₁,B₂。

Then B₁=[0 b₃ 0 b₁]^T B₂=[b₄ 0 0 b₂]^T。

It should be appreciated that for those of ordinary skills, can according to the above description be improved or be become Change, and all these modifications and variations should all belong to the protection domain of appended claims of the present invention.

Claims (4)

1. it is a kind of that two dimension H is realized based on digraph opinion_∞The method of wave filter FM-II state-space models, it is characterised in that including Following steps：

1) to the H of uncertain system_∞Filtering is analyzed, and builds two dimension H_∞The FM models of filtering, the two-dimentional H_∞The FM moulds of filtering Type is as follows：

X (i, j)=A₁x(i-1,j)+A₂x(i,j-1)+B₁u(i-1,j)+B₂u(i,j-1) (1)

Y (i, j)=Cx (i, j)+Du (i, j) (2)

Wherein x (i, j) represents state vector, and u (i, j) represents external disturbance input, and y (i, j) represents controlled output；A₁,A₂,B₁, B₂, C, D are real number matrix, and the transmission function of system (1) and (2) is

$G(z_1,z_2)=\frac{n(z_1,z_2)}{d(z_1,z_2)}=C{(I_r-\underset{i=1}{\overset{2}{\Σ}}{A}_i{z}_i)}^{-1}\underset{i=1}{\overset{2}{\Σ}}{B}_i{z}_i+D---\left(3\right)$

Wherein d (z₁,z₂) represent proper polynomial, z₁,z₂Represent and postpone operation；I_rIt is r rank unit matrixs；

$D=\underset{z_1\→0,z_2\→0}{\lim }G(z_1,z_2)---\left(4\right)$

2) the Two-Dimensional Discrete Systems G (z for giving₁,z₂), by proper polynomial d (z₁,z₂) a series of monomial is resolved into, Then one digraph circulation of each monomial correspondence；

3) the digraph circulation for combining all monomials obtains proper polynomial d (z₁,z₂) circulation, by two-dimentional digraph arc collection State matrix A is obtained with vertex set₁,A₂；

4) using in ψ matrix methods：A₁=A₁₀+D_HT1C,A₂=A₂₀+D_HT2C, determines matrix A₁₀,A₂₀, by ψ=C (I-A₁₀z₁- A₂₀z₂)^-1, obtain ψ matrixes；

5) for transmission function molecule n (z₁,z₂)=ψ (B₁z₁+B₂z₂), accomplished matrix B₁,B₂。

2. method according to claim 1, it is characterised in that the step 2) in represent the individual event that proper polynomial is decomposed Formula digraph circulation specific practice is, by d (z₁,z₂) decompose as follows：

To a monomial d_i(z₁,z₂) the digraph circulation of construction monomial, vertex set number M=degd_i(z₁,z₂), i=1 ... p； Wherein degd_i(z₁,z₂) it is 2-d polynomial d (z₁,z₂) in monomial number of times maximum.

3. method according to claim 1, it is characterised in that the step 3) in combination monomial digraph circulation d_i (z₁,z₂) obtain proper polynomial d (z₁,z₂) circular treatment detailed process it is as follows：

One digraph D is made up of nonempty finite set V and S, and wherein V and S is respectively the vertex set and arc collection of digraph D, summit Each element for collecting V is referred to as the summit of digraph D, and each element of S is referred to as the arc of digraph D；One two-dimentional digraph D⁽²⁾=(V, S), wherein V={ v₁,v₂,…,v_nAnd S={ ξ₁,ξ₂Respectively represent digraph D vertex set and arc collection；Digraph D⁽²⁾Rank be summit in D number, digraph D⁽²⁾Scale be D⁽²⁾The number of middle arc；If from vertex v_iTo v_jIn the presence of (z₁,z₂) Arc, then corresponding matrix (A₁,A₂) jth row, the i-th row are not 0, wherein i, j=1 ..., n.

4. method according to claim 1, it is characterised in that the step 3) in the digraph circulation of combination monomial follow Following principle：The digraph that combination obtains proper polynomial will not produce new circulation；All of monomial digraph meets at most Latter summit；The coefficient of the arc that last summit is gone out is characterized polynomial coefficient in proper polynomial digraph.