CN102929141B - Aircraft time lag varying model approximation and controller designing method - Google Patents

Abstract

The invention discloses an aircraft time lag varying model approximation and controller designing method, and aims to solve the technical problem that the existing robust control theory is lack of the design steps and an aircraft controller cannot be directly designed. The technical scheme is as follows: the robust stability solvable condition of a time varying approximation system is given, a closed loop expecting pole fed back by a linear system state is directly utilized for selection, and a given conditional inequality is given to design a feedback matrix directly according to the characteristic that the real part of all closed loop expecting poles are negative numbers. By utilizing the method, the aircraft model containing time lag varying uncertainty are obtained by engineering technicians in the research field during wind tunnel or flight test, and an aircraft controller can be designed directly, so that the technical problem in the current research that only the robust stability inequality is given and the aircraft controller cannot be designed directly is solved.

Description

Aircraft time-varying model time lag approaches and controller design method

Technical field

The present invention relates to a kind of controller design method, particularly relate to a kind of aircraft time-varying model time lag and approach and controller design method.

Background technology

Aircraft robust control is one of emphasis problem of current international airline circle research, when high performance airplane Controller gain variations, must consider robust stability and kinds of robust control problems; Practical flight device model is the non-linear differential equation of very complicated Unknown Model structure, and in order to describe the non-linear of this complexity, people adopt wind-tunnel and flight test to obtain the test model described by discrete data usually; In order to reduce risks and reduce experimentation cost, usually carry out flight maneuver test according to differing heights, Mach number, like this, the discrete data describing aircraft test model is not a lot, and this model is very practical to the good aircraft of static stability.But the modern and following fighter plane all relaxes restriction to static stability to improve " agility ", and fighter plane requires to work near open loop critical temperature rise usually; So just require that flight control system can transaction module uncertain problem well; Following subject matter to be considered: test is obtained a certain approximate model of discrete data and describes by (1), there is Unmarried pregnancy in model in practical flight Control System Design; (2) wind tunnel test can not be carried out full scale model free flight, there is constraint, the input action selections of the selection of flight test discrete point, initial flight state, maneuvering flight etc. can not, by all non-linear abundant excitations, adopt System Discrimination gained model to there is various error; (3) flight environment of vehicle and experimental enviroment are had any different, and flow field change and interference etc. make actual aerodynamic force, moment model and test model have any different; (4) there is fabrication tolerance in execution unit and control element, also there is the phenomenons such as aging, wearing and tearing in system operation, not identical with the result of flight test; (5) in Practical Project problem, need controller fairly simple, reliable, usually need to simplify with being mathematics model person, remove the factor of some complexity; Therefore, when studying the control problem of present generation aircraft, just robustness problem must be considered; Particularly the aircraft angle of attack, yaw angle measures and much physics, in chemical process the time lag of various degrees uncertain, if ignore these time lags in the analysis of system or design process, just may there is the result of mistake or cause the instability of system.

After 1980, carry out the control theory research of multiple uncertain system in the world, the H-infinit particularly proposed by Canadian scholar Zames is theoretical, Zames thinks, based on the LQG method of state-space model, why robustness is bad, mainly because represent that uncertain interference is unpractical with White Noise Model; Therefore, when supposing that interference belongs to a certain known signal collection, Zames proposes by the norm of its corresponding sensitivity function as index, design object is under contingent worst interference, make the error of system be issued to minimum in this norm meaning, thus AF panel problem is converted into solve closed-loop system is stablized; From then on, lot of domestic and international scholar expands the research of H-infinit control method; At aeronautical chart, the method is in the exploratory stage always, U.S. NASA, and the states such as German aerospace research institute, Holland are all studied robust control method, achieves a lot of emulation and experimental result; Domestic aviation universities and colleges have also carried out a series of research to aircraft robust control method, as document (Shi Zhongke, Wu Fangxiang etc., " robust control theory ", National Defense Industry Press, in January, 2003; Su Hongye. " robust control basic theory ", Science Press, in October, 2010) introduce, but these results and the distance of practical application also differ very large, are difficult to directly design practical flight controller and apply; Particularly a lot of research only gives uncertain time delay system Robust Stability according to Lyapunov theorem, but relate to less for the problem such as existence condition of these solution of inequality, specific implementation robust Controller Design time lag step can not be obtained, there is no to solve the technical matters of directly design robust flight controller.

Summary of the invention

Being difficult to directly design the technical deficiency of flight controller in order to overcome existing robust control theory shortage design procedure, the invention provides a kind of aircraft time-varying model time lag and approaching and controller design method; This method provide the piecewise approximation design conditions of real time delayed time-varying system Robust Stability Controller, the closed loop of State Feedback for Linear Systems is directly utilized to expect the selection of poles, and expect that the real part of limit is all the feature of negative according to all closed loops, give qualifications inequality direct design of feedback matrix, what can obtain wind-tunnel or flight test directly designs flight controller containing time-varying Hurst index dummy vehicle time lag, solves current research and only provides robust stability inequality and the technical matters that directly cannot design flight controller.

The technical solution adopted for the present invention to solve the technical problems is: a kind of aircraft time-varying model time lag approaches and controller design method, is characterized in comprising the following steps:

Step one, obtained by wind-tunnel or flight test under assigned altitute, Mach number condition containing probabilistic aircraft time-varying model time lag be:

$\stackrel{\·}{x}\left(t\right)=[A_0\left(t\right)+\mathrm{\Δ}{A}_0\left(t\right)]x+A_{\mathrm{\τ}}\left(t\right)x(t-\mathrm{\τ})+B\left(t\right)u\left(t\right)---\left(1\right)$

In formula, x (t) ∈ R ⁿ, u (t) ∈ R ^mbe respectively state and input vector, A ₀(t), A _τ(t), B (t) for known constant coefficient matrix, τ be the unknown delays time, Δ A ₀t () is matrix of coefficients unknown portions; Symbol is identical in full;

Simultaneous $\left\{\begin{array}{c}N(t,\mathrm{\τ})=\mathrm{\Φ}(t-\mathrm{\τ},t,\mathrm{\τ})\\ \min {\{x_{\mathrm{real}}\left(t\right)-\mathrm{\Φ}(t,t_0,\mathrm{\τ})x\left(t_0\right)-{\∫}_{t_0}^t\mathrm{\Φ}(t,\mathrm{\ζ},\mathrm{\τ})B\left(\mathrm{\ζ}\right)u\left(\mathrm{\ζ}\right)\mathrm{d\ζ}\}}^2\end{array}\right.$ Iterative N (t, τ),

In formula: Φ (t, t ₀, τ) and be state-transition matrix

$\stackrel{\·}{\mathrm{\Φ}}(t,t_0,\mathrm{\τ})=[A_0\left(t\right)+A_{\mathrm{\τ}}\left(t\right)N(t,\mathrm{\τ})]\mathrm{\Φ}(t,t_0,\mathrm{\τ})$

X _realt () is real system condition responsive;

After of equal value, the matrix of coefficients of system is

A(t)=A ₀(t)+A _τ(t)N(t，τ)+ΔA(t)

In formula: Δ A (t) is matrix of coefficients unknown portions;

According to known A (t), the variation range classification of B (t), is namely expressed as A (t), B (t) in different time sections:

$\left\{\begin{array}{c}A\left(t\right)=A_{0i}+{\mathrm{\&Delta;A}}_{0i}\\ B\left(t\right)=B_{0i}+{\mathrm{\&Delta;B}}_{0i}\end{array}\right.,t_{\mathrm{ij}}\&le;{t<t}_{\mathrm{ij}}+T_{\mathrm{ij}}(i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,r,j=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,p)$

In formula, A _0i, B _0ifor known constant matrices, Δ A _0iΔ B _0ifor unknown matrix, t _ij, T _ijfor time constant, r, p are positive integer, and i, j are that the A (t) of subscript, different time sections, B (t) expression formula form can be identical; In time period t _ij≤ t<t _ij+ T _ijin, flight controller is: u (t)=-K _ix (t)

In formula, K _ifor constant feedback matrix;

Bring in (1) formula, have:

$\stackrel{\&CenterDot;}{x}\left(t\right)=[(A_{0i}-B_{0i}{K}_i)+({\mathrm{\&Delta;A}}_{0i}-\mathrm{\&Delta;}{B}_{0i}{K}_i)]x\left(t\right)+\mathrm{\&Delta;}{A}_{\mathrm{\&tau;}}\left(t\right)x(t-\mathrm{\&tau;})$

In formula: Δ A _τt () is nubbin unknown after system equivalence;

Step 2, choose (A _0i-B _0ik _0i) eigenwert different and real part is negative, design of feedback matrix K _imake to satisfy condition:

${\mathrm{\&Lambda;}}_i>M_i^T[{(\mathrm{\&Delta;}{A}_{0i}-\mathrm{\&Delta;}{B}_{0i}{K}_i)}^T{M}_i^{-T}{M}_i^{-1}(\mathrm{\&Delta;}{A}_{0i}-\mathrm{\&Delta;}{B}_{0i}{K}_i)+\mathrm{\&Delta;}{A}_{\mathrm{\&tau;}}^T\left(t\right)M_i^{-T}{M}_i^{-1}\mathrm{\&Delta;}{A}_{\mathrm{\&tau;}}\left(t\right)]M_i;$

This controller makes

$\stackrel{\&CenterDot;}{x}\left(t\right)=[(A_{0i}-B_{0i}{K}_i)+(\mathrm{\&Delta;}{A}_{0i}-\mathrm{\&Delta;}{B}_{0i}{K}_i)]x\left(t\right)+{\mathrm{\&Delta;A}}_{\mathrm{\&tau;}}\left(t\right)x(t-\mathrm{\&tau;})$ Robust stability;

In formula, M _ifor the matrix of a linear transformation,

$M_i^{-1}(A_{0i}-B_{0i}{K}_i)M_i=\mathrm{diag}[{\mathrm{\&sigma;}}_{i1}+j{\mathrm{\&omega;}}_{i1},{\mathrm{\&sigma;}}_{i2}+j{\mathrm{\&omega;}}_{i2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&sigma;}}_{\mathrm{in}}+j{\mathrm{\&omega;}}_{\mathrm{in}}],$

σ _ik, ω _ik(k=1,2 ..., n) be real number, j ω _ik(k=1,2 ..., n) represent imaginary number, diag is diagonal matrix symbol,

${\mathrm{\&Lambda;}}_i=\mathrm{diag}[{\mathrm{\&sigma;}}_{i1}^2,{\mathrm{\&sigma;}}_{i2}^2,\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&sigma;}}_{\mathrm{in}}^2];$

Δ A _0i-Δ B _0ik _iusually Δ A is assumed to be _0i-Δ B _0ik _i=H _if _iw _i, H _i, W _iall be assumed to be matrix, 0<F _i≤ I, I=diag [1,1 ..., 1] and be unit battle array.

The invention has the beneficial effects as follows: approach system segment robust stability solution conditions by becoming time provided by the invention, the closed loop of State Feedback for Linear Systems is directly utilized to expect the selection of poles, and expect that the real part of limit is all the feature of negative according to all closed loops, give qualifications inequality direct design of feedback matrix.What the engineering technical personnel of this research field were obtained wind-tunnel or flight test directly designs flight controller containing time-varying Hurst index dummy vehicle time lag, solves current research and only provides robust stability inequality and the technical matters that directly cannot design flight controller.

Below in conjunction with embodiment, the present invention is elaborated.

Embodiment

Aircraft time-varying model time lag of the present invention approach and controller design method concrete steps as follows:

1, obtained by wind-tunnel or flight test under assigned altitute, Mach number condition containing probabilistic aircraft time-varying model time lag be:

$\stackrel{\&CenterDot;}{x}\left(t\right)=[A_0\left(t\right)+\mathrm{\&Delta;}{A}_0\left(t\right)]x+A_{\mathrm{\&tau;}}\left(t\right)x(t-\mathrm{\&tau;})+B\left(t\right)u\left(t\right)---\left(1\right)$

In formula, x (t) ∈ R ⁿ, u (t) ∈ R ^mbe respectively state and input vector, A ₀(t), A _τ(t), B (t) for known constant coefficient matrix, τ be the unknown delays time, Δ A ₀t () is matrix of coefficients unknown portions; Symbol is identical in full; Simultaneous $\left\{\begin{array}{c}N(t,\mathrm{\&tau;})=\mathrm{\&Phi;}(t-\mathrm{\&tau;},t,\mathrm{\&tau;})\\ \min {\{x_{\mathrm{real}}\left(t\right)-\mathrm{\&Phi;}(t,t_0,\mathrm{\&tau;})x\left(t_0\right)-{\&Integral;}_{t_0}^t\mathrm{\&Phi;}(t,\mathrm{\&zeta;},\mathrm{\&tau;})B\left(\mathrm{\&zeta;}\right)u\left(\mathrm{\&zeta;}\right)\mathrm{d\&zeta;}\}}^2\end{array}\right.$ Iterative N (t, τ),

In formula: Φ (t, t ₀, τ) and be state-transition matrix

$\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}(t,t_0,\mathrm{\&tau;})=[A_0\left(t\right)+A_{\mathrm{\&tau;}}\left(t\right)N(t,\mathrm{\&tau;})]\mathrm{\&Phi;}(t,t_0,\mathrm{\&tau;})$

X _realt () is real system condition responsive;

After of equal value, the matrix of coefficients of system is

A(t)=A ₀(t)+A _τ(t)N(t，τ)+ΔA(t)

In formula: Δ A (t) is matrix of coefficients unknown portions;

According to known A (t), the variation range classification of B (t), is namely expressed as A (t), B (t) in different time sections:

$\left\{\begin{array}{c}A\left(t\right)=A_{0i}+{\mathrm{\&Delta;A}}_{0i}\\ B\left(t\right)=B_{0i}+{\mathrm{\&Delta;B}}_{0i}\end{array}\right.,t_{\mathrm{ij}}\&le;{t<t}_{\mathrm{ij}}+T_{\mathrm{ij}}(i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,r,j=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,p)$

In formula, A _0i, B _0ifor known constant matrices, Δ A _0iΔ B _0ifor unknown matrix, t _ij, T _ijfor time constant, r, p are positive integer, and i, j are that the A (t) of subscript, different time sections, B (t) expression formula form can be identical;

In time period t _ij≤ t<t _ij+ T _ijin, flight controller is: u (t)=-K _ix (t)

In formula, K _ifor constant feedback matrix;

Bring in (1) formula, have:

$\stackrel{\&CenterDot;}{x}\left(t\right)=[(A_{0i}-B_{0i}{K}_i)+({\mathrm{\&Delta;A}}_{0i}-\mathrm{\&Delta;}{B}_{0i}{K}_i)]x\left(t\right)+\mathrm{\&Delta;}{A}_{\mathrm{\&tau;}}\left(t\right)x(t-\mathrm{\&tau;})$

In formula: Δ A _τt () is nubbin unknown after system equivalence;

2, (A is chosen _0i-B _0ik _0i) eigenwert different and real part is negative, design of feedback matrix K _imake to satisfy condition:

${\mathrm{\&Lambda;}}_i>M_i^T[{(\mathrm{\&Delta;}{A}_{0i}-\mathrm{\&Delta;}{B}_{0i}{K}_i)}^T{M}_i^{-T}{M}_i^{-1}(\mathrm{\&Delta;}{A}_{0i}-\mathrm{\&Delta;}{B}_{0i}{K}_i)+\mathrm{\&Delta;}{A}_{\mathrm{\&tau;}}^T\left(t\right)M_i^{-T}{M}_i^{-1}\mathrm{\&Delta;}{A}_{\mathrm{\&tau;}}\left(t\right)]M_i;$

This controller makes

$\stackrel{\&CenterDot;}{x}\left(t\right)=[(A_{0i}-B_{0i}{K}_i)+(\mathrm{\&Delta;}{A}_{0i}-\mathrm{\&Delta;}{B}_{0i}{K}_i)]x\left(t\right)+{\mathrm{\&Delta;A}}_{\mathrm{\&tau;}}\left(t\right)x(t-\mathrm{\&tau;})$ Robust stability;

In formula, M _ifor the matrix of a linear transformation,

$M_i^{-1}(A_{0i}-B_{0i}{K}_i)M_i=\mathrm{diag}[{\mathrm{\&sigma;}}_{i1}+j{\mathrm{\&omega;}}_{i1},{\mathrm{\&sigma;}}_{i2}+j{\mathrm{\&omega;}}_{i2},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&sigma;}}_{\mathrm{in}}+j{\mathrm{\&omega;}}_{\mathrm{in}}],$

σ _ik, ω _ik(k=1,2 ..., n) be real number, j ω _ik(k=1,2 ..., n) represent imaginary number, diag is diagonal matrix symbol,

${\mathrm{\&Lambda;}}_i=\mathrm{diag}[{\mathrm{\&sigma;}}_{i1}^2,{\mathrm{\&sigma;}}_{i2}^2,\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&sigma;}}_{\mathrm{in}}^2];$

Δ A _0i-Δ B _0ik _iusually Δ A is assumed to be _0i-Δ B _0ik _i=H _if _iw _i, H _i, W _iall be assumed to be matrix, 0<F _i≤ I, I=diag [1,1 ..., 1] and be unit battle array;

Getting Flight Altitude Moving state variable is input variable is u=δ _e, wherein q is rate of pitch, and α is the air-flow angle of attack, for the angle of pitch, δ _efor elevating rudder drift angle; In time period 20≤t<100, State Equation Coefficients matrix is:

$A_{01}=\left[\begin{array}{ccc}-0.5000& -8.6500& 0\\ 1.0000& -0.3800& 0\\ 1.0000& 0& 0\end{array}\right],$ $B_{01}=\left[\begin{array}{c}-6.5000\\ -0.1000\\ 0\end{array}\right],$

Uncertain part is:

$\mathrm{\&Delta;}{A}_{01}=\left[\begin{array}{ccc}0.1000& -0.6000& 0\\ -0.3000& 0.4000& 0\\ 0& 0& 0\end{array}\right]F_1,$ $\mathrm{\&Delta;}{B}_{01}={\mathrm{\&lambda;}}_1\left[\begin{array}{c}2.3500\\ 0.0500\\ 0\end{array}\right],$ $\mathrm{\&Delta;}{A}_{\mathrm{\&tau;}}\left(t\right)=\left[\begin{array}{ccc}0& 0.1& 0\\ 0& 0.05& 0\\ 0& 0& 0\end{array}\right],$

0<F ₁≤I，0≤λ ₁<1，

Closed loop is selected to expect limit and A ₀₁-B ₀₁k ₁eigenwert σ (A ₀₁-B ₀₁k ₁)=diag [-0.5 ,-1,2], can obtain:

$A_{01}-B_{01}{K}_1=\left[\begin{array}{ccc}-3.2738& 1.3482& -4.0502\\ 0.9573& -0.2262& -0.0623\\ 1.0000& 0& 0\end{array}\right],$ $M_1=\left[\begin{array}{ccc}-0.8005& -0.5173& 0.2203\\ 0.4461& 0.6817& -0.8703\\ 0.4003& 0.5173& -0.4406\end{array}\right]$

Controller is: K ₁=[-0.3794 1.5382-0.6231].

Claims (1)

1. aircraft time-varying model time lag approaches and a controller design method, it is characterized in that comprising the following steps:

Step one, obtained by wind-tunnel or flight test under assigned altitute, Mach number condition containing probabilistic aircraft time-varying model time lag be:

$\stackrel{.}{x}\left(t\right)=[A_0\left(t\right)+\mathrm{\&Delta;}{A}_0\left(t\right)]x+A_{\mathrm{\&tau;}}\left(t\right)x(t-\mathrm{\&tau;})+B\left(t\right)u\left(t\right)$

In formula, x (t) ∈ R ⁿ, u (t) ∈ R ^mbe respectively state and input vector, A ₀(t), A _τ(t), B (t) for known constant coefficient matrix, τ be the unknown delays time, Δ A ₀t () is matrix of coefficients unknown portions; Symbol is identical in full;

Simultaneous iterative N (t, τ),

In formula: Φ (t, t ₀, τ) and be state-transition matrix

$\stackrel{.}{\mathrm{\&Phi;}}(t,t_0,\mathrm{\&tau;})=[A_0\left(t\right)+A_{\mathrm{\&tau;}}\left(t\right)N(t,\mathrm{\&tau;})]\mathrm{\&Phi;}(t,t_0,\mathrm{\&tau;})$

X _realt () is real system condition responsive;

After of equal value, the matrix of coefficients of system is

A(t)＝A ₀(t)+A _τ(t)N(t,τ)+ΔA(t)

In formula: Δ A (t) is matrix of coefficients unknown portions;

According to known A (t), the variation range classification of B (t), is namely expressed as A (t), B (t) in different time sections:

$\left\{\begin{array}{c}A\left(t\right)=A_{0i}+{\mathrm{\&Delta;A}}_{0i}\\ B\left(t\right)=B_{0i}+{\mathrm{\&Delta;B}}_{0i}\end{array}\right.t_{\mathrm{ij}}\&le;t<t_{\mathrm{ij}}{+T}_{\mathrm{ij}},i=\mathrm{1,2},...,r,j=\mathrm{1,2},...,p$

In formula, A _0i, B _0ifor known constant matrices, Δ A _0iΔ B _0ifor unknown matrix, t _ij, T _ijfor time constant, r, p are positive integer, and i, j are that the A (t) of subscript, different time sections, B (t) expression formula form is identical;

In time period t _ij≤ t<t _ij+ T _ijin, flight controller is: u (t)=-K _ix (t)

In formula, K _ifor constant feedback matrix;

Bring in (1) formula, have:

$\stackrel{.}{x}\left(t\right)=[(A_{0i}-B_{0i}{K}_i)+({\mathrm{\&Delta;A}}_{0i}-{\mathrm{\&Delta;B}}_{0i}{K}_i)]x\left(t\right)+{\mathrm{\&Delta;A}}_{\mathrm{\&tau;}}\left(t\right)x(t-\mathrm{\&tau;})$

In formula: Δ A _τt () is nubbin unknown after system equivalence;

Step 2, choose (A _0i-B _0ik _i) eigenwert different and real part is negative, design of feedback matrix K _imake to satisfy condition:

${\mathrm{\&Lambda;}}_i>M_i^T[{({\mathrm{\&Delta;A}}_{0i}-{\mathrm{\&Delta;B}}_{0i}{K}_i)}^T{M}_i^{-T}{M}_i^{-1}({\mathrm{\&Delta;A}}_{0i}-{\mathrm{\&Delta;B}}_{0i}{K}_i)+{\mathrm{\&Delta;A}}_{\mathrm{\&tau;}}^T\left(t\right)M_i^{-T}\mathrm{\&Delta;}{A}_{\mathrm{\&tau;}}\left(t\right)]M_i;$ This controller makes

$\stackrel{.}{x}\left(t\right)=[(A_{0i}-B_{0i}{K}_i)+({\mathrm{\&Delta;A}}_{0i}-{\mathrm{\&Delta;B}}_{0i}{K}_i)]x\left(t\right)+{\mathrm{\&Delta;A}}_{\mathrm{\&tau;}}\left(t\right)x(t-\mathrm{\&tau;})$ Robust stability;

In formula, M _ifor the matrix of a linear transformation,

$M_i^{-1}\left(A_{0i}{-B}_{0i}{K}_i\right)M_i=\mathrm{diag}\left[\begin{array}{cccc}{\mathrm{\&sigma;}}_{i1}+{\mathrm{j\&omega;}}_{i1},& {\mathrm{\&sigma;}}_{i2}+{\mathrm{j\&omega;}}_{i2},& ...,& {\mathrm{\&sigma;}}_{\mathrm{in}}+,{\mathrm{j\&omega;}}_{\mathrm{in}}\end{array}\right],$

σ _ik, ω _ik, k=1,2 ..., n is real number, j ω _ik, k=1,2 ..., n represents imaginary number, and diag is diagonal matrix symbol,

${\mathrm{\&Lambda;}}_i=\mathrm{diag}\left[\begin{array}{cccc}{\mathrm{\&sigma;}}_{i1}^2,& {\mathrm{\&sigma;}}_{i2}^2,& ...,& {\mathrm{\&sigma;}}_{\mathrm{in}}^2\end{array}\right];$

Δ A _0i-Δ B _0ik _iusually Δ A is assumed to be _0i-Δ B _0ik _i=H _if _iw _i, H _i, W _iall be assumed to be known matrix, 0<F _i≤ I, I=diag [1,1 ..., 1] and be unit battle array.