CN1707467A - Model order reducing method for a parameter system - Google Patents

Abstract

The present invention belongs to the field of microelectromechanical MEMS and electronic technology, and is especially one kind of order reducing method for system model with parameters. The model order reducing technology can raise simulating and verifying speed of system model effectively and improve the design scheme of circuit and device timely. The present invention establishes model order reducing method to parameter system. Transfer function of parameter system is multiple series developed to constitute son projection matrix, and the son projection matrix is orthogonalized to constitute projection matrix. The projection matrix is used in reducing order of the original parameter system and the obtained reduced-order system is independent on some special values of the system parameters and can maintain order reducing precision on different values. The present invention has greatly raised order reducing precision and efficiency.

Description

A kind of model order reducing method of parameter system

Technical field

The invention belongs to micro electronmechanical MEMS and electronic technology field, be specifically related to a kind of model order reducing method of parameter system.

Technical background

The calorifics phenomenon is all being played the part of very important role always in many micro electro mechanical devices, microhotplate sensor for example, microfluidics, electro-thermal micromotors[2] etc.Therefore electricity heat analogy is the important component part in the modern project design.Usually need come a thermal modeling is accurately described with finite element analysis method, what obtain thus be a large-scale ordinary differential equation group.The method of traditional direct modeling is very consuming time, and design and system level simulation have been caused very big difficulty.Recently, in the simulation field of electrothermics, the foundation of compact model (compact model) has become the focal issue [3] [4] of discussion.In actual applications, be exactly that it must be independent of boundary condition to one of compact models important requirement.This means that engineering design person can change the environment (device environment) of device with same compact models, thereby obtain the design result under the varying environment.And no matter how the device environment changes, the error between this compact models and the original model can both keep within the acceptable range.

The method of model reduction has obtained development [1] rapidly in the quick simulation field of heavy construction system.By model reduction, people can obtain a compact models (reduced-order model) of original system.Thereby in the short period of time to circuit or device function and performance verify fast so that its design proposal is in time improved.But traditional order reducing method can only be handled the system model of not being with parameter, because of rather than boundary condition method independently, promptly can not be independent of systematic parameter.And for some physical systems of more accurate description, the model that many application problems are set up all has some variable parameters, for example following system:

$C\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}+(G+\mathrm{kD})x\left(t\right)=\mathrm{Bu}\left(t\right)$

y(t)＝Ex(t) (1)

Wherein k is a variable parameter.The value of k can be adjusted as required at any time.X (t) is the N dimension state variable of system, as node voltage in the circuit or branch current, and the temperature of device in micro electronmechanical (MEMS) field.Matrix C, D, G ∈ R ^{N * N}, E ∈ R ^{P * N}, B ∈ R ^{N * l}Be system matrix, they all are constant matricess, normally by to circuit or device disperse that modeling obtains.N is the node number.U (t) ∈ R ^l, in circuit the input signal variable that enters circuit, the number of input port in the l indication circuit refers to that l signal enters circuit from l port simultaneously.At micro electronmechanical field u (t) normally 1, just right-hand member has only matrix B usually.Y (t) ∈ R ^p, in circuit the variable of describing the output signal of circuit, s is the number of output port, the expression signal is exported from p port through oversampling circuit; Can the value of the temperature at some position in the device in micro electronmechanical.

For observing system corresponding to the different qualities under the value condition of different parameters (referring to accompanying drawing 2), the every change of parameter once must be simulated once again to original system, the output of the system after the calculating parameter value changes, thereby the different characteristic of verification system.Verify the value of many kinds of different parameters if desired, so just must many times simulate original large scale system.Obviously can cause very big calculated amount.The purpose of parameter model depression of order is exactly to replace original parameter system with the system of a small-scale band parameter, make all parameters of original system all be retained in this minisystem, and the precision of the parameter system that this is little can not degenerate with the variation of parameter value, and promptly little parameter system can both guarantee its precision within the acceptable range to the value of any parameter.Yet, traditional model order reducing method can not obtain such parameter reduced-order model, because these methods can only be carried out man-to-man depression of order to any one group of fixing parameter, this just shows if we need verify many groups parameter value, so traditional model order reducing method must calculate many reduced-order models, the value of corresponding each the group parameter of each reduced-order model.Such order reducing method is very unpractical.

The deficiencies in the prior art part

The state equation of traditional linear system has following form:

$C\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}+\mathrm{Gx}\left(t\right)=\mathrm{Bu}\left(t\right)$

y(t)＝Ex(t) (2)

Wherein, the implication homologous ray (1) of various parameters.

Traditional model order reducing method [1] at first carries out the Laplace conversion to linear system (2), obtains frequency-domain expression:

sCX(s)+GX(s)＝BU(s)

Y(s)＝EX(s) (3)

Calculate the transport function of linear system (3) then

H(s)＝Y(s)/U(s)＝E(sC+G) ^-1B (4)

Transport function is at s ₀=0 carries out obtaining after the series expansion

$H\left(s\right)=\underset{l=0}{\overset{\∞}{\∑}}E{M}_i{s}^i---\left(5\right)$

Wherein: EM _l=E (G ^-1C) ⁱG ^-1B is the square of transport function.According to the square of transport function, can construct following projection matrix:

spancol{ V}＝spancol{M ₀，M ₁，M ₂，…M _m} (6)

Utilize projection matrix V can obtain the system of depression of order, concrete steps are as follows, and the state variable to original system is similar to x ≈ Vz earlier, obtains:

$C\stackrel{\‾}{V}\frac{\mathrm{dz}\left(t\right)}{\mathrm{dt}}+G\stackrel{\‾}{V}z\left(t\right)=\mathrm{Bu}\left(t\right)$

y(t)＝E Vz(t) (7)

Then system (7) is projected on the subspace at V place, obtains following reduced order system:

${\stackrel{\‾}{V}}^{T}C\stackrel{\‾}{V}\frac{\mathrm{dz}\left(t\right)}{\mathrm{dt}}+{\stackrel{\‾}{V}}^{T}G\stackrel{\‾}{V}z\left(t\right)={\stackrel{\‾}{V}}^T\mathrm{Bu}\left(t\right)$

y(t)＝E Vz(t) (8)

There is following shortcoming in this technology aspect realization parameter system model reduction:

1, the system matrix C in the system (2), G is a constant matrices, does not comprise any variable parameter.If traditional order reducing method is applied to parameter system (1), the transport function that we obtain (1) is:

H(s)＝E(sC+G+kD) ^-1B＝E[I-(-(G+kD) ^-1Cs)] ^-1(G+kD) ^-1B

$=E\underset{j=0}{\overset{\∞}{\∑}}{[-{(G+\mathrm{kD})}^{-1}C]}^j{(G+\mathrm{kD})}^{-1}B{s}^j=E\underset{j=0}{\overset{\∞}{\∑}}{\tilde{M}}_j{s}^j$

The square of this transport function is: $E{\tilde{M}}_i=E[-{(G+\mathrm{kD})}^{-1}C{]}^j{(G+\mathrm{kD})}^{-1}B,{}^{}$ j＝0，1，2...。Obviously they depend on parameter k, if use The projection matrix of constructing variable system (1), a fixing projection matrix that just can't obtain is because projection matrix is by right

Carry out that orthogonalization procedure obtains.In this process, must be constant vector by orthogonalized vector, rather than the function of certain parameter.Therefore if obtain projection matrix, just must be fixed up the value of parameter k, that is to say the value k of a first given parameter ₀, basis is corresponding to k then ₀The square structure projection matrix of transport function, that is:

spancol{ V}＝spancol{ M ₀， M ₁， M ₂，… M _m} (9)

Wherein, M _i=E[-(G+k ₀D) ^-1C] ^j(G+k ₀D) ^-1B, j=0,1,2 ... m.But depend on fixing parameter k by following reduced order system (10) strictness that such projection matrix obtains ₀, when the value of parameter changed, the precision of reduced order system will become very poor (referring to accompanying drawing 3).

${\stackrel{\‾}{V}}^{T}C\stackrel{\‾}{V}\frac{\mathrm{dz}\left(t\right)}{\mathrm{dt}}+{\stackrel{\‾}{V}}^{T}G\stackrel{\‾}{V}z\left(t\right)+{\stackrel{\‾}{V}}^T\mathrm{kD}\stackrel{\‾}{V}z\left(t\right)={\stackrel{\‾}{V}}^{T}B$

y(t)＝E Vz(t) (10)

If 2 in order to guarantee the precision of reduced order system, when the every change of parameter is once just carried out depression of order one time to the raw parameter system, the cost of depression of order will be very high so, special when the many different value of parameter needs, just must carry out many times depression of order to original system, if whenever carry out time that one time depression of order spent and original system to be carried out the time phase difference that direct modeling spent little, traditional order reducing method has just lost the meaning of its depression of order.

List of references

[1]R.W.Freund(2000)Krylov-subspace?methods?for?reduced?order?modeling?in?circuitsimulation.Journal?of?Computational?and?Applied?Mathematics?vol.123，pp.395-421.

[2]D.L?De?Voe(2002)Thermal?issues?in?MEMS?and?microscale?systems.IEEE?Transactionson?Components?and?Packaging?Technologies，vol.25，pp.576-583.

[3]C.J.M.Lasance(2003)Recent?progress?in?compact?thermal?models，SemiconductorThermal?Measurement?and?Management?Symposium，2003.Ninteenth?Annual?IEEE，11-13March.

[4]M.N.Sabry(2003a)Compact?thermal?models?for?electronic?systems.IEEE?Transactions?onComponents?and?Packaging?Technologies，vol.26，pp.179-185.

Summary of the invention

The objective of the invention is to propose a kind of precision height, the order reducing method of province consuming time at parameter system.

The model order reducing method at parameter system that the present invention proposes is a kind of order reducing method that the frequency domain transfer function of parameter system (1) is carried out multistage number expansion.Concrete steps are as follows.

The first step is carried out the Laplace conversion to parameter system (1), obtains:

sCX(s)+(G+kD)X(s)＝BU(s)

Y (s)=EX (s) s here is the frequency domain variable;

Second step was obtained the transport function of parameter system (1):

H(s)＝y(s)/U(s)＝E(sC+G+kD) ^-1B

The 3rd step was carried out series expansion with transfer function H (s) about s and k:

(1), earlier transfer function H (s) about frequency domain s at s ₀=0 is carried out series expansion:

H(s)＝E(sC+G+kD) ^-1B＝E[I-(-(G+kD) ^-1Cs)] ^-1(G+kD) ^-1B

$=E\underset{j=0}{\overset{\∞}{\∑}}{[-{(G+\mathrm{kD})}^{-1}C]}^j{(G+\mathrm{kD})}^{-1}B{s}^j=E\underset{j=0}{\overset{\∞}{\∑}}{\tilde{M}}_j{s}^j$

Wherein

K is relevant with parameter, by following formula frequency domain state variable X (s)=(sC+G+kD) as can be known ^-1BU (s) is

Linear combination.

(2), right ${\tilde{M}}_j=E{[-{(G+\mathrm{kD})}^{-1}C]}^j{(G+\mathrm{kD})}^{-1}B$ J=0,1,2 ... m about parameter k at k ₀=0 is carried out series expansion.Wherein choosing of m can be chosen according to the scale of original system and the precision of desired reduced order system.Generally smaller for the scale that guarantees reduced order system, be raw parameter system about N=5000 for scale, the desirable 4-6 of m, for example m=5.When N increased, m also can correspondingly increase, to guarantee precision.

When j=0,

${\tilde{M}}_0={(G+\mathrm{kD})}^{-1}B={(I+G^{-1}\left(\mathrm{kD}\right))}^{-1}{G}^{-1}B$

$={(I-(-G^{-1}\left(\mathrm{kD}\right))}^{-1}{G}^{-1}B=\underset{i_0=0}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_0}{G}^{-1}B{k}^{i_0}$

When j=1,

${\tilde{M}}_1=-{(G+\mathrm{kD})}^{-1}C{(G+\mathrm{kD})}^{-1}B$

$=-{(G+\mathrm{kD})}^{-1}C{\tilde{M}}_0=-{(I+k{G}^{-1}D)}^{-1}{G}^{-1}C{\tilde{M}}_0$

$=-{(I-(-k{G}^{-1}D))}^{-1}{G}^{-1}C{\tilde{M}}_0=-\underset{i_1=0}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_1}{G}^{-1}C{k}^{i_1}{\tilde{M}}_0$

$=[-\underset{i_1=0}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_1}{G}^{-1}C{k}^{i_1}]\left[\underset{i_0=0}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_0}{G}^{-1}B{k}^{i_0}\right]$

Have for any j,

${\tilde{M}}_j={[-{(G+\mathrm{kD})}^{-1}C]}^j{(G+\mathrm{kD})}^{-1}B=[-{(G+\mathrm{kD})}^{-1}C]{\tilde{M}}_{j-1}$

$=[-\underset{i_j}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_j}{G}^{-1}C{k}^{i_j}]{\tilde{M}}_{j-1}$

$=[-\underset{i_j=0}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_j}{G}^{-1}C{k}^{i_j}][-\underset{i_{j-1}}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_{j-1}}{G}^{-1}C{k}^{i_{j-1}}]{\tilde{M}}_{j-2}$

$=\left[\underset{i=1}{\overset{j}{\mathrm{\Π}}}(-\underset{i_j=0}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_j}{G}^{-1}C{k}^{i_j})\right]\left[\underset{i_0=0}{\overset{\∞}{\∑}}{(-G^{-1}D)}^{i_0}{G}^{-1}B{k}^{i_0}\right]$

Can further be organized into ${\tilde{M}}_1=-\underset{i_1=0}{\overset{\&infin;}{\&Sum;}}\underset{i_0=0}{\overset{\&infin;}{\&Sum;}}-{\left(G^{-1}D\right)}^{i_1}{G}^{-1}C{(-G^{-1}D)}^{i_0}{G}^{-1}B{k}^{i_1}{\mathrm{<figure-callout\; id="0"\; label="k\; i"\; filenames="A20051002527000161.png"\; state="\{\{state\}\}">k</figure-callout>}}^{i_{}}$

Correspondingly,

Can be organized into

${\tilde{M}}_j={(-1)}^j\underset{i_j=0}{\overset{\&infin;}{\&Sum;}}\underset{i_{j-1}=0}{\overset{\&infin;}{\&Sum;}}\&CenterDot;\&CenterDot;\&CenterDot;\underset{i_1=0}{\overset{\&infin;}{\&Sum;}}\underset{i_0=0}{\overset{\&infin;}{\&Sum;}}[{(-G^{-1}D)}^{i_j}{G}^{-1}C{(-G^{-1}D)}^{i_{j-1}}$

$G^{-1}C\&CenterDot;\&CenterDot;\&CenterDot;{(-G^{-1}D)}^{i_j}{G}^{-1}C{(-G^{-1}D)}^{i_0}{G}^{-1}B]k^{i_j}{k}^{i_{j-1}}\&CenterDot;\&CenterDot;\&CenterDot;k^{{\mathrm{<figure-callout\; id="0"\; label="i"\; filenames="A20051002527000161.png"\; state="\{\{state\}\}">i</figure-callout>}}_0}$

As can be seen from the above equation

J=0,1 ... m is the linear combination of the column vector in the following column matrix:

${(-G^{-1}D)}^{i_j}{G}^{-1}C\&CenterDot;\&CenterDot;\&CenterDot;{(-G^{-1}D)}^{i_1}{G}^{-1}C{(-G^{-1}D)}^{i_0}{G}^{-1}B$

(i _j，…，i ₁，i ₀＝0，1，…，∞)

The 4th step structure projection matrix V, concrete steps are:

(1), basis

Structure projection matrix V ₀As follows:

$\mathrm{spancol}\left\{V_0\right\}=K_{i_0}(-G^{-1}D,G^{-1}B)$

(2), basis Structure projection matrix V ₁As follows:

${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">B</figure-callout>}}_1=C[G^{-1}B,(-G^{-1}D)G^{-1}B,\&CenterDot;\&CenterDot;\&CenterDot;{(-G^{-1}D)}^{q_{i_0}}{G}^{-1}B]$

$\mathrm{spancol}\left\{V_1\right\}=K_{i_1}({-G}^{-1}D,G^{-1}{B}_1)$

(3), similarly, can construct V ₂, V ₃, V _i, that is:

${\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">B</figure-callout>}}_2=C[G^{-1}{\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">B</figure-callout>}}_1,-G^{-1}D{G}^{-1}{\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">B</figure-callout>}}_1,\&CenterDot;\&CenterDot;\&CenterDot;{(-G^{-1}D)}^{q_{i_j}}{G}^{-1}{B}_1]$

$\mathrm{spancol}\left\{V_2\right\}=K_{i_2}(-G^{-1}D,G^{-1}{B}_2)$

$B_j=C[(G^{-1}{B}_{j-1},-G^{-1}D{G}^{-1}{B}_{j-1},\&CenterDot;\&CenterDot;\&CenterDot;,{(-G^{-1}D)}^{q_{i_{j-1}}}{G}^{-1}B]$

$\mathrm{spancol}\left\{V_j\right\}=K_{i_j}(-G^{-1}D,G^{-1}{B}_j)---j=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,m$

For the raw parameter system about the N=5000 rank, wherein i _jGenerally get number, i.e. i less than 10 _m, i _M-1... i ₀≤ 10, and $q_{i_j}\&le;i_j,j=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;m-1.$ Choose on the one hand not too largely like this, to be in order can covering the square of high-order in the projection matrix on the other hand, thereby can to improve the precision of depression of order for the scale that guarantees reduced order system.

(4), construct a unified projection matrix V at last and calculate V earlier ₁, V ₂..., V _mUnion: $\hat{V}={\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">V</figure-callout>}}_1\&cup;{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">V</figure-callout>}}_2\&cup;...\&cup;V_m,$ Make up V again: to the column vector among the V by In all column vectors after orthogonalization process, obtain the column vector of V.In step (1), (2), (3), K _Ij, j=0,1 ... m refers to i _jKrylov subspace, rank, Krylov subspace, n rank K _n(A B) is defined as: K _n(A, B)=spancol (B, AB, A ²B ..., A ^N-1B).

The 5th step projected to the raw parameter system on the subspace at V place, obtained the system of depression of order, and the same formula of method (7), (8) are promptly introduced new approximate state variable x ≈ Vz earlier, obtain:

$\mathrm{CV}\frac{\mathrm{dz}\left(t\right)}{\mathrm{dt}}+\mathrm{GVz}\left(t\right)+\mathrm{kDVz}\left(t\right)=\mathrm{Bu}\left(t\right)$

y(t)＝Ez(t) (11)

V is taken advantage of on both sides respectively ^T, last projection obtains reduced order system

$\hat{C}\frac{\mathrm{dz}\left(t\right)}{\mathrm{dt}}+\hat{G}z\left(t\right)+k\hat{D}z\left(t\right)=\hat{B}u\left(t\right)$

$\hat{y}\left(t\right)=\hat{E}z\left(t\right)$ (12)

Wherein $\hat{C}=V^T\mathrm{CV},\hat{G}=V^T\mathrm{GV},\hat{D}=V^T\mathrm{DV},\hat{B}=V^{T}B,\hat{E}=\mathrm{EV}.$

A kind of model order reducing method at parameter system that the present invention proposes to different parameter values, can obtain a unified reduced-order model.And guaranteeing value for different parameters, this reduced-order model can both keep higher precision.The present invention utilizes the multistage number of the transport function of parameter system to launch, transport function is launched into progression about s and k respectively, thereby obtain the projection matrix irrelevant with parameter k, make the precision of the reduced order system that obtains by this projection matrix can be because of the change of parameter value variation.

The present invention has following advantage:

1, high depression of order precision

The present invention has obtained a reduced order system that does not rely on certain fixed value of parameter to the raw parameter system, when the value of parameter changes, the precision of reduced order system remains in certain higher scope, thereby has improved the strict precision that depends on traditional order reducing method of certain parameter fixed value greatly.

2, low calculated amount and storage capability

The present invention has saved calculated amount and storage capability greatly.If with traditional order reducing method, in order to guarantee the precision of reduced order system, when the value of parameter changes, must carry out depression of order again to the raw parameter system, just obtained different reduced order systems for different parameter values like this.Especially when needing to analyze the value of many parameters, just need the many reduced order systems of calculating, very complicated thereby the process of model reduction will become, its calculated amount even will be more than directly parameter system being simulated required calculated amount.What the present invention obtained is a unified reduced order system, it remains into the parameter of raw parameter system in the system behind the depression of order as symbol, make when analyzing the value of different parameters, only need the value of the parameter in the reduced order system is changed, this unified reduced order system is simulated, and the analog result that obtains is all very accurate for different parameters.

3, can be applied to the system of multiparameter

Description of the invention is to launch at the system that contains a parameter (1), goes on foot the model reduction can be easy to be generalized to multiparameter system by the 3rd step and the 4th of invention.If promptly also have other parameter, after the 3rd step finished so, continue to repeat for the 3rd step, other parameters are further carried out series expansion, this is very natural popularization process, does not do detailed derivation here.

Description of drawings

Fig. 1 is a MEMS micro-thruster computing unit.

Fig. 2 serves as reasons the output (temperature) of the parameter system that this computing unit obtains corresponding to the curve map of different parameters.

The error curve diagram that Fig. 3 obtains for traditional order reducing method, wherein solid line is made up of the error of many reduced order systems, promptly when the every change of parameter k one time, just correspondingly the raw parameter system is carried out depression of order one time, obtain the error of a plurality of corresponding reduced order systems by k having been got a plurality of different values, these errors are linked up constitute this error line.After the selected k value of methods that other six error line are corresponding traditional respectively, system is carried out the result that depression of order obtains.

The Error Graph of the parameter reduced order system that Fig. 4 obtains for the present invention.

Number in the figure: 1 is SOG, and 2 is polysilicon, and 3 is SiNx, and 4 is silicon dioxide,

5 is silicon substrate, and 6 is fuel.

Embodiment

Further specify the present invention below by specific embodiment.

Fig. 1 is the computing unit of a MEMS micro-thruster.Wherein the small rectangular block in the upper left corner is represented heating radiator (heater). and obtain a reduced order system (1) of being with parameter by this computing unit being carried out finite element discretization, wherein parameter k represents convection coefficient.The output y (t) of this parameter system represents the temperature of heating element central authorities, and temperature is directly related with convection coefficient k, when the value of convection coefficient k changes, central authorities temperature can great changes will take place thereupon, see Fig. 2.The exponent number of this parameter system is n=4257

This parameter system is carried out depression of order, and at first the multistage number of the transport function of computing system launches, and calculates projection matrix according to series expansion then.The compute matrix V of elder generation ₀, get i ₀=15, promptly we carry out orthogonalization process to following 15 vectors,

G ^-1B, (G ^-1D) G ^-1B, (G ^-1D) ²G ^-1B ..., (G ^-1D) ¹⁴G ^-1The orthogonalized technology of B (13) adopts existed algorithms Anorldi process to realize.We obtain V after the orthogonalization ₀

For V ₁, 8 groups of vectors at first getting in (13) constitute B ₁, promptly ${\mathrm{<figure-callout\; id="0"\; label="q\; i"\; filenames="A20051002527000161.png"\; state="\{\{state\}\}">q</figure-callout>}}_{i_{}}=7$

B ₁＝C[G ^-1B，(-G ^-1D)G ^-1B，…，(-G ^-1D) ⁷G ^-1B]

Get i then ₁=4, i.e. V ₁Undertaken by following column vector that orthogonalization obtains,

G ^-1B ₁，-G ^-1DG ^-1B ₁，(-G ^-1D) ²G ^-1B ₁，(-G ^-1D) ³G ^-1B ₁ (14)

For V ₂, from (14), take out 4 groups of vectors similarly and constitute B ₂, promptly ${\mathrm{<figure-callout\; id="1"\; label="q\; i"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">q</figure-callout>}}_{i_{}}=3$

B ₂＝C[G ^-1B ₁，-G ^-1DG ^-1B ₁，…，(-G ^-1D) ³G ^-1B ₁]

Get i then ₂=3, i.e. V ₂Undertaken by following column vector that orthogonalization obtains,

G ^-1B ₂，-G ^-1DG ^-1B ₂，(-G ^-1D) ²G ^-1B ₂

Obtain V ₀, V ₁, V ₂The back just can get $\hat{V}={\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="A20051002527000161.png"\; state="\{\{state\}\}">V</figure-callout>}}_0\&cup;{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">V</figure-callout>}}_1\&cup;{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="A20051002527000151.png,A20051002527000152.png"\; state="\{\{state\}\}">V</figure-callout>}}_2,$ Right at last

Carry out orthogonalization process and obtain projection matrix V, have 49 column vectors among the V this moment.After obtaining matrix V, carry out variable and replace x ≈ Vz, carry out depression of order according to (11) (12) formula, owing among the V 49 column vectors are arranged, thus z is the vector of one 49 dimension, the exponent number of the reduced-order model that obtains so is exactly q=49.Fig. 2, Fig. 3, Fig. 4 are experimental results.

Fig. 2 is the situation of change of the output of system corresponding to the value of different parameters k, promptly corresponding to the temperature variation of the heating element central authorities of different parameters k.Transverse axis is time (second), and the longitudinal axis is the logarithm value corresponding to the temperature of different k.As seen from the figure, when parameter k changed, variation of temperature was very big.This just means if just must carry out depression of order many times to the raw parameter system with traditional model order reducing method.

In Fig. 3, provided the error of traditional model order reducing method, the model order q=33 behind the depression of order.Wherein solid line is the error that obtains of depression of order many times, promptly whenever gets a different k and just carries out depression of order again one time, and therefore such depression of order effect is fine, and error is very little.But as preceding surface analysis, such depression of order has lost the meaning of depression of order.After the selected k value of the methods that other six error line are corresponding traditional respectively, system is carried out the result that depression of order obtains, the i.e. error of reduced order system (10).No matter how can see the value of k chooses, if have to a reduced-order model with traditional order reducing method, this reduced-order model is for less near near the error ratio the selected k value so, if still the value of parameter is away from selected value, error is will very fast change big.

In Fig. 4, provided the error of the parameter model order reducing method of the present invention's proposition.What we obtained here is a reduced-order model, and its exponent number also is q=33.Parameter k is retained in the reduced-order model.Fig. 4 is the situation of change of the error of parameter reduced-order model corresponding to the value of different k.How to change no matter can see k, error remains at below 1%, and this is enough accurate in commercial Application.

This example shows that the present invention can obtain the reduced-order model of high-precision parameter system, is a kind of reducing technique of parameter model efficiently, can be applied to the depression of order of the system model of mimic channel, Digital Analog Hybrid Circuits and band parameter such as micro electronmechanical.

Claims (2)

1, a kind of model order reducing method of parameter system, this parameter system has following form:

$\begin{array}{c}C\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}+(G+\mathrm{kD})x\left(t\right)=\mathrm{Bu}\left(t\right)\\ y\left(t\right)=\mathrm{Ex}\left(t\right)\end{array}---\left(1\right)$

Wherein, x (t) is the N dimension state variable of system, Matrix C, D, G ∈ R ^{N * N}, E ∈ R ^{P * N}, B ∈ R ^{N * l}Be system matrix, obtain u (t) ∈ R by modeling that circuit or device are dispersed ^l, in circuit the input signal variable that enters circuit, the number of input port in the l indication circuit is 1 at micro electronmechanical field u (t); Y (t) ∈ R ^p, in circuit the variable of describing the output signal of circuit, p represents the number of output port, is the value of the temperature at some position in the device in micro electronmechanical, k is variable parameter; It is characterized in that the frequency domain transfer function of parameter system (1) is carried out multistage number expansion, concrete steps are as follows:

The first step is carried out the Laplace conversion to parameter system (1), obtains:

sCX(s)+(G+kD)X(s)＝BU(s)

Y(s)＝EX(s)

Here s is the frequency domain variable.

Second step: obtain system transter:

H(s)＝E(sC+G+kD) ^-1B

The 3rd step: transfer function H (s) is carried out series expansion about s and k:

(1), with H (s) at s ₀=0 progression that is launched into s:

$H\left(s\right)=E{(\mathrm{sC}+G+\mathrm{kD})}^{-1}B=E{[I-(-{(G+\mathrm{kD})}^{-1}\mathrm{Cs})]}^{-1}{(G+\mathrm{kD})}^{-1}B$

$=E\underset{j=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}[-{(G+\mathrm{kD})}^{-1}C{]}^j{(G+\mathrm{kD})}^{-1}B{s}^j=E\underset{j=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\tilde{M}}_j{s}^j$

Wherein

K is relevant with parameter;

(2), further will

At k ₀=0 progression that is launched into about k:

${\tilde{M}}_j={(-1)}^j\underset{i_j=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\underset{i_{j-1}=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\&CenterDot;\&CenterDot;\&CenterDot;\underset{i_1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\underset{=0i_0=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}[{(-G^{-1}D)}^{i_j}{G}^{-1}C{\left({-G}^{-1}D\right)}^{i_{j-1}}$

$G^{-1}C\&CenterDot;\&CenterDot;\&CenterDot;{(-G^{-1}D)}^{i_1}{G}^{-1}C{\left({-G}^{-1}D\right)}^{i_0}{G}^{-1}B]k^{i_j}{k}^{i_{j-1}}\&CenterDot;\&CenterDot;\&CenterDot;k^{i_0}$ j＝0，1，…，m；

The 4th step: make up projection matrix V:

(1), basis Make up projection matrix V ₀:

$\mathrm{spanol}\left\{V_0\right\}=K_{i_0}({-G}^{-1}D,G^{-1}B)$

(2), basis Make up projection matrix V ₁:

$B_1=C[G^{-1}B,\left({-G}^{-1}D\right)G^{-1}B,\&CenterDot;\&CenterDot;\&CenterDot;{\left({-G}^{-1}D\right)}^{q_{i_0}}{G}^{-1}B]$

$\mathrm{spancol}\left\{V_1\right\}=K_{i_1}({-G}^{-1}D,G^{-1}{B}_1)$

(3), basis

Make up projection matrix V ₂:

$B_2=C[G^{-1}{B}_1,{-G}^{-1}D{G}^{-1}{B}_1,\&CenterDot;\&CenterDot;\&CenterDot;,{\left({-G}^{-1}D\right)}^{q_{i_1}}{G}^{-1}{B}_1]$

$\mathrm{spancol}\left\{V_2\right\}=K_{i_2}({-G}^{-1}D,G^{-1}{B}_2)$



(4), make up matrix V similarly _j:

$B_j=C\left[\right(G^{-1}{B}_{j-1},-G^{-1}{\mathrm{DG}}^{-1}{B}_{j-1},\&CenterDot;\&CenterDot;\&CenterDot;,{\left({-G}^{-1}D\right)}^{q_{i_{j-1}}}{G}^{-1}B]$

$\mathrm{spancol}\left\{V_j\right\}=K_{i_j}({-G}^{-1}D,G^{-1}{B}_j)$

j＝1，2，…，m

(5), make up unified projection matrix V:

Calculate V ₁, V ₂..., V _mUnion: $\hat{V}=V_{1}U{V}_{2}U...{\mathrm{UV}}_m,$ Make up again among the V:V column vector by In all column vectors after orthogonalization process, obtain;

The 5th step: parameter system (1) is carried out the projection depression of order, at first make approximate variable and replace x ≈ Vz, obtain

$\begin{array}{c}C\frac{\mathrm{dVz}\left(t\right)}{\mathrm{dt}}+\mathrm{GVz}\left(t\right)+\mathrm{kDVz}\left(t\right)=\mathrm{Bu}\left(t\right)\\ y\left(t\right)=\mathrm{EVz}\end{array}---\left(11\right)$

V is taken advantage of on both sides respectively ^T, get reduced order system to the end:

$\begin{array}{c}\hat{C}\frac{\mathrm{dz}\left(t\right)}{\mathrm{dt}}+\hat{G}z\left(t\right)+k\hat{D}z\left(t\right)=\hat{B}u\left(t\right)\\ \hat{y}\left(t\right)=\widehat{\mathrm{Ez}}\left(t\right)\end{array}---\left(12\right)$

Wherein $\hat{C}=V^T\mathrm{CV},\hat{G}=V^T\mathrm{GV},\hat{D}=V^T\mathrm{DV},\hat{B}=V^{T}B,\hat{E}=\mathrm{EV}.$

2, according to the order reducing method of the described parameter system of claim, it is characterized in that in the step 3, for the parameter system about N=5000, get m=4,5 or 6.

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