CN105304920B

CN105304920B - A kind of flat solid oxide fuel cell local temperature method of estimation

Info

Publication number: CN105304920B
Application number: CN201510811980.8A
Authority: CN
Inventors: 罗慧; 李曦; 尹泉
Original assignee: Huazhong University of Science and Technology
Current assignee: Huazhong University of Science and Technology
Priority date: 2015-11-20
Filing date: 2015-11-20
Publication date: 2017-11-07
Anticipated expiration: 2035-11-20
Also published as: CN105304920A

Abstract

The present invention relates to a kind of flat solid oxide fuel cell local temperature method of estimation, comprise the following steps：S1, set up non-linear stack temperature model according to mass conservation law and law of conservation of energy, and finite element processing is carried out to the stack temperature model；S2, near stable operating point, to stack temperature model described in step S1 carry out linearization process obtain linear stack temperature model；S3, based on above-mentioned linear stack temperature model, the imperial Burger observer designed for observing stack temperature；S4, estimation SOFC pile interior temperature distribution situation.The present invention can ensure that the safety and stably of pile work, while having good robustness to electric loading disturbance and inlet air flow rate disturbance.

Description

A kind of flat solid oxide fuel cell local temperature method of estimation

Technical field

The present invention relates to temperature inside fuel cell field, more particularly to a kind of flat solid oxide fuel cell pile Spend distribution estimation method.

Background technology

SOFC (SOFC, Solid Oxide Fuel Cell) can be direct by chemical energy as one kind The system of electric energy is converted into, because it has the remarkable advantages, quilt such as high energy conversion rate, low emission and flexible fuel injection method The green energy resource electricity generation system of 21 century most prospect is described as, is increasingly paid close attention in recent years by people.In different type SOFC in, flat solid oxide fuel cell is most attractive.Flat solid is aoxidized Thing fuel cell easy to manufacture, assemble and higher power efficiency can be provided.Although flat solid oxide fuel is electric Pond technically makes great progress, but still also has many difficult points to need to overcome to be allowed to extensive use and business Change.The difficult point of wherein most critical is the temperature and thermograde in detection and control fuel cell pile.

It is a kind of high temperature fuel electricity between 600~1000 DEG C to be due to the running temperature of SOFC Pond, it is contemplated that the security of battery material, it is necessary to be controlled to the maximum temperature in pile.Furthermore, solid oxide fuel The core component PEN of battery pile is to be stacked together what is constituted in the way of similar sandwich by three layers of solid.And group This three layers of solids into PEN have different thermal expansion characters again, therefore the temperature distributing disproportionation in pile is even, that is, works as temperature When gradient is excessive, PEN may produce deformation and be even broken because thermal stress is excessive.Therefore, for solid oxide fuel Battery can be stablized, long-life operation, it is necessary to which maximum temperature and the maximum temperature gradient control of pile can be born in material Within the scope of.

Maximum temperature and maximum temperature gradient are pacified as two in SOFC pile most important temperature All referring to mark, to control it, only need to grasp the Temperature Distribution in pile.However, difficulty based on practical operation and Temperature Distribution in consideration in terms of cost, SOFC pile is difficult to direct measurement and obtained.Because solid oxygen Compound fuel cell pile is operated under hot environment, and the requirement to air-tightness is very high, it is impossible to beaten on pile Too many hole is put into thermocouple, carrys out direct measurement temperature.Except on testing stand to the temperature of SOFC monocell Monitoring may make outside some thermocouple direct measurements, typically for solid oxide fuel battery system, stack temperature Index is the gas temperature for considering pile entrance and exit.

Chinese patent (CN201410184688) is estimated there is provided a kind of local temperature of SOFC Meter method, the contribution of this method is essentially consisted in there is provided a kind of synovial membrane observer of good performance.But, although sliding mode observer The characteristics of with quick response input with good robustness, but it can come the problem of one, band is serious, i.e. system controller Output has shake.This is due to sliding mode observer according to system state at that time, purposefully super flat in sliding formwork in transition mode Constantly converted near face, force system to be moved by the state trajectory of predetermined " sliding mode ", purposefully existed in transition mode Constantly converted near sliding formwork hyperplane, result in the generation of shake.

The content of the invention

It is flat that the technical problems to be solved by the invention are to provide a kind of anti-electric loading disturbance and anti-inlet air flow rate is disturbed Plate type solid oxide fuel cell local temperature method of estimation.

The technical scheme that the present invention solves above-mentioned technical problem is as follows：A kind of flat solid oxide fuel cell pile Temperature Distribution method of estimation, comprises the following steps：

S1, non-linear stack temperature model set up according to mass conservation law and law of conservation of energy, and to the pile Temperature model carries out finite element processing；

S2, near stable operating point, to stack temperature model described in step S1 carry out linearization process obtain linearly Stack temperature model；

S3, based on above-mentioned linear stack temperature model, the imperial Burger observer designed for observing stack temperature；

S4, using the reality output of SOFC as stack temperature observer input, by described imperial primary Difference between the output of lattice observer and the reality output of SOFC is acted on as observation error feedback quantity In observer, until observation error converges to zero, SOFC pile interior temperature distribution can be now estimated Situation.

Further, non-linear stack temperature mould is set up according to mass conservation law and law of conservation of energy described in step S1 Type, is specifically referred to：

According to the molar fraction of mass balance calculation air and fuel channel,

Wherein,WithOxygen in respectively k-th node, nitrogen, mole of hydrogen and vapor Fraction,WithAir mole flow velocity and fuel mole flow velocity at respectively k-th node exit,WithRespectively For the number of moles of gas of k-th of node in negative electrode and anode, i^kFor the electric current of k-th of node；

The heat conservation of each node can be described with following formula：

The solid layer heat energy conservation of k-th of node can be described with following formula：

The dynamic characteristic of pile end solid layer：

Electrochemical model：

The operating voltage of the SOFC is equal to energy nernst voltage and subtracts three polarizing voltages：

Wherein V_cellFor the voltage of monocell piece,For the energy nernst voltage of k-th of node,For k-th node etc. Imitate resistance,WithIt is expressed as follows:

Further, it is to the concrete mode that stack temperature model carries out linearization process in step S2：According to quasistatic matter The hypothesis for measuring conservation carries out model simplification, it is assumed that heat energy dynamical equation (10) is into (13) It is constant, then SOFC meets the quasistatic conservation of mass, then new penetration quality dynamic equilibrium equation is：

From equation (14)-(17) it can be seen that the operating voltage of single SOFC cell piece be onEquation

From equation (17)-(18) if can be seen thatWithIt is known, variable Can be byIndividually determine.Due to 15 variables It is to be described by 15 algebraic equation (17)-(18).15 variables may be considered constant in heat energy dynamic equilibrium equation And SOFC model, which can simplify, to be as follows：

Wherein,

ω=I_tot,

State variable x is the temperature of air and fuel, and y is the temperature output undetermined that can be surveyed；

Following inearized model can be obtained after being linearized at stable operating point x0 to simplified nonlinear model：

Wherein Δ is the deviation of stable operating point, i.e. Δ x=x-x0, Δ u=u-u0, Δ ω=ω-ω 0.

Further, imperial Burger observer concrete mode is designed in the step S3 is, based on the linear stack temperature mould Type, the imperial Burger observer of design：

Evaluated error equation is：

Wherein

Further, the evaluated error of the imperial Burger observer designed in the step S3 is divided into air themperature evaluated error With solid temperature evaluated error, particularly：

The evaluated error of air themperature subsystem：

Wherein

The evaluated error of solid temperature subsystem：

Wherein,

Sub- observer A is configured according to direct Method of Pole Placement₁-L₁C₁And A₂-L₂C₂Characteristic value, defined feature value ratio, α is A₁-L₁C₁Characteristic value and A the ratio between characteristic value,

Wherein eig (M) representing matrixs M characteristic value, after eigenvalue ratio α is determined, can obtain L₁And L₂.Observer Gain L can be by L₁And L₂Constitute:

Wherein, [L₁₁ L₁₂ L₁₃ L₁₄ L₁₅]^T=L₁；

[L₂₁ L₂₂ L₂₃ L₂₄ L₂₅]^T=L₂

The beneficial effects of the invention are as follows：The present invention is directed to inside the SOFC pile using hydrogen as fuel Temperature Distribution is difficult to this problem measured directly, by setting up finite element mechanism mould to SOFC pile Type, and using the model as references object, set up the non-linear imperial Burger observer to pile interior temperature distribution.By selecting to close Suitable eigenvalue ratio, non-linear Luenberger observer can obtain suitable Temperature Distribution estimation performance.Present invention only requires obtain Know the entrance and outlet temperature of air and fuel, you can accurately to estimate the Temperature Distribution in pile, and then be soild oxide Fuel cell pile temperature control provides foundation, it is ensured that pile can safely and steadly work.Moreover, the present invention is to electric loading Disturbance and inlet air flow rate disturbance have good robustness.

Brief description of the drawings

Fig. 1 is signal flow diagram of the invention；

Fig. 2 is the overall structure and its operation principle schematic diagram of SOFC of the present invention；

Fig. 3 is SOFC pile finite element segmentation schematic diagram in the first embodiment of the invention；

Fig. 4 is the output pattern emulated on matlab/simulink platforms in the first of the invention embodiment.

Embodiment

The principle and feature of the present invention are described below in conjunction with accompanying drawing, the given examples are served only to explain the present invention, and It is non-to be used to limit the scope of the present invention.

As shown in figure 1, a kind of flat solid oxide fuel cell local temperature method of estimation, including following step Suddenly：

Wherein V_airAnd V_solThe air capacity and fuel quantity of respectively each node,WithIn respectively k-th node The temperature of air and fuel,WithThe air pressure of respectively k-th node negative electrode and anode.

One operation SOFC pile comprising anode, negative electrode, dielectric substrate, connector, fuel and Air, its temperature is all different.The heat transfer coefficient of solid structure, i.e. PEN and connector is much smaller than the heat transfer system of air Number.It can be considered that the temperature of PEN and connector is identical.The flow velocity of fuel is much smaller than the flow velocity of air, so fuel There is time enough to be exchanged heat with solid layer.Therefore, the temperature of fuel and solid layer is considered as identical.Using these it is assumed that The solid layer temperature of each node is not changed significantly.

The heat conservation of each node can be described with following formula：

The dynamic characteristic of pile end solid layer：

Electrochemical model：

Wherein,For the energy nernst voltage of k-th of node,For the equivalent resistance of k-th of node,WithRespectively ohmic polarization resistance, activation polarization resistance and concentration polarization resistance, can nernst voltage, be monolithic battery voltage.

Based on above-mentioned complicated and nonlinear SOFC model, design one is to estimate stack temperature The non-linear imperial Burger observer of distribution is extremely difficult.

Further, it is to the concrete mode that stack temperature model carries out linearization process in step S2：According to quasistatic matter The hypothesis for measuring conservation carries out model simplification, it is assumed that in heat energy dynamical equation (10)-(13) It is constant, then SOFC meets the quasistatic conservation of mass, then new penetration quality dynamic equilibrium equation is

Then SOFC model simplification is：

Wherein,

ω=I_tot,

Wherein Δ is the deviation of stable operating point, that is, Δ x=x-x0, Δ u=u-u0, Δ ω=ω-ω 0.

Evaluated error equation is：

Wherein

Because SOFC is many time scale systems, the gain that direct POLE PLACEMENT USING is obtained may be uncomfortable With.The exponent number of L each elements differs greatly in observer gain.Big impact it is possible that, this gain can cause this to see Survey device unstable.

Comparatively,It is rightInfluence compareInfluence more directly effectively.ThereforeEvaluated error ratioEvaluated error be more suitable for adjustmentConvergence rate.Similar Conclusion pairFor be also rational.So the error system in equation (23) can be decomposed into air themperature subsystem and Solid temperature subsystem.

The evaluated error of air themperature subsystem：

Wherein

The evaluated error of solid temperature subsystem：

Wherein,

Wherein eig (M) representing matrixs M characteristic value.After eigenvalue ratio α is determined, L can be obtained₁And L₂.Observer Gain L can be by L₁And L₂Constitute:

Wherein, [L₁₁ L₁₂ L₁₃ L₁₄ L₁₅]^T=L₁；

[L₂₁ L₂₂ L₂₃ L₂₄ L₂₅]^T=L₂

In Fig. 1, input refers to fuel and air themperature, the flow velocity of porch；Feedback oscillator is solved by imperial Burger observer； Reality output is the fuel in exit and the temperature of air；The fuel in output estimation exit and the Temperature estimate of air；Other Quantity of state is estimated as the fuel of its elsewhere and the Temperature estimate of air.

During Fig. 4 is the first of the invention embodiment, the output figure emulated on matlab/simulink platforms Shape.

The condition of emulation is：

I_load=30A.

In 2000s, the current loading of pile steps to 35A from 30A, and Fig. 4 is illustrated to air themperature distribution and fuel The estimation accuracy of Temperature Distribution.

The foregoing is only presently preferred embodiments of the present invention, be not intended to limit the invention, it is all the present invention spirit and Within principle, any modification, equivalent substitution and improvements made etc. should be included in the scope of the protection.

Claims

1. a kind of flat solid oxide fuel cell local temperature method of estimation, it is characterised in that including following step Suddenly：

S1, non-linear stack temperature model set up according to mass conservation law and law of conservation of energy, and to the stack temperature Model carries out finite element processing；

S2, near stable operating point, to stack temperature model described in step S1 carry out linearization process obtain linear pile Temperature model；

S4, using the reality output of SOFC as stack temperature observer input, by the Long Baigeguan The difference surveyed between the output of device and the reality output of SOFC acts on sight as observation error feedback quantity Device is surveyed, until observation error converges to zero, SOFC pile interior temperature distribution situation can be now estimated； Non-linear stack temperature model is set up according to mass conservation law and law of conservation of energy described in step S1, specifically referred to：

The molar fraction equation of air and fuel channel is set up according to the conservation of mass:

Wherein,WithOxygen in respectively k-th node, nitrogen, hydrogen and vapor mole point Number,WithAir mole flow velocity and fuel mole flow velocity at respectively k-th node exit,WithRespectively The number of moles of gas of k-th of node, i in negative electrode and anode^kFor the electric current of k-th of node, F is mole flow velocity；

The air layer energy conservation equation of each node is set up according to the conservation of energy：

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Pile is operated under heat-insulated environment, so provide two ends respectively and middle energy conservation equation, for intermediate node, the The solid layer energy conservation equation of k node：

Wherein,WithThe temperature of air and fuel in respectively k-th node,WithRespectively oxygen and nitrogen Specific heat capacity, C₁-C₅It is the constant coefficient determined by battery material and electrode characteristic respectively,WithRespectively hydrogen and water Enthalpy, ρ_penAnd ρ_icRespectively PEN and connector density, c_penAnd c_icRespectively PEN and connector specific heat capacity, V_cellFor monolithic Cell voltage, τ_penAnd τ_icRespectively PEN and connector thickness, V_cellFor monolithic battery voltage, R is desired air constant；

The dynamic characteristic of pile two ends solid layer：

The electric current i of k-th of node is obtained by electrochemical model^kWith monolithic battery voltage V_cellBetween functional relation；

<mrow> <msub> <mi>V</mi> <mrow> <mi>c</mi> <mi>e</mi> <mi>l</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>E</mi> <mi>N</mi> <mi>k</mi> </msubsup> <mo>-</mo> <msup> <mi>i</mi> <mi>k</mi> </msup> <mrow> <mo>(</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>h</mi> <mi>m</mi> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>a</mi> <mi>c</mi> <mi>t</mi> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>n</mi> </mrow> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>E</mi> <mi>N</mi> <mi>k</mi> </msubsup> <mo>-</mo> <msup> <mi>i</mi> <mi>k</mi> </msup> <msubsup> <mi>R</mi> <mi>s</mi> <mi>k</mi> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow> 2

Wherein,For the energy nernst voltage of k-th of node,For the equivalent resistance of k-th of node,WithIt is expressed as follows:

Wherein,For the energy nernst voltage of k-th of node,For the equivalent resistance of k-th of node,With Respectively ohmic polarization resistance, activation polarization resistance and concentration polarization resistance, can nernst voltage, be monolithic battery voltage.

2. a kind of flat solid oxide fuel cell local temperature method of estimation according to claim 1, it is special Levy and be, be to the concrete mode that stack temperature model carries out linearization process in step S2：According to the quasistatic conservation of mass Assuming that carrying out model simplification, it is assumed that heat energy dynamical equation (10) is into (13)It is constant , then SOFC meets the quasistatic conservation of mass, then new penetration quality dynamic equilibrium equation is

<mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>F</mi> <mrow> <mi>c</mi> <mi>a</mi> </mrow> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <msub> <mi>O</mi> <mn>2</mn> </msub> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>-</mo> <mrow> <mo>(</mo> <msubsup> <mi>F</mi> <mrow> <mi>c</mi> <mi>a</mi> </mrow> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>F</mi> </mrow> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </munderover> <msup> <mi>i</mi> <mi>j</mi> </msup> <mo>)</mo> </mrow> <msubsup> <mi>x</mi> <msub> <mi>O</mi> <mn>2</mn> </msub> <mi>k</mi> </msubsup> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>F</mi> </mrow> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </munderover> <msup> <mi>i</mi> <mi>j</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>F</mi> <mrow> <mi>a</mi> <mi>n</mi> </mrow> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <msub> <mi>H</mi> <mn>2</mn> </msub> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>F</mi> <mrow> <mi>a</mi> <mi>n</mi> </mrow> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <msub> <mi>H</mi> <mn>2</mn> </msub> <mi>k</mi> </msubsup> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>F</mi> </mrow> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </munderover> <msup> <mi>i</mi> <mi>j</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

Then SOFC model simplification is：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>&omega;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> <mo>=</mo> <mi>C</mi> <mi>x</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>

Wherein,

ω=I_tot,

<mrow> <mfrac> <mrow> <mi>d</mi> <mi>&Delta;</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>A</mi> <mi>&Delta;</mi> <mi>x</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>&Delta;</mi> <mi>u</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <mi>&Delta;</mi> <mi>&omega;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>

3. a kind of flat solid oxide fuel cell local temperature method of estimation according to claim 2, it is special Levy and be, imperial Burger observer concrete mode is designed in the step S3 is, based on the linear stack temperature model, design dragon Burger observer：

<mrow> <mfrac> <mrow> <mi>d</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>A</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <mi>&omega;</mi> <mo>+</mo> <mi>L</mi> <mrow> <mo>(</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>-</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>

Evaluated error equation is：

WhereinFor x evaluated error,For x estimate,For y estimate, L is gain matrix.

4. a kind of flat solid oxide fuel cell local temperature method of estimation according to claim 3, it is special Levy and be, the evaluated error of the imperial Burger observer designed in the step S3 is divided into air themperature evaluated error and solid temperature Evaluated error is spent, particularly：

The observer evaluated error of air themperature subsystem：

Wherein

The observer evaluated error of solid temperature subsystem：

Wherein,L₁And L₂The respectively observer gain square of air themperature subsystem and solid temperature subsystem Battle array, A₁And A₂The respectively state matrix of air themperature subsystem and solid temperature subsystem, C₁And C₂Respectively air themperature is sub The output matrix of system and solid temperature subsystem；

Sub- observer A is configured according to direct Method of Pole Placement₁-L₁C₁And A₂-L₂C₂Characteristic value, defined feature value ratio, α is A₁- L₁C₁Characteristic value and A the ratio between characteristic value,

<mrow> <mi>&alpha;</mi> <mo>=</mo> <mfrac> <mrow> <mi>e</mi> <mi>i</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>e</mi> <mi>i</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>e</mi> <mi>i</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>e</mi> <mi>i</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>

Wherein eig (M) representing matrixs M characteristic value, after eigenvalue ratio α is determined, can obtain L₁And L₂, observer gain Matrix L is by L₁And L₂Constitute:

Wherein, [L₁₁ L₁₂ L₁₃ L₁₄ L₁₅]^T=L₁；

[L₂₁ L₂₂ L₂₃ L₂₄ L₂₅]^T=L₂。