CN105371872B

CN105371872B - Gyroscope flywheel system disturbance method of estimation based on extension High-gain observer

Info

Publication number: CN105371872B
Application number: CN201510990028.9A
Authority: CN
Inventors: 刘晓坤; 赵辉; 马克茂; 霍鑫; 史维佳; 姚郁
Original assignee: Harbin Institute of Technology
Current assignee: Harbin Institute of Technology
Priority date: 2015-12-24
Filing date: 2015-12-24
Publication date: 2017-11-14
Anticipated expiration: 2035-12-24
Also published as: CN105371872A

Abstract

The present invention is the gyroscope flywheel system disturbance method of estimation based on extension High-gain observer, belongs to inertial navigation field.The present invention proposes the gyroscope flywheel system disturbance method of estimation based on extension High-gain observer to solve dynamic disturbances estimation problem of the gyroscope flywheel in the big angle of heel working condition of rotor.The inventive method includes：Step 1: according to the kinetics equation of gyroscope flywheel system, the gyroscope flywheel system state equation containing unknown disturbance is established；Step 2: according to the gyroscope flywheel system state equation containing unknown disturbance, design extension High-gain observer；Step 3: the checking of observation error convergence and Design of Observer parameter ε regulations；Step 4: gyroscope flywheel system disturbance is estimated.The present invention is applied to gyroscope flywheel system disturbance and estimated.

Description

Gyroscope flywheel system disturbance method of estimation based on extension High-gain observer

Technical field

The present invention is the gyroscope flywheel system disturbance method of estimation based on extension High-gain observer, and in particular to inertia is led Boat field.

Background technology

Gyroscope flywheel is a kind of to have actuator and the electromechanical servo device of sensor function, its base concurrently applied to spacecraft Developed in the physical arrangement of traditional inertial instruments-dynamically tuned gyro, DTG.But gyroscope flywheel and dynamic tuning top Spiral shell instrument functionally significantly different is that gyroscope flywheel not only realizes the two-dimensional carrier angular speed as dynamically tuned gyro, DTG Measure function, moreover it is possible to realize Three dimensions control torque output function.And gyroscope flywheel is to realize three-dimensional moment output function, gyro flies Wheel rotor need to radially produce the rolling motion of wide-angle in bidimensional, need to axially produce speed governing campaign, these motion states show Difference is write, gyroscope flywheel is result in and realizes that two-dimentional spacecraft angular rate measurement function is adjusted compared with power on the basis of three-dimensional moment output Humorous gyroscope is increasingly complex.

It is required to realize drift error compensation in dynamically tuned gyro, DTG development, to overcome the influence of undesirable factor, from And ensure measurement accuracy.And the static drift error compensating method that dynamically tuned gyro, DTG is commonly used is servo turntable method and power Square feedback transmitter, both static drift error compensating methods can be to caused by the factors such as such as different elasticity, mass unbalance Systematic error carries out calibration compensation；But because this method is a kind of static scaling method, can not be to as gyroscope flywheel The system that works long hours in big rolling motion state of rotor carry out complete calibration compensation, it is therefore desirable to further research is dynamic The disturbance method of estimation of state is estimated to being estimated due to drift error caused by the big rolling motion of gyroscope flywheel rotor using experiment Count obtained test data sequence and compensation is modeled to disturbance.

Kalman Filter Technology is a kind of optimal State Estimation method, and it is by recursive algorithm, by the presence obtained in real time The discrete experimental data of noise pollution, the optimal estimation of unbiased and minimum variance is carried out to system mode.But Kalman filtering Dependent on accurate complete system mathematic model, and to gyroscope flywheel system, the disturbance caused by undesirable factor is unknown, and It is difficult to it is modeled, so the disturbance estimation for carrying out gyroscope flywheel system using Kalman filtering has larger difficulty.Base It is a kind of Nonlinear Observer to develop extended mode observer in active disturbance rejection thought, such a Nonlinear Observer Disturbance expansion in gyroscope flywheel system can be single order state by method, be fed back using specific nonsmooth nonlinearities error, knot The design parameter properly worked as, realize to stateful observation.Although numerous scholars are to such a Nonlinear Observer method Have made intensive studies, still, Nonlinear Observer method causes parameter tuning difficult because design parameter is numerous, and Nonlinear Observer method uses continuous nonsmooth nonlinearities structure, it is difficult to is received with traditional Design of Observer theory Holding back property and observation error analysis, are not well solved especially yet to the error convergence problem of high-order (more than second order) system. To gyroscope flywheel system, if carrying out disturbance estimation, the selection of relatively reasonable design parameter using Nonlinear Observer method And error convergence proves the relatively complicated complexity that then seems, realizes that difficulty is larger.

The content of the invention

The present invention proposes base to solve dynamic disturbances estimation problem of the gyroscope flywheel in big angle of heel working condition In the gyroscope flywheel system disturbance method of estimation of extension High-gain observer, comprise the following steps：

Step 1: according to the kinetics equation of gyroscope flywheel system, the gyroscope flywheel system influenceed containing unknown disturbance is built System state equation；

The radially movable bidimensional angle of heel φ of tilt sensor direct measurement rotor of gyroscope flywheel system_x,φ_y, utilize The gyroscope flywheel system dynamics equation that two class Lagrangian methods are established, by corresponding coordinate transform, by not considering for system The ideal kinetics equation of disturbance is converted to the φ that can be obtained by sensor direct measurement_x,φ_yUnder the housing coordinate system at place, So as to obtain considering the gyroscope flywheel system state equation for not modeling disturbance；

From gyroscope flywheel rotor two-dimensional direction angle of heel (φ_x,φ_y) and tilt angular speedAs state Variable：Then containing the gyroscope flywheel system mode for not modeling disturbance Shown in equation such as formula (1)：

Shown in measurement equation such as formula (2)：

Wherein, f₁(x,t),f₂(x, t) is represented ideally, the Nonlinear Mechanism item of gyroscope flywheel；u_x,u_yRepresent two Tie up the control moment of torquer, g_x1(x,t),g_x2(x,t),g_y1(x,t),g_y2(x, t) represents the nonlinear factor of bidimensional torquer ；

σ_x(x,t),σ_y(x, t) expression system does not model disturbance term；y₁,y₂Represent that measurable gyroscope flywheel turns respectively Sub- bidimensional angle of heel (φ_x,φ_y)；

After being arranged by formula (1) (2), formula (3) can be obtained：

Wherein, σ_d(x, t) does not model nonlinear disturbance item for the continuous bounded of gyroscope flywheel system；U is that bidimensional continuously has Boundary's control input, i.e. bidimensional torquer export；F (x, t), g (x, t) are nominal model, and are twice continuously differentiable bounded Nonlinear function；

Wherein,

Wherein, C_θWith S_θExpression formula be respectively rotational angle theta cosine value cos θ and sine value sin θ；

I_rx,I_ry,I_rzRespectively rotor is known quantity in the principal axis of inertia direction rotary inertia of rotor block coordinate system three；

I_gx,I_gy,I_gzRespectively balance ring is balancing the principal axis of inertia direction rotary inertia of ring body coordinate system three, is known quantity；

k_x,k_yRespectively known flexible support torsion bar torsional rigidity；c_gx,c_gyRespectively known flexible support damping system Number；

T_cx=k_tyi_y,T_cy=k_txi_xRespectively bidimensional torquer is exported to the control moment of rotor, i.e., in equation (1) u_x,u_y；

k_tx,k_tyThe respectively scale factor of known sensor, i_x,i_yThe respectively electric current of bidimensional torquer, it is sensor It is measurable；

θ_z,Gyroscope flywheel motor Shaft angle and rotating speed are represented respectively, are that sensor is measurable；

θ in equation_x,θ_y,φ_z,I₁,I₂,β₁,β₂, η is intermediate variable, and concrete form difference is as follows：

I₁=I_gx+I_rxcos²θ_y+I_rzsin²θ_y

I₂=I_gz-I_gy-I_ry+I_rxsin²θ_y+I_rzcos²θ_y；

Step 2: according to the gyroscope flywheel system state equation containing unknown disturbance, can be surveyed using bidimensional tilt sensor Measure φ_x,φ_y, design extension High-gain observer；

To utilize measurement equation y=Cx, realize to state variable x and nonlinear disturbance item σ_dThe accurate estimation of (x, t), if Count High-gain observer extended below：

Wherein,For High-gain observer state variable；To extend High-gain observer state variable；

H (ε), F (ε) are the gain matrix of observer, and its concrete form is as follows：

Wherein, design parameter ε>0 is small design parameter；Design parameter α_ij, i=1,2,3, j=1,2 are selected as reality Number；

Step 3: the checking of observation error convergence and Design of Observer parameter ε regulations；

The observation error convergence of the designed gyroscope flywheel extension High-gain observer of analysis, according to accuracy of observation need Ask, adjust and provide applicable extension High-gain observer design parameter；

Define error vectorI.e.By the first formula in formula (3) and the formula of formula (4) first Make the difference, and it is state that the nonlinear terms made the difference in rear gained equation are carried out into integral extensionFormula is obtained after arrangement (6)：

Wherein,

For the nonlinear terms of formula (7)；

By the formula of equation (4) secondThe 3rd formula and the 6th formula of equation (6) are brought into, and is organized into matrix form, Such as formula (7)：

Formula (7) can be further abbreviated as：

Wherein,

According to state equation (8), nonlinear terms δ can be considered the disturbance input of system, stateIt is considered as system output, then the phase HopeMiddle h_i, i=1,2...6's is designed to δ pairs of counteractingInfluence, realize the asymptotic convergence of state observation error, consider by Disturbance input δ is to state outputTransmission function, to (8) carry out Laplace transformation, obtain formula (9)：

Formula (9) further spread out for：

Wherein,

Remember respectivelyAccording to equation (9), if transmission function G₁(s),G₂(s) equal identically vanishing, then It is completely counterbalanced by nonlinear disturbance and inputs δ to state output errorInfluence, accurately realize estimating for gyroscope flywheel system total state Meter；Select design parameter h₁~h₆, to ω ∈ R, make the Infinite Norm shown in formula (11) simultaneously arbitrarily small；

IfChoose

Wherein, α_ij, i=1,2,3；J=1,2 meets the Hurwitz multinomials shown in following (12)；ε is normal number, and ε ＜＜ 1；

s³+α_1js²+α_2js+α_3j, j=1,2 (12)

By h₁~h₆Bring into G_j(s) in, can obtain：

Wherein, P (s)=(ε s)³+α_1j(εs)²+α_2j(εs)+α_3j, j=1,2；According to formula (13), The value for extending High-gain observer by reducing ε is designed, disturbance estimated accuracy is improved, realizes required precision index；

Step 4: realize that gyroscope flywheel system disturbance is estimated；

The disturbance observation of gyroscope flywheel system is carried out using the High-gain observer after extension, with reference to step 3 adjusted design Parameter ε, the disturbance characterized using multivariate regression models is estimated, until observation data meet expectation estimation precision index, Disturbance estimation test data sequence caused by the big rolling motion of gyroscope flywheel system is obtained, realizes that gyroscope flywheel system does not model Disturbance term σ_dx(x,t),σ_dyThe estimation of (x, t).

Beneficial effect of the present invention：

1st, the inventive method utilizes designed extension High-gain observer, fully utilizes bidimensional tilt sensor measurement letter Breath and the error of observation information, largely weaken and adversely affected caused by calculus of differences, improve observation accuracy；

2nd, the inventive method is compared with traditional mechanical gyroscope uses static error method, the present invention is directed by top Spiral shell flywheel rotor it is big tilt angular movement caused by disturbance estimated, be a kind of dynamic error estimation, available for Disturbance estimation and compensation further are carried out to gyroscope flywheel measurement equation on the basis of static error demarcation, are existing error calibration skill A kind of complementary technology of art.

Brief description of the drawings

Fig. 1 is the schematic process flow diagram of the gyroscope flywheel system disturbance method of estimation based on extension High-gain observer；

When Fig. 2 is design parameter ε=0.1, disturbance estimate figure of the gyroscope flywheel in x-axis；

When Fig. 3 is design parameter ε=0.1, disturbance estimate figure of the gyroscope flywheel in y-axis；

When Fig. 4 is design parameter ε=0.1, disturbance evaluated error figure of the gyroscope flywheel in x-axis；

When Fig. 5 is design parameter ε=0.1, disturbance evaluated error figure of the gyroscope flywheel in y-axis；

When Fig. 6 is design parameter ε=0.001, disturbance estimate figure of the gyroscope flywheel in x-axis；

When Fig. 7 is design parameter ε=0.001, disturbance estimate figure of the gyroscope flywheel in y-axis；

When Fig. 8 is design parameter ε=0.001, gyroscope flywheel disturbs evaluated error figure in x-axis；

When Fig. 9 is design parameter ε=0.001, disturbance evaluated error figure of the gyroscope flywheel in y-axis.

Embodiment

Embodiment one：Gyroscope flywheel system disturbance estimation side of the present embodiment based on extension High-gain observer Method is realized according to following steps：

Step 1: according to the kinetics equation of gyroscope flywheel system, the gyroscope flywheel system shape containing unknown disturbance is established State equation；

Step 2: according to the gyroscope flywheel system state equation containing unknown disturbance, design extension High-gain observer；

Step 3: observation error convergence and Design of Observer parameter ε regulations；

Step 4: realize that gyroscope flywheel system disturbance is estimated.

Embodiment two：Present embodiment is unlike embodiment one：Characterized in that, described step The rapid one gyroscope flywheel system state equation containing unknown disturbance is realized according to following steps：

Angle of heel (φ of the gyroscope flywheel rotor in two-dimensional direction_x,φ_y) and tilt angular speedAs state variable x：Then containing the gyroscope flywheel system state equation for not modeling disturbance As shown in formula (1)：

Shown in measurement equation such as formula (2)：

After being arranged by formula (1) (2), formula (3) can be obtained：

Wherein, σ_d(x, t) does not model nonlinear disturbance item for the continuous bounded of gyroscope flywheel system；U is that bidimensional continuously has Boundary's control input, i.e. bidimensional torquer export；F (x, t), g (x, t) are nominal model, and non-for twice continuously differentiable bounded Linear function；

Wherein,

I₁=I_gx+I_rxcos²θ_y+I_rzsin²θ_y。

I₂=I_gz-I_gy-I_ry+I_rxsin²θ_y+I_rzcos²θ_y；

Embodiment three：Present embodiment is unlike embodiment one or two：It is characterized in that, described The step of two design extension High-gain observers realized according to following steps：

Using measurement equation y=Cx, realize to state variable x and nonlinear disturbance item σ_dThe estimation of (x, t), design are as follows Extend High-gain observer：

Wherein, design parameter ε>0 is small design parameter；Design parameter α_ij, i=1,2,3, j=1,2 are selected as reality Number, and following Hurwitz multinomials should be met：

s³+α_1js²+α_2js+α_3j, j=1,2

Other steps and parameter are identical with one of embodiment one to two.

Embodiment four：Unlike one of present embodiment and embodiment one to three：Characterized in that, Described step three observation error convergence and Design of Observer parameter ε regulations are realized according to following steps：

Wherein,

For the nonlinear terms of formula (7)；

Formula (7) can be further abbreviated as：

Wherein,

According to state equation (8), nonlinear terms δ can be considered the disturbance input of system, stateIt is considered as system output, then the phase HopeMiddle h_i, i=1,2...6's is designed to δ pairs of counteractingInfluence, realize the asymptotic convergence of state observation error, consider By disturbance input δ to state outputTransmission function, to (8) carry out Laplace transformation, obtain formula (9)：

Formula (9) further spread out for：

Wherein,

Remember respectivelyAccording to equation (9), if transmission function G₁(s),G₂(s) equal identically vanishing, then it is complete Full nonlinear disturbance of offsetting inputs δ to state output errorInfluence, accurately realize the estimation of gyroscope flywheel system total state； Select design parameter h₁~h₆, to ω ∈ R, make the Infinite Norm shown in formula (11) simultaneously arbitrarily small；

IfChoose

s³+α_1js²+α_2js+α_3j, j=1,2 (12)

By h₁~h₆Bring into G_j(s) in, can obtain：

Wherein P (s)=(ε s)³+α_1j(εs)²+α_2j(εs)+α_3j, j=1,2；According to formula (13), The value for extending High-gain observer by reducing ε is designed, disturbance estimated accuracy is improved, realizes required precision index；

Other steps and parameter are identical with one of embodiment one to three.

Embodiment five：Unlike one of present embodiment and embodiment one to four：Characterized in that, Described step four gyroscope flywheel system disturbance estimation is realized according to following steps：

The disturbance observation of gyroscope flywheel system is carried out using the High-gain observer after the extension designed by step 2, with reference to Step 3 adjusted design parameter ε, the disturbance of the gyroscope flywheel system to being characterized using multivariate regression models is estimated, makes observation Data meet expectation estimation precision index, obtain disturbance estimation test data sequence caused by the big rolling motion of gyroscope flywheel system Row, realize the disturbance term σ that gyroscope flywheel system does not model_dx(x,t),σ_dyThe estimation of (x, t)；

Other steps and parameter are identical with one of embodiment one to four.

Embodiment

If gyroscope flywheel system disturbance σ_kShown in mathematical description such as formula (14)：

Wherein, a, b_i,c_jIt is constant value undetermined coefficient, m, n are the exponent number of the selection of multivariate regression models, f=ω_m/(2π) For time-varying motor rotation frequency, ω_mFor motor speed；σ in gyroscope flywheel physical system mathematical modeling such as formula (1) during emulation_x, σ_yIt is expressed as shown in formula (15)：

Formula (15) extends the estimation target of High-gain observer in testing；

The rotary inertia J of gyroscope flywheel system rotor, balance ring_r,J_gIt is given as respectively：

Wherein, J_rxy,J_rxz,J_ryzThe disturbance factor as caused by rotor unbalance is represented, it is influenceed in gyroscope flywheel system side σ is presented as in journey such as formula (1)_x(x,t),σ_y(x, t) unknown disturbance item, disturbing influence is characterized by formula (15) in emulation；

The torsional rigidity k and damped coefficient c of two pairs of torsion bars in gyroscope flywheel system_gIt is set to：

K=0.092Nm/rad；c_g=2.5e-4Nms/rad

If gyroscope flywheel is in torque output state, the axle input instruction of gyroscope flywheel three difference in the direction of principal axis of spacecraft three For motor driving shaft rotary speed instruction：ω'_m=23sin (0.2 π t)+157rad/s, rotor radially x-axis, the bidimensional of y-axis Angle of heel instructs：

φ'_y=1sin (0.2 π t) °

Extend High-gain observer design parameter α_ij, i=1,2,3, j=1,2 are as follows：

α_1j=12.6；α_2j=312.1；α_3j=3225.3；J=1,2

When not considering to measure noise according to gyroscope flywheel parameter setting and input instruction, in sensor, ε takes 0.1 respectively, When 0.001, system disturbance (σ_x,σ_y) observation effect respectively as shown in Fig. 2-Fig. 9.

Understood according to Fig. 2-Fig. 9, as ε=0.1, observation error is located at 10^-3In the order of magnitude；As ε=0.001, observation Error is located at 10^-4In the order of magnitude, observation error improves an order of magnitude；Gyro is flown using designed High-gain observer Take turns the disturbance estimation of two radial axlesReach preferable observation effect, the estimated accuracy of disturbance with the reduction of ε values and Improve.

Claims

1. the gyroscope flywheel system disturbance method of estimation based on extension High-gain observer, it is characterised in that described based on expansion The gyroscope flywheel system disturbance method of estimation for opening up High-gain observer is realized according to following steps：

Step 1: according to the kinetics equation of gyroscope flywheel system, the gyroscope flywheel system mode side containing unknown disturbance is established Journey；

Angle of heel (φ of the gyroscope flywheel rotor in bidimensional radial direction_x,φ_y) and tilt angular speedAs state variable x：Then containing not modeling the gyroscope flywheel system state equation of disturbance such as Shown in formula (1)：

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Shown in system measurements equation such as formula (2)：

Wherein, f₁(x,t),f₂(x, t) is represented ideally, the Nonlinear Mechanism item of gyroscope flywheel；u_x,u_yRepresent bidimensional torque The control moment of device, g_x1(x,t),g_x2(x,t),g_y1(x,t),g_y2(x, t) represents that the nonlinear system of bidimensional torquer is several；σ_x (x,t),σ_y(x, t) expression system does not model disturbance term；y₁,y₂Measurable gyroscope flywheel rotor bidimensional tilt is represented respectively Angle (φ_x,φ_y)；

After being arranged by formula (1) (2), formula (3) can be obtained：

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Wherein, σ_d(x, t) does not model nonlinear disturbance item for the continuous bounded of gyroscope flywheel system；U is the continuous bounded control of bidimensional Input, i.e. bidimensional torquer export；F (x, t), g (x, t) are nominal model, and are the non-linear letter of twice continuously differentiable bounded Number；

Wherein,

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&sigma;</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&sigma;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&sigma;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </mtd> <mtd> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>g</mi> <mrow> <mi>x</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>g</mi> <mrow> <mi>y</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>g</mi> <mrow> <mi>x</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>g</mi> <mrow> <mi>y</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> 1

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> </msub> </mrow> <msubsup> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> <mn>2</mn> </msubsup> </mfrac> <mo>&lsqb;</mo> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>+</mo> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>)</mo> </mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> </msub> </mfrac> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <mo>)</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>-</mo> <mo>(</mo> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>+</mo> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>)</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>z</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>-</mo> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>(</mo> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>-</mo> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>)</mo> </mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <msub> <mi>&beta;</mi> <mn>1</mn> </msub> <msub> <mi>I</mi> <mn>1</mn> </msub> </mfrac> <mo>(</mo> <mo>-</mo> <msub> <mi>c</mi> <mrow> <mi>g</mi> <mi>x</mi> </mrow> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> <mo>+</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>I</mi> <mn>2</mn> </msub> <msub> <mi>S</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </mrow> </msub> <mo>&CenterDot;</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>S</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>&rsqb;</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>-</mo> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> <mo>&rsqb;</mo> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mi>&eta;</mi> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> </mfrac> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>c</mi> <mrow> <mi>g</mi> <mi>y</mi> </mrow> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>-</mo> <msub> <mi>k</mi> <mi>y</mi> </msub> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> <mo>-</mo> <mo>&lsqb;</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msubsup> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> <mn>2</mn> </msubsup> <msub> <mi>S</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>&rsqb;</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mo>&lsqb;</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>S</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>&rsqb;</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> <mo>&rsqb;</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>;</mo> </mrow>

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>)</mo> </mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>-</mo> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>)</mo> </mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>z</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mrow> <mo>&lsqb;</mo> <mrow> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>+</mo> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>-</mo> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <msub> <mi>&beta;</mi> <mn>2</mn> </msub> <msub> <mi>I</mi> <mn>1</mn> </msub> </mfrac> <mo>(</mo> <mo>-</mo> <msub> <mi>c</mi> <mrow> <mi>g</mi> <mi>x</mi> </mrow> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>I</mi> <mn>2</mn> </msub> <msub> <mi>S</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </mrow> </msub> <mo>&CenterDot;</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>S</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>&rsqb;</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>-</mo> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> <mo>&rsqb;</mo> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> </mfrac> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>c</mi> <mrow> <mi>g</mi> <mi>y</mi> </mrow> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>-</mo> <msub> <mi>k</mi> <mi>y</mi> </msub> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> <mo>-</mo> <mo>&lsqb;</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msubsup> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> <mn>2</mn> </msubsup> <msub> <mi>S</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>&rsqb;</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> <mn>2</mn> </msubsup> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&lsqb;</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>S</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>&rsqb;</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mrow> <mn>2</mn> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> <mo>&rsqb;</mo> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>;</mo> </mrow>

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>g</mi> <mrow> <mi>x</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msub> <mi>&beta;</mi> <mn>1</mn> </msub> <msub> <mi>I</mi> <mn>1</mn> </msub> </mfrac> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <mo>+</mo> <mfrac> <mi>&eta;</mi> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> </mfrac> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>g</mi> <mrow> <mi>y</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msub> <mi>&beta;</mi> <mn>1</mn> </msub> <msub> <mi>I</mi> <mn>1</mn> </msub> </mfrac> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <mo>-</mo> <mfrac> <mi>&eta;</mi> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> </mfrac> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>g</mi> <mrow> <mi>x</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msub> <mi>&beta;</mi> <mn>2</mn> </msub> <msub> <mi>I</mi> <mn>1</mn> </msub> </mfrac> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <mo>-</mo> <mfrac> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> </mfrac> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>g</mi> <mrow> <mi>y</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msub> <mi>&beta;</mi> <mn>2</mn> </msub> <msub> <mi>I</mi> <mn>1</mn> </msub> </mfrac> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <mo>+</mo> <mfrac> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>y</mi> </mrow> </msub> </mfrac> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>

k_x,k_yRespectively known flexible support torsion bar torsional rigidity；c_gx,c_gyRespectively known flexible support damped coefficient；

T_cx=k_tyi_y,T_cy=k_txi_xRespectively bidimensional torquer is exported to the control moment of rotor, i.e. u in equation (1)_x,u_y；

k_tx,k_tyThe respectively scale factor of known sensor, i_x,i_yThe respectively electric current of bidimensional torquer, it can be surveyed for sensor Amount；

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> <mo>=</mo> <mi>arcsin</mi> <mrow> <mo>(</mo> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>+</mo> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> <mo>=</mo> <mi>arcsin</mi> <mrow> <mo>(</mo> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <mo>-</mo> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>z</mi> </msub> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> </mfrac> <mrow> <mo>(</mo> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>C</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>y</mi> </msub> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </mtd> <mtd> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>=</mo> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>-</mo> <msub> <mi>C</mi> <msub> <mi>&phi;</mi> <mi>y</mi> </msub> </msub> <msub> <mi>S</mi> <msub> <mi>&phi;</mi> <mi>z</mi> </msub> </msub> <msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>S</mi> <msub> <mi>&theta;</mi> <mi>x</mi> </msub> </msub> <msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> 2

I₁=I_gx+I_rxcos²θ_y+I_rzsin²θ_y

I₂=I_gz-I_gy-I_ry+I_rxsin²θ_y+I_rzcos²θ_y；

Using measurement equation y=Cx, realize to state variable x and nonlinear disturbance item σ_dThe estimation of (x, t), design are extended below High-gain observer：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mi>A</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>+</mo> <mi>B</mi> <mo>&lsqb;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>u</mi> <mo>-</mo> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mo>&rsqb;</mo> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <mi>&epsiv;</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <mi>C</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>&epsiv;</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <mi>C</mi> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>&epsiv;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>h</mi> <mn>1</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>h</mi> <mn>2</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>h</mi> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>h</mi> <mn>4</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <msub> <mi>&alpha;</mi> <mn>11</mn> </msub> <mi>&epsiv;</mi> </mfrac> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <msub> <mi>&alpha;</mi> <mn>21</mn> </msub> <msup> <mi>&epsiv;</mi> <mn>2</mn> </msup> </mfrac> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mfrac> <msub> <mi>&alpha;</mi> <mn>12</mn> </msub> <mi>&epsiv;</mi> </mfrac> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mfrac> <msub> <mi>&alpha;</mi> <mn>22</mn> </msub> <msup> <mi>&epsiv;</mi> <mn>2</mn> </msup> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </mtd> <mtd> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mi>&epsiv;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>5</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>6</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&alpha;</mi> <mn>31</mn> </msub> <msup> <mi>&epsiv;</mi> <mn>3</mn> </msup> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mfrac> <msub> <mi>&alpha;</mi> <mn>32</mn> </msub> <msup> <mi>&epsiv;</mi> <mn>3</mn> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

Wherein, design parameter ε>0 is small design parameter；Design parameter α_ij, i=1,2,3, j=1,2 are selected as real number；

The observation error convergence of the gyroscope flywheel extension High-gain observer of Observation Design, according to accuracy of observation demand, adjustment Obtain applicable extension High-gain observer design parameter；

Define error vectorI.e.First formula in formula (3) and the formula of formula (4) first are made the difference, And it is state that the nonlinear terms made the difference in rear gained equation are carried out into integral extensionFormula (6) is obtained after arrangement：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&eta;</mi> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mi>x</mi> </msub> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mi>x</mi> </msub> </msub> <mo>=</mo> <msub> <mover> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>+</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>f</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>f</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mover> <mi>g</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>e</mi> <mi>x</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>+</mo> <msubsup> <mover> <mi>g</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>e</mi> <mi>y</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>3</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>h</mi> <mn>3</mn> </msub> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>4</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>h</mi> <mn>4</mn> </msub> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>&eta;</mi> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mi>y</mi> </msub> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mi>y</mi> </msub> </msub> <mo>=</mo> <msub> <mover> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>+</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>f</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>f</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mover> <mi>g</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>e</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>+</mo> <msubsup> <mover> <mi>g</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>e</mi> <mi>y</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> 3

Wherein,

For the nonlinear terms of formula (7)；

By the formula of equation (4) secondThe 3rd formula and the 6th formula of equation (6) are brought into, and is organized into matrix form, it is such as public Formula (7)：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>4</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&eta;</mi> <mo>&CenterDot;</mo> </mover> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>5</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>3</mn> </msub> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>4</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>6</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&eta;</mi> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&eta;</mi> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&delta;</mi> <mn>1</mn> </msub> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&delta;</mi> <mn>2</mn> </msub> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

Formula (7) can be further abbreviated as：

<mrow> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>A</mi> <mo>~</mo> </mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>+</mo> <mover> <mi>B</mi> <mo>~</mo> </mover> <mi>&delta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

Wherein,

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&eta;</mi> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mi>x</mi> </msub> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&eta;</mi> <msub> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mi>y</mi> </msub> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </mtd> <mtd> <mrow> <mover> <mi>A</mi> <mo>~</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>5</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>3</mn> </msub> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>4</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>h</mi> <mn>6</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </mtd> <mtd> <mrow> <mover> <mi>B</mi> <mo>~</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </mtd> <mtd> <mrow> <mi>&delta;</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&delta;</mi> <mn>1</mn> </msub> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&delta;</mi> <mn>2</mn> </msub> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>&sigma;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>f</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>f</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>g</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>e</mi> <mi>x</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>+</mo> <msubsup> <mover> <mi>g</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>e</mi> <mi>y</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>&sigma;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>f</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>f</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>g</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>e</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>+</mo> <msubsup> <mover> <mi>g</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>e</mi> <mi>y</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>

According to state equation (8), nonlinear terms δ can be considered the disturbance input of system, stateIt is considered as system output, then it is expected Middle h_i, i=1,2...6's is designed to δ pairs of counteractingInfluence, realize the asymptotic convergence of state observation error, consider by disturbing δ is inputted to state outputTransmission function, to (8) carry out Laplace transformation, obtain formula (9)：

<mrow> <mfrac> <mrow> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mi>s</mi> <mi>I</mi> <mo>-</mo> <mover> <mi>A</mi> <mo>~</mo> </mover> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mi>B</mi> <mo>~</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

Formula (9) further spread out for：

<mrow> <mfrac> <mrow> <mover> <mi>x</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>G</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>G</mi> <mn>21</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>G</mi> <mn>31</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>G</mi> <mn>42</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>G</mi> <mn>52</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>G</mi> <mn>62</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

Wherein,

Remember respectivelyAccording to equation (9), if transmission function G₁(s),G₂(s) equal identically vanishing, then support completely The nonlinear disturbance that disappears inputs δ to state output errorInfluence, accurately realize the estimation of gyroscope flywheel system total state；Selection Design parameter h₁~h₆, to ω ∈ R, make the Infinite Norm shown in formula (11) simultaneously arbitrarily small；

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>&omega;</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mi>&infin;</mi> </msub> <mo>=</mo> <munder> <mi>max</mi> <mrow> <mn>1</mn> <mo><</mo> <mi>i</mi> <mo>&le;</mo> <mn>3</mn> </mrow> </munder> <munder> <mi>sup</mi> <mi>&omega;</mi> </munder> <mo>|</mo> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>&omega;</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>&omega;</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mi>&infin;</mi> </msub> <mo>=</mo> <munder> <mi>max</mi> <mrow> <mn>4</mn> <mo><</mo> <mi>i</mi> <mo>&le;</mo> <mn>6</mn> </mrow> </munder> <munder> <mi>sup</mi> <mi>&omega;</mi> </munder> <mo>|</mo> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mi>&omega;</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

IfChoose

s³+α_1js²+α_2js+α_3j, j=1,2 (12)

By h₁~h₆Bring into G_j(s) in, can obtain：

<mrow> <msub> <mi>G</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>&epsiv;</mi> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mi>&epsiv;</mi> <mn>2</mn> </msup> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&epsiv;</mi> <mi>s</mi> <mo>+</mo> <msub> <mi>&alpha;</mi> <mrow> <mn>1</mn> <mi>j</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mi>&epsiv;</mi> <mi>s</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>&alpha;</mi> <mrow> <mn>1</mn> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&epsiv;</mi> <mi>s</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&alpha;</mi> <mrow> <mn>2</mn> <mi>j</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>

Wherein P (s)=(ε s)³+α_1j(εs)²+α_2j(εs)+α_3j, j=1,2；According to formula (13), design expands Exhibition High-gain observer improves disturbance estimated accuracy, realizes required precision index by reducing ε value；

Step 4: realize that gyroscope flywheel system disturbance is estimated；

The disturbance observation of gyroscope flywheel system is carried out using the High-gain observer after the extension of step 2 design, with reference to step 3 Adjusted design parameter ε, the disturbance of the gyroscope flywheel system to being characterized using multivariate regression models is estimated, expires observation data Sufficient expectation estimation precision index, disturbance estimation test data sequence caused by the big rolling motion of gyroscope flywheel system is obtained, it is real The disturbance term σ that existing gyroscope flywheel system does not model_dx(x,t),σ_dyThe estimation of (x, t).