CN104793645B - A kind of magnetic levitation ball position control method - Google Patents
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- 238000005339 levitation Methods 0.000 title claims abstract description 34
- 238000000034 method Methods 0.000 title claims abstract description 21
- 229910000831 Steel Inorganic materials 0.000 claims abstract description 44
- 239000010959 steel Substances 0.000 claims abstract description 44
- 238000004804 winding Methods 0.000 claims abstract description 21
- 238000012886 linear function Methods 0.000 claims abstract description 5
- 238000005457 optimization Methods 0.000 claims description 12
- 239000011159 matrix material Substances 0.000 claims description 8
- 238000005312 nonlinear dynamic Methods 0.000 abstract 1
- 230000006641 stabilisation Effects 0.000 abstract 1
- 238000011105 stabilization Methods 0.000 abstract 1
- 239000000725 suspension Substances 0.000 description 9
- 230000005672 electromagnetic field Effects 0.000 description 8
- 238000005516 engineering process Methods 0.000 description 5
- 238000004364 calculation method Methods 0.000 description 3
- 230000000694 effects Effects 0.000 description 3
- 230000003044 adaptive effect Effects 0.000 description 2
- 238000013528 artificial neural network Methods 0.000 description 2
- 230000005674 electromagnetic induction Effects 0.000 description 2
- 230000005484 gravity Effects 0.000 description 2
- 241000287196 Asthenes Species 0.000 description 1
- 230000002411 adverse Effects 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 238000011217 control strategy Methods 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 238000010586 diagram Methods 0.000 description 1
- 230000004907 flux Effects 0.000 description 1
- 230000006698 induction Effects 0.000 description 1
- 230000010354 integration Effects 0.000 description 1
- 238000005070 sampling Methods 0.000 description 1
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Abstract
The invention discloses a kind of magnetic levitation ball position control method, the shortcoming of precise physical model is difficult to set up for maglev ball system, sets up tape function weight coefficient type autoregression model to describe the non-linear dynamic characteristic between electromagnetism winding input voltage and steel ball position using system identifying method.The model weight coefficient of the once linear function of steel ball position as Gauss RBF networks, and with function type coefficient of the RBF networks as non linear autoregressive model, the model is preferably portrayed the dynamic characteristic of maglev ball system.At a time regression coefficient is constant to the model, similar to a linear ARX model.Based on this, the present invention one time-varying of design, local linear predictive controller, by being controlled in each moment line solver quadratic programming come the quick optimal location for realizing steel ball, meets maglev ball system stabilization, quickly required.
Description
Technical Field
The invention relates to the technical field of automatic control, in particular to a magnetic levitation ball position control method.
Background
The magnetic suspension technology is a typical electromechanical integration technology integrating electromagnetism, electronics, control engineering, signal processing, mechanics and dynamics. Magnetic levitation technology has been widely used in the engineering fields of magnetic levitation trains, magnetic levitation bearings, magnetic levitation motors and the like because of its characteristics of no contact, no friction, low noise and the like. The magnetic suspension ball system mainly generates electromagnetic force by electrifying a certain current to an electromagnetic winding, so that the electromagnetic winding is balanced with the gravity of the steel ball, the steel ball is suspended in the air and is in a balanced state, and the purpose of stable operation of the system is achieved. The magnetic levitation ball system with the single direction has the characteristics of intrinsic nonlinearity, open loop instability and quick response, is easily influenced by a power supply and an external environment, and certain parameters have strong uncertainty and cannot be accurately measured. The electromagnetic field magnetic saturation phenomenon enables the input current in the electromagnetic field to be not in direct proportion to the magnetic induction intensity and the flux linkage of the electromagnetic winding, the nonlinearity of the system is increased, and the electromagnetic force model of the system cannot be expressed by a simple mathematical equation; meanwhile, the steel ball in the electromagnetic field generates eddy current, which adversely affects the inductance of the electromagnetic winding, so that the inductance of the electromagnetic winding is not constant but is a function of the air gap g from the steel ball to the magnetic pole surface of the electromagnet and has a nonlinear relationship with the air gap g. Therefore, it is very difficult to establish an accurate physical model of the magnetic levitation ball system, which is a difficult point for the magnetic levitation ball system to realize stable control.
The PID control structure is simple, input current/voltage can be adjusted to enable the steel ball to be suspended, a physical model of a magnetic suspension system is not required to be established, control parameters need to be manually set, the adaptability is poor, the effective control range of the nonlinear magnetic suspension system is small, overshoot is large when the position of the steel ball changes rapidly, and steel ball shaking is large. The fuzzy reasoning is used for automatically adjusting the control parameters of the PID, so that the capability of changing the parameters along with the change of the air gap between the steel ball and the electromagnet can be improved. But this method relies on a fuzzy rule base, the establishment of which is subject to the experience of the designer. On the other hand, by analyzing the working principle of the magnetic suspension system, a physical model is established on the basis of certain assumed conditions, and then the steel ball can be subjected to adaptive control, sliding film control, predictive control and the like. The biggest difficulty in implementing these methods is that it is difficult to obtain an accurate physical model that accurately describes the dynamic characteristics of a magnetic levitation system, because certain assumptions are difficult to satisfy in engineering applications. In addition, the physical model is subjected to linearization processing by utilizing a linearization technology, and then a linear control strategy is designed, so that the online control optimization speed can be accelerated, but the nonlinear characteristic of the system is lost, and the description capacity of the model on the magnetic suspension system is weakened, thereby reducing the control effect.
Disclosure of Invention
The invention aims to solve the technical problem of providing a magnetic levitation ball position control method aiming at the defects of the prior art.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a magnetic levitation ball position control method is suitable for a single-degree-of-freedom magnetic levitation ball system moving up and down, wherein the single-degree-of-freedom magnetic levitation ball system comprises a coil winding generating an electromagnetic field and a photoelectric sensor detecting the position of a steel ball; the input voltage of the electromagnetic winding is given by a controller control driving circuit; and the position signals of the steel balls acquired by the photoelectric sensors are transmitted to a control computer through a data acquisition card. Establishing an autoregressive model for the magnetic levitation ball system:
wherein g (t) is the position of the magnetic levitation ball, u (t) is the input voltage of the electromagnetic winding, and ξ (t) is white noise;V0、Vi gand Vj uThe weight coefficient of the RBF network is a linear function of the position of the steel ball;the coefficient is a constant coefficient, is identified by an SNPOM optimization method, and is obtained by calculation of a least square method on the basis of obtaining nonlinear parameters.
Then, designing a predictive controller based on the autoregressive model, optimizing the following quadratic programming function J to obtain the optimal control quantity, and controlling the position g (t) of the magnetic levitation ball:
wherein,predicting variables of the steel ball position forward by steps at t moment, and obtaining an expression of step I prediction according to the autoregressive model at t moment, namelyAndobtaining a system prediction coefficient matrix by the autoregressive model and the multistep prediction variables at the moment t;gr(t + l) is a forward reference position given by step l at time t;u (t + p), wherein p is 0,1,2 and 3 are input voltages of the electromagnetic winding to be optimized at the time t, and only the first term u (t) is taken to act on the controlled magnetic levitation ball;△ u (t) u (t-1) is the input voltage increment;is a state vector; φ0、φi gand phij uIs the coefficient of the regression function of the autoregressive model, phi7 u0; and I is an identity matrix.
Compared with the prior art, the invention has the beneficial effects that: the invention adopts the idea of system identification modeling, establishes the dynamic model of the magnetic levitation ball system by utilizing the input/output data of the actual operation of the system, including the relevant information of the dynamic characteristics of the system, and can describe the dynamic characteristics of the system to the greatest extent without considering the influence of the magnetic saturation phenomenon and the eddy current effect on the model (the influence is included in the identification data). The method is suitable for the complex systems with strong nonlinearity and strong uncertainty, and can be popularized to other similar systems; the coefficient of the RBF network of the RBF-ARX model with the function weight established by the invention depends on the position of the steel ball in the electromagnetic field, so that the function approximation capacity of the RBF network is improved, and the obtained group of pseudo-linear ARX models can better describe the dynamic characteristic of the nonlinear magnetic levitation ball system. The modeling error of one-step prediction output of the system is within +/-0.4%; the time-varying and local linear predictive controller is designed based on the RBF-ARX model with the function weight, the optimal control quantity can be rapidly optimized and calculated, the online optimization time is reduced, the optimization calculation can be completed in the sampling period (5 milliseconds) of the magnetic suspension ball system, and the rapid and stable control on the position of the steel ball is realized.
Drawings
FIG. 1 is a diagram of a magnetic levitation ball system according to the present invention.
Detailed Description
The system structure of the magnetic suspension ball system is shown in figure 1, and is a single-degree-of-freedom system which can only control the steel ball to move up and down. The PC 9 outputs control voltage through a controller, the control voltage is transmitted to the electromagnetic winding driving circuit 6 through the D/A converter 8, the electromagnetic winding 2 generates electromagnetic induction under the condition of being electrified with corresponding current, an electromagnetic field is formed below the winding, electromagnetic induction force F is applied to the steel ball 1 in the field, the steel ball moves up/down, and an air gap G between the electromagnet and the steel ball (namely the position of the steel ball) is adjusted until the electromagnetic force F is balanced with the gravity G of the steel ball; meanwhile, the photoelectric sensor composed of the LED light source 3 and the photoelectric plate 4 is used for detecting the position of the steel ball, and a corresponding voltage signal is transmitted back to the PC for output through the processing circuit 5 and the A/D converter 7. In the system shown in fig. 1, the radius of the steel ball 1 is 12.5 mm, the mass is 22 g, the number of turns of the electromagnetic winding 2 is 2450, and the equivalent resistance is 13.8 ohms.
The magnetic suspension ball system is influenced by magnetic saturation and eddy effect, and is also influenced by power supply and external interference. Therefore, a dynamic model of the relation between the input voltage of the electromagnetic winding and the position of the steel ball in the electromagnetic field is constructed by adopting a modeling method based on a RBF-ARX model with function weights. In the invention, a data identification technology is utilized, and a linear function of the position of the steel ball is used as a weighting coefficient, and a RBF network of a Gaussian kernel is used as a function coefficient in a nonlinear ARX model. The model is a nonlinear time-varying model with a linear ARX model structure, the independent variables of the model are the input voltage of an electromagnetic winding and the regression quantity of the position of a steel ball, the position of the steel ball is the semaphore representing the state of a system, a function linearly related to the position of the steel ball is adopted to approximate the constant weight of an RBF neural network, and then the RBF structure is used for carrying out real-time online adjustment on the model parameters. The RBF-ARX model with the function weight is very similar to the linear ARX model in a local linear interval, and in addition, the parameters of the RBF-ARX model can be automatically updated and automatically adjusted along with the nonlinear state of the system, so that the RBF-ARX model has good global adaptive characteristics.
Constructing a function weight coefficient type RBF-ARX model of a dynamic characteristic model between a steel ball position and an input voltage of an electromagnetic winding, and identifying model parameters by adopting an SNPOM optimization Method (details: Peng H, Ozaki T, Haggan-Ozaki V, Toyoda Y.2003, A parameter optimization Method for thermal basic function types) combining a Levenberg-Marquardt Method (LMM) and a linear Least Square Method (LSM) to obtain the following structure:
wherein,
g (t) is the position of the steel ball and is also the air gap between the steel ball and the electromagnet; u (t) is the input voltage of the electromagnetic winding; phi is a0、φi gAnd phij uAre respectively the regression function coefficients of the autoregressive model; w (t-1) is the system working point state, and the steel ball position g (t-1) is used for representing the system working point state; v0、Andthe function weight coefficient of the RBF neural network is a linear function of the position of the steel ball, the | | · | | is a 2 norm, ξ (t) is white noise;the linear constant coefficient is identified by an SNPOM optimization method and obtained by LSM calculation on the basis of obtaining nonlinear parameters. Order to Formula (1) is rewritten asThen
Wherein,the SNPOM carries out model identification and uses 4000 observation data in total as observation data of the position of the steel ball.
A time-varying and linear predictive controller is designed by utilizing the local linear characteristic of a magnetic levitation ball system. Rewriting formula (1) to a polynomial
Wherein,
state variables defining the system:
the state space model of equation (1) is then:
here, the
Middle phi of the above formula0、φi gAnd phij uNot constant, but varies with the steel ball position g (t). Defining the relevant predictor variables:
wherein,is a multi-step forward prediction of the state vector,is a multi-step forward prediction vector of the position of the steel ball,is a multi-step forward prediction vector of the winding input voltage. Assuming that u (t + j) is u (t +3) (j ≧ 4), from (5) to (7), it is possible to obtain:
here, ,andthe coefficient matrix for system prediction can be obtained by the equations (5) - (8), and all values are values which are changed along with the steel ball position g (t) depending on the position of the steel ball in the electromagnetic field at the time t.
Therefore, a linear predictive controller which changes along with t can be designed according to the local linear characteristic of the RBF-ARX model with the function weight of the magnetic levitation ball system at the time t.
Defining control increasesMeasurement ofAnd desired output variable
Then there is quadratic programming optimization function with local linear predictive control
Wherein,and I is an identity matrix.
For the magnetic levitation ball system, the formula (13) is an optimization problem of quadratic programming, and the optimal control quantity can be obtained through online optimization. Therefore, the predictive control of the nonlinear magnetic levitation ball system is simplified into linear predictive control which changes along with the position state of the steel ball, the online optimization time of the optimal control quantity can be greatly saved, and the steel ball can quickly reach a stable state.
Claims (2)
1. A magnetic levitation ball position control method is characterized by comprising the following steps:
1) establishing an autoregressive model for the magnetic levitation ball system:
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wherein g (t) is the position of the magnetic levitation ball at the time t; u (t) is the electromagnetic winding input voltage; ξ (t) is white noise;
is a linear function of the position of the steel ball;the constant coefficient is identified by an SNPOM optimization method; g (t-1) is the position of the magnetic levitation ball at the time of t-1;
2) designing a predictive controller based on the autoregressive model, optimizing the following quadratic programming function J to obtain the optimal control quantity, and controlling the position g (t) of the magnetic levitation ball:
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wherein,gr(t + l) is the given forward reference position for l steps at time t, l being 1,2, …, 12;u (t + p) is the input voltage of the electromagnetic winding to be optimized at the moment t, only the first term u (t) is taken to act on the controlled magnetic levitation ball, and p is 0,1,2 and 3;Δ u (t) u (t-1) is the input voltage increment; andpredicting a coefficient matrix for the system;is a state vector;φ0、andare the coefficients of the regression function of the autoregressive model,i is an identity matrix; i is12Is an identity matrix of 12 orders; i is4Is an identity matrix of order 4.
2. The magnetic levitation ball position control method as recited in claim 1,
<mrow> <msub> <mover> <mi>B</mi> <mo>&OverBar;</mo> </mover> <mi>t</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>B</mi> <mi>t</mi> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>t</mi> </msub> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <msub> <mi>B</mi> <mi>t</mi> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>3</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>A</mi> <mi>t</mi> </msub> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <msub> <mi>B</mi> <mi>t</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>4</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>3</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> <mn>4</mn> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mrow> <mn>4</mn> <mo>-</mo> <mi>i</mi> </mrow> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>11</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>10</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>9</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> <mn>11</mn> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mrow> <mn>11</mn> <mo>-</mo> <mi>i</mi> </mrow> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
wherein:
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>&phi;</mi> <mn>1</mn> <mi>g</mi> </msubsup> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&phi;</mi> <mn>2</mn> <mi>g</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&phi;</mi> <mn>6</mn> <mi>g</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&phi;</mi> <mn>7</mn> <mi>g</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>B</mi> <mi>t</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>&phi;</mi> <mn>1</mn> <mi>u</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&phi;</mi> <mn>2</mn> <mi>u</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&phi;</mi> <mn>6</mn> <mi>u</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>2
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