CN105242573A - Satellite attitude controlled ground full-physical simulation intelligent control system - Google Patents

Abstract

The invention provides a satellite attitude controlled ground full-physical simulation intelligent control system, comprising a pressure regulating valve, a pressure sensor, a spirit level, a weighing sensor and a controller, wherein the control end of the pressure regulating valve and the output end of the pressure sensor are connected with the controller, and the spirit level and the output end of the weighing sensor are connected with the controller; the spirit level measures the levelness of a base and outputs data to the controller, and the weighing sensor measures the weight of a test bed in real time and outputs data to the controller; based on the thought of fussy intelligent control, a dependent variable is selected for establishing a system model, the controller enabling the system to be asymptotically stable is designed according to the Lyapunov stability analysis theory, the conservatism of the system design is lower by adopting the thought of piecewise linearization, and the description rules of the controlled object and the controller are not limited so as to improve the flexibility of the system design. The system is simple in principle and convenient to maintain.

Description

Satellite gravity anomaly ground full physical simulation intelligent control system

Technical field

The present invention relates to measuring technique, be specifically related to a kind of satellite gravity anomaly ground full physical simulation intelligent control system.

Background technology

The special running environment of spacecraft makes its ground simulation test seem particularly important, nucleus equipment for the full physics ground artificial system of spacecraft development is the air supporting emulation platform built based on air-floating ball bearing, rely on the air film that pressurized air is formed between air floatation ball and ball bearing housing, simulation stage body is floated, thus realize approximate friction free relative motion condition, with the analog satellite mechanical environment that disturbance torque is very little suffered by outer space.Wherein vortex torque and bearing capacity are the indexs of two most criticals for these, different supply gas pressures, bearing capacity can show different vortex torques, the simulation performance of vortex torque direct relation platform, therefore study this bearing capacity, vortex torque automatic stability control there is important Research Significance.

Find through searching document, application for a patent for invention number: 201410398151, patent name is the micro-disturbance torque measuring system of full-automatic three-axis air-bearing table and method, comprise air floatation ball and ball-and-socket, and the system and method provided, when utilization is rotated entirely, the nutating that symmetrical structure is formed, effectively can solve the interference problem between air-floating ball bearing and ball-and-socket, can be implemented in the disturbance torque measuring air-floating ball bearing in low-angle scope.But the method does not relate to the problem such as bearing capacity and supply gas pressure, the adjustment of aircraft manufacturing technology ground artificial system stability can not be directly used in.

Chinese invention patent application number: CN201410390718, patent name is a kind of test macro and method thereof of adopting the coupled motions of single-axle air bearing table simulation flexible satellite three-axis attitude, adopt the attitude motion of a certain axle of single-axle air bearing table analog satellite, by carrying out Function Extension to single-axle air bearing table, take the mode that physical characteristics and mathematical model combine, resolving by mathematical model, by the three axle coupling torques be subject in celestial body single-axis attitude motion process, flexible appendage disturbance torque and space environment disturbance torque are realized by moment output unit, the validity adopting the motion of single-axle air bearing table checking flexible satellite single-axis attitude can be improved, reflect satellite single-axis attitude motion in-orbit more really, but this invention does not relate to the contents such as the Commissioning Analysis of air floating table stability.

Summary of the invention

The object of the present invention is to provide a kind of satellite gravity anomaly ground full physical simulation system intelligence control system, principle is simple, and cost is low, engineering is easy to realize, and is convenient to safeguard.

The present invention is achieved by the following technical solutions:

A kind of satellite gravity anomaly ground full physical simulation intelligent control system, comprises pressure regulator valve, pressure transducer, level meter, LOAD CELLS and controller; Pressure regulator valve is connected with the supply air line of air-floating ball bearing, pressure transducer is connected with the cushion chamber of air-floating ball bearing, the control end of pressure regulator valve is all connected with controller with the output terminal of pressure transducer, level meter and LOAD CELLS are all installed on ball bearing housing lower end, and level meter is all connected with controller with the output terminal of LOAD CELLS; Data are also exported to controller by the levelness of level measurement pedestal, and data are also exported to controller by the weight of the real-time measuring test-bed body of LOAD CELLS.

The control method of controller as above is as follows:

Choose supply gas pressure P, actual bearer amount G, foundation level degree L, motor-driven angle A be dependent variable, wherein supply gas pressure P, actual bearer amount G, foundation level degree L can detect in real time and feed back to controller, the controlled quentity controlled variable W of the controlled quentity controlled variable C and pressure adjustmenting mechanism that choose attitude-adjusting system is output variable, set up word set and control rule base according to fuzzy theory, in modeling, regular i be described below:

Rule i: if P belongs to and G belongs to and L belongs to

So $\stackrel{\·}{x}\left(t\right)=A_{i}x\left(t\right)+A_{di}x(t-d(t\left)\right)+B_{i}u\left(t\right),i=1,2,...,p---\left(1\right)$

Wherein, x (t) is system state variables, and x (t-d (t)) is the delaying state variable of system, and d (t) is retardation, and u (t) is control variable, the derivative of system state variables x (t), A _i, A _di, B _ibe respectively state matrix, delaying state matrix and gating matrix, be i-th fuzzy subset of the variablees such as supply gas pressure P, actual bearer amount G, foundation level degree L, motor-driven angle A respectively, then the state equation of system is:

$\stackrel{\·}{x}\left(t\right)=\underset{i=1}{\overset{p}{\Σ}}{\mathrm{\ω}}_i\left(x\right(t\left)\right)\[A_{i}x\left(t\right)+A_{di}x(t-d(t\left)\right)+B_{i}u\left(t\right)\]---\left(2\right)$

Wherein, ω _i(x (t))>=0 is membership function, and p is the regular quantity set up.

Retardation τ (t) meets following constraint:

$0\&le;d\left(t\right)<h,a\&le;\stackrel{\&CenterDot;}{d}\left(t\right)<b---\left(3\right)$

Wherein, h is the upper bound postponed, the derivative that a and b postpones respectively lower bound and the upper bound.

The regular j of controller is:

Rule j: if P belongs to and G belongs to and L belongs to and A belongs to

So u (t)=K _jx (t), j=1,2 ..., c (4)

Wherein, K _jit is gating matrix to be designed.Then controller is:

$u\left(t\right)=\underset{j=1}{\overset{c}{\&Sigma;}}{m}_j\left(x\right(t\left)\right)K_{j}x\left(t\right)---\left(5\right)$

Wherein m _j(x (t))>=0 is membership function, and

Therefore, the state equation of closed-loop system is:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{\mathrm{\&omega;}}_i\left(x\right(t\left)\right)m_j\left(x\right(t\left)\right)\&lsqb;{\stackrel{\&OverBar;}{A}}_{i}x\left(t\right)+A_{di}x(t-d(t\left)\right)\&rsqb;---\left(6\right)$

Wherein, ${\stackrel{\&OverBar;}{A}}_i=A_i+B_i{K}_j.$

Definition h _ij(x (t)) ≡ ω _i(x (t)) m _j(x (t)), for convenience of description, by ω simultaneously _i(x (t)), m _j(x (t)), h _ij(x (t)), Δ h _ij(x (t)), uses ω respectively _i, m _j, h _ij, Δ h _ij, substitute, as follows.

Definition for h _ijlinear description, and wherein Δ h _ijfor h _ijlinearly describe with it the bound of difference.

For the system of foregoing description, have to draw a conclusion:

State equation is met to the closed-loop system of formula (6), simultaneously deferred gratification formula (3), if there is matrix $\stackrel{\&OverBar;}{P}>0,\stackrel{\&OverBar;}{R}>0,{\stackrel{\&OverBar;}{Q}}_i>0,(i=1,2,3,4),{\stackrel{\&OverBar;}{Y}}_{ij}={{\stackrel{\&OverBar;}{Y}}_{ij}}^T\&GreaterEqual;0,\stackrel{\&OverBar;}{L},$ Following LMI group formula (7)-(12) are set up, and so this closed-loop system is asymptotically stability, and system feasible gating matrix is $K_j={\stackrel{\&OverBar;}{K}}_j{\stackrel{\&OverBar;}{P}}^{-1}.$

$\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}\left(\right({\tilde{h}}_{ij}+\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}){\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}+(\mathrm{\&Delta;}{\stackrel{\&OverBar;}{h}}_{ij}-\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}\left)\left[\begin{array}{cc}{\stackrel{\&OverBar;}{Y}}_{ij}& 0\\ 0& 0\end{array}\right]\right)<0---\left(7\right)$

${\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}-\left[\begin{array}{cc}{\stackrel{\&OverBar;}{Y}}_{ij}& 0\\ 0& 0\end{array}\right]\&le;0---\left(8\right)$

$\left[\begin{array}{cc}-\stackrel{\&OverBar;}{R}& \stackrel{\&OverBar;}{L}\\ {\stackrel{\&OverBar;}{L}}^T& -\stackrel{\&OverBar;}{R}\end{array}\right]\&le;0---\left(9\right)$

${\stackrel{\&OverBar;}{Y}}_{ij}={{\stackrel{\&OverBar;}{Y}}_{ij}}^T\&GreaterEqual;0---\left(10\right)$

Wherein

${\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}=\left[\begin{array}{cccccc}{\mathrm{\&Gamma;}}_{11}^{\&prime;}& 0& A_{di}\stackrel{\&OverBar;}{P}+\frac{1}{h}\stackrel{\&OverBar;}{R}+\frac{1}{h}\stackrel{\&OverBar;}{L}& 0& -\frac{1}{h}\stackrel{\&OverBar;}{L}& \stackrel{\&OverBar;}{P}{A}_i^T+{\stackrel{\&OverBar;}{K}}_j^T{B}_i^T\\ *& {\mathrm{\&Gamma;}}_{22}& 0& 0& 0& 0\\ *& *& {\mathrm{\&Gamma;}}_{33}^{\&prime;}& 0& \frac{1}{h}\stackrel{\&OverBar;}{R}+\frac{1}{h}\stackrel{\&OverBar;}{L}& \stackrel{\&OverBar;}{P}{A}_{di}^T\\ *& *& *& {\mathrm{\&Gamma;}}_{44}& 0& 0\\ *& *& *& *& -{\stackrel{\&OverBar;}{Q}}_4-\frac{1}{h}\stackrel{\&OverBar;}{R}& 0\\ *& *& *& *& *& \frac{1}{h}(\stackrel{\&OverBar;}{R}-2\stackrel{\&OverBar;}{P})\end{array}\right]---\left(11\right)$

$\begin{array}{c}{\mathrm{\&Gamma;}}_{11}^{\&prime;}=\stackrel{\&OverBar;}{P}{(A_i+B_i{K}_j)}^T+(A_i+B_i{K}_j)\stackrel{\&OverBar;}{P}+{\stackrel{\&OverBar;}{Q}}_1-\frac{1}{h}\stackrel{\&OverBar;}{R}\\ =\stackrel{\&OverBar;}{P}{A}_i^T+{\stackrel{\&OverBar;}{K}}_j^T{B}_i^T+A_i\stackrel{\&OverBar;}{P}+B_i{\stackrel{\&OverBar;}{K}}_j+{\stackrel{\&OverBar;}{Q}}_1-\frac{1}{h}\stackrel{\&OverBar;}{R}\\ {\mathrm{\&Gamma;}}_{22}=(\frac{b}{2}-1){\stackrel{\&OverBar;}{Q}}_1+(1-\frac{a}{2}){\stackrel{\&OverBar;}{Q}}_2\\ {\mathrm{\&Gamma;}}_{33}^{\&prime;}=-(1-b){\stackrel{\&OverBar;}{Q}}_2+(1-a){\stackrel{\&OverBar;}{Q}}_3-\frac{2}{h}\stackrel{\&OverBar;}{R}-\frac{1}{h}\stackrel{\&OverBar;}{L}-\frac{1}{h}{\stackrel{\&OverBar;}{L}}^T\\ {\mathrm{\&Gamma;}}_{44}=(\frac{b}{2}-1){\stackrel{\&OverBar;}{Q}}_3+(1-\frac{a}{2}){\stackrel{\&OverBar;}{Q}}_4\end{array}---\left(12\right)$

The principle of the invention is simple, and improvement cost is low, engineering is easy to realize, and is convenient to safeguard, can realizes the accurate control of system.

Accompanying drawing explanation

Fig. 1 is satellite gravity anomaly ground full physical simulation intelligent control system schematic diagram.

Embodiment

Below in conjunction with accompanying drawing citing, the invention will be further described.

Embodiment 1:

Composition graphs 1, a kind of satellite gravity anomaly ground full physical simulation intelligent control system, comprises pressure regulator valve 1, pressure transducer 2, level meter 3, LOAD CELLS 4 and controller 5; It is characterized in that, pressure regulator valve 1 is connected with the supply air line of air-floating ball bearing, pressure transducer 2 is connected with the cushion chamber of air-floating ball bearing, the control end of pressure regulator valve 1 is all connected with controller 5 with the output terminal of pressure transducer 2, level meter 3 and LOAD CELLS 4 are all installed on ball bearing housing lower end, and level meter 3 is all connected with controller 5 with the output terminal of LOAD CELLS 4; Level meter 3 is measured the levelness of pedestal and data is exported to controller 5, LOAD CELLS 4 in real time measuring test-bed body weight and data are exported to controller 5.

The control principle of controller is as follows:

Choose supply gas pressure P, actual bearer amount G, foundation level degree L, motor-driven angle A be dependent variable, wherein supply gas pressure P, actual bearer amount G, foundation level degree L can detect in real time and feed back to controller, the controlled quentity controlled variable W of the controlled quentity controlled variable C and pressure adjustmenting mechanism that choose attitude-adjusting system is output variable, set up word set and control rule base according to fuzzy theory, in modeling, regular i be described below:

Rule i: if P belongs to and G belongs to and L belongs to

So $\stackrel{\&CenterDot;}{x}\left(t\right)=A_{i}x\left(t\right)+A_{di}x(t-d(t\left)\right)+B_{i}u\left(t\right),i=1,2,...,p---\left(1\right)$ Wherein, x (t) is system state variables, and x (t-d (t)) is the delaying state variable of system, and d (t) is retardation, and u (t) is control variable, the derivative of system state variables x (t), A _i, A _di, B _ibe respectively state matrix, delaying state matrix and gating matrix, be i-th fuzzy subset of the variablees such as supply gas pressure P, actual bearer amount G, foundation level degree L, motor-driven angle A respectively, then the state equation of system is:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\underset{i=1}{\overset{p}{\&Sigma;}}{\mathrm{\&omega;}}_i\left(x\right(t\left)\right)\&lsqb;A_{i}x\left(t\right)+A_{di}x(t-d(t\left)\right)+B_{i}u\left(t\right)\&rsqb;---\left(2\right)$

Wherein, ω _i(x (t))>=0 is membership function, and p is the regular quantity set up.

Retardation d (t) meets following constraint:

$0\&le;d\left(t\right)<h,a\&le;\stackrel{\&CenterDot;}{d}\left(t\right)<b---\left(3\right)$

Wherein, h is the upper bound postponed, the derivative that a and b postpones respectively lower bound and the upper bound.

The regular j of controller is:

Rule j: if P belongs to and G belongs to and L belongs to and A belongs to

So u (t)=K _jx (t), j=1,2 ..., c (4)

Wherein, K _jit is gating matrix to be designed.Then controller is:

$u\left(t\right)=\underset{j=1}{\overset{c}{\&Sigma;}}{m}_j\left(x\right(t\left)\right)K_{j}x\left(t\right)---\left(5\right)$

Wherein m _j(x (t))>=0 is membership function, and

Therefore, the state equation of closed-loop system is:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{\mathrm{\&omega;}}_i\left(x\right(t\left)\right)m_j\left(x\right(t\left)\right)\&lsqb;{\stackrel{\&OverBar;}{A}}_{i}x\left(t\right)+A_{di}x(t-d(t\left)\right)\&rsqb;---\left(6\right)$

Wherein, ${\stackrel{\&OverBar;}{A}}_i=A_i+B_i{K}_j.$

Definition h _ij(x (t)) ≡ ω _i(x (t)) m _j(x (t)), for convenience of description, by ω simultaneously _i(x (t)), m _j(x (t)), h _ij(x (t)), Δ h _ij(x (t)), uses ω respectively _i, m _j, h _ij, Δ h _ij, substitute, as follows.

Definition for h _ijlinear description, and wherein Δ h _ijfor h _ijlinearly describe with it the bound of difference.

For the system of foregoing description, have to draw a conclusion:

State equation is met to the closed-loop system of formula (6), simultaneously deferred gratification formula (3), if there is matrix $\stackrel{\&OverBar;}{P}>0,\stackrel{\&OverBar;}{R}>0,{\stackrel{\&OverBar;}{Q}}_j>0,(i=1,2,3,4),{\stackrel{\&OverBar;}{Y}}_{ij}={{\stackrel{\&OverBar;}{Y}}_{ij}}^T\&GreaterEqual;0,\stackrel{\&OverBar;}{L},$ Following LMI group formula (7)-(12) are set up, and so this closed-loop system is asymptotically stability, and system feasible gating matrix is $K_j={\stackrel{\&OverBar;}{K}}_j{\stackrel{\&OverBar;}{P}}^{-1};$

$\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}\left(\right({\tilde{h}}_{ij}+\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}){\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}+(\mathrm{\&Delta;}{\stackrel{\&OverBar;}{h}}_{ij}-\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}\left)\left[\begin{array}{cc}{\stackrel{\&OverBar;}{Y}}_{ij}& 0\\ 0& 0\end{array}\right]\right)<0---\left(7\right)$

${\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}-\left[\begin{array}{cc}{\stackrel{\&OverBar;}{Y}}_{ij}& 0\\ 0& 0\end{array}\right]\&le;0---\left(8\right)$

$\left[\begin{array}{cc}-\stackrel{\&OverBar;}{R}& \stackrel{\&OverBar;}{L}\\ {\stackrel{\&OverBar;}{L}}^T& -\stackrel{\&OverBar;}{R}\end{array}\right]\&le;0---\left(9\right)$

${\stackrel{\&OverBar;}{Y}}_{ij}={{\stackrel{\&OverBar;}{Y}}_{ij}}^T\&GreaterEqual;0---\left(10\right)$

Wherein

${\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}=\left[\begin{array}{cccccc}{\mathrm{\&Gamma;}}_{11}^{\&prime;}& 0& A_{di}\stackrel{\&OverBar;}{P}+\frac{1}{h}\stackrel{\&OverBar;}{R}+\frac{1}{h}\stackrel{\&OverBar;}{L}& 0& -\frac{1}{h}\stackrel{\&OverBar;}{L}& \stackrel{\&OverBar;}{P}{A}_i^T+{\stackrel{\&OverBar;}{K}}_j^T{B}_i^T\\ *& {\mathrm{\&Gamma;}}_{22}& 0& 0& 0& 0\\ *& *& {\mathrm{\&Gamma;}}_{33}^{\&prime;}& 0& \frac{1}{h}\stackrel{\&OverBar;}{R}+\frac{1}{h}\stackrel{\&OverBar;}{L}& \stackrel{\&OverBar;}{P}{A}_{di}^T\\ *& *& *& {\mathrm{\&Gamma;}}_{44}& 0& 0\\ *& *& *& *& -{\stackrel{\&OverBar;}{Q}}_4-\frac{1}{h}\stackrel{\&OverBar;}{R}& 0\\ *& *& *& *& *& \frac{1}{h}(\stackrel{\&OverBar;}{R}-2\stackrel{\&OverBar;}{P})\end{array}\right]---\left(11\right)$

$\begin{array}{c}{\mathrm{\&Gamma;}}_{11}^{\&prime;}=\stackrel{\&OverBar;}{P}{(A_i+B_i{K}_j)}^T+(A_i+B_i{K}_j)\stackrel{\&OverBar;}{P}+{\stackrel{\&OverBar;}{Q}}_1-\frac{1}{h}\stackrel{\&OverBar;}{R}=\stackrel{\&OverBar;}{P}{A}_i^T+{\stackrel{\&OverBar;}{K}}_j^T{B}_i^T+A_i\stackrel{\&OverBar;}{P}+B_i{\stackrel{\&OverBar;}{K}}_j+{\stackrel{\&OverBar;}{Q}}_1-\frac{1}{h}\stackrel{\&OverBar;}{R}\\ {\mathrm{\&Gamma;}}_{22}=(\frac{b}{2}-1){\stackrel{\&OverBar;}{Q}}_1+(1-\frac{a}{2}){\stackrel{\&OverBar;}{Q}}_2\\ {\mathrm{\&Gamma;}}_{33}^{\&prime;}=-(1-b){\stackrel{\&OverBar;}{Q}}_2+(1-a){\stackrel{\&OverBar;}{Q}}_3-\frac{2}{h}\stackrel{\&OverBar;}{R}-\frac{1}{h}\stackrel{\&OverBar;}{L}-\frac{1}{h}{\stackrel{\&OverBar;}{L}}^T\\ {\mathrm{\&Gamma;}}_{44}=(\frac{b}{2}-1){\stackrel{\&OverBar;}{Q}}_3+(1-\frac{a}{2}){\stackrel{\&OverBar;}{Q}}_4\end{array}---\left(12\right)$

Embodiment 2:

The principle of analyzer-controller design below:

According to the closed-loop system described in embodiment 1, in order to the stability of analytic system, choose Lyapunov function as follows:

$\begin{array}{c}V(t,x_t)=x^T\left(t\right)Px\left(t\right)+{\&Integral;}_{t-\frac{d\left(t\right)}{2}}^t{x}^T\left(s\right)Q_{1}x\left(s\right)ds+{\&Integral;}_{t-d\left(t\right)}^{t-\frac{d\left(t\right)}{2}}{x}^T\left(s\right)Q_{2}x\left(s\right)ds\\ +{\&Integral;}_{t-\frac{d\left(t\right)+h}{2}}^{t-d\left(t\right)}{x}^T\left(s\right)Q_{3}x\left(s\right)ds+{\&Integral;}_{t-h}^{t-\frac{d\left(t\right)+h}{2}}{x}^T\left(s\right)Q_{4}x\left(s\right)ds+{\&Integral;}_{-h}^0{\&Integral;}_{t+\mathrm{\&theta;}}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)R\stackrel{\&CenterDot;}{x}\left(s\right)dsd\mathrm{\&theta;}\end{array}---\left(13\right)$

Above formula is carried out differentiate,

$\begin{array}{c}\stackrel{\&CenterDot;}{V}(t,x_t)=2x^T\left(t\right)P\stackrel{\&CenterDot;}{x}\left(t\right)+x^T\left(t\right)Q_{1}x\left(t\right)-(1-\frac{\stackrel{\&CenterDot;}{d}\left(t\right)}{2})x^T(t-\frac{d\left(t\right)}{2})Q_{1}x(t-\frac{d\left(t\right)}{2})\\ +(1-\frac{\stackrel{\&CenterDot;}{d}\left(t\right)}{2})x^T(t-\frac{d\left(t\right)}{2})Q_{2}x(t-\frac{d\left(t\right)}{2})-(1-\stackrel{\&CenterDot;}{d}(t\left)\right)x^T(t-d(t\left)\right)Q_{2}x(t-d(t\left)\right)\\ +(1-\stackrel{\&CenterDot;}{d}(t\left)\right)x^T(t-d(t\left)\right)Q_{3}x(t-d(t\left)\right)\\ -(1-\frac{\stackrel{\&CenterDot;}{d}\left(t\right)}{2})x^T(t-\frac{d\left(t\right)+h}{2})Q_{3}x(t-\frac{d\left(t\right)+h}{2})\\ +(1-\frac{\stackrel{\&CenterDot;}{d}\left(t\right)}{2})x^T(t-\frac{d\left(t\right)+h}{2})Q_{4}x(t-\frac{d\left(t\right)+h}{2})-x^T(t-h)Q_{4}x(t-h)\\ +h{\stackrel{\&CenterDot;}{x}}^T\left(t\right)R\stackrel{\&CenterDot;}{x}\left(t\right)-{\&Integral;}_{t-h}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)R\stackrel{\&CenterDot;}{x}\left(s\right)ds\end{array}$

Utilize $\underset{i=1}{\overset{p}{\&Sigma;}}{\mathrm{\&omega;}}_i=\underset{j=1}{\overset{c}{\&Sigma;}}{m}_j=\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{\mathrm{\&omega;}}_i{m}_j=1$ And following lemma:

Lemma 1 (lemma of Schuler benefit): for symmetric matrix $L=\left[\begin{array}{cc}{L}_{11}& L_{12}\\ L_{12}^T& L_{22}\end{array}\right],$ Following description is of equal value:

1)L＜0，

$2)L_{11}<0,L_{22}-L_{12}^T{L}_{11}^{-1}{L}_{12}<0,---\left(15\right)$

$3)L_{22}<0,L_1-L_{12}{L}_{22}^{-1}{L}_{12}^T<0.$

Lemma 2: for matrix $N=\left[\begin{array}{cc}-R& L\\ L^T& -R\end{array}\right]\&le;0,d\left(t\right)\&Element;(0,h\&rsqb;,$ And vector equation make following integration have definition, so just have inequality below to set up:

$-h\underset{t-h}{\overset{t}{\&Integral;}}{\stackrel{\&CenterDot;}{x}}^T\left(t\right)R\stackrel{\&CenterDot;}{x}\left(t\right)\&le;{\mathrm{\&eta;}}^T\left(t\right)\mathrm{\&Xi;}\mathrm{\&eta;}\left(t\right)---\left(16\right)$

Wherein

$\mathrm{\&Xi;}=\left[\begin{array}{ccc}-R& R+L& -L\\ *& -2R-L-L^T& R+L\\ *& *& -R\end{array}\right]---\left(17\right)$

η(t)＝[x ^T(t)x ^T(t-d(t))x ^T(t-h)] ^T(18)

Can following formula be obtained:

$\stackrel{\&CenterDot;}{V}\&le;\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{\mathrm{\&omega;}}_i{m}_j\mathrm{\&Theta;}=\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{\mathrm{\&omega;}}_i{m}_j{\mathrm{\&Phi;}}^T\left(t\right){\mathrm{\&Omega;}}_{ij}\mathrm{\&Phi;}\left(t\right)---\left(19\right)$

Wherein

$\begin{array}{c}\mathrm{\&Theta;}=x^T\left(t\right){\stackrel{\&OverBar;}{A}}_i^{T}Px\left(t\right)+x^T(t-d(t\left)\right)A_{di}^{T}Px\left(t\right)+x^T\left(t\right)P{\stackrel{\&OverBar;}{A}}_{i}x\left(t\right)\\ +x^T\left(t\right){\mathrm{PA}}_{di}x(t-d(t\left)\right)+x^T\left(t\right)Q_{1}x\left(t\right)\\ -(1-\frac{b}{2})x^T(t-\frac{d\left(t\right)}{2})Q_{1}x(t-\frac{d\left(t\right)}{2})\\ +(1-\frac{a}{2})x^T(t-\frac{d\left(t\right)}{2})Q_{2}x(t-\frac{d\left(t\right)}{2})-(1-b)x^T(t-d(t\left)\right)Q_{2}x(t-d(t\left)\right)\\ +(1-\frac{a}{2})x^T(t-\frac{d\left(t\right)}{2})Q_{2}x(t-\frac{d\left(t\right)}{2})-(1-b)x^T(t-d(t\left)\right)Q_{2}x(t-d(t\left)\right)\\ +(1-a)x^T(t-d(t\left)\right)Q_{3}x(t-d(t\left)\right)\\ -(1-\frac{b}{2})x^T(t-\frac{d\left(t\right)+h}{2})Q_{3}x(t-\frac{d\left(t\right)+h}{2})\\ +(1-\frac{a}{2})x^T(t-\frac{d\left(t\right)+h}{2})Q_{4}x(t-\frac{d\left(t\right)+h}{2})-x^T(t-h)Q_{4}x(t-h)\\ +x^T\left(t\right){\stackrel{\&OverBar;}{A}}_i^{T}hR{\stackrel{\&OverBar;}{A}}_{i}x\left(t\right)+x^T(t-d(t\left)\right)A_{di}^{T}hR{\stackrel{\&OverBar;}{A}}_{i}x\left(t\right)\\ +x^T\left(t\right){\stackrel{\&OverBar;}{A}}_i^T{\mathrm{hRA}}_{di}x(t-d(t\left)\right)+x^T(t-d(t\left)\right)A_{di}^{T}hR{\stackrel{\&OverBar;}{A}}_{di}x(t-d(t\left)\right)\\ +\frac{1}{h}{\mathrm{\&eta;}}^T\left(t\right)\mathrm{\&Xi;}\mathrm{\&eta;}\left(t\right)\end{array}---\left(20\right)$

$\mathrm{\&Phi;}\left(t\right)={\left[\begin{array}{ccccc}{x}^T\left(t\right)& x^T(t-\frac{d\left(t\right)}{2})& x^T(t-d(t\left)\right)& x^T(t-\frac{d\left(t\right)+h}{2})& x^T(t-h)\end{array}\right]}^T---\left(21\right)$

${\mathrm{\&Omega;}}_{ij}=\left[\begin{array}{ccccc}{\mathrm{\&Gamma;}}_{11}& 0& {\mathrm{\&Gamma;}}_{13}& 0& -\frac{1}{h}L\\ *& {\mathrm{\&Gamma;}}_{22}& 0& 0& 0\\ *& *& {\mathrm{\&Gamma;}}_{33}& 0& \frac{1}{h}R+\frac{1}{h}L\\ *& *& *& {\mathrm{\&Gamma;}}_{44}& 0\\ *& *& *& *& -Q_4-\frac{1}{h}R\end{array}\right]---\left(22\right)$

$\begin{array}{c}{\mathrm{\&Gamma;}}_{11}={\stackrel{\&OverBar;}{A}}_i^{T}P+P{\stackrel{\&OverBar;}{A}}_i+h{\stackrel{\&OverBar;}{A}}_i^{T}R{\stackrel{\&OverBar;}{A}}_i+Q_1-\frac{1}{h}R\\ {\mathrm{\&Gamma;}}_{22}=(\frac{b}{2}-1)Q_1+(1-\frac{a}{2})Q_2\\ {\mathrm{\&Gamma;}}_{33}={\mathrm{hA}}_{di}^T{\mathrm{RA}}_{di}-(1-b)Q_2+(1-a)Q_3-\frac{2}{h}R-\frac{1}{h}L-\frac{1}{h}{L}^T\\ {\mathrm{\&Gamma;}}_{44}=(\frac{b}{2}-1)Q_3+(1-\frac{a}{2})Q_4\\ {\mathrm{\&Gamma;}}_{13}={\mathrm{PA}}_{di}+{\mathrm{hA}}_i^T{\mathrm{RA}}_{di}+\frac{1}{h}R+\frac{1}{h}L\end{array}---\left(23\right)$

Continue process, Ke Yiyou:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{w}_i{m}_j{\mathrm{\&Phi;}}^T{\mathrm{\&Omega;}}_{ij}\mathrm{\&Phi;}\\ =\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{\tilde{h}}_{ij}{\mathrm{\&Phi;}}^T{\mathrm{\&Omega;}}_{ij}\mathrm{\&Phi;}+\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}(h_{ij}-{\tilde{h}}_{ij}){\mathrm{\&Phi;}}^T{\mathrm{\&Omega;}}_{ij}\mathrm{\&Phi;}\\ \&le;\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}({\tilde{h}}_{ij}+\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}){\mathrm{\&Phi;}}^T{\mathrm{\&Omega;}}_{ij}\mathrm{\&Phi;}+\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}(h_{ij}-{\tilde{h}}_{ij}-\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}){\mathrm{\&Phi;}}^T{\mathrm{\&Omega;}}_{ij}\mathrm{\&Phi;}\\ \&le;\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}({\tilde{h}}_{ij}+\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}){\mathrm{\&Phi;}}^T{\mathrm{\&Omega;}}_{ij}\mathrm{\&Phi;}+\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}(\mathrm{\&Delta;}{\stackrel{\&OverBar;}{h}}_{ij}-\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}){\mathrm{\&Phi;}}^T{Y}_{ij}\mathrm{\&Phi;}\\ =\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{\mathrm{\&Phi;}}^T\left(\right({\tilde{h}}_{ij}+\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}){\mathrm{\&Omega;}}_{ij}+(\mathrm{\&Delta;}{\stackrel{\&OverBar;}{h}}_{ij}-\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}\left)Y_{ij}\right)\mathrm{\&Phi;}\end{array}---\left(24\right)$

Wherein h _ija piecewise linearity describes, h _ij≡ w _im _j, y _ij>=Ω _ijand $Y_{ij}={Y_{ij}}^T\&GreaterEqual;0.$

Continue below to provide gating matrix K _jmethod for designing:

Because above-mentioned (22)-(23) can be treated to:

${\mathrm{\&Omega;}}_{ij}^{\&prime;}=\left[\begin{array}{cccccc}{\mathrm{\&Gamma;}}_{11}^{\&prime;}& 0& {\mathrm{PA}}_{di}+\frac{1}{h}R+\frac{1}{h}L& 0& -\frac{1}{h}L& {(A_i+B_i{K}_j)}^T\\ *& {\mathrm{\&Gamma;}}_{22}& 0& 0& 0& 0\\ *& *& {\mathrm{\&Gamma;}}_{33}^{\&prime;}& 0& \frac{1}{h}R+\frac{1}{h}L& A_{di}^T\\ *& *& *& {\mathrm{\&Gamma;}}_{44}& 0& 0\\ *& *& *& *& -Q_4-\frac{1}{h}R& 0\\ *& *& *& *& *& -\frac{1}{h}{R}^{-1}\end{array}\right]---\left(25\right)$

${\mathrm{\&Gamma;}}_{11}^{\&prime;\&prime;}={(A_i+B_i{K}_j)}^{T}P+P(A_i+B_i{K}_j)+Q_1-\frac{1}{h}R$

${\mathrm{\&Gamma;}}_{22}=(\frac{b}{2}-1)Q_1+(1-\frac{a}{2})Q_2$

${\mathrm{\&Gamma;}}_{33}^{\&prime;\&prime;}=-(1-b)Q_2+(1-a)Q_3-\frac{2}{h}R-\frac{1}{h}L-\frac{1}{h}{L}^T$

${\mathrm{\&Gamma;}}_{44}=(\frac{b}{2}-1)Q_3+(1-\frac{a}{2})Q_4$

At formula (25) premultiplication diag{P respectively ^-T, P ^-T, P ^-T, P ^-T, P ^-T, { P is taken advantage of on I}, the right side ^-1, P ^-1, P ^-1, P ^-1, P ^-1, I}, and introduce variable-definition below

$\stackrel{\&OverBar;}{P}=P^{-1},\stackrel{\&OverBar;}{R}=P^{-1}{\mathrm{RP}}^{-1},\stackrel{\&OverBar;}{L}=P^{-1}{\mathrm{LP}}^{-1},{\stackrel{\&OverBar;}{Q}}_i=P^{-1}{Q}_i{P}^{-1},(i=1,2,3,4),{\stackrel{\&OverBar;}{K}}_j=K_j{P}^{-1}$

So can obtain:

${\mathrm{\&Omega;}}_{ij}=\left[\begin{array}{cccccc}{\mathrm{\&Gamma;}}_{11}^{\&prime;}& 0& A_{di}\stackrel{\&OverBar;}{P}+\frac{1}{h}\stackrel{\&OverBar;}{R}+\frac{1}{h}\stackrel{\&OverBar;}{L}& 0& -\frac{1}{h}\stackrel{\&OverBar;}{L}& \stackrel{\&OverBar;}{P}{A}_i^T+{\stackrel{\&OverBar;}{K}}_j^T{B}_i^T\\ *& {\mathrm{\&Gamma;}}_{22}& 0& 0& 0& 0\\ *& *& {\mathrm{\&Gamma;}}_{33}^{\&prime;}& 0& \frac{1}{h}\stackrel{\&OverBar;}{R}+\frac{1}{h}\stackrel{\&OverBar;}{L}& \stackrel{\&OverBar;}{P}{A}_{di}^T\\ *& *& *& {\mathrm{\&Gamma;}}_{44}& 0& 0\\ *& *& *& *& -{\stackrel{\&OverBar;}{Q}}_4-\frac{1}{h}\stackrel{\&OverBar;}{R}& 0\\ *& *& *& *& *& -\frac{1}{h}{R}^{-1}\end{array}\right]---\left(26\right)$

$\left[\begin{array}{cc}-\stackrel{\&OverBar;}{R}& \stackrel{\&OverBar;}{L}\\ {\stackrel{\&OverBar;}{L}}^T& -\stackrel{\&OverBar;}{R}\end{array}\right]\&le;0---\left(27\right)$

$\begin{array}{c}{\mathrm{\&Gamma;}}_{11}^{\&prime;}=\stackrel{\&OverBar;}{P}{A}_i^T+{\stackrel{\&OverBar;}{K}}_j^T{B}_i^T+A_i\stackrel{\&OverBar;}{P}+B_i{\stackrel{\&OverBar;}{K}}_j+{\stackrel{\&OverBar;}{Q}}_1-\frac{1}{h}\stackrel{\&OverBar;}{R}\\ {\mathrm{\&Gamma;}}_{22}=(\frac{b}{2}-1){\stackrel{\&OverBar;}{Q}}_1+(1-\frac{a}{2}){\stackrel{\&OverBar;}{Q}}_2\\ {\mathrm{\&Gamma;}}_{33}^{\&prime;}=-(1-b){\stackrel{\&OverBar;}{Q}}_2+(1-a){\stackrel{\&OverBar;}{Q}}_3-\frac{2}{h}\stackrel{\&OverBar;}{R}-\frac{1}{h}\stackrel{\&OverBar;}{L}-\frac{1}{h}{\stackrel{\&OverBar;}{L}}^T\\ {\mathrm{\&Gamma;}}_{44}=(\frac{b}{2}-1){\stackrel{\&OverBar;}{Q}}_3+(1-\frac{a}{2}){\stackrel{\&OverBar;}{Q}}_4\end{array}---\left(28\right)$

Because inequality perseverance is below set up:

So have: conclusion below can be obtained thus:

For the closed-loop system (6) of deferred gratification (3) formula constraint, if there is matrix $\stackrel{\&OverBar;}{P}>0,\stackrel{\&OverBar;}{R}>0,{\stackrel{\&OverBar;}{Q}}_i>0,(i=1,2,3,4),{\stackrel{\&OverBar;}{Y}}_{ij}={{\stackrel{\&OverBar;}{Y}}_{ij}}^T\&GreaterEqual;0,\stackrel{\&OverBar;}{L},$ Formula (7)-(12) are set up, and so closed-loop system (6) is asymptotically stability, and system feasible gating matrix is

Embodiment 3:

Can find out in superincumbent analysis, containing membership function in the controller design method obtained, that is the method is that degree of membership relies on, this effectively can reduce the conservative property of system, in addition, time lag treatment adopts the thought that time lag is cut, by time delay interval [0, h] be divided into [0, d (t)/2], [d (t)/2, d (t)], [d (t), [d (t)+h]/2] and [[d (t)+h]/2, h] four sub-ranges, the degree of Delay-Dependent can be improved on the one hand, avoid again on the other hand increasing overweight computation burden.

Claims (1)

1. a satellite gravity anomaly ground full physical simulation intelligent control system, comprises pressure regulator valve (1), pressure transducer (2), level meter (3), LOAD CELLS (4) and controller (5); Pressure regulator valve (1) is connected with the supply air line of air-floating ball bearing, pressure transducer (2) is connected with the cushion chamber of air-floating ball bearing, it is characterized in that, the control end of pressure regulator valve (1) is all connected with controller (5) with the output terminal of pressure transducer (2), level meter (3) and LOAD CELLS (4) are all installed on ball bearing housing lower end, and level meter (3) is all connected with controller (5) with the output terminal of LOAD CELLS (4); Level meter (3) is measured the levelness of pedestal and data is exported to controller (5), LOAD CELLS (4) in real time measuring test-bed body weight and data are exported to controller (5); The control method of controller is as follows:

Choose supply gas pressure P, actual bearer amount G, foundation level degree L, motor-driven angle A be dependent variable, wherein supply gas pressure P, actual bearer amount G, foundation level degree L detect in real time and feed back to controller, the controlled quentity controlled variable W of the controlled quentity controlled variable C and pressure adjustmenting mechanism that choose attitude-adjusting system is output variable, set up word set and control rule base according to fuzzy theory, in modeling, regular i be described below:

Rule i: if P belongs to and G belongs to and L belongs to

So $\stackrel{\&CenterDot;}{x}\left(t\right)=A_{i}x\left(t\right)+A_{di}x(t-d(t\left)\right)+B_{i}u\left(t\right),i=1,2,...,p---\left(1\right)$

Wherein, x (t) is system state variables, and x (t-d (t)) is the delaying state variable of system, and d (t) is retardation, and u (t) is control variable, the derivative of system state variables x (t), A _i, A _di, B _ibe respectively state matrix, delaying state matrix and gating matrix, be i-th fuzzy subset of the variablees such as supply gas pressure P, actual bearer amount G, foundation level degree L, motor-driven angle A respectively, then the state equation of system is:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\underset{i=1}{\overset{p}{\&Sigma;}}{\mathrm{\&omega;}}_i\left(x\right(t\left)\right)\&lsqb;A_{i}x\left(t\right)+A_{di}x\left(t-d\left(t\right)\right)+B_{i}u\&rsqb;---\left(2\right)$

Wherein, ω _i(x (t))>=0 is membership function, and p is the regular quantity set up;

Retardation d (t) meets following constraint:

$0\&le;d\left(t\right)<h,a\&le;\stackrel{\&CenterDot;}{d}\left(t\right)<b---\left(3\right)$

Wherein, h is the upper bound postponed, the derivative that a and b postpones respectively lower bound and the upper bound;

The regular j of controller is:

Rule j: if P belongs to and G belongs to and L belongs to and A belongs to

So u (t)=K _jx (t), j=1,2 ..., c (4)

Wherein, K _jbe gating matrix to be designed, then controller is:

$u\left(t\right)=\underset{j=1}{\overset{c}{\&Sigma;}}{m}_j\left(x\left(t\right)\right)K_{j}x\left(t\right)---\left(5\right)$

Wherein m _j(x (t))>=0 is membership function, and

Therefore, the state equation of closed-loop system is:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}{\mathrm{\&omega;}}_i\left(x\right(t\left)\right)m_j\left(x\right(t\left)\right)\&lsqb;{\stackrel{\&OverBar;}{A}}_{i}x\left(t\right)+A_{di}x(t-d(t\left)\right)\&rsqb;---\left(6\right)$

Wherein, ${\stackrel{\&OverBar;}{A}}_i=A_i+B_i{K}_j;$

Definition h _ij(x (t)) ≡ ω _i(x (t)) m _j(x (t)), for convenience of description, by ω simultaneously _i(x (t)), $m_j\left(x\right(t\left)\right),h_{ij}\left(x\right(t\left)\right),{\tilde{h}}_{ij}\left(x\right(t\left)\right),\mathrm{\&Delta;}{\stackrel{\&OverBar;}{h}}_{ij}\left(x\left(t\right)\right),\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}\left(x\left(t\right)\right),$ Use ω respectively _i, m _j, h _ij, Δ h _ij, substitute, as follows;

Definition for h _ijlinear description, and wherein Δ h _ijfor h _ijlinearly describe with it the bound of difference;

For the system of foregoing description, have to draw a conclusion:

State equation is met to the closed-loop system of formula (6), simultaneously deferred gratification formula (3), if there is matrix $\stackrel{\&OverBar;}{P}>0,\stackrel{\&OverBar;}{R}>0,{\stackrel{\&OverBar;}{Q}}_i>0,(i=1,2,3,4),{\stackrel{\&OverBar;}{Y}}_{ij}={{\stackrel{\&OverBar;}{Y}}_{ij}}^T\&GreaterEqual;0,\stackrel{\&OverBar;}{L},$ Following LMI group formula (7)-(12) are set up, and so this closed-loop system is asymptotically stability, and system feasible gating matrix is $K_j={\stackrel{\&OverBar;}{K}}_j{\stackrel{\&OverBar;}{P}}^{-1};$

$\underset{i=1}{\overset{p}{\&Sigma;}}\underset{j=1}{\overset{c}{\&Sigma;}}\left(\right({\tilde{h}}_{ij}+\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}){\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}+(\mathrm{\&Delta;}{\stackrel{\&OverBar;}{h}}_{ij}-\mathrm{\&Delta;}{\underset{\&OverBar;}{h}}_{ij}\left)\left[\begin{array}{cc}{\stackrel{\&OverBar;}{Y}}_{ij}& 0\\ 0& 0\end{array}\right]\right)<0---\left(7\right)$

${\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}-\left[\begin{array}{cc}{\stackrel{\&OverBar;}{Y}}_{ij}& 0\\ 0& 0\end{array}\right]\&le;0---\left(8\right)$

$\left[\begin{array}{cc}-\stackrel{\&OverBar;}{R}& \stackrel{\&OverBar;}{L}\\ {\stackrel{\&OverBar;}{L}}^T& -\stackrel{\&OverBar;}{R}\end{array}\right]\&le;0---\left(9\right)$

${\stackrel{\&OverBar;}{Y}}_{ij}={{\stackrel{\&OverBar;}{Y}}_{ij}}^T\&GreaterEqual;0---\left(10\right)$

Wherein

${\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_{ij}=\left[\begin{array}{cccccc}{\mathrm{\&Gamma;}}_{11}^{\&prime;}& 0& A_{di}\stackrel{\&OverBar;}{P}+\frac{1}{h}\stackrel{\&OverBar;}{R}+\frac{1}{h}\stackrel{\&OverBar;}{L}& 0& -\frac{1}{h}\stackrel{\&OverBar;}{L}& \stackrel{\&OverBar;}{P}{A}_i^T+{\stackrel{\&OverBar;}{K}}_j^T{B}_i^T\\ *& {\mathrm{\&Gamma;}}_{22}& 0& 0& 0& 0\\ *& *& {\mathrm{\&Gamma;}}_{33}^{\&prime;}& 0& \frac{1}{h}\stackrel{\&OverBar;}{R}+\frac{1}{h}\stackrel{\&OverBar;}{L}& \stackrel{\&OverBar;}{P}{A}_{di}^T\\ *& *& *& {\mathrm{\&Gamma;}}_{44}& 0& 0\\ *& *& *& *& -{\stackrel{\&OverBar;}{Q}}_4-\frac{1}{h}\stackrel{\&OverBar;}{R}& 0\\ *& *& *& *& *& \frac{1}{h}\left(\stackrel{\&OverBar;}{R}-2\stackrel{\&OverBar;}{P}\right)\end{array}\right]---\left(11\right)$

$\begin{array}{c}{\mathrm{\&Gamma;}}_{11}^{\&prime;}=\stackrel{\&OverBar;}{P}{\left(A_i+B_i{K}_j\right)}^T+\left(A_i+B_i{K}_j\right)\stackrel{\&OverBar;}{P}+{\stackrel{\&OverBar;}{Q}}_1-\frac{1}{h}\stackrel{\&OverBar;}{R}\\ =\stackrel{\&OverBar;}{P}{A}_i^T+{\stackrel{\&OverBar;}{K}}_j^T{B}_i^T+A_i\stackrel{\&OverBar;}{P}+B_i{\stackrel{\&OverBar;}{K}}_j+{\stackrel{\&OverBar;}{Q}}_1-\frac{1}{h}\stackrel{\&OverBar;}{R}\\ {\mathrm{\&Gamma;}}_{22}=\left(\frac{b}{2}-1\right){\stackrel{\&OverBar;}{Q}}_1+\left(1-\frac{a}{2}\right){\stackrel{\&OverBar;}{Q}}_2\\ {\mathrm{\&Gamma;}}_{33}^{\&prime;}=-\left(1-b\right){\stackrel{\&OverBar;}{Q}}_2+\left(1-a\right){\stackrel{\&OverBar;}{Q}}_3-\frac{2}{h}\stackrel{\&OverBar;}{R}-\frac{1}{h}\stackrel{\&OverBar;}{L}-\frac{1}{h}{\stackrel{\&OverBar;}{L}}^T\\ {\mathrm{\&Gamma;}}_{44}=\left(\frac{b}{2}-1\right){\stackrel{\&OverBar;}{Q}}_3+\left(1-\frac{a}{2}\right){\stackrel{\&OverBar;}{Q}}_4\end{array}---\left(12\right).$