CN110377044B - Finite time height and attitude tracking control method of unmanned helicopter - Google Patents
Finite time height and attitude tracking control method of unmanned helicopter Download PDFInfo
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Abstract
The invention discloses a finite time height and attitude tracking control method of an unmanned helicopter, which comprises the following steps: firstly, considering the flapping models of a main rotor and an aileron, and establishing a height and attitude comprehensive model; then, constructing a finite time interference observer to estimate unknown time-varying interference, and acquiring an interference estimation value; then, designing a height and posture composite anti-interference tracking controller based on the observed interference estimation value by combining an exponentiation integration method; and finally, selecting proper controller gain and observer gain according to a design rule to realize the limited time tracking of the height and the attitude. The invention improves the accuracy, rapidity and anti-interference of the height and attitude closed-loop tracking system of the unmanned helicopter.
Description
Technical Field
The invention relates to a finite time height and attitude tracking control method of an unmanned helicopter, belonging to the technical field of flight control of unmanned helicopters.
Background
In recent years, unmanned helicopters have attracted much attention due to their unique advantages, such as vertical take-off and landing, low-altitude flight, flight in narrow and complex environments such as mountainous areas and buildings. Based on these advantages, unmanned helicopters are widely used for aerial photography, rescue, detection, and other tasks. However, unmanned helicopters are a highly nonlinear, under-actuated, strongly coupled system. Furthermore, its flight performance is susceptible to disturbances, such as wind gusts. Therefore, designing a high-performance flight controller becomes a challenging research topic.
At present, the control method of the unmanned helicopter system mainly comprises linear control and nonlinear control. Traditional control methods are based primarily on linearized mathematical models. However, the linearization of the model occurs at the equilibrium point, and the controller performance will be greatly degraded when the system state deviates from the equilibrium point. In order to overcome the defects of the linear control method, more and more nonlinear control methods are used for unmanned helicopter control, such as sliding mode control, backstepping control, model prediction control and the like. However, these methods only guarantee asymptotic stability of the closed loop system.
The document (i.a. raptis, k.p. valavanis, w.a. Moreno, a novel nonlinear feedback Control system for using the rotation matrix, IEEE Transactions on Control Systems Technology, vol.19, No.2,2011,465 and 473) proposes to introduce a nested saturation feedback function in the backstepping design, and to Control the attitude by using the structural characteristics of the rotation matrix, so that the helicopter tracks the predetermined position and yaw reference trajectory. The document (K.Yan, Q.Wu, M.Chen, Robust adaptive backstepping Control for autonomous helicopter with flapping dynamics, 201713 th IEEE International reference on Control & Automation (ICCA 2017, pp.1027-1032.) proposes a Robust adaptive backstepping controller of an unmanned autonomous helicopter with flapping dynamics, and designs a nonlinear disturbance observer to estimate external unknown disturbance. The controller designed by the article deals with the influence of external unknown interference. But only demonstrates that all signals of the closed loop system are consistently bounded and stable.
Disclosure of Invention
In consideration of the characteristics of high nonlinearity, under-actuation and strong coupling of the unmanned helicopter system, the invention provides a finite time height and attitude tracking control method of the unmanned helicopter, which is a composite anti-interference height and attitude finite time tracking control method combining a finite time interference observer technology and an exponentiation integration method, so that the unmanned helicopter system has finite time tracking performance and strong anti-interference performance.
The invention adopts the following technical scheme for solving the technical problems:
the invention provides a finite time height and attitude tracking control method of an unmanned helicopter, which comprises the following steps:
the method comprises the following steps: considering a waving model and time-varying interference, and establishing a height and posture comprehensive model;
the height and attitude comprehensive model is as follows:
wherein: Θ ═ ηT z]T, Uz=-CθCφTmM is the mass of the unmanned helicopter, g, eta ═ phi theta psi]T∈R3Z and VzRespectively representing the gravity acceleration, the attitude, the height and the linear velocity, omega, corresponding to the height of the unmanned helicopter under an inertial coordinate systemb=[p q r]T∈R3Representing the attitude angular velocity of the unmanned helicopter under a body coordinate system, wherein p, q and r are respectively a rolling angular velocity, a pitch angular velocity and a yaw angular velocity;is an attitude vector transformation matrix, phi, theta and psi are respectively the roll angle, the pitch angle and the yaw angle of the unmanned helicopter, SψDenotes sin psi, Sφ、Cφ、Cθ、SθAnd TθDenotes sin phi, cos theta, sin theta and tan theta, respectively, and J ═ diag { Ixx Iyy IzzIs an inertia matrix, Ixx,Iyy,IzzThe rotational inertia of the unmanned helicopter around the x axis, the y axis and the z axis respectively; a and b are respectively the longitudinal direction and the transverse direction of the main rotorWaving angles, respectively, from the longitudinal input signal deltalonAnd a transverse input signal deltalatControlling; a. thelonAnd BlatEffective gains, A, for controlling the input signal to the longitudinal and transverse flapping angles of the main rotor, respectivelycAnd BdCoupling coefficients of the main rotor and the ailerons, C and d being longitudinal and transverse flapping angles of the ailerons, respectively, ClonAnd DlatEffective gains for controlling input signals to the longitudinal and lateral flap angles of the aileron, respectively; tau isfAnd τsAre all time constants; tau isbThe moment generated for the main rotor and the tail wing,Tmand TtThe resultant force generated by the main rotor and the resultant force generated by the empennage are respectively generated; cma,CmbAre physical parameters related to the stiffness of the main rotor; z is a radical ofmAnd ztThe z-axis axial distance, x, from the main rotor and the tail rotor rotation axis to the center of gravity of the helicoptertThe x-axis axial distance from the empennage rotating shaft to the center of gravity of the helicopter; qm=CMQTm 3/2+DMQTotal moment generated for rotation of main rotor, CMQAnd DMQAre all normal numbers related to the generation of reactive torque of the main rotor; d ═ Jd [ [ (Jd)ω)T mdvz]T,dω=[dp dq dr]TRepresenting time-varying disturbances acting on the attitude channel of the unmanned helicopter system, dvzRepresenting time-varying disturbances acting on the vertical direction speed path of the unmanned helicopter system, dp、dq、drRespectively, the interference acting on the channels of the roll angle, the pitch angle and the yaw rate;
step two: constructing a finite time interference observer to estimate unknown time-varying interference, and acquiring an interference estimation value;
let p be χ1,q=χ2,r=χ3,Vz=χ4,dp=d1,dq=d2,dr=d3,dvz=d4To n orderInterference d that can be minimizediThere is a known Lipschitz constant LiAnd > 0, the finite time disturbance observer is:
wherein, when i is 1,2,3,4, λi,jFor a positive observer gain, j is 0,1,2,are respectively chii、di、Is determined by the estimated value of (c),sgn (. cndot.) is a sign function, y1=[Tmbzm-Ttzt+Cmbb-qr(Izz-Iyy)]/Ixx+d1,y2=[Tmazm+Cmaa-pr(Ixx-Izz)]/Iyy+d2,y3=[Ttxt-Qm-pq(Iyy-Ixx)]/Izz+d3,y4=(mg-CθCφTm)/m+d4;
Step three: designing a height and attitude composite anti-interference finite time tracking controller based on the interference estimated value obtained in the step two, and carrying out interference compensation and height and attitude tracking;
the height and attitude composite anti-interference finite time tracking controller comprises:
wherein α ═ α1/α2∈(1,2),α1、α2Are all positive odd numbers, thetaref=[φref θref ψref zref]T,φref、θrefAnd psirefRespectively desired roll, pitch and yaw angles, zrefA desired height; e.g. of the type1=Θ-Θref=[eφ eθ eψez]T,eφFor roll angle tracking error, eθFor pitch angle tracking error, eψFor yaw angle tracking error, ezIs the height tracking error; e.g. of the typea=a-arefAnd eb=b-brefErrors of longitudinal and transverse waving angles, arefAnd brefRespectively desired longitudinal and transverse flapping angles, k, of the main rotor1,1,k1,2,k1,3,k1,4,k2,1,k2,2,k2,3,k2,4,k3,k4Is the controller gain;
step four: and selecting observer gain and controller gain, and realizing the finite time stability of the unmanned helicopter height and attitude closed-loop tracking system through the height and attitude composite anti-interference finite time tracking controller.
As a further technical solution of the present invention, the observer gain and the controller gain selected in step four are:
k1,1>0,k1,2>0,k1,3>0,k1,4>0;
order to h1,1=(2-1/α)2(1-1α)k1,1 1+α,h1,2=(2-1/α)2(1-1α)k1,2 1+α,h1,3=(2-1/α)2(1-1α)k1,3 1+α,h1,4=(2-1/α)2(1-1α)k1,4 1+α),
Then k is2,1,k2,2,k2,3,k2,4Satisfy k2,1/h1,1-ρ1,1>0,k2,2/h1,2-ρ1,2>0,k2,3/h1,3-ρ1,3>0,k2,4/h1,4-ρ1,4>0,k3,k4Satisfy k3>0,k4>0,i=1,2,3,4,j=0,1,2,λi,j>0。
As a further technical scheme of the invention, a in the third steprefAnd brefAre respectively:
as a further technical solution of the present invention, the observation error system corresponding to the observer in the step two is:
Technical effects
Compared with the prior art, the technical scheme provided by the invention has the following technical effects:
the designed high-order differential finite time disturbance observer can be used for observing various forms of disturbance such as constant disturbance, slope disturbance, high-order disturbance, sinusoidal disturbance and the like and various derivatives thereof, enables an observation error to be converged to 0 in finite time, and has universality.
And (II) the finite time observer technology and the power integration method are combined, and the designed composite anti-interference finite time controller can effectively process the negative influence of interference on the system, so that the height and attitude tracking errors are converged to 0 in finite time.
And thirdly, the composite anti-interference finite time control technical idea provided by the invention is suitable for system control design in other technical fields, and has wide application prospect.
Drawings
FIG. 1 is a control block diagram of a closed loop system for altitude and attitude of an unmanned helicopter;
FIG. 2 is a flow chart of the steps of the solution;
FIG. 3 is a coordinate system diagram of the unmanned helicopter;
FIG. 4 is a plot of attitude and altitude tracking error responses, wherein (a) is roll angle, (b) is pitch angle, (c) is yaw angle, and (d) is altitude;
FIG. 5 is a plot of attitude and altitude actual position versus desired position, wherein (a) is roll angle, (b) is pitch angle, (c) is yaw angle, and (d) is altitude;
FIG. 6 is a plot of observed error response, where (a) is the disturbance for the roll rate channel, (b) is the disturbance for the pitch rate channel, (c) is the disturbance for the yaw rate channel, and (d) is the disturbance for the altitude rate channel;
FIG. 7 is a comparison graph of an actual disturbance value and an observed disturbance value, wherein (a) is a disturbance of a roll rate channel, (b) is a disturbance of a pitch rate channel, (c) is a disturbance of a yaw rate channel, and (d) is a disturbance of a altitude rate channel;
FIG. 8 is a graph showing the variation of control signals, wherein (a) is TmAnd (b) is TtAnd (c) is deltalonAnd (d) is deltalat。
Detailed Description
The technical scheme of the invention is further explained in detail by combining the attached drawings:
the method comprises the following steps: and sequentially calculating a 6-degree-of-freedom model, a flapping model and force and moment of the unmanned helicopter, and then comprehensively obtaining a height and attitude model considering interference.
(a) The unmanned helicopter model with 6 degrees of freedom is as follows:
wherein m represents the mass of the unmanned helicopter, g, P ═ x y z]T∈R3、η=[φ θ ψ]T∈R3And V ═ VxVy Vz]T∈R3Respectively representing it in an inertial coordinate systemAcceleration of gravity, position, attitude and linear velocity; omegab=[p q r]T∈R3And fb=[fx fy fz]T∈R3、τb=[τx τy τz]T∈R3Respectively representing the attitude angular velocity, the force and the moment under the body coordinate system; p, q and r are respectively a rolling angular velocity, a pitching angular velocity and a yaw angular velocity. And a transformation matrix from the body coordinate system to the inertial coordinate system is defined asPhi, theta and psi, respectively, the roll angle, pitch angle and yaw angle of the unmanned helicopter, Sψ、Cψ、Sφ、Cφ、Cθ、SθAnd TθDenoted sin ψ, cos ψ, sin φ, cos θ, sin θ and tan θ, respectively. Attitude vector transformation matrixThe pitch angle being maintained in normal flightTherefore H is not singular. J ═ diag { I ═ Ix Iy IzIs an inertia matrix, Ixx,Iyy,IzzThe rotary inertia of the unmanned helicopter around the x, y and z axes.
(b) Waving model
The main rotor flap model is as follows:
the flap model is:
wherein, tauf,τsAre all time constants, a, bLongitudinal and transverse flapping angles of the main rotor are respectively provided by a longitudinal input signal deltalonAnd a transverse input signal deltalatAnd (5) controlling. A. thelon,BlatFor controlling the effective gain of the input signal to the longitudinal and transverse flapping angles of the main rotor, Ac,BdIs the coupling coefficient of the main rotor and the ailerons, C, d are the longitudinal and transverse flapping angles of the ailerons, Clon,DlatTo control the effective gain of the input signal to the longitudinal and lateral flap angles of the aileron.
(c) Calculation of force and moment
Because a and b are very small, the requirements of sina ≈ a, sinb ≈ 1, cosa ≈ 1 and cosb ≈ 1 are met, and the force of the unmanned helicopter in the x and y directions is far less than the force in the z direction during height control, so that f and b are very smallb=[0 0 -Tm]TThe main rotor and the tail wing generate a moment ofTm,TtRespectively the resultant force generated by the main rotor and the resultant force generated by the empennage. Cma,CmbIs a physical parameter related to the stiffness of the main rotor. z is a radical ofm,ztThe Z-axis axial distances from the main rotor and the tail wing rotating shaft to the center of gravity of the helicopter respectively; x is the number oftThe x-axis axial distance from the empennage rotation axis to the center of gravity of the helicopter. Qm=CMQ·Tm 3/2+DMQTotal moment generated for rotation of main rotor, CMQ,DMQIs a normal number related to the generation of the reaction torque of the main rotor.
(d) Height and attitude integrated model
Considering the flapping dynamic characteristics of the main rotor and the ailerons, the height and attitude comprehensive model of the unmanned helicopter is established as follows:
wherein D ═ Jd [ (-)ω)T mdvz]T,dω=[dp dq dr]TRepresenting time-varying disturbances acting on the attitude channel of the unmanned helicopter system, dvzRepresenting time-varying disturbances acting on the vertical direction speed path of the unmanned helicopter system, dp、dq、drRespectively, the disturbances acting on the channels of roll angle, pitch angle, yaw rate.
Step two: and designing a finite time interference observer to obtain an interference estimation value and a derivative estimation value thereof.
For convenience of illustration, let p ═ χ1,q=χ2,r=χ3,Vz=χ4,dp=d1,dq=d2,dr=d3,dvz=d4. For differentiable d of n orderi(i ═ 1,2,3,4) has a known Lipschitz constant LiAnd > 0, designing the following finite time disturbance observer:
description of the drawings: when i is 1,2,3,4,are respectively chii,di,Is estimated value, functionAnd sgn (·) is a standard sign function. y is1=[Tmbzm-Ttzt+Cmbb-qr(Izz-Iyy)]/Ixx+d1,y2=[Tmazm+Cmaa-pr(Ixx-Izz)]/Iyy+d2,y3=[Ttxt-Qm-pq(Iyy-Ixx)]/Izz+d3,y4=(mg-CθCφTm)/m+d4。λi,j(j ═ 0,1,2) is the positive observer gain to be designed.
The corresponding observation error system is:
wherein e isi,0=χi-ξi,0,ei,1=di-ξi,1,Selecting proper observer gain lambda for observing errori,j(j is 0,1,2), the observation error can be made in a finite time T1Converging to zero. The finite time observer of the form of equation (5) is suitable for disturbances in the form of constant values, ramps, higher orders, sinusoids, etc.
Step three: and (4) applying the interference and derivative estimation values thereof obtained in the step two to feed-forward compensation, and designing a composite anti-interference height and attitude finite time tracking controller by combining a finite time observer technology and an exponentiation integration method to ensure that the height and attitude tracking errors have finite time convergence to 0.
Control signal Tm,Tt,δlon,δlatThe specific form of (A) is as follows:
wherein α ═ α1/α2∈(1,2),α1、α2Is positive odd number, thetaref=[φref θref ψref zref]TTo a desired attitude and height, phiref、θrefAnd psirefRespectively desired roll, pitch and yaw angles, zrefA desired height; e.g. of the type1=Θ-Θref=[eφ eθ eψ ez]TTo track errors, eφFor roll angle tracking error, eθFor pitch angle tracking error, eψFor yaw angle tracking error, ezIs the height tracking error; e.g. of the typea=a-aref,eb=b-brefLongitudinal flap angle and transverse flap angle errors, respectively. a isref,brefRespectively designed as follows:
k1,1,k1,2,k1,3,k1,4,k2,1,k2,2,k2,3,k2,4,k3,k4is the gain to be designed.
Step four: and selecting proper controller gain and observer gain according to a design rule to realize the finite time stability of the unmanned helicopter height and attitude closed-loop tracking system.
The observer gain and the controller gain selected in the fourth step are as follows:
k1,1>0,k1,2>0,k1,3>0,k1,4>0;
order to h1,1=(2-1/α)2(1-1/α)k1,1 1+α,h1,2=(2-1/α)2(1-1/α)k1,2 1+α,h1,3=(2-1/α)2(1-1/α)k1,3 1+α,h1,4=(2-1/α)2(1-1/α)k1,4 1+α,
Then k is2,1,k2,2,k2,3,k2,4Satisfy k2,1/h1,1-ρ1,1>0,k2,2/h1,2-ρ1,2>0,k2,3/h1,3-ρ1,3>0,k2,4/h1,4-ρ1,4>0,k3,k4Satisfy k3>0,k4>0,i=1,2,3,4,j=0,1,2,λi,j>0。
For height and attitude models with unknown disturbances, finite time stabilization, i.e. height tracking error and attitude tracking error, can be achieved if the controller is designed as in equations (7) to (10) for a finite time T2Converging to zero.
In order to verify the effectiveness of the height and attitude limited time tracking control method proposed by the invention, a numerical simulation was performed with MATLAB.
Initial state of unmanned helicopter is eta0=[0.50.50.5]Trad,z02m, the desired attitude angle and height ηref=[0.5sint 0.5sint 0.5sint]Trad,zref5m, the interference expression is set to d1=2.5sin(0.3t+30),d2=1.2sin(0.5t+15),d3=1.8sin(0.2t+30),d41.8sin (0.9t + 15). Fig. 4 shows (a) to (d) that attitude and altitude tracking errors can be converged to 0 relatively quickly in the presence of disturbance, fig. 5 shows (a) to (d) that are graphs comparing actual positions of attitude and altitude with desired positions, fig. 6 shows (a) to (d) that track performance of the observation error system, and disturbance observation errors can be converged to 0 quickly and accurately, fig. 7 shows (a) to (d) that compare actual values of disturbance with observed values, and fig. 8 shows (a) to (d) that are graphs showing changes of control input signals. Simulation results show that the height and attitude composite anti-interference finite time tracking controller designed by the invention is effective.
The above embodiments are merely illustrative of the technical ideas of the present invention, and do not limit the scope of the present invention. It should be noted that any improvement made to the technical solution on the technical idea of the present invention belongs to the protection scope of the present invention.
Claims (4)
1. A finite time altitude and attitude tracking control method of an unmanned helicopter is characterized by comprising the following steps:
the method comprises the following steps: considering a waving model and time-varying interference, and establishing a height and posture comprehensive model;
the height and attitude comprehensive model is as follows:
wherein: Θ ═ ηT z]T, Uz=-CθCφTmM is the mass of the unmanned helicopter, g, eta ═ phi theta psi]T∈R3Z and VzRespectively representing the gravity acceleration, the attitude, the height and the linear velocity, omega, corresponding to the height of the unmanned helicopter under an inertial coordinate systemb=[p q r]T∈R3Representing the attitude angular velocity of the unmanned helicopter under a body coordinate system, wherein p, q and r are respectively a rolling angular velocity, a pitch angular velocity and a yaw angular velocity;is an attitude vector transformation matrix, phi, theta and psi are respectively the roll angle, the pitch angle and the yaw angle of the unmanned helicopter, SψDenotes sin psi, Sφ、Cφ、Cθ、SθAnd TθDenotes sin phi, cos theta, sin theta and tan theta, respectively, and J ═ diag { Ixx Iyy IzzIs an inertia matrix, Ixx,Iyy,IzzThe rotational inertia of the unmanned helicopter around the x axis, the y axis and the z axis respectively; a and b are respectively the longitudinal and transverse flapping angles of the main rotor, and are respectively provided by a longitudinal input signal deltalonAnd a transverse input signal deltalatControlling; a. thelonAnd BlatEffective gains, A, for controlling the input signal to the longitudinal and transverse flapping angles of the main rotor, respectivelycAnd BdCoupling coefficients of the main rotor and the ailerons, C and d being longitudinal and transverse flapping angles of the ailerons, respectively, ClonAnd DlatEffective gains for controlling input signals to the longitudinal and lateral flap angles of the aileron, respectively; tau isfAnd τsAre all time constants; tau isbThe moment generated for the main rotor and the tail wing,Tmand TtThe resultant force generated by the main rotor and the resultant force generated by the empennage are respectively generated; cma,CmbAre physical parameters related to the stiffness of the main rotor; z is a radical ofmAnd ztThe z-axis axial distance, x, from the main rotor and the tail rotor rotation axis to the center of gravity of the helicoptertThe x-axis axial distance from the empennage rotating shaft to the center of gravity of the helicopter; qm=CMQTm 3/2+DMQTotal moment generated for rotation of main rotor, CMQAnd DMQIs a normal number related to the generation of the reaction torque of the main rotor; d ═ Jd [ [ (Jd)ω)Tmdvz]T,dω=[dp dq dr]TRepresenting time-varying disturbances acting on the attitude channel of the unmanned helicopter system, dp、dq、drRespectively, disturbances acting on the channels of roll, pitch and yaw rates, dvzRepresenting time-varying interference acting on a vertical direction speed channel of the unmanned helicopter system;
step two: constructing a finite time interference observer to estimate unknown time-varying interference, and acquiring an interference estimation value;
let p be χ1,q=χ2,r=χ3,Vz=χ4,dp=d1,dq=d2,dr=d3,dvz=d4For a negligible interference d of order niThere is a known Lipschitz constant LiAnd > 0, the finite time disturbance observer is:
wherein, when i is 1,2,3,4, λi,jFor a positive observer gain, j is 0,1,2,are respectively chii、di、Is determined by the estimated value of (c),sgn (. cndot.) is a sign function, y1=[Tmbzm-Ttzt+Cmbb-qr(Izz-Iyy)]/Ixx+d1,y2=[Tmazm+Cmaa-pr(Ixx-Izz)]/Iyy+d2,y3=[Ttxt-Qm-pq(Iyy-Ixx)]/Izz+d3,y4=(mg-CθCφTm)/m+d4;
Step three: designing a height and attitude composite anti-interference finite time tracking controller based on the interference estimated value obtained in the step two, and carrying out interference compensation and height and attitude tracking;
the height and attitude composite anti-interference finite time tracking controller comprises:
wherein α ═ α1/α2∈(1,2),α1、α2Are all positive odd numbers, thetaref=[φref θref ψref zref]T,φref、θrefAnd psirefRespectively desired roll, pitch and yaw angles, zrefA desired height; e.g. of the type1=Θ-Θref=[eφ eθ eψez]T,eφFor roll angle tracking error, eθFor pitch angle tracking error, eψFor yaw angle tracking error, ezIs the height tracking error; e.g. of the typea=a-arefAnd eb=b-brefErrors of longitudinal and transverse waving angles, arefAnd brefRespectively desired longitudinal and transverse flapping angles, k, of the main rotor1,1,k1,2,k1,3,k1,4,k2,1,k2,2,k2,3,k2,4,k3,k4Is the controller gain;
step four: and selecting observer gain and controller gain, and realizing the finite time stability of the unmanned helicopter height and attitude closed-loop tracking system through the height and attitude composite anti-interference finite time tracking controller.
2. The finite time altitude and attitude tracking control method of an unmanned helicopter of claim 1, wherein the observer gain and the controller gain selected in step four are:
k1,1>0,k1,2>0,k1,3>0,k1,4>0;
order to h1,1=(2-1/α)2(1-1/α)k1,1 1+α,h1,2=(2-1/α)2(1-1/α)k1,2 1+α,h1,3=(2-1/α)2(1-1/α)k1,3 1+α,h1,4=(2-1/α)2(1-1/α)k1,4 1+α,
Then k is2,1,k2,2,k2,3,k2,4Satisfy k2,1/h1,1-ρ1,1>0,k2,2/h1,2-ρ1,2>0,k2,3/h1,3-ρ1,3>0,k2,4/h1,4-ρ1,4>0,k3,k4Satisfy k3>0,k4>0,i=1,2,3,4,j=0,1,2,λi,j>0。
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