CN107256028A - Lost-control protection control algolithm under the diagonal power loss state of quadrotor - Google Patents

Lost-control protection control algolithm under the diagonal power loss state of quadrotor Download PDF

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Publication number
CN107256028A
CN107256028A CN201710606622.2A CN201710606622A CN107256028A CN 107256028 A CN107256028 A CN 107256028A CN 201710606622 A CN201710606622 A CN 201710606622A CN 107256028 A CN107256028 A CN 107256028A
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CN107256028B (en
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胡凯健
吴玉虎
孙希明
汪锐
吴振宇
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Liaoning Lucheng Intelligent Manufacturing Co ltd
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Dalian University of Technology
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    • GPHYSICS
    • G05CONTROLLING; REGULATING
    • G05DSYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/08Control of attitude, i.e. control of roll, pitch, or yaw
    • G05D1/0808Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft
    • GPHYSICS
    • G05CONTROLLING; REGULATING
    • G05DSYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/10Simultaneous control of position or course in three dimensions
    • G05D1/101Simultaneous control of position or course in three dimensions specially adapted for aircraft

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  • Aviation & Aerospace Engineering (AREA)
  • Radar, Positioning & Navigation (AREA)
  • Remote Sensing (AREA)
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  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
  • Other Investigation Or Analysis Of Materials By Electrical Means (AREA)
  • Medicines Containing Antibodies Or Antigens For Use As Internal Diagnostic Agents (AREA)
  • Control Of Position, Course, Altitude, Or Attitude Of Moving Bodies (AREA)

Abstract

The invention discloses the lost-control protection control algolithm under a kind of diagonal power loss state of quadrotor, its step is:S1, hypothesis quadrotor 1, No. 3 motors run out of steam, then abandon driftage and the control of luffing angle;S2, based on τθ=0 and the angle of pitch and roll angle have vital effect to quadrotor smooth flight;Choose Fr, F 'input=G ' Fr, solve;S3, determination are after 2 diagonal motor damages, the mathematical modeling of quadrotor;S4, order expect that posture is φdAnd θd, desired position is xd、ydAnd zd;S5, set up error system.

Description

Out-of-control protection control algorithm of four-rotor aircraft in diagonal power loss state
Technical Field
The invention relates to the technical field of four-rotor aircrafts, in particular to an out-of-control protection control algorithm of a four-rotor aircraft in a diagonal power loss state.
Background
The Quadrotor aircraft (Quadrotor) is an Unmanned Aerial Vehicle (UAV), which has 4 motors, the rotating directions of adjacent motors are opposite, the rotating directions of opposite motors are the same, and the rolling, pitching, yawing and vertical upward control effects can be realized by changing the rotating speed of the motors. A quad-rotor aircraft typically includes a frame, 4 brushless dc motors, 2 pairs of propellers, a flight control panel, an electronic governor, a power battery, a GPS and other components, the flight control panel includes high performance microprocessors, sensors, power management and other modules, and the common sensors include a three-axis accelerometer, a magnetometer, a gyroscope, a barometer, and the like. Therefore, the four-rotor aircraft has the advantages of simple mechanical structure, easiness in design, convenience in carrying and the like. Owing to the design characteristics of the four-rotor aircraft, the four-rotor aircraft can realize actions such as vertical take-off and landing, hovering in the air, high-maneuverability flight and the like in a complex environment. In addition, the four-rotor aircraft is also a good aerial platform, for example, the four-rotor aircraft carries a camera for aerial photography, short-distance aerial transportation or carries out tasks of climbing wall surfaces and grabbing heavy objects after being equipped with mechanical arms.
With the development of society and the improvement of economy, the development speed of the four-rotor aircraft is changing day by day and is advancing at an explosive speed. With the increase of the number of the four-rotor aircrafts, the corresponding problems caused by flight faults of the four-rotor aircrafts are more and more, wherein the highest frequency is faults such as aircraft diagonal power loss, and the causes of aircraft diagonal power loss are many, for example: damage to the propeller, motor or electronic governor, etc. Under the state that the propeller is damaged, the aircraft has diagonal power loss, and the quad-rotor aircraft can have flight faults and even fall, so that how to control the flight of the quad-rotor aircraft under the diagonal power loss becomes an urgent need to be considered in the industry.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide an out-of-control protection control algorithm of a four-rotor aircraft in a diagonal power loss state, so that the problem of controlling the balanced flight of the four-rotor aircraft in the diagonal power loss state of the aircraft caused by propeller damage or other faults is solved.
In order to realize the purpose, the invention is realized by the following technical scheme:
a four-rotor aircraft out-of-control protection control algorithm under a diagonal power loss state comprises the following steps:
s1, if two symmetric propellers of the four-rotor aircraft are damaged, the power of the opposite angle is lost; suppose four-rotor aircraft No. 1, 3When the motor loses power, the lift force F1=F3=0;
Wherein F represents the lift force generated by the cooperation of a single motor and a propeller, and F1And F3Respectively representing the lifting force generated by No. 1 and No. 3 motors.
The relationship between the model control quantity and the motor lift force is:
Finput=GFr,\*MERGEFORMAT(0.15)
wherein:
wherein,coefficient of lift of propeller cTThe coefficient of propeller torque is cQ(ii) a l is the distance from the motor to the center of mass of the aircraft;
because rank (G: F)input) 3 > rank (g) 2, equation without solution
In order to be able to solve the equation, the control of the yaw and pitch angles is abandoned;
s2 based on tauθThe pitch angle and the roll angle are 0, and the four-rotor aircraft has a vital effect on the stable flight; selecting the following formula to solve Fr
Finput=G′Fr,\*MERGEFORMAT(0.16)
Wherein:
the following can be obtained:
from the above formula, τ is knownψ=-σFlθ=0;τφIs roll torque, FlIs the total lift, τ, of the aircraftΨIs yaw torque, τθIs the pitch torque;
s3, determining that after the 2 diagonal motors are damaged, the mathematical model of the four-rotor aircraft is expressed as follows:
s*=sin*,c*=cos*,t*=tan*
η=[φ θ ψ]Tthe Euler angle is used for describing the attitude information of the four-rotor aircraft; the Euler angle is called the Kaerdan angle (Tait-Bryan) and is defined by sequential rotations of the z-y-x axis, which are respectively a yaw angle (yaw) around the z axis and denoted by psi; rotation about the y-axis is the pitch angle (pitch), denoted by θ; rotation about the x-axis is the roll angle (roll), denoted by φ;
ξ=[x y z]Tthe position of the quadrotor aircraft under an inertial coordinate system; upsilon [ ]xυyυz]TThe speed of the quad-rotor aircraft in the inertial coordinate system; i ═ diag (I) for moment of inertiaxx,Iyy,Izz) The aircraft mass is denoted by m, ω ═ pqr]TThe angular speed of the four-rotor aircraft around the shaft under a body coordinate system;
s4, let the expected attitude be phidThe desired position is xd、ydAnd zd
The following error system was established:
from the above formula, one can obtain:
the position errors in equations (1.7) and (1.8) are recombined as shown below:
thus, can obtain
As an improvement to the above solution, the target (x) is controlled for the horizontal position control system (1.4)d,yd) Obtaining a horizontal position control type for a second-order micro horizontal position track;
if the virtual control input phidGiven as follows:
wherein
The horizontal position tracking error eh1,eh2The index stabilizes at zero equilibrium point.
Compared with the prior art, the invention has the following beneficial effects:
the out-of-control protection control algorithm of the four-rotor aircraft in the diagonal power loss state solves the problem of controlling the flight of the four-rotor aircraft in the diagonal power loss state when a propeller is damaged or a motor is damaged. Under the condition that 2 opposite propellers of the four-rotor aircraft are damaged or motors are damaged, namely, the diagonal power loss is caused, the balanced flight of the four-rotor aircraft can still be controlled by the method.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
Fig. 1 is a control structure diagram of the present invention in a state where a diagonal motor is damaged.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived from the embodiments of the present invention by a person skilled in the art without any creative effort, should be included in the protection scope of the present invention.
As shown in fig. 1, this figure depicts the controller structure under diagonal power failure. The controller is divided into an inner ring and an outer ring, wherein the inner ring is controlled by a horizontal position, and the outer ring is controlled by a height and a roll angle. Red crosses represent the amount by which the aircraft is not given in this state; arrows represent the direction of information flow; the symbol above the line is the name of the information.
The invention relates to an out-of-control protection control algorithm of a four-rotor aircraft in a diagonal power loss state, which comprises the following steps:
s1, if two symmetric propellers of the four-rotor aircraft are damaged, the power of the opposite angle is lost; if the No. 1 and No. 3 motors of the four-rotor aircraft lose power, the lift force F is generated1=F3=0;
Wherein F represents the lift force generated by the cooperation of a single motor and a propeller, and F1And F3Respectively representing the lifting force generated by No. 1 and No. 3 motors.
The relationship between the model control quantity and the motor lift force is:
Finput=GFr,\*MERGEFORMAT(0.28)
wherein:
wherein,coefficient of lift of propeller cTThe coefficient of propeller torque is cQ(ii) a l is the distance from the motor to the center of mass of the aircraft;
because rank (G: F)input) 3 > rank (g) 2, equation without solution
In order to be able to solve the equation, the control of the yaw and pitch angles is abandoned;
s2 based on tauθThe pitch angle and the roll angle are 0, and the four-rotor aircraft has a vital effect on the stable flight; selecting the following formula to solve Fr
Finput=G′Fr,\*MERGEFORMAT(0.29)
Wherein:
the following can be obtained:
from the above formula, τ is knownψ=-σFlθ=0;τφIs roll torque, FlIs the total lift, τ, of the aircraftΨIs yaw torque, τθIs the pitch torque;
s3, determining that after the 2 diagonal motors are damaged, the mathematical model of the four-rotor aircraft is expressed as follows:
s*=sin*,c*=cos*,t*=tan*
η=[φ θ ψ]Tfor Euler angles, for describing quad-rotor aircraftThe attitude information of (a); the Euler angle is called the Kaerdan angle (Tait-Bryan) and is defined by sequential rotations of the z-y-x axis, which are respectively a yaw angle (yaw) around the z axis and denoted by psi; rotation about the y-axis is the pitch angle (pitch), denoted by θ; rotation about the x-axis is the roll angle (roll), denoted by φ;
ξ=[x y z]Tthe position of the quadrotor aircraft under an inertial coordinate system; upsilon [ ]xυyυz]TThe speed of the quad-rotor aircraft in the inertial coordinate system; i ═ diag (I) for moment of inertiaxx,Iyy,Izz) The aircraft mass is denoted by m, ω ═ pqr]TThe angular speed of the four-rotor aircraft around the shaft under a body coordinate system;
s4, let the expected attitude be phidThe desired position is xd、ydAnd zd
The following error system was established:
from the above formula, one can obtain:
the position errors in equations (1.7) and (1.8) are recombined as shown below:
thus, can obtain
As an improvement to the above solution, the target (x) is controlled for the horizontal position control system (1.4)d,yd) Obtaining a horizontal position control type for a second-order micro horizontal position track;
if the virtual control input phidGiven as follows:
wherein
The horizontal position tracking error eh1,eh2The index stabilizes at zero equilibrium point.
Compared with the prior art, the invention has the following beneficial effects:
the out-of-control protection control algorithm of the four-rotor aircraft in the diagonal power loss state solves the problem of controlling the flight of the four-rotor aircraft in the diagonal power loss state when a propeller is damaged or a motor is damaged. Under the condition that 2 opposite propellers of the four-rotor aircraft are damaged or motors are damaged, namely, the diagonal power loss is caused, the balanced flight of the four-rotor aircraft can still be controlled by the method.

Claims (3)

1. The utility model provides a four rotor crafts protection control algorithm out of control under diagonal power loss state which characterized in that: the four-rotor aircraft out-of-control protection control algorithm comprises the following steps:
s1, if two symmetric propellers of the four-rotor aircraft are damaged, the power of the opposite angle is lost; if the No. 1 and No. 3 motors of the four-rotor aircraft lose power, the lift force F is generated1=F3=0;
Wherein F represents the lift force generated by the cooperation of a single motor and a propeller, and F1And F3Respectively represent the lifting force generated by No. 1 and No. 3 motors;
The relationship between the model control quantity and the motor lift force is:
Finput=GFr,\*MERGEFORMAT (0.1)
wherein:
wherein,coefficient of lift of propeller cTThe coefficient of propeller torque is cQ(ii) a l is the distance from the motor to the center of mass of the aircraft;
because rank (G: F)input) 3 > rank (g) 2, equation is not solved;
in order to be able to solve the equation, the control of the yaw and pitch angles is abandoned;
s2 based on tauθThe pitch angle and the roll angle are 0, and the four-rotor aircraft has a vital effect on the stable flight; selecting the following formula to solve Fr
F′input=G′Fr,\*MERGEFORMAT (0.2)
Wherein:
the following can be obtained:
from the above formula, τ is knownψ=-σFlθ=0;τφIs roll torque, FlIs the total lift, τ, of the aircraftΨIs yaw torque, τθIs the pitch torque;
s3, determining that after the 2 diagonal motors are damaged, the mathematical model of the four-rotor aircraft is expressed as follows:
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s*=sin*,c*=cos*,t*=tan*
η=[φ θ ψ]Tthe Euler angle is used for describing the attitude information of the four-rotor aircraft; the Euler angle is called Kaldo angle, and is defined by sequential rotation of z-y-x axes, and is respectively a yaw angle which is rotated around the z axis and is expressed by psi; the rotation around the y axis is a pitch angle and is represented by theta; the rotation around the x axis is a roll angle and is represented by phi;
ξ=[x y z]Tthe position of the quadrotor aircraft under an inertial coordinate system; upsilon [ ]xυyυz]TThe speed of the quad-rotor aircraft in the inertial coordinate system; i ═ diag (I) for moment of inertiaxx,Iyy,Izz) The mass of the aircraft is denoted by m,
ω=[p q r]Tthe angular speed of the four-rotor aircraft around the shaft under a body coordinate system;
s4, let the expected attitude be phidThe desired position is xd、ydAnd zd
The following error system was established:
e1=φ-φd,
e7=x-xd,
e9=y-yd,
e11=z-zd,
from the above formula, one can obtain:
the position errors in equations (1.7) and (1.8) are recombined as shown below:
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thus, can obtain
2. The algorithm for runaway protection control under a diagonal power loss condition in a quad-rotor aircraft according to claim 1, wherein: controlling the target (phi) for a posture and height control system (1.5)d,zd) Obtaining a roll angle and height control type for a second-order differentiable roll angle and height track;
if the control input is
F′input=(Q′+NQ″)-1(X-M-NP),\*MERGEFORMAT (0.11)
Tracking error e of roll anglei(i ═ 1,2) and height tracking error ei(i-5, 6) is exponentially stable at zero equilibrium;
wherein:
<mrow> <msup> <mi>Q</mi> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>&amp;sigma;</mi> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mfrac> <mfrac> <mrow> <mi>s</mi> <mi>&amp;theta;</mi> </mrow> <mrow> <mi>c</mi> <mi>&amp;theta;</mi> </mrow> </mfrac> <mi>c</mi> <mi>&amp;phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msup> <mi>Q</mi> <mi>n</mi> </msup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mfrac> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <mi>c</mi> <mi>&amp;phi;</mi> <mi>c</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</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>
<mrow> <mi>X</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mi>k</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>6</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mn>6</mn> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mi>k</mi> <mn>5</mn> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>P</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mo>-</mo> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> <mo>)</mo> <mi>q</mi> <mi>r</mi> <mo>-</mo> <msub> <mi>D</mi> <mi>&amp;phi;</mi> </msub> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>g</mi> <mo>-</mo> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msubsup> <mi>D</mi> <mi>z</mi> <mi>i</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mi>z</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
<mrow> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi>s</mi> <mi>&amp;theta;</mi> </mrow> <mrow> <mi>c</mi> <mi>&amp;theta;</mi> </mrow> </mfrac> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>q</mi> <mi>c</mi> <mi>&amp;phi;</mi> <mo>-</mo> <mi>r</mi> <mi>s</mi> <mi>&amp;phi;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <mi>&amp;theta;</mi> </mrow> </mfrac> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>q</mi> <mi>s</mi> <mi>&amp;phi;</mi> <mo>+</mo> <mi>r</mi> <mi>c</mi> <mi>&amp;phi;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mrow> <mi>s</mi> <mi>&amp;theta;</mi> </mrow> <mrow> <mi>c</mi> <mi>&amp;theta;</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mo>)</mo> <mi>p</mi> <mi>r</mi> <mo>+</mo> <msub> <mi>D</mi> <mi>&amp;theta;</mi> </msub> <mi>q</mi> </mrow> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mfrac> <mi>s</mi> <mi>&amp;phi;</mi> <mo>+</mo> <mfrac> <msub> <mi>D</mi> <mi>&amp;psi;</mi> </msub> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mfrac> <mi>r</mi> <mi>c</mi> <mi>&amp;phi;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>
3. the quad-rotor aircraft diagonal power loss of claim 1The out-of-control protection control algorithm under the state is characterized in that: controlling the target (x) for a horizontal position control system (1.4)d,yd) Obtaining a horizontal position control type for a second-order micro horizontal position track;
if the virtual control input phidGiven as follows:
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;phi;</mi> <mi>d</mi> </msub> <mo>=</mo> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mfrac> <mi>&amp;beta;</mi> <msqrt> <mrow> <msubsup> <mi>&amp;alpha;</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;alpha;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </msqrt> </mfrac> <mo>-</mo> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mfrac> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> <msqrt> <mrow> <msubsup> <mi>&amp;alpha;</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;alpha;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </msqrt> </mfrac> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mo>\</mo> <mo>*</mo> <mi>M</mi> <mi>E</mi> <mi>R</mi> <mi>G</mi> <mi>E</mi> <mi>F</mi> <mi>O</mi> <mi>R</mi> <mi>M</mi> <mi>A</mi> <mi>T</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>0.12</mn> <mo>)</mo> </mrow> </mrow>
wherein
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msub> <mi>F</mi> <mi>l</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mn>7</mn> </msub> <mi>s</mi> <mi>&amp;theta;</mi> <mi>c</mi> <mi>&amp;psi;</mi> <mo>+</mo> <msub> <mi>e</mi> <mn>9</mn> </msub> <mi>s</mi> <mi>&amp;theta;</mi> <mi>s</mi> <mi>&amp;psi;</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msub> <mi>F</mi> <mi>l</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mn>7</mn> </msub> <mi>s</mi> <mi>&amp;psi;</mi> <mo>-</mo> <msub> <mi>e</mi> <mn>9</mn> </msub> <mi>c</mi> <mi>&amp;psi;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>\</mo> <mo>*</mo> <mi>M</mi> <mi>E</mi> <mi>R</mi> <mi>G</mi> <mi>E</mi> <mi>F</mi> <mi>O</mi> <mi>R</mi> <mi>M</mi> <mi>A</mi> <mi>T</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>0.13</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>&amp;beta;</mi> <mo>=</mo> <mo>-</mo> <msubsup> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>7</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>e</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>9</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>e</mi> <mn>7</mn> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>+</mo> <msub> <mi>e</mi> <mn>9</mn> </msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mi>k</mi> <mrow> <mi>h</mi> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mrow> <mi>h</mi> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mrow> <mi>h</mi> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>k</mi> <mrow> <mi>h</mi> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mrow> <mi>h</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mn>7</mn> </msub> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msubsup> <mi>D</mi> <mi>x</mi> <mi>i</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>e</mi> <mn>9</mn> </msub> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msubsup> <mi>D</mi> <mi>y</mi> <mi>i</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>\</mo> <mo>*</mo> <mi>M</mi> <mi>E</mi> <mi>R</mi> <mi>G</mi> <mi>E</mi> <mi>F</mi> <mi>O</mi> <mi>R</mi> <mi>M</mi> <mi>A</mi> <mi>T</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>0.14</mn> <mo>)</mo> </mrow> </mrow>
The horizontal position tracking error eh1,eh2The index stabilizes at zero equilibrium point.
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