CN111948935A - Self-coupling PD control theory method of under-actuated VTOL aircraft - Google Patents

Self-coupling PD control theory method of under-actuated VTOL aircraft Download PDF

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CN111948935A
CN111948935A CN202010765670.8A CN202010765670A CN111948935A CN 111948935 A CN111948935 A CN 111948935A CN 202010765670 A CN202010765670 A CN 202010765670A CN 111948935 A CN111948935 A CN 111948935A
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曾喆昭
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Abstract

Aiming at the control problem of a non-minimum phase under-actuated vertical take-off and landing (VTOL) aircraft, the invention provides an auto PD (ACPD) control theory method. Firstly, mapping the mass center of the VTOL aircraft into a Huygens vibration center through coordinate transformation; secondly, designing ACPD controllers at longitudinal and transverse positions for a Huygens vibration center respectively, obtaining virtual instructions of the bottom thrust and the rolling attitude angle of the VTOL aircraft respectively, and designing an auto-coupling PD controller of the rolling attitude angle to form a rolling torque so as to realize position tracking control of the VTOL aircraft system; and finally, proving the robust stability and the disturbance resistance robustness of the closed-loop control system through a complex frequency domain analysis theory. Theoretical analysis and simulation results show the effectiveness of the method, and the method has important scientific significance and wide application prospect in the field of non-minimum phase under-actuated VTOL control systems.

Description

Self-coupling PD control theory method of under-actuated VTOL aircraft
Technical Field
Non-minimum phase under-actuated nonlinear system control and aircraft control.
Background
Because of its short take-off distance and vertical take-off and landing performance, the VTOL (vertical take-off and landing) aircraft plays an extremely important role in military and civil fields, and has become an object of aircraft in controversy and research in various countries. The VTOL vehicle is a typical under-actuated system because there are three degrees of freedom of motion but only two control inputs. Since the VTOL aircraft not only has the underactuation characteristic and the nonlinear coupling characteristic, but also is a non-minimum phase system, the control problem becomes more complex, and the VTOL aircraft draws wide attention of many domestic and foreign scholars in the control field. So far, the control methods related to VTOL aircrafts mainly include: a state feedback control method, an inversion (backstepping) control method, a dynamic surface control method, a sliding mode control method, an inverse optimal feedback control method, a double closed-loop PID control method, a nonlinear information fusion control method and the like. Although the existing control methods all obtain effective control results, the state feedback control method has the limitation of poor dynamic quality and steady-state performance; the Backstepping control method has the limitation of differential explosion; the control methods such as dynamic surface control, sliding mode control, inverse optimal feedback control, nonlinear information fusion control and the like have the limitations of complex structure and large calculated amount; the double closed-loop PID control method has the limitations of poor gain robustness and poor disturbance resistance robustness.
In order to effectively improve the dynamic quality and the steady-state performance of an under-actuated VTOL aircraft control system, reduce the calculated amount, improve the real-time performance and enhance the robust stability and the disturbance resistance robustness of the control system, the urgent need is to invent a control theory method which has the advantages of simple controller structure, easy parameter setting, good dynamic quality and steady-state performance of the control system and strong disturbance resistance robustness.
Disclosure of Invention
The invention provides an Auto-Coupling probability-Differential (ACPD) control theory method of an under-actuated VTOL aircraft, which is characterized by comprising the following steps of:
1) expected trajectory x from VTOL aerial vehicle centroid coordinates1d、x3dAnd desired roll attitude angle x5dThe expected trajectory of the available Huygens vibration center coordinates is 0:
z1d＝x1d- sin x5d＝x1d，z3d＝x3d+ cos x5d＝x3d+
wherein,
Figure BDA0002614503660000011
0the coupling effect coefficient of the roll moment on the transverse and longitudinal motion is shown, and m and J are the mass of the VTOL aircraft and the moment of inertia along the longitudinal axis respectively;
2) calculating the lateral position tracking error and its error differential according to step 1):
e11＝z1d-z1，
Figure BDA0002614503660000021
wherein z is1＝x1- sin x5，
Figure BDA0002614503660000022
x1And x5Respectively the lateral position and roll attitude angle of the VTOL vehicle,
Figure BDA0002614503660000023
3) defining the lateral position ACPDy controller according to step 2) as:
Figure BDA0002614503660000024
wherein z iscy>0 is the speed factor of the ACPDy controller;
4) calculating the longitudinal position tracking error and its error differential according to step 1):
e31＝z3d-z3，
Figure BDA0002614503660000025
wherein z is3＝x3+ cos x5,
Figure BDA0002614503660000026
x3Is the longitudinal position of the VTOL aerial vehicle,
Figure BDA0002614503660000027
5) defining the longitudinal position ACPDz controller according to step 4) as:
Figure BDA0002614503660000028
wherein z iscz>0 is the velocity factor of the ACPDz controller, g is the gravitational acceleration;
6) forming a control force according to steps 3) and 5)u1And the bottom thrust u of the VTOL aerial vehiclet：
Figure BDA0002614503660000029
ut＝mu1
7) Forming a rolling attitude angle virtual command of the VTOL aircraft according to the steps 3) and 5):
x5r＝-atan(uy/uz)
8) calculating the tracking error and the differential of the roll attitude angle according to the step 7):
e51＝x5r-x5，
Figure BDA00026145036600000210
9) ACPD2 controller u defining roll attitude angle according to step 8)2And forming a roll moment u of the VTOL aerial vehiclem：
Figure BDA00026145036600000211
um＝Ju2
Wherein z isc2>0 is the speed factor of the ACPD2 controller.
According to the method, the mass center of the VTOL aircraft is mapped into a Huygens vibration center through coordinate transformation, so that not only can the control input decoupling of a new system be realized, but also the problem of zero dynamic instability of the VTOL aircraft with a non-minimum phase can be solved; secondly, designing ACPD controllers at longitudinal and transverse positions for a Huygens vibration center respectively, obtaining virtual instructions of bottom thrust and rolling attitude angle of the VTOL aircraft respectively, and designing an auto-coupling PD controller of the rolling attitude angle to form rolling torque; and finally, proving the robust stability and the disturbance resistance robustness of the closed-loop control system through a complex frequency domain analysis theory. Theoretical analysis and simulation results show the effectiveness of the method, and the method has important scientific significance and wide application prospect in the field of non-minimum phase under-actuated VTOL control systems.
Drawings
Fig. 1 under-actuated VTOL aircraft model.
FIG. 2 shows control results of the VTOL aircraft in working procedures of vertical takeoff, hovering, landing, etc., (a) transverse position tracking trajectory, (b) longitudinal position tracking trajectory, (c) rolling attitude angle tracking trajectory, (d) bottom thrust control input, and (e) rolling torque control input.
Detailed Description
1. Under-actuated VTOL aircraft model and mapping thereof
1.1 under-actuated VTOL aircraft model
Consider a certain under-actuated VTOL aircraft system:
Figure BDA0002614503660000031
the state variables y, z and theta are respectively the horizontal and longitudinal positions and the roll attitude angle of the mass center of the VTOL aircraft in a reference coordinate; m and J are the mass and moment of inertia along the longitudinal axis of the aircraft, respectively; g is the acceleration of gravity; control input utAnd umThe bottom vertical thrust and the rolling torque are respectively;0is the coupling coefficient of roll torque to lateral and longitudinal motion.
Since the VTOL aerial vehicle (1) has three degrees of freedom y, z and theta, but only two control inputs utAnd umAnd thus is a typical under-actuated nonlinear coupling system.
For the convenience of analysis, the state variables are respectively: x is the number of1＝y，
Figure BDA0002614503660000032
x3＝z，
Figure BDA0002614503660000033
x5＝θ，
Figure BDA0002614503660000034
The system (1) may be represented in the form of a system (2); is then provided with
Figure BDA0002614503660000035
The system (2) can be abbreviated in the form of a system (3).
As known from the system (3), u2Not only with u1There is a coupling effect between the two and the roll attitude angle x of the system5There is also a coupling effect, which makes the control of the VTOL aircraft system more complex.
Figure BDA0002614503660000041
Figure BDA0002614503660000042
1.2. Under-actuated VTOL aircraft model mapping
And (3) performing the following coordinate transformation on the system (3):
z1＝x1- sin x5，
Figure BDA0002614503660000043
z3＝x3+ cos x5，
Figure BDA0002614503660000044
the system (3) can be represented in the form of equation (4).
Figure BDA0002614503660000045
Obviously, the coordinate transformation not only realizes the input decoupling effect of the new VTOL aircraft system (4), but also enables the unstable centroid position (x) of the VTOL aircraft system (3)1,x3) Mapping to Huygens vibration center (z) with flat output1,z3)。
Since the output of the new VTOL aircraft system (4) is flat (z)1And z3Are all not affected by u2Is thus effectively avoidedThe problem of zero dynamic instability of the minimum phase VTOL aircraft. Since the system (4) is an equivalent mapping to the system (3) or system (2), the controller u designed by the VTOL aerial vehicle system (4)1And u2Effective control of the system (3) or the system (2) can be achieved. How to apply effective control to the system (4) is the core technology of the invention, namely ACPD control technology.
VTOL aircraft control System design
2.1 basic control concept for VTOL aerial vehicles
For the convenience of analysis, let uyAnd uzRespectively as follows:
Figure BDA0002614503660000051
Figure BDA0002614503660000052
the VTOL aircraft system (4) can be simplified to the form of equation (7):
Figure BDA0002614503660000053
according to the formulae (5) and (6), u can be obtained1And x5Respectively as follows:
Figure BDA0002614503660000054
x5＝-atan(uy/uz) (9)
obviously, as long as the controller u for the transverse and longitudinal positions is designedyAnd uzTo obtain a control force u1So as to obtain the vertical thrust u of the bottom of the VTOL aircraftt＝mu1(ii) a If the rolling attitude angle virtual instruction is set as follows:
x5r＝-atan(uy/uz) (10)
then the roll attitude angle controller u of the VTOL aircraft can be designed2Further obtain the rolling torque um＝Ju2。
VTOL aircraft bottom thrust controller design
Let the expected trajectory of VTOL aircraft centroid coordinate be x1dAnd x3dAnd setting the expected rolling attitude angle as x5d＝θdThe expected trajectory of the Huygens vibration center coordinates is thus available as:
z1d＝x1d- sin x5d＝x1d (11)
z3d＝x3d+ cos x5d＝x3d+ (12)
1) lateral position ACPDy controller design
Let the lateral position tracking error be:
e11＝z1d-z1 (13)
wherein z is1d＝x1d，z1＝x1- sin x5。
The derivative of the lateral tracking error from equation (13) and system (7) is:
Figure BDA0002614503660000061
in combination with the system (7), the lateral position controlled error system is obtained as follows:
Figure BDA0002614503660000062
in the case of neglecting the integral element, the transverse position ACPDy controller is designed as follows:
Figure BDA0002614503660000063
wherein z iscy>0 is the speed factor of the ACPDy controller.
2) Longitudinal position ACPDz controller design
Let the longitudinal position tracking error be:
e31＝z3d-z3 (16)
wherein z is3d＝x3d+，z3＝x3+ cos x5。
The differential of the longitudinal position tracking error from equation (16) and system (7) is:
Figure BDA0002614503660000064
in combination with the system (7), the available longitudinal position controlled error system is:
Figure BDA0002614503660000065
in the case of neglecting the integral element, the longitudinal position ACPDz controller is designed as:
Figure BDA0002614503660000066
wherein z iscz>0 is the velocity factor of the ACPDz controller and g is the gravitational acceleration.
3) VTOL aerial vehicle bottom thrust control input
Obtaining the control force u of the transverse and longitudinal positions according to the equations (15) and (18), respectivelyyAnd uzThen, the control force u can be obtained according to the formula (8)1Comprises the following steps:
Figure BDA0002614503660000067
and further can obtain the bottom vertical thrust of the under-actuated VTOL aircraft: u. oft＝mu1。
VTOL aircraft roll torque controller design
Obtaining the lateral and longitudinal position controller u according to the equations (15) and (18), respectivelyyAnd uzThen, the roll attitude angle can be obtained according to the formula (10)Virtual instructions: x is the number of5r＝-atan(uy/uz). The rolling attitude angle tracking error is set as follows:
e51＝x5r-x5 (20)
the derivative of the roll angle tracking error from equation (20) and system (7) is:
Figure BDA0002614503660000071
in combination with the system (7), the attitude angle controlled error system is obtained by:
Figure BDA0002614503660000072
in the case of neglecting the integral element, the roll attitude angle ACPD2 controller is designed as follows:
Figure BDA0002614503660000073
wherein z isc2>0 is the speed factor of the ACPD2 controller.
The roll moment of the under-actuated VTOL aircraft can be obtained by the formula (22): u. ofm＝Ju2。
2.4 closed-loop control System analysis
Theorem 1 setting
Figure BDA0002614503660000074
Then if and only if zcy>At 0, the VTOL aerial vehicle lateral position closed loop control system is bounded in input and output stable.
And (3) proving that: a lateral position control force u defined by the formula (15)ySubstituting the controlled error system (14) results in a closed loop control system for the lateral position as follows:
Figure BDA0002614503660000075
considering the error initial state:
Figure BDA0002614503660000076
therefore, the system (23) is subjected to a single-sided laplace transform and arranged to obtain:
Figure BDA0002614503660000077
wherein the first term of the system (24) is a zero-input response:
Figure BDA0002614503660000078
the second term is the zero state response:
Figure BDA0002614503660000079
the transfer function defining the lateral position closed loop system is:
Figure BDA0002614503660000081
when z iscy>At 0, due to Hy(s) double real poles s in the left half-plane of the complex frequency domainp＝-zcy<0, and thus system (25) or (24) is bounded input and bounded output stable, after certification.
Theorem 2 setting
Figure BDA0002614503660000082
The method has the following steps: | dy|≤y<Infinity, then if and only if zcy>At 0, the closed-loop control system for the transverse position of the VTOL aerial vehicle has good disturbance resistance robustness.
And (3) proving that: according to the system transfer function (25), the closed loop system (24) can be simplified as:
E11(s)＝E11x(s)+Hy(s)Dy(s) (26)
wherein,
Figure BDA0002614503660000083
from the system (25), the unit impulse response is obtained as:
hy＝t exp(-zcyt) (27)
the time domain solution of the closed loop system (26) can thus be expressed as:
e11＝e11x+hy*dy (28)
wherein,
Figure BDA0002614503660000084
"+" indicates convolution integral operation.
The differential of the error can be obtained from equation (28)
Figure BDA0002614503660000085
The following were used:
Figure BDA0002614503660000086
wherein,
Figure BDA0002614503660000087
when z iscy>At the time of 0, the number of the first,
Figure BDA0002614503660000088
therefore, as long as | dy|≤y<Infinity, then must be:
Figure BDA0002614503660000089
Figure BDA00026145036600000810
the above analysis shows that: when z iscy>0, as long as the bounded condition is satisfied: | dy|≤y<Infinity, the lateral position closed loop system (23) can be set from any initial error state that is not zero
Figure BDA00026145036600000811
Approaching to the stable balance point origin (0,0), zero error tracking control can be realized theoretically. When z iscy>At 0, due to e11→ 0 and e12→ 0 only with | dy|≤y<Infinity is related to
Figure BDA00026145036600000812
The specific model is irrelevant, so the ACPDy closed-loop control system has good disturbance resistance robustness after verification.
For the sake of space saving, the analysis of the longitudinal position closed-loop control system and the roll attitude angle closed-loop control system of the under-actuated VTOL aircraft is not repeated, and the robust stability analysis can be performed with reference to theorems 1 and 2.
2.5 adaptive speed factor
From theorems 1 and 2, it can be seen that: if and only if zcy>At 0, the transverse position control system of the under-actuated VTOL aircraft is in bounded input and bounded output stability, which shows that the speed factor of the ACPDy controller has a large setting margin, so the transverse position control system is stable in a large range. Similarly, the longitudinal position control system and the rolling attitude angle control system are stable in a large range. However, the velocity factor z when three variac PD controllers ACPDy, ACPDz and ACPD2 are usedcy、zczAnd zc2When the proportional control force and the differential control force are both large, although the response speed of the control system can be increased and the disturbance resistance can be enhanced, the output of the system is possibly subjected to overshoot, the control force is subjected to oscillation, and the actuator is not facilitated; when the speed factors are small, the proportional control force and the differential control force are small, so that the response speed of the control system is reduced, and the steady-state control accuracy and the disturbance rejection capability are also reduced. Therefore, in order to obtain good dynamic quality and steady state performance of the control system, an adaptive speed factor should be used.
Considering the sensitivity characteristic of error differentiation, the adaptive speed factor models of three self-coupling PD controllers, namely ACPDy, ACPDz and ACPD2, are defined as follows:
zcy＝zcmy exp(-|e12|) (30)
zcz＝zcmz exp(-|e32|) (31)
zc2＝zcm2 exp(-|e52|) (32)
wherein,
Figure BDA0002614503660000091
zcmy、zcmzand zcm2The maximum velocity factors of the three controllers ACPDy, ACPDz and ACPD2, respectively, for the lateral-longitudinal position and roll attitude angle.
Furthermore, considering that the lateral-longitudinal position should reach the coordinate point of the desired trajectory at the same time, the maximum speed factor of its controller should be the same: z is a radical ofcmy＝zcmz(ii) a The virtual command, considering again the roll attitude angle, is formed by the transverse and longitudinal control forces, i.e. x5r＝-atan(uy/uz) Thus, the maximum velocity factor z of the roll attitude controller ACPD2 is requiredcm2The following inequality conditions should be satisfied:
5zcmy≤zcm2≤10zcmy (33)
3. simulation results and analysis
In order to verify the effectiveness of the invention, relevant parameters of the under-actuated VTOL aircraft are set as follows: aircraft mass m 5 × 104kg, coefficient of coupling00.05, moment of inertia J2 × 105kg·m2. Setting the sampling frequency fs1000Hz, and the gravity acceleration g is 9.8 m.s-2(ii) a The Huygens vibration center coordinates are: z is a radical of1＝x1- sin x5，z3＝x3+ cos x5。
In the following experiments, the maximum speed factors of the three controllers were set to: z is a radical ofcmy＝zcmz＝10，zcm2Therefore, the overall composition structure of the under-actuated VTOL aircraft control system is as follows:
1) an ACPDy controller:
Figure BDA0002614503660000101
wherein z iscy＝10exp(-|e12|)；e11＝z1d-z1，
Figure BDA0002614503660000102
z1＝x1- sin x5，z1d＝x1d；x1dAnd x1Respectively a VTOL aircraft center of mass transverse coordinate expected value and an actual value.
2) ACPDz controller:
Figure BDA0002614503660000103
wherein z iscz＝10exp(-|e32|)，e31＝z3d-z3，
Figure BDA0002614503660000104
z3＝x3+ cos x5,z3d＝x3d+；x3dAnd x3Respectively a desired value and an actual value of the mass center longitudinal coordinate of the VTOL aircraft,
Figure BDA0002614503660000105
respectively obtain uyAnd uzAfter that, a control force u can be formed1Comprises the following steps:
Figure BDA0002614503660000106
considering the input limited case, it is required that: u is not less than 61Not more than 12, and then the vertical thrust of the bottom of the under-actuated VTOL aircraft can be obtained as follows: u. oft＝mu1。
3) ACPD2 controller:
Figure BDA0002614503660000107
wherein z isc2＝80exp(-|e52|)；e51＝x5r-x5，
Figure BDA0002614503660000108
x5r＝-atan(uy/uz) And | x5r|≤0.3rad。
Considering the input limited case, it is required that: | u2And | is less than or equal to 80, so that the roll torque of the under-actuated VTOL aircraft can be obtained: u. ofm＝Ju2。
To avoid differential operations of the speed measurement (without speed sensor) and the desired command, the error differential involved by the three controllers is approximated using a difference quotient:
Figure BDA0002614503660000109
wherein,
Figure BDA00026145036600001010
tsis the sampling period; in the same way, e32≈(e31-e31-1)/ts，e52≈(e51-e51-1)/ts。
Let VTOL aircraft's horizontal expected trajectory be x1d10 meters, desired roll attitude angle x5dThe desired longitudinal trajectory is 0:
Figure BDA00026145036600001011
setting the initial state of the VTOL aerial vehicle as
Figure BDA00026145036600001012
The simulation results using the present invention are shown in fig. 2. As can be seen from FIG. 2, the control method provided by the invention controls the whole working condition processes of vertical takeoff, hovering and landing of the under-actuated VTOL aircraft, obtains good dynamic quality and steady-state performance, and shows the effectiveness of the control method provided by the invention.
4. Conclusion
Aiming at the control problem of an under-actuated VTOL aircraft, the self-coupling PID control method is invented, the robustness stability of a closed-loop control system is analyzed in a complex frequency domain, and a simulation result shows the effectiveness of the control method, so that the control method has good dynamic quality and steady-state performance, and the control system has a simple structure, small calculated amount and is convenient for practical application. Simulation experiments also find that: maximum speed is more than or equal to 2 and less than or equal to zcmy＝zcmzLess than or equal to 14 and 5zcmy≤zcm2≤10zcmyWithin the range of the invention, the control method can realize effective control, thus being a control method with wide range, robust and stable.
The method has important scientific significance and wide application value in the field of under-actuated VTOL aircraft control.

Claims (1)

1. An auto PD control theory method of an under-actuated VTOL aircraft is characterized by comprising the following steps:
1) expected trajectory x from VTOL aerial vehicle centroid coordinates1d、x3dAnd desired roll attitude angle x5dThe expected trajectory of the available Huygens vibration center coordinates is 0:
z1d＝x1d-sin x5d＝x1d，z3d＝x3d+cos x5d＝x3d+
wherein,
Figure FDA0002614503650000011
0the coupling effect coefficient of the roll moment on the transverse and longitudinal motion is shown, and m and J are the mass of the VTOL aircraft and the moment of inertia along the longitudinal axis respectively;
2) calculating the lateral position tracking error and its error differential according to step 1):
e11＝z1d-z1，
Figure FDA0002614503650000012
wherein z is1＝x1-sin x5，
Figure FDA0002614503650000013
x1And x5Respectively the lateral position and roll attitude angle of the VTOL vehicle,
Figure FDA0002614503650000014
3) defining the lateral position ACPDy controller according to step 2) as:
Figure FDA0002614503650000015
wherein z iscy>0 is the speed factor of the ACPDy controller;
4) calculating the longitudinal position tracking error and its error differential according to step 1):
e31＝z3d-z3，
Figure FDA0002614503650000016
wherein z is3＝x3+cos x5,
Figure FDA0002614503650000017
x3Is the longitudinal position of the VTOL aerial vehicle,
Figure FDA0002614503650000018
5) defining the longitudinal position ACPDz controller according to step 4) as:
Figure FDA0002614503650000019
wherein z iscz>0 is the velocity factor of the ACPDz controller, g is the gravitational acceleration;
6) forming the control force u according to steps 3) and 5)1And the bottom vertical thrust u of the VTOL aircraftt：
Figure FDA00026145036500000110
ut＝mu1
7) Forming a rolling attitude angle virtual command of the VTOL aircraft according to the steps 3) and 5):
x5r＝-atan(uy/uz)
8) calculating the tracking error and the differential of the roll attitude angle according to the step 7):
e51＝x5r-x5，
Figure FDA0002614503650000021
9) ACPD2 controller u defining roll attitude angle according to step 8)2And forming a roll moment u of the VTOL aerial vehiclem：
Figure FDA0002614503650000022
um＝Ju2
Wherein z isc2>0 is the speed factor of the ACPD2 controller.
CN202010765670.8A 2020-08-03 2020-08-03 Self-coupling PD control theory method of under-actuated VTOL aircraft Pending CN111948935A (en)

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