CN112764426A - Internal model anti-interference control method of quad-rotor unmanned aerial vehicle system - Google Patents
Internal model anti-interference control method of quad-rotor unmanned aerial vehicle system Download PDFInfo
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Abstract
The invention provides an unmanned aerial vehicle anti-interference control method based on an internal model control principle, and a physical system of an unmanned aerial vehicle is considered. The intelligent control method comprises the steps of establishing an unmanned aerial vehicle dynamic model with external disturbance, building an external disturbance system, designing an unmanned aerial vehicle distributed control algorithm with the external disturbance based on theoretical knowledge such as graph theory, convex analysis and the like, wherein the algorithm is required to meet constraint conditions of the system and balance points exist, further designing a Lyapunov function, proving the convergence of the set calculation method by Lyapunov theorem, ensuring that state variables of the unmanned aerial vehicle converge to a target position, and finally verifying the effectiveness and feasibility of the method again by a numerical simulation example.
Description
Technical Field
The invention belongs to the field of combination of an unmanned aerial vehicle technology and an internal model control technology under consideration of a physical system, and relates to a distributed method for internal model disturbance rejection when an unmanned aerial vehicle is disturbed by the outside.
Background
Four rotor crafts, also known as four rotor unmanned aerial vehicle, four rotor helicopter, four shaft air vehicle etc. are called four rotors for short, are the most typical unmanned aerial vehicle among the many rotor crafts. The quad-rotor unmanned aerial vehicle has the dynamic characteristics of nonlinearity, underactuation and strong coupling, and is often used as an experimental carrier for theoretical research and method verification by researchers. Aiming at the control problem of a quadrotor unmanned aerial vehicle system with external disturbance, a plurality of control methods exist, such as PID control, adaptive control, sliding mode control, robust control and the like, but the control effect brought by each control mode is not completely the same, so that on the basis of combining the characteristics of the unmanned aerial vehicle system, a control method which enables the unmanned aerial vehicle to have the best autonomous flight effect is needed to be invented.
Internal Model Control (IMC) is a Control strategy for controller design based on an object mathematical model, and an important feature of the IMC structure is that robustness can be used as a design target of a system in a simple manner, and the IMC has the advantages of simple design, good Control performance, easy on-line analysis and the like. The method is not only a practical advanced control algorithm, but also an important theoretical basis for researching model-based control strategies such as predictive control and the like, and is a powerful tool for improving the design level of a conventional control system. The research on internal model control is continuously perfected, and the internal model control is developed from a conventional internal model control design method to an intelligent internal model control method. Along with the development of unmanned aerial vehicle technique in recent years, the application of unmanned aerial vehicle has all received people's extensive attention in the aspect of civilian or military, and the research to unmanned aerial vehicle technique is also more and more. Because the unmanned aerial vehicle system itself has the problems of poor robustness, low control precision and the like, the research on the stable control of the unmanned aerial vehicle becomes one of the key problems in the unmanned aerial vehicle autonomous flight technology. Although the traditional PID control algorithm can well complete the stable control of the unmanned aerial vehicle system under the condition that the unmanned aerial vehicle is not considered to be subjected to external disturbance, the traditional PID control algorithm has some limitations, and can not meet ideal control requirements under the condition that an actuator has extreme disturbance. Therefore, the internal model control strategy can stably control the unmanned aerial vehicle under the condition that the actuator has abnormal value disturbance, and has no good inhibition effect on dynamic disturbance. Compared with classical control strategies such as PID (proportion integration differentiation), the method can improve the control precision of the unmanned aerial vehicle system, has better robustness and has more practical value.
Disclosure of Invention
In order to overcome the defect that a quad-rotor unmanned aerial vehicle system can stably execute flight tasks under the condition of external disturbance, the invention provides an internal model control method based on the unmanned aerial vehicle system, which can effectively compensate the disturbance caused by the external system of the unmanned aerial vehicle and ensure the stable flight of the unmanned aerial vehicle.
The technical scheme proposed for solving the technical problems is as follows:
step 1, converting a matrix R from a body coordinate system (B system) to an inertial coordinate system (E system)BEComprises the following steps:
where xi is [ x, y, z ]]T、η＝[φ,θ,ψ]TThe position and the direction of the quad-rotor unmanned aerial vehicle are shown, and phi, theta and psi respectively show the roll angle, the pitch angle and the yaw angle of the unmanned aerial vehicle; coordinate system E system (E, X)E,YE,ZE) Representing an inertial frame, frame B (B, X)B,YB,ZB) Is a coordinate system of the body.
And 2, before establishing the mathematical model, assuming that the quad-rotor unmanned aerial vehicle is a rigid body, and the origin of the body coordinate system is coincident with the mass center of the quad-rotor unmanned aerial vehicle, analyzing the dynamic model of the quad-rotor unmanned aerial vehicle by adopting an Euler-Lagrange formula. The four-rotor unmanned mobile energy comprises two parts of translation and rotation, and the translation kinetic energy is
Rotational kinetic energy
Potential energy of the quad-rotor unmanned aerial vehicle is U-mgz, wherein m is the mass of the quad-rotor unmanned aerial vehicle, I-diag { I ═x,Iy,IzIf the moment of inertia matrix is used, g is the gravity coefficient, then:
the quad-rotor unmanned aerial vehicle model can be obtained from the following euler-lagrange equation:
wherein (F)ξτ) denotes generalized force, FξIs the translation force under the inertial coordinate system, tau ═ tau1 τ2 τ3]T，τ1For roll moment, τ2Representing the pitching moment, τ3Is the yaw moment.
Substituting equation (3) into equation (2) yields:
the force generated by the rotation of the rotor wing under the coordinate system of the body is FR＝[0 0 ut]T，ut＝f1+f2+f3+f4If the rotor force is F under the inertial coordinate systemξ1＝RBEFR(ii) a Four rotor unmanned aerial vehicle receive the influence of air resistance and external disturbance at flight in-process, and air resistance is
Wherein Kξ＝diag{K1,K2,K3The is an air resistance coefficient matrix; external disturbance dξ＝[dx dydz]TAnd the translational force borne by the quad-rotor unmanned aerial vehicle is Fξ＝Fξ1+Fξ2+dξ。
To four rotor unmanned aerial vehicle carry out the atress analysis can know, the moment that the rotor is rotatory to produce is
b is the lift coefficient of the rotor, d is the drag moment coefficient, l is the center of the rotor to four rotorsLength, w, of center of mass of unmanned aerial vehiclei(i ═ 1,2,3,4) represents the individual rotor speeds; four rotor unmanned aerial vehicle receive the influence of air drag moment and external disturbance at flight in-process, and the air drag moment is
Wherein Kη＝diag{K4,K5,K6The is an air resistance coefficient matrix; external disturbance dη＝[dφ dθ dψ]TThen τ is τ11+τ22+dη。
Combining formula (4) and formula (5), when the structural symmetry of four rotor unmanned aerial vehicle or control accuracy require to be lower, neglect the simplified model of four rotor unmanned aerial vehicle that the Coriolis term can be got:
step 3, determining a system model of the unmanned aerial vehicle and a concrete expression form of external disturbance, wherein a dynamic model of the quadrotor unmanned aerial vehicle i subjected to the external disturbance can be rewritten into the following form:
wherein q isi＝(xi,yi,zi,φi,θi,ψi)T；
Is a symmetric matrix composed of inertia and mass;
is a Coriolis force matrix;
representing a gravitational potential energy matrix;
for control input to the system, Fi＝RBE(0 0 ui)T。
It can be easily demonstrated that the system (7) has the following properties:
1) matrix array
Is obliquely symmetrical;
2)Mi(qi) Is about qiHas a positive definite matrix, and
about
Is linearly bounded, i.e. there is a normal number k m i、
And
make inequality
And
if true;
3) matrix array
With respect to qiIs bounded and about
Linearly bounded, i.e. the presence of a positive constant kciMake inequality
This is true.
4)For any one
Is provided with
Wherein
Is a regression matrix composed of measurable state variables of the system,
is a constant vector made up of uncertainty parameters in the system.
An external disturbance in the system (7) having the form:
wherein the content of the first and second substances,
all eigenvalues of (a) are all located on the imaginary axis, which means that the perturbation is bounded.
Secondly, we first give a rationale about external disturbances.
Introduction 1: p (λ) ═ λs+p1λs-1+…+psIs the minimum polynomial of S, the perturbation (8) can be rewritten as:
wherein the content of the first and second substances,
Ψ＝[1|01×(s-1)]and
obviously, there is one vector G such that F Φ + G Ψ is the Hurwitz matrix. Furthermore, there is a positive definite symmetric matrix P such that FTP+PF＝-2IsThis is true.
And 4, designing an unmanned aerial vehicle distributed control algorithm for inhibiting external disturbance based on an internal model control principle. The algorithm is expressed as follows:
wherein u isiIs the control input of the ith drone, biIs the local resource constraint of the ith drone, qiIs the location information of the ith drone,
is a gradient term used to guide the agent to search for the optimal solution;
is a consistent item to ensure that the agent can share some necessary identical information with neighbors;
is an internal model term used to suppress external interference. It is required to satisfy
And the cost function of each drone is denoted fi(q) the communication topology of the drone is a weighted balanced directed graph, using
Expressed, the optimization problem can be expressed as follows:
the objective of the whole algorithm is to find a balance point, and the balance point not only enables the unmanned aerial vehicle to reach the expected position under the condition of external disturbance, but also minimizes the cost of the whole unmanned aerial vehicle cluster.
And step 5, proving that the algorithm is converged, and can well inhibit external disturbance and search a balance point.
The first step is as follows: order to
As a compensation disturbance; this is the case:
the algorithm can be rewritten into the following compact form:
the second step is that: assuming a balance point
There exists, i.e. one nash equilibrium point can be found so that the drone system can reach the desired formation under external disturbance, the equilibrium point is set
Can be substituted into formula (12) to obtain
There is such an equilibrium point.
The third step: performing orthogonal transformation, splitting the unmanned aerial vehicle system into two subsystems, selecting Lyapunov function V for convergence verification, and respectively selecting V1,V2The following were used:
the fourth step: proves that the Lyapunov function is negative definite, and is obtained by derivation
Respectively as follows:
wherein
Then selecting V as V1+V2Is provided with
Obtaining the syndrome.
In practice, the flight of the unmanned aerial vehicle is always disturbed by the external environment (mainly caused by wind speed and disturbance torque generated by the wind speed), so that in order to design an unmanned aerial vehicle controller resisting external disturbance, the robust controller based on distributed algorithm design provided by the patent is a quad-rotor unmanned aerial vehicle controller combining an internal model principle and an unmanned aerial vehicle dynamic model, and the correctness and the effectiveness of the method are proved on mathematical proofs and numerical simulation.
The invention has the advantages and innovation that: the unmanned aerial vehicle control method has the advantages that robustness of the unmanned aerial vehicle system is enhanced, especially in a disturbed environment, dynamic tracking control of a single unmanned aerial vehicle and formation of unmanned aerial vehicle control formation can be achieved in a complex environment, and rapid and stable convergence to a target value can be achieved. Innovations are that the unmanned aerial vehicle system is known to be a characteristic of an Euler Lagrange system, so that the unmanned aerial vehicle is modeled into an Euler Lagrange dynamics model, an external disturbance term is added, and disturbance of an external system is suppressed through an internal model principle.
Drawings
The table is the system parameters of the drone.
Fig. 1 is a diagram of a communication topology between drones.
Fig. 2 drone X-axis position under external disturbance condition.
Fig. 3Y-axis position of the drone under external disturbance conditions.
Fig. 4 the Z-axis position of the drone under external disturbance conditions.
Detailed Description
The invention is further described with reference to the accompanying drawings.
With reference to table one and fig. 1 to 4, an internal model disturbance rejection control method for a quad-rotor unmanned aerial vehicle system includes the following steps:
step 1, converting a matrix R from a body coordinate system (B system) to an inertial coordinate system (E system)BEComprises the following steps:
where xi is [ x, y, z ]]T、η＝[φ,θ,ψ]TThe position and the direction of the quad-rotor unmanned aerial vehicle are shown, and phi, theta and psi respectively show the roll angle, the pitch angle and the yaw angle of the unmanned aerial vehicle; coordinate system E system (E, X)E,YE,ZE) Representing an inertial frame, frame B (B, X)B,YB,ZB) Is a coordinate system of the body.
And 2, before establishing the mathematical model, assuming that the quad-rotor unmanned aerial vehicle is a rigid body, and the origin of the body coordinate system is coincident with the mass center of the quad-rotor unmanned aerial vehicle, analyzing the dynamic model of the quad-rotor unmanned aerial vehicle by adopting an Euler-Lagrange formula. The four-rotor unmanned mobile energy comprises two parts of translation and rotation, and the translation kinetic energy is
Rotational kinetic energy
Potential energy of the quad-rotor unmanned aerial vehicle is U-mgz, wherein m is the mass of the quad-rotor unmanned aerial vehicle, I-diag { I ═x,Iy,IzIf the moment of inertia matrix is used, g is the gravity coefficient, then:
the quad-rotor unmanned aerial vehicle model can be obtained from the following euler-lagrange equation:
wherein (F)ξτ) denotes generalized force, FξIs the translation force under the inertial coordinate system, tau ═ tau1 τ2 τ3]T，τ1For roll moment, τ2Representing the pitching moment, τ3Is the yaw moment.
Substituting equation (3) into equation (2) yields:
the force generated by the rotation of the rotor wing under the coordinate system of the body is FR＝[0 0 ut]T，ut＝f1+f2+f3+f4If the rotor force is F under the inertial coordinate systemξ1＝RBEFR(ii) a Four rotor unmanned aerial vehicle receive the influence of air resistance and external disturbance at flight in-process, and air resistance is
Wherein Kξ＝diag{K1,K2,K3The is an air resistance coefficient matrix; exterior partDisturbance is dξ＝[dx dydz]TAnd the translational force borne by the quad-rotor unmanned aerial vehicle is Fξ＝Fξ1+Fξ2+dξ。
To four rotor unmanned aerial vehicle carry out the atress analysis can know, the moment that the rotor is rotatory to produce is
b is the lift coefficient of the rotor, d is the drag moment coefficient, l is the length from the center of the rotor to the center of mass of the quad-rotor unmanned aerial vehicle, wi(i ═ 1,2,3,4) represents the individual rotor speeds; four rotor unmanned aerial vehicle receive the influence of air drag moment and external disturbance at flight in-process, and the air drag moment is
Wherein Kη＝diag{K4,K5,K6The is an air resistance coefficient matrix; external disturbance dη＝[dφ dθ dψ]TThen τ is τ11+τ22+dη。
Combining formula (4) and formula (5), when the structural symmetry of four rotor unmanned aerial vehicle or control accuracy require to be lower, neglect the simplified model of four rotor unmanned aerial vehicle that the Coriolis term can be got:
step 3, determining a system model of the unmanned aerial vehicle and a concrete expression form of external disturbance, wherein a dynamic model of the quadrotor unmanned aerial vehicle i subjected to the external disturbance can be rewritten into the following form:
wherein q isi＝(xi,yi,zi,φi,θi,ψi)T；
Is a symmetric matrix composed of inertia and mass;
is a Coriolis force matrix;
representing a gravitational potential energy matrix;
for control input to the system, Fi＝RBE(0 0 ui)T。
It can be easily demonstrated that the system (7) has the following properties:
5) matrix array
Is obliquely symmetrical;
6)Mi(qi) Is about qiHas a positive definite matrix, and
about
Is linearly bounded, i.e. there is a normal number k m i、
And
make inequality
And
if true;
7) matrix array
With respect to qiIs bounded and about
Linearly bounded, i.e. the presence of a positive constant kciMake inequality
This is true.
8) For any one
Is provided with
Wherein
Is a regression matrix composed of measurable state variables of the system,
is a constant vector made up of uncertainty parameters in the system.
An external disturbance in the system (7) having the form:
wherein the content of the first and second substances,
all eigenvalues of (a) are all located on the imaginary axis, which means that the perturbation is bounded.
Secondly, we first give a rationale about external disturbances.
Introduction 1: p (λ) ═ λs+p1λs-1+…+psIs the minimum polynomial of S, the perturbation (8) can be rewritten as:
wherein the content of the first and second substances,
Ψ＝[1|01×(s-1)]and
obviously, there is one vector G such that F Φ + G Ψ is the Hurwitz matrix. Furthermore, there is a positive definite symmetric matrix P such that FTP+PF＝-2IsThis is true.
And 4, designing an unmanned aerial vehicle distributed control algorithm for inhibiting external disturbance based on an internal model control principle. The algorithm is expressed as follows:
wherein u isiIs the control input of the ith drone, biIs the local resource constraint of the ith drone, qiIs the location information of the ith drone,
is a gradient term used to guide the agent to search for the optimal solution;
is a consistent item to ensure that the agent can share some necessary identical information with neighbors;
is an internal model term to prevent external interference. It is required to satisfy
And the cost function of each drone is denoted fi(q) the communication topology of the drone is weighted balancingDirected graph, using
Expressed, the optimization problem can be expressed as follows:
the objective of the whole algorithm is to find a balance point, and the balance point not only enables the unmanned aerial vehicle to reach the expected position under the condition of external disturbance, but also minimizes the cost of the whole unmanned aerial vehicle cluster.
And step 5, proving that the algorithm is converged, and can well inhibit external disturbance and search a balance point.
The first step is as follows: order to
As a compensation disturbance; this is the case:
the algorithm can be rewritten into the following compact form:
the second step is that: assuming a balance point
There exists, i.e. one nash equilibrium point can be found so that the drone system can reach the desired formation under external disturbance, the equilibrium point is set
Can be substituted into formula (12) to obtain
There is such an equilibrium point.
Third stepThe method comprises the following steps: performing orthogonal transformation, splitting the unmanned aerial vehicle system into two subsystems, selecting Lyapunov function V for convergence verification, and respectively selecting V1,V2The following were used:
the fourth step: proves that the Lyapunov function is negative definite, and is obtained by derivation
Respectively as follows:
wherein
Then selecting V as V1+V2Is provided with
Obtaining the syndrome.
Numerical simulation selects
C＝[1 0]Wherein p is a number 1,
Ψ＝[1 0].
that is, F Φ + G Ψ satisfies the herwitz condition.
And di(t)＝Aisin(pt+ci) The simulation is selected
x1(0)＝5,x2(0)＝2,x3(0)＝-2,x4(0)＝-0.5,x5(0)＝4,x6(0) 3.5. Finally, the convergence to the target value can be known through experiments.
From the table one we can get the specific parameters of the drone system in order to calculate specific values. From fig. 1, we can know the communication topology structure between the unmanned aerial vehicles, and from the numerical simulation experiment in fig. 2, it can be known that the state change of the unmanned aerial vehicles is obvious when five unmanned aerial vehicles are subjected to external disturbance at the beginning, and when the method designed by the invention is adopted, the fluctuation caused by the disturbance can be obviously inhibited, and the target convergence value can be quickly reached.
Claims (1)
1. The invention provides an internal model control method based on an unmanned aerial vehicle system, which can effectively compensate disturbance caused by an external system of the unmanned aerial vehicle and ensure stable flight of the unmanned aerial vehicle. The invention mainly comprises the following steps:
step 1, converting a matrix R from a body coordinate system (B system) to an inertial coordinate system (E system)BEComprises the following steps:
where xi is [ x, y, z ]]T、η＝[φ,θ,ψ]TThe position and the direction of the quad-rotor unmanned aerial vehicle are shown, and phi, theta and psi respectively show the roll angle, the pitch angle and the yaw angle of the unmanned aerial vehicle; coordinate system E system (E, X)E,YE,ZE) Representing an inertial frame, frame B (B, X)B,YB,ZB) Is made into a machineAnd (4) a body coordinate system.
And 2, before establishing the mathematical model, assuming that the quad-rotor unmanned aerial vehicle is a rigid body, and the origin of the body coordinate system is coincident with the mass center of the quad-rotor unmanned aerial vehicle, analyzing the dynamic model of the quad-rotor unmanned aerial vehicle by adopting an Euler-Lagrange formula. The four-rotor unmanned mobile energy comprises two parts of translation and rotation, and the translation kinetic energy is
Rotational kinetic energy
Potential energy of the quad-rotor unmanned aerial vehicle is U-mgz, wherein m is the mass of the quad-rotor unmanned aerial vehicle, I-diag { I ═x,Iy,IzIf the moment of inertia matrix is used, g is the gravity coefficient, then:
the quad-rotor unmanned aerial vehicle model can be obtained from the following euler-lagrange equation:
wherein (F)ξτ) denotes generalized force, FξIs the translation force under the inertial coordinate system, tau ═ tau1 τ2 τ3]T，τ1For roll moment, τ2Representing the pitching moment, τ3Is the yaw moment.
Substituting equation (3) into equation (2) yields:
the force generated by the rotation of the rotor wing under the coordinate system of the body is FR＝[0 0 ut]T，ut＝f1+f2+f3+f4If the rotor force is F under the inertial coordinate systemξ1＝RBEFR(ii) a Four rotor unmanned aerial vehicle receive the influence of air resistance and external disturbance at flight in-process, and air resistance is
Wherein Kξ＝diag{K1,K2,K3The is an air resistance coefficient matrix; external disturbance dξ＝[dx dy dz]TAnd the translational force borne by the quad-rotor unmanned aerial vehicle is Fξ＝Fξ1+Fξ2+dξ。
To four rotor unmanned aerial vehicle carry out the atress analysis can know, the moment that the rotor is rotatory to produce is
b is the lift coefficient of the rotor, d is the drag moment coefficient, l is the length from the center of the rotor to the center of mass of the quad-rotor unmanned aerial vehicle, wi(i ═ 1,2,3,4) represents the individual rotor speeds; four rotor unmanned aerial vehicle receive the influence of air drag moment and external disturbance at flight in-process, and the air drag moment is
Wherein Kη＝diag{K4,K5,K6The is an air resistance coefficient matrix; external disturbance dη＝[dφ dθ dψ]TThen τ is τ11+τ22+dη。
Combining formula (4) and formula (5), when the structural symmetry of four rotor unmanned aerial vehicle or control accuracy require to be lower, neglect the simplified model of four rotor unmanned aerial vehicle that the Coriolis term can be got:
step 3, determining a system model of the unmanned aerial vehicle and a concrete expression form of external disturbance, wherein a dynamic model of the quadrotor unmanned aerial vehicle i subjected to the external disturbance can be rewritten into the following form:
wherein q isi＝(xi,yi,zi,φi,θi,ψi)T；
Is a symmetric matrix composed of inertia and mass;
is a Coriolis force matrix;
representing a gravitational potential energy matrix;
for control input to the system, Fi＝RBE(0 0 ui)T。
It can be easily demonstrated that the system (7) has the following properties:
1) matrix array
Is diagonally symmetrical.
2)Mi(qi) Is about qiHas a positive definite matrix, and
about
Is linearly bounded, i.e. there is a normal number k m i、
And kmiMake inequality
And
this is true.
3) Matrix array
With respect to qiIs bounded and about
Linearly bounded, i.e. the presence of a positive constant kciMake inequality
This is true.
4) For any one
Is provided with
Wherein
Is a regression matrix composed of measurable state variables of the system,
is a constant composed of uncertainty parameters in the systemAnd (5) vector quantity.
An external disturbance in the system (7) having the form:
wherein the content of the first and second substances,
all eigenvalues of (a) are all located on the imaginary axis, which means that the perturbation is bounded.
Secondly, we first give a rationale about external disturbances.
Introduction 1: p (λ) ═ λs+p1λs-1+…+psIs the minimum polynomial of S, the perturbation (8) can be rewritten as:
wherein the content of the first and second substances,
Ψ＝[1|01×(s-1)]and
obviously, there is one vector G such that F Φ + G Ψ is the Hurwitz matrix. Furthermore, there is a positive definite symmetric matrix P such that FTP+PF＝-2IsThis is true.
And 4, designing an unmanned aerial vehicle distributed control algorithm for inhibiting external disturbance based on an internal model control principle. The algorithm is expressed as follows:
wherein u isiIs the control input for the ith drone,biis the local resource constraint of the ith drone, qiIs the location information of the ith drone,
is a gradient term used to guide the agent to search for the optimal solution;
is a consensus item to ensure that the agent can share some of the same information necessary with the neighbors,
is an internal model term used to suppress external interference. It is required to satisfy
And the cost function of each drone is denoted fi(q) the communication topology of the drone is a weighted balanced directed graph, using
Expressed, the optimization problem can be expressed as follows:
the objective of the whole algorithm is to find a balance point, and the balance point not only enables the unmanned aerial vehicle to reach the expected position under the condition of external disturbance, but also minimizes the cost of the whole unmanned aerial vehicle cluster.
And step 5, proving that the algorithm is converged, and can well inhibit external disturbance and search a balance point.
The first step is as follows: order to
As a compensation disturbance; this is the case:
the algorithm can be rewritten into the following compact form:
the second step is that: assuming a balance point
There exists, i.e. one nash equilibrium point can be found so that the drone system can reach the desired formation under external disturbance, the equilibrium point is set
Can be substituted into formula (12) to obtain
There is such an equilibrium point.
The third step: performing orthogonal transformation, splitting the unmanned aerial vehicle system into two subsystems, selecting Lyapunov function V for convergence verification, and respectively selecting V1,V2The following were used:
the fourth step: proves that the Lyapunov function is negative definite, and is obtained by derivation
Respectively as follows:
wherein
(normal number),
then selecting V as V1+V2Is provided with
Obtaining the syndrome.
Therefore, the unmanned aerial vehicle can quickly and accurately reach the expected target position under the condition of external disturbance.
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