A PDE fault-tolerant formation control method for aircraft swarms under dynamic topology

Abstract

The present invention discloses a PDE fault-tolerant formation control method for an aircraft cluster under a dynamic topology. First, a motion model of each aircraft in the coordinate direction is constructed, and a communication edge dynamic development law is established under the model. Secondly, a repulsion function and a switching attraction function between aircraft are constructed. Then, the dynamic characteristics of the switching ODE model of the cluster under the dynamic communication edge are analyzed, and then a PDE model is established, and the equilibrium curve under any given boundary input is used to generate a target deployment curve. Finally, the PDE model boundary conditions are used to construct a navigator controller and a follower controller in the aircraft cluster. The present invention only requires fewer calculation parameters, and the follower controller does not need information on the expected deployment curve, and generates an expected deployment profile according to the equilibrium curve of the PDE model, which solves the fault-tolerant formation control problem of the aircraft cluster under dynamic topology and actuator failure.

Description

Technical Field

[0001] The present invention belongs to the field of aircraft formation control, and in particular relates to a PDE fault-tolerant formation control method for an aircraft cluster under a dynamic topology. Background Art

[0002] Aircraft swarm is a system that completes various tasks through the mutual cooperation of a large number of aircraft. Therefore, the swarm can complete control tasks that a single aircraft cannot complete, and it has lower cost, better performance, stronger adaptability and robustness. It has been widely used in fields such as planning and regulation of smart microgrids, networking of wireless sensors, multi-aircraft formations in aerospace, and collaboration of multiple robots. In the process of aircraft cooperation, due to the limited perception capabilities of the sensors carried by the aircraft itself and the mobility of the aircraft itself, the aircraft may exceed the perception range of its neighbors, so it is necessary to consider the effective communication range. For example, only when the aircraft is within the maximum perception distance range of its neighbors can the two establish a communication edge, which leads to a dynamically changing communication topology, that is, the communication edges between aircraft change over time. Dynamic topology can better reflect the communication situation of aircraft swarms in practical applications, so it has been paid attention to and studied by many scholars. In recent years, the research on fault-tolerant formation control of aircraft swarm systems based on ordinary differential models (ODEs) has achieved fruitful results. In these results, the method of converging all ODE solutions to the expected values ​​of parameters such as position, displacement, and distance is usually used to obtain the desired deployment. Therefore, in the controller design process, not only the expected deployment information of each aircraft is needed, but also the expected deployment of the cluster as a whole. When considering the formation control of a large-scale cluster system, it is not easy to analyze the controller design.

[0003] Using the idea of ​​continuum, the entire cluster is regarded as the research object, and all aircraft can be deployed into the desired target formation under the technology of partial differential model (PDE). The main control idea is: first, in each coordinate direction of the plane, two decoupled partial differential equations or one coupled partial differential equation are used to model the dynamic behavior of all aircraft; secondly, the leader controller is designed using the boundary control idea to control the partial differential equation model to track the desired deployment curve; finally, the controllers of all follower aircraft are obtained by discretizing the partial differential equations.

[0004] Although cluster control technology based on PDE model has been developed, the established PDE model is based on the cluster ODE model under static topology, and there is almost no research result on the fault-tolerant formation control of clusters under dynamic topology based on PDE model. This is because, in the process of PDE modeling, it is challenging to use mathematical expressions to measure the relationship between aircraft under dynamic topology. Summary of the invention

[0005] In view of this, an embodiment of the present invention provides a PDE fault-tolerant formation control technology for aircraft clusters under dynamic topology. According to the conditions of dynamic addition of communication edges in the topology of the aircraft cluster, its dynamic development law is established; according to the dynamic development law, combined with the artificial potential function method, the switching attraction function and repulsion function generated by the dynamic addition of communication edges between aircraft are constructed to measure the deployment requirements of the cluster and the obstacle avoidance requirements between aircraft; according to the switching ODE model under the action of the potential function, the Taylor expansion technology and the continuous approximation idea are used, and the switching signal is introduced to construct the aircraft cluster PDE model with dynamic topology and actuator failure; according to the switching PDE model, the equilibrium curve and formation target information are used to generate the desired target deployment, and under the Lyapunov control technology, a navigator controller based on boundary control is designed; according to the controlled switching PDE model, the difference method is used for discretization to obtain the follower controller.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] S1. Consider a finite number of aircraft with horizontal and vertical coordinates (z *1 ,z *2 ) plane, where the state of aircraft i Indicates the horizontal position. is the vertical position, The motion model of each aircraft in the coordinate direction is:

[0008]

[0009] in is the control input, and the nth aircraft is marked as the leader aircraft. Define the complex variable is the position of aircraft i at time t, where the real component and imaginary component represent the position of aircraft i at time z *1 and z *2 The topology of the aircraft cluster is constructed as an undirected graph G(V,E(t)), where V is a set of vertices, each vertex represents an aircraft, and the follower aircraft are numbered from 1 to n-1, using (i,j). represents the communication edge between aircraft i and j, then the set of communication edges is There exists (i,j)∈E(t) if and only if (j,i)∈E(t). The connectivity of G(V,E(t)) means that there is a communication edge between any two aircraft at time t.

[0010] To illustrate the evolution of G(V,E(t)), consider two communication sets for each aircraft i at time t. One is the perception range set of aircraft i, which includes all aircraft that aircraft i can perceive, that is, the neighbor set The other is a subset of its neighbor set Based on this, the following communication edge dynamic development law is constructed:

[0011] E(t)=E(t - )∪E * (t),

[0012] t - Indicates that at the moment before t, aircraft j only has access to a subset of the neighbor set of aircraft i The communication edge is formed with it only within the specified domain. Considering the initial connected topology satisfies E(0)={(i,j):j∈N i (0)}.

[0013] S2. Based on the dynamic development law of the communication edge and combined with the artificial potential function method, the repulsion function and attraction function between the aircraft are constructed respectively. The repulsion function is used to prevent collisions between aircraft due to close distances, as shown below:

[0014]

[0015] Where d1 is used to define the set of ranges within which the repulsive force exists: Represents the distance between aircraft i and j. The attraction function is used to ensure the formation deployment between aircraft. Due to the dynamic addition of communication edges, when 0≤α ij When <d-∈, the attractive potential function is:

[0016]

[0017] in constant and To ensure that the communication distance meets a ij =d-∈, function W ij Smoothness; when d-∈≤α ij When ≤d, the attractive potential function is:

[0018]

[0019] in constant and Make function The switching at d-∈ is smooth and Μ is a given constant.

[0020] S3. According to the attraction function and repulsion function, under the switching concept, the dynamic characteristics of the switching ODE model of the aircraft under the dynamic communication edge are as follows:

[0021]

[0022] in, constant W ij , and V ij The weights on the horizontal and vertical coordinates are considered to be known, F i (t) is the actuator failure of aircraft i, which causes uncertainty to the system.

[0023] First, using the analytical nature of the aircraft state variables, the Taylor expansion technique is used to obtain:

[0024]

[0025] The relationship between the continuous spatial variable x and the discrete index of the aircraft is x i =iΔx, Δx represents the step size of the spatial variable.

[0026] Secondly, we use the Taylor expansion of the state quantity to approximate z j (t),z l (t),z k (t)Get

[0027] Finally, according to the Zeno-free switching behavior of the attractive potential function and a finite number of aircraft, a switching law based on the aircraft state is designed. Here, the state space is divided into a finite number of subintervals, each of which is controlled by a mode, that is, the threshold set of the mode r1 is the control gain, and the specific switching process is as follows:

[0028] Consider the switching law as

[0029]

[0030]

[0031] in is the norm α ij Representation under spatial variables.

[0032] Finally, the PDE model of the aircraft cluster under dynamic topology and actuator failure is:

[0033] z t (t,x)=z x (t,x)P σ(t) (t,x)+z xx (t,x)Q σ(t) (t,x)+F(t,x)

[0034] in, The initial state is z(t0,x)=z0(x), and the boundary condition is z t (t,0)=0,z t (t,1)=U(t), where U(t) is the boundary controller.

[0035] S4. Based on the constructed PDE model, use its equilibrium curve At its boundary conditions Next, according to any given boundary input Different target deployment curves can be generated.

[0036] S5. According to Lyapunov stability theory, construct a boundary controller, that is, the navigator controller in the aircraft cluster under the PDE model; design the boundary controller of the PDE model as shown below:

[0037]

[0038] The tracking error is (t,x)∈[t0,T]×[0,1], Superscript (·) * Representing the conjugate of this variable, the dynamic characteristics of the tracking error are expressed as:

[0039]

[0040] ζ t (t,0)=0,

[0041]

[0042] The control gain satisfies r1>0, and the parameter satisfies k1>0. According to the PDE modeling process, the boundary controller performs the role of the leader controller in the aircraft cluster.

[0043] S6. According to the PDE model, the central difference method is used to discretize it and obtain the follower controller:

[0044]

[0045] Beneficial effects of the present invention: The present invention takes into account the characteristics of aircraft clusters having a large number of dynamic monomers and inevitable dynamic topology in actual scenarios, and utilizes the macro control concept of the cluster to perform formation fault-tolerant control analysis on a large number of monomers, thereby obtaining a distributed fault-tolerant formation controller for each aircraft to ensure that the aircraft cluster achieves the target deployment under dynamic topology and actuator failures. The PDE-based formation control method requires only a small number of calculation parameters, and the follower controller does not require information on the desired deployment curve, and the desired deployment profile can be generated according to the equilibrium curve of the PDE model. The present invention effectively solves the problem of fault-tolerant formation control of aircraft clusters under dynamic topology and actuator failures. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 Schematic diagram of ODE and PDE conversion;

[0047] Figure 2 It is the implementation flow chart of the PDE fault-tolerant formation control method for aircraft clusters under dynamic topology;

[0048] Figure 3 The flowchart for constructing the PDE model of aircraft cluster under dynamic topology;

[0049] Figure 4 It is the initial deployment diagram of the aircraft cluster;

[0050] Figure 5 This is a deployment diagram of the aircraft cluster of the present invention at 4 seconds;

[0051] Figure 6 This is a deployment diagram of the aircraft cluster of the present invention at 8 seconds;

[0052] Figure 7 This is a deployment diagram of the aircraft cluster of the present invention at 15 seconds;

[0053] Figure 8 This is the deployment diagram of the aircraft cluster under the ODE model at 15s. DETAILED DESCRIPTION

[0054] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. The components of the embodiments of the present invention generally described and shown in the drawings here can be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents the selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative work belong to the scope of protection of the present invention.

[0055] Embodiment 1:

[0056] Figure 1 The mutual conversion relationship between the cluster ODE model and PDE model is given. Figure 2 As shown, an embodiment of the present invention provides a PDE fault-tolerant formation control method for an aircraft cluster under a dynamic topology, comprising the following steps:

[0057] Step S100: Consider a finite number of aircraft with horizontal and vertical coordinates (z *1 ,z *2 ) plane, where the state of aircraft i Indicates its horizontal position, is the vertical position, The motion model of each aircraft in the coordinate direction is:

[0058]

[0059] in is the control input, and the nth aircraft is marked as the leader aircraft. Define the complex variable is the position of aircraft i at time t, where the real component and imaginary component represent the position of aircraft i at time z *1 and z *2 The topology of the aircraft cluster is considered to be an undirected graph G(V,E(t)), where V is a set of vertices, each vertex represents an aircraft, and the follower aircraft are numbered from 1 to n-1, using (i,j). represents the communication edge between aircraft i and j, then the set of communication edges is There exists (i,j)∈E(t) if and only if (j,i)∈E(t). The connectivity of G(V,E(t)) means that there is a communication edge between any two aircraft at time t.

[0060] To illustrate the evolution of G(V,E(t)), consider two communication sets for each aircraft i at time t. One is the perception range set of aircraft i, which contains all aircraft that aircraft i can perceive, that is, the neighbor set The other is a subset of its neighbor set Based on this, the following communication edge dynamic development law is constructed:

[0061]

[0062] t - Indicates that at the moment before t, aircraft j only has access to a subset N of the neighbor set of aircraft i i *(t) only forms a communication edge with it within the domain specified by (t), and the initial connected topology satisfies E(0) = {(i,j):j∈N i (0)}.

[0063] Step S110: constructing attraction function and repulsion function between aircrafts according to the dynamic development law and in combination with the artificial potential function method;

[0064] Specifically, the rejection function is: Based on the development law of communication edges, its attraction function is a piecewise function:

[0065]

[0066] Among them, d1 is used to define the set of ranges within which the repulsive force exists: represents the distance between aircraft i and j; constant and Make function The switching at d-∈ is smooth and Μ is a given constant.

[0067] Step S120: According to the attraction function and the repulsion function, under the switching concept, the dynamic characteristics of the switching ODE model of the aircraft under the dynamic communication edge are:

[0068]

[0069] in, constant W ij , and V ij The weights on the horizontal and vertical coordinates, F i (t) is the actuator failure of vehicle i, which causes uncertainty in the system and affects the stability of the system and the achievement of its goals. Therefore, fault-tolerant control design is needed.

[0070] S130: From the perspective of cluster macro control, based on the switching ODE model of the aircraft cluster under dynamic topology and actuator failure, the PDE model is established. The process is as follows: Figure 3 As shown in the figure, a boundary-based leader controller is designed to solve the fault-tolerant formation control problem of aircraft clusters.

[0071] S131: First, using the analytical nature of the aircraft state variables, the Taylor expansion technique is used to obtain:

[0072]

[0073] The relationship between the continuous spatial variable x and the discrete index of the aircraft is x i=iΔx, Δx represents the step size of the spatial variable. Then, the Taylor expansion of the state quantity is used to approximate z j (t),z l (t),z k (t) obtains:

[0074]

[0075] in is the norm α ij Representation under spatial variables.

[0076] S132: Finally, according to the Zeno-free switching behavior of the attractive potential function and a finite number of aircraft, a switching law based on the aircraft state is designed. In this case, the state space is divided into a finite number of subintervals, each of which is controlled by a mode, i.e., the threshold set of the mode The specific switching process is as follows:

[0077] Consider the switching law as

[0078] in,

[0079] (1) If Choose the initial state σ(t0) = q 0 .

[0080] (2) For t>t0, if σ(t - )=q and z(t0,x)∈Ω q , let σ(t)=q. If σ(t - ) = q and The next mode is selected using the above switching law.

[0081] S133: Further sorting yields:

[0082]

[0083] in:

[0084]

[0085]

[0086]

[0087]

[0088] S134: Therefore, the PDE model of the aircraft cluster under dynamic topology and actuator failure is:

[0089] z t(t,x)=z x (t,x)P σ(t) (t,x)+z xx (t,x)Q σ(t) (t,x)+F(t,x)

[0090] in, The initial state is z(t0,x)=z0(x) and the boundary condition is z t (t,0)=0,z t (t,1)=U(t), where U(t) is the boundary controller.

[0091] Step S140: Based on the constructed PDE model, use its equilibrium curve At its boundary conditions Next, according to any given boundary input Different target deployment curves can be generated.

[0092] Step S150: According to the PDE model, Lyapunov control technology is applied to design a navigator controller based on boundary control:

[0093] Specifically, the navigator controller is:

[0094]

[0095] The tracking error is (t,x)∈[t0,T]×[0,1], Superscript (·) * Representing the conjugate of this variable, the dynamic characteristics of the tracking error are expressed as:

[0096]

[0097] ζ t (t,0)=0,

[0098]

[0099] The control gain satisfies r1>0, and the parameter satisfies k1>0; the PDE boundary controller performs the role of the leader controller in the aircraft cluster.

[0100] Step S160: According to the PDE model under boundary control, a follower controller is obtained by discretizing it using a difference method.

[0101] Specifically, the follower controller is:

[0102] Embodiment 2:

[0103] An embodiment of the present invention provides a PDE fault-tolerant formation control technology for an aircraft cluster under a dynamic topology, including: establishing a dynamic development law of topological communication edges to determine the dynamic addition conditions of topological communication edges; constructing attraction functions and repulsion functions between aircraft to constrain the distance between aircraft; analyzing the ODE dynamic characteristics under the aircraft potential function under dynamic communication edges to adopt the continuous approximation idea and introduce the switching law to construct a PDE model; constructing a target deployment to obtain a deployment curve of a desired aircraft cluster system; designing a navigator controller based on boundary control to control the mobile deployment of the navigator aircraft; and designing a follower controller to control the mobile deployment of the follower aircraft.

[0104] In this embodiment, the target deployment curve of the aircraft cluster system consisting of 7 aircraft is:

[0105]

[0106] The sensing radius of each aircraft is d=6. When the distance between aircraft is less than 4, a communication edge (∈=2) is established. When the distance between aircraft is less than d1=0.3, a repulsive force is generated. The actuator failure is F(t)=0.01sin(-0.17t).

[0107] Figure 4 The initial deployment of the aircraft is given. Figure 5 As shown, in the present invention, at t=4s, aircraft 2, 3 and aircraft 1 each establish a communication edge. Figure 6 As shown, at t=8s, the present invention adds communication edges between aircraft 2 and 4, and aircraft 3 and 5, while the established communication edges remain unchanged. Figure 7 It shows that after 15 seconds from the initial arc-shaped deployment, the cluster is deployed into a heptagon. Figure 8 The curves of actual deployment and expected deployment of aircraft under the ODE model are shown. It can be seen that the fault-tolerant formation control results of the aircraft cluster based on the PDE model are consistent with the control results based on the ODE model. The designed PDE fault-tolerant formation control method realizes the target deployment of the flight cluster under dynamic topology and actuator failure.

Claims

1. A PDE fault-tolerant formation control method for aircraft clusters under dynamic topology, characterized in that: The steps include: S1. Construct a motion model of each aircraft in the coordinate direction, and use the model to establish the dynamic development law of the communication edge; The construction of the motion model is as follows: The aircraft is in horizontal and vertical coordinates (z *1 ,z *2 ) plane, where the state of aircraft i Indicates the horizontal position. is the vertical position, Then the motion model of each aircraft in the coordinate direction is: in is the control input, and the nth aircraft is marked as the leader aircraft; The specific process of establishing the dynamic development law of the communication edge is as follows: Defining complex variables is the position of aircraft i at time t, where the real component and imaginary component represent the position of aircraft i at time z *1 and z *2 The topology of the aircraft cluster is constructed as an undirected graph G(V,E(t)), where V is a set of vertices, each vertex represents an aircraft, and the follower aircraft are numbered from 1 to n-1, using (i,j). represents the communication edge between aircraft i and j, then the set of communication edges is t<∞, there exists (i,j)∈E(t) if and only if (j,i)∈E(t); the connectivity of G(V,E(t)) means that there is a communication edge between any two aircraft at time t; Consider two communication sets for each aircraft i at time t. One is the perception range set of aircraft i, which includes all aircraft perceived by aircraft i, that is, the neighbor set The other is a subset of its neighbor set Based on this, the following communication edge dynamic development law is constructed: t - Indicates that at the moment before t, aircraft j only has access to a subset N of the neighbor set of aircraft i i * (t) only forms a communication edge with it within the domain specified by (t), and the initial connected topology satisfies E(0) = {(i,j):j∈N i (0)}; S2. According to the dynamic development law of the communication edge and combined with the artificial potential function method, the repulsion function and attraction function between the aircraft are constructed respectively; S3. According to the attraction function and repulsion function, under the switching idea, the dynamic characteristics of the switching ODE model of the aircraft under the dynamic communication edge are obtained, and then its PDE model is established; S4. Based on the PDE model, use its equilibrium curve to generate different target deployment curves under any given boundary input; S5. According to Lyapunov stability theory, a boundary controller is constructed, i.e., the navigator controller in the aircraft cluster under the PDE model; S6. According to the PDE model, the central difference algorithm is used to discretize the PDE model and obtain the follower controller in the aircraft cluster under the PDE model.

2. The PDE fault-tolerant formation control method for a cluster of aircraft under a dynamic topology according to claim 1, characterized in that: In S2, the rejection function is as follows: Where d1 defines the set of repulsive force existence range represents the distance between aircraft i and j; When 0≤α ij When <d-∈, the attraction function is: in, constant and To ensure that the communication distance meets a ij =d-, function W ij Smoothness; when d-∈≤α ij When ≤d, the attraction function is: in constant and Make function The switching at d-∈ is smooth and Μ is a given constant.

3. The PDE fault-tolerant formation control method for a cluster of aircraft under a dynamic topology according to claim 2, characterized in that: The specific process of step S3 is as follows: S3.

1. Based on the attraction function and repulsion function, under the switching idea, the dynamic characteristics of the switching ODE model of the aircraft under the dynamic communication edge are obtained as follows: in, constant W ij , and V ij The weights on the horizontal and vertical coordinates, F i (t) The actuator failure of aircraft i brings uncertainty to the system; S3.

2. Using the analytic property of the aircraft state variables, Taylor expansion is used to obtain: The relationship between the continuous spatial variable x and the discrete index of the aircraft is x i =iΔx, Δx represents the step size of the spatial variable; Then, the Taylor expansion of the state quantity is used to approximate z j (t),z l (t),z k (t)Get Finally, according to the Zeno-free switching behavior of the attractive potential function and a finite number of aircraft, a switching law based on the aircraft state is set, where the state space is divided into a finite number of subintervals, each of which is controlled by a mode, that is, the threshold set of the mode: Set the switching law to in, r1 is the control gain; The PDE model of aircraft clusters under dynamic topology and actuator failure is: z t (t,x)=z x (t,x)P σ(t) (t,x)+z xx (t,x)Q σ(t) (t,x)+F(t,x) in, The initial state is z(t0,x)=z0(x) and the boundary condition is z t (t,0)=0,z t (t,1)=U(t), where U(t) is the leader controller.

4. The PDE fault-tolerant formation control method for a cluster of aircraft under a dynamic topology according to claim 3, characterized in that: In S4, the equilibrium curve is: With boundary conditions in is any given boundary input.

5. The PDE fault-tolerant formation control method for a cluster of aircraft under a dynamic topology according to claim 4, characterized in that: In S5, the specific process of constructing the navigator controller is as follows: According to Lyapunov stability theory, a boundary controller of the PDE model is designed as shown below: The tracking error is (t,x)∈[t0,T]×[0,1], Superscript (·) * Representing the conjugate of this variable, the dynamic characteristics of the tracking error are expressed as: The control gain satisfies r1>0, and the parameter satisfies k1>0; the boundary controller performs the role of the navigator controller in the aircraft cluster, that is, the navigator controller in the aircraft cluster of the PDE model.

6. The PDE fault-tolerant formation control method for a cluster of aircraft under a dynamic topology according to claim 5, characterized in that: The follower controller in S6 is as follows:

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