Pseudo-spectral differential dynamic planning method and related device

Abstract

The invention discloses a pseudo-spectral differential dynamic planning method and a related device, and relates to the technical field of nonlinear optimal control solving.The method comprises the steps that firstly, a nonlinear optimal guidance problem is constructed, and the optimal value function of the nonlinear optimal guidance problem meets an HJB equation; the method comprises the following steps: performing second-order expansion on an HJB equation based on a specified reference trajectory to obtain a differential equation about a partial derivative of an optimal value function, then determining and solving a first two-point boundary value problem to determine an optimal control quantity variation, and determining a second two-point boundary value problem based on the optimal control quantity variation to update the trajectory; and finally, based on a pseudo-spectral method, Lagrange interpolation is adopted to process the trajectory and the constructed matrix, and a second two-point boundary value problem is converted and iteratively solved. According to the scheme, the method is based on second-order expansion of the Hamiltonian-Jacobian-Bellman equation, the pseudo-spectral method is embedded into the differential dynamic programming method framework, and compared with the prior art, the method has better calculation efficiency and iteration convergence.

Description

Technical Field

[0001] The present application relates to the technical field of nonlinear optimal control solution, and in particular to a pseudo-spectral differential dynamic programming method and related devices. Background Art

[0002] Differential dynamic programming is a commonly used method for solving nonlinear optimal control problems. Based on the Bellman optimality principle and the second-order expansion of the optimal value function, the nonlinear optimal control problem is transformed into a sequence of differential equation sub-problems, and these differential equations are iteratively solved to obtain the optimal trajectory and the local optimal feedback guidance law. In the existing differential dynamic programming methods, the methods for solving differential equations mainly include the sweep method and the multiple target method, which have limitations in terms of computational efficiency and numerical sensitivity. Summary of the invention

[0003] The purpose of the present application is to provide a pseudo-spectral differential dynamic programming method and related devices, which can improve the computational efficiency and iterative convergence of the differential dynamic programming method.

[0004] To achieve the above objectives, this application provides the following solutions:

[0005] In a first aspect, the present application provides a pseudo-spectral differential dynamic programming method, comprising the following steps:

[0006] Based on the initial constraints, terminal constraints and dynamic model, a nonlinear optimal guidance problem is constructed; the optimal value function of the nonlinear optimal guidance problem satisfies the Hamilton-Jacobi-Bellman equation.

[0007] Based on the specified reference trajectory, the Hamilton-Jacobi-Bellman equation is expanded to the second order to obtain the differential equation of the partial derivative of the optimal value function.

[0008] According to the differential equation about the partial derivative of the optimal value function, the derivative of the construction matrix is ​​obtained, and the first two-point boundary value problem is determined; the first two-point boundary value problem is the homogeneous form of the trajectory two-point boundary value problem.

[0009] Solve the first two-point boundary value problem, determine the optimal control quantity variation, and determine the second two-point boundary value problem based on the optimal control quantity variation. Solve the second two-point boundary value problem to update the trajectory.

[0010] Based on the pseudo-spectral method, Lagrange interpolation is used to process the trajectory and construction matrix, the second two-point boundary value problem is transformed and solved iteratively to obtain the discrete optimal trajectory and discrete construction matrix.

[0011] In a second aspect, the present application provides a pseudo-spectral differential dynamic programming device, comprising the following modules:

[0012] The nonlinear optimal guidance problem construction module is used to construct a nonlinear optimal guidance problem based on initial constraints, terminal constraints and dynamic models; the optimal value function of the nonlinear optimal guidance problem satisfies the Hamilton-Jacobi-Bellman equation.

[0013] The module for determining the partial derivative differential equation of the value function is used to perform a second-order expansion of the Hamilton-Jacobi-Bellman equation based on a specified reference trajectory to obtain a differential equation of the partial derivative of the optimal value function.

[0014] The first two-point boundary value problem determination module is used to obtain the derivative of the construction matrix according to the differential equation of the partial derivative of the optimal value function, and determine the first two-point boundary value problem; the first two-point boundary value problem is a homogeneous form of the trajectory two-point boundary value problem.

[0015] The second two-point boundary value problem determination module is used to solve the first two-point boundary value problem, determine the optimal control quantity variation, and determine the second two-point boundary value problem based on the optimal control quantity variation; solving the second two-point boundary value problem realizes the update of the trajectory.

[0016] The pseudo-spectral differential dynamic programming module is used to process the trajectory and construction matrix based on the pseudo-spectral method using Lagrange interpolation, transform the second two-point boundary value problem and solve it iteratively to obtain the discrete optimal trajectory and discrete construction matrix.

[0017] In a third aspect, the present application provides a computer device, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the pseudo-spectral differentiation dynamic programming method described above.

[0018] In a fourth aspect, the present application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the pseudo-spectral differentiation dynamic programming method described above.

[0019] In a fifth aspect, the present application provides a computer program product, comprising a computer program, which, when executed by a processor, implements the steps of the pseudo-spectral differentiation dynamic programming method described above.

[0020] According to the specific embodiments provided in this application, this application discloses the following technical effects:

[0021] The present application provides a pseudo-spectral differential dynamic programming method and related devices, in which a nonlinear optimal guidance problem is first constructed based on initial constraints, terminal constraints and a dynamic model, and its optimal value function satisfies the Hamilton-Jacobi-Bellman equation; based on a specified reference trajectory, the Hamilton-Jacobi-Bellman equation is expanded to a second order to obtain a differential equation for the partial derivative of the optimal value function, and then the first two-point boundary value problem is determined and solved, the optimal control quantity variation is determined, and the second two-point boundary value problem is determined based on the optimal control quantity variation, and the second two-point boundary value problem is solved to update the trajectory; finally, based on the pseudo-spectral method, the trajectory and the construction matrix are processed by Lagrange interpolation, the second two-point boundary value problem is transformed and iteratively solved, and a discrete optimal trajectory and a discrete construction matrix are obtained. The above-mentioned scheme of the present application is based on the second-order expansion of the Hamilton-Jacobi-Bellman equation. Different from the existing differential dynamic programming method, it constructs two two-point boundary value problems with the same homogeneous solutions, which are used to calculate the control quantity variation and perform trajectory update respectively, and embeds the pseudo-spectral method into the framework of the differential dynamic programming method, which has better computational efficiency and iterative convergence. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0023] Figure 1 A flowchart of a pseudo-spectral differentiation dynamic programming method provided in one embodiment of the present application.

[0024] Figure 2 A schematic diagram of an x-axis position curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0025] Figure 3 A schematic diagram of a y-axis position curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0026] Figure 4 A schematic diagram of a z-axis position curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0027] Figure 5 A schematic diagram of an x-axis velocity curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0028] Figure 6A schematic diagram of a y-axis velocity curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0029] Figure 7 A schematic diagram of a z-axis velocity curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0030] Figure 8 A schematic diagram of a pitch angle curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0031] Fig. 9 A schematic diagram of a yaw angle curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0032] Fig.10 A schematic diagram of a pitch angle rate curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0033] Fig.11 A schematic diagram of a yaw rate curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0034] Fig.12 A schematic diagram of a throttling rate curve during a digital simulation iteration of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0035] Fig.13 A schematic diagram of a quality curve in a digital simulation iteration process of a pseudo-spectral differential dynamic programming method provided in one embodiment of the present application.

[0036] Fig.14 A schematic diagram of functional modules of a pseudo-spectral differentiation dynamic programming device provided in one embodiment of the present application.

[0037] Fig.15 A schematic diagram of the structure of a computer device provided in one embodiment of the present application. DETAILED DESCRIPTION

[0038] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0039] In order to make the above-mentioned objects, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0040] The embodiment of the present application provides a pseudo-spectral differential dynamic programming method, such as Figure 1 As shown, the following steps are included:

[0041] A1. Based on the initial constraints, terminal constraints and dynamic model, a nonlinear optimal guidance problem is constructed; the optimal value function of the nonlinear optimal guidance problem satisfies the Hamilton-Jacobi-Bellman equation (HJB equation). In this embodiment, the nonlinear optimal guidance problem is shown as follows:

[0042]

[0043] Among them, J is the overall performance index; φ[x(1),a] is the terminal performance index; is the integral performance index; a is the total flight time; τ is the normalized time between 0 and 1; x(τ) is the state quantity corresponding to the normalized time τ; u(τ) is the control quantity corresponding to the normalized time τ; x(1) is the terminal state quantity; c f is the terminal constraint value; f[τ,x(τ),u(τ),a] is the dynamic model; x(0) is the state bar corresponding to the initial moment; x0 is the given initial moment constraint value; C is the terminal constraint matrix.

[0044] The optimal value function of the nonlinear optimal guidance problem is as follows:

[0045]

[0046] Where V[τ,x(τ),a,v] is the optimal value function; Φ[x(1),a,v] is the terminal augmented performance index related to the terminal performance index and terminal constraints, Φ[x(1),a,v]=φ[x(1),a]+v T [Cx(1)-c f ]; v is the multiplier vector corresponding to the terminal constraint; s is the normalized time between τ-1. According to the Bellman optimality principle, V[τ,x(τ),a,v] satisfies the Hamilton-Jacobi-Bellman equation.

[0047] A2. Based on the specified reference trajectory, the Hamilton-Jacobi-Bellman equation is expanded to a second order to obtain a differential equation for the partial derivative of the optimal value function. In this embodiment, the partial derivative of the optimal value function includes the partial derivative of the optimal value function with respect to the normalized time τ, the partial derivative of the optimal value function with respect to the state quantity x, and the partial derivative of the optimal value function with respect to the total flight time a.

[0048]

[0049] At τ = 1, V[1,x(1),a,v] = Φ[x(1),a,v] satisfies the following equation:

[0050]

[0051] Among them, V τ [τ,x(τ),a,v] is the partial derivative of the optimal value function with respect to the normalized time τ; H[τ,x(τ),u(τ),a,v] is the Hamiltonian function; V x [1,x(1),a,v] is the partial derivative of the optimal value function with respect to the state x; the superscript T is the transposition symbol; V a [1,x(1),a,v] is the partial derivative of the optimal value function with respect to the total flight time a. The Hamiltonian function H[τ,x(τ),u(τ),a,v] can be expressed as follows:

[0052] H[τ,x(τ),u(τ),a,v]=L[τ,x(τ),u(τ),a]+V x [τ,x(τ),a,ν]f[τ,x(τ),u(τ),a].

[0053] At τ = 0, the initial state is given, so the total flight time a should be chosen to minimize V[0,x(0),a], resulting in the following equation:

[0054]

[0055] Given a fixed x r (τ),u r (τ) and a r , select the reference trajectory The Hamiltonian function H can be expanded to the second order as:

[0056]

[0057] in, The horizontal line on the symbol represents the value of that variable along the reference trajectory; The subscripts in represent the partial derivatives with respect to the corresponding variables.

[0058] V τ The left-hand side of (τ) is expanded to the second order as shown below:

[0059]

[0060] in, The subscripts in represent the partial derivatives with respect to the corresponding variables.

[0061] Substituting the above formula into the HJB equation, we get the following differential equation:

[0062]

[0063]

[0064] Through the above differential equation, the variational form of the optimal control quantity is preliminarily determined as follows:

[0065]

[0066] The parameters with wavy lines are defined as follows:

[0067]

[0068]

[0069] A3. According to the differential equation of the partial derivative of the optimal value function, the derivative of the construction matrix is ​​obtained, and the first two-point boundary value problem is determined; the first two-point boundary value problem is the homogeneous form of the trajectory two-point boundary value problem.

[0070] The feedback control law of the above nonlinear optimal guidance problem can be expressed as follows:

[0071]

[0072] In order to obtain the optimal feedback control law, it is necessary to solve the Riccati equation. First, according to the terminal conditions, define the following matrix:

[0073]

[0074] The matrices X(τ), A, Λ(τ), M(τ), and P are all construction matrices that satisfy the following equation at the terminal moment:

[0075]

[0076] The following equations are obtained by differentiating Λ(τ) and M(τ):

[0077]

[0078] According to the differential equation obtained in step A2, the derivatives of matrices X(τ), Λ(τ) and M(τ) can be obtained, which are expressed as:

[0079]

[0080] Select the current time τ c The boundary conditions of are as follows:

[0081] X(τ c )=I,M(τc )=0.

[0082] In summary, the first two-point boundary value problem is determined. In this embodiment, the first two-point boundary value problem is shown as follows:

[0083]

[0084] Among them, Λ(τ), X(τ), M(τ), P and A are construction matrices; the points on the symbol are derivatives; I is the identity matrix; τ c is the current normalized time; is φ x The value of the partial derivative of the state quantity along the reference trajectory; is φ x The value of the partial derivative of the total flight time a along the reference trajectory; φ x is the partial derivative of the terminal performance index with respect to the state quantity x; is φ a The value of the partial derivative of the state quantity along the reference trajectory; is φ a The value of the partial derivative of the total flight time a along the reference trajectory; φ a is the partial derivative of the terminal performance index with respect to the total flight time a; Λ(1), X(1) and M(1) are the values ​​of Λ(τ), X(τ) and M(τ) at the terminal moment. It can be seen that the above two-point boundary value problem is a homogeneous form of the trajectory two-point boundary value problem.

[0085] A4. Solve the first two-point boundary value problem, determine the optimal control quantity variation, and determine the second two-point boundary value problem based on the optimal control quantity variation. Solve the second two-point boundary value problem to update the trajectory.

[0086] Solving the first two-point boundary value problem, we can obtain the construction matrix X(τ), A, Λ(τ), M(τ) and P, where X(τ) means the state trajectory x(τ) for the current state x c The sensitivity of A is the total flight time a to the current state x. c The sensitivity of Λ(τ) is the trajectory V x (τ) for the current state x c The sensitivity of M(τ) is the trajectory V a (τ) for the current state x c The sensitivity of P is the multiplier v to the current state x. c In this embodiment, the optimal control amount is varied as follows:

[0087]

[0088] Among them, δu(τ) is the optimal control quantity variation; is the value of the partial derivative of the Hamiltonian function to the control quantity along the reference trajectory; H u (τ) is the value of the partial derivative of the control quantity along the reference trajectory; the superscript -1 is the inverse of the matrix; is the value of the partial derivative of the control quantity of the dynamic model along the reference trajectory; V x (τ) The value of the partial derivative of the state quantity along the reference trajectory; V x (τ) is the partial derivative of the optimal value function with respect to the state quantity; Λ(τ), X(τ) and A are construction matrices; is the state quantity of the reference trajectory; V x (τ) The value of the partial derivative of the total flight time a along the reference trajectory.

[0089] Next, V x (τ) and V a (τ) along the reference Expanding to first order, we get:

[0090]

[0091] Taking the derivative of the above formula, we get:

[0092]

[0093] Where: It can be expressed as:

[0094]

[0095] Substituting the obtained optimal control quantity variation, we can get:

[0096]

[0097] The optimal control quantity is varied Substitution and In the formula, we get:

[0098]

[0099]

[0100] Reference Trajectory given, V x The perturbation value of (τ) can be expressed as:

[0101]

[0102] Place the terminal condition along the reference and Expanding to first order, we get:

[0103]

[0104] According to the above formula, the second two-point boundary value problem can be determined as shown below:

[0105]

[0106] Among them, x c is the current state quantity; V x (1) V x The value of [1,x(1),a,v] at the terminal time; V a (1) V a The value of [1,x(1),a,v] at the terminal moment; G1(τ), G2(τ), G3(τ), G4(τ), G5(τ), G6(τ), g1(τ), g2(τ), g3(τ), p1(τ) and p2(τ) are all intermediate variables, which are defined by the following formulas:

[0107]

[0108]

[0109] Solving the second two-point boundary value problem obtained above can realize the trajectory x(τ), V a (τ), updates of a and v.

[0110] A5. Based on the pseudo-spectral method, Lagrange interpolation is used to process the trajectory and construction matrix, the second two-point boundary value problem is transformed and solved iteratively to obtain the discrete optimal trajectory and discrete construction matrix.

[0111] Introducing normalized time in the interval [-1,1] It satisfies the following formula:

[0112]

[0113] Use Lagrange interpolation to convert x(τ), V a (τ), X(τ), Λ(τ) and M(τ) are approximated as follows:

[0114]

[0115] in, is the Lagrange interpolation basis function. The Lagrange interpolation node contains the LGR integration point and boundary nodes Then the two-point boundary value problem can be transformed into the following formula:

[0116]

[0117]

[0118] Among them, D ki is the element in the kth row and ith column of the differential matrix D.

[0119] The two-point boundary value problem can be transformed into the following form:

[0120]

[0121] The nonhomogeneous and homogeneous solutions to the two-point boundary value problem are defined by:

[0122]

[0123] Then the above equations can be expressed as follows:

[0124] A s Y s0 =B s .

[0125] A s Y s =B s x c +B s .

[0126] Solving the above equation yields the following:

[0127]

[0128] Solve for Y by iteration s , and obtain the discrete optimal trajectory a and v, and solve for Y in the last iteration s0 , and obtain the discrete matrix A and P. Continuous trajectory x(τ), V a (τ), a and v, as well as the matrices X(τ), Λ(τ), M(τ), A and P can be expressed in analytical form through Lagrange interpolation.

[0129] The existing differential dynamic programming method focuses on iterative solution of the problem, lacks quantification and analysis of iterative convergence, and cannot guarantee reliable convergence. However, due to the scheme proposed in the above embodiment of the present application, the numerical calculation of iterative solution is carried out under the pseudo-spectral framework, so an analytical convergence criterion is given as follows:

[0130] Assume that the solution obtained in the i-th iteration is as follows:

[0131]

[0132] Then the i+1th guidance calculation process can be expressed as the following simple iterative form:

[0133]

[0134] in, is the linear equation to be solved in the i+1th iteration, A S , B S and b S is the coefficient matrix that constitutes the above linear equation.

[0135] In order to describe the convergence of the iterative solution process, the iterative convergence index is defined as the matrix The maximum eigenvalue of is determined by the following formula to determine the iterative convergence index:

[0136]

[0137] Among them, ρ c is the iterative convergence index; max is the maximum eigenvalue; is the solution obtained at the i-th iteration.

[0138] According to the convergence of simple iteration, when ρ c <1, the iterative solution process is convergent. Therefore, ρ c Describes the convergence of the iterative solution process, that is, ρ c The smaller it is, the stronger the convergence.

[0139] Since the iterative convergence is mainly related to the initial guess and the dynamic nonlinearity, when ρ c >1, ρ can be reduced by modifying the kinetic model c :In the early stage of iteration, when the initial guess deviates greatly from the optimal trajectory, the key parameters in the dynamic model are corrected to reduce the nonlinearity, thereby reducing ρ c ; When the reference trajectory is close to the optimal trajectory in the late iteration, while ensuring ρ c <1, the dynamic model is gradually restored to solve the original problem.

[0140] Next, a digital simulation case will be used as a method to demonstrate the effectiveness of the solution proposed in this application, not an actual flight mission. The simulation condition selects a certain type of reusable rocket-powered descent flight mission. Using the pseudo-spectral differential dynamic programming method of this application, after digital simulation, the iterative optimization process of the x-axis position curve is as follows Figure 2 As shown in the figure, the iterative optimization process of the y-axis position curve is as follows Figure 3 As shown in Figure 2, the iterative optimization process of the z-axis position curve is as follows: Figure 4 As shown in the figure, the iterative optimization process of the x-axis speed curve is as follows Figure 5 As shown in Figure 2, the iterative optimization process of the y-axis speed curve is as follows: Figure 6 As shown in Figure 2, the iterative optimization process of the z-axis velocity curve is as follows: Figure 7 As shown in Figure 2, the iterative optimization process of the pitch angle curve is as follows: Figure 8 As shown in Figure 2, the iterative optimization process of the yaw angle curve is as follows: Fig. 9 As shown in the figure, the iterative optimization process of the pitch rate curve is as follows: Fig.10 As shown in the figure, the iterative optimization process of the yaw rate curve is as follows: Fig.11 As shown in Figure 2, the iterative optimization process of the throttling rate curve is as follows: Fig.12 As shown in Figure 2, the iterative optimization process of the quality curve is as follows: Fig.13 shown.

[0141] The method provided in the above embodiment of the present application is based on the second-order expansion of the Hamilton-Jacobi-Bellman equation. Different from the existing differential dynamic programming method, it constructs two two-point boundary value problems with the same homogeneous solution, which are used to calculate the control quantity variation and perform trajectory update respectively, and embeds the pseudo-spectral method into the framework of the differential dynamic programming method, which has better computational efficiency and iterative convergence. In addition, based on the fixed point iterative convergence theorem, a quantized pseudo-spectral differential dynamic programming method convergence criterion is proposed, and a model correction method is designed based on the quantized convergence criterion to improve the reliability of algorithm convergence.

[0142] Based on the same inventive concept, the embodiment of the present application also provides a device for implementing the pseudo-spectral differential dynamic programming method involved above. The implementation scheme for solving the problem provided by the device is similar to the implementation scheme recorded in the above method, so the specific limitations in the embodiments of one or more pseudo-spectral differential dynamic programming devices provided below can refer to the limitations of the pseudo-spectral differential dynamic programming method above, and will not be repeated here.

[0143] In an exemplary embodiment, Fig.14 As shown, a pseudo-spectral differential dynamic programming device is provided, comprising the following modules:

[0144] The nonlinear optimal guidance problem construction module is used to construct a nonlinear optimal guidance problem based on initial constraints, terminal constraints and dynamic models; the optimal value function of the nonlinear optimal guidance problem satisfies the Hamilton-Jacobi-Bellman equation.

[0145] The module for determining the partial derivative differential equation of the value function is used to perform a second-order expansion of the Hamilton-Jacobi-Bellman equation based on a specified reference trajectory to obtain a differential equation of the partial derivative of the optimal value function.

[0146] The first two-point boundary value problem determination module is used to obtain the derivative of the construction matrix according to the differential equation of the partial derivative of the optimal value function, and determine the first two-point boundary value problem; the first two-point boundary value problem is a homogeneous form of the trajectory two-point boundary value problem.

[0147] The second two-point boundary value problem determination module is used to solve the first two-point boundary value problem, determine the optimal control quantity variation, and determine the second two-point boundary value problem based on the optimal control quantity variation; solving the second two-point boundary value problem realizes the update of the trajectory.

[0148] The pseudo-spectral differential dynamic programming module is used to process the trajectory and construction matrix based on the pseudo-spectral method using Lagrange interpolation, transform the second two-point boundary value problem and solve it iteratively to obtain the discrete optimal trajectory and discrete construction matrix.

[0149] certainly, Fig.14 The architecture shown is only exemplary and can be omitted according to actual needs when implementing different functions. Fig.14 One or at least two components of the system shown.

[0150] In an exemplary embodiment, a computer device is provided. The computer device may be a server or a terminal. The internal structure diagram thereof may be as follows: Fig.15 As shown. The computer device includes a processor, a memory, an input / output interface (Input / Output, referred to as I / O) and a communication interface. Wherein, the processor, the memory and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. Wherein, the processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The input / output interface of the computer device is used to exchange information between the processor and an external device. The communication interface of the computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor, a pseudo-spectral differential dynamic programming method provided in the above embodiment can be implemented.

[0151] Those skilled in the art will understand that Fig.15 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine certain components, or have a different arrangement of components.

[0152] In an exemplary embodiment, a computer device is further provided, including a memory and a processor, wherein a computer program is stored in the memory, and the processor implements the steps in the above-mentioned method embodiments when executing the computer program.

[0153] In an exemplary embodiment, a computer-readable storage medium is provided, storing a computer program, and when the computer program is executed by a processor, the steps in the above method embodiments are implemented.

[0154] In an exemplary embodiment, a computer program product is provided, including a computer program, and when the computer program is executed by a processor, the steps in the above method embodiments are implemented.

[0155] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use and processing of relevant data must comply with relevant regulations.

[0156] Those of ordinary skill in the art can understand that all or part of the processes in the above-mentioned embodiment methods can be completed by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to the memory, database or other medium used in the embodiments provided in the present application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetoresistive random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM may be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM).

[0157] The database involved in each embodiment provided in this application may include at least one of a relational database and a non-relational database. The non-relational database may include a distributed database based on blockchain, etc., but is not limited thereto. The processor involved in each embodiment provided in this application may be a general-purpose processor, a central processing unit, a graphics processor, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., but is not limited thereto.

[0158] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0159] This article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. A pseudo-spectral differential dynamic programming method, characterized in that: include: Based on the initial constraints, the terminal constraints and the dynamic model, a nonlinear optimal guidance problem is constructed; the optimal value function of the nonlinear optimal guidance problem satisfies the Hamilton-Jacobi-Bellman equation; Based on the specified reference trajectory, the Hamilton-Jacobi-Bellman equation is expanded to the second order to obtain the differential equation of the partial derivative of the optimal value function; According to the differential equation of the partial derivative of the optimal value function, the derivative of the construction matrix is ​​obtained, and the first two-point boundary value problem is determined; the first two-point boundary value problem is a homogeneous form of the trajectory two-point boundary value problem; Solving the first two-point boundary value problem, determining the optimal control amount variation, and determining the second two-point boundary value problem based on the optimal control amount variation, and solving the second two-point boundary value problem to update the trajectory; Based on the pseudo-spectral method, Lagrange interpolation is used to process the trajectory and the construction matrix, the second two-point boundary value problem is transformed and solved iteratively to obtain a discrete optimal trajectory and a discrete construction matrix.

2. The pseudo-spectral differentiation dynamic programming method according to claim 1, characterized in that: The nonlinear optimal guidance problem is expressed as follows: Among them, J is the overall performance index; φ[x1),a] is the terminal performance index; is the integral performance index; a is the total flight time; τ is the normalized time between 0 and 1; x(τ) is the state quantity corresponding to the normalized time τ; u(τ) is the control quantity corresponding to the normalized time τ; x(1) is the terminal state quantity; c f is the given terminal constraint value; f[τ,x(τ),u(τ),a] is the dynamic model; x(0) is the state bar corresponding to the initial moment; x0 is the given initial moment constraint value; C is the terminal constraint matrix; the optimal value function of the nonlinear optimal guidance problem is as follows: Where V[τ,x(τ),a,v] is the optimal value function; φ[x(1),a,v]=φ[x(1),a]+v T [Cx(1)-c f ]; ν is the multiplier vector corresponding to the terminal constraint; s is the normalized time between τ-1.

3. The pseudo-spectral differentiation dynamic programming method according to claim 2, characterized in that: The partial derivatives of the optimal value function include the partial derivative of the optimal value function with respect to the normalized time τ, the partial derivative of the optimal value function with respect to the state quantity x, and the partial derivative of the optimal value function with respect to the total flight time a: Among them, V τ [τ,x(τ),a,v] is the partial derivative of the optimal value function with respect to the normalized time τ; H[τ,x(τ),u(τ),a,v] is the Hamiltonian function; V x [1,x(1),a,v] is the partial derivative of the optimal value function with respect to the state x; the superscript T is the transposition symbol; V a [1,x(1),a,v] is the partial derivative of the optimal value function with respect to the total flight time a.

4. The pseudo-spectral differentiation dynamic programming method according to claim 3, characterized in that: The optimal control quantity variation is shown as follows: Among them, δu(τ) is the optimal control variable; is the value of the partial derivative of the Hamiltonian function to the control quantity along the reference trajectory; H u (τ) is the value of the partial derivative of the control quantity along the reference trajectory; the superscript -1 is the inverse of the matrix; is the value of the partial derivative of the control quantity of the dynamic model along the reference trajectory; V x (τ) The value of the partial derivative of the state quantity along the reference trajectory; V x (τ) is the partial derivative of the optimal value function with respect to the state quantity; Λ(τ), X(τ) and A are construction matrices; is the state quantity of the reference trajectory; V x (τ) The value of the partial derivative of the total flight time a along the reference trajectory.

5. The pseudo-spectral differentiation dynamic programming method according to claim 4, characterized in that: The first two-point boundary value problem is expressed as follows: Among them, Λ(τ), X(τ), M(τ), P and A are construction matrices; the points on the symbol are derivatives; I is the identity matrix; τ c is the current normalized time; is φ x The value of the partial derivative of the state quantity along the reference trajectory; is φ x The value of the partial derivative of the total flight time a along the reference trajectory; φ x is the partial derivative of the terminal performance index with respect to the state quantity x; is φ a The value of the partial derivative of the state quantity along the reference trajectory; is φ a The value of the partial derivative of the total flight time a along the reference trajectory; φ a is the partial derivative of the terminal performance index with respect to the total flight time a; Λ(1), X(1) and M(1) are the values ​​of Λ(τ), X(τ) and M(τ) at the terminal time; The second two-point boundary value problem is expressed as follows: Among them, x c is the current state quantity; V x (1) V x The value of [1,x(1),a,v] at the terminal time; V a (1) V a The values ​​of [1,x(1),a,v] at the terminal moment; G1(τ), G2(τ), G3(τ), G4(τ), G5(τ), G6(τ), g1(τ), g2(τ), g3(τ), p1(τ) and p2(τ) are all intermediate variables.

6. The pseudo-spectral differentiation dynamic programming method according to claim 5, characterized in that: The iterative convergence index is determined by the following formula: Among them, ρ c is the iterative convergence index; max is the maximum eigenvalue; is the solution obtained in the i-th iteration; is the linear equation solved for the i+1th iteration, A S , B S 、b S To form the above linear equation The coefficient matrix of .

7. A pseudo-spectral differential dynamic programming device, characterized in that: include: A nonlinear optimal guidance problem construction module is used to construct a nonlinear optimal guidance problem based on initial constraints, terminal constraints and a dynamic model; the optimal value function of the nonlinear optimal guidance problem satisfies the Hamilton-Jacobi-Bellman equation; An optimal value function partial derivative differential equation determination module is used to perform a second-order expansion of the Hamilton-Jacobi-Bellman equation based on a specified reference trajectory to obtain a differential equation about the partial derivative of the optimal value function; A first two-point boundary value problem determination module, used to obtain the derivative of the construction matrix according to the differential equation of the partial derivative of the optimal value function, and determine the first two-point boundary value problem; the first two-point boundary value problem is a homogeneous form of the trajectory two-point boundary value problem; A second two-point boundary value problem determination module is used to solve the first two-point boundary value problem, determine the optimal control amount variation, and determine the second two-point boundary value problem based on the optimal control amount variation; solve the second two-point boundary value problem to update the trajectory; The pseudo-spectral differential dynamic programming module is used to process the trajectory and the construction matrix based on the pseudo-spectral method by using Lagrange interpolation, transform the second two-point boundary value problem and solve iteratively to obtain a discrete optimal trajectory and a discrete construction matrix.

8. A computer device comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the pseudo-spectral differentiation dynamic programming method according to any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the pseudo-spectral differentiation dynamic programming method according to any one of claims 1 to 6 is implemented.

10. A computer program product, comprising a computer program, characterized in that When the computer program is executed by a processor, the pseudo-spectral differentiation dynamic programming method according to any one of claims 1 to 6 is implemented.

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