Method, device and equipment for transforming singular optimal control problem and storage medium

Abstract

This application provides a method, apparatus, device, and storage medium for transforming a singular optimal control problem, relating to the field of automatic control technology. The method includes: obtaining a mathematical model of the singular optimal control problem; ensuring that the switching function of the control variable in the mathematical model is always zero on the singular arcs; establishing supplementary conditions on the singular arcs based on the switching function; constructing mixed control and state constraints dependent on state variables based on the supplementary conditions; adding the mixed control and state constraints as path constraints to the singular optimal control problem to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem remain equivalent in terms of optimal solution. This method achieves the effect of transforming a singular optimal control problem into a general optimal control problem with lower computational difficulty.

Description

Technical Field

[0001] This application relates to the field of automatic control technology, and more specifically, to a method, apparatus, device, and storage medium for transforming singular optimal control problems. Background Technology

[0002] Currently, the optimal control problem aims to find the optimal control strategy that satisfies the system's dynamic constraints, boundary conditions, and path limitations, minimizing or maximizing performance indicators. It has wide applications in aerospace, automation, and chemical engineering. Pontryagin's minimum principle provides the theoretical basis for handling variational problems with limited control variables, typically yielding optimal solutions in Bang-Bang form. However, when the Hamiltonian function does not explicitly contain control variables within a finite time interval, the optimal solution may involve singular arcs. In such cases, the minimum principle cannot directly determine the specific expression for optimal control; these problems are called singular optimal control problems. Typical engineering examples include the Goddard problem for optimizing the trajectory during the rocket's ascent phase and terminal guidance scenarios for spacecraft.

[0003] In existing technologies for solving singular optimal control problems, there are two main categories: analytical methods and numerical methods. Regarding analytical methods, Kelley proposed the generalized Legendre-Clebsch condition for the optimality of singular arcs in 1964. Subsequently, scholars such as Kopp, Moyer, Goh, and Robbins further developed this theory, providing a theoretical basis for screening candidate singular extrema. Bryson and Ho proposed singular surface equations to describe the relationship between state variables on singular arcs, and Tsiotras and Kelly further extended this concept. However, while analytical methods can provide new necessary conditions, they are difficult to directly obtain complete numerical solutions, limiting their application in complex engineering problems. In numerical methods, Jacobson et al. proposed a regularization method, which transforms the original problem into a series of limit solutions to non-singular problems by introducing an integral quadratic control function; Maurer proposed a multi-shot method to calculate the segmentation points of non-singular and singular sub-arcs; Aronna et al. proposed a shot method for optimal control problems with linear control variables; Rao and Benson developed the GPOPS software using the pseudospectral method, which was successfully applied to solving the one-dimensional Goddard problem; Pager established the BBSOC framework for handling non-smooth and singular optimal control problems. However, all of the above numerical methods require special treatment of singular arcs, treating singular optimal control problems as a special type of problem different from general optimal control problems, and lack a unified solution framework.

[0004] However, the shortcomings of existing technologies are as follows: on the one hand, although analytical methods provide theoretical conditions, they are difficult to implement numerically; on the other hand, although numerical methods can solve the problem, they require special design for singular arcs and cannot directly apply mature general optimal control solvers. This approach of separating singular problems from general problems leads to a complex solution process, poor versatility, and high requirements for the user's theoretical foundation and programming skills.

[0005] Therefore, existing solution steps and algorithms usually require special design for singularities, which increases the complexity of singular optimal control problems. Summary of the Invention

[0006] The purpose of this application is to provide a method, apparatus, device, and storage medium for transforming singular optimal control problems, so as to solve the above-mentioned problems existing in the prior art. It can transform singular optimal control problems into general optimal control problems with lower computational difficulty, thereby reducing the computational complexity of singular optimal control problems.

[0007] Firstly, a transformation method for the singular optimal control problem is provided, which may include: Obtain a mathematical model for the singular optimal control problem; wherein, in the mathematical model, the switching function of the control variable is always zero on the singular arc; Based on the switching function, supplementary conditions are established on the singular arc; based on the supplementary conditions, control and state hybrid constraints dependent on state variables are constructed. The control and state hybrid constraints are added as path constraints to the singular optimal control problem to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem are equivalent in terms of optimal solution.

[0008] Secondly, a transformation apparatus for a singular optimal control problem is provided, the apparatus comprising: An acquisition module is used to acquire the mathematical model of the singular optimal control problem; wherein, in the mathematical model, the switching function of the control variable is always zero on the singular arc; A construction module is used to establish supplementary conditions on the singular arc based on the switching function; and to construct control and state hybrid constraints dependent on state variables based on the supplementary conditions. The transformation module is used to add the control and state hybrid constraints as path constraints to the singular optimal control problem to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem are equivalent in terms of optimal solution.

[0009] Thirdly, an electronic device is provided, which includes a processor, a communication interface, a memory, and a communication bus, wherein the processor, the communication interface, and the memory communicate with each other through the communication bus; Memory, used to store computer programs; When a processor executes a program stored in memory, it implements any of the steps described in the first aspect above.

[0010] Fourthly, a computer-readable storage medium is provided, wherein a computer program is stored therein, and when executed by a processor, the computer program implements the steps of any of the methods described in the first aspect above.

[0011] This application provides a method, apparatus, device, and storage medium for transforming a singular optimal control problem (SOCP). The method obtains a mathematical model of the SOCP; in the mathematical model, the switching function of the control variable is always zero on the singular arcs. Based on the switching function, supplementary conditions are established on the singular arcs; based on the supplementary conditions, control and state hybrid constraints dependent on state variables are constructed. These control and state hybrid constraints are added as path constraints to the SOCP, resulting in a transformed general optimal control problem. The general optimal control problem and the SOCP remain equivalent in terms of optimal solution. In this scheme, by introducing necessary conditions to derive the singular segment control-state hybrid constraints, the SOCP is transformed into a general optimal control problem (OCP), thereby reducing the computational complexity of the SOCP. Attached Figure Description

[0012] To more clearly illustrate the technical solutions of the embodiments of this application, the accompanying drawings used in the embodiments of this application will be briefly introduced below. It should be understood that the following drawings only show some embodiments of this application and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0013] Figure 1 A flowchart illustrating a method for transforming a singular optimal control problem, provided in an embodiment of this application; Figure 2 A flowchart illustrating a method for transforming a singular optimal control problem, provided in an embodiment of this application; Figure 3 A schematic diagram of the structure of a transformation device for a singular optimal control problem provided in an embodiment of this application; Figure 4 This is a schematic diagram of the structure of an electronic device provided in an embodiment of this application. Detailed Implementation

[0014] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of the embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application. Unless otherwise defined, the technical or scientific terms used in this application should have the ordinary meaning understood by those skilled in the art. The words "first," "second," and similar terms used in this application do not indicate any order, quantity, or importance, but are only used to distinguish different components. The words "comprising" or "including," etc., mean that the element or object preceding the word covers the element or object listed after the word and its equivalents, but do not exclude other elements or objects. The words "connected," "coupled," or "connected," etc., are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. "Up," "down," "left," "right," etc., are only used to indicate relative positional relationships. When the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0015] The method for transforming the singular optimal control problem provided in this application can be applied to electronic devices, terminal devices, devices or apparatuses for transforming singular optimal control problems, or other devices or apparatuses capable of executing this embodiment, and there are no limitations on this application. In this embodiment, the execution subject is described as an electronic device.

[0016] The preferred embodiments of this application are described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit this application. Furthermore, the embodiments and features in the embodiments of this application can be combined with each other without conflict.

[0017] Figure 1 This is a flowchart illustrating a transformation method for a singular optimal control problem provided in an embodiment of this application. Figure 1 As shown, the method may include: Step S101: Obtain the mathematical model of the singular optimal control problem; wherein, in the mathematical model, the switching function of the control variable is always zero on the singular arc.

[0018] For example, a mathematical model for a singular optimal control problem is obtained. In optimal control, the mathematical model typically includes system state equations, performance index functions, boundary conditions, constraints, and control variable inequality constraints. On singular arcs, the switching function of the control variables is always zero.

[0019] Step S102: Based on the switching function, establish supplementary conditions on the singular arc; based on the supplementary conditions, construct control and state hybrid constraints that depend on the state variables.

[0020] For example, the switching function is continuously differentiated until the singular control variable appears explicitly, yielding supplementary conditions. These supplementary conditions are represented as a system of homogeneous linear equations about the costate variables, including the relationships between the singular control variable and the state and costate variables. Based on these supplementary conditions, control and state hybrid constraints that depend only on the state variables are constructed.

[0021] Step S103: Add the mixed control and state constraints as path constraints to the singular optimal control problem to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem are equivalent in terms of optimal solution.

[0022] For example, by using mixed control and state constraints as path constraints, the original finite-order singular optimal control problem is introduced. While maintaining the performance index function and system dynamic equations unchanged, the singular optimal control problem is transformed into a general optimal control problem. This facilitates the use of a standard numerical solver suitable for general optimal control problems to solve the transformed problem and obtain the optimal control strategy containing singular control variables, where the state variables are determined by the system dynamic equations and the evolution of the control variables. Therefore, by introducing supplementary conditions in the singular segment, we successfully transformed the Singular Optimal Control Problem (SOCP) into a Two-Point Boundary Value Problem (TBVP), thus enabling the use of indirect methods to solve the SOCP.

[0023] The method provided in this application obtains a mathematical model of a singular optimal control problem; wherein, in the mathematical model, the switching function of the control variable is always zero on the singular arcs. Based on the switching function, supplementary conditions are established on the singular arcs; based on the supplementary conditions, control and state hybrid constraints dependent on state variables are constructed. The control and state hybrid constraints are added as path constraints to the singular optimal control problem to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem remain equivalent in terms of optimal solution. In this scheme, by introducing necessary conditions to derive the singular segment control-state hybrid constraints, the singular optimal control problem is transformed into a general optimal control problem with lower computational difficulty, thereby reducing the computational complexity of the singular optimal control problem.

[0024] Figure 2 A flowchart illustrating a transformation method for a singular optimal control problem provided in this application is shown below. Figure 2 As shown, in this embodiment... Figure 1 Based on the embodiments, the method is described in detail below, and the method includes: Step S201: Obtain the mathematical model of the singular optimal control problem; wherein, in the mathematical model, the switching function of the control variable is always zero on the singular arc.

[0025] In one example, the singular optimal control problem is a finite-order problem, which includes singularities of even order and satisfies the generalized Legendre-Clebsch conditions.

[0026] In one example, the path constraints are enforced within the time interval corresponding to the singular arcs, and the transformed general optimal control problem has equivalent first-order necessary conditions to the singular optimal control problem.

[0027] For example, a mathematical model of the singular optimal control problem is obtained. Specifically, for the general optimal control problem (OCP) in which some control variables appear linearly in a dynamic system, the performance index function is of the Mayer type, as shown in the following formula (1): --(1) Dynamic constraints: --(2) Boundary conditions: --(3) Inequality constraints: --(4) in, For state variables; Both are control variables, and have ; t is a time variable, t0 and t f Let be the initial and final values ​​of time. These functions are defined as follows:

[0028] Will Defined as a costate variable of state variable x, the Hamiltonian function of the aforementioned OCP(1)-(4) can be expressed as: --(10) because It appears linearly in the Hamiltonian function, Rewritten as: --(11) According to the variational method theory, the optimal solution to problems (1)-(4) needs to satisfy the following conditions:

[0029]

[0030] and These are the vector multipliers associated with the boundary conditions (3) and the inequality constraints (4), respectively.

[0031] For OCP, solving the nonlinear algebraic equations (12)-(22) is an indirect solution to the problem. In this way, the solution to OCP is transformed into solving a continuous TVBP. TVBP requires that the state and costate variables at the initial and terminal time points satisfy specific boundary conditions and inequality constraints, while satisfying the constraints and control strategies of the dynamic system throughout the entire time range.

[0032] The optimal control problem in this application considers control variables. Bounded: --(twenty three).

[0033] The optimal control can be derived by applying the Pontryagin minimum principle. The analytical structure. To minimize the Hamiltonian function, the optimal control exhibits a switching characteristic, and the solution structure is as follows: , in, It is called a switching function.

[0034] Optionally, the singular optimal control problem is a finite-order problem, with an even number of singular orders, and satisfies the generalized Legendre-Clebsch conditions. Path constraints are enforced within the time interval corresponding to the singular arcs, and the transformed general optimal control problem has equivalent first-order necessary conditions to the singular optimal control problem.

[0035] Optionally, for singular arcs, consider a subinterval. The switching function is always 0: --(25) Unless otherwise specified, all the following discussions are within subintervals. superior.

[0036] The classic Legendre-Clebsch condition requires that the Hessian matrix of the Hamiltonian is positive semi-definite along the optimal trajectory: --(26) In the above formula , , . Notice exist Linearity occurs in the middle, therefore we have Combining equation (26), we clearly obtain: --(27) Therefore, determinant In the strange subspace Every point on it is always 0: --(28)

[0037] The above equation defines the singular arc [Bliss, 1946; Bryson and Ho, 1975; Bell and Jacobson, 1975; Goh], where the control variable... Also known as a singular control variable, denoted as Therefore, it can be said that in the subinterval superior It is strange.

[0038] Step S202: Based on the switching function, establish supplementary conditions on the singular arc.

[0039] In one example, S202 includes: continuously differentiating the switching function until the singular control variable appears explicitly, thus obtaining supplementary conditions; the supplementary conditions are expressed as a homogeneous linear system of equations about the costate variables, and the supplementary conditions include the relationship between the singular control and the state variables and the costate variables.

[0040] For example, when discussing SOCP, formula (15) always holds within the singular segment, and the singular control variable... It does not appear explicitly in the switching function. This characteristic means that in singular cases, it cannot be uniquely determined solely by the nonlinear algebraic equations (12)-(22). The value of makes the problem unsolvable by closed-loop solution methods. To overcome this challenge, an additional is proposed to be added to the singular segment. Dimensional condition, which is directly related to the singular control variable. By combining this new supplementary condition with the existing nonlinear algebraic equations (12)-(22), a complete solution framework for TBVP can be constructed, thereby effectively solving SOCP.

[0041] Step S203: Based on the supplementary conditions, construct control and state hybrid constraints that depend on state variables.

[0042] In one example, S203 includes: combining the relation with the homogeneous linear equations of costate variables in the first-order necessary conditions of the singular optimal control problem, eliminating the costate variables, and generating a mixed control and state constraint; wherein the mixed control and state constraint is a function vector that depends on time and state variables.

[0043] In one example, the mixed control and state constraints are equivalent to the supplementary conditions provided that the coefficient matrix is ​​invertible.

[0044] For example, for the order of singular optimal control, following the generalized Legendre-Clebsch conditions, the following is defined: To determine the optimal control strategy at the extreme points of the objective function, There exists a smallest integer with at least one non-zero element. This means that, when the switching function is... After the second derivative, the singular control variable In This is evident in the equations (13) and (14). It is worth noting that, according to equations (13) and (14), and They are all costate variables It is a homogeneous linear function. This application assumes that the control variable is a homogeneous linear function. The expression depends only on the state and costate, therefore Also costate variables The homogeneous function. At this point, =0 constitutes a relationship with costate variables A set of homogeneous linear equations: --29 And it satisfies the following sub-conditions: 1. Integer It must be an even number. 2. If ,but It must be negative half-definite.

[0045] Where, integer The order is called SOCP. Note that it is possible for any Singular control variables None of them are displayed in the present In this case, it is impossible to obtain effective information about the singular control variable from (29), so the order of SOCP is defined as infinite.

[0046] Equation (29) for singular optimal control Added one Given the dimensional conditions and the nonlinear algebraic equation system (12)-(19), TBVP can achieve a closed-loop solution. Thus, the originally complex SOCP is transformed into a solvable TBVP, providing feasibility for solving it using the indirect method.

[0047] Optionally, in the subsequent discussion of this paper, for the SOCP composed of (1)-(4), consider the following assumptions: 1. Assumption 1 Assuming SOCP is a finite-order problem, then the singular control variables... It can be obtained from (29).

[0048] 2. Assumption 2 Consider a problem with a finite number of singular arcs and switching points. That is, assume that "flutter" does not occur.

[0049] 3. Assumption 3 The prior structure of SOCP is assumed to include the total number of switching points, as well as the number and specific locations of singular and non-singular arcs.

[0050] 4. Assumption 4 Imagine solving TBVP to control variables The expression can depend only on the state variable. and costate variables .

[0051] Singular optimal control is given in (29). The supplementary condition is that if a theoretical correspondence between SOCP and general OCP can be established, transforming finite-order SOCP into typical OCP, the singular particularity can be eliminated. At this point, all methods applicable to general OCP (including direct methods such as pseudospectral methods) can be directly applied to solve SOCP without any modification to the algorithm structure, thus significantly improving the solver's adaptability and flexibility.

[0052] Alternatively, similar to (29), differentiate the switching function until the singular control appears explicitly in the expression for the first time, to obtain: --(30) Note that (14), (25), (27), (29), and (30) are all about costate variables. The homogeneous linear equation, through simplification, yields the following about... , , , of One relation:

[0053] Performing a linear transformation on the above equation, we obtain:

[0054] Observational expression (32), definition: (33) It can be seen that the matrix Includes The information, and the matrix It does not contain .

[0055] From the matrix Winning column vectors ( It can be 1 to (any value of ), each column vector is denoted as ,matrix Winning There are n column vectors, each denoted as . The following relationship is formed: --(34) Step S204: Add the mixed control and state constraints as path constraints to the singular optimal control problem to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem are equivalent in terms of optimal solution.

[0056] For example, according to the nontriviality condition of Pontryagin's minimum principle, for the optimal solution, the costate variable cannot be identically equal to zero. Therefore, in the linear system of equations (35), only when Only then can a non-trivial solution for the costate variables be obtained. In this case, the costate variables... This can be viewed as "disappearance," thus yielding a result regarding... The expression.

[0057] Then retrieve according to the list above. Next, and guarantee Each component in the equation is taken, resulting in: --(35) this The equation is denoted as:

[0058] Observed Only with This is relevant, thus avoiding the coupling between singular control and costate variables. If in the singular segment, ... Treating it as a control-state hybrid constraint can supplement SOCP's discussion of singular control. When information is missing, SOCP is converted to general OCP.

[0059] It is particularly important to emphasize that, although the specific expressions for matrices A and B have been explicitly given in (33), any discussion of costate variables... Homogeneous linear equations, as long as they can provide effective information for singular control, can be used as effective extensions to matrix A. Similarly, homogeneous linear equations with costate variables that do not contain effective information for singular control can also be used as extensions to matrix B. However, the current challenge is that it is impossible to provide effective information for singular control to matrix B. The provision of further supplementary information limits our ability to develop the expression for matrix A. Nevertheless, in certain specific cases, we are able to provide additional information for matrix B, which allows for greater flexibility and depth in its expression.

[0060] For example, it is well known that when the Hamiltonian function does not explicitly include a time variable and the terminal time is free, This is true. Combining (25), we can deduce that: --(37) In this case, the Hamiltonian function is about the costate variables. A homogeneous linear equation that does not contain Any information. At this point, (37) can be used as a valid supplementary component of matrix B.

[0061] Optionally, for the SOCP composed of (1)-(4), in the singular interval Supplementary control - state hybrid constraints It is converted into a general OCP:

[0062] Considering that the problem introduces only an additional control-state hybrid constraint (43) in the singular segment, the Hamiltonian function of the transformed typical OCP remains unchanged, i.e. (10). Accordingly, (14), (25), (27) and (30) also maintain their validity.

[0063] In formula (36), it is already known that the function can be... Each component is expanded into a matrix and The determinant of a matrix formed by column vectors. Specifically, the matrix... The column vectors are all column vectors of matrix A. The column vectors of A and B are all column vectors of matrix B. The specific definitions of matrices A and B have been explained in detail in (33).

[0064] because Each line can be expanded into indivual The determinant of a matrix composed of column vectors can be determined using relevant knowledge of matrix theory. indivual If the column vectors are linearly independent, then there exists... Coefficients of groups not all being zero , so that: (43) Given that (14), (25), (27), and (30) hold true, we can obtain: --(44) Then multiply both ends of (43) It can be deduced that: (45) Right now:

[0065] Rearrange (46) and introduce As a matrix The coefficient matrix of the column vectors is obtained as follows:

[0066] Obviously, when the coefficient matrix When reversible, Established, combined with matrix The description in (33) proves this. .

[0067] Therefore, this application demonstrates that in SOCP, supplementary conditions can be used. Combining (14), (25), (27) and (30), the control-state hybrid constraint can be derived. Conversely, when the coefficient matrix... When reversible, combining (14), (25), (27), and (30), it is also possible to... Derivation This conclusion is summarized by the following theorem: Theory 1 (Supplementary Conditions and Control-State Hybrid Constraint Equivalence Theory): In the coefficient matrix Under reversible conditions, supplementary conditions and control-state hybrid constraints Completely equivalent.

[0068] Theorem 1 shows that a finite-order SOCP can be transformed into a general OCP, and the solution to the problem remains unchanged during this transformation. That is, the general optimal control problem and the singular optimal control problem are equivalent in terms of optimal solution.

[0069] Step S205: Solve the transformed general optimal control problem using the standard numerical solver corresponding to the general optimal control problem to obtain a control strategy containing singular control variables.

[0070] For example, the steps above illustrate a transformation strategy for converting a finite-order SOCP into a general OCP. This transformation strategy not only simplifies the solution process but also significantly expands the application scope of existing solution strategies. Specifically, it allows all solution methods applicable to general OCP, whether direct or indirect, to be applied to the solution of SOCP without any adjustments to the algorithmic framework.

[0071] The Radau pseudospectral method (RPM), as an efficient direct method, is favored for its fast computational efficiency, good convergence properties, and low sensitivity to initial conditions. Furthermore, the existence of mature optimization toolkits for RPM greatly simplifies the solution process and lowers the technical threshold for users. The following details the consistency between the KKT conditions of the nonlinear programming problem (NLP) obtained after RPM discretization and the discrete form of the first-order necessary conditions of the original problem. Combining the solution of the original SOCP indirectly transformed into TBVP, the optimality of using RPM to solve the SOCP problem will be proven.

[0072] Optionally, for the Radau pseudospectral method, problems (38)-(42) are defined in the time interval as However, in order to apply RPM, the time interval must be... Convert to time interval The transformation is as follows: --(48) The continuous OCP expression after time-domain transformation is as follows:

[0073] Like other pseudospectral methods, RPM approximates state variables using a global interpolation polynomial, with interpolation points being discrete points over the time interval. A key difference between different pseudospectral methods lies in the choice of these discrete points. For RPM, the polynomial is chosen... of The roots are taken as discrete points, denoted as . ,in for Legendre polynomial, , The following is called a set. The elements in are LGR point.

[0074] Define a new node ,set up for The extended set. As interpolation nodes, perform state variable... The Lagrange interpolation is as follows: --(51) in, These are the basis functions for Lagrange interpolation. Based on the properties of Lagrange interpolation, the state approximation... truth value of the state At the interpolation point They are exactly equal, that is It is important to note that... It is not an LGR point, but is used in state approximation.

[0075] The state interpolation described above is used to approximate the derivative of the state variable at the LGR point: (52) Among them, the differential of the Lagrange interpolation basis function Radau pseudospectral differential matrix The elements are unrelated to the problem itself, but only to the selection of LGR points.

[0076] The performance metrics have been restated as follows: --(53) definition Combining the state differential approximation of (53), the orthogonal configuration of the dynamic equations at the LGR point is obtained:

[0077] in, It is a differential matrix The OK.

[0078] Discrete boundary conditions are expressed in general form as follows: --(55) Finally, inequality equation (41) and path constraint (42) are enforced at the LGR point as follows:

[0079] in, This represents all time mapping intervals that fall within the singular arcs. The LGR points within the range. Thus, an NLP is defined by the performance index function (53) and constraints (54)-(57) after Radau pseudospectral discretization. The solution of this NLP is an approximate solution of the continuous OCP.

[0080] Optionally, for the KKT conditions in NLP, the optimality conditions of NLP are called KKT conditions, which can be obtained from the augmented performance function (Lagrange function). The augmented performance function utilizes Lagrange multipliers. Construction combining all constraints in NLP: (58) In the above formula, , , , and , , .

[0081] By setting the derivative of the augmented Lagrange function with respect to each variable to zero, we obtain the KKT optimality conditions:

[0082]

[0083]

[0084] in, The meaning is as follows: Assuming ,So yes The Jacobian matrix, its first... Action is .

[0085] Optionally, for the first necessary condition of the general OCP after transformation, for the general OCP after transformation (38)-(42), the augmented Hamiltonian Defined as:

[0086] In the above formula , , Furthermore, dynamic costate variables Inequality-bound Lagrange multipliers Control-state hybrid constraint Lagrange multipliers .

[0087] Using the variational principle to derive the enhanced Hamiltonian, we obtain the first-order necessary conditions for OCP(38)-(42):

[0088]

[0089]

[0090] in, Lagrange multipliers related to boundary conditions.

[0091] The augmented Hamiltonian function at the initial time With terminal time The values ​​can be represented as follows:

[0092] The above integral is calculated in the time variable. This is performed over an interval that is transformed by an affine transformation from the physical time interval. It is derived from mapping. In fact, according to the theorem 1... and The complete equivalence of the equations shows that the first necessary condition of the general OCP after the transformation, consisting of (75)-(89), is completely consistent with the solution of the nonlinear algebraic equation system (12)-(19) and (29) of the TBVP indirectly transformed from SOCP.

[0093] Optionally, for comorphic mappings, define the adjoint matrix. :

[0094] in, and These are the orthogonal weights at the corresponding LGR points.

[0095] For the purpose of proving the equivalence theorem of the KKT conditions and the first-order necessary conditions in the subsequent text, the relevant information is given here without proof. , Two lemmas: Lemma 1: Matrix for The differential matrix of a polynomial space of degree 1. That is, if It is the highest Polynomials of order, vectors The Each component is Then we have:

[0096] Lemma 2: ,in For components that are all 1s Dimensional column vector.

[0097] Note that the values ​​of the costate variables at the initial time, the terminal time, and the LGR point can be obtained through the KKT multiplier mapping, and the relationship is as follows:

[0098] Based on the costate mappings of (94)-(98), and combining Lemma 1 and Lemma 2, through the derivation process

[15] , the KKT conditions (59)-(73) are transformed into:

[0099]

[0100] Meanwhile, the discrete augmented Hamiltonian function is defined as: -(114) Thus, it has been proven that when the general OCP is discretized using RPM, the resulting KKT conditions (99)-(113) of the NLP are completely equivalent to the discrete forms of the first-order optimal necessary conditions (75)-(89). Furthermore, since the first-order necessary conditions of the general OCP after the transformation are completely consistent with the solution of the TBVP indirectly transformed from SOCP, it can be concluded that the solution obtained by RPM is equivalent to the solution of the complete TBVP.

[0101] Optionally, for the solution strategy, by introducing control-state hybrid constraints in the singular segments, the finite-order SOCP is transformed into a typical OCP, thus theoretically establishing the correspondence between SOCP and general OCP. This transformation strategy allows standard numerical solvers to be directly applied to solving SOCP without any adjustments to the algorithm framework. This improvement significantly enhances the adaptability and flexibility of the solver, providing an efficient and direct method for solving SOCP. Thus, by adding control-state hybrid constraints to the singular segments, the Goddard problem can be transformed into a general OCP and solved using the Radau pseudospectral method.

[0102] The method provided in this application derives singular segment control-state hybrid constraints by introducing necessary conditions, thereby transforming the singular optimal control problem (SOCP) into a general optimal control problem (OCP). This, in turn, reduces the computational complexity of the singular optimal control problem by converting the SOCP into an OCP. By transforming the SOCP into an OCP, the algorithm library applicable to SOCP is expanded, including but not limited to pseudospectral methods with mature software toolkits. The Radau pseudospectral method is used to discretize the transformed OCP, and it is proven that the discretized form of the KKT conditions in NLP is completely consistent with the first-order necessary conditions, and also matches the optimality conditions of TBVP, demonstrating the optimality of RPM in solving the SOCP. Furthermore, the Radau pseudospectral method can be directly applied to the transformed OCP without any preprocessing, which not only simplifies the solution process but also significantly improves the solution efficiency.

[0103] Corresponding to the above method, embodiments of this application also provide a transformation device for singular optimal control problems, such as... Figure 3 As shown, the device includes: Module 41 is used to obtain the mathematical model of the singular optimal control problem; wherein, in the mathematical model, the switching function of the control variable is always zero on the singular arc; Module 42 is used to establish supplementary conditions on singular arcs based on switching functions; and to construct control and state hybrid constraints that depend on state variables based on the supplementary conditions. The transformation module 43 is used to add the mixed control and state constraints as path constraints to the singular optimal control problem, so as to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem are equivalent in terms of optimal solution.

[0104] The functions of each functional unit in the singular optimal control problem transformation device provided in the above embodiments of this application can be implemented through the above method steps. Therefore, the specific working process and beneficial effects of each unit in the singular optimal control problem transformation device provided in the embodiments of this application will not be repeated here.

[0105] This application also provides an electronic device, such as... Figure 4 As shown, it includes a processor 510, a communication interface 520, a memory 530, and a communication bus 540, wherein the processor 510, the communication interface 520, and the memory 530 communicate with each other through the communication bus 540.

[0106] Memory 530 is used to store computer programs; The processor 510 performs the above steps when executing the program stored in the memory 530.

[0107] The communication bus mentioned above can be a Peripheral Component Interconnect (PCI) bus or an Extended Industry Standard Architecture (EISA) bus, etc. This communication bus can be divided into address bus, data bus, control bus, etc. For ease of illustration, only one thick line is used to represent it in the diagram, but this does not mean that there is only one bus or one type of bus.

[0108] The communication interface is used for communication between the aforementioned electronic devices and other devices.

[0109] The memory may include random access memory (RAM) or non-volatile memory (NVM), such as at least one disk storage device. Optionally, the memory may also be at least one storage device located remotely from the aforementioned processor.

[0110] The processors mentioned above can be general-purpose processors, including central processing units (CPUs), network processors (NPs), etc.; they can also be digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components.

[0111] The implementation methods and beneficial effects of the various components of the electronic device in the above embodiments for solving the problem can be found in [reference needed]. Figure 1 The steps in the illustrated embodiments are used to implement the electronic device. Therefore, the specific working process and beneficial effects of the electronic device provided in this application will not be repeated here.

[0112] In another embodiment provided in this application, a computer-readable storage medium is also provided, which stores instructions that, when executed on a computer, cause the computer to perform a transformation method for any of the singular optimal control problems in the above embodiments.

[0113] In another embodiment provided in this application, a computer program product containing instructions is also provided, which, when run on a computer, causes the computer to perform a transformation method for any of the singular optimal control problems in the above embodiments.

[0114] Those skilled in the art will understand that the embodiments in this application can be provided as methods, systems, or computer program products. Therefore, the embodiments in this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the embodiments in this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0115] This application describes embodiments of methods, apparatus (systems), and computer program products according to embodiments of this application with reference to flowchart illustrations and / or block diagrams. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0116] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0117] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0118] Although preferred embodiments have been described in this application, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the embodiments of this application.

[0119] Obviously, those skilled in the art can make various modifications and variations to the embodiments of this application without departing from the spirit and scope of the embodiments of this application. Therefore, if these modifications and variations to the embodiments of this application fall within the scope of the claims in this application and their equivalents, then this application also intends to include these modifications and variations.

Claims

1. A method for transforming a singular optimal control problem, characterized in that, The method includes: Obtain a mathematical model for the singular optimal control problem; wherein, in the mathematical model, the switching function of the control variable is always zero on the singular arc; Based on the switching function, supplementary conditions are established on the singular arc; based on the supplementary conditions, control and state hybrid constraints dependent on state variables are constructed. The control and state hybrid constraints are added as path constraints to the singular optimal control problem to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem are equivalent in terms of optimal solution.

2. The method as described in claim 1, characterized in that, Based on the switching function, supplementary conditions are established on the singular arc, including: The switching function is continuously differentiated until the singular control variable appears explicitly, thus obtaining supplementary conditions. The supplementary conditions are expressed as a homogeneous linear equation system about the costate variable, and include the relationship between the singular control variable and the state variable and the costate variable.

3. The method as described in claim 2, characterized in that, The construction of control and state hybrid constraints dependent on state variables based on the supplementary conditions includes: The aforementioned relation is combined with the homogeneous linear equation of the costate variable in the first-order necessary condition of the singular optimal control problem to eliminate the costate variable and generate a mixed control and state constraint; wherein the mixed control and state constraint is a function vector that depends on time and state variables.

4. The method as described in claim 3, characterized in that, The control and state hybrid constraints and the supplementary conditions are equivalent to each other under the condition that the coefficient matrix is ​​invertible.

5. The method as described in claim 1, characterized in that, The singular optimal control problem is a finite-order problem, the singular order of which is even, and it satisfies the generalized Legendre-Clebsch condition.

6. The method as described in claim 1, characterized in that, The path constraint is enforced within the time interval corresponding to the singular arc, and the transformed general optimal control problem has equivalent first-order necessary conditions to the singular optimal control problem.

7. The method according to any one of claims 1-6, characterized in that, The method further includes: The transformed general optimal control problem is solved using the standard numerical solver corresponding to the general optimal control problem to obtain a control strategy containing singular control variables.

8. A transformation device for a singular optimal control problem, characterized in that, The device includes: An acquisition module is used to acquire the mathematical model of the singular optimal control problem; wherein, in the mathematical model, the switching function of the control variable is always zero on the singular arc; A construction module is used to establish supplementary conditions on the singular arc based on the switching function; and to construct control and state hybrid constraints dependent on state variables based on the supplementary conditions. The transformation module is used to add the control and state hybrid constraints as path constraints to the singular optimal control problem to obtain the transformed general optimal control problem; the general optimal control problem and the singular optimal control problem are equivalent in terms of optimal solution.

9. An electronic device, characterized in that, The electronic device includes a processor, a communication interface, a memory, and a communication bus, wherein the processor, the communication interface, and the memory communicate with each other through the communication bus; Memory, used to store computer programs; A processor, when executing a program stored in memory, implements the method of any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the method described in any one of claims 1-7.

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