A probabilistic uncertain closed-loop robust trajectory planning method considering actuator saturation

Abstract

The present application relates to a kind of probability uncertain closed-loop robust trajectory planning method considering actuator saturation, belong to the field of vehicle control technology, to reduce the sensitivity of guided ammunition scheme trajectory to uncertain factors, and while avoiding actuator saturation as far as possible, give full play to its limited control ability.First, the control quantity of the optimal trajectory of the determined performance is obtained as the initial value of the subsequent algorithm;Second, the main uncertain factors are modeled, and the uncertainty quantification module is built based on PCE method;Then, the corresponding constraint condition and objective function are designed, and the open-loop robust planning model is established, the algorithm accuracy and the time-consuming of calculation are considered, and the closed-loop tracking PID controller is designed and coupled with the open-loop planning model;Finally, the direct shooting method is used for closed-loop robust planning.The present application fully considers the matching of scheme trajectory and guidance control system, and overcomes the problems of insufficient precision of nonlinear model description and initial value sensitivity in traditional method.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of aircraft control, and particularly relates to a probabilistic uncertain closed-loop robust trajectory planning method considering actuator saturation. BACKGROUND

[0002] Among the key technologies in guided missile and rocket research, scheme trajectory planning has always been a core and hot issue. Limited by cost and the harsh mechanical environment in the launch process, guided ammunition is usually small in size and small in aerodynamic rudder, and the navigation and control devices that can be applied to the missile and the overall missile structural layout all have special requirements, and the projectile is usually in unpowered flight in the controlled trajectory segment. Therefore, guided ammunition has the characteristics of small lift-drag ratio, limited maneuverability and anti-interference capability, and the rationality of the scheme trajectory design is particularly critical under such conditions.

[0003] Most of the current research on scheme trajectory design discusses deterministic optimal control problems and does not consider the influence of uncertain factors, so the designed trajectory is too ideal. In the actual flight process of the projectile, the dynamics model, aerodynamic and environmental parameters, and the state at the start control point all have uncertainties, and the planning design variables, performance indicators, and constraint conditions are closely related to these parameters. The planning results based on the deterministic model may cause the design variables to exceed the range, the performance indicators to have deviations, and the constraint conditions to break the boundary, etc. This problem can be alleviated by robust planning.

[0004] In the process of robust planning, it is crucial to obtain the propagation rule of random disturbances along the trajectory under the action of the current control variable. Linear analysis of covariance (LAC) method can obtain similar statistical results as Monte Carlo simulation (MCS) by integrating the error propagation equation only once. Due to its high efficiency, LAC has been widely used in robust planning problems. However, LAC requires linearization of the model, which will result in large approximation errors when dealing with highly nonlinear problems. At the same time, the robust planning based on LAC method is sensitive to initial values, as it is based on the reference trajectory for shaping, and the robust optimal trajectory is often not within the neighborhood of the reference trajectory, which may cause the algorithm to fall into local optimum and terminate prematurely.

[0005] The current available technical achievements all belong to open-loop robust planning methods, which ignore the coupling between the scheme trajectory and the guidance control system. In the actual environment, the deviation caused by various uncertainties is usually eliminated by the guidance control system. Therefore, the matching between the scheme trajectory and the guidance control system has an important influence on the final task execution effect. For the UAV with sufficient maneuvering ability, the open-loop planning has no big problem with performance loss, but for the guided ammunition with limited rudder control ability and unpowered flight, it is very urgent to reduce the loss as much as possible and fully tap its maneuvering performance. Therefore, it is necessary to carry out closed-loop robust trajectory planning considering the feedback of the guidance control system. Up to now, the discussion on such problems is still insufficient. SUMMARY

[0006] The technical problem solved by the present application is:

[0007] In order to avoid the shortcomings of the prior art, the present application provides a probability uncertain closed-loop robust trajectory planning method considering actuator saturation, which aims to reduce the sensitivity of the scheme trajectory to uncertain factors and fully exert the limited control ability of the actuator while avoiding actuator saturation as much as possible.

[0008] In order to solve the above technical problems, the technical scheme adopted by the present application is:

[0009] A probability uncertain closed-loop robust trajectory planning method considering actuator saturation, characterized in that it comprises the following steps:

[0010] S1: For a simplified three-degree-of-freedom dynamics model of guided ammunition, ignoring the influence of random factors, a deterministic trajectory planning is carried out with the control energy optimization as the objective function, and the optimal control quantity is obtained as the initial value of the open-loop robust planning model;

[0011] S2: Considering the main uncertain factors in the flight process of the projectile, and modeling the deviation of the main uncertain factors from the nominal value as a form subject to a specific probability distribution to obtain an uncertain factor model; introducing the uncertain factors into the dynamics model to obtain a random dynamics system;

[0012] S3: Based on the polynomial chaos expansion (PCE) method, the uncertainty of the random dynamics system is quantified to obtain the statistical characteristics of the uncertain factors under the action of the specified control quantity, including:

[0013] S3.1: According to the specific expression of the robust planning problem, if the problem is determined at the end time, the random dynamics system is retained; if the problem is free at the end time, the substitution method is used to change the independent variable from time t to other state variables which change monotonically and are bounded in the random dynamics system or define auxiliary variables;

[0014] S3.2: According to the generalized PCE theory, the WSRSM based on Guass integral node sampling is used to carry out the dimension expansion conversion on the random dynamic system, so that a high-dimensional deterministic dynamic model with PCE coefficients as state variables is obtained;

[0015] S3.3: According to the maximum interaction limit truncation strategy, the high-order cross terms of the second order and above in the high-dimensional deterministic dynamic model are truncated to obtain a truncated model;

[0016] S3.4: The truncated model obtained in step S3.3 is integrated under the specified control quantity, the variation law of all PCE coefficients with respect to the independent variable is obtained, and the statistical characteristics of the state variables in the random dynamic system are obtained according to the generalized PCE theory;

[0017] S4: Based on the statistical characteristics of the state variables obtained in step S3, corresponding constraint conditions and objective functions are designed, and an open-loop robust planning model is established; the optimal control problem of the open-loop robust planning model is converted into a nonlinear programming problem for solving by using a direct method, so that the open-loop robust optimal trajectory of the projectile is obtained;

[0018] S5: Based on the open-loop robust optimal trajectory of step S4, a trajectory tracking algorithm is designed, the coupling between the scheme trajectory and the guidance control system is considered, and a closed-loop robust planning model is established;

[0019] S6: The uncertain factors in the uncertain factor model in step S2 are pulled apart, the trajectory tracking algorithm designed in step S5 is used to track the open-loop robust optimal trajectory obtained in step S4 to obtain the closed-loop control quantity of the projectile, the closed-loop control energy loss is introduced into the objective function in a weighted form, and the closed-loop robust optimal trajectory of the projectile is iteratively obtained.

[0020] Further technical solutions of the application: in step S1, for the simplified three-degree-of-freedom dynamic model of the guided projectile:

[0021]

[0022] In the formula, F x , F y , F z are respectively the total drag, the total lift and the total lateral force acting on the projectile, m is the mass of the projectile, g is the gravitational acceleration, θ and ψ are respectively the trajectory inclination angle and the trajectory deflection angle, x, y and z are respectively the range, the height and the lateral deviation of the projectile; ρ is the air density, V is the projectile speed, q=ρV 2 / 2 is the dynamic pressure, S is the characteristic area of the projectile, C x0 is the zero-lift-drag coefficient, k c is the total induced drag coefficient, are respectively the total lift coefficient and the total lateral force coefficient with respect to the angle of attack α and the angle of side slip β, for the axisymmetric shape,

[0023] Ignoring the influence of random factors, a deterministic trajectory planning is determined with an optimal energy control objective function:

[0024]

[0025] Where b and p are the corresponding boundary constraints and path constraints, t0 and t f are the initial and terminal time, respectively;

[0026] The optimal control problem of the above equation is solved by using a sequential quadratic programming method to obtain the optimal control quantity U * as the initial value of the robust planning model.

[0027] A further technical solution of the present application: in step S2, the random factors in the real flight environment are abstracted as aerodynamic parameters, air density and start-up point state, and are modeled as the deviation of the real value and the nominal value, and are assumed to be subject to mutually independent normal distribution:

[0028]

[0029] Where, respectively represent the real value of the zero-lift-drag coefficient, the total lift coefficient derivative and the total lateral force coefficient derivative, N d , N l , N l′ are random numbers subject to standard normal distribution N(0, 1), σ d , σ l , σ l′ respectively represent the degree of deviation of the above parameters from the reference value, N ρ is a random number subject to standard normal distribution N(0, 1), N V , N θ , N ψ , N x , N y , N z are random numbers subject to standard normal distribution N(0, 1), σ V , σ θ , σ ψ , σ x , σ y , σ z respectively represent the standard deviation of each state at the start-up point, V0, θ0, ψ0, t0, y0, z0 are the initial values of the projectile speed, trajectory inclination angle, trajectory deflection angle, time, projectile height and projectile lateral deviation, is the real value of the air density considering the uncertainty factor, are the real values of the start-up point state considering the uncertainty factor.

[0030] The further technical scheme of the present application is that in the step S3.1, the independent variables of the dynamic model are converted into the range x by considering the free end flight time moment:

[0031]

[0032] In the step S3.2, according to the generalized PCE theory, the WSRSM based on Guass integral node sampling is used to expand and convert the dynamic model, so as to obtain a high-dimensional deterministic dynamic model with PCE coefficients as state variables:

[0033]

[0034] In the formula, S PCE ∈R 396 is the PCE coefficient of all state variables, P PCE ∈R 8 is the PCE coefficient of the random parameter; H=(Ψ T W s Ψ) -1 Ψ T W∈R 396×792 is the regression coefficient matrix, W s ∈R 792×792 =diag(ω s ) is the weight matrix.

[0035] In the step S3.3, according to the maximum interaction limit truncation strategy:

[0036]

[0037] In the formula, λ=[λ1, λ2,..., λ d ] is a multi-exponent, A d，p is a set of multi-exponents, and ||λ||0 represents the rank of the multi-exponent.

[0038] The high-order cross terms of two orders or more in the full-order PCE model are truncated, and the model calculation amount is reduced.

[0039] In the step S3.4, the truncated model obtained in the step S3.3 is integrated under the specified control quantity, the change rule of all PCE coefficients with the independent variable is obtained, and the statistical characteristics of the state variables in the original random dynamic model are obtained according to the generalized PCE theory:

[0040]

[0041] In the formula, ξ∈R 10 is a d-dimensional cell array, Φ j (ξ) is an orthogonal polynomial function obtained by tensor product of φ j (ξ k ), and φj (ξ k ),k=1,2,...,d for each dimension of the uncertainty factor corresponding to the orthogonal basis function, S ij is the jth PCE coefficient of the ith state variable.

[0042] Further technical solutions of the application: in step S4, the corresponding constraint condition and objective function are designed, and an open-loop robust programming model is established:

[0043]

[0044] minJ o =D(y f )+D(z f ) (13)

[0045] In the formula, α max , β max are the maximum values of the attack angle α and the side slip angle β, V fmin is the minimum value of the terminal velocity, y T , z T are the y-axis and z-axis coordinate values of the target position; the operator E represents the mean value, and the operator D represents the variance;

[0046] The optimal control problem is converted into a nonlinear programming problem by using the direct shooting method, and the open-loop robust optimal trajectory of the projectile is obtained.

[0047] Further technical solutions of the application: in step S5, a trajectory tracking algorithm is designed:

[0048]

[0049] In the formula, and are the height and side deflection at the range x on the reference trajectory, is the open-loop control amount, k P , k I , k D respectively represent the proportional parameter, the integral parameter and the differential parameter, α c , β c are the control values of the attack angle α and the side slip angle β.

[0050] Further technical solutions of the application: in step S6, the closed-loop control energy loss is introduced into the objective function in a weighted form, and the closed-loop robust optimal trajectory of the projectile is obtained iteratively:

[0051]

[0052] In the formula, ω D is the weight of the terminal position scattered in the closed-loop performance index.

[0053] A computer system comprising: one or more processors; and computer readable storage media storing one or more programs for execution by the one or more processors, wherein the one or more programs, when executed by the one or more processors, cause the one or more processors to carry out the method described above.

[0054] A computer readable storage medium storing computer executable instructions which, when executed, perform the method described above.

[0055] A computer program product comprising computer executable instructions which, when executed, perform the method described above.

[0056] The present application has the following advantages:

[0057] The method for probabilistic uncertainty closed-loop robust trajectory planning considering actuator saturation provided by the present application has the following advantages compared with the prior art:

[0058] 1. The present application uses an accurate and efficient PCE method to quantify uncertainty, overcoming the problems of insufficient precision of nonlinear model description and sensitivity to initial value of traditional LAC methods; at the same time, the method of the present application performs maximum interaction limit base truncation on the full-order PCE model, which greatly reduces the dimension and calculation time of the problem while ensuring the precision of the model.

[0059] 2. The present application uses WSRSM based on Guass integral node sampling to transform the dynamic model. In the case of a certain number of sampling points, the Guass integral node and the corresponding weight are uniquely determined, avoiding the additional randomness introduced by the traditional Latin Hypercube Design (LHD) sampling; at the same time, unlike the traditional SRSM, WSRSM fully considers the difference in the influence of sample points on the regression process, which improves the accuracy of the PCE model without increasing the substantial calculation cost.

[0060] 3. The closed-loop robust trajectory planning method proposed by the present application couples the planning and tracking control of the scheme trajectory, fully considers the limited rudder control ability of guided ammunition, and reduces the sensitivity of the scheme trajectory to uncertain factors while avoiding actuator saturation as much as possible. BRIEF DESCRIPTION OF DRAWINGS

[0061] The accompanying drawings are included to provide a further understanding of the application and are incorporated in and constitute a part of this specification, illustrate embodiments of the application and together with the description serve to explain the principles of the application. In the drawings:

[0062] Figure 1is a probability uncertainty closed-loop robust trajectory planning method considering actuator saturation provided by the application.

[0063] Figure 2 is a probability uncertainty closed-loop robust trajectory planning method considering actuator saturation of the application. DETAILED DESCRIPTION

[0064] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application is further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application and do not limit the present application. In addition, the technical features involved in each embodiment of the application described below can be combined with each other as long as they do not conflict with each other.

[0065] The present application provides a probability uncertainty closed-loop robust trajectory planning method considering actuator saturation, as shown in Figure 1 The method comprises the following steps:

[0066] S1: For a simplified three-degree-of-freedom dynamic model of guided ammunition, the influence of random factors is ignored, and a deterministic trajectory planning is carried out with the optimal control energy as the objective function, and the optimal control quantity is obtained as the initial value of the subsequent robust planning algorithm.

[0067] S2: Considering the main uncertain factors in the flight process of the projectile, and modeling the deviation of the nominal value as a form subject to a specific probability distribution; the uncertain factors are introduced into the dynamic model to obtain a random dynamic system. For uncertain factors that are not suitable for probability description, they are regarded as unmodeled disturbances, which are compensated by the guidance control system.

[0068] S3: Considering the uncertain factor model established in step S2, the random dynamic system is quantified based on the Polynomial Chaos Expansion (PCE) method, and the statistical characteristics under the action of the specified control quantity are obtained, mainly including:

[0069] S3.1: According to the specific expression of the robust planning problem, if the problem is determined at the end time, the random dynamic system is retained; if the problem is free at the end time, the substitution method is used to change the independent variable from time t to other state variables that change monotonically and are bounded in the random dynamic system or define auxiliary variables.

[0070] S3.2: According to the generalized PCE theory, the weighted stochastic response surface method (WSRSM) based on Guass integral node sampling is used to expand the dimension of the random dynamic system, and a high-dimensional deterministic dynamic model with PCE coefficients as state variables is obtained.

[0071] S3.3: According to the maximum interaction limit truncation strategy, the high-order cross terms with less influence (second order and above) in the full-order PCE model are truncated to reduce the model calculation amount.

[0072] S3.4: The truncated model obtained in step S3.3 is integrated under the specified control quantity to obtain the variation law of all PCE coefficients with respect to the independent variable, and the statistical characteristics of the state variables in the random dynamic system are obtained according to the generalized PCE theory.

[0073] S4: On the basis of step S3, the corresponding constraint conditions and objective function are designed according to the problem requirements, and an open-loop robust planning model is established. The optimal control problem is converted into a nonlinear programming problem for solving by using the direct method, and the open-loop robust optimal trajectory of the projectile is obtained.

[0074] S5: Trade-off between accuracy and calculation cost, design a simple and effective trajectory tracking algorithm, on the basis of step S4, consider the coupling of the scheme trajectory and the guidance control system, establish a closed-loop robust planning model.

[0075] S6: The uncertainty factors modeled in step S2 are pulled apart to simulate a relatively harsh real environment, and the algorithm designed in step S5 is used to track the open-loop robust optimal trajectory obtained in step S4 to obtain the closed-loop control quantity of the projectile. The closed-loop control energy loss is introduced into the objective function in a weighted form, and the closed-loop robust optimal trajectory of the projectile is iteratively obtained.

[0076] In order to make the person skilled in the art better understand the present application, the present application will be described in detail below in conjunction with specific examples.

[0077] Example 1

[0078] The embodiment of the present application provides a probability uncertain closed-loop robust trajectory planning method considering actuator saturation, and the specific steps are as follows:

[0079] S1: For the simplified three-degree-of-freedom dynamic model of guided ammunition:

[0080]

[0081] In the formula, F x , F y , F zrespectively, m is the mass of the projectile, g is the gravitational acceleration, θ and ψ are the trajectory inclination and trajectory deflection, respectively, x, y and z are the range, altitude and lateral deviation of the projectile, ρ is the air density, V is the projectile velocity, q = ρV 2 is the dynamic pressure, S is the characteristic area of the projectile, C x0 is the zero-lift drag coefficient, k c is the total induced drag coefficient, respectively, are the partial derivatives of the total lift coefficient and the total side force coefficient with respect to the angle of attack α and the angle of sideslip β, for axisymmetric shapes,

[0082] Ignoring the influence of random factors, the deterministic trajectory planning is determined with the optimal control energy as the objective function:

[0083] Let the combination of state variables be X = [V, θ, ψ, x, y, z] T , and the combination of control variables be U = [α, β] T , then equation (17) can be expressed as:

[0084]

[0085] where t is time.

[0086] The optimal control problem equation (19) is solved by using the sequential quadratic programming method to obtain the optimal control variable U * as the initial value of the subsequent robust planning algorithm.

[0087]

[0088] In the formula, b and p are the corresponding boundary constraints and path constraints, respectively, t0 and t f are the initial and terminal time, respectively.

[0089] S2: Considering the main uncertain factors in the flight process of the projectile, and modeling the deviation from the nominal value as a form subject to a certain probability distribution. For uncertain factors that are not suitable for probability description, they are regarded as unmodeled disturbances, which are compensated by the guidance control system.

[0090] S2.1: Aerodynamic parameter deviation

[0091] Considering the machining error of the projectile body and the difference of the trajectory environment, it is assumed that the deviation of the aerodynamic parameters is subject to independent normal distribution:

[0092]

[0093] In the formula, respectively, represent the true value of the zero-lift drag coefficient, the derivative of the total lift coefficient, and the derivative of the total side force coefficient, Nd , N l , N l′ are random numbers obeying standard normal distribution N(0, 1), and σ d , σ l , σ l′ respectively represent the degree of deviation of the above parameters from the reference value.

[0094] S2.2: Meteorological environment deviation

[0095] The difference between the real meteorological environment and the standard meteorological environment mainly lies in the deviation of the wind field, air temperature and atmospheric density. The influence of the wind field on trajectory calculation is not suitable for probability description, which can be regarded as an unmodeled random force and compensated by the guidance control system. The air temperature mainly affects the calculation of sound speed, and then affects the flight Mach number, and the change of the Mach number will lead to the change of the aerodynamic parameters. Therefore, the deviation of the air temperature can be attributed to the deviation of the aerodynamic parameters. In summary, the meteorological environment deviation is modeled as the deviation of the atmospheric density. The statistical characteristics of the deviation of the real atmospheric density from the standard value are related to the altitude y, and the ratio of the standard deviation to the standard atmospheric density can be approximated as:

[0096]

[0097] Therefore, the real atmospheric density can be represented as

[0098]

[0099] In the formula, N ρ are random numbers obeying standard normal distribution N(0, 1).

[0100] S2.3: Deviation of the state at the start point

[0101] After the projectile is launched, the state value at the start point will inevitably deviate from the design value of the scheme trajectory under the action of various uncertain factors. The present application assumes that the state deviation obeys a normal distribution independent of each other, and the scheme design value at the start point is X0=[V0, θ0, ψ0, x0, y0, z0] T The real state at the start point can be represented as:

[0102]

[0103] In the formula, N V , N θ , N ψ , N x , N y , N z are random numbers obeying standard normal distribution N(0, 1), and σ V , σ θ , σ ψ , σ x,σ y ,σ z denote the standard deviation of each state at the activation point, respectively.

[0104] S3: Considering the uncertainty model established in step S2, the uncertainty of the random dynamic system is quantified based on the Polynomial Chaos Expansion (PCE) method, and the statistical characteristics under the action of the specified control quantity are obtained, including:

[0105] S3.1: According to the specific expression of the robust planning problem, if the end time is determined, the original dynamic model is retained; if the problem is free at the end time, the substitution method is used to replace the independent variable from time t to other state variables that monotonically change and are bounded in the system or define auxiliary variables.

[0106] For generality, it is assumed that the end time of the trajectory is free and unknown, and the target position to be attacked is fixed. During flight, the range x of the projectile monotonically increases and is bounded. Therefore, the dynamic model (17) can be transformed into:

[0107]

[0108] Let the combination of the transformed system state variables be S = [V, θ, ψ, t, y, z] T , and the combination of the control variables still be U = [α, β] T , then equation (24) can be expressed as:

[0109]

[0110] Similarly, equation (23) can be transformed into:

[0111]

[0112] S3.2: According to the generalized PCE theory, the Weighted Stochastic Response Surface Method (WSRSM) based on Guass integral node sampling is used to expand the dimension of the random trajectory dynamic equation set, and a high-dimensional deterministic dynamic model with PCE coefficients as state variables is obtained.

[0113] The main uncertainty factors during the flight of the projectile can be summarized into two categories: parameter uncertainty and initial state uncertainty. Among them, the dimension of parameter uncertainty d P = 4, and the dimension of initial state uncertainty d S = 6, therefore, the uncertainty factor dimension of the entire random dynamic system is d = 10. Let the random parameters be P = [σ d ,σ l,σ l′ ,σ ρ ] T and introduce it into the kinetic system shown in equation (25), the stochastic kinetic model can be expressed as:

[0114]

[0115] The non-intrusive PCE method based on Guass integral node sampling and WSRSM is adopted to transform equation (27) into a high-dimensional deterministic kinetic model with PCE coefficients as state variables. Taking the PCE order p = 2, the number of PCE coefficients corresponding to each state variable after expansion is:

[0116]

[0117] The number of sampling points for regression is:

[0118] n s = 2(P+1) = 132 (29)

[0119] According to the generalized chaotic polynomial theory, each state variable in equation (27) can be expanded into a polynomial form:

[0120]

[0121] where ξ ∈ R 10 is a d-dimensional cell array, where each element is a combination of n s Gaussian integral sampling points corresponding to uncertain factors. Let the monomial orthogonal basis function corresponding to each dimension of the uncertain factor be φ j (ξ k ), k = 1, 2,..., d, then Φ j (ξ) is the orthogonal polynomial function obtained by taking the tensor product of φ j (ξ k ). S ij is the jth PCE coefficient of the ith state variable, and the total number is:

[0122] n PCE = d S (P+1) = 396 (31)

[0123] For random parameters subject to normal distribution, a 1st order, 1-dimensional orthogonal polynomial model can accurately describe its uncertainty. Therefore, each random parameter can be expanded into the form of the sum of 2 terms:

[0124] P i = p i0 Φ0(ξ P )+p i1 Φ1(ξ P), i = 1, 2, ..., d P (32)

[0125] In the formula, ξ P ∈R 4 For d P An array of dimensional cells, where each element is the n-th degree of uncertainty. s A combination of Gaussian integral sampling points.

[0126] Thus, the stochastic system shown in equation (27) is transformed into a high-dimensional deterministic system:

[0127]

[0128] In the formula, S PCE ∈R 396 P represents the PCE coefficients for all state variables. PCE ∈R 8 Let F be the PCE coefficients of the random parameters, F∈R 792 To make n s Substituting each sample point into the true response value in the original random system equation (27), Ψ∈R 792×396 The coefficient matrix of the transformed deterministic system is described in detail below:

[0129] Let Ω(ξ) i )=[Φ0(ξ i ),Φ1(ξ i ),...Φ P (ξ i )], i=1,2,...,n s Define the intermediate matrix:

[0130]

[0131] In the formula, the symbol Let the Kronecker product be represented, then:

[0132]

[0133] Considering the different importance of each sample point, the weight vector is denoted as:

[0134]

[0135] In the formula, ω is n s The weights corresponding to each sample point.

[0136] Therefore, according to WSRSM, the overdetermined system of equations (17) can be solved by weighted least quadratic regression:

[0137]

[0138] H = (Ψ T W s Ψ) -1 Ψ T W∈R 396×792 is the regression coefficient matrix, W s ∈R 792×792 = diag(ω s ) is the weight matrix.

[0139] S3.3: According to the maximum interaction limit truncation strategy, the high-order cross terms with less influence (two orders or more) in the full-order PCE model are truncated to reduce the model calculation amount.

[0140] The converted deterministic system state variables are as high as 396, and the number of Gaussian integral sampling points used for regression is also as high as 132. Under this problem scale, the calculation time of a single uncertainty propagation is barely acceptable, however, in the subsequent robust planning of the gliding trajectory, the iterative process will repeatedly call the uncertainty propagation module, and the convergence speed of the optimization algorithm will obviously decrease when the number of variables is too large, which will cause a significant reduction in calculation efficiency. It is known that the total number of orthogonal polynomials increases exponentially with the increase of the dimension d of the uncertainty factor, but in many problems, the terms of the orthogonal polynomials are not equally important, and in most cases, the important terms in the expansion only involve a few variables. In most actual engineering systems, the output response is mainly affected by the input variables and the low-order cross terms, and the influence of the high-order cross terms is small. Based on the above analysis, the maximum interaction limit truncation strategy is adopted to directly remove the high-order polynomials, so as to reduce the calculation amount.

[0141] In the full-order PCE model formula (30), the multivariate orthogonal polynomial is constructed by direct tensor product, and when the PCE order is p, the following needs to be met:

[0142]

[0143] In the formula, λ = [λ1, λ2,..., λ d ] is a multi-index, A d,p is a set of multi-indices, λ i is the order of the monomial orthogonal polynomial basis corresponding to the j-th variable in the multivariate orthogonal polynomial, and the operator |·|1 represents the 1-norm.

[0144] The main idea of the maximum interaction limit truncation is to select low-rank λ from the multi-index set A d,p defined in formula (38) to form a subset A d,p,r , so that λ not only satisfies ||λ||1≤p, but also satisfies ||λ||0≤r.

[0145]

[0146] where || λ ||0represents the rank of the multi-exponential.

[0147] For the present application, d = 10 and p = 2. Therefore, r = 1 is taken according to equation (23). After removing the high-order cross terms, the number of PCE coefficients P + 1 of each dimension random variable is reduced from 66 in full order to 11. Therefore, the total number of state variables of the deterministic dynamic system after the expansion transformation is reduced from 396 to 66, and the number of Gaussian integral sampling points for regression is reduced from 132 to 22, with a reduction of 83.3%.

[0148] S3.4: Integrate the truncated model obtained in step S3.3 under the specified control quantity, obtain the variation of all PCE coefficients with the independent variable, and obtain the statistical characteristics of the state variables in the original random dynamic model according to the generalized PCE theory.

[0149] Numerical integral operation is performed on equation (37) to obtain the variation of all PCE coefficients with the range x. The mean μ and standard deviation σ of each state variable in the original random system can be directly obtained through the PCE coefficients:

[0150]

[0151] S4: On the basis of step S3, combine the problem requirements to design corresponding constraint conditions and objective functions, and establish an open-loop robust programming model. The optimal control problem is converted into a nonlinear programming problem for solution by using the direct method, and the open-loop robust optimal trajectory of the projectile is obtained.

[0152] S4.1: Constraint conditions

[0153] After considering the influence of uncertain factors, there are countless results (bounded) of the flight trajectory under the same open-loop control input. Therefore, the focus of open-loop robust programming is not the specific value of the state, but its statistical law, and the statistical moments of the state variables should also be considered in the constraint conditions.

[0154] S4.1.1: Boundary constraints

[0155] The state deviation at the starting point obeys a normal distribution, and therefore the constraint of the mean of the initial state is:

[0156]

[0157] To ensure the final striking accuracy and effect, the constraint of the mean of the terminal state is:

[0158]

[0159] S4.1.2: Process constraints

[0160] Considering the limited steering capability of the projectile, the control variable constraints during flight are:

[0161]

[0162] S4.2: Performance index

[0163] For the scheme trajectory, the dispersion of the terminal position is usually taken as the standard to measure its sensitivity to uncertainty, and less attention is paid to the dispersion of other terminal states and the state envelope during the flight of the projectile. Therefore, the performance index of the open-loop robust planning is designed as:

[0164] minJ o = D(y f ) + D(z f ) (44)

[0165] In the above formula, the operator E represents the mean value, and the operator D represents the variance.

[0166] S5: Trade off accuracy and computational cost, design a simple and effective trajectory tracking algorithm, on the basis of step S4, consider the coupling of the scheme trajectory and the guidance control system, and establish a closed-loop robust planning model.

[0167] Considering that the guidance module will be called repeatedly during the iterative process of the closed-loop planning algorithm, a simple and effective trajectory tracking algorithm is needed. The traditional LQR algorithm needs to linearize the model near the reference trajectory, and when the deviation is large, a large tracking error will be generated. At the same time, the LQR algorithm needs to solve the Riccati equation, which increases the difficulty of solving the algorithm. The present application adopts a PID controller with an analytical expression and independent of the model to track the scheme trajectory. Let the height and lateral deviation at the range x on the reference trajectory be and The control variable is The position error between the projectile and the reference trajectory can be expressed as:

[0168]

[0169] The closed-loop control variable after feedback correction can be expressed as:

[0170]

[0171] In the formula, k P , k I , k D represent the proportional parameter, integral parameter and differential parameter respectively.

[0172] The dynamic model and related constraints of the closed-loop robust planning are the same as those in the open-loop planning, which will not be repeated here.

[0173] The energy loss of the closed-loop control during the projectile's flight is:

[0174]

[0175] The performance metrics for closed-loop robust programming can then be designed as follows:

[0176] minJ c =ω D J o +E c (48)

[0177] In the formula, ω D The weights of the end positions distributed in the closed-loop performance index.

[0178] Observing equation (48), it can be seen that closed-loop robust trajectory planning is a multi-objective optimal control problem. Considering J o With E c The magnitudes of these differences are significant. To balance the planning focus, this invention uses the deterministic energy-optimal trajectory without considering random disturbances as the benchmark trajectory, and renders the performance indicators dimensionless. Let the terminal position dispersion of the benchmark trajectory after uncertainty propagation be denoted as... Controlling energy loss to The performance indicators then become:

[0179]

[0180] S6: The uncertainties modeled in step S2 are biased to simulate a relatively harsh real environment. The algorithm designed in step S5 is used to track the open-loop robust optimal trajectory obtained in step S4 to obtain the closed-loop control quantity of the projectile. The closed-loop control energy loss is introduced into the objective function in a weighted form, and the projectile's closed-loop robust optimal trajectory is obtained iteratively.

[0181] like Figure 2 As shown, considering the case where all uncertainties are skewed to the half-limit (±1.5σ values ​​of each random parameter in this invention), tracking the robust optimal trajectory can obtain closed-loop feedback of the control quantity, thereby obtaining the control energy loss during the projectile's flight (according to simulation experience and measured data, the case where all uncertainties are skewed to the half-limit is quite demanding; at this point, trajectory tracking is relatively difficult, and control energy consumption is also large. Adopting such a conservative approach in the planning process can basically ensure the smooth operation of closed-loop guidance during actual projectile flight). Introducing the control energy loss as a penalty function into the planning performance index allows us to consider the coupling and matching characteristics of the proposed trajectory and guidance system during the iteration process, thus planning a closed-loop robust optimal trajectory that balances robustness and feasibility.

[0182] The above merely illustrates the specific embodiments of the present application, but the protection scope of the present application is not limited thereto, and any skilled person in the art can easily think of various equivalent modifications or replacements within the technical scope disclosed by the present application, and these modifications or replacements shall be covered within the protection scope of the present application.

Claims

1. A probabilistic uncertain closed-loop robust trajectory planning method considering actuator saturation, characterized in that, The method comprises the following steps: S1: for a simplified three-degree-of-freedom dynamic model of guided ammunition, ignoring the influence of random factors, performing deterministic trajectory planning with optimal control energy as an objective function to obtain optimal control as an initial value of an open-loop robust planning model; For a simplified three-degree-of-freedom dynamic model of guided ammunition: (1) wherein, , , are the total drag, total lift and total side force acting on the projectile, is the mass of the projectile, is the acceleration of gravity, , are the trajectory angle of attack and the trajectory angle of sideslip, , , are the range, the height and the sideslip of the projectile; is the air density, is the velocity of the projectile, is the dynamic pressure, is the characteristic area of the projectile, is the zero-lift-drag coefficient, is the total induced-drag coefficient, , are the partial derivatives of the total-lift and total-side-force coefficients with respect to the angle of attack and the angle of sideslip , respectively, for axisymmetric shapes, ; Ignoring the influence of random factors, performing deterministic trajectory planning with optimal control energy as an objective function: (2) wherein and are the respective boundary constraints and path constraints, and are the initial and final times, respectively, is the combination of state variables, is the combination of control variables; The optimal control problem is solved by using the sequential quadratic programming method, and the optimal control variable is obtained as the initial value of the robust programming model; S2: considering main uncertain factors in the flight process of the projectile, modeling deviations of the main uncertain factors from nominal values as normal distribution forms; and introducing the uncertain factors into the dynamic model to obtain a random dynamic system; S3: based on a polynomial chaos expansion (PCE) method, quantifying uncertainty of the random dynamic system to obtain statistical characteristics of the uncertain factors under the action of specified control, including: S3.1: According to the specific expression of the robust planning problem, if the end time of the problem is determined, the stochastic dynamic system is retained; if the end time of the problem is free, the substitution method is adopted, and the independent variable is changed from time t to other state variables which are monotonically changing and bounded in the stochastic dynamic system or auxiliary variables are defined; S3.2: according to generalized PCE theory, performing dimension expansion conversion on the random dynamic system by using WSRSM based on Guass integral node sampling to obtain a high-dimensional deterministic dynamic model with PCE coefficients as state variables; S3.3: according to a maximum interaction limit truncation strategy, truncating high-order cross terms of the second order and above in the high-dimensional deterministic dynamic model to obtain a truncated model; S3.4: integrating the truncated model obtained in step S3.3 under specified control to obtain variation rules of all PCE coefficients with respect to independent variables, and obtaining statistical characteristics of state variables in the random dynamic system according to generalized PCE theory; S4: based on the statistical characteristics of the state variables obtained in step S3, designing corresponding constraint conditions and an objective function to establish an open-loop robust planning model; converting the optimal control problem of the open-loop robust planning model into a nonlinear programming problem for solving by using a direct method to obtain an open-loop robust optimal trajectory of the projectile; S5: designing a trajectory tracking algorithm based on the open-loop robust optimal trajectory in step S4, considering coupling between a scheme trajectory and a guidance control system to establish a closed-loop robust planning model; Designing a trajectory tracking algorithm: (14) wherein and are the height and the sideslip at the range on the reference trajectory, respectively, x are the height and the sideslip at the range on the reference trajectory, respectively, is the open-loop control, , , are the proportional, integral and derivative parameters, respectively, , are the control values of the calibrated angle of attack and the calibrated sideslip angle , respectively. S6: pulling the uncertain factors in the uncertain factor model in step S2, tracking the open-loop robust optimal trajectory obtained in step S4 by using the trajectory tracking algorithm designed in step S5 to obtain closed-loop control of the projectile, introducing closed-loop control energy loss into the objective function in a weighted form, and iteratively obtaining a closed-loop robust optimal trajectory of the projectile.

2. The probabilistic uncertain closed-loop robust trajectory planning method considering actuator saturation according to claim 1, wherein, In step S2, random factors in a real flight environment are abstracted as aerodynamic parameters, air density and start control points, and are modeled as deviations of real values from nominal values, and are assumed to be subject to mutually independent normal distributions. (3) (4) (5) wherein, , , respectively represent the true value of zero-lift coefficient, total lift coefficient derivative and total side force coefficient derivative, , , are random numbers obeying standard normal distribution N(0, 1), , , respectively represent the degree of deviation of the above parameters from the reference value, are random numbers obeying standard normal distribution N(0, 1), , , , , , are random numbers obeying standard normal distribution N(0, 1), , , , , , respectively represent the standard deviation of each state at the starting point, , , , , , are respectively the initial value of the projectile velocity, the trajectory inclination angle, the trajectory deflection angle, the time, the projectile height, and the projectile side deviation, is the true value of air density considering the uncertainty factor, , , , , , are respectively the true value of the starting point state considering the uncertainty factor. 3.The probabilistic uncertain closed-loop robust trajectory planning method considering actuator saturation according to claim 2, wherein, In said step S3.1, considering the end flight time instant free, the arguments of the dynamics model are transformed into range x : (6) In step S3.2, according to generalized PCE theory, WSRSM based on Guass integral node sampling is used to perform dimension expansion conversion on the dynamic model to obtain a high-dimensional deterministic dynamic model with PCE coefficients as state variables. (7) wherein PCE for all state variables, PCE for random parameters; regression coefficient matrix, weight matrix; In step S3.3, according to a maximum interaction limit truncation strategy: (8) wherein is a polyexponent, is a set of polyexponents, denotes the rank of the polyexponent; Truncating high-order cross terms of the second order and above in the full-order PCE model to reduce model calculation amount; In the step S3.4, the truncated model obtained in the step S3.3 is integrated under the specified control amount to obtain the variation law of all PCE coefficients with the independent variable, and the statistical characteristics of the state variable in the original random dynamic model are obtained according to the generalized PCE theory: (9) In the formula, for d dimensional cell array, To Find the orthogonal polynomial function obtained by the tensor product. These are the univariate orthogonal basis functions corresponding to the uncertainties in each dimension. For the first i The first state variable j PCE coefficients. 4.The probabilistic uncertain closed-loop robust trajectory planning method considering actuator saturation according to claim 3, wherein, In the step S4, corresponding constraint conditions and objective functions are designed to establish an open-loop robust programming model: (10) (11) (12) (13) wherein , are maximum values of the effective angle of attack and the effective side slip angle , is the minimum value of the terminal speed, , are the y-axis and z-axis coordinate values of the target position; the operator E represents the mean value, and the operator D represents the variance. The optimal control problem is converted into a nonlinear programming problem for solving by using a direct shooting method, and an open-loop robust optimal trajectory of the projectile is obtained.

5. The probabilistic uncertain closed-loop robust trajectory planning method considering actuator saturation according to claim 4, wherein, In the step S6, the closed-loop control energy loss is introduced into the objective function in a weighted form, and an iterative closed-loop robust optimal trajectory of the projectile is obtained: (15) (16) In the formula, is the weight of the end position scattered in the closed-loop performance index.

6. A computer system, characterized by The method comprises the following steps: One or more processors, a computer readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the method of claim 1.

7. A computer-readable storage medium, characterized in that Computer executable instructions are stored, and the instructions are executed to implement the method of claim 1.

8. A computer program product, characterised in that Computer executable instructions are included, and the instructions are executed to implement the method of claim 1.

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