A time delay optimization method for nonlinear time-delay control system of aircraft

Abstract

The application discloses a time delay optimization method for a nonlinear time delay control system of an aircraft, and the method comprises the following steps: constructing a dynamic equation of a general nonlinear time delay system of the aircraft, introducing a Fourier transform to fit a time delay term, and obtaining a time delay term with a truncated series of N; performing two-angle sum transformation on the time delay term with the truncated series of N and solving by using a time domain minimum residual method to obtain a semi-analytical form solution; introducing Tikhonov regularization and a confidence domain limit, performing time delay iterative adjustment on an objective function of a minimum value optimization problem, and outputting an optimal time delay. The application can control a gap / time delay wing flutter system to a steady state in the shortest time, improve instantaneous stability of the aircraft, and reduce loss of the aircraft. The application can be widely applied to the field of aircraft control technology as a time delay optimization method for a nonlinear time delay control system of an aircraft.

Description

Technical Field

[0001] This invention relates to the field of aircraft control technology, and in particular to a time delay optimization method for nonlinear time-delay control systems for aircraft. Background Technology

[0002] As modern aircraft develop towards higher maneuverability, higher integration, and multi-functionality, the nonlinear characteristics of their flight control systems are becoming increasingly prominent. Simultaneously, time delays (time lags) arising from sensors, processors, and actuators are prevalent. The coupling effect of nonlinearity and time delays poses a severe challenge to the control performance and stability of aircraft, potentially leading to performance degradation, oscillations, or even flutter and other catastrophic consequences. Delay feedback control, as an effective control strategy, suppresses chaos or improves dynamic performance by stabilizing the unstable periodic trajectory of the system. However, its success hinges on the precise determination of the optimal time delay parameters. Existing methods all have inherent limitations in solving such problems: numerical simulation methods (such as the Runge-Kutta method) rely on inefficient trial-and-error or grid search, resulting in high computational costs and difficulty in guaranteeing global optimality; approximate analytical methods, including traditional perturbation methods (such as the LP method and multi-scale methods) and homotopy analysis (HAM), also have inherent flaws. While homotopy analysis theoretically overcomes the dependence of perturbation methods on "small parameters," thus enabling the handling of some strongly nonlinear problems, a common drawback of these methods is that their mathematical construction requires the system function to have good smoothness. Therefore, when dealing with non-smooth nonlinear elements such as gaps and hysteresis commonly found in aircraft, they must also undergo approximation, introducing model errors. Furthermore, these methods still rely on external iteration or trial and error to determine the optimal time delay, resulting in low efficiency. Frequency domain analysis methods (such as the harmonic balance method (HBM) and incremental harmonic balance method (IHB)) are not only cumbersome in derivative calculations when dealing with complex models, but also require smoothing approximations when dealing with non-smooth nonlinear elements such as gaps and hysteresis, introducing model errors and affecting the accuracy of the solution. In summary, existing technologies suffer from narrow applicability, computational complexity, or insufficient accuracy when dealing with strongly nonlinear time-delay control systems for aircraft containing non-smooth elements. Summary of the Invention

[0003] To address the aforementioned technical problems, the present invention aims to provide a time delay optimization method for a nonlinear time-delay control system for aircraft, which enables the gap / hysteresis wing flutter system to be controlled to a steady state in the shortest possible time, thereby improving the instantaneous stability of the aircraft and reducing aircraft losses.

[0004] The first technical solution adopted in this invention is: a time delay optimization method for a nonlinear time-delay control system for aircraft, comprising the following steps:

[0005] The dynamic equations of a general nonlinear time-delay system for an aircraft are constructed, and the time-delay term is fitted using Fourier transform, resulting in...

[0006] In the above equation, R represents the residual, N represents the cutoff series, and M represents the mass matrix of the dynamic system. Let N represent the acceleration vector with a cutoff order of N, and C represent the damping matrix of the dynamic system. Let x represent the velocity vector with truncation order N, K represent the stiffness matrix of the dynamic system, and x represent the velocity vector with truncation order N. N K represents a displacement vector with a truncation order of N. 3 This represents the cubic nonlinear stiffness coefficient matrix of the dynamic system. Let x represent the cubes of the three components of a displacement vector with a truncation series of N, respectively, and let F represent the time delay coefficient matrix. N (t-τ) represents the time delay term with truncation series N.

[0007] Furthermore, the specific expression of the objective function of the minimum optimization problem is as follows:

[0008]

[0009] In the above formula, a * This represents the optimal value of the variable representing the objective function. Let R(a,t) represent the nonlinear time-delay optimization objective function, R(a,t) represent the residual vector of the system, A represent the feasible region of the optimal variable value a of the unknown objective function, and t represent time.

[0010] Furthermore, the step of introducing Tikhonov regularization and confidence region constraints to iteratively adjust the time delay of the objective function of the minimum optimization problem and output the optimal time delay specifically includes:

[0011] The objective function of the minimum optimization problem is linearized at discrete time points to obtain an approximate linear objective function;

[0012] Tikhonov regularization is introduced to regularize the approximate linear objective function, resulting in a regularized objective function.

[0013] The regularization parameters of the regularized objective function are determined by the L-curve curve method, and a confidence region constraint is introduced. The time delay of the general nonlinear time delay system of the aircraft is iteratively adjusted in combination with the regularization parameters until the preset conditions are met, and the optimal cutoff coefficient is obtained.

[0014] The optimal cutoff coefficient is used as a semi-analytical solution for the general nonlinear time-delay system of the aircraft to optimize the dynamic indicators of the aircraft system and output the optimal time delay.

[0015] Furthermore, the expression for the confidence region restriction is as follows:

[0016]

[0017] In the above formula, The above formula represents the consistency index. Indicators of consistency Let represent a set of known parameters that satisfy the feasible region A, and let δa represent the reasonable iteration update amount found. Indicates substituting known parameters The original nonlinear time-delay objective function, Indicates substituting known parameters And the original nonlinear time-delay objective function is improved by adding a reasonable number of iterations. This represents the linearized objective function approximated at discrete time points with zero iterations. This represents the linearized objective function approximated at discrete time points. Residual increment in the following form This represents the residual increment after increasing the number of reasonable iterations. This represents the first-order response sensitivity matrix.

[0018] The beneficial effects of the method of this invention are as follows: This invention constructs the dynamic equation of a general nonlinear time-delay system of an aircraft, introduces Fourier transform to fit the time-delay term, obtains a time-delay term with a truncated series of N, and uses the time-domain minimum residual method to transform the solution problem of the nonlinear time-delay differential equation into an optimization problem. The time-delay parameter itself is used as the optimization variable and iteratively solved together with the system response. Furthermore, the time-delay term with a truncated series of N is subjected to a two-angle sum transformation and solved by the time-domain minimum residual method to obtain a semi-analytical solution. Finally, Tikhonov regularization and confidence region constraints are introduced to iteratively adjust the time delay of the objective function of the minimum optimization problem, outputting the optimal time delay. Combined with Tikhonov regularization to handle ill-conditioned problems, and the introduction of a consistency index to control linearization error, the gap / hysteresis wing flutter system is controlled to a steady state in the shortest time, improving the instantaneous stability of the aircraft and reducing aircraft losses. Attached Figure Description

[0019] Figure 1 This is a flowchart illustrating the steps of a time delay optimization method for a nonlinear time delay control system for aircraft according to the present invention.

[0020] Figure 2 This is a schematic diagram of an aircraft model provided in a specific embodiment of the present invention. Detailed Implementation

[0021] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments. The step numbers in the following embodiments are only for ease of explanation and do not limit the order of the steps. The execution order of each step in the embodiments can be adapted according to the understanding of those skilled in the art.

[0022] Reference Figure 1 as well as Figure 2 This invention provides a time delay optimization method for nonlinear time-delay control systems for aircraft, the method comprising the following steps:

[0023] S100. Construct the dynamic equations of a general nonlinear time-delay system for an aircraft, and introduce Fourier transform to fit the time-delay term to obtain a time-delay term with a truncated series of N.

[0024] In this embodiment, the dynamic equations of the gap / hysteresis wing flutter system are as follows:

[0025]

[0026] In the above system of equations:

[0027] x=[ξ,α,y] T

[0028]

[0029]

[0030] In the above formula, e j f j g j Let ξ and γ represent system parameters, and x = [ξ, α, y]. T Let τ represent a variable with three degrees of freedom. j This represents the time delay parameter, with an initial value of x(0) = x0. It has already been provided.

[0031] As a preferred embodiment of this method, a Fourier series expansion is introduced to fit the time delay term x(t-τ), and the fitting formula is as follows:

[0032]

[0033] In the above formula, x j (t-τ) represents the j-th term of the time delay term vector. b jk ,c jk Let ω(t-τ) represent the coefficients of the trigonometric infinite series after Fourier transform, k represent the term number after the trigonometric series expansion, τ represent the time delay, t represent time, and ω(t-τ) represent the unknown frequency of the system containing τ.

[0034] in, T is x j The minimum period of (t-τ), b jk ,c jk It can be calculated using the following integral formula:

[0035]

[0036] Furthermore, a set of cutoff coefficients and time delay terms with a cutoff series number of N are obtained, which are expressed as:

[0037]

[0038] In the above formula, Let j represent the j-th term of the time-delay term vector with truncation series N, where N represents the truncation series. b jk ,c jk Let represent the coefficients of the truncated series, τ represent the time delay, t represent time, k represent the ordinal number of the truncated series, and ω(t-τ) represent the unknown frequency of the system containing τ.

[0039] S200. The solution in semi-analytic form is obtained by performing a two-angle sum transformation on the time delay term with truncation series N and solving it using the time-domain minimum residual method.

[0040] S210. Perform a two-angle sum transformation on the time delay term with truncation series N to obtain the time delay term with separated time delay parameters;

[0041] In this embodiment, the time delay term x with a series number of N is truncated. N Performing a two-angle sum transformation on (t-τ) yields the time-delay term with separated time-delay parameters. An example operation is shown below:

[0042]

[0043] The above equation α k ,β k This represents the coefficients of the Fourier series containing the time delay τ.

[0044] Clearly, this method has a significant advantage in separating time-delay parameters. Other methods, such as the perturbation method, require many complex steps and operations to achieve the same goal.

[0045] S220. Substitute the time delay term obtained by separating the time delay parameters into the dynamic equation of the general nonlinear time delay system of the aircraft to construct the time delay dynamic equation obtained by separating the time delay parameters.

[0046] In this embodiment, the Time-Domain Minimum Residual Method (TMRM) is used to solve the time-delay dynamic equations with separated time-delay parameters. The specific solution process is as follows:

[0047]

[0048] In the above formula, M represents the mass matrix of the dynamic system, and C represents the damping matrix of the dynamic system.

[0049] S230. Based on the separation of time delay parameters, the solution of the time delay dynamic equation is set to be expandable into an infinite series form, and an infinite series solution is established.

[0050] S240. Expand the infinite series solution by differential operations to obtain the series expressions for velocity and acceleration of the time-delay dynamic equation;

[0051] S250. The infinite series solution is truncated and combined with the series expressions of velocity and acceleration in the time-delay dynamics equation. The residual is then defined to construct the objective function of the minimum optimization problem.

[0052] In this embodiment, the above equations are assumed to have solutions that can be expanded into an infinite series of the following form:

[0053]

[0054] Where {χ k (t)|k=0,1,2…} is a basis function and {a k By solving for the first and second derivatives of the infinite series with respect to time, corresponding to the coefficients of the basis functions |k=0,1,2…}, the velocity and acceleration of the system can be expressed as:

[0055]

[0056] Let x(t) be the above. Substituting these into the dynamic equations, the original equations are zero over the entire time domain t∈[0,+∞]. However, the series solutions cannot take infinite terms. Therefore, the above solutions are truncated, resulting in a set of finite series solutions with a truncated series of number N. These solutions are then substituted into the original dynamic equations. The residual is defined as:

[0057]

[0058] Choose an appropriate basis function χ k (t) and corresponding coefficient a k Afterwards, the residuals will be as small as possible in the time domain, so the residual R is the unknown coefficient a. kThe time delay is a function of time t, i.e., R = (a, t). Therefore, the analytical solution problem of the system can be transformed into a minimum optimization problem. This method has significant advantages. For any nonlinear time delay problem, since this method is insensitive to the initial value of the iteration, the convergence domain of the unknown parameters is very large, and the main requirement is to find suitable basis functions, which greatly improves the computational efficiency. Other algorithms, such as the HAM method, can also solve this problem, but the HAM method requires deriving the 0th-order deformation equation and higher-order deformation equations, selecting auxiliary parameters and linear auxiliary operators, etc. At the same time, symbolic integration is required in each iteration, and the computational efficiency of the HAM method increases cubically with the number of series terms. It is worth noting that since the time delay term has been decomposed, the time delay is temporarily treated as a constant coefficient in the optimization problem.

[0059] As a preferred embodiment of this method, the objective function of the minimum optimization problem is expressed as follows:

[0060]

[0061] In the above formula, a * This represents the optimal value of the variable representing the objective function. Let R(a,t) represent the nonlinear time-delay optimization objective function, R(a,t) represent the residual vector of the system, A represent the feasible region of the optimal variable value a of the unknown objective function, and t represent time.

[0062] S300 introduces Tikhonov regularization and confidence region constraints to iteratively adjust the time delay of the objective function of the minimum optimization problem and outputs the optimal time delay.

[0063] S310. Linearize the objective function of the minimum optimization problem at discrete time nodes to obtain an approximate linear objective function;

[0064] S320. Tikhonov regularization is introduced to regularize the approximate linear objective function, resulting in a regularized objective function.

[0065] In this embodiment, the objective function is... Linearization is performed at discrete time points to make the original nonlinear objective function... It is transformed into an approximate linear objective function. Right now:

[0066]

[0067] In the above formula The first-order response sensitivity matrix is ​​of the following form, and its expression is:

[0068]

[0069] The approximate linear objective function is solved by introducing Tikhonov regularization. This may be an ill-conditioned problem, therefore the objective function equation becomes as follows:

[0070]

[0071] In the above formula, I is the identity matrix, and λ represents the regularization parameter with λ≥0.

[0072] S330. The regularization parameter of the regularized objective function is determined by the L-curve curve method, and the confidence region constraint is introduced. The time delay of the general nonlinear time delay system of the aircraft is iteratively adjusted in combination with the regularization parameter until the preset condition is met, and the optimal cutoff coefficient is obtained.

[0073] S340. Using the optimal cutoff coefficient as a semi-analytical solution for the general nonlinear time-delay system of the aircraft, the dynamic indicators of the aircraft system are optimized, and the optimal time delay is output.

[0074] In this embodiment, the regularization parameter λ is determined using the L-curve curve method, which corresponds to the maximum curvature of the L-curve curve. Since the L-curve curve method only applies to the linearized objective function... Instead of the original nonlinear objective function, a confidence region constraint, or "consistency index," needs to be introduced, which is expressed as:

[0075]

[0076] When the consistency index is satisfied If the result is positive, it means that the linearized objective function is sufficiently close to the original nonlinear objective function.

[0077] As a preferred embodiment of this method, the time delay τ is iteratively adjusted, and the above steps are repeated to minimize the time required to converge to the target period.

[0078] When the iteration conditions are met, the calculation is completed, and the optimal truncation coefficient a is obtained. k This yields the semi-analytical solution x(t) of the nonlinear time-delay dynamic equation. By using the optimal time delay τ, the optimal control function can be planned to optimize the system's dynamic indicators, enabling the gap / hysteresis wing flutter system to reach steady state in the shortest time, thereby improving the aircraft's instantaneous stability and reducing aircraft losses. This is the optimal time-delay feedback control method.

[0079] In summary, the embodiments of the present invention address the problems of low accuracy and poor computational efficiency of existing time-delay control methods in strongly nonlinear systems. First, the time-delay term x(t-τ) is expanded into a truncated Fourier series of series N. Then, the time-delay parameter τ is separated through trigonometric transformation to obtain the explicit coefficients α containing τ.k (ω),β k (ω); The time-domain minimum residual method (TMRM) is used to transform the dynamic equations into a semi-analytical optimization problem, in order to Using Tikhonov regularization and L-curve method as the objective function, ill-conditioned problems are handled; a consistency index is introduced. Controlling linearization error, when The time delay τ is iteratively updated until it converges to the target period. This method is applicable to wing flutter systems with gap / hysteresis nonlinearity, improving control accuracy and computational efficiency. By optimizing τ, the aircraft's stabilization time can be shortened and energy consumption reduced.

[0080] The above is a detailed description of the preferred embodiments of the present invention. However, the present invention is not limited to the embodiments described. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the present invention. All such equivalent modifications or substitutions are included within the scope defined by the claims of this application.

Claims

1. A time delay optimization method for a nonlinear time-delay control system of an aircraft, characterized in that, The method comprises the following steps: The dynamic equation of the general nonlinear time-delay system of the aircraft is constructed, the Fourier transform is introduced to fit the time-delay term, and the time-delay term with a truncated series of N is obtained; The two-angle sum transform is performed on the time-delay term with a truncated series of N, and the time-delay term with separated time-delay parameters is obtained; The time-delay term with separated time-delay parameters is substituted into the dynamic equation of the general nonlinear time-delay system of the aircraft, and the time-delay dynamic equation with separated time-delay parameters is constructed; Based on the time-delay dynamic equation with separated time-delay parameters, it is assumed that the solution of the time-delay dynamic equation can be expanded into an infinite series form, and an infinite series solution is established; The infinite series solution is expanded by differential operation, and the series expression of the velocity and acceleration of the time-delay dynamic equation is obtained; The infinite series solution is truncated and substituted into the time-delay dynamic equation with separated time-delay parameters, and a residual error is defined, and the objective function of the minimum optimization problem is constructed; The Tikhonov regularization and the confidence domain restriction are introduced, the objective function of the minimum optimization problem is adjusted by time delay iteration, and the optimal time delay is output.

2. The time delay optimization method for a nonlinear time-delay aircraft control system according to claim 1, wherein The expression of the Fourier transform introduced to fit the time-delay term is specifically as follows: In the above formula, represents the first term of the time delay term vector, represents the coefficient of each term of the triangular infinite series after Fourier transform, represents the term number after the triangular series expansion, represents the time delay, represents the time, represents that the system contains unknown frequencies.

3. The time delay optimization method for a nonlinear time-delay aircraft control system of claim 2, wherein The expression of the time-delay term with a truncated series of N is specifically as follows: In the above formula, represents the first term of the time delay term vector with a truncated series of , represents the term number of the truncated series, represents the coefficient of each term of the truncated series, represents the time delay, represents the time, represents the term number of the truncated series, represents that the system contains an unknown frequency, .​ 4. The time delay optimization method for a nonlinear time-delay aircraft control system of claim 3, wherein, The expression of the definition of the residual error is specifically as follows: in the above formulae, denotes the residual, denotes the truncated series, denotes the mass matrix of the dynamic system, denotes the acceleration vector of the truncated series of order denotes the damping matrix of the dynamic system, denotes the velocity vector of the truncated series of order denotes the stiffness matrix of the dynamic system, denotes the displacement vector of the truncated series of order denotes the cubic nonlinear stiffness coefficient matrix of the dynamic system, denotes the cubic of the three components of the displacement vector of the truncated series of order denotes the time delay term coefficient matrix, denotes the time delay term of the truncated series of order 5. The method for time-delay optimization of a nonlinear time-delay aircraft control system of claim 4, wherein The expression of the objective function of the minimum optimization problem is specifically as follows: In the above formulae, denotes the optimal variable value of the objective function, denotes the nonlinear time-delay optimization objective function, denotes the residual vector of the system, denotes the feasible region of the optimal variable value of the unknown objective function denotes the time.​ 6. The method for time-delay optimization of a nonlinear time-delay aircraft control system of claim 5, wherein The step of introducing the Tikhonov regularization and the confidence domain restriction to adjust the objective function of the minimum optimization problem by time delay iteration and outputting the optimal time delay specifically comprises the following steps: The objective function of the minimum optimization problem is linearized at discrete time nodes to obtain an approximate linear objective function; The Tikhonov regularization is introduced to regularize the approximate linear objective function to obtain a regularized objective function; The regularization parameter of the regularized objective function is determined by the L-curve curve method, and the confidence domain restriction is introduced; the time delay of the general nonlinear time-delay system of the aircraft is adjusted by iteration combined with the regularization parameter until a preset condition is met, and the optimal truncation coefficient is obtained; The optimal truncation coefficient is used as the semi-analytical solution of the general nonlinear time-delay system of the aircraft, the dynamic index of the aircraft system is optimized, and the optimal time delay is output.

7. The method for time-delay optimization of a nonlinear time-delay aircraft control system of claim 6, wherein The expression of the confidence domain restriction is specifically as follows: wherein denotes the consistency index, wherein denotes the consistency index, denotes the set of known parameters satisfying the feasible region , denotes the found reasonable iteration update, denotes the original nonlinear time-delay objective function with the known parameters , denotes the original nonlinear time-delay objective function with the known parameters and the added reasonable iteration update, denotes the linearized objective function at the discrete time node with zero iteration update, denotes the linearized objective function at the discrete time node, denotes the residual increment of the form , denotes the residual increment with the added reasonable iteration update, denotes the first order response sensitivity matrix.

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