Vibration Optimization Control Method for Hypersonic Vehicles Based on Time-Variation Reliability

Abstract

This invention discloses a vibration optimization control method for hypersonic vehicles based on time-varying reliability, belonging to the field of hypersonic vehicle vibration control. This method decomposes the hypersonic vehicle into a cantilever beam structure of the forebody and aftbody, and establishes a vibration control system. Uncertainty and random noise are quantified using interval variables and Gaussian noise, and the Taylor sub-interval method is employed to analyze the dynamic response of the vibration control system. This method designs a non-probabilistic time-varying reliability analysis method based on sub-intervals to evaluate the reliability of vibration control systems with multi-source uncertainties, improving computational accuracy and efficiency. The method of this invention can effectively control airframe vibration and ensure the structural reliability of the aircraft airframe.

Description

Technical Field

[0001] This invention relates to the field of vibration control technology for hypersonic vehicles, and in particular to a vibration optimization control method for hypersonic vehicles based on time-varying reliability. Background Technology

[0002] Because hypersonic vehicles operate across a wide range of altitudes and speeds and must perform critical missions in complex and ever-changing flight environments, their structures inevitably withstand complex aerodynamic forces and random disturbances. These complex aerodynamic forces and random disturbances cause vibrations in the fuselage, affecting the aerodynamic distribution, controllability, and structural reliability. In some extreme cases, vibrations can even lead to structural damage. Therefore, designing appropriate vibration control methods is essential to ensure the flight safety and structural reliability of hypersonic vehicles.

[0003] Currently, active and passive control of structural vibration is widely used in vibration control across various fields such as aerospace, automotive engineering, and civil engineering. Compared to passive control, active control methods are more economical and practical. Depending on the control objectives and performance requirements of the actual problem, different control methods can be employed, such as proportional-integral-derivative (PID) control, linear quadratic regulator (LQR) control, robust control, and stochastic optimal control. Traditional control methods can effectively control deterministic systems. However, real-world engineering systems often involve uncertainties, such as discrete material parameters and random noise, which can lead to structural failure. While robust control addresses uncertainties, it is often overly conservative, resulting in high power consumption. Therefore, reliability-based optimization control methods offer a more reasonable approach to handling system uncertainties.

[0004] Due to the complex flight structure of hypersonic vehicles, material parameters exhibit discreteness, non-uniform mass distribution, and random noise interference, resulting in multi-source uncertainties in the vibration system. Therefore, to improve the control performance and structural reliability of the vehicle's vibration system, it is necessary to consider the effects of parameter uncertainties and random noise and design appropriate optimization control algorithms. Currently, research on reliability-based vibration optimization control of hypersonic vehicles is still in its early stages. Summary of the Invention

[0005] This invention provides a vibration optimization control method for hypersonic vehicles based on time-varying reliability to ensure the safety and reliability of the vehicle during flight.

[0006] A first aspect of the present invention provides a vibration optimization control method for hypersonic vehicles based on time-varying reliability, comprising the following steps:

[0007] Step 1), establish a vibration control system for the hypersonic aircraft;

[0008] Step 2) Quantify the uncertainty of the vibration control system, establish an uncertain vibration control model, and analyze the uncertain dynamic response;

[0009] Step 3), initialize the weight matrix Q of LQR control;

[0010] Step 4) Calculate the minimum value of the performance function and solve for the weight matrix Q and the state feedback gain.

[0011] Step 5), using state feedback gain The vibration control system is controlled, and the upper and lower bounds of the dynamic response of the vibration control system under closed-loop control are calculated using Taylor's sub-interval algorithm.

[0012] Step 6): Evaluate the time-varying reliability of the vibration control system based on the non-probabilistic time-varying reliability analysis method of the design, and determine whether the reliability meets the constraints. If it does, proceed to step 7); otherwise, update the search range of the weight matrix and return to step 4.

[0013] Step 7) Output the reliability of the vibration control system and the optimal weight matrix Q.

[0014] Optionally, in one embodiment of the present invention, step 1) further includes:

[0015] Step 1-1): Vibration Analysis of Hypersonic Vehicles

[0016] Considering the free vibration of the hypersonic vehicle's fuselage, the equation for the transverse vibration of the cantilever beam is established:

[0017]

[0018] In the formula: E is Young's modulus, I is the moment of inertia, and y(x,t) is the elastic vibration displacement. Mass density;

[0019] Modal analysis shows that the lateral displacement of a cantilever beam is the product of the vibration mode Φ(x) and the generalized coordinate η(t):

[0020] y(x,t)=Φ(x)η(t) (2)

[0021] Steps 1-2): Modal Analysis

[0022] Substituting equation (2) into equation (1), we get:

[0023]

[0024] In the formula: ω represents the natural frequency;

[0025] The i-th mode shape function of a cantilever beam Write it in the following form:

[0026]

[0027] In the formula: i = 1, 2, ..., n m A i Let L be a constant, and L be the position of the free end;

[0028] The solution to the frequency equation is expressed as:

[0029]

[0030] In the formula: n is the number of mode shapes;

[0031] Since the vibration modes are mutually orthogonal, the coefficient A is calculated by the following normalization method. i Value:

[0032]

[0033] Steps 1-3): Generalized coordinate modeling

[0034] Ignoring the effect of gravity, by the definition of the Lagrange function, we obtain the following relationship:

[0035]

[0036] In the formula: T and V are the total kinetic energy and potential energy of the vibration control system, respectively, and η i For generalized coordinates, The derivative of the generalized coordinates, n m The number of mode orders to be selected;

[0037] Since the vibration control system is analyzed near the equilibrium point, coupling and nonlinear terms are ignored, and a state feedback control law u(t) = -K is introduced. c X(t), the vibration control system is represented in the following form:

[0038]

[0039] In the formula: Let represent the mass matrix, and Represents the stiffness matrix, and C1 is the damping matrix, calculated using the Rayleigh damping formula, ζ is the Rayleigh damping ratio, and K... cTo control the feedback gain, B0 is the matrix of the control force, reflecting the position and quantity information of the control force, E0 is the position matrix of the generalized force, and Φ(L) is the mode shape function value of the free end.

[0040] Optionally, in one embodiment of the present invention, step 2) further includes:

[0041] Step 2-1): Modeling Uncertain Parameters

[0042] Practical vibration control systems contain uncertain parameters. Considering the uncertainties in parameter mass density and stiffness, the uncertain parameter vector of the vibration control system is... Represented by the following interval variables:

[0043]

[0044] In the formula: b, The upper and lower bounds of the parameter b are uncertain. b is the m-th uncertain parameter. m , These are parameters b. m The lower and upper bounds, l=2 is the number of uncertain parameters, c m It is the coefficient of variation of the m-th uncertain parameter;

[0045] Step 2-2): Dynamic response analysis with multi-source uncertainties

[0046] Considering the presence of uncertain parameters in the vibration control system, the vibration control equation of the vibration control system can be written in the following form:

[0047]

[0048] If Δb is small enough, A(b) I ), B(b) I ), E(b I The first-order Taylor expansion of () is shown below:

[0049]

[0050] The Taylor first-order expansion of the vibration control system state is as follows:

[0051]

[0052] Therefore, according to Taylor's interval algorithm, the state response of the vibration control system can be expressed in the following form:

[0053]

[0054] In the formula: Let ξ represent the partial derivative of the vibration control system state with respect to the m-th uncertain parameter, ξ∈[-1,1]. When random noise exists in the vibration control system, a Kalman filter is used for processing.

[0055] Optionally, in one embodiment of the present invention, step 3) further includes:

[0056] Assuming the vibration system of the machine body is controllable and observable, the state feedback gain of LQR control is expressed as K. c =R -1 B T P is the state covariance matrix, which satisfies the following Riccati equation:

[0057] PA+A T P-PBR -1 B T P+Q=0 (16)

[0058] In the formula: Q is the weight matrix of the vibration control system state, and R is the weight matrix of the control input, both of which are symmetric matrices;

[0059] The initial search range of the weight matrix Q for the vibration control system state is given by the following equation, while the weight matrix R = βI is determined:

[0060]

[0061] In the formula: q i =0, It is q i The upper and lower bounds of C, and C q β is a constant.

[0062] Optionally, in one embodiment of the present invention, step 4) further includes:

[0063] Step 4-1): Generate the weight matrix Q j

[0064] N is randomly generated within the search range. Q Weight matrix Q j j = 1, 2, ..., N Q The weight matrix R = βI is fixed;

[0065] Step 4-2): Calculate the weight matrix Q j Performance function value under R = βI

[0066] When the vibration control system has uncertain parameters, its Riccati equation is extended to an interval form. The interval Riccati equation can be decomposed into a deterministic part and an uncertain part:

[0067] Pc A c +(A c ) T P c -P c B c R -1 (B c ) T P c +Q j =0(18)

[0068]

[0069] Due to matrix Q j R and P are symmetric, therefore A c +B c R -1 (B c ) T P c =[(A c ) T +P c B c R -1 (B c ) T ] T P I The uncertain part δP I It can be calculated using the Lyapunov equation, δP equals δP I The absolute value;

[0070] State feedback gain decomposed into deterministic part and uncertain parts ΔK c Let be the radius of the uncertain part. The performance function is decomposed into a deterministic part and an uncertain part, which are calculated by the following formulas:

[0071]

[0072] In the formula: ΔJ j The uncertain part of the performance function The radius is calculated using the natural expansion method of the interval;

[0073] Step 4-2): Calculate the minimum performance function The optimal weight matrix Q is obtained by inverse solving:

[0074]

[0075] The optimal weight matrix Q is derived from The inverse solution yields the determining part of the optimal state feedback gain, denoted as:

[0076] Optionally, in one embodiment of the present invention, step 5) further includes:

[0077] Step 5-1): Adjust the state feedback gain Substitute into the vibration control equation:

[0078]

[0079] Step 5-2): Calculate the dynamic response interval based on Taylor's subinterval algorithm.

[0080] The Taylor sub-interval method is introduced, where k m Let S represent the number of subintervals for the m-th uncertain parameter. Subintervals with different uncertain parameters can be freely combined, and the total number of subintervals is S = k1…k m …k l ;

[0081] For the s-th (s=1,2,…,S) subinterval combination, the state response interval can be expressed as:

[0082]

[0083] In the formula: This represents the radius of the subinterval divided by the m-th uncertain parameter;

[0084] Considering the dynamic response of all sub-interval combinations, the upper and lower bounds of the dynamic response of the vibration control system are described as follows:

[0085] y(t,[b] s )=CX(t,[b] s (25)

[0086]

[0087] Optionally, in one embodiment of the present invention, step 6) further includes:

[0088] Step 6-1): Definition of the limit function

[0089] For a vibration system with uncertain parameters, the limit state function is defined as:

[0090] g(t,b)=y cr -y(t,b)(27)

[0091] In the formula: y(t,b) represents the amplitude response of the vibration control system, y cr This represents the allowable boundary for the amplitude response;

[0092] Step 6-2): Nonprobabilistic Time-Varying Reliability Analysis Method

[0093] When the vibration system has uncertain parameters, the amplitude responses y(t1,b) and y(t2,b) at two adjacent times t1 and t2 are in interval form:

[0094] y(t,b)=y c (t,b)-y r (t,b),

[0095] In the formula: y(t,b), These are the lower and upper bounds of the amplitude response y(t,b), respectively. c (t,b),y r (t, b) are the center and radius of the response interval, respectively;

[0096] The normalized interval response is described as a standard square region containing multiple slanted rectangles. The aspect ratio of the rectangles indicates the correlation between the amplitude responses of the vibration control system. The normalized relationship is as follows:

[0097]

[0098] In the formula: ξ1 and ξ2 are the standardized variables of y(t1,b) and y(t2,b), respectively;

[0099] The correlation coefficients between the amplitude responses y(t1,b) and y(t2,b) are expressed as follows:

[0100]

[0101] In the formula: d is half the length of one side of the skew matrix;

[0102] Considering the uncertainties of the vibration control system, the limit state equation is standardized to the following form:

[0103] g(t,b)=g c (t,b)+Δg(t,b)ξ

[0104] g c (t,b)=y cr -y c (t,b) (31)

[0105] Δg(t,b)=y r (t,b)

[0106] Where: g c (t,b) and Δg(t,b) are the center and radius of the limit function interval, respectively;

[0107] Because the failure boundary g(t,b)=0, the boundary conditions are... and The expression is as follows:

[0108]

[0109] The span rate from time t1 to time t2 can be expressed as:

[0110]

[0111] Where: Δt is the time interval, and Let these represent the area of ​​the failure domain and the total area of ​​the tilted triangle, respectively.

[0112] The nonprobabilistic time-varying reliability of a vibration control system is calculated by the following formula:

[0113]

[0114] In the formula: P f Let Pos(t0) represent the failure probability at initial time t0, and R be the failure probability. T For the reliability of the vibration control system;

[0115] Step 6-3): Nonprobabilistic time-varying reliability analysis method based on sub-intervals

[0116] When the uncertain parameters are divided into multiple sub-intervals, the reliability of the vibration control system can be calculated by the following formula:

[0117]

[0118] In the formula: [P f ] s It is the failure probability of the s-th sub-interval combination;

[0119] Step 6-4): Evaluate the time-varying reliability of the body vibration system based on the non-probabilistic time-varying reliability analysis method, and determine whether the reliability meets the reliability constraint R. T >R cr R cr It is the minimum acceptable reliability;

[0120] If the reliability satisfies R T >R cr Then proceed to step 7);

[0121] If not satisfied, then update the search range of the weight matrix Q, where Return to step 4).

[0122] This invention presents a time-varying reliability-based vibration optimization control method for hypersonic vehicles. The method decomposes the hypersonic vehicle into a forebody and aftbody cantilever beam structure and establishes a vibration control system. Uncertainty and random noise are quantified using interval variables and Gaussian noise, and the Taylor sub-interval method is employed to analyze the dynamic response of the vibration control system. A non-probabilistic time-varying reliability analysis method based on sub-intervals is designed to evaluate the reliability of the vibration control system with multi-source uncertainties. This method effectively controls the airframe vibration and ensures the structural reliability of the aircraft.

[0123] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0124] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein:

[0125] Figure 1 A flowchart of a vibration optimization control method for hypersonic vehicles based on time-varying reliability provided in an embodiment of the present invention;

[0126] Figure 2 This is a longitudinal model of the hypersonic vehicle of the present invention;

[0127] Figure 3 This is the cantilever beam structure equivalent to the hypersonic vehicle of the present invention;

[0128] Figure 4 This is a schematic diagram of the response interval between two adjacent time points in this invention;

[0129] Figure 5 This is a schematic diagram of the first channel theory of the present invention;

[0130] Figure 6 This is a flowchart illustrating the calculation process of the hypersonic vehicle vibration optimization control method based on time-varying reliability according to the present invention.

[0131] Figure 7 These are schematic diagrams showing the amplitude under both uncontrolled and controlled conditions in this invention.

[0132] Figure 8 This is a schematic diagram illustrating the time-varying reliability of the present invention with and without control. Detailed Implementation

[0133] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0134] Figure 1 This is a flowchart of a vibration optimization control method for hypersonic vehicles based on time-varying reliability, provided according to an embodiment of the present invention.

[0135] like Figure 1 As shown, the vibration optimization control method for hypersonic vehicles based on time-varying reliability includes the following steps:

[0136] Step 1), establish a vibration control system for the hypersonic aircraft;

[0137] Step 2) Quantify the uncertainty of the vibration control system, establish an uncertain vibration control model, and analyze the uncertain dynamic response;

[0138] Step 3), initialize the weight matrix Q of LQR control;

[0139] Step 4) Calculate the minimum value of the performance function and solve for the weight matrix Q and the state feedback gain.

[0140] Step 5), using state feedback gain The vibration control system is controlled, and the upper and lower bounds of the dynamic response of the vibration control system under closed-loop control are calculated using Taylor's sub-interval algorithm.

[0141] Step 6): Evaluate the time-varying reliability of the system based on the non-probabilistic time-varying reliability analysis method of the design, and determine whether the reliability meets the constraints. If it does, proceed to step 7); otherwise, update the search range of the weight matrix and return to step 4.

[0142] Step 7) Output the reliability of the vibration control system and the optimal weight matrix Q.

[0143] Figure 2 For the longitudinal model of the hypersonic vehicle, the research object of this invention, this invention models it as two cantilever beams fixed at the center of mass. Figure 3 This is the equivalent cantilever beam structure for a hypersonic vehicle. Table 1 shows the basic parameters and values ​​of the equivalent cantilever beam for the vehicle.

[0144] Table 1 Basic parameters of the cantilever beam structure of the aircraft

[0145]

[0146] Optionally, in one embodiment of the present invention, step 1) further includes:

[0147] Step 1-1): Vibration Analysis of Hypersonic Vehicles

[0148] Considering the free vibration of the hypersonic vehicle's fuselage, the equation for the transverse vibration of the cantilever beam is established:

[0149]

[0150] In the formula: E is Young's modulus, I is the moment of inertia, and y(x,t) is the elastic vibration displacement. Mass density;

[0151] Modal analysis shows that the lateral displacement of a cantilever beam is the product of the vibration mode Φ(x) and the generalized coordinate η(t):

[0152] y(x,t)=Φ(x)η(t) (2)

[0153] Steps 1-2): Modal Analysis

[0154] Substituting equation (2) into equation (1), we get:

[0155]

[0156] In the formula: ω represents the natural frequency;

[0157] The i-th mode shape function of a cantilever beam Write it in the following form:

[0158]

[0159] In the formula: i = 1, 2, ..., n m A i Let L be a constant, and L be the position of the free end;

[0160] The solution to the frequency equation is expressed as:

[0161]

[0162] In the formula: n is the number of mode shapes;

[0163] Since the vibration modes are mutually orthogonal, the coefficient A is calculated by the following normalization method. i Value:

[0164]

[0165] Steps 1-3): Generalized coordinate modeling

[0166] Ignoring the effect of gravity, by the definition of the Lagrange function, we obtain the following relationship:

[0167]

[0168] In the formula: T and V are the total kinetic energy and potential energy of the vibration control system, respectively, and η i For generalized coordinates, The derivative of the generalized coordinates, n m The number of modes in a vibration mode;

[0169] Since the vibration control system is analyzed near the equilibrium point, its coupling and nonlinear terms are ignored. This invention only discusses the vibration amplitude at the free end x = L, and introduces a state feedback control law u(t) = -K. c X(t), the vibration control system is represented in the following form:

[0170]

[0171] In the formula: Let represent the mass matrix, and Represents the stiffness matrix, and C1 is the damping matrix, calculated using the Rayleigh damping formula, ζ is the Rayleigh damping ratio, and K... c To control the feedback gain, B0 is the matrix of the control force, reflecting the position and quantity information of the control force, E0 is the position matrix of the generalized force, and Φ(L) is the mode shape function value of the free end.

[0172] Optionally, in one embodiment of the present invention, step 2) further includes:

[0173] Step 2-1): Modeling Uncertain Parameters

[0174] Practical vibration control systems contain uncertain parameters. Considering the uncertainties in parameter mass density and stiffness, the uncertain parameter vector of the vibration control system is... Represented by the following interval variables:

[0175]

[0176] In the formula: b, The upper and lower bounds of the parameter b are uncertain. b is the m-th uncertain parameter. m , These are parameters b. m The lower and upper bounds, l=2 is the number of uncertain parameters, c m It is the coefficient of variation of the m-th uncertain parameter;

[0177] Step 2-2): Dynamic response analysis with multi-source uncertainties

[0178] Considering the presence of uncertain parameters in the vibration control system, the vibration control equation of the vibration control system can be written in the following form:

[0179]

[0180] If Δb is small enough, A(b) I ), B(b) I ), E(b I The first-order Taylor expansion of () is shown below:

[0181]

[0182] The Taylor first-order expansion of the vibration control system state is as follows:

[0183]

[0184] Therefore, according to Taylor's interval algorithm, the state response of the vibration control system can be expressed in the following form:

[0185]

[0186] In the formula: Let ξ represent the partial derivative of the vibration control system state with respect to the m-th uncertain parameter, ξ∈[-1,1]. When random noise exists in the vibration control system, a Kalman filter is used for processing.

[0187] Optionally, in one embodiment of the present invention, step 3) further includes:

[0188] Assuming the vibration control system is controllable and observable, the state feedback gain of LQR control is expressed as K. c =R - 1 B T P is the state covariance matrix, which satisfies the following Riccati equation:

[0189] PA+A T P-PBR -1 B T P+Q=0(16)

[0190] In the formula: Q is the weight matrix of the vibration control system state, and R is the weight matrix of the control input, both of which are symmetric matrices;

[0191] The initial search range of the weight matrix Q for the vibration control system state is given by the following equation, while the weight matrix R = βI is determined:

[0192]

[0193] In the formula: q i =0, It is q i The upper and lower bounds of C, and C q β is a constant.

[0194] Optionally, in one embodiment of the present invention, step 4) further includes:

[0195] Step 4-1): Generate the weight matrix Q j

[0196] N is randomly generated within the search range. Q Weight matrix Q j j = 1, 2, ..., N Q The weight matrix R = βI is fixed; Step 4-2): Calculate the weight matrix Q. j Performance function value under R = βI

[0197] When the vibration control system has uncertain parameters, the Riccati equation of the system is extended to an interval form. The interval Riccati equation can be decomposed into a deterministic part and an uncertain part:

[0198] P c A c +(A c ) T P c -P c B c R -1 (B c ) T P c +Q j =0(18)

[0199]

[0200] P I The uncertain part δP I It can be calculated using the Lyapunov equation, and by the natural expansion of the interval, the radius δP of the uncertain part is equal to δP. I The absolute value of.

[0201] Due to matrix Q j R and P are symmetric, therefore A c +B c R -1 (B c ) T P c =[(A c ) T +P c B c R-1 (B c ) T ] T P I The uncertain part δP I It can be calculated using the Lyapunov equation, δP equals δP I The absolute value;

[0202] The state feedback gain can be extended to an uncertain form, and the state feedback gain can be decomposed into a deterministic part. and uncertain parts ΔK c Let be the radius of the uncertain part. The performance function is decomposed into a deterministic part and an uncertain part, which are calculated by the following formulas:

[0203]

[0204] In the formula: ΔJ j The uncertain part of the performance function The radius is calculated using the natural expansion method of the interval;

[0205] Step 4-2): Calculate the minimum performance function The optimal weight matrix Q is obtained by inverse solving:

[0206]

[0207] The optimal weight matrix Q is derived from The inverse solution yields the determining part of the optimal state feedback gain, denoted as:

[0208] Optionally, in one embodiment of the present invention, step 5) further includes:

[0209] Step 5-1): Adjust the state feedback gain Substitute into the vibration control equation:

[0210]

[0211] Step 5-2): Calculate the dynamic response interval based on Taylor's subinterval algorithm.

[0212] To reduce errors, this invention introduces the Taylor sub-interval method, where k m Let S represent the number of subintervals for the m-th uncertain parameter. Subintervals with different uncertain parameters can be freely combined, and the total number of subintervals is S = k1…k m …k l ;

[0213] For the s-th (s=1,2,…,S) subinterval combination, the state response interval can be expressed as:

[0214]

[0215] In the formula: This represents the radius of the subinterval divided by the m-th uncertain parameter;

[0216] Considering the dynamic response of all sub-interval combinations, the upper and lower bounds of the dynamic response of the vibration control system are described as follows:

[0217] y(t,[b] s )=CX(t,[b] s (25)

[0218]

[0219] Optionally, in one embodiment of the present invention, step 6) further includes:

[0220] Step 6-1): Definition of the limit function

[0221] For a vibration control system with uncertain parameters, the limit state function is defined as:

[0222] g(t,b)=y cr -y(t,b)(27)

[0223] In the formula: y(t,b) represents the amplitude response of the vibration control system, y cr This represents the allowable boundary for the amplitude response;

[0224] Step 6-2): Nonprobabilistic Time-Varying Reliability Analysis Method

[0225] When the vibration control system has uncertain parameters, the amplitude responses y(t1,b) and y(t2,b) at two adjacent times t1 and t2 can be expressed as: Figure 4 The rectangle shown in (a) has an amplitude interval form as follows:

[0226] y(t,b)=y c (t,b)-y r (t,b),

[0227] In the formula: y(t,b), These are the lower and upper bounds of the amplitude response y(t,b), respectively. c (t,b),y r (t, b) are the center and radius of the response interval, respectively;

[0228] The normalized interval response is described as a standard square region, such as Figure 4As shown in (b), this region contains multiple slanted rectangles. The aspect ratios of these rectangles indicate the correlation between the amplitude responses of the vibration control system. After normalization, the following relationship is obtained:

[0229]

[0230] In the formula: ξ1 and ξ2 are the standardized variables of y(t1,b) and y(t2,b), respectively;

[0231] The correlation coefficients between the amplitude responses y(t1,b) and y(t2,b) are expressed as follows:

[0232]

[0233] In the formula: d is Figure 4 In (b), half the length of one side of the skewed matrix;

[0234] Considering the uncertainties of the vibration control system, the limit state equation is standardized to the following form:

[0235] g(t,b)=g c (t,b)+Δg(t,b)ξ

[0236] g c (t,b)=y cr -y c (t,b) (31)

[0237] Δg(t,b)=y r (t,b)

[0238] Where: g c (t,b) and Δg(t,b) are the center and radius of the limit function interval, respectively;

[0239] Figure 5 This is a schematic diagram of the first channel theory. Since the failure boundary g(t,b)=0, the boundary conditions are... and The expression is as follows:

[0240]

[0241] The span rate from time t1 to time t2 can be expressed as:

[0242]

[0243] Where: Δt is the time interval, and Let g(t1,b) represent the area of ​​the failure domain and the total area of ​​the tilted triangle, respectively, which are determined by the limit function g(t1,b) at time t1 and the limit function g(t2,b) at time t2.

[0244] The nonprobabilistic time-varying reliability of a vibration control system is calculated by the following formula:

[0245]

[0246] In the formula: P f Let Pos(t0) represent the failure probability at initial time t0, and R be the failure probability. T For the reliability of the vibration control system;

[0247] Step 6-3): Nonprobabilistic time-varying reliability analysis method based on sub-intervals

[0248] To more accurately assess the time-varying reliability of vibration control systems, this invention proposes a nonprobabilistic time-varying reliability analysis method based on sub-intervals. When the uncertain parameters are divided into multiple sub-intervals, the reliability of the vibration control system can be calculated by the following formula:

[0249]

[0250] In the formula: [P f ] s It is the failure probability of the s-th sub-interval combination;

[0251] Step 6-4): Evaluate the time-varying reliability of the vibration control system based on the nonprobabilistic time-varying reliability analysis method, and determine whether the reliability meets the reliability constraint R. T >R cr R cr It is the minimum acceptable reliability;

[0252] If the reliability satisfies R T >R cr Then proceed to step 7);

[0253] If not satisfied, then update the search range of the weight matrix Q, where Return to step 4).

[0254] like Figure 6 The diagram shown is a calculation flowchart of the hypersonic vehicle vibration optimization control method based on time-varying reliability according to the present invention. The non-probabilistic time-varying reliability analysis method and the hypersonic vehicle vibration optimization control method based on time-varying reliability of the present invention will be described below through a specific embodiment.

[0255] In the following examples, the amplitude refers to the amplitude of the aircraft forebody x = x f The amplitude at the point; since lower-order modes play a decisive role, only the first three modes of vibration are considered in this example, i.e., the number of vibration modes n. m =3.

[0256] Step 1-1): Vibration Analysis of Hypersonic Vehicles

[0257] Considering the free vibration of the hypersonic vehicle's fuselage, modal analysis shows that the lateral displacement of the cantilever beam can be expressed as the product of the vibration mode Φ(x) and the generalized coordinate η(t).

[0258] y(x,t)=Φ(x)η(t)

[0259] Steps 1-2): Modal Analysis

[0260] The i-th mode shape function of a cantilever beam It can be written in the following form

[0261]

[0262] In the formula: i = 1, 2, 3, A i x is a constant f The specific values ​​are shown in Table 1.

[0263] The solution to the frequency equation can be expressed as:

[0264] β1x f =1.875β2x f =4.694β3x f =7.855

[0265] Since the vibration modes are mutually orthogonal, the coefficient A can be calculated using the following normalization method. i value

[0266]

[0267] Steps 1-3): Generalized coordinate modeling

[0268] Since the vibration system is analyzed near its equilibrium point, coupling and nonlinear terms are ignored. It is assumed that the aircraft forebody undergoes free vibration from the initial moment, with initial states η1(0) = 0.3, η2(0) = -0.03, and η3(0) = -0.003. The vibration equations for the aircraft forebody are established as follows:

[0269]

[0270] In the formula: Let represent the mass matrix, and Represents the stiffness matrix, and C1 is the damping matrix, and ζ = 0.05 is the Rayleigh damping ratio. Given that the cantilever beam undergoes free vibration for one second, f(t) = 160 * e is applied to all three generalized coordinates. -50(t-1)The external force N.

[0271] Step 2) The specific process is as follows:

[0272] Practical vibration control systems have uncertain parameters. This invention considers the uncertainties in the mass density and stiffness of the parameter precursors.

[0273]

[0274] Precursor mass density in this example The coefficient of variation is c1 = 0.01, and the coefficient of variation of the bending stiffness EI is c2 = 0.011.

[0275] Considering the presence of uncertain parameters and random noise in the vibration control system, the vibration control equation can be written as follows:

[0276]

[0277] In the formula: the noise v(t) and w(t) are Gaussian noises with zero mean, and their standard deviations are σ1 = 2 × 10⁻⁶. -5 and σ² = 1 × 10 -4 Random noise is processed using Kalman filtering.

[0278] The specific process of step 3) is as follows:

[0279] Assuming the aircraft vibration control system is controllable and observable, the state feedback gain of LQR control is expressed as K. c =R - 1 B T P is the state covariance matrix, satisfying the following Riccati equation.

[0280] PA+A T P-PBR -1 B T P+Q=0

[0281] In the formula: Q is the state weight matrix of the vibration control system, and R is the weight matrix of the control input, both of which are symmetric matrices.

[0282] The initial search range for the weight matrix Q is given by the following formula, while the weight matrix R = βI is determined.

[0283]

[0284] In the formula: q i =0, It is q i The upper and lower bounds of C, and C q β is a constant, and C in this example q =50, β=0.05.

[0285] Step 4-1): Generate the weight matrix Q j

[0286] N is randomly generated within the search range. Q = 200 weight matrices Q j j = 1, 2, ..., N Q The weight matrix R = 0.05I is fixed.

[0287] Step 4-2): Calculate the weight matrix Q j Performance function value under R = βI

[0288] As can be seen from the preceding analysis, the state feedback gain can be decomposed into a deterministic component. and uncertain parts ΔK c The radius of the uncertain part can be calculated using the natural expansion method of the interval. In this case, the performance function can be decomposed into a deterministic part and an uncertain part, calculated by the following formula.

[0289]

[0290] ΔJ j The uncertain part of the performance function The radius can be calculated using the natural expansion method of the interval.

[0291] Step 4-2): Calculate the minimum performance function J min The optimal weight matrix Q is obtained by inverse solving.

[0292]

[0293] Therefore, the optimal weight matrix Q can be derived from J. min The optimal state feedback gain is obtained by inverse solving and denoted as .

[0294] The specific process of step 5) is as follows:

[0295] Step 5-1): Calculate the optimal feedback control gain. Substitute into the vibration control equation

[0296]

[0297] Step 5-2): Calculate the dynamic response interval based on Taylor's subinterval algorithm.

[0298] In this example, the number of uncertain parameters is l = 2, the number of subintervals for mass density and bending stiffness is k1 = k2 = 6, and the total number of subintervals is S = 36.

[0299] For the s-th (s=1,2,…,36) subinterval combination, its state response interval can be expressed as:

[0300]

[0301] In the formula: This represents the radius of the subinterval divided by the m-th uncertain parameter.

[0302] Considering the dynamic response of all sub-interval combinations, the interval of the dynamic response of the vibration control system can be described as follows:

[0303]

[0304] The specific process of step 6) is as follows:

[0305] Step 6-1): Definition of the limit function

[0306] y in this example cr Indicates the uncontrolled amplitude boundary. The amplitude boundary of the vibration control system under closed-loop control is represented as follows:

[0307]

[0308] Step 6-2): Nonprobabilistic Time-Varying Reliability Analysis Method

[0309] Figure 4 The side length of the tilt matrix in (b) can be calculated from the amplitude response correlation coefficient.

[0310]

[0311] At this point, the span rate from time t1 to time t2 can be expressed as:

[0312]

[0313] In the formula: Δt = 0.001 time interval, and Let represent the area of ​​the failure domain and the total area of ​​the tilted triangle, respectively.

[0314] Step 6-3): A nonprobabilistic time-varying reliability analysis method based on sub-intervals

[0315] The reliability calculation formula for the vibration control system based on the nonprobabilistic time-varying reliability analysis method proposed in this invention is as follows:

[0316]

[0317] In the formula: [P f ] sIt is the failure probability of the s-th subinterval combination. In this example, the number of uncertain parameters is l = 2, the number of subintervals for mass density and bending stiffness is k1 = k2 = 6, and the total number of subintervals is S = 36.

[0318] Step 6-4): Evaluate the time-varying reliability of the vibration control system based on the nonprobabilistic time-varying reliability analysis method, and determine whether the reliability meets the reliability constraint R. T >R cr R cr This is the minimum permissible reliability; in this example, R... cr =0.96.

[0319] If the reliability satisfies R T >R cr If the condition is not met, proceed to step 7). If not, update the search range of the weight matrix Q, where... Return to step 4).

[0320] The specific process of step 7) is as follows:

[0321] Output the optimal weight matrix Q = dia(g[6.3, 46.2, 94.3, 2.3, 19.8 and 8 vibration]) and the reliability R of the control system. T =0.9644.

[0322] Finally, the simulation results are as follows: Figure 7-8 As shown.

[0323] like Figure 7 The diagram shows a controlled ( Figure 7 (a) and uncontrolled ( Figure 7 (b) The change in fuselage amplitude, controlling the time-varying reliability of the aircraft forebody, such as Figure 8 As shown in the simulation diagram, the amplitude of the controlled system is significantly reduced, and the reliability remains within an acceptable range compared to the uncontrolled fuselage.

[0324] This invention proposes a vibration optimization control method for hypersonic vehicles based on nonprobabilistic reliability. This method combines the advantages of sub-interval algorithms and first-channel theory, improving the computational efficiency and accuracy of time-varying reliability. By using reliability as a constraint, it effectively controls the vehicle's vibration while ensuring high airframe reliability. This method not only effectively controls the airframe vibration of hypersonic vehicles but also keeps the structural reliability of the airframe within the constraints.

[0325] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0326] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "N" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0327] Any process or method description in the flowchart or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or N executable instructions for implementing custom logic functions or processes, and the scope of preferred embodiments of the invention includes additional implementations in which functions may be performed not in the order shown or discussed, including substantially simultaneously or in reverse order depending on the functions involved, as will be understood by those skilled in the art to which embodiments of the invention pertain.

Claims

1. A vibration optimization control method for hypersonic vehicles based on time-varying reliability, characterized in that, Includes the following steps: Step 1), establish a vibration control system for the hypersonic aircraft; Step 2) Quantify the uncertainty of the vibration control system, establish an uncertain vibration control model, and analyze the uncertain dynamic response; Step 3), initialize the weight matrix Q of LQR control; Step 4) Calculate the minimum value of the performance function and solve for the weight matrix Q and the state feedback gain. Step 5), using state feedback gain The vibration control system is controlled, and the upper and lower bounds of the dynamic response of the vibration control system under closed-loop control are calculated using Taylor's sub-interval algorithm. Step 6): Evaluate the time-varying reliability of the vibration control system based on the non-probabilistic time-varying reliability analysis method of the design, and determine whether the reliability meets the constraints. If it does, proceed to step 7); otherwise, update the search range of the weight matrix and return to step 4. Step 7) Output the reliability of the vibration control system and the optimal weight matrix Q; Step 2) further includes: Step 2-1): Modeling Uncertain Parameters Practical vibration control systems contain uncertain parameters. Considering the uncertainties in parameter mass density and stiffness, the uncertain parameter vector of the vibration control system is... Represented by the following interval variables: In the formula: b , It is the upper and lower bounds of the uncertain parameter b. It is the m-th uncertain parameter. b m , These are parameters b. m The lower and upper bounds, l=2 is the number of uncertain parameters, c m It is the coefficient of variation of the m-th uncertain parameter; Step 2-2): Dynamic response analysis with multi-source uncertainties Considering the presence of uncertain parameters in the vibration control system, the vibration control equation of the vibration control system can be written in the following form: A(b I ), B(b) I ), E(b I The first-order Taylor expansion of () is shown below: The Taylor first-order expansion of the vibration control system state is as follows: Therefore, according to Taylor's interval algorithm, the state response of the vibration control system can be expressed in the following form: In the formula: Let ξ represent the partial derivative of the vibration control system state with respect to the m-th uncertain parameter, ξ∈[-1,1]. When random noise exists in the vibration control system, a Kalman filter is used for processing. Step 5) further includes: Step 5-1): Adjust the state feedback gain Substitute into the vibration control equation: Step 5-2): Calculate the dynamic response interval based on Taylor's subinterval algorithm. The Taylor sub-interval method is introduced, where k m Let S represent the number of subintervals for the m-th uncertain parameter. Subintervals with different uncertain parameters can be freely combined, and the total number of subintervals is S = k1…k m …k l ; For the s-th (s=1,2,…,S) subinterval combination, the state response interval can be expressed as: In the formula: This represents the radius of the subinterval divided by the m-th uncertain parameter; Considering the dynamic response of all sub-interval combinations, the upper and lower bounds of the dynamic response of the vibration control system are described as follows: y(t,[b] s )=CX(t,[b] s )(25) 2. The method according to claim 1, characterized in that, Step 1) further includes: Step 1-1): Vibration Analysis of Hypersonic Vehicles Considering the free vibration of the hypersonic vehicle's fuselage, the equation for the transverse vibration of the cantilever beam is established: In the formula: E is Young's modulus, I is the moment of inertia, and y(x,t) is the elastic vibration displacement. Mass density; Modal analysis shows that the lateral displacement of a cantilever beam is the product of the vibration mode Φ(x) and the generalized coordinate η(t): y(x,t)=Φ(x)η(t) (2) Steps 1-2): Modal Analysis Substituting equation (2) into equation (1), we get: In the formula: ω is the natural frequency; The i-th mode shape function of a cantilever beam Write it in the following form: In the formula: i = 1, 2, ..., n m A i Let L be a constant and L be the free end. The solution to the frequency equation is expressed as: In the formula: n is the order of the mode shape; Since the vibration modes are mutually orthogonal, the coefficient A is calculated by the following normalization method. i Value: Steps 1-3): Generalized coordinate modeling Ignoring the effect of gravity, by the definition of the Lagrange function, we obtain the following relationship: In the formula: T and V are the total kinetic energy and potential energy of the vibration control system, respectively, and η i For generalized coordinates, The derivative of the generalized coordinates, n m This represents the number of mode shapes. Since the vibration control system is analyzed near the equilibrium point, coupling and nonlinear terms are ignored, and a state feedback control law u(t) = -K is introduced. c X(t), the vibration control system is represented in the following form: In the formula: Let represent the mass matrix, and Represents the stiffness matrix, and C1 is the damping matrix, calculated using the Rayleigh damping formula, ζ is the Rayleigh damping ratio, and K... c To control the feedback gain, B0 is the matrix of the control force, reflecting the position and quantity information of the control force, E0 is the position matrix of the generalized force, and Φ(L) is the mode shape function value of the free end.

3. The method according to claim 1, characterized in that, Step 3) further includes: Assuming the vibration system of the machine body is controllable and observable, the state feedback gain of LQR control is expressed as K. c =R -1 B T P is the state covariance matrix, which satisfies the following Riccati equation: PA+A T P-PBR -1 B T P+Q=0(16) In the formula: Q is the weight matrix of the vibration control system state, and R is the weight matrix of the control input, both of which are symmetric matrices; The initial search range of the weight matrix Q for the vibration control system state is given by the following equation, while the weight matrix R = βI is determined: In the formula: q i =0, It is q i The upper and lower bounds of C, and C q β is a constant.

4. The method according to claim 1, characterized in that, Step 4) further includes: Step 4-1): Generate the weight matrix Q j N is randomly generated within the search range. Q Weight matrix Q j j = 1, 2, ..., N Q The weight matrix R = βI is fixed; Step 4-2): Calculate the weight matrix Q j Performance function value under R = βI When the vibration control system has uncertain parameters, its Riccati equation is extended to an interval form. The interval Riccati equation can be decomposed into a deterministic part and an uncertain part: P c A c +(A c ) T P c -P c B c R -1 (B c ) T P c +Q j =0(18) Due to matrix Q j R and P are symmetric, therefore A c +B c R -1 (B c ) T P c =[(A c ) T +P c B c R -1 (B c ) T ] T , P I The uncertain part δP I It can be calculated using the Lyapunov equation, δP equals δP I The absolute value; State feedback gain decomposed into deterministic part and uncertain parts ΔK c Let be the radius of the uncertain part. The performance function is decomposed into a deterministic part and an uncertain part, which are calculated by the following formulas: In the formula: ΔJ j The uncertain part of the performance function The radius is calculated using the natural expansion method of the interval; Step 4-2): Calculate the minimum performance function The optimal weight matrix Q is obtained by inverse solving: The optimal weight matrix Q is derived from The inverse solution yields the determining part of the optimal state feedback gain, denoted as:

5. The method according to claim 1, characterized in that, Step 6) further includes: Step 6-1): Definition of the limit function For a vibration system with uncertain parameters, the limit state function is defined as: g(t,b)=y cr -y(t,b)(27) In the formula: y(t,b) represents the amplitude response of the vibration control system, y cr This represents the allowable boundary for the amplitude response; Step 6-2): Nonprobabilistic Time-Varying Reliability Analysis Method When the vibration system has uncertain parameters, the amplitude responses y(t1,b) and y(t2,b) at two adjacent times t1 and t2 are in interval form: In the formula: y (t,b), These are the lower and upper bounds of the amplitude response y(t,b), respectively. c (t,b),y r (t, b) are the center and radius of the response interval, respectively; The normalized interval response is described as a standard square region containing multiple slanted rectangles. The aspect ratio of the rectangles indicates the correlation between the amplitude responses of the vibration control system. The normalized relationship is as follows: In the formula: ξ1 and ξ2 are the standardized variables of y(t1,b) and y(t2,b), respectively; The correlation coefficients between the amplitude responses y(t1,b) and y(t2,b) are expressed as follows: In the formula: d is half the length of one side of the skew matrix; Considering the uncertainties of the vibration control system, the limit state equation is standardized to the following form: Where: g c (t,b) and Δg(t,b) are the center and radius of the limit function interval, respectively; Because the failure boundary g(t,b)=0, the boundary conditions are... and The expression is as follows: The span rate from time t1 to time t2 can be expressed as: Where: Δt is the time interval, and Let these represent the area of ​​the failure domain and the total area of ​​the tilted triangle, respectively. The nonprobabilistic time-varying reliability of a vibration control system is calculated by the following formula: In the formula: P f Let Pos(t0) represent the failure probability at initial time t0, and R be the failure probability. T For the reliability of the vibration control system; Step 6-3): Nonprobabilistic time-varying reliability analysis method based on sub-intervals When the uncertain parameters are divided into multiple sub-intervals, the reliability of the vibration control system can be calculated by the following formula: In the formula: [P f ] s It is the failure probability of the s-th sub-interval combination; Step 6-4): Evaluate the time-varying reliability of the vibration control system based on the nonprobabilistic time-varying reliability analysis method, and determine whether the reliability meets the reliability constraint R. T >R cr R cr It is the minimum acceptable reliability; If the reliability satisfies R T >R cr Then proceed to step 7); If not satisfied, then update the search range of the weight matrix Q, where Return to step 4).

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