Elastic constraint self-adaptive relative motion control method adaptive to spacecraft input saturation

Abstract

The invention relates to an elastic constraint self-adaptive relative motion control method adaptive to spacecraft input saturation, and belongs to the technical field of autonomous rendezvous and docking of a ring fire orbit. According to the invention, rapid and high-precision relative motion control capable of presetting the dynamic state of the system is realized, and the contradiction between performance constraint control and actual input saturation is reconciliated. Firstly, an elastic constraint is designed, and the elastic constraint is mathematically characterized as a monotonous curve of which the convergence rate and the convergence time can be defined by parameter setting; secondly, designing a scaling constraint type Lyapunov function, and deriving a basic structure of a control algorithm by taking the scaling constraint type Lyapunov function as a stability analysis framework; and finally, designing a scaling factor, which is nested in a scaling constraint type Lyapunov function, and is independent of a design mode of a controller, so that the complex back-stepping method derivative operation can be effectively avoided. The algorithm is simple to realize and easy to solve, and asymptotic stability and finite time stability of the system can be realized.

Description

Technical Field

[0001] This invention belongs to the field of autonomous rendezvous and docking technology in Mars orbit, and relates to an elastic constraint adaptive relative motion control method that adapts to spacecraft input saturation. Background Technology

[0002] Close-range relative motion control of spacecraft is one of the hottest issues in the aerospace field. The mission scenarios involved include, but are not limited to, autonomous rendezvous and spacecraft formation, and therefore have received widespread attention from scientists around the world.

[0003] Achieving high-performance control of the relative motion of multiple spacecraft in close proximity requires overcoming the effects of disturbances such as orbital perturbations and solar radiation pressure, while achieving centimeter-level control accuracy. Besides the high precision requirements, relative motion control within 100 meters, typically in rendezvous missions, requires a relative position convergence time of no more than 10 minutes, meaning there is a time constraint on reaching the target position. However, classical control algorithms typically rely on extensive parameter design and tuning to achieve high-precision system stability, and it is difficult to determine a precise settling time through design. Furthermore, from a mathematical perspective, the dynamics of a system subjected to external disturbances often enter a limit cycle after being processed with high gain (such as increasing the proportional term), ultimately only achieving bounded stability.

[0004] Adaptive control, which enables online automatic updating of control parameters to adapt to disturbances, unmodeled dynamics, and input saturation, is an effective solution. Based on the deterministic equivalence principle, the adaptive strategy can be viewed as a perfect cancellation approach, converging the estimated values ​​to a manifold surface determined by the system error and regression matrix, thus providing asymptotically stable control for the controlled object.

[0005] The development of constraint control methods aims to solve the problem of "how to ensure that the relative motion state evolves within a preset range while also guaranteeing a designable convergence time?" In other words, constraint control enables the complete dynamic process of relative motion between spacecraft to be pre-defined by the algorithm user in terms of time span, state overshoot, and convergence performance, preventing violations. However, the realization of constraint characteristics relies on the rapid increase of control output to suppress the trend of system state motion, inevitably creating a contradiction with the upper limit of the execution capability of actual spacecraft. Summary of the Invention

[0006] The technical problem solved by this invention is to overcome the shortcomings of the prior art and propose an elastic constraint adaptive relative motion control method that adapts to spacecraft input saturation. This method can achieve asymptotically stable and actually finite-time stable results under the constraints of preset constraints for the relative motion control of multiple spacecraft. The constraint boundary can automatically absorb the influence of input saturation, automatically relax in a short time, and converge to the original constraint range as the input saturation disappears.

[0007] The solution of the present invention is:

[0008] An elastically constrained adaptive relative motion control method for adapting to spacecraft input saturation includes:

[0009] Step 1: Establish the orbital coordinate system {O} l ,x l ,y l ,z l}, i.e., LVLH coordinate system; establish the dynamic equations of relative motion of the spacecraft;

[0010] Step 2: Based on the spacecraft relative motion dynamics equations obtained in Step 1, define the system error. The control objective is to make the system error approach zero.

[0011] Step 3: Based on the system error determined in Step 2, design constraints on the error evolution during the dynamic process of the system; these constraints should be able to adapt to input saturation and prevent the system error from exceeding the constraint range due to input saturation.

[0012] Step 4: Define a scaling-constrained Lyapunov function, incorporate the elastic constraints determined in Step 3 into the Lyapunov function for stability analysis, and use it as the mathematical basis for the design of adaptive control.

[0013] Step 5: Based on the Lyapunov function proposed in Step 4, design an adaptive controller;

[0014] Step 6: Design scaling factor dynamics to provide a feedforward effect independent of the controller design, eliminating the influence of filtering error introduced by the first-order filtering process in the backstepping method on the asymptotic stability of the system; based on the self-adjusting elastic bound, scaling constraint type Lyapunov function, adaptive controller, and scaling factor dynamics designed above, and based on the spacecraft relative motion dynamics model, the system achieves asymptotic stability under the influence of input saturation and uncertainty, and the system dynamics always meet the preset elastic constraints.

[0015] In the aforementioned elastically constrained adaptive relative motion control method for adapting to spacecraft input saturation, in step one, the orbital coordinate system {O} l ,x l ,y l ,zl The method for creating} is as follows:

[0016] Origin O l A reference point located in the space orbit; connecting the Earth's center of mass O and O l The position vector is R c , then y l The axis starts from O l And extend R c The direction, z l The axis and the position vector R c The normal to the plane defined by the velocity direction of the reference spacecraft coincides with x. l The axes are respectively with y l axis and z l The axes are perpendicular and form a right-handed coordinate system.

[0017] In the aforementioned adaptive relative motion control method for spacecraft input saturation with elastic constraints, the relative motion dynamics of the multi-spacecraft system are established on the reference point and its LVLH coordinate system that satisfy the orbital operation law. The control method realizes the coordinated motion of multiple spacecraft about this point. By applying the principle of virtual work to introduce the external force on the dynamic system, the nonlinear dynamics of any tracking spacecraft relative to the reference point are integrated into the following Euler-Lagrange form:

[0018]

[0019] In the formula, To track the mass diagonal matrix of a spacecraft, the diagonal elements are the mass m of the spacecraft. f ; This represents the position vector of the tracking spacecraft relative to a reference point, with the subscript f representing the tracking spacecraft and the subscripts x, y, and z indicating the position of the vector within {x, y, z}. l ,y l ,z l} Components of the coordinate axes; Indicates the control input for tracking spacecraft; F df The vector is formed by the projection of the perturbation onto the LVLH coordinate axes. It is a Coriolis matrix; Represented as:

[0020]

[0021] Represented as:

[0022]

[0023] Linear parameterization of the uncertainty in formula (1) yields:

[0024]

[0025] Regression Matrix and parameter vector θ f They are respectively:

[0026]

[0027] in, All are matrix coefficients.

[0028] In the above-mentioned elastic constraint adaptive relative motion control method for spacecraft input saturation, formula (1) is extended to the case of n spacecraft. Let the subscripts be represented as f = 1, ..., n, and the position of the relative reference point of each spacecraft be represented as q1, ..., q n ;

[0029] Further define the state vector of the multi-spacecraft system The second-order differential equation can be rearranged according to formula (1) as follows:

[0030]

[0031] In the formula, All are diagonal block matrices; their diagonal parts are the H values ​​of each of the n spacecraft. f C f matrix;

[0032] It consists of n spacecraft, each with its own N f ,F df ,u f The vector formed by the arrangement;

[0033] The constraint of the mass matrix H in formula (4) is: for any real vector satisfy in These are the constant value limits;

[0034] Let the velocity vector Substituting the linear parameterized expression (4) for each spacecraft into equation (6), the multi-spacecraft relative motion dynamics system is transformed into a canonical form:

[0035]

[0036] in,

[0037] The input saturation of the relative motion dynamics equations of multiple spacecraft, i.e., equation (7), is expressed as the control vector. Each element u in i ={u xi,u yi ,u zi All are limited to:

[0038]

[0039] In the formula, This represents the input signal calculated by the controller;

[0040] and u imin yes The upper and lower limits.

[0041] In the aforementioned elastically constrained adaptive relative motion control method for spacecraft input saturation, in step two, since multiple spacecraft do not necessarily maintain pairwise information exchange, but rather maintain directed full coverage of information, algebraic graph theory is used to establish the system error; the graph is a data structure composed of vertex sets and edge sets, denoted by G(V,E,A); where V={v1,v2,…,v…} n} represents a set of n nodes; each v i Corresponding to one node; Represents an edge set; e ij This corresponds to an edge that connects a pair of nodes; A represents the mapping from a set of edges to pairs of nodes, that is, the correspondence between each edge and its two endpoints, written as A:E→{{v i ,v j}|v i ,v j ∈V}, its mathematical description is an adjacency matrix, that is, when there is directed communication between the i-th spacecraft and the j-th spacecraft, the element a in the i-th row and j-th column of matrix A is... ij =1, otherwise 0;

[0042] Based on algebraic graph theory, the motion error signal of multiple spacecraft relative to a reference point is represented as... A single element expands as follows:

[0043]

[0044] Among them, b i =1 indicates the target spacecraft; otherwise, it is b. i =0;

[0045] q0 is the expected trajectory of the reference point; if it is 0, it means the reference point is fixed; if it is a time-varying function, it means the trajectory of the reference point.

[0046] In the above-mentioned adaptive relative motion control method with elastic constraints to adapt to spacecraft input saturation, in step three, the constraint condition is named the self-adjusting elastic bound and is represented as a smooth function h(t).

[0047] The smooth function h(t) satisfies:

[0048] S1. For all t≥0, h(t)>0;

[0049] S2. If input saturation does not occur, then And when t≥t f hour, Keep constant, where t f It is the preset stabilization time;

[0050] S3, when t s When saturation occurs, in It is a nonnegative scalar function that satisfies and as well as Where a1, a2 > 0 and

[0051] In the above-mentioned adaptive relative motion control method for spacecraft input saturation with elastic constraints, the self-adjusting elastic boundary is an elastic constraint. The time-varying state constraint effect expected by the control algorithm is satisfied as follows: when the system inevitably experiences input saturation, causing the actual control quantity to be unable to suppress the growth trend of system error in time, the constraint range automatically increases until the system error decreases and the input saturation subsides, and then the constraint automatically returns to the originally set range.

[0052] From a mathematical perspective, the expression for the self-adjusting elastic bound h(t) contains... It is a second-order homogeneous differential equation with two real roots; The general solution is a superposition of exponential functions; let and ensure The curve starts growing from zero and then gradually decays back to zero; therefore, when When stimulated, the value of the self-adjusting elastic bound automatically increases and returns to its original value when the stimulus disappears, exhibiting elastic change behavior.

[0053] Considering the motion of any spacecraft in any direction, the constraint is achieved by limiting the change of system error within a preset range. The state constraint characteristics of the system are obtained by inversely deriving equation (9). Setting the self-adjusting elastic bound in an asymmetric manner can increase the flexibility of the constraint, that is, the control error in each direction can be constrained to different limits in the positive and negative intervals. The asymmetric constraint is to arrange the self-adjusting elastic bound into two sets of vectors. The subscripts u and l represent the upper and lower bounds, respectively, and the specific expressions for each element are as follows:

[0054]

[0055] In the formula, ρ iu ρ il These represent the preset positive and negative direction constraints for monotonically convergent convergence, respectively.

[0056] This represents the elastic constraint effect of a self-adjusting elastic boundary that can adapt to input saturation;

[0057] For ease of description, let the subscript *∈{iu,il} and the second-level subscript j=1,2,3 represent the element order of the vector. The first half of the self-adjusting elastic bound is designed as follows:

[0058]

[0059] In the formula, and All are non-negative constants;

[0060] when When the value is zero, expression (10) contains only the first half of the monotonically non-increasing signal; otherwise, the unreachable quantity of the calculated required input signal exceeding the actual input limit will be included in the expression. and In the middle; the dynamic description of both is as follows:

[0061]

[0062] In the formula, Λ il and Λ iu It is a negative diagonal gain matrix;

[0063] Γ1 and Γ2 are positive diagonal gain matrices, and all others are non-negative;

[0064]

[0065] Therefore, for a given constant vector If the unreachable quantity of the control input satisfies This holds true for all t≥0, then This holds true for all k∈{1,2,3}; where Γ3=I3 is a 3-dimensional identity matrix;

[0066] The initial value of the self-adjusting elastic bound at time zero Greater than the initial value of the system error e 1i (0).

[0067] In the aforementioned adaptive relative motion control method with elastic constraints to adapt to spacecraft input saturation, the specific method for incorporating the elastic constraint conditions into the stability analysis and using it as the mathematical basis for the design of the adaptive control in step four is as follows:

[0068] Define a scaling-constrained Lyapunov function, which contains a scaling factor r. 1i (t), whose complete expression is:

[0069]

[0070] In the formula, the second-level subscript j∈{1,2,3} represents e 1i The order of elements;

[0071] For all All satisfy V1≥0;

[0072] Taking the derivative of (14) gives:

[0073]

[0074] That is, by

[0075]

[0076] The dominant part is through design This improves the derivative form of the Lyapunov function, thereby enhancing the system's control performance.

[0077] In the above-described adaptive relative motion control method for spacecraft input saturation under elastic constraints, the design method of the adaptive controller in step five is as follows:

[0078] The virtual control signal for the first-order system is designed using the backstepping method; a first-order low-pass filter is used. output signal The virtual control quantity ξ that replaces the traditional backstep design i Where the constant coefficient c > 0; let the virtual control signal be:

[0079]

[0080] In the formula, It is the adjacency matrix element a corresponding to the i-th spacecraft. ij The sum of; The input is:

[0081]

[0082] In the formula, k 1i It is a positive control gain;

[0083] k ri It is a positive feedback coefficient;

[0084] remember This is the filtering error;

[0085] Regarding the input saturation problem, there is a [condition / condition] regardless of whether input saturation occurs. The expected control input without amplitude limitation is:

[0086]

[0087] Where, δ 1i (t), δ 2i (t) are all self-definable positive scalar functions; and satisfy boundedness. and

[0088] k 2i >0 controls the gain; It is an adaptive estimator;

[0089] A control component is designed based on the following inequality: for any real vector x, there exists a positive scalar real variable function δ(t) that satisfies:

[0090]

[0091] Furthermore, the adaptive estimator updates the unknown parameter vector θ in real time based on the linear parameterized expression in step one, i.e., formula (4). i The estimated value Its dynamic representation is as follows:

[0092]

[0093] Where σ is a positive correction coefficient, e 2i It is e 1i The derivative of .

[0094] In the above-mentioned elastically constrained adaptive relative motion control method for adapting to spacecraft input saturation, in step six, the scaling factor in formula (16) dynamically... Independent design forms a feedforward, eliminating the filtering error α introduced by combining virtual control quantities with filters. i The effect; let r 1i (0)≥1, design for:

[0095]

[0096] Where, k ri It is a positive feedback coefficient;

[0097] ||M i e 1i || is in formula (16) The modulus of the coefficient matrix, and:

[0098]

[0099] The advantages of this invention compared to the prior art are:

[0100] (1) The method proposed in this invention for the control problem in the key technology breakthrough achieves fast, high-precision, and preset system dynamic relative motion control, and reconciles the contradiction between performance-constrained control and actual input saturation.

[0101] (2) The present invention designs an elastic constraint, which is mathematically represented as a monotonic curve whose convergence rate and convergence time can be determined by parameter settings;

[0102] (3) The present invention designs a scaling constraint type Lyapunov function, which can be used as a stability analysis framework to derive the basic structure of the control algorithm;

[0103] (4) This invention implements the design of the scaling factor, which is nested in a scaling-constrained Lyapunov function. The controller-independent design effectively avoids the cumbersome backstepping derivative operation. The algorithm is simple to implement, easy to solve, and can achieve asymptotic and finite-time stability of the system. Attached Figure Description

[0104] Figure 1 This is a flowchart of the elastic constraint adaptive relative motion control of the present invention;

[0105] Figure 2 This is a schematic diagram illustrating the definition of the coordinate system for the close-range relative motion of the spacecraft in this invention. Detailed Implementation

[0106] The present invention will be further described below with reference to the embodiments.

[0107] This invention provides an elastically constrained adaptive relative motion control method to adapt to spacecraft input saturation. It achieves fast, high-precision, and pre-programmable relative motion control of the system dynamics, and reconciles the contradiction between performance-constrained control and actual input saturation. The algorithm is simple to implement, easy to solve, and can achieve asymptotic and finite-time stability of the system.

[0108] Elastically constrained adaptive relative motion control methods that can adapt to spacecraft input saturation, such as Figure 1 As shown, the specific steps include the following:

[0109] 1. Establish the orbital coordinate system {O l ,x l ,y l ,z l}, that is, the LVLH coordinate system, such as Figure 2 As shown. Establish the dynamic equations of relative motion of the spacecraft;

[0110] Orbital coordinate system {Ol ,x l ,y l ,z l The method for creating} is as follows:

[0111] Origin O l A reference point located in the space orbit; connecting the Earth's center of mass O and O l The position vector is R c , then y l The axis starts from O l And extend R c The direction, z l The axis and the position vector R c The normal to the plane defined by the velocity direction of the reference spacecraft coincides with x. l The axes are respectively with y l axis and z l The axes are perpendicular and form a right-handed coordinate system. The relative motion dynamics of the multi-spacecraft system are established on the aforementioned reference point that satisfies the orbital laws and its LVLH coordinate system. The coordinated motion of multiple spacecraft about this point is achieved through control methods.

[0112] Based on the definition of the LVLH coordinate system, the relative motion dynamics of the spacecraft are established. The generalized coordinates are established on the LVLH coordinate system of the reference point. Using the principle of virtual work, the external forces acting on the dynamic system are introduced. The nonlinear dynamics of any tracking spacecraft relative to the reference point can be integrated into the following Eulerian-Lagrange form:

[0113]

[0114] in, This represents the position vector of the tracking spacecraft relative to a reference point, with the subscript f representing the tracking spacecraft and the subscripts x, y, and z indicating the position of the vector within {x, y, z}. l ,y l ,z l} Components of the coordinate axes. Indicates the control input for tracking spacecraft. Let be the true anomaly angular velocity, μ be the gravitational coefficient of the central body, and r be the angular velocity at the true anomaly. l =||R c ||(Not in the formula). This is a diagonal matrix representing the mass of a spacecraft, where the diagonal elements are the mass m of the spacecraft. f The vector formed by the projection of the perturbation onto the LVLH coordinate axes; It is a Coriolis matrix, represented as:

[0115]

[0116] It includes nonlinear terms, namely:

[0117]

[0118] as well as It is the vector formed by the projection of the disturbance onto the LVLH coordinate axes. The uncertainty in equation (1) is linearly parameterized, i.e.:

[0119]

[0120] The regression matrix and parameter vector are as follows:

[0121]

[0122] in:

[0123]

[0124] All are matrix coefficients;

[0125] Extending equation (1) to the case of n spacecraft, let the subscripts be f = 1, ..., n, and the position of each spacecraft relative to the reference point can be represented as q1, ..., q1. n Further define the state vector of the multi-spacecraft system. The second-order differential equation can then be rearranged according to equation (1) as follows:

[0126]

[0127] in, All are diagonal block matrices, with their diagonal parts being the H values ​​of each of the n spacecraft. f C f matrix, It consists of n spacecraft, each with its own N f ,F df ,u f The vectors formed by the arrangement. The mass matrix H in equation (4) is unknown in the control method of this patent because the actual mass of the spacecraft in orbit is uncertain due to fuel consumption. In engineering, a rough range is obtained by calculating the fuel consumption in stages. Therefore, the boundedness assumption is always satisfied, that is, for any real vector satisfy in These are the constant upper and lower limits, respectively.

[0128] Let the velocity vector Substituting the linear parameterized expression (4) for each spacecraft into equation (6), the multi-spacecraft relative motion dynamics system can be transformed into a canonical form:

[0129]

[0130] in as well as

[0131] The input saturation of the multi-spacecraft relative motion dynamics equation (7) is expressed as the control vector. Each element u in i ={u xi ,u yi ,u zi All are limited to:

[0132]

[0133] in, This represents the input signal calculated by the controller, while and u imin yes The upper and lower limits.

[0134] 2. Based on the system dynamics equations derived in step 1, define the system error. The control objective is to make this error approach zero. Since multiple spacecraft do not necessarily need to maintain pairwise information exchange, but only need to maintain directed information coverage across all spacecraft, it is necessary to establish the system error using algebraic graph theory. A graph is a data structure composed of a set of nodes (or simply a set of nodes) and a set of edges (or simply a set of edges), denoted by G(V,E,A), where V={v1,v2,…,v…} n} represents a set of n nodes, where each v i Corresponding to one node; Denotes the edge set, e ij This corresponds to an edge that connects a pair of nodes; This represents the mapping from a set of edges to pairs of nodes, that is, the correspondence between each edge and its two endpoints, written as... Its mathematical description is an adjacency matrix, that is, when there is directed communication between the i-th spacecraft and the j-th spacecraft, the element a in the i-th row and j-th column of matrix A is... ij =1, otherwise 0. Based on algebraic graph theory, the motion error signal of multiple spacecraft relative to a reference point is represented as: A single element expands as follows:

[0135]

[0136] Where b i =1 indicates the target spacecraft (or virtual center point); otherwise, it is b. i =0, q0 is the expected trajectory of the reference point. If it is 0, it means that the reference point is fixed. If it is a time-varying function, it means that the reference point is moving.

[0137] 3. Based on the system error determined in step 2, design constraints for the error evolution during the system's dynamic process. These constraints need to be able to adapt to input saturation, preventing the system error from exceeding the constraint range due to input saturation. The constraint is named the self-adjusting elastic bound, which is one of the core technical points of this patent. It is represented by a smooth function h(t) and must satisfy: 1) For all t≥0, h(t)>0; 2) If input saturation does not occur, then... And when t≥t f hour, Keep constant, where t f It is the preset settling time; 3) when t s When saturation occurs, in It is a nonnegative scalar function that satisfies and as well as Where a1, a2 > 0 and

[0138] Self-adjusting elastic bound is a type of elastic constraint. Specifically, it refers to the constraint effect that the control algorithm expects to achieve: when the system inevitably experiences input saturation, causing the actual control quantity to be unable to suppress the growth trend of system error in time, the constraint range automatically increases until the system error decreases and the input saturation subsides, at which point the constraint automatically returns to the originally set range.

[0139] From a mathematical perspective, the expression for the self-adjusting elastic bound h(t) contains... It is a second-order homogeneous differential equation with two real roots as its characteristic equation. This means The general solution is a superposition of exponential functions. Let and ensure The curve, after increasing from zero, can gradually decay back to zero. Therefore, when When stimulated, the value of the self-adjusting elastic bound automatically increases and returns to its original value when the stimulus disappears, exhibiting elastic change behavior.

[0140] Considering the motion of any spacecraft in any direction, the constraint is achieved by limiting the variation of system error within a preset range. The state constraint characteristics of the system are obtained by inversely applying equation (9). Using an asymmetric approach to set the self-adjusting elastic bounds can increase the flexibility of the constraints, meaning that the control error in each direction can be constrained to different degrees in both positive and negative intervals. Asymmetric constraints arrange the self-adjusting elastic bounds into two sets of vectors. The subscripts u and l represent the upper and lower bounds, respectively, and the specific expressions for each element are as follows:

[0141]

[0142] The first half of the expression on the right side of the equal sign in the above equation is ρ iu (or ρ) il The ) indicates a pre-defined positive (or negative) directional constraint condition for monotonically convergent convergence; the latter part... (or ) represents the elastic constraint effect of a self-adjusting elastic boundary that can adapt to input saturation.

[0143] For ease of description, let the subscript *∈{iu,il} and the second-level subscript j=1,2,3 represent the element order of the vector. The first half of the self-adjusting elastic bound is designed as follows:

[0144]

[0145] in, and All are nonnegative constants. When When the value is zero, expression (10) contains only the first half of the monotonically non-increasing signal; otherwise, the calculated required input signal exceeding the actual input limit's "unreachable quantity" (i.e., the portion of the theoretically required input amplitude exceeding the upper and lower saturation limits under infinite amplitude conditions) will be included in... and The dynamic descriptions of both are as follows:

[0146]

[0147] Among them, Λ il and Λ iu Γ1 and Γ2 are negative diagonal gain matrices, while Γ1 and Γ2 are positive diagonal gain matrices. Furthermore:

[0148] All are non-negative. Therefore, for a given constant vector... If the "unreachable" condition of the control input is satisfied This holds true for all t≥0, then This holds true for all k∈{1,2,3}, where Γ3=I3 is a 3D identity matrix. Note the initial value of the self-adjusting elastic bound at time zero. It needs to be greater than the initial value of the system error e. 1i (0).

[0149] 4. Step 3 determined the expression for the elastic constraint, and its effect needs to be reflected in the stability analysis, serving as the mathematical basis for the adaptive controller design. Defining a scaling constraint-type Lyapunov function is the second core technical point of this patent; this function contains a scaling factor r. 1i (t), whose complete expression is:

[0150]

[0151] Where the second-level subscript j∈{1,2,3} represents e 1i The order of elements. For all All satisfy V1≥0. The typical characteristic of the scaling factor is that after dividing a variable by the scaling factor, the resulting new variable can express the dynamic characteristics of the original variable, and can also extract the component dominated by the scaling factor during the dynamic analysis (and derivative calculation) of the original variable. The scaling factor constructs a designable signal independent of the controller to provide additional feedforward for the system. The scaling-constrained Lyapunov function takes advantage of the above advantages to provide favorable conditions for simplifying the design of the control algorithm. Taking the derivative of (14) yields:

[0152]

[0153] It can be seen from:

[0154]

[0155] The dominant part can be designed This improves the derivative form of the Lyapunov function, thereby enhancing the system's control performance.

[0156] 5. Building upon the scaling-constrained Lyapunov function proposed in the previous step, an adaptive controller is designed. The virtual control signal for the first-order system is designed using the backstepping method, but its computational complexity needs to be avoided (i.e., calculating the derivative becomes extremely difficult as the system's nonlinearity increases). The classical backstepping method involves calculating the dynamics of the acceleration hierarchy; considering the self-adjusting elastic bound constraints of asymmetric pairs, especially the characteristics of the second-order homogeneous equations, differentiating the system expressions one by one is not optimal. Therefore, a first-order low-pass filter is used. output signal The virtual control quantity ξ that replaces the traditional backstep design i (The meaning of the virtual control quantity in the backstepping method is a well-known concept in the field and will not be elaborated further.) The constant coefficient c > 0, which is the solution for the filtered output. Since it already includes an integral operation, there's no need to calculate complex derivative terms during the design of the backstep feedforward term. The mathematical expression of a first-order low-pass filter is a first-order constant-coefficient differential equation. Thanks to its stable tracking characteristics, the virtual control signal can be:

[0157]

[0158] in It is the adjacency matrix element a corresponding to the i-th spacecraft. ij The sum of, and the input is:

[0159]

[0160] Where, k 1i It is a positive control gain, k ri It is a positive feedback coefficient, denoted as... This represents the filtering error. For the input saturation problem, this applies regardless of whether input saturation occurs. Therefore, the expected control input without amplitude limitation is:

[0161]

[0162] Where, δ 1i (t), δ 2i (t) are all self-defined positive scalar functions, which only need to satisfy boundedness. and (in and δ 1i (It is a positive constant), such as a fractional function or a negative power of the natural exponent e, k 2i >0 is the control gain. It is an adaptive estimator. The reason why the above controller contains three sets of fractions is to ensure the asymptotic stability of the system. The control component is designed based on the following inequality: for any real vector x, there exists a positive scalar real-variable function δ(t) that satisfies:

[0163]

[0164] Furthermore, the adaptive estimator updates the unknown parameter vector θ in real time based on the linear parameterization expression (4) in step (1). i The estimated value Its dynamic representation is as follows:

[0165]

[0166] 6. Based on the design of the scaling factor dynamics in steps 4 and 5, a feedforward effect independent of the controller design is provided, eliminating the influence of filtering error introduced by the first-order filtering process in the backstepping method on the asymptotic stability of the system. The dynamics of the scaling factor in equation (16) were mentioned in step 4. Independent design to form feedforward, eliminating the filtering error α introduced by combining virtual control quantity with filter. i The effect of r. 1i (0)≥1, design the following dynamics:

[0167]

[0168] Where, k ri It is a positive feedback coefficient. For ease of expression, ||M i e 1i || is in equation (16) The modulus of the coefficient matrix, and

[0169]

[0170] Through the above calculations and designs, including the self-adjusting elastic bound, the scaling constraint-type Lyapunov function, the adaptive controller, and the scaling factor dynamics, the system can achieve asymptotic stability under the influence of input saturation and uncertainty based on the spacecraft relative motion dynamics model, and the system dynamics always meet the preset elastic constraints.

[0171] This invention first designs an elastic constraint, mathematically represented as a monotonic curve whose convergence rate and time can be clearly defined by parameter settings. Second, it designs a scaling constraint-type Lyapunov function, which serves as the framework for stability analysis, allowing the derivation of the basic structure of the control algorithm. Finally, it designs a scaling factor nested within the scaling constraint-type Lyapunov function. This controller-independent design effectively avoids the cumbersome backstepping derivative calculation.

[0172] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention by utilizing the methods and techniques disclosed above without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

Claims

1. An elastically constrained adaptive relative motion control method for spacecraft input saturation, characterized in that: include: Step 1: Establish the orbital coordinate system {O} l ,x l ,y l ,z l }, i.e., LVLH coordinate system; establish the dynamic equations of relative motion of the spacecraft; Step 2: Based on the spacecraft relative motion dynamics equations obtained in Step 1, define the system error. The control objective is to make the system error approach zero. Step 3: Based on the system error determined in Step 2, design constraints on the error evolution during the dynamic process of the system; these constraints should be able to adapt to input saturation and prevent the system error from exceeding the constraint range due to input saturation. Step 4: Define a scaling-constrained Lyapunov function, incorporate the elastic constraints determined in Step 3 into the Lyapunov function for stability analysis, and use it as the mathematical basis for the design of adaptive control. Step 5: Based on the Lyapunov function proposed in Step 4, design an adaptive controller; Step 6: Design scaling factor dynamics to provide a feedforward effect independent of the controller design, eliminating the influence of filtering error introduced by the first-order filtering process in the backstepping method on the asymptotic stability of the system; based on the self-adjusting elastic bound, scaling constraint type Lyapunov function, adaptive controller, and scaling factor dynamics designed above, and based on the spacecraft relative motion dynamics model, the system achieves asymptotic stability under the influence of input saturation and uncertainty, and the system dynamics always meet the preset elastic constraints.

2. The elastically constrained adaptive relative motion control method for spacecraft input saturation according to claim 1, characterized in that: In step one, the orbital coordinate system {O l ,x l ,y l ,z l The method for creating} is as follows: Origin O l A reference point located in the space orbit; connecting the Earth's center of mass O and O l The position vector is R c , then y l The axis starts from O l And extend R c The direction, z l The axis and the position vector R c The normal to the plane defined by the velocity direction of the reference spacecraft coincides with x. l The axes are respectively with y l axis and z l The axes are perpendicular and form a right-handed coordinate system.

3. The elastic constraint adaptive relative motion control method for spacecraft input saturation according to claim 2, characterized in that: The relative motion dynamics of the multi-spacecraft system are established on the aforementioned reference point that satisfies the orbital laws and its LVLH coordinate system. Coordinated motion of multiple spacecraft about this point is achieved through control methods. The principle of virtual work is used to introduce external forces acting on the dynamic system, and the nonlinear dynamics of any tracking spacecraft relative to the reference point are integrated into the following Eulerian-Lagrange form: In the formula, To track the mass diagonal matrix of a spacecraft, the diagonal elements are the mass m of the spacecraft. f ; This represents the position vector of the tracking spacecraft relative to a reference point, with the subscript f representing the tracking spacecraft and the subscripts x, y, and z indicating the position of the vector within {x, y, z}. l ,y l ,z l } Components of the coordinate axes; Indicates the control input for tracking spacecraft; F df The vector is formed by the projection of the perturbation onto the LVLH coordinate axes. It is a Coriolis matrix; Represented as: Represented as: Linear parameterization of the uncertainty in formula (1) yields: Regression Matrix and parameter vector θ f They are respectively: in, All are matrix coefficients.

4. The elastically constrained adaptive relative motion control method for spacecraft input saturation according to claim 3, characterized in that: Extending formula (1) to the case of n spacecraft, let the subscripts be f = 1, ..., n, and the position of each spacecraft relative to the reference point be q1, ..., q1. n ; Further define the state vector of the multi-spacecraft system The second-order differential equation can be rearranged according to formula (1) as follows: In the formula, All are diagonal block matrices; their diagonal parts are the H values ​​of each of the n spacecraft. f C f matrix; It consists of n spacecraft, each with its own N f ,F df ,u f The vector formed by the arrangement; The constraint of the mass matrix H in formula (4) is: for any real vector satisfy in These are the constant value limits; Let the velocity vector Substituting the linear parameterized expression (4) for each spacecraft into equation (6), the multi-spacecraft relative motion dynamics system is transformed into a canonical form: in, The input saturation of the relative motion dynamics equations of multiple spacecraft, i.e., equation (7), is expressed as the control vector. Each element u in i ={u xi ,u yi ,u zi All are limited to: In the formula, τ *i This represents the input signal calculated by the controller; and u imin is u *i The upper and lower limits.

5. The elastically constrained adaptive relative motion control method for spacecraft input saturation according to claim 3, characterized in that: In step two, since multiple spacecraft do not necessarily maintain pairwise information exchange, but rather ensure directed full coverage of information, algebraic graph theory is used to establish the system error. The graph is a data structure composed of a set of vertices and a set of edges, denoted by G(V,E,A); where V={v1,v2,…,v…} n } represents a set of n nodes; each v i Corresponding to one node; Represents an edge set; e ij This corresponds to an edge that connects a pair of nodes; A represents the mapping from a set of edges to pairs of nodes, that is, the correspondence between each edge and its two endpoints, written as A:E→{{v i ,v j }|v i ,v j ∈V}, its mathematical description is an adjacency matrix, that is, when there is directed communication between the i-th spacecraft and the j-th spacecraft, the element a in the i-th row and j-th column of matrix A is... ij =1, otherwise 0; Based on algebraic graph theory, the motion error signal of multiple spacecraft relative to a reference point is represented as... A single element expands as follows: Among them, b i =1 indicates the target spacecraft; otherwise, it is b. i =0; q0 is the expected trajectory of the reference point; if it is 0, it means the reference point is fixed; if it is a time-varying function, it means the trajectory of the reference point.

6. The elastically constrained adaptive relative motion control method for spacecraft input saturation according to claim 5, characterized in that: In step three, the constraint condition is named the self-adjusting elastic bound and is represented by the smooth function h(t). The smooth function h(t) satisfies: S1. For all t≥0, h(t)>0; S2. If input saturation does not occur, then And when t≥t f hour, Keep constant, where t f It is the preset stabilization time; S3, when t s When saturation occurs, in It is a nonnegative scalar function that satisfies and as well as Where a1, a2 > 0 and 7. The elastically constrained adaptive relative motion control method for spacecraft input saturation according to claim 6, characterized in that: The self-adjusting elastic boundary is an elastic constraint that the time-varying state constraint effect expected by the control algorithm is satisfied as follows: when the system inevitably experiences input saturation, causing the actual control quantity to be unable to suppress the growth trend of system error in time, the constraint range automatically increases until the system error decreases and the input saturation subsides, and then the constraint automatically returns to the originally set range. From a mathematical perspective, the expression for the self-adjusting elastic bound h(t) contains... It is a second-order homogeneous differential equation with two real roots; The general solution is a superposition of exponential functions; let and ensure The curve starts growing from zero and then gradually decays back to zero; therefore, when When stimulated, the value of the self-adjusting elastic bound automatically increases and returns to its original value when the stimulus disappears, exhibiting elastic change behavior. Considering the motion of any spacecraft in any direction, the constraint is achieved by limiting the change of system error within a preset range. The state constraint characteristics of the system are obtained by inversely deriving equation (9). Setting the self-adjusting elastic bound in an asymmetric manner can increase the flexibility of the constraint, that is, the control error in each direction can be constrained to different limits in the positive and negative intervals. The asymmetric constraint is to arrange the self-adjusting elastic bound into two sets of vectors. The subscripts u and l represent the upper and lower bounds, respectively, and the specific expressions for each element are as follows: In the formula, ρ iu ρ il These represent the preset positive and negative direction constraints for monotonically convergent convergence, respectively. This represents the elastic constraint effect of a self-adjusting elastic boundary that can adapt to input saturation; For ease of description, let the subscript *∈{iu,il} and the second-level subscript j = 1,2,3 represent the element order of the vector. The first half of the self-adjusting elastic bound is designed as follows: In the formula, and All are non-negative constants; when When the value is zero, expression (10) contains only the first half of the monotonically non-increasing signal; otherwise, the unreachable quantity of the calculated required input signal exceeding the actual input limit will be included in the expression. and In the middle; the dynamic description of both is as follows: In the formula, Λ il and Λ iu It is a negative diagonal gain matrix; Γ1 and Γ2 are positive diagonal gain matrices, and all others are non-negative; Therefore, for a given constant vector If the unreachable quantity of the control input satisfies This holds true for all t≥0, then This holds true for all k∈{1,2,3}; where Γ3=I3 is a 3-dimensional identity matrix; The initial value of the self-adjusting elastic bound at time zero Greater than the initial value of the system error e 1i (0).

8. The elastically constrained adaptive relative motion control method for spacecraft input saturation according to claim 6, characterized in that: In step four, the specific method for incorporating elastic constraints into stability analysis and using them as the mathematical basis for adaptive control design is as follows: Define a scaling-constrained Lyapunov function, which contains a scaling factor r. 1i (t), whose complete expression is: In the formula, the second-level subscript j∈{1,2,3} represents e 1i The order of elements; For all All satisfy V1≥0; Taking the derivative of (14) gives: That is, by The dominant part is through design This improves the derivative form of the Lyapunov function, thereby enhancing the system's control performance.

9. The elastically constrained adaptive relative motion control method for spacecraft input saturation according to claim 8, characterized in that: In step five, the design method for the adaptive controller is as follows: The virtual control signal for the first-order system is designed using the backstepping method; a first-order low-pass filter is used. output signal The virtual control quantity ξ that replaces the traditional backstep design i Where the constant coefficient c > 0; let the virtual control signal be: In the formula, It is the adjacency matrix element a corresponding to the i-th spacecraft. ij The sum of; The input is: In the formula, k 1i It is a positive control gain; k ri It is a positive feedback coefficient; remember This is the filtering error; Regarding the input saturation problem, there is a [condition / condition] regardless of whether input saturation occurs. The expected control input without amplitude limitation is: Where, δ 1i (t), δ 2i (t) are all self-definable positive scalar functions; and satisfy boundedness. and k 2i >0 controls the gain; It is an adaptive estimator; A control component is designed based on the following inequality: for any real vector x, there exists a positive scalar real variable function δ(t) that satisfies: Furthermore, the adaptive estimator updates the unknown parameter vector θ in real time based on the linear parameterized expression in step one, i.e., formula (4). i The estimated value Its dynamic representation is as follows: Where σ is a positive correction coefficient, e 2i It is e 1i The derivative of .

10. The elastically constrained adaptive relative motion control method for spacecraft input saturation according to claim 9, characterized in that: In step six, the scaling factor in formula (16) is dynamically... Independent design forms a feedforward, eliminating the filtering error α introduced by combining virtual control quantities with filters. i The effect; let r 1i (0)≥1, design for: Where, k ri It is a positive feedback coefficient; ||M i e 1i || is in formula (16) The modulus of the coefficient matrix, and:

Images