A method for determining the optimal posture of a rotatable panel under multi-target DOA estimation

Abstract

The application discloses a kind of optimal posture determination methods of rotatable panel under multi-target DOA estimation, belong to array signal processing and the field of integrated technology of perception.Firstly, the array of two degrees of freedom rotatable panel and multi-target signal model are constructed, the initial DOA estimation value of multi-target is obtained by MUSIC algorithm, the explicit mapping relationship of panel posture angle and DOA estimation Cramer-Lo lower bound CRB is deduced to establish, with the sum of multi-target CRB trace minimization as goal to construct with constraint optimization problem, finally optimal posture is solved using projection gradient descent method combined with Armijo backtracking line search.The application forms complete technical closed loop, solves the precision imbalance of traditional fixed array multi-target estimation, the problem that existing rotatable scheme has no global precision optimization mechanism, can significantly improve the overall precision and robustness of multi-target DOA estimation, widely applicable to unmanned aerial vehicle cluster monitoring, low-altitude security and other multi-target perception scene.

Description

Technical Field

[0001] This invention belongs to the fields of array signal processing and integrated sensing technology, specifically involving a method for determining the optimal attitude of a rotatable panel under multi-target DOA estimation, which can be applied to intelligent sensing scenarios such as precision monitoring of UAV swarms, airspace safety early warning, and multi-target positioning and tracking in low-altitude economic scenarios. Background Technology

[0002] With the rapid development of integrated sensing and computing technology, application scenarios such as low-altitude economy, drone swarm operations, and airspace safety management are constantly emerging. Base stations, sensing nodes, or integrated sensing and communication equipment need to estimate the Direction of Arrival (DOA) of multiple spatial targets simultaneously in order to achieve target discovery, angle positioning, trajectory tracking, and risk warning.

[0003] Most existing DOA estimation techniques are based on the premise that the array geometry is fixed. Traditional fixed arrays can only achieve good spatial resolution performance in a narrow beam range near the normal. When multiple targets are distributed in space and have a large directional span, fixed arrays cannot form the optimal observation geometry for all targets. Edge targets that are far from the normal will have problems such as reduced equivalent aperture, decreased steering vector sensitivity, and weakened parameter discriminability, which ultimately leads to a significant deterioration in DOA estimation accuracy.

[0004] Existing rotatable antenna or reconfigurable array solutions are mostly geared towards single-target alignment, link gain optimization, or communication performance improvement. They only increase the directional gain in the direction of a single target by adjusting the antenna orientation, without establishing a global perception accuracy optimization mechanism for multi-target scenarios. In multi-target DOA estimation scenarios, single-target alignment will cause the observation performance of other targets to deteriorate sharply, making it impossible to balance the estimation accuracy of multiple targets, resulting in insufficient overall system robustness.

[0005] More importantly, existing technologies have not formed a complete closed-loop technical logic: either they only stay at the array model construction and theoretical performance analysis without establishing a quantitative mapping between attitude adjustment and estimation accuracy; or they only propose a theoretical framework for attitude optimization without combining engineering constraints to provide a feasible solution method and application closed loop, and cannot be directly applied to actual multi-target perception scenarios. Summary of the Invention

[0006] The purpose of this invention is to provide a method for determining the optimal attitude of a rotatable panel under multi-target DOA estimation, which solves the problems of existing fixed arrays in multi-target perception scenarios, such as difficulty in balancing the estimation accuracy of all targets, significant performance degradation when some targets deviate from the array normal, and lack of an attitude optimization mechanism based on the overall CRB of multiple targets.

[0007] To achieve the above objectives, the technical solution of the present invention is implemented as follows: A method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation includes the following steps: S1: Construct the array geometry model, spatial rotation model and multi-target array signal receiving model of the two-DOF rotatable panel, and clarify the mapping relationship between the panel attitude angle, the target incident direction and the array guidance vector, so as to provide a physical and mathematical basis for subsequent DOA estimation and attitude optimization. S2: Based on the multi-target array received signal model constructed by S1, the received signals of multiple snapshot arrays are collected, and the covariance matrix of the received signals is decomposed into subspaces using the MUSIC algorithm. The two-dimensional DOA estimation results of all observed targets are obtained through two-dimensional spectral peak search, providing target direction input for subsequent accuracy quantization. S3: Based on the spatial rotation model of S1 and the target DOA estimation results obtained in S2, under the observation model of deterministic signal plus complex Gaussian white noise, the likelihood function and log-likelihood function of the array observation vector are constructed. By taking the partial derivative of the DOA parameter to be estimated, the Fisher information matrix FIM corresponding to the two-dimensional DOA parameter of each target is derived. The inverse of FIM is used to obtain the Cramer-Rao lower bound CRB of the DOA estimation of each target. The functional mapping relationship between the panel attitude angle and the lower bound of the multi-target DOA estimation accuracy is established. S4: Based on the multi-target CRB obtained from S3 and associated with the panel attitude angle, the sum of the traces of all target CRB matrices is used as the global cost function. Combined with the angular boundary constraints of the panel mechanical rotation and the feasibility constraints of the positive definiteness of the FIM of all targets, a constrained optimization problem with the best overall perception accuracy for multi-target is constructed. S5: The projected gradient descent method is used to solve the constrained optimization problem constructed in S4. The gradient expression of the global cost function with respect to the attitude angle of the panel is derived. In each iteration, the attitude angle is updated along the negative gradient direction. The updated attitude angle is mapped back to the preset feasible region through the projection operator. At the same time, the Armijo backtracking line search is combined to ensure the iterative descent property, and the FIM positive definiteness check is combined to ensure the feasibility of the iterative solution. After the iteration converges, the optimal attitude angle of the rotatable panel is output.

[0008] Furthermore, in step S1, the spatial rotation model is as follows: the panel first rotates clockwise by an angle β around the Z-axis of the global coordinate system, and then rotates counterclockwise by an angle α around the X-axis of the global coordinate system. After the rotation, the direction vector of the k-th target in the initial state panel coordinate system satisfies:

[0009] in, Let be the initial incident direction unit vector of target k. Let Z be the rotation matrix about the Z-axis. The rotation matrix about the X-axis is expressed as follows: ,

[0010] The expanded form of the direction vector of target k after rotation is:

[0011] in, These are the elevation and azimuth angles of the target k at its initial incident direction, respectively.

[0012] Furthermore, in step S1, in the multi-target array receiving signal model, the expression for the received signal of the (n,m)th array element to the kth target is:

[0013] Among them, s k Let λ be the transmitted signal of the k-th target, and x be the signal wavelength. n z is the distance from the origin to the nth element distributed along the x-axis. m It is the distance from the origin to the m-th element distributed along the z-axis, n n，m Let be the noise signal received by the (n,m)th antenna.

[0014] Furthermore, in step S2, the MUSIC algorithm execution process is as follows: constructing a reception matrix from the array received signals of L snapshots. Calculate the sample covariance matrix of the received signal. , for R Y Singular value decomposition yields the signal subspace U. s With noise subspace U n Construct the MUSIC spectral function:

[0015] Searching for the angles corresponding to the K largest spectral peaks in the two-dimensional angular domain yields the two-dimensional DOA estimation results for the K targets, where Nx and Nz are the number of array elements along the X and Z axes, respectively, A is the array manifold matrix, S is the signal matrix, and Z is the noise matrix. is the array guide vector, and K is the total number of targets.

[0016] Furthermore, in step S3, the FIM sub-block corresponding to the two-dimensional DOA parameters of the k-th target is a 2×2 real symmetric matrix, specifically expressed as:

[0017] in, For noise power, S represents the partial derivatives of the array steering vector corresponding to the k-th target with respect to the elevation and azimuth angles, respectively.k Let K be the transmitted signal corresponding to target k, and let CRB be the CRB matrix of the k-th target. k =J k -1 (α,β).

[0018] Furthermore, in step S3, the partial derivative expression of the steering vector component corresponding to the (n,m)th array element with respect to the DOA parameter is:

[0019]

[0020] in, Let be the steering vector component of the (n,m)th array element corresponding to the kth target. Let be the position vector of the (n,m)th array element in the global coordinate system after the panel is rotated.

[0021] Furthermore, in step S4, the standard form of the constrained optimization problem is:

[0022] in, , pitch angle The upper and lower limits of mechanical rotation, , Azimuth The upper and lower limits of mechanical rotation, This indicates that the FIM matrix of the k-th target is positive definite.

[0023] Furthermore, in step S5, the gradient expression of the global cost function with respect to the attitude angle is:

[0024] in, For the global cost function, , The FIM matrix of the k-th target versus the pitch angle Azimuth The partial derivatives of .

[0025] Furthermore, in step S5, the expression for the projection operator is:

[0026] Where C is the set of feasible regions for attitude angles.

[0027] Furthermore, in step S5, the execution process of Armijo backtracking search and FIM positive definiteness check is as follows: starting from the initial step size u=u0, the step size decay operation is repeatedly executed. This continues until the updated attitude angles simultaneously satisfy the positive definiteness of all target FIM matrices and the descent condition:

[0028]

[0029] Where c is the descent control parameter, with a value of 10. -4 , Let be the attitude angle vector of the t-th iteration. Let be the gradient vector for the t-th iteration.

[0030] Beneficial effects: (1) Complete logical closed loop and strong engineering feasibility. This invention forms a complete progressive logical chain from modeling, sensing, quantification, optimization, solution to application. Each step has a clear connection between the preceding and following steps, with no logical gaps. All technical features have clear mathematical and physical support and can be directly transplanted to actual rotatable array sensing systems.

[0031] (2) Optimal overall accuracy across multiple targets while maintaining balance. This invention optimizes the sum of the CRB traces of all targets rather than optimizing a single target. It can automatically match the spatial distribution of multiple targets and seek the optimal trade-off in accuracy among multiple targets, effectively alleviating the problem of accuracy degradation at the edge of fixed arrays. Simulation results show that in multi-target scenarios, this scheme reduces MSE by more than 40% compared to fixed panels and single-target alignment panels, and is closer to the theoretical lower bound of CRB.

[0032] (3) The optimization objective is clear and the solution is stable and reliable. This invention uses the theoretical accuracy lower bound CRB of DOA estimation as the optimization basis, rather than empirical indicators, and the physical meaning of the optimization objective is clear. At the same time, through projection operators, Armijo backtracking search and FIM positive definiteness check, the descent of the iterative process and the feasibility of the solution are guaranteed, avoiding local optima and algorithm divergence problems, and the convergence stability is significantly improved.

[0033] (4) High versatility and wide applicability. This invention is not limited to specific DOA estimation algorithms. In addition to the MUSIC algorithm, it can be extended to adapt to various DOA estimation frameworks such as ESPRIT, maximum likelihood estimation, and sparse reconstruction. At the same time, it can be extended to optimization objectives such as weighted CRB trace and minimum and maximum CRB minimization according to actual needs, and adapt to multi-target perception scenarios with different priorities. Attached Figure Description

[0034] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings: Figure 1This is a flowchart of the method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to an embodiment of the present invention; Figure 2 This is a diagram of a two-degree-of-freedom rotatable panel model in the method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to an embodiment of the present invention. Figure 3 This is an analysis diagram of a two-degree-of-freedom rotatable panel in the method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to an embodiment of the present invention. Figure 4 This is a schematic diagram of the target spatial distribution in the performance verification experiment of this invention, with the global coordinate system and the incident direction and angle parameters of the two test targets marked. Figure 5 This is a two-dimensional cross-sectional curve of the cost function in the performance verification experiment of this invention, including the α-dimensional cross-section when β=-0.00° and the β-dimensional cross-section when α=45.00°, with the optimal attitude point marked; Figure 6 This is a three-dimensional surface plot of the cost function in the performance verification experiment of this invention, with the coordinates of the global optimal attitude point and the corresponding cost function value marked. Figure 7 This is a graph comparing the MSE performance under different numbers of snapshots in the performance verification experiment of this invention. The horizontal axis is the number of snapshots T, and the vertical axis is MSE. It includes curves for 4 comparison schemes. Figure 8 This is a graph comparing the MSE performance under different target numbers in the performance verification experiment of this invention. The horizontal axis represents the target number K, and the vertical axis represents MSE. It includes curves for 4 comparison schemes. Figure 9 This is a graph comparing the MSE performance of different array element numbers in the performance verification experiment of this invention. The horizontal axis represents the number of array elements N, and the vertical axis represents MSE. It includes curves for 4 comparison schemes. Detailed Implementation

[0035] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0036] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0037] Example 1 See Figures 1-3 A method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation, comprising the following steps: S1: Construct the array geometry model, spatial rotation model and multi-target array signal receiving model of the two-DOF rotatable panel, and clarify the mapping relationship between the panel attitude angle, the target incident direction and the array guidance vector, so as to provide a physical and mathematical basis for subsequent DOA estimation and attitude optimization. S2: Based on the multi-target array received signal model constructed by S1, the received signals of multiple snapshot arrays are collected, and the covariance matrix of the received signals is decomposed into subspaces using the MUSIC algorithm. The two-dimensional DOA estimation results of all observed targets are obtained through two-dimensional spectral peak search, providing target direction input for subsequent accuracy quantization. S3: Based on the spatial rotation model of S1 and the target DOA estimation results obtained in S2, under the observation model of deterministic signal plus complex Gaussian white noise, the likelihood function and log-likelihood function of the array observation vector are constructed. By taking the partial derivative of the DOA parameter to be estimated, the Fisher information matrix FIM corresponding to the two-dimensional DOA parameter of each target is derived. The inverse of FIM is used to obtain the Cramer-Rao lower bound CRB of the DOA estimation of each target. The functional mapping relationship between the panel attitude angle and the lower bound of the multi-target DOA estimation accuracy is established. S4: Based on the multi-target CRB obtained from S3 and associated with the panel attitude angle, the sum of the traces of all target CRB matrices is used as the global cost function. Combined with the angular boundary constraints of the panel mechanical rotation and the feasibility constraints of the positive definiteness of the FIM of all targets, a constrained optimization problem with the best overall perception accuracy for multi-target is constructed. S5: The projected gradient descent method is used to solve the constrained optimization problem constructed in S4. The gradient expression of the global cost function with respect to the attitude angle of the panel is derived. In each iteration, the attitude angle is updated along the negative gradient direction. The updated attitude angle is mapped back to the preset feasible region through the projection operator. At the same time, the Armijo backtracking line search is combined to ensure the iterative descent property, and the FIM positive definiteness check is combined to ensure the feasibility of the iterative solution. After the iteration converges, the optimal attitude angle of the rotatable panel is output.

[0038] This embodiment constructs a spatial rotation model of a rotatable planar array, establishes the functional relationship between the rotational attitude and the array guidance vector, FIM (Fisher Information Matrix), and CRB (Cramer-Rao bound), and thus formulates the panel attitude optimization problem as a constrained non-convex optimization problem. It then solves the problem using methods such as projective gradient descent and backtracking search, thereby obtaining the optimal panel attitude that minimizes the overall DOA estimation error for multiple targets. This fundamentally solves the core problems of the imbalance in multi-target estimation accuracy of traditional fixed arrays and the lack of a global accuracy optimization mechanism in existing rotatable schemes. It achieves active matching between the panel attitude and the spatial distribution of multiple targets, significantly improving the overall accuracy and robustness of multi-target DOA estimation.

[0039] In a specific example, in step S1, the spatial rotation model is as follows: the panel first rotates clockwise by an angle β around the Z-axis of the global coordinate system, and then rotates counterclockwise by an angle α around the X-axis of the global coordinate system. After the rotation, the direction vector of the k-th target in the initial state panel coordinate system satisfies:

[0040] in, Let be the initial incident direction unit vector of target k. Let Z be the rotation matrix about the Z-axis. The rotation matrix about the X-axis is expressed as follows: ,

[0041] The expanded form of the direction vector of target k after rotation is:

[0042] in, These are the elevation and azimuth angles of the target k at its initial incident direction, respectively.

[0043] This embodiment accurately describes the spatial attitude change of the panel through a two-degree-of-freedom rotation matrix, establishes the coordinate transformation relationship of the target incident direction before and after the panel rotation, clarifies the regulating effect of the panel attitude angle on the target equivalent direction, and provides the core geometric basis for the construction of the array guide vector.

[0044] In a specific example, in step S1, the expression for the received signal of the (n,m)th array element to the kth target in the multi-target array receiving signal model is:

[0045] Among them, s k Let λ be the transmitted signal of the k-th target, and x be the signal wavelength. n z is the distance from the origin to the nth element distributed along the x-axis. m It is the distance from the origin to the m-th element distributed along the z-axis, n n，m Let be the noise signal received by the (n,m)th antenna.

[0046] This embodiment starts with the received signal of a single array element, constructs the quantization relationship between the target transmitted signal, the array spatial position, the target incident direction, and noise, establishes the basic observation model for array signal processing, and provides the original data model for subsequent covariance matrix calculation and DOA estimation.

[0047] In a specific example, in step S2, the MUSIC algorithm is executed as follows: a reception matrix is ​​constructed from the array received signals of L snapshots. Calculate the sample covariance matrix of the received signal. , for R Y Singular value decomposition yields the signal subspace U. s With noise subspace U n Construct the MUSIC spectral function:

[0048] Searching for the angles corresponding to the K largest spectral peaks in the two-dimensional angular domain yields the two-dimensional DOA estimation results for the K targets, where Nx and Nz are the number of array elements along the X and Z axes, respectively, A is the array manifold matrix, S is the signal matrix, and Z is the noise matrix. is the array guide vector, and K is the total number of targets.

[0049] It should be noted that, assuming the number of received signal snapshots is L, the received signal under L snapshots ,in:

[0050]

[0051]

[0052]

[0053]

[0054] The sample covariance matrix of the received signal is Perform singular value decomposition on it to obtain , , These are the singular vectors of the signal subspace and the noise subspace, respectively. , These are the singular values ​​in the signal subspace and noise subspace, respectively. Then, the MUSIC spectrum corresponding to each target is searched. By identifying the K largest spectral peaks, we can obtain estimates of the K angles of arrival.

[0055] This embodiment utilizes the orthogonality of the array signal subspace and noise subspace, separates the signal and noise components through covariance matrix decomposition, constructs a spectral function to achieve the search and identification of two-dimensional angles for multiple targets, and completes the accurate estimation of the initial DOA for multiple targets, providing a clear target direction input for the subsequent derivation of FIM and CRB. This embodiment achieves stable and high-precision initial estimation of two-dimensional DOA for multiple targets, solves the core problem of "no clear optimization object" before attitude optimization, ensures that subsequent accuracy quantization and attitude optimization can be carried out for actual observation targets, and is adaptable to complex scenarios such as multiple targets and low signal-to-noise ratio.

[0056] In a specific example, in step S3, the FIM sub-block corresponding to the two-dimensional DOA parameter of the k-th target is a 2×2 real symmetric matrix, specifically expressed as:

[0057] in, For noise power, S represents the partial derivatives of the array steering vector corresponding to the k-th target with respect to the elevation and azimuth angles, respectively. k Let K be the transmitted signal corresponding to target k, and let CRB be the CRB matrix of the k-th target. k =J k -1 (α,β).

[0058] It should be noted that the FIM sub-block corresponding to the two-dimensional DOA parameter of the k-th target in this embodiment is a 2×2 real symmetric matrix. The derivation process is as follows: The Fisher information matrix and CRB are derived. Assume the transmitted signal s corresponds to target k. k Completely known. The array observation model can be represented as: (6) The array observation vectors follow a joint complex Gaussian distribution, i.e., satisfy... ,in It is a mean vector that satisfies The likelihood function of the signal distribution is obtained from the array observation vector, depending on the parameter vector η: (7) Log-likelihood function: (8) The following is the detailed derivation of the FIM. Taking real parameters ηp and ηq as examples, the elements of the Fisher information matrix are defined as follows: (9) In the joint complex Gaussian model, the second derivative form is more direct. The derivation will be discussed in detail below.

[0059] Find the first derivative, and denote it as... Taking the derivative of equation (8) with respect to ηp, and using the rules for differentiation of complex vectors, we obtain: (10) Next, we need to find the second derivative and take the expectation. We need to differentiate equation (10) with respect to ηq: (11) Based on the expanded derivative The above equation can be simplified to: (12) The expectation of the second term in the above formula ,so

[0060] Substituting the above back into equation (9), noting the conjugate relationship and symmetry, the FIM element can be represented as: (13) The classic form of FIM can be obtained by organizing it: (14) Next, we consider expanding the partial derivatives of the parameters to be estimated for the k-th target: (15) According to the definition of a steering vector: (16) For the k-th target, when only considering its own 2×2 sub-block, the sub-block elements are:

[0061] (17)

[0062] The two-dimensional DOA parameter FIM sub-block of the k-th target can be written as: (18).

[0063] This embodiment starts from the likelihood function of array observations, derives the Fisher information matrix of DOA parameters through second-order differentiation, quantifies the amount of information about the parameters to be estimated contained in the observation data, and then obtains the theoretical lower bound (CRB) of DOA estimation through matrix inversion, establishing a quantitative relationship of "observation data → amount of parameter information → limit of estimation accuracy".

[0064] This embodiment transforms the abstract "DOA estimation accuracy" into a differentiable and optimizable mathematical matrix, clarifies the theoretical limit of estimation accuracy, and establishes a direct correlation between accuracy indicators and array steering vectors, providing a core and quantifiable evaluation basis for subsequent attitude optimization.

[0065] In a specific example, in step S3, the partial derivative expression of the steering vector component corresponding to the (n,m)th element with respect to the DOA parameter is:

[0066]

[0067] in, Let be the steering vector component of the (n,m)th array element corresponding to the kth target. Let be the position vector of the (n,m)th array element in the global coordinate system after the panel is rotated.

[0068] It should be noted that the specific derivation process of the partial derivative expression of the steering vector component corresponding to the (n,m)th element with respect to the DOA parameter is as follows: Let the (n,m)th component of ak be denoted as... The corresponding derivative is When the coordinates of the array elements and the target position are given, the guiding component corresponding to the (n,m)th array element is usually written as: (19) in, It is the position vector of the (n,m)th array element in global coordinates. It is a unit direction vector pointing to target k, satisfying ,in The panel normal vector is The initial position vector of the (n,m)th array element.

[0069] Differentiating equation (19) yields: (20) (twenty one) Each antenna Vectorization Substituting back into equation (18), we obtain the FIM of the panel's DOA estimate for the k-th target pair in the current sampling, denoted as... It is about The function.

[0070] And since the CRB for target k is the inverse matrix of its FIM, then: (twenty two) Considering that this scheme only concerns the estimation of two angles of arrival, i.e., the angle of arrival for each target. The accuracy of the estimation of the angle of arrival of multiple targets can be achieved using... To reflect this. Subsequently, by finding the panel rotation angle, so that... Minimize the minimum value to find the optimal panel posture.

[0071] This embodiment derives the partial derivative expressions of the steering vector components with respect to the DOA parameters, incorporating the element position vectors after panel rotation into the partial derivative calculation. This establishes a complete mapping chain: "panel attitude angle → element position → steering vector partial derivative → FIM matrix → CRB lower bound," making FIM and CRB explicit functions of the attitude angle. This embodiment achieves quantifiable and differentiable calculation of the impact of panel attitude on DOA estimation accuracy, solving the core quantification problem of "how attitude changes affect estimation accuracy," and providing crucial mathematical support for subsequent gradient calculation of the objective function and gradient descent iteration.

[0072] In a specific example, in step S4, the standard form of the constrained optimization problem is:

[0073] in, , pitch angle The upper and lower limits of mechanical rotation, , Azimuth The upper and lower limits of mechanical rotation, This indicates that the FIM matrix of the k-th target is positive definite.

[0074] It should be noted that, to solve this non-convex constraint optimization problem, the gradient descent method will be used below. Let... First, note that for any invertible matrix function J(θ) (where θ is a scalar variable), the differential identity holds. Therefore, we can conclude that: (twenty four) Therefore, the objective function of this problem is related to The gradient can be written as: (25) For ease of representation, let's remember... They are all about The function vectors. Let them be related to... Taking the partial derivative, we can obtain the vectorized mixed derivative: (26) (27) Define the Jacobian guided derivative matrix: (28) According to J k The common form can be written as: (29) Then regarding attitude derivative: (30) (31) in,

[0075] Substituting equations (30) and (31) back into equation (25), we can obtain the gradient. The specific expression.

[0076] This embodiment uses minimizing the sum of the traces of all target CRB matrices as the global optimization objective, while incorporating boundary constraints for panel mechanical rotation and the feasibility constraint of FIM positive definiteness. It transforms the engineering requirement of "optimal overall perception accuracy for multiple targets" into a standard constrained non-convex optimization problem, clearly defining the optimization objective, boundaries, and feasible region. This embodiment ensures that the physical meaning of the optimization objective—minimizing the lower bound of the overall estimation error for multiple targets—is clear, while also taking into account the mechanical rotation constraints and algorithmic feasibility requirements in practical engineering. It avoids obtaining theoretically unattainable solutions and provides a standard and legitimate optimization proposition for subsequent solutions.

[0077] In a specific example, in step S5, the gradient expression of the global cost function with respect to the attitude angle is:

[0078] in, For the global cost function, , The FIM matrix of the k-th target versus the pitch angle Azimuth The partial derivatives of .

[0079] This embodiment, based on matrix differential identities, derives the gradient expression of the global cost function with respect to two attitude angles, clarifying the fastest descent direction of the cost function as the attitude angle changes. This provides a precise update direction for the projected gradient descent iteration, enabling directional optimization during the iterative process. This embodiment yields a closed-form gradient calculation expression, avoiding errors and computational overhead associated with numerical differentiation, ensuring the accuracy of the update direction during iteration, and enabling the iteration to steadily progress towards minimizing the cost function. This significantly improves the efficiency and accuracy of solving for the optimal attitude.

[0080] In a specific example, in step S5, the expression for the projection operator is:

[0081] Where C is the set of feasible regions for attitude angles.

[0082] This embodiment addresses the boundary constraints of attitude angles by designing a corresponding amplitude-limiting projection operator. After each gradient update, attitude angles exceeding the mechanical rotation range are mapped back to the preset feasible region, ensuring that the attitude angles in each iteration conform to the actual physical rotation capability of the panel. This embodiment fundamentally avoids mechanically unrealizable attitude angles during iteration, ensuring the engineering feasibility of all iterative solutions. It also solves the problem of out-of-bounds divergence in gradient descent iterations, improving the stability and convergence of the iteration process.

[0083] In a specific example, the execution process of Armijo backtracking search and FIM positive definiteness check in step S5 is as follows: starting from the initial step size u=u0, the step size decay operation is repeatedly performed. This continues until the updated attitude angles simultaneously satisfy the positive definiteness of all target FIM matrices and the descent condition:

[0084]

[0085] Where c is the descent control parameter, with a value of 10. -4 , Let be the attitude angle vector of the t-th iteration. Let be the gradient vector for the t-th iteration.

[0086] This embodiment dynamically adjusts the iteration step size through Armijo backtracking search, ensuring effective reduction of the cost function in each iteration. Simultaneously, it incorporates a positive definiteness check of the FIM matrix to ensure the invertibility of the FIM matrix corresponding to the updated attitude angle and the clear physical meaning of the CRB, thus solving the problems of gradient descent with fixed step sizes easily diverging and the infeasibility of iterative solutions. This embodiment significantly improves the robustness and convergence stability of the algorithm, avoiding both iterative divergence caused by excessively large step sizes and slow convergence caused by excessively small step sizes. It also fundamentally guarantees the theoretical validity of the iterative solution (only positive definiteness of the FIM matrix allows for the inversion of the CRB), ensuring the algorithm can stably converge to an engineering-feasible and theoretically optimal panel attitude.

[0087] Performance verification experiment: To verify the effectiveness and technical advantages of the method proposed in this invention, two sets of verification experiments were set up: an optimal attitude solution verification experiment and a multi-scenario performance comparison verification experiment. The experimental environment was the MATLAB 2023b simulation platform, and the experimental object was a uniform planar array rotatable panel with an array element spacing of half the signal wavelength.

[0088] Experiment 1: Optimal Attitude Solution Verification Experiment 1.1 Experimental Setup Two far-field UAV targets are selected in space, with their true two-dimensional DOAs being (15°, 90°) and (75°, 90°) respectively. The two-degree-of-freedom rotation range of the panel is: pitch angle azimuth The signal-to-noise ratio (SNR) was set to 10 dB, the number of snapshots (T) to 32, and the array element size to be a 4×4 uniform planar array. Using the method of this invention, an array model was constructed, initial DOA estimation of the target was completed, attitude-related functional methods (FIM) and critical attitude basis (CRB) were derived, an optimization problem was constructed, and the optimal attitude was solved using the projective gradient descent method.

[0089] 1.2 Experimental Results 1. Optimal attitude solution result: After iterative convergence, the global optimal attitude of the panel is obtained as the pitch angle. azimuth Under this pose, the global cost function (the sum of the CRB traces of all targets) reaches its minimum value f = 1.407 × 10⁻⁶. 6 .

[0090] 2. Verification results shown in the attached diagram: Figure 4 This is a schematic diagram of the spatial distribution of the targets, clearly showing the incident directions and angular spans of the two targets; Figure 5 The two-dimensional cross-sectional curves are shown below. Time cost function varies change curve The curve of the cost function as a function of β shows that the lowest point of the curve coincides completely with the optimal attitude point obtained by the solution. Figure 6 The three-dimensional surface plot of the cost function visually illustrates how the cost function changes. , The variation pattern is such that the optimal attitude point obtained by solving is located at the global minimum point of the surface, and there is no local optimum trap.

[0091] 1.3 Experimental Conclusions The optimization modeling and solution method proposed in this invention can stably and accurately solve for the global optimal pose of the panel in a multi-objective scene. The solution results completely match the theoretical optimal value, verifying the accuracy and convergence of the method.

[0092] Experiment 2: Performance Comparison and Verification Experiment in Multiple Scenarios 2.1 Experimental Setup Four comparison schemes were set up as follows: FPA-0: A panel antenna that is fixed in its initial state and never rotates (baseline scheme 1). FPA-center: A panel antenna that is always pointed at a single target and whose attitude remains unchanged (baseline scheme 2). Proposed: The optimal posture method for the rotatable panel proposed in this invention (the present invention solution). CRB lowerfimum: The theoretical lowest error lower bound for DOA unbiased estimation (performance ceiling reference).

[0093] The mean squared error (MSE) was used as the evaluation metric. A smaller MSE value indicates higher DOA estimation accuracy and better performance. Comparative experiments were conducted from three core dimensions to verify the performance advantages of this scheme.

[0094] 2.2 Experiment 1: Performance Comparison of Different Shot Counts T Fixed experimental parameters: number of targets K=2, number of array elements N=16 (4×4 array), signal-to-noise ratio SNR=10dB, and the number of snapshots T ranges from 4, 8, 16, 32, to 64.

[0095] Experimental results (corresponding) Figure 7 As the number of snapshots T increases from 4 to 64, the MSE of all four schemes shows a decreasing trend; among them, the MSE of the scheme of this invention decreases the most significantly, and under all the number of snapshots, the MSE is always significantly lower than that of the FPA-0 fixed panel and the FPA-center single target alignment panel, and the value is closest to the CRB theoretical lower bound.

[0096] Conclusion: This invention breaks through the accuracy limitations of traditional fixed arrays in scenarios with limited snapshots and low sampling rates, and can achieve DOA estimation performance closer to the theoretical optimal level, making it suitable for low-altitude sensing scenarios with high real-time requirements.

[0097] 2.3 Experiment 2: Performance Comparison of Different Target Numbers K Fixed experimental parameters: number of snapshots T=32, number of array elements N=16 (4×4 array), signal-to-noise ratio SNR=10dB, and the number of targets K ranges from 2, 3, 4, to 5.

[0098] Experimental results (corresponding) Figure 8 As the number of targets K increases from 2 to 5, the MSE of traditional fixed panels and single-target alignment panels shows a sharp upward trend, and the multi-target resolution deteriorates rapidly. However, the MSE of the present invention increases significantly less. Under all target numbers, the MSE is always lower than that of the two baseline schemes, and maintains the smallest gap with the CRB theory's lower bound.

[0099] Conclusion: This invention can maintain stable low MSE performance in multi-target, large-angle-span scenarios, and has better multi-target DOA estimation robustness than traditional schemes, effectively balancing the estimation accuracy of multiple targets.

[0100] 2.4 Experiment 3: Performance Comparison of Different Numbers of Array Elements N Fixed experimental parameters: number of snapshots T=32, number of targets K=2, signal-to-noise ratio SNR=10dB, and the number of array elements N ranges from 8, 12, 16, 20, to 24.

[0101] Experimental results (corresponding) Figure 9 As the number of array elements N increases from 8 to 24, the MSE of all four schemes shows a decreasing trend; the MSE of the scheme of this invention has the largest decrease. Under all array element numbers, the MSE is always significantly lower than that of the two baseline schemes and is closest to the lower bound of the CRB theory.

[0102] Conclusion: Under the same accuracy requirements, the solution of this invention can achieve the target performance with fewer array elements compared with the traditional solution, effectively reducing array hardware costs and system complexity, and has higher engineering application value.

[0103] Overall experimental conclusion: Two sets of verification experiments fully demonstrate that the method for determining the optimal attitude of a rotatable panel under multi-target DOA estimation proposed in this invention can accurately solve for the global optimal attitude of the panel. Compared with traditional fixed panels and single-target aligned panels, it has better DOA estimation accuracy and robustness in scenarios with different number of snapshots, different number of targets, and different number of array elements, achieving a significant improvement in the overall perception performance of multi-targets and fully achieving the technical effect expected by the invention.

[0104] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation, characterized in that, Includes the following steps: S1: Construct the array geometry model, spatial rotation model and multi-target array signal receiving model of the two-DOF rotatable panel, and clarify the mapping relationship between the panel attitude angle, the target incident direction and the array guidance vector, so as to provide a physical and mathematical basis for subsequent DOA estimation and attitude optimization. S2: Based on the multi-target array received signal model constructed by S1, the received signals of multiple snapshot arrays are collected, and the covariance matrix of the received signals is decomposed into subspaces using the MUSIC algorithm. The two-dimensional DOA estimation results of all observed targets are obtained through two-dimensional spectral peak search, providing target direction input for subsequent accuracy quantization. S3: Based on the spatial rotation model of S1 and the target DOA estimation results obtained in S2, under the observation model of deterministic signal plus complex Gaussian white noise, the likelihood function and log-likelihood function of the array observation vector are constructed. By taking the partial derivative of the DOA parameter to be estimated, the Fisher information matrix FIM corresponding to the two-dimensional DOA parameter of each target is derived. The inverse of FIM is used to obtain the Cramer-Rao lower bound CRB of the DOA estimation of each target. The functional mapping relationship between the panel attitude angle and the lower bound of the multi-target DOA estimation accuracy is established. S4: Based on the multi-target CRB obtained from S3 and associated with the panel attitude angle, the sum of the traces of all target CRB matrices is used as the global cost function. Combined with the angular boundary constraints of the panel mechanical rotation and the feasibility constraints of the positive definiteness of the FIM of all targets, a constrained optimization problem with the best overall perception accuracy for multi-target is constructed. S5: The projected gradient descent method is used to solve the constrained optimization problem constructed in S4. The gradient expression of the global cost function with respect to the attitude angle of the panel is derived. In each iteration, the attitude angle is updated along the negative gradient direction. The updated attitude angle is mapped back to the preset feasible region through the projection operator. At the same time, the Armijo backtracking line search is combined to ensure the iterative descent property, and the FIM positive definiteness check is combined to ensure the feasibility of the iterative solution. After the iteration converges, the optimal attitude angle of the rotatable panel is output.

2. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 1, characterized in that, In step S1, the spatial rotation model is as follows: the panel first rotates clockwise by an angle β around the Z-axis of the global coordinate system, and then rotates counterclockwise by an angle α around the X-axis of the global coordinate system. After the rotation, the direction vector of the k-th target in the initial state panel coordinate system satisfies: in, Let be the initial incident direction unit vector of target k. Let Z be the rotation matrix about the Z-axis. The rotation matrix about the X-axis is expressed as follows: ， The expanded form of the direction vector of target k after rotation is: in, These are the elevation and azimuth angles of the target k at its initial incident direction, respectively.

3. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 1, characterized in that, In step S1, in the multi-target array receiving signal model, the expression for the received signal of the (n,m)th array element to the kth target is: Among them, s k Let λ be the transmitted signal of the k-th target, and x be the signal wavelength. n z is the distance from the origin to the nth element distributed along the x-axis. m It is the distance from the origin to the m-th element distributed along the z-axis, n n，m Let be the noise signal received by the (n,m)th antenna.

4. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 1, characterized in that, In step S2, the MUSIC algorithm is executed as follows: a receiving matrix is ​​constructed from the array received signals of L snapshots. Calculate the sample covariance matrix of the received signal. , for R Y Singular value decomposition yields the signal subspace U. s With noise subspace U n Construct the MUSIC spectral function: Searching for the angles corresponding to the K largest spectral peaks in the two-dimensional angular domain yields the two-dimensional DOA estimation results for the K targets, where Nx and Nz are the number of array elements along the X and Z axes, respectively, A is the array manifold matrix, S is the signal matrix, and Z is the noise matrix. is the array guide vector, and K is the total number of targets.

5. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 1, characterized in that, In step S3, the FIM sub-block corresponding to the two-dimensional DOA parameters of the k-th target is a 2×2 real symmetric matrix, specifically expressed as: in, For noise power, S represents the partial derivatives of the array steering vector corresponding to the k-th target with respect to the elevation and azimuth angles, respectively. k Let K be the transmitted signal corresponding to target k, and let CRB be the CRB matrix of the k-th target. k =J k -1 (α,β).

6. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 5, characterized in that, In step S3, the partial derivative expression of the steering vector component corresponding to the (n,m)th array element with respect to the DOA parameter is: in, Let be the steering vector component of the (n,m)th array element corresponding to the kth target. Let be the position vector of the (n,m)th array element in the global coordinate system after the panel is rotated.

7. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 1, characterized in that, In step S4, the standard form of the constrained optimization problem is: in, , pitch angle The upper and lower limits of mechanical rotation, , Azimuth The upper and lower limits of mechanical rotation, The FIM matrix of the k-th target is positive definite.

8. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 1, characterized in that, In step S5, the gradient expression of the global cost function with respect to the attitude angle is: in, For the global cost function, , The FIM matrix of the k-th target versus the pitch angle Azimuth The partial derivatives of .

9. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 1, characterized in that, In step S5, the expression for the projection operator is: Where C is the set of feasible regions for attitude angles.

10. The method for determining the optimal attitude of a rotatable panel under multi-objective DOA estimation according to claim 1, characterized in that, In step S5, the Armijo backtracking search and FIM positive definiteness check are performed as follows: starting from the initial step size u=u0, the step size decay operation is repeatedly executed. This continues until the updated attitude angles simultaneously satisfy the positive definiteness of all target FIM matrices and the descent condition: Where c is the descent control parameter, with a value of 10. -4 , Let be the attitude angle vector of the t-th iteration. Let be the gradient vector for the t-th iteration.

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