Interval Kalman filtering design method based on internal positive characterization technology

Abstract

The invention belongs to the technical field of signal processing and state estimation, discloses an interval Kalman filtering design method based on an internal positive characterization technology, and mainly aims at an uncertain system in which a system matrix, an observation matrix and a noise covariance matrix are in an interval form. Developing an internal positive representation theory oriented to an interval uncertain linear system; and designing an interval Kalman filter for realizing full surrounding of a target state by utilizing interval system matrix boundary information in combination with an extended internal positive characterization technology and a symmetric positive semidefinite characteristic of an interval error covariance matrix. The method provided by the invention can effectively eliminate the dependence of the filter on the complete information of the interval matrix of the target system, reduce the complexity of operation, effectively suppress the interval expansion degree in the operation process, and improve the precision of state estimation.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing and state estimation, and specifically relates to a method for designing an interval Kalman filter based on an internal positive representation technique. Background Art

[0002] With the development of intelligent navigation, autonomous driving, and multi-agent systems, state estimation technology, as the basic support for system perception and decision-making control, has received extensive attention. In tasks such as navigation and positioning and trajectory tracking of mobile intelligent platforms such as driverless vehicles, unmanned aerial vehicles, and autonomous underwater vehicles in the ocean, the accurate estimation of the system state is of great significance for ensuring its operation safety and task execution ability.

[0003] Due to its theoretical completeness and excellent performance in Gaussian linear systems, the traditional Kalman filter has been widely used in practical engineering. However, in actual systems, there are usually various uncertainty factors such as inaccurate modeling, changing sensor noise, and system parameter perturbations, resulting in the inability to accurately obtain the system state transition matrix, observation matrix, and noise covariance matrix, affecting the accuracy of state estimation and even causing filtering failure.

[0004] To address the above problems, in recent years, researchers have proposed the interval Kalman filter method. By introducing interval mathematics theory and establishing an uncertain model including upper and lower bounds to describe the interval variation range of system parameters, the robustness of system state estimation under uncertain conditions is improved. This method is widely applicable to uncertain environments where parameters are difficult to accurately model or the system has external perturbations. For example, in the navigation and target tracking tasks of unmanned systems in complex scenarios, it has good application prospects.

[0005] However, most of the existing interval Kalman filtering methods rely on the complete prior information of the system interval matrix, resulting in high computational complexity and prone to the problem of interval expansion during the iteration process, that is, the interval boundaries of state estimation continuously expand over time, ultimately leading to a decrease in estimation accuracy. In addition, due to the possible non-invertibility of the interval matrix, the traditional filtering framework may fail during the filtering process and is difficult to meet the requirements of real-time performance and stability.

[0006] Therefore, there is an urgent need for a state estimation method applicable to uncertain system models, which can effectively solve the state estimation problem under the condition of lacking complete system interval information, suppress interval expansion, reduce computational complexity, and improve the robust estimation ability of the system in complex environments, especially for the navigation and positioning and trajectory tracking data processing scenarios of intelligent unmanned systems in dynamic and perturbed environments. Summary of the Invention

[0007] To solve the above technical problems, the present invention provides a design method for interval Kalman filtering based on internal positive representation technology, which can effectively eliminate the dependence of the filter on the complete information of the interval matrix of the target system, reduce the computational complexity, and at the same time, effectively suppress the degree of interval expansion during the operation and improve the accuracy of state estimation.

[0008] The design method for interval Kalman filtering based on internal positive representation technology according to the present invention includes the following steps: Step 1: Introduce the internal positive representation technology into the linear system with an interval matrix, and convert the linear system with an interval matrix into a corresponding non-negative system; Step 2: Design an interval Kalman filter that depends on the interval boundary information in combination with the internal positive representation technology; Step 3: Analyze the interval error covariance matrix of the interval Kalman filter, and deduce the optimal boundary of the interval error covariance matrix and the corresponding optimal Kalman gain matrix; Step 4: Design an optimized interval Kalman filter that combines the internal positive representation technology based on the optimal boundary of the interval error covariance matrix and the optimal Kalman gain matrix.

[0009] Further, Step 1 is specifically as follows: Step 1-1: Based on the interval analysis theory, represent the uncertainty of the target system with an interval matrix, and construct the following linear system with an interval matrix, which is observable: , where, represents the discrete-time index, and respectively represent the state vector and the observation output vector of the target system at discrete time k, represents the state vector of the target system at discrete time k + 1, and are independent input noise and observation noise, and respectively represent the interval state matrix and the interval observation matrix with bounded uncertainty, and are respectively the interval covariance matrix of the input noise with bounded uncertainty and the interval covariance matrix of the observation noise; and respectively represent real vectors with dimensions of and in the Euclidean space, , and respectively represent real vectors with dimensions of , and Sum interval matrix; the matrix with superscript " " is the interval matrix; these interval matrices together describe the uncertainty of the system, and the specific description is as follows; , where, , , , and , , , respectively represent the lower bound matrix and the upper bound matrix of the interval matrix , , , ; in a linear system with interval matrices, the matrix with subscript "m" is the midpoint matrix (real matrix) composed of the midpoints of all interval elements of the interval matrix, and the matrix with subscript "r" is the radius matrix (real matrix) composed of the radii of all interval elements of the interval matrix; Step 1-2, to introduce the internal positive representation technology into the system described in Step 1-1, define the matrix as follows: Arbitrarily given a -dimensional real vector and a -dimensional interval matrix , where n, p, and q are arbitrary real numbers, define: , where, and respectively represent the non-negative interval vector and the non-negative interval matrix corresponding to the real vector w and the interval matrix ; represents -dimensional identity matrix; is the -dimensional transformation matrix for realizing the inverse transformation in the internal positive representation technology, and its function is to restore the decomposed non-negative vector to the original real vector; and respectively represent the non-negative vector with dimension and the non-negative interval with dimension in the Euclidean space; the positive part and the negative part of w are both non-negative vectors, the positive part and the negative part of , where, and respectively represent vectors and the elements of the s-th row, and respectively represent the intervals and the element of the u-th row and v-th column, and respectively represent the lower and upper bounds of the element of the u-th row and v-th column of the interval matrix ; the symbol " " represents the relationship "and", , , ; According to the newly defined vectors and matrices above, the following equalities hold: , and ; Step 1-3: Based on the matrix defined by the real vector w and the interval matrix in Step 1-2, obtain the state vector , , the observation vector , the input noise , the observation noise , the input noise interval covariance matrix and the observation noise interval covariance matrix corresponding non-negative vectors or non-negative matrices are respectively , , , , , and . Construct the following non-negative linear system with an interval matrix from these new matrices, which may not be observable: , Two formulas in this system are also called the forward transformation in the internal positive representation technique, where, and are independent Gaussian white noises; and by combining the formulas and , the above non-negative linear system can be transformed back to the original system; and are called two forward transformations of the internal positive representation technique, and are called two inverse transformations of the internal positive representation technique. These four transformations together constitute the internal positive representation technique for linear systems with interval matrices, where , .

[0010] Furthermore, step 2 is specifically as follows: Step 2-1: Since the system described in step 1-3 does not retain the observable property of the system described in step 1-1, an effective interval Kalman filter cannot be designed based on this system and further discussion is required; To ensure that the system after applying the internal positive representation technique remains observable, a non-singular matrix is defined. This non-singular matrix can transform the midpoint system corresponding to the system described in step 1-1 into the observable canonical form. Then, combining this non-singular matrix T, the system described in step 1-1 will be transformed into: , where, , , and this system is observable; Step 2-2: According to the two forward transformation equations in the internal positive representation technique obtained in step 1-3, the non-negative linear system equations corresponding to the linear system with interval matrices described in step 2-1 can be obtained as follows, and it is observable: , where, , , , , and respectively represent , , , , and the non-negative vectors and matrices obtained by defining vectors and matrices according to T in step 1-2; it should be noted that the system described in step 2-2 is observable; Step 2-3: Since the system described in step 2-2 is observable, therefore, the correction step of the interval Kalman filter for such a linear system with interval matrices is expressed as: , where, , and respectively represent the interval Kalman gain matrix, the interval prior error covariance matrix, and the interval error covariance matrix at time k; since , and are all interval matrices, to ensure the existence of the interval inverse operation , the real matrix inverse operation is used to replace the original interval matrix inverse operation ; Define the estimation interval of the target state at time k as , then the interval Kalman filter based on the interval matrix boundary information of the linear system with interval matrix described in step 1-1 is designed as follows: , where and represent the state estimation interval and the prior estimation interval respectively. The vector and matrix with an overline represent the upper bounds of the corresponding interval vector and interval matrix, and the vector and matrix with an underline represent the lower bounds of the corresponding interval vector and interval matrix. , .

[0011] This interval Kalman filter makes full use of the characteristics of non-negative interval matrix operations and greatly improves the execution and computational efficiency of the algorithm.

[0012] Furthermore, step 3 is specifically as follows: Step 3-1: Combine the definition of the interval error covariance matrix in step 2-3 to obtain the expression of the definite error covariance matrix corresponding to the non-negative interval system based on the internal positive representation technology described in step 2-2 as follows: , where , , , , , , and represent the definite real matrices corresponding to , , , , , and respectively. The meaning of the definite real matrix is that for the linear system with interval matrix itself, although the relevant matrices in its actual operation process cannot be accurately obtained from the outside, they are actually definite. The matrices corresponding to these systems in the actual operation process are called definite real matrices. Define the optimal boundary of the interval error covariance matrix as , and clarify that the design goal of is the minimum error covariance matrix that satisfies . Step 3-2: Define the prior error covariance matrix as follows: , where , The element in the \(i\)-th row and \(j\)-th column is , and the remaining elements are 0; , , and are the orthogonal matrices corresponding to the spectral decompositions of the respective symmetric matrices, , , and are the diagonal matrices corresponding to the spectral decompositions of the respective symmetric matrices, satisfying , ; Step 3-3. According to the result of Step 3-2, design satisfying the condition as: , where is the number of elements in matrix , that is, ; is an arbitrary real number; The element in the \(i\)-th row and \(j\)-th column is , and the remaining elements are 0; The element in the \(m\)-th row and \(l\)-th column is , and the remaining elements are 0; , and are respectively the orthogonal matrix and the diagonal matrix after the spectral decomposition of . The symbol “ ” represents the relationship “or”; is to derive the Kalman gain matrix satisfying the optimal property. Take the derivative of the expression of : , and , Let , and we get: , where , , The Kalman gain matrix clarifies the minimality, that is, according to this Kalman gain matrix, is the minimum error covariance matrix.

[0013] Further, step 4 is specifically as follows: Simplify the designed error covariance boundary matrix in step 3-3 and substitute the optimal Kalman gain matrix to obtain: , Therefore, the interval Kalman filter for the linear system with interval matrix described in step 1-1 finally optimized is designed as follows: .

[0014] Using the interval Kalman filter designed with the optimized error covariance boundary matrix , compared with the algorithm designed in step 2, it can provide a target state estimation interval with a smaller width and higher accuracy.

[0015] The beneficial effects of the present invention are as follows: For an uncertain system with interval forms of system matrix, observation matrix and noise covariance matrix, the internal positive representation technology is introduced to transform the uncertain system into a corresponding non-negative system, weakening the dependence of the interval Kalman filter on the complete information of the system interval matrix; and based on the symmetric and positive semi-definite characteristics of the interval error covariance matrix, the accuracy of the interval Kalman filter is optimized; the optimal boundary of the interval error covariance matrix and the corresponding optimal Kalman gain matrix are derived, suppressing the rate of interval expansion, so as to achieve high-precision estimation of the state of the uncertain system. Description of the Drawings

[0016] Figure 1 is a schematic flow chart of the method of the present invention; Figure 2 is a comparison diagram of the state trajectory of the first dimension (the angle of change of the slip angle) of the uncertain system and the upper and lower bounds of the state estimation interval provided by the interval Kalman filter designed in step 2-3; Figure 3 is a comparison diagram of the state trajectory of the first dimension (the angle of change of the slip angle) of the uncertain system and the upper and lower bounds of the optimized estimation interval provided by the optimized interval Kalman filter designed in step 4; Figure 4 is a comparison diagram of the state trajectory of the second dimension (the acceleration in the vehicle yaw direction) of the uncertain system and the upper and lower bounds of the state estimation interval provided by the interval Kalman filter designed in step 2-3; Figure 5 is a comparison diagram of the state trajectory of the second dimension (the acceleration in the vehicle yaw direction) of the uncertain system and the upper and lower bounds of the optimized estimation interval provided by the optimized interval Kalman filter designed in step 4. Detailed Embodiment

[0017] To make the content of the present invention more clearly understood, the following further detailed description of the present invention is provided according to specific embodiments in conjunction with the accompanying drawings.

[0018] As Figure 1 shown, the method for realizing interval state estimation of the target system facing model uncertainty according to the present invention includes the following steps: Step 1: Introduce the internal positive representation technology into the linear system with interval matrices, and convert the linear system with interval matrices into a corresponding non - negative system; Step 2: Design an interval Kalman filter that depends on interval boundary information in combination with the internal positive representation technology; Step 3: Analyze the interval error covariance matrix of the interval Kalman filter, and derive the optimal boundary of the interval error covariance matrix and the corresponding optimal Kalman gain matrix; Step 4: Based on the optimal boundary of the interval error covariance matrix and the optimal Kalman gain matrix, design an optimized interval Kalman filter that combines the internal positive representation technology.

[0019] The following uses a specific example from the automotive field to illustrate the specific steps of the design method and verify the effectiveness of the design method. This example is based on the dynamic model of a two - wheel vehicle. After linearization and discretization, the system states obtained include the angular velocity of the change in the slip angle and the acceleration in the vehicle yaw direction (yaw acceleration). This model is applicable to the design and analysis of state - estimation filters under uncertain systems.

[0020] In this example, take the system state matrix and the observation matrix respectively as: , The input noise covariance matrix and the observation noise covariance matrix are: , .

[0021] Define the initial state of the target system as: , and select the initial value of the interval Kalman filter as: , where and respectively represent the lower bound and the upper bound of the initial value of the interval Kalman filter.

[0022] In this example is a two - dimensional variable, where the first dimension represents the angular velocity of the change in the slip angle, and the second dimension Represents the acceleration in the vehicle yaw direction, Figure 2 , Figure 3 respectively describe the state of the uncertain system in the first dimension and the upper bounds of the state estimation intervals provided by the two types of interval Kalman filters designed in Steps 2-3 and 4 and the lower bounds . Figure 4 , Figure 5 respectively describe the state of the uncertain system in the second dimension and the upper bounds of the state estimation intervals provided by the two types of interval Kalman filters designed in Steps 2-3 and 4 and the lower bounds .

[0023] Figures 2 - 5 In [reference], the abscissa of the coordinate system is the time series, and the ordinate represents the system state estimation interval, where each dimension variable of the system state is represented by a trajectory in the figure, and the upper bound of the state estimation interval and the lower bound of each dimension are represented by a trajectory respectively. Figure 2 Plots the angular velocity trajectory of the change in the slip angle of the car, the lower bound trajectory and the upper bound trajectory of the angular velocity interval estimation of the interval Kalman filter designed in Step 2-3; Figure 3 Plots the angular velocity trajectory of the change in the slip angle of the car, the lower bound trajectory and the upper bound trajectory of the optimized angular velocity interval estimation of the interval Kalman filter designed in Step 4; Figure 4 Plots the acceleration trajectory in the vehicle yaw direction, the lower bound trajectory and the upper bound trajectory of the acceleration interval estimation of the interval Kalman filter designed in Step 2-3, Figure 5 Plots the acceleration trajectory in the vehicle yaw direction, the lower bound trajectory and the upper bound trajectory of the optimized acceleration interval estimation of the interval Kalman filter designed in Step 4.

[0024] It can be clearly seen from Figures 2 - 3 that all the estimation intervals can completely envelope the true state trajectory . It can be clearly seen from Figures 4 - 5 that all the estimation intervals can completely envelope the true state trajectory , which indicates that the two sets of filters have good inclusiveness characteristics. Although the traditional interval filter can maintain the integrity of the state interval in the initial estimation, as time progresses, it lacks effective constraints on the error covariance, resulting in a tendency for the estimation interval to expand with time in both dimensions. Among them, the state component has its upper and lower bounds rapidly widened during sharp vehicle direction changes, There are obvious fluctuations during the turning acceleration stage, and the estimated interval expands more significantly. This typical "interval expansion effect" directly leads to conservative state estimation and reduces the perception accuracy of the system for vehicle dynamic changes. In contrast, the optimized interval Kalman filter shows better convergence and compactness in the state estimation processes of and . Specifically, Figure 3 The optimized interval estimation trajectory of the angular velocity plotted by Figure 2 is narrower than the interval estimation trajectory of the angular velocity plotted by Figure 5 . The optimized interval estimation trajectory of the acceleration plotted by Figure 4 is narrower than the interval estimation trajectory of the acceleration plotted by

[0025] . This indicates that the interval Kalman filter designed in step 4 has higher estimation accuracy than the interval Kalman filters designed in steps 2 - 3.

[0026] The above are only the preferred solutions of the present invention and are not intended to further limit the present invention. All equivalent changes made by using the content of the specification and drawings of the present invention are within the protection scope of the present invention.

Claims

1. An interval Kalman filter design method based on internal positive characterization technology, characterized in that: The following steps are involved: Step 1, introducing the internal positive representation technology into the linear system with interval matrix, and transforming the linear system with interval matrix into the corresponding non-negative system; Step 2: Combine the internal positive characterization technology to design an interval Kalman filter that relies on interval boundary information; Step 3, analyzing the interval error covariance matrix of the interval Kalman filter, deriving the optimal boundary of the interval error covariance matrix and the corresponding optimal Kalman gain matrix; Step 4: Based on the optimal boundary of the interval error covariance matrix and the optimal Kalman gain matrix, an optimized interval Kalman filter combined with the internal positive characterization technology is designed.

2. The interval Kalman filter design method based on internal positive characterization technology according to claim 1 is characterized in that: Step 1 is as follows: Step 1-1: Based on interval analysis theory, the uncertainty of the target system is represented by an interval matrix, and the following linear system with an interval matrix is ​​constructed, which is observable: ， in, represents a discrete time index, and They represent the state vector and observation output vector of the target system at discrete time k, respectively. represents the state vector of the target system at discrete time k+1, and are independent input noise and observation noise, and denote the bounded uncertain interval state matrix and interval observation matrix respectively, and are the bounded uncertain input noise interval covariance matrix and observation noise interval covariance matrix respectively; and They represent the dimensions of the Euclidean space. and A real vector of , , and They represent the dimensions of the Euclidean space. , and and interval matrix; containing superscript " "The matrix is ​​an interval matrix; ， in, , , , and , , , Represents the interval matrix , , , The lower bound matrix and upper bound matrix of the interval matrix; in a linear system with interval matrices, the matrix with subscript "m" is the midpoint matrix composed of the midpoints of all interval elements of the interval matrix; the matrix with subscript "r" is the radius matrix composed of the radii of all interval elements of the interval matrix; Step 1-2: To introduce the internal positive characterization technique into the system described in step 1-1, define the matrix as follows: Any given one dimensional real vector and a dimensional interval matrix , where n, p and q are arbitrary real numbers, define: ， in, and Represent the real vector w and interval matrix respectively The corresponding non-negative interval vectors and non-negative interval matrices; express dimensional identity matrix; It is the inverse transformation achieved in the internal forward characterization technology. dimensional transformation matrix, which is used to restore the decomposed non-negative vector to the original real vector; and They represent the dimensions of the Euclidean space. The non-negative vector and dimension of The non-negative interval of ; the positive part of w and negative part are all non-negative vectors, The positive part and negative part are all non-negative interval matrices, and their respective definitions are as follows: ， in, and Represents vectors and The elements of row s, and Respectively represent the interval and The element at row u and column v, and Represents the interval matrix The lower and upper bounds of the element in the uth row and vth column, symbol " " indicates the relationship "and", , , ; According to the new vectors and matrices defined above, the following equation is established: , and ; Step 1-3: Based on step 1-2, the real vector w and the interval matrix The defined matrix obtains the state vector in step 1-1 , , observation vector , Input Noise , observation noise , input noise interval covariance matrix and the observation noise interval covariance matrix The corresponding non-negative vectors or non-negative matrices are , , , , , and , from these new matrices we construct the following non-negative linear system with interval matrices, which are not necessarily observable: ， in, and are independent Gaussian white noises; and The two internal positive characterization techniques are called forward transformations, and These four transformations together constitute the internal positive representation technique for linear systems with interval matrices, which are called the two inverse internal positive representation techniques. , .

3. The interval Kalman filter design method based on internal positive characterization technology according to claim 2 is characterized in that: Step 2 is as follows: Step 2-1: In order to make the system still observable after the internal positive characterization technique is applied, define a non-singular matrix , combined with this non-singular matrix T, the system described in step 1-1 will be transformed into: ， in, , , and the system is observable; Step 2-2, according to the two forward transformation equations in the internal positive characterization technique obtained in step 1-3, the non-negative linear system equation corresponding to the linear system with interval matrix described in step 2-1 is obtained, which is observable: ， in, , , , , and Respectively , , , , and T are non-negative vectors and matrices obtained by defining vectors and matrices according to steps 1-2; Step 2-3: Since the system described in step 2-2 is observable, the correction steps of the interval Kalman filter for this type of linear system with an interval matrix are expressed as: ， in, , and They represent the interval Kalman gain matrix, interval prior error covariance matrix and interval error covariance matrix at time k respectively; since , and They are all interval matrices. In order to ensure the interval inverse operation Exists, using real matrix inversion Replace the original interval matrix inverse operation ; Define the estimated interval of the target state at time k as , then the interval Kalman filter that depends on the interval matrix boundary information for the linear system with interval matrix described in step 1-1 is designed as follows: ， in, and Respectively represent the state estimation interval and the prior estimation interval, the underlined vectors and matrices represent the upper bounds of the corresponding interval vectors and interval matrices, and the underlined vectors and matrices represent the lower bounds of the corresponding interval vectors and interval matrices, , .

4. The interval Kalman filter design method based on internal positive characterization technology according to claim 3 is characterized in that: Step 3 is as follows: Step 3-1, combined with the definition of the interval error covariance matrix in step 2-3, the expression of the deterministic error covariance matrix corresponding to the non-negative linear system based on the internal positive characterization technology described in step 2-2 is as follows: ， in, , , , , , , and Respectively , , , , , and The corresponding real number matrix is ​​determined; the optimal boundary of the interval error covariance matrix is ​​defined as ,clear The design goal is to meet The minimum error covariance matrix of ; Step 3-2: Define the prior error covariance matrix as follows: ， in, ， in, The element in row i and column j is , a matrix with the rest of the elements being 0; , , and are the orthogonal matrices corresponding to the spectral decomposition of each symmetric matrix, , , and are the diagonal matrices corresponding to the spectral decomposition of each symmetric matrix, satisfying , ; Step 3-3: Based on the results of step 3-2, design conditional for: ， in, For the matrix The number of elements in ; is an arbitrary real number; The element in row i and column j is , a matrix with the rest of the elements being 0; The element in row m and column l is , a matrix with the rest of the elements being 0; , and They are Orthogonal matrix and diagonal matrix after spectral decomposition, symbol" " indicates the relationship "or"; To derive satisfaction Kalman gain matrix of optimal characteristics ,right The expression is derivatized: ， and , make ,get: ， in, ， ， The Kalman gain matrix specifies The minimum property of for The minimum error covariance matrix.

5. The interval Kalman filter design method based on internal positive characterization technology according to claim 4 is characterized in that: Step 4 is as follows: For the error covariance boundary matrix designed in step 3-3 Simplify and substitute into the optimal Kalman gain matrix ,have to: ， Therefore, the interval Kalman filter for the linear system with interval matrix described in step 1-1 is finally optimized as follows: 。

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