Envelope constrained collaborative filtering method and device for nonlinear cluster system under communication restriction

Abstract

The application discloses an envelope constraint cooperative filtering method and device for a nonlinear cluster system under communication restriction, comprising the following steps: approximating unknown dynamics of a nonlinear system by using a neural network; for the problem of limited communication bit rate of a cluster topology, designing a double-description coding scheme to cope with possible data packet loss in information interaction; establishing a sufficient condition for ensuring that a filtering error system simultaneously satisfies preset average performance and envelope constraints in probability; converting a filter design problem into a convex optimization problem, and obtaining the filter by solving a set of recursive linear matrix inequalities. The application solves the problem that existing filtering methods are difficult to simultaneously process unknown nonlinear dynamics and guarantee envelope constraint performance under limited communication bit rate, and can be applied to the field of multi-agent cluster cooperative situation awareness and estimation with unknown dynamics and communication constraints.

Description

Technical Field

[0001] This invention relates to the field of multi-agent cluster cooperative control and state estimation technology, and in particular to an envelope-constrained cooperative filtering method and apparatus for nonlinear cluster systems under communication constraints. Background Technology

[0002] Multi-agent swarm systems (such as UAV swarms) are widely used in tasks such as cooperative reconnaissance and high-maneuverability target tracking. In these highly dynamic scenarios, agents can only exchange information through local topology communication; therefore, achieving accurate distributed cooperative state estimation (filtering) is the core foundation of the system. However, existing swarm distributed filtering techniques still face the following challenges in practical applications:

[0003] 1. Strong nonlinearity of unknown targets: Highly maneuverable targets in swarm collaborative tracking often have complex unknown dynamics. Traditional filtering algorithms based on linearized models suffer significant performance degradation or even filter divergence when faced with severe model mismatch, making it difficult to meet the requirements of high-precision collaborative sensing.

[0004] 2. Limited Wireless Communication Bandwidth (Bit Rate): Clusters typically rely on ad hoc wireless networks, where channel bandwidth resources are extremely limited. High-frequency information exchange between nodes can easily lead to network congestion and packet loss, thereby disrupting the cluster's communication topology. Existing methods often simplify packet loss as a random event, ignoring the physical nature of the limited bit rate, resulting in poor performance in real-world constrained networks.

[0005] 3. Conservatism in performance guarantees: To ensure system security, traditional filtering methods (such as...) Filtering, including cluster filtering, is often designed deterministically based on the "worst-case" scenario. This design leads to overly conservative filters when dealing with extremely low-probability extreme conditions, severely wasting the limited computing and communication resources of cluster nodes and reducing the system's agility.

[0006] To address the aforementioned problems, this application proposes an envelope-constrained collaborative filtering method and apparatus for nonlinear trunking systems under communication constraints. Summary of the Invention

[0007] The purpose of this invention is to provide an envelope-constrained collaborative filtering method and apparatus for nonlinear trunking systems under communication constraints, thereby solving the filtering problem of channel-constrained trunking systems with unknown nonlinearity.

[0008] The technical solution to achieve the purpose of this invention is as follows:

[0009] An envelope-constrained collaborative filtering method for a nonlinear trunking system under communication constraints includes the following steps:

[0010] Step 1: Establish a mathematical model for an intelligent agent swarm system with unknown nonlinear characteristics;

[0011] Step 2: Design a dual-description coding communication scheme to handle the encoding, transmission, and decoding of measurement values ​​of the clustered intelligent agents;

[0012] Step 3: Based on the mathematical model, construct a distributed filter structure based on a neural network. The distributed filter uses the information decoded by the dual description coding scheme to estimate the state of the nonlinear target.

[0013] Step 4: Design the corresponding averaging filter based on its structure. Performance metrics and probabilistic envelope constraint performance metrics;

[0014] Step 5: Based on average The performance index and probabilistic envelope constraint performance index are used to derive sufficient conditions for the existence of the filter and solve for the filter parameters.

[0015] Furthermore, the mathematical model in step 1 is specifically as follows:

[0016]

[0017] in, , To estimate the period, and Let represent the state and actual measured value of swarm agent i at time k, respectively; , , and It is a known time-varying matrix of appropriate dimension; the process noise is... This indicates that the measurement noise is caused by Representation; function It is an unknown smooth nonlinear function on a compact set; , , They represent , , A set of Vie Euclidean space; N is the number of swarm agents; It is a square-integrable space.

[0018] Furthermore, step 2 involves designing a dual-description coding communication scheme, specifically including:

[0019] Step 2.1: Design a mathematical model for a rate-constrained trunking communication network; the rate-constrained mathematical model is as follows:

[0020]

[0021] Among them, This represents the total available bit rate of a given network, while This represents the bit rate allocated to agent i; and They represent the set of integers and the set of positive integers, respectively.

[0022] Step 2.2: Design the encoder and decoder to obtain the actual measurement value received by the filter; the actual measurement value received by the filter is:

[0023]

[0024] in, It is the decoded measurement value received by the filter at time k. It is the decoded measurement value received by the filter at time k-1. This is the decoding error, and it satisfies... ; It consists of two independent Bernoulli random variables , The determined random variable, let , , ;when hour, ;when hour, ;also, ,in, , , , , , Given a positive number; Represents the mathematical expectation. This indicates rounding down to the nearest integer.

[0025] Furthermore, step 3, which involves constructing a distributed filter structure based on a neural network, specifically includes the following steps:

[0026] Step 3.1: Design a neural network to approximate the nonlinear function of the system;

[0027] Step 3.2: Design a distributed filter structure based on neural networks;

[0028] Step 3.3: Design the neural network weight update law.

[0029] Furthermore, the nonlinearity of the neural network approximation system is:

[0030]

[0031] in, It is the weight matrix of the neural network. This represents the activation function. Describes the approximation error, and satisfies , Given integers; Let Frobenius norm be the matrix. Let be the Euclidean norm of the vector.

[0032] Furthermore, the distributed filter structure is as follows:

[0033]

[0034] in, This represents the state estimate at time k. This represents the measurement estimate at time k. This represents the information at time k. Represents the weight estimation matrix. , and It is the filter parameter matrix that needs to be solved; These are elements within the adjacency matrix of the cluster communication topology. It is the set of neighboring nodes of agent i.

[0035] Furthermore, the neural network weight update law is as follows:

[0036]

[0037] in, and To tune the positive scalar, .

[0038] Furthermore, in step 4, the corresponding average filter is designed. Performance metrics and probabilistic envelope constraint performance metrics, specifically including:

[0039] 1) Average Performance metrics are determined by the following formula:

[0040]

[0041] Wherein, the state estimation error is ; ; Given an integer, Given a positive definite matrix;

[0042] 2) The method for determining the performance index of probabilistic envelope constraints is as follows:

[0043] Given the following input with zero initial conditions:

[0044]

[0045] in, Represents an n-dimensional column vector consisting entirely of 1s. Indicates input The value under zero initial conditions, and the estimation error under zero initial conditions. satisfy:

[0046]

[0047] in, and Represent the given Expected output and tolerance band; For a pre-specified positive scalar, and satisfying ; , This indicates the probability of an event occurring.

[0048] Furthermore, the filter parameters in step 5 are obtained by solving a linear matrix inequality, which is:

[0049]

[0050] The matrix parameters are as follows:

[0051]

[0052]

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[0055]

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[0058]

[0059]

[0060]

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[0065]

[0066] in, Let be a predefined positive definite matrix, and , , ; and For the sequence of positive scalar parameters that need to be solved, ; It is an n-dimensional identity matrix. Indicates the Kronecker product. This represents a block diagonal matrix, where the diagonal blocks are respectively... abbreviated as ; Represents column vectors .

[0067] An envelope-constrained collaborative filtering device for a communication-limited nonlinear cluster system includes a nonlinear system mathematical model building unit, a dual-description coding communication unit, a filter structure construction unit, a constraint performance index design unit, and a solution unit. The nonlinear system mathematical model building unit is used to build a mathematical model of the intelligent agent cluster system with unknown nonlinear characteristics. The dual-description coding communication unit processes the encoding, transmission, and decoding of intelligent agent measurements through a designed dual-description coding communication scheme. The filter structure construction unit is used to construct a distributed filter structure based on a neural network to collaboratively estimate the state of the nonlinear system. The constraint performance index design unit is used to design corresponding probabilistic performance indices in conjunction with the filter structure. The solution unit is used to derive sufficient conditions for the existence of the filter and solve for the filter parameters.

[0068] Compared with the prior art, the present invention has the following significant advantages: (1) For systems with unknown nonlinearity, a neural network is introduced for online approximation, and a distributed filter with probabilistic envelope constraints is designed to ensure the estimation accuracy and reliability under complex dynamics; (2) In order to overcome the bit rate limitation of the communication channel, a dual description coding scheme is proposed, which effectively combats the problem of data packet loss caused by insufficient bandwidth and significantly improves the communication robustness of the cluster system in harsh network environments; (3) The proposed filtering framework integrates neural network, dual description coding and probabilistic performance indexes, solves the problem that existing methods cannot handle unknown dynamics and communication constraints at the same time, and can be widely applied to the field of collaborative estimation of cluster systems with complex constraints. Attached Figure Description

[0069] Figure 1 This is a flowchart illustrating the filtering design method of the present invention.

[0070] Figure 2 These are state components in the embodiments of the present invention. and its estimates Line graph.

[0071] Figure 3 These are state components in the embodiments of the present invention. and its estimates Line graph.

[0072] Figure 4 This is the filtering error in the embodiments of the present invention. Line graph.

[0073] Figure 5 This is the filtering error in the embodiments of the present invention. Line graph.

[0074] Figure 6 These are the actual measured values ​​in the embodiments of the present invention. and its decoded value The curve graph. Detailed Implementation

[0075] To make the advantages of the algorithm proposed in this application clearer, the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0076] Combination Figure 1 This invention discloses an envelope-constrained collaborative filtering method for a nonlinear trunking system under communication constraints, comprising the following steps:

[0077] Step 1: Establish a mathematical model for a class of cluster systems with unknown nonlinear characteristics;

[0078] Step 2: To address the limited communication bit rate in cluster communication networks, a dual-description coding communication scheme is designed and adopted to handle the encoding, transmission, and decoding of agent measurement values.

[0079] Step 3: Based on the mathematical model, construct a distributed filter structure based on a neural network. The filter uses the information decoded by the dual description coding scheme to perform a collaborative estimation of the state of the nonlinear system.

[0080] Step 4: Design the corresponding averaging filter based on its structure. Performance metrics and probabilistic envelope constraint performance metrics;

[0081] Step 5: Based on average The performance index and probabilistic envelope constraint performance index are used to derive sufficient conditions for the existence of the filter and solve for the filter parameters.

[0082] As a specific implementation method, step 1 establishes a mathematical model of the cluster system with unknown nonlinear characteristics, as follows:

[0083] (1)

[0084] in, , To estimate the period, and Let represent the state and actual measured value of swarm agent i at time k, respectively; , , and It is a known time-varying matrix of appropriate dimension; the process noise is... This indicates that the measurement noise is caused by Representation; function It is an unknown smooth nonlinear function on a compact set; Let N represent the set of m-dimensional Euclidean space; N is the number of swarm agents. It is a square-integrable space.

[0085] As one specific implementation method, step 2 designs a dual-description coding communication scheme, as follows:

[0086] Step 2.1: To address the bit rate limitation problem in trunked communication networks, a bit rate limitation mathematical model of the following form is adopted:

[0087] (2)

[0088] Among them, This represents the total available bit rate of a given network, while This represents the bit rate allocated to agent i; and They represent the set of integers and the set of positive integers, respectively.

[0089] Step 2.2: Design the encoder and decoder in the dual-description coding scheme to obtain the actual measurement value received by the filter.

[0090] Measurement data Before transmission, the data needs to be encoded by an encoder to generate two transmission symbols. The encoder used is...

[0091] (3)

[0092] For agent i, and To correspond to dual encoders, For quantizer, For index generation functions, and Assign a function to the index. Quantizer. The format is as follows:

[0093] (4)

[0094] in, Given the quantization range of agent i, For the highest quantization level, Indicates rounding down. This corresponds to the quantization region, specifically:

[0095] (5)

[0096] The index generation function takes the following form:

[0097] (6)

[0098] The index allocation function takes the following form:

[0099] (7)

[0100] Among them, transmission symbol , , This indicates the remainder.

[0101] After encoding and transmission, the decoder receives the transmission symbol. and Decoding is performed to obtain the decoded measurement value and decoding error. The decoder used is as follows:

[0102] (8)

[0103] in, and Indicates the side decoder, It is the central decoder. For the center index estimation function, and For the side index estimation function, It is the inverse quantization function. It is the decoded measurement value received by the filter at time k. , Let be two independent Bernoulli random variables, and , The index estimation function takes the following form:

[0104] (9)

[0105] (10)

[0106] (11)

[0107] The inverse quantization function takes the following form:

[0108] (12)

[0109] Therefore, the decoding error can be obtained as:

[0110] (13)

[0111] in, , , , , .

[0112] Furthermore, the decoded measurement value is:

[0113] (14)

[0114] in, It is the decoded measurement value received by the filter at time k-1. It is by , The determined random variable, and ;when hour, ;when hour, .

[0115] As one specific implementation method, the distributed filter based on neural networks constructed in step 3 is as follows:

[0116] Step 3.1: Design a neural network to approximate the nonlinear function of the system, and the system state equation can be obtained as follows:

[0117] (15)

[0118] in, It is the weight matrix of the neural network. This represents the activation function. Describes the approximation error, and satisfies , Given integers; Let Frobenius norm be the matrix. Let be the Euclidean norm of the vector.

[0119] Step 3.2: Design a distributed filter structure based on a neural network. Based on the actual decoded measurement values ​​received by the filter, a filter with the following form can be constructed:

[0120] (16)

[0121] in, This represents the state estimate at time k. This represents the measurement estimate at time k. This represents the information at time k. Represents the weight estimation matrix. , and It is the filter parameter matrix that needs to be solved; These are elements within the adjacency matrix of the cluster communication topology. It is the set of neighboring nodes of agent i.

[0122] Step 3.3: Design the neural network weight update law. Based on the gradient descent method, the neural network weight update law can be designed in the following form:

[0123] (17)

[0124] in, and To tune the positive scalar, .

[0125] As one specific implementation method, the average value set in step 4 The performance metrics and probabilistic envelope constraint performance metrics are as follows:

[0126] 1) Average Performance metrics are determined using the following methods

[0127] (18)

[0128] Wherein, the state estimation error is ; ; Given an integer, Given a positive definite matrix.

[0129] 2) The performance index of the probabilistic envelope constraint is determined by the following method.

[0130] Given the following input with zero initial conditions:

[0131] (19)

[0132] in, Represents an n-dimensional column vector consisting entirely of 1s. Indicates input The value under zero initial conditions, and the estimation error under zero initial conditions. satisfy

[0133] (20)

[0134] in, and Represent the given Expected output and tolerance band; For a pre-specified positive scalar, and satisfying ; , This indicates the probability of an event occurring.

[0135] As a specific implementation method, the sufficient condition for the existence of the filter and the solution of the filter parameters in step 5 are as follows:

[0136] (twenty one)

[0137] The matrix parameters are as follows:

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[0157] in, Let be a predefined positive definite matrix, and , , ; and ( () represents the sequence of positive scalar parameters to be solved; It is an n-dimensional identity matrix. Indicates the Kronecker product. This represents a block diagonal matrix, where the diagonal blocks are respectively... abbreviated as ; Represents column vectors .

[0158] The present invention also provides an envelope-constrained collaborative filtering device for a communication-limited nonlinear trunking system, comprising:

[0159] The system comprises a nonlinear system mathematical model building unit, a dual-description coding communication unit, a filter structure construction unit, a constraint performance index design unit, and a solution unit. The nonlinear system mathematical model building unit is used to build a mathematical model of a cluster system with unknown nonlinear characteristics. The dual-description coding communication unit is used to process the encoding, transmission, and decoding of agent measurement values. The filter structure construction unit is used to construct a distributed filter structure based on a neural network to collaboratively estimate the state of the nonlinear system. The constraint performance index design unit is used to design corresponding probabilistic performance indices in conjunction with the filter structure. The solution unit is used to derive sufficient conditions for the existence of the filter and solve for the filter parameters.

[0160] Example

[0161] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0162] In this embodiment, the Duffing equation of the following form is regarded as a kinematic model of a highly maneuverable target (such as a non-cooperative target) subjected to complex external perturbations:

[0163]

[0164] in, Indicates displacement. Indicates speed, Indicates acceleration. Parameter , and These represent the damping coefficient, linear stiffness, and nonlinear stiffness, respectively. and These represent the amplitude and angular frequency of the externally driven vibration, respectively.

[0165] By defining the state vector as Using a sampling period of The forward Euler method yields the discrete-time representation:

[0166]

[0167] And vector It includes nonlinear and external action terms. To model a system that is initially driven by external forces and subsequently evolves according to its own dynamic laws, we will... Defined as a piecewise function:

[0168]

[0169] in, and They represent The components (i.e., the displacement and velocity obtained from sampling).

[0170] The system parameters are set as follows:

[0171]

[0172] The number of cluster nodes performing the collaborative tracking task is set to N=3, and its adjacency matrix is... for:

[0173]

[0174] The activation function of the neural network is chosen as The remaining parameters are set as follows:

[0175]

[0176] The initial conditions are set as follows:

[0177]

[0178] In this simulation, we assume that the total bit rate is R = 21 bps, which is evenly distributed among the three cluster agents, i.e., R1 = R2 = R3 = 7 bps. Figures 2-6 This is a simulation result diagram of an embodiment of the present invention. The trajectory of the system state. and And their estimated values ​​are respectively seen in Figures 2-3 The corresponding estimation error and Then in Figures 4-5 Displayed in China. By Figure 2-5 It can be seen that the filtering algorithm proposed in this invention is applicable and performs well. Furthermore, Figure 6 Actual measured value With decoded measurement value Comparisons showed that the dual-description coding scheme can still successfully track the original measurement value even under bit rate constraints, thus verifying the effectiveness of the dual-description coding scheme proposed in this invention.

[0179] In summary, this invention employs a filtering strategy that integrates neural network approximation and dual-description coding, solving the problem that existing filtering methods struggle to simultaneously handle unknown nonlinear dynamics and guarantee envelope constraint performance under limited communication bit rates. This approach can be applied to multi-agent cluster collaborative situational awareness and estimation domains with unknown dynamics and communication constraints. To handle unknown nonlinear dynamics in the system, a neural network is used for online approximation; to enhance communication robustness under constrained wireless topologies, a dual-description coding scheme is employed to combat packet loss. Subsequently, a mechanism is established to ensure that the filtering error meets preset probability envelope constraints and average... Sufficient conditions for performance are determined, and filter gain parameters are obtained by solving a set of recursive linear matrix inequalities. Furthermore, this invention provides co-design criteria for filter parameters and coding schemes, ensuring the overall co-sensing performance of the system.

[0180] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be within the scope of protection of the present invention.

Claims

1. An envelope-constrained collaborative filtering method for a nonlinear trunking system under communication constraints, characterized in that, Includes the following steps: Step 1: Establish a mathematical model for an intelligent agent swarm system with unknown nonlinear characteristics; Step 2: Design a dual-description coding communication scheme to handle the encoding, transmission, and decoding of measurement values ​​of the clustered intelligent agents; Step 3: Based on the mathematical model, construct a distributed filter structure based on a neural network. The distributed filter uses the information decoded by the dual description coding scheme to estimate the state of the nonlinear target. Step 4: Design the corresponding averaging filter based on its structure. Performance metrics and probabilistic envelope constraint performance metrics; Step 5: Based on average The performance index and probabilistic envelope constraint performance index are used to derive sufficient conditions for the existence of the filter and solve for the filter parameters.

2. The envelope-constrained collaborative filtering method for communication-constrained nonlinear trunking systems according to claim 1, characterized in that, The mathematical model in step 1 is as follows: in, , To estimate the period, and Let represent the state and actual measured value of swarm agent i at time k, respectively; , , and It is a known time-varying matrix of appropriate dimension; the process noise is... This indicates that the measurement noise is caused by Representation; function It is an unknown smooth nonlinear function on a compact set; , , They represent , , A set of Vie Euclidean space; N is the number of swarm agents; It is a square-integrable space.

3. The envelope-constrained collaborative filtering method for communication-constrained nonlinear trunking systems according to claim 2, characterized in that, Step 2, which designs a dual-description coding communication scheme, specifically includes: Step 2.1: Design a mathematical model for a rate-constrained trunking communication network; the rate-constrained mathematical model is as follows: Among them, This represents the total available bit rate of a given network, while This represents the bit rate allocated to agent i; and They represent the set of integers and the set of positive integers, respectively. Step 2.2: Design the encoder and decoder to obtain the actual measurement value received by the filter; the actual measurement value received by the filter is: in, It is the decoded measurement value received by the filter at time k. It is the decoded measurement value received by the filter at time k-1. This is the decoding error, and it satisfies... ; It consists of two independent Bernoulli random variables , The determined random variable, let , , ;when hour, ;when hour, ;also, ,in, , , , , , Given a positive number; Represents the mathematical expectation. This indicates rounding down to the nearest integer.

4. The envelope-constrained collaborative filtering method for communication-constrained nonlinear trunking systems according to claim 3, characterized in that, Step 3, which involves constructing a distributed filter structure based on a neural network, specifically includes the following steps: Step 3.1: Design a neural network to approximate the nonlinear function of the system; Step 3.2: Design a distributed filter structure based on neural networks; Step 3.3: Design the neural network weight update law.

5. The envelope-constrained collaborative filtering method for communication-constrained nonlinear trunking systems according to claim 6, characterized in that, The nonlinearity of the neural network approximation system is: in, It is the weight matrix of the neural network. This represents the activation function. Describes the approximation error, and satisfies , Given integers; Let Frobenius norm be the matrix. Let be the Euclidean norm of the vector.

6. The envelope-constrained collaborative filtering method for communication-constrained nonlinear trunking systems according to claim 5, characterized in that, The distributed filter structure is as follows: in, This represents the state estimate at time k. This represents the measurement estimate at time k. This represents the information at time k. Represents the weight estimation matrix. , and It is the filter parameter matrix that needs to be solved; These are elements within the adjacency matrix of the cluster communication topology. It is the set of neighboring nodes of agent i.

7. The envelope-constrained collaborative filtering method for communication-constrained nonlinear trunking systems according to claim 6, characterized in that, The neural network weight update law is as follows: in, and To tune the positive scalar, .

8. The envelope-constrained collaborative filtering method for communication-constrained nonlinear trunking systems according to claim 9, characterized in that, In step 4, the corresponding average is designed in conjunction with the filter. Performance metrics and probabilistic envelope constraint performance metrics, specifically including: 1) Average Performance metrics are determined by the following formula: Wherein, the state estimation error is ; ; Given an integer, Given a positive definite matrix; 2) The method for determining the performance index of probabilistic envelope constraints is as follows: Given the following input with zero initial conditions: in, Represents an n-dimensional column vector consisting entirely of 1s. Indicates input The value under zero initial conditions, and the estimation error under zero initial conditions. satisfy: in, and Represent the given Expected output and tolerance band; For a pre-specified positive scalar, and satisfying ; , This indicates the probability of an event occurring.

9. The envelope-constrained collaborative filtering method for communication-constrained nonlinear trunking systems according to claim 8, characterized in that, The filter parameters in step 5 are obtained by solving a linear matrix inequality, which is: The matrix parameters are as follows: in, Let be a predefined positive definite matrix, and , , ; and For the sequence of positive scalar parameters that need to be solved, ; It is an n-dimensional identity matrix. Indicates the Kronecker product. This represents a block diagonal matrix, where the diagonal blocks are respectively... abbreviated as ; Represents column vectors .

10. An envelope-constrained collaborative filtering device for implementing a communication-constrained nonlinear trunking system as described in any one of claims 1-9, characterized in that, The system includes a nonlinear system mathematical model building unit, a dual-description coding communication unit, a filter structure construction unit, a constraint performance index design unit, and a solution unit. The nonlinear system mathematical model building unit is used to build a mathematical model of an intelligent agent cluster system with unknown nonlinear characteristics. The dual-description coding communication unit processes the encoding, transmission, and decoding of intelligent agent measurement values ​​through a designed dual-description coding communication scheme. The filter structure construction unit is used to construct a distributed filter structure based on a neural network to collaboratively estimate the state of the nonlinear system. The constraint performance index design unit is used to design corresponding probabilistic performance indices in conjunction with the filter structure. The solution unit is used to derive sufficient conditions for the existence of the filter and solve for the filter parameters.

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