Distributed robust observer design method based on non-peak high-gain observation technology

Abstract

The present invention belongs to the technical field of distributed observers, and discloses a distributed robust observer design method based on peak-free high-gain observation technology. For a continuous-time linear time-invariant complex system affected by external disturbances, under the condition that the external disturbance is unknown but bounded and the sensor topology of local measurement satisfies strong connectivity, the target system is observable decomposed to obtain an observable subsystem and an unobservable subsystem; then, the observable subsystem is observer structure decomposed, and a peak-free high-gain observer is designed for the decomposed subsystem; and the unobservable subsystem is compensated by neighbor sensor measurement information and an adaptive strategy is introduced. The method of the present invention gets rid of the dependence of observer design on system and communication topology information, eliminates the peak phenomenon of high-gain observers, and provides the specific form and sufficient conditions of the distributed robust observer design based on peak-free high-gain observation technology.

Description

Technical Field

[0001] The present invention belongs to the technical field of distributed observers, and in particular relates to a distributed robust observer design method based on a peak-free high-gain observation technology. Background Art

[0002] With the development of swarm intelligence, distributed state estimation has attracted widespread attention. For complex target systems with high-dimensional, wide-area and decentralized characteristics, the use of distributed observers to estimate the state of the target system not only solves the shortcomings of the traditional Lumberg observer, but also promotes the development of swarm intelligence engineering applications.

[0003] The influence of external disturbances is ubiquitous and unavoidable in practical engineering applications, and high-gain observers have developed into an important tool for system output feedback control design and are a widely used type of robust nonlinear observer. High-gain observers have the characteristics of fast convergence speed for target system state estimation, and can improve the accuracy of target system state estimation by reducing high-gain parameters; however, traditional high-gain observers have the disadvantage of transient peaks, which is not conducive to practical engineering applications.

[0004] Existing distributed state estimation technologies include distributed observers and distributed Kalman filters. Distributed Kalman filters are mainly used to solve the state estimation when the uncertainty of the system's initial error and the noise intensity are completely known, while distributed observers do not need to know the system's initial error and have better stability; however, the existing distributed observer design method does not consider the impact of external disturbances, a common factor, on system operation, and the observer design is based on the design method of the traditional Lumberg observer. It does not use high-gain technology combined with adaptive strategies, still relies on system topology information, and has a slow convergence speed, and cannot handle the impact of disturbances on state estimation. Summary of the invention

[0005] In order to solve the above technical problems, the present invention provides a distributed robust observer design method based on peak-free high-gain observation technology, which introduces high-gain observation technology and adaptive strategy into the design of distributed observers, solves the state estimation problem of large-scale complex target systems under the influence of external disturbances, and reduces the dependence on system and communication topology information; at the same time, the introduced peak-free high-gain observer design method eliminates the disadvantages of transient peaks in traditional high-gain observers while ensuring fast state estimation.

[0006] The distributed robust observer design method based on the peak-free high-gain observation technology of the present invention comprises the following steps:

[0007] Step 1: Consider a continuous-time linear time-invariant target system affected by external disturbances, and calculate the system matrix (C i , A) perform observability decomposition to obtain the corresponding observable subsystems and unobservable subsystems;

[0008] Step 2: Decompose the observer structure of the observable subsystem and design a peak-free high-gain observer;

[0009] Step 3: For the unobservable subsystem, use the neighbor sensor measurement information to compensate and introduce an adaptive strategy;

[0010] Step 4: Give sufficient conditions for the observer to successfully estimate the target system.

[0011] Furthermore, in step 1, consider a continuous-time linear time-invariant target system subject to external disturbances:

[0012]

[0013] in They are the status information and output information of the target system. is the derivative of the state variable, are the system matrix and the output matrix respectively, Indicates the external disturbance to the target system; Indicates that the vector dimension is n×1, using represents a matrix of dimension n×m in Euclidean space; for external perturbations Existence constant So that for all times t ≥ 0 we have

[0014] Consider a sensor network composed of N agents responsible for measuring the output information of the target system. The measurement information of the i-th agent is:

[0015] y i (t) = C i x(t),

[0016] in And its dimension is p i The dimension p of y(t) satisfies the relation p i ≤p, is a partial row of C; each agent measures local output information and exchanges information with neighboring nodes.

[0017] Furthermore, for (C i , A) Decompose observability, specifically:

[0018] Define the system matrix (C i , the unobservable subspace of A) Where l = 1, 2, ..., n represents the lth subspace, represents the intersection operation of the subspace, ker represents the kernel of the subspace, and The dimension of Representation subspace Orthogonal complement of The column vector of The orthogonal basis of The column vector of Orthogonal basis, construct the orthogonal transformation matrix Make C i T i =[C id 0], where is the matrix T i The transposed matrix of is a matrix with matching dimensions;

[0019] definition in is the state variable of the observable subsystem, is the state variable of the unobservable subsystem, so the observable subsystem is:

[0020]

[0021] y i (t) = C id x id (t),

[0022] in For x id The derivative of (t), is the matrix D i The transposed matrix of

[0023] The unobservable subsystems are:

[0024]

[0025] in For x id The derivative of (t), is the matrix U i The transposed matrix of .

[0026] Furthermore, step 2 is specifically as follows:

[0027] Step 2-1: Calculate the observable subsystem matrix (C id , A id ) is used to decompose the observer structure:

[0028] Step 2-1-1: There is a non-singular matrix So that the matrix A id and C id There exist the following non-singular transformations:

[0029]

[0030]

[0031] in is the matrix of the corresponding dimension after transformation, Here I id，l-1 represents the l-1 dimensional unit matrix of the i-th observable subsystem, and 0 represents the zero matrix of the corresponding dimension or the constant zero; it can be seen that the system matrix after decomposition is is observable;

[0032] Step 2-1-2, for convenience, remember represents a function of disturbance and state variables. Use φ id In short, the observable subsystem after decomposition is:

[0033]

[0034]

[0035]

[0036] in is the state variable of the observable subsystem after the observer structure decomposition transformation, yes The lth element of φ id，l Yes id The lth element of It is the output information of the observable subsystem after the observer structure decomposition and transformation;

[0037] Step 2-2, design a peak-free high-gain observer:

[0038] Step 2-2-1: The target system state satisfies the existence of a compact set So that for all times t ≥ 0, x(t) ∈ χ, the high-gain observer estimates the variables of the observable subsystem as follows:

[0039]

[0040]

[0041] in The actual estimate of the observer is yes The lth element of , col represents a column vector, The intermediate variable of the observer is yes The specific form of the high gain observer is as follows:

[0042]

[0043]

[0044]

[0045]

[0046]

[0047]

[0048]

[0049] Where ∈ is the high gain parameter to be determined, α i，l and β i，l is the observer parameter to be selected, and the function φ is defined id The threshold value in Represents φ id，l The boundary value of yes The l+1th element of yes The derivative of express No. The derivative of an element, yes The derivative of yes No. The derivative of an element, yes The l-1th element of id，l is an intermediate variable; saturation function in is the sign function, r i，l is saturation;

[0050] Step 2-2-2, arbitrarily select ∈∈(0,1);

[0051] Step 2-2-3, randomly select a pair of positive numbers (α i，1 , β i，1 );when When α is selected as positive i，l and β i，l satisfy where γ i，l-1 is the L2 gain of the observable subsystem; select Any positive number.

[0052] Furthermore, step 3 is specifically as follows:

[0053] Step 3-1: For the unobservable subsystem, the designed observer adopts the method of neighbor sensor measurement information compensation. The observer form is as follows:

[0054]

[0055] where γ i +ρ i is the feedback gain, l ij is the i-th row and j-th column element in the Laplacian matrix of the sensor-agent communication topology;

[0056] Step 3-2: Design feedback gain γ i +ρ i The adaptive strategy is as follows:

[0057]

[0058]

[0059] Where k is a positive constant.

[0060] Furthermore, step 4 is specifically as follows:

[0061] Step 4-1: The information interaction topology between sensor agents adopts a directed graph containing N nodes. To represent, define its adjacency matrix and Laplace matrix as W = (w ij ) and L=(l ij ), a directed graph The strong connectivity condition must be met, that is, starting from any node in the directed graph, there is a directed path to reach all other nodes;

[0062] Step 4-2, Adaptive term γ in step 3-2 i The initial value of γ i (0) must satisfy the condition that it is greater than 0;

[0063] Step 4-3: Select the initial value of the observer and

[0064] The beneficial effects of the present invention are as follows: the method described in the present invention is mainly aimed at a continuous-time linear time-invariant complex target system with the influence of external disturbances. Under the condition that the external disturbance is unknown but bounded and the sensor topology of local measurement satisfies strong connectivity, the target system is observable decomposed to obtain an observable subsystem and an unobservable subsystem; then, the observable subsystem is decomposed by the observer structure, and a peak-free high-gain observer of the decomposed subsystem is designed by introducing a saturation function, and the observer instantaneous peak phenomenon is eliminated under the premise of ensuring rapid convergence; the unobservable subsystem is compensated by using the neighbor sensor measurement information and introducing an adaptive strategy to avoid dependence on the target system information and communication topology information. The method of the present invention introduces a high-gain observer into the design of a distributed observer, solves the state estimation problem of a large-scale complex target system under the influence of external disturbances, and eliminates the drawback of transient peaks in traditional high-gain observers by introducing saturation technology under the condition of ensuring rapid convergence; the designed distributed robust observer does not depend on the target system and communication topology information, and has strong robustness to external disturbances. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 It is a schematic diagram of the process of the present invention;

[0066] Figure 2 Schematic diagram of the error of the observer in estimating the state of the target system when the high gain parameter ∈ = 0.01;

[0067] Figure 3 Schematic diagram of the error of the observer in estimating the state of the target system when the high gain parameter ∈ = 0.001;

[0068] Figure 4 is a schematic diagram of the trajectory of the adaptive variable;

[0069] Figure 5 Schematic diagram of state estimation error based on traditional high-gain observer. DETAILED DESCRIPTION

[0070] In order to make the contents of the present invention more clearly understood, the present invention is further described in detail below based on specific embodiments in conjunction with the accompanying drawings.

[0071] like Figure 1-2 As shown, the distributed robust observer design method based on the peak-free high-gain observation technology of the present invention comprises the following steps:

[0072] Step 1: Consider a continuous-time linear time-invariant target system affected by external disturbances, and calculate the system matrix (C i , A) perform observability decomposition to obtain the corresponding observable subsystems and unobservable subsystems;

[0073] Step 2: Decompose the observer structure of the observable subsystem and design a peak-free high-gain observer;

[0074] Step 3: For the unobservable subsystem, use the neighbor sensor measurement information to compensate and introduce an adaptive strategy;

[0075] Step 4: Give sufficient conditions for the observer to successfully estimate the target system.

[0076] The following is a detailed description with reference to specific examples. The following examples are only used to illustrate the present invention and do not limit the scope of application of the present invention.

[0077] Consider a target system, the system matrix is ​​as follows:

[0078]

[0079] C1=[1 0 -1 0 1 1], C2=[1 1 0 -1 0 0],

[0080] C3=[1 0 1 0 -1 0], C4=[0 1 0 -1 0 0],

[0081] C5=[-1 0 2 0 1 1],

[0082] A sensor network consisting of five intelligent agents is adopted, and its communication topology is expressed by the adjacency matrix as follows:

[0083]

[0084] The external disturbance is selected as

[0085] First, for each (C i , A) Perform observability decomposition to obtain the observable subsystem and unobservable subsystem of the system. Construct the transformation matrix T i =[D i U i ], i=1,...,5 as follows:

[0086]

[0087]

[0088]

[0089]

[0090]

[0091] The corresponding subsystem matrix is ​​as follows:

[0092]

[0093]

[0094] C 1d =[0 0 2], C 2d =[0 0 0 1.7321], C 3d =[0 0 1.7321], C 4d =[0 0 1.4142], C 5d = [0 0 2.6458];

[0095]

[0096]

[0097]

[0098]

[0099]

[0100] Then, for the observable subsystem (C id , A id ) to decompose the observer structure; select a non-singular matrix and as follows:

[0101]

[0102]

[0103]

[0104]

[0105]

[0106]

[0107] The decomposition results are as follows:

[0108]

[0109] The high gain parameters are selected as ∈ = 0.01 and ∈ = 0.001 respectively, k = 20 in the adaptive strategy, and the initial state of observer i is Select as r i =3 (i=1, ..., 5). For observer i, select α i，1=3,α i，2 =9,α i，3 =2,β i，1 =4,β i，2 =5.

[0110] In order to illustrate that the observation error of the distributed robust observer designed by the present invention can approach 0 as the high gain parameter ∈ decreases, ∈ of different sizes are selected for comparison in the examples. Figure 2 and Figure 3 They respectively describe the state estimation error of the target system by the distributed observer when ∈=0.01 and ∈=0.001. Since the target system state is 6-dimensional, the observer is also 6-dimensional, and there are 6 observers in total, so Figure 2 and Figure 3 There are 36 error trajectories in each. By comparison, it can be shown that the observation error can tend to 0 as the high gain parameter ∈ decreases; and, Figure 3 It is shown that the observer can robustly monitor the impact of external disturbances on the system over time to achieve a fast estimation of the target state, and the observer does not have transient peak phenomena. Figure 4 describes the adaptive parameter γ i +ρ i Since there are 6 observers, Figure 4 There are 6 tracks of adaptive parameters.

[0111] Furthermore, in order to illustrate the innovativeness of the present invention, the comparative experimental results of the traditional high-gain observer are added to the example. Figure 5 As shown, the high gain parameter ∈ = 0.001. Figure 3 contrast, Figure 5 The observer error shows a large deviation near t = 0, and the transient peak is obvious, which is not conducive to practical engineering applications. In contrast, the present invention eliminates the peak phenomenon while ensuring rapid estimation of the target system state, laying a theoretical foundation for practical applications.

[0112] The above description is only a preferred embodiment of the present invention and is not intended to be a further limitation of the present invention. All equivalent changes made using the contents of the present specification and drawings are within the protection scope of the present invention.

Claims

1. A distributed robust observer design method based on non-peak high-gain observation technology is characterized in that: The following steps are involved: Step 1: Consider a continuous-time linear time-invariant target system affected by external disturbances, and calculate the system matrix (C i , A) perform observability decomposition to obtain the corresponding observable subsystems and unobservable subsystems; Step 2: Decompose the observer structure of the observable subsystem and design a peak-free high-gain observer; Specifically: Step 2-1: Calculate the observable subsystem matrix (C id , A id ) is used to decompose the observer structure: Step 2-1-1: There is a non-singular matrix So that the matrix A id and C id There exist the following non-singular transformations: Where n is the dimension of the system state vector, is the system matrix (C i , the unobservable subspace of A) The dimension of is the matrix of the corresponding dimension after transformation, Here represents the i-th observable subsystem dimensional unit matrix, 0 represents the zero matrix of the corresponding dimension or the constant zero; it can be seen that the system matrix after decomposition is observable; Step 2-1-2, for convenience, remember represents a function of disturbance and state variables. Use φ id In short, the observable subsystem after decomposition is: in is the state variable of the observable subsystem after the observer structure decomposition transformation, yes The lth element of φ id，l Yes id The lth element of It is the output information of the observable subsystem after the observer structure decomposition and transformation; Step 2-2, design a peak-free high-gain observer: Step 2-2-1: The target system state satisfies the existence of a compact set So that for all times t ≥ 0, The high-gain observer estimates the variables of the observable subsystem as follows: in The actual estimate of the observer is yes The lth element of , col represents a column vector, The intermediate variable of the observer is yes The lth element of The specific form of the high gain observer is as follows: Where ∈ is the high gain parameter to be determined, α i，l and β i，l is the observer parameter to be selected, and the function φ is defined id The threshold value in Represents φ id，l The boundary value of yes The l+1th element of yes The derivative of express No. The derivative of an element, yes The derivative of yes No. The derivative of an element, yes The l-1th element of id，l is an intermediate variable; saturation function in is the sign function, r i，l is saturation; Step 2-2-2, arbitrarily select ∈∈(0,1); Step 2-2-3, randomly select a pair of positive numbers (α i，1 , β i，1 );when When α is selected as positive i，l and β i，l satisfy where γ i，l-1 is the L2 gain of the observable subsystem; select is any positive number; Step 3: For the unobservable subsystem, use the neighbor sensor measurement information to compensate and introduce an adaptive strategy; Step 4: Give sufficient conditions for the observer to successfully estimate the target system.

2. The distributed robust observer design method based on non-peak high-gain observation technology according to claim 1 is characterized in that: In step 1, consider a continuous-time linear time-invariant target system subject to external disturbances: in They are the status information and output information of the target system. is the derivative of the state variable, are the system matrix and the output matrix respectively, Indicates the external disturbance to the target system; Indicates that the vector dimension is n×1, using represents a matrix of dimension n×m in Euclidean space; for external perturbations Existence constant So that for all times t ≥ 0 we have Consider a sensor network composed of N agents responsible for measuring the output information of the target system. The measurement information of the i-th agent is: y i (t)=C i x(t), in And its dimension is p i The dimension p of y(t) satisfies the relation p i ≤p, is a partial row of C; each agent measures local output information and exchanges information with neighboring nodes.

3. The distributed robust observer design method based on non-peak high-gain observation technology according to claim 2 is characterized in that: Yes (C i , A) Decompose observability, specifically: Define the system matrix (C i , the unobservable subspace of A) in Indicates subspaces, ∩ represents the intersection operation of subspaces, ker represents the kernel of subspaces, and subspaces The dimension of Representation subspace Orthogonal complement of The column vector of The orthogonal basis of The column vector of Orthogonal basis, construct the orthogonal transformation matrix Make C i T i =[C id 0], where is the matrix T i The transposed matrix of is a matrix with matching dimensions; definition in is the state variable of the observable subsystem, is the state variable of the unobservable subsystem, so the observable subsystem is: y i (t)=C id x id (t), in For x id The derivative of (t), is the matrix D i The transposed matrix of The unobservable subsystems are: in For x id The derivative of (t), is the matrix U i The transposed matrix of .

4. The distributed robust observer design method based on non-peak high-gain observation technology according to claim 3 is characterized in that: Step 3 is as follows: Step 3-1: For the unobservable subsystem, the designed observer adopts the method of neighbor sensor measurement information compensation. The observer form is as follows: where γ i +ρ i is the feedback gain, l ij is the i-th row and j-th column element in the Laplacian matrix of the sensor-agent communication topology; Step 3-2: Design feedback gain γ i +ρ i The adaptive strategy is as follows: Where k is a positive constant.

5. The distributed robust observer design method based on non-peak high-gain observation technology according to claim 4 is characterized in that: Step 4 is as follows: Step 4-1: The information interaction topology between sensor agents adopts a directed graph containing N nodes. To represent, define its adjacency matrix and Laplace matrix as W = (w ij ) and L=(l ij ), a directed graph The strong connectivity condition must be met, that is, starting from any node in the directed graph, there is a directed path to reach all other nodes; Step 4-2, Adaptive term γ in step 3-2 i The initial value of γ i (0) must satisfy the condition that it is greater than 0; Step 4-3: Select the initial value of the observer and

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