A Fault Estimation Method Based on a Finite-Time Generalized Unknown-Input Observer

Abstract

The present invention belongs to the field of robust fault estimation for linear systems, and particularly relates to a fault estimation method based on a finite-time generalized unknown input observer. The method comprises the following steps: 1. Establish a linear system model including disturbances and faults; 2. Judge the property of the linear system established in step 1. If the fault is reconstructible or asymptotically reconstructible, then continue with this method; 3. Decompose the linear system model established in step 1 to decompose out a strongly observable subsystem; 4. Construct a finite-time generalized unknown input observer for the strongly observable subsystem decomposed in step 3 for state estimation; 5. Estimate the fault based on the state values obtained in step 4. The present invention can quickly estimate the fault.

Description

Technical Field

[0001] The present invention relates to a fault estimation method based on a finite-time generalized unknown input observer, belonging to the technical field of robust fault estimation for linear systems. Background Art

[0002] In terms of improving the fault estimation speed, the UIO method based on a finite-time observer can accurately estimate the system state and faults within any preset time, but it requires that the matrix composed of the respective distribution matrices of the faults and disturbances is column full rank and satisfies the observer matching condition. The conditions are strict and it cannot solve the problem of fast estimation of mismatched faults in a linear system under the influence of disturbances. Summary of the Invention

[0003] The purpose of the present invention is to solve the problem of fast estimation of mismatched faults in a linear system under the influence of disturbances in the above-mentioned existing technologies, and provide a fast and robust fault estimation method based on a finite-time generalized unknown input observer. One aspect of the present invention relaxes the strict conditions that the matrix composed of the respective distribution matrices of the faults and disturbances is column full rank and the observer matching required by the traditional finite-time fault estimation observer to the fault and disturbance reconstructible conditions; the other is to define an equivalent condition for a fault asymptotically reconstructible system, further relaxing the observer existence condition.

[0004] A fault estimation method based on a finite-time generalized unknown input observer of the present invention is characterized in that it includes the following steps:

[0005] Step 1: Establish a linear system model including disturbances and faults;

[0006] Step 2: Judge the properties of the linear system established in Step 1. If the fault is reconstructible or asymptotically reconstructible, continue with this method;

[0007] Step 3: Decompose the linear system model established in Step 1 to decompose out a strongly observable subsystem;

[0008] Step 4: Construct a finite-time generalized unknown input observer for the strongly observable subsystem decomposed in Step 3 for state estimation;

[0009] Step 5: Estimate the fault according to the state value obtained in Step 4.

[0010] Further, in the above Step 1, the established linear system model is:

[0011] (1);

[0012] Wherein, , , , , The vectors respectively composed of the states of the system, the control inputs, the faults, the disturbances, and the outputs is a constant matrix of appropriate dimension. Since the control input plays no role during the fault estimation process, for convenience, in the present invention, it is assumed to be 0;

[0013] Let , , .

[0014] Furthermore, the process of judging the properties of the linear system established in step 1 in step 2 is as follows:

[0015] Make the following judgment on the linear system model (1):

[0016] ;

[0017] where is the basis of the weakly unobservable subspace of the linear system , satisfies ; is the basis of the weakly unobservable subspace of the subsystem , and , the dimension .

[0018] If the above equation holds, it is judged that the linear system model (1) is fault - reconstructible.

[0019] Furthermore, if the linear system model (1) is fault - reconstructible, in step 3, the system is decomposed as follows:

[0020] Define the non - singular transformation matrix

[0021]

[0022] where, .

[0023] Let , the linear system model (1) can be transformed into:

[0024] (2);

[0025] where, is the transformed system state, and ; the state matrix , the matrix is calculated from , , is an arbitrary matrix of appropriate dimension; the matrix , , and the subsystem is strongly observable;

[0026] Find a non - singular matrix , and its inverse , such that , where has full column rank. Taking the transpose of gives , then the subsystem is transformed into the following strongly observable subsystem :

[0027] (3);

[0028] Among them, the matrix , the matrix , the matrix , is the defined intermediate vector, is obtained by solving the following equation:

[0029] , is obtained by solving .

[0030] Furthermore, in the fourth step, first construct a generalized unknown - input observer for the strongly observable subsystem as follows:

[0031] Use the derivatives of each order of the output in the system as additional signals to obtain a new output equation: (4);

[0032] Among them, is the -th derivative of the output , is the -th derivative of , represents the vector composed of and its derivatives of each order (see the vector above the symbol in Equation (4)), represents the matrix above the symbol in Equation (4), represents the matrix above the symbol in Equation (4), represents the vector above the symbol in Equation (4);

[0033] The strongly observable subsystem The generalized unknown input observer is designed as follows:

[0034] (5);

[0035] Wherein, and are the vectors to be estimated, the matrix is the observer parameter, and it is ensured that .

[0036] Furthermore, the design process of the generalized unknown input observer parameter is as follows:

[0037] Decompose the matrices and into the following forms:

[0038] , ;

[0039] According to the matrices and , define ,

[0040] Wherein, must be stable, and it is ensured that and hold.

[0041] Furthermore, in the fourth step, establish an invertible matrix:

[0042] (6);

[0043] Wherein, and The calculation methods of the two symbols are the same as those of the foregoing ; and , .

[0044] Furthermore, in the fourth step, calculate , , where is rows.

[0045] Furthermore, in the fourth step, according to , select a matrix that ensures the stability of , and solve an observer matrix .

[0046] Furthermore, in the fourth step, according to the above-obtained , where .

[0047] Furthermore, in step 4, according to , the observer and are obtained by solving.

[0048] Furthermore, in step 4, the finite-time generalized unknown input observer designed for the strongly observable subsystem is:

[0049] (7);

[0050] where is abbreviated as , is the process estimation vector, is the estimated value of the system state , are the parameters of the observer to be designed; the delay is an arbitrarily preset convergence time in advance, , ,

[0051] .

[0052] Furthermore, in step 4, the process of designing the parameters of the finite-time generalized unknown input observer is:

[0053] Design two generalized unknown input observers:

[0054] (8);

[0055] Let

[0056] , and some parameters in the observer (7) can be formed.

[0057] where , , and the symbols with all represent the corresponding parameters of the th observer.

[0058] Furthermore, in step 4, according to the solving method in the solution of the parameters of the generalized unknown input observer, solve the parameters of the th observer, and select such that:

[0059] (i) is stable, (ii) .

[0060] Furthermore, in the fifth step, the fault estimation value can converge to the true value within a finite time, and the fault estimation process is as follows:

[0061] (9);

[0062] where, is solved by .

[0063] Furthermore, in the second step, if the fault reconfiguration condition cannot be satisfied, then according to the third step, the system is decomposed, and the process is as follows:

[0064] Find a non-singular matrix with its inverse such that , where has full column rank, then the system is decomposed into:

[0065] (10);

[0066] where the state vector , the state vector , the state vector , the vector ; is the fault vector; , , , , , , , , , , , , , , , are all matrices after system transformation, and the system is strongly observable , and the estimated value of its corresponding system state can be solved by the finite-time generalized unknown input observer (7).

[0067] Perform the following equality judgment on the linear system model (1):

[0068] (11);

[0069] If the equality (11) holds and is Hurwitz, then it is judged that the system is asymptotically reconstructible for faults.

[0070] Furthermore, for a system that is asymptotically reconstructible for faults, the fault estimate value can asymptotically converge to the true value. In step 5, the fault estimation expression is:

[0071] (12);

[0072] where is solved by .

[0073] In the above fault estimation expression, is obtained by the following observer:

[0074] (13).

[0075] The working principle of the present invention is: decompose the system into a strongly observable subsystem, design a finite-time generalized unknown input observer for the state of the strongly observable subsystem, and then use the estimated system state to estimate the fault.

[0076] The beneficial effects of the present invention are:

[0077] 1. Relax the finite-time fault estimation conditions affected by disturbances. Traditional finite-time fault estimation observers require strict conditions such as the matrix composed of the respective distribution matrices of faults and disturbances being column full rank and the observer matching. The present invention relaxes the observer conditions to fault and disturbance reconstructibility. As long as the fault and disturbance distribution matrices meet the fault reconstructibility conditions, the fault estimation error can be bounded within a finite time.

[0078] 2. Define the equivalent conditions for a system that is asymptotically reconstructible for faults, further relaxing the system conditions for the existence of the observer; due to the action of the finite-time generalized unknown input observer, the speed of fault estimation is accelerated. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 is the flowchart of the method of the present invention;

[0080] Figure 2 is the fault estimation diagram based on the generalized unknown input observer in Embodiment 2;

[0081] Figure 3 is the fault estimation diagram based on the generalized unknown input observer in Embodiment 2;

[0082] Figure 4 is the state estimation error diagram based on the finite-time generalized unknown input observer in Embodiment 2;

[0083] Figure 5It is the fault of the finite-time generalized unknown input observer in Embodiment 2 estimation diagram;

[0084] Figure 6 It is the fault of the finite-time generalized unknown input observer in Embodiment 2 estimation diagram. Specific implementation manners

[0085] To make the objectives, technical solutions and advantages of the present invention clearer, the technical solutions of the present invention will be further described below with reference to the accompanying drawings.

[0086] Embodiment 1

[0087] As Figure 1 shown is the flowchart of the method of the present invention, and the method of the present invention includes the following steps:

[0088] Step 1: Establish a linear system model, considering disturbances and faults.

[0089] The linear system model is:

[0090] (1);

[0091] Wherein, , , , , are respectively the vectors composed of the states of the system, the vectors composed of the control inputs, the vectors composed of the faults, the vectors composed of the disturbances, and the vectors composed of the outputs. is a proper-dimensional constant matrix. Since the control input does not play any role during the process of estimating the faults, for the sake of convenience, in the present invention, it is assumed to be 0. Let , , .

[0092] Step 2: Judge whether the linear system model (1) is fault-reconfigurable.

[0093] The following judgment is made on the linear system model (1):

[0094] ,

[0095] Wherein is the basis of the weakly unobservable subspace of the system , ; is the basis of the weakly unobservable subspace of the subsystem , and , the dimension .

[0096] If the equation holds, determine whether the linear system model (1) is fault-reconfigurable.

[0097] Step 3: If the linear system model (1) is fault-reconfigurable, perform the following decomposition:

[0098] Define the non-singular transformation matrix

[0099] ,

[0100] where .

[0101] Let , and the linear system model (1) can be transformed into:

[0102] (2);

[0103] where is the transformed system state, and ; the state matrix , the matrix is calculated from , , is an arbitrary matrix of appropriate dimension; the matrix , , and the subsystem is strongly observable.

[0104] Find a non-singular matrix , the inverse , such that , where has full column rank. Taking the transpose of gives , then the subsystem is transformed into the strongly observable subsystem :

[0105] (3);

[0106] where the matrix , the matrix , the matrix , is the defined intermediate vector, is solved from the following equation:

[0107] . is solved from .

[0108] Step 4: Construct a finite-time generalized unknown input observer for the strongly observable subsystem The process is as follows:

[0109] Use the system The output in The derivatives of each order are used as additional signals to obtain a new output equation:

[0110] (4);

[0111] The design of the generalized unknown input observer is designed as:

[0112] (5);

[0113] Wherein, And Are the vectors to be estimated, and the matrix Is the observer parameter, and it is ensured that .

[0114] The design process of the generalized unknown input observer parameters is:

[0115] Decompose the matrix And Into the following form:

[0116] , .

[0117] Define ,

[0118] Wherein, Must be stable, and it is ensured that , Holds.

[0119] Establish an invertible matrix:

[0120] (6);

[0121] Wherein, And The calculation methods of the two symbols are the same as those of the foregoing , and , .

[0122] Let , , where Is Rows.

[0123] According to , select a matrix That ensures the stability of , and solve an observer matrix .

[0124] Then , where .

[0125] According to , the observer and are obtained by solving.

[0126] For the strongly observable subsystem , the finite-time generalized unknown input observer designed is:[[]]

[0127] (7);

[0128] Among them, for the sake of writing convenience, is abbreviated as , is the process estimation vector, is the estimated value of the system state , are the parameters of the observer to be designed; the delay is an arbitrarily pre-set convergence time, , .

[0129] The process of designing the parameters of the finite-time generalized unknown input observer is as follows:

[0130] Design two generalized unknown input observers:

[0131] (8);

[0132] Let , and some parameters in the observer (7) can be formed.

[0133] Among them, , , and the symbols with all represent the corresponding parameters of the th observer.

[0134] Select and such that

[0135] (i) is stable, (ii) .

[0136] Step 5: The fault estimated value can converge to the true fault value within the finite time , and the fault estimation process is:[[]]

[0137] (9);

[0138] Among them, is obtained from Solve.

[0139] Step 6: If the fault reconfiguration condition cannot be satisfied, then decompose the system The process is as follows:

[0140] Find a non-singular matrix , and its inverse , such that , where has full column rank. Then, the state in system (2) can be decomposed into two parts. After decomposition, the system becomes:

[0141] (10);

[0142] Among them, the state vector , the state vector , the state vector , and the vector ; is the fault vector; , , , , , , , , , , , , , , , are all matrices after system transformation; and the system is strongly observable , and the estimated value of its corresponding system state can be solved by the finite-time generalized unknown input observer (7).

[0143] Perform the following equality judgment on the linear system model (1):

[0144] (11);

[0145] If equation (11) holds and is Hurwitz, then it is judged that the system is fault asymptotically reconstructible.

[0146] Step 7: For a fault asymptotically reconstructible system, its fault estimate value can asymptotically converge to the true value, and the fault estimate expression is:

[0147] (12);

[0148] Among them, 。

[0149] The above fault estimation expression is obtained by the following observer:

[0150] (13).

[0151] Embodiment 2

[0152] Considering that the fault asymptotical reconstructible system is a more general system than the fault reconstructible system, taking an aircraft flight control system as an example, the effectiveness of the proposed fault estimation method based on the finite-time generalized unknown input observer is further demonstrated. The relevant parameters of the mathematical model of this system can be expressed as:

[0153] , , , , , , 。

[0154] Assume that the actuator fault is:

[0155] ,

[0156] The disturbance is 。

[0157] Calculate the basis of the weakly unobservable subspace , 。

[0158] According to the judgment method of system properties, it is judged that this system is fault asymptotical reconstructible.

[0159] First, use the generalized unknown input observer to estimate the fault, and the pole configuration is , and the fault estimation is shown in Figure 2 and Figure 3 . The abscissa in the figure represents the simulation time, the ordinate represents the fault amplitude, the solid line represents the true value of the fault, and the dashed line represents the estimated value of the fault. It can be seen from the simulation diagram that the fault can be asymptotically estimated, and the generalized unknown input observer designed in this paper is effective.

[0160] In order to improve the speed of fault estimation, use the method of the finite-time generalized unknown input observer. The poles of the two observers are respectively configured as and , the delay time is set to 0.1 s, then the system state estimation error is shown in Figure 4 , and the fault estimated value is shown in Figure 5 and Figure 6 。

[0161] As can be seen from Figure 4 , the three states of the system converge rapidly, while the other state converges asymptotically. This is because the rapidly converging states belong to the strongly observable part of the linear system model (1) and can be calculated by a finite-time observer, while the asymptotically converging state belongs to the weakly unobservable part, and its dynamic convergence characteristics are determined by . As can be seen from Figure 5 and Figure 6 , the fault can converge rapidly because the strongly observable states converge in finite time; however, when the fault error converges to a very small neighborhood, it then turns to asymptotic convergence, which is caused by the asymptotically converging state. Nevertheless, the fault estimation algorithm designed in this paper still exhibits the advantage of rapidity.

[0162] The above numerical examples of the present invention are only for illustrating in detail the calculation model and calculation process of the present invention, rather than limiting the implementation manners of the present invention. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is impossible to list all the implementation manners here. Any obvious changes or modifications derived from the technical solutions of the present invention still fall within the protection scope of the present invention.

Claims

1. A fault estimation method based on a finite-time generalized unknown input observer, characterized in that: The following steps are involved: Step 1: Establish a linear system model including interference and faults; Step 2: determine the properties of the linear system established in step 1. If the fault is reconfigurable or asymptotically reconfigurable, continue with the method. Step 3: Decompose the linear system model established in step 1 to obtain strongly observable subsystems; Step 4: Construct a finite-time generalized unknown input observer for state estimation for the strongly observable subsystem decomposed in step 3; Step 5: Estimate the fault according to the state value obtained in step 4; In step 2, if the linear system model (1) is fault reconfigurable, the system is decomposed as follows: Define a nonsingular transformation matrix: P -1 =[V ⊥+ V] Among them, V + =(V T V) -1 V T ,V ⊥+ =V ⊥T (V ⊥ V ⊥T ) -1 ; make The linear system model (1) can be transformed into: in, is the system state after the conversion, and State Matrix Matrix K * Depend on Calculated, Υ2 is any matrix of suitable dimension; the matrix C1=CV ⊥+ , and the subsystem (A 11 ,C1,B1,D) strong and considerable; Find a non-singular matrix H = [H1 H2], inverse Make in The column is full rank, Find the transposed matrix and we get Then the subsystem (A 11 ,C1,B1,D) is transformed into the following strongly observable subsystem Among them, the matrix matrix matrix ζ(t) is the intermediate vector defined, Solve the following equation to get: E1, F2 Solved to get; In step 4, for the strongly observable subsystem The designed finite-time generalized unknown input observer is: in, Abbreviated as is the process estimation vector, System Status The estimated value of H * are the observer parameters to be designed; the delay τ>0 is the convergence time arbitrarily set in advance, (t∈[t0-τ,t0]), In step 4, the parameter design process of the finite-time generalized unknown input observer is: Design two generalized unknown input observers: make It can form part of the parameters in the observer (7); in, The symbols with i all represent the corresponding parameters of the i-th observer; In step 2, if the fault reconfiguration condition cannot be met, the system ∑(A, B, C, D) is decomposed according to step 3. The process is: Non-singular matrix H = [H1 H2], inverse Make in If the column is full rank, then the system ∑(A,B,C,D) can be decomposed into: Among them, the state vector State Vector State Vector vector f is the fault vector; A 11 , A 211 , A 221 , A 212 , A 222 , A 224 , B 11 , B 12 , B 21 , B 22 , B 31 , B 32 , B 33 , are all matrices after system transformation, and the system Strong and impressive The corresponding system status Estimated value of It can be solved by the finite-time generalized unknown input observer (7); The linear system model (1) is judged as follows: If equation (11) holds, and A 221 is Hurwitz, then the system ∑(A,B,C,D) is fault-asymptotically reconfigurable.

2. The fault estimation method based on finite-time generalized unknown input observer according to claim 1, characterized in that: In step 1, the linear system model is: in, are the vectors composed of the system state, control input, fault, interference and output respectively. C. M, N, E, and F are constant matrices of suitable dimensions. In the process of estimating faults, the control input u(t) has no effect and is assumed to be 0; Let ω(t) = [f(t) T , ξ(t) T T , B = [M, E], D = [N, F],​ Where B is the matrix composed of M and E, and D is the matrix composed of N and F.

3. The fault estimation method based on finite-time generalized unknown input observer according to claim 2 is characterized in that: In step 2, the process of determining the properties of the linear system established in step 1 is: The linear system model (1) is judged as follows: where V is the basis of the weakly unobservable subspace of the linear system ∑(A,B,C,D), V ⊥ Meet V ⊥ V = 0; It is a subsystem The basis of the weakly unobservable subspace of Dimensions if If the equation holds, then the linear system model (1) is fault reconfigurable.

4. The fault estimation method based on finite-time generalized unknown input observer according to claim 1, characterized in that: In step 4, firstly, the strongly observable subsystem Constructing a generalized unknown input observer, the process is as follows: Using the system The derivatives of each order of output y(t) are used as additional signals to obtain a new output equation: Among them, y (α) is the α-order derivative of the output y(t), ζ (α) is the α-order derivative of ζ(t), Y [0:α] Represents the vector composed of y(t) and its derivatives of various orders, see the symbol Y in equation (4) [0:α] The vector above, Φ α The symbol Φ in equation (4) α The matrix above, M α The symbol M in equation (4) α The matrix above, Γ [0：α] The symbol M in equation (4) α The vector above; Strongly observable subsystem The generalized unknown input observer is designed as: in, and is the estimated vector, matrix N * ,L,E * is the observer parameter, and ensure 5. The fault estimation method based on finite-time generalized unknown input observer according to claim 4 is characterized in that: In step 4, the parameter design process of the generalized unknown input observer is: The matrices L and E * Decomposed into the following form: L=[L0 L1 LL α-1 ], According to the matrix L and E * , define R = [LN * E * 0]+[0 E * ], Among them, N * Must be stable and ensure N * -A 11 +RΦ α =0 and Established; Create the reversibility matrix: The calculation methods of the two symbols ⊥ and + are the same as those of V ⊥ 、V + is calculated in the same way; and calculate Where J2 is m rows; according to Choose a guaranteed N * Stable matrix R1, and solve an observer matrix N * ; Then R=[R1 R2]G, where According to R = [LN * E * 0]+[0 E * ], and solve for the observer L and E * .

6. The fault estimation method based on finite-time generalized unknown input observer according to claim 1, characterized in that: In step 4, the parameters of the i-th observer are solved according to the R1 solution method in the generalized unknown input observer parameter solution. And choose τ so that: (i) Stable, (ii)

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