Method for designing interval function observer of nonlinear interconnection system

Abstract

The invention discloses a design method for an interval function observer of a nonlinear interconnection system, relates to the technical field of nonlinear interconnection systems, and aims to solve the problem of interval estimation of a state function of the nonlinear interconnection system. The method comprises the following steps: establishing a nonlinear interconnection system model of which the state varies discretely or continuously along with time; designing an interval function observer for the nonlinear interconnection system model; aiming at the interval function observer, considering nonlinear items and interconnection items existing in the nonlinear interconnection system, and designing a group of constraint conditions as sufficient conditions for existence of the interval function observer based on the stability of the designed interval function observer and the convergence of state function estimation errors of the nonlinear interconnection system; calculating a gain matrix of the interval function observer based on the sufficient condition of the existence of the interval function observer; and based on the gain matrix, completing the design of the nonlinear interconnection system interval function observer.

Description

Technical Field

[0001] The present invention relates to the technical field of nonlinear interconnected systems, and in particular to a design method of an interval function observer for a nonlinear interconnected system. Background Art

[0002] The physical state of the system, as a reflection of the system characteristics, cannot be obtained by direct observation in many practical situations. The interval function observer is a kind of interval observer that reconstructs the linear function of the state variable instead of the entire state variable. This design can reduce the order and complexity of the observer and is suitable for more complex systems. The interval function observer is more targeted in the interval estimation of the state function and has gradually become an important research topic in control theory. The interval function observer has certain advantages in system fault diagnosis. By using an interval function observer with higher observation accuracy, the sensitivity of the residual interval to faults can be improved, thereby more effectively performing fault detection and fault estimation. In addition, the interval function observer can also be applied to fields such as system control and state estimation to help achieve accurate control and prediction of system states.

[0003] As a kind of complex large system, interconnected systems are widely used in communications, electricity, biology, petrochemicals and other fields. Research on the design method of interval function observers for interconnected systems can not only enrich the design theory of interval observers for control systems, but also has more important practical application significance. Summary of the invention

[0004] The present invention proposes a method for designing an interval function observer for a nonlinear interconnected system, which comprises the following steps: S1: Establish a nonlinear interconnected system model whose state changes discretely or continuously over time; S2: Designing an interval function observer for the nonlinear interconnected system model in step S1; S3: for the interval function observer in step S2, considering the nonlinear terms and interconnected terms existing in the nonlinear interconnected system, based on the stability of the designed interval function observer and the convergence of the estimation error of the state function of the nonlinear interconnected system, a set of constraints are designed as sufficient conditions for the existence of the interval function observer; S4: Based on the sufficient conditions for the existence of the interval function observer in step S3, a gain matrix of the interval function observer is calculated; S5: Based on the gain matrix in step S4, the design of the interval function observer for the nonlinear interconnected system is completed.

[0005] The present invention is further configured as follows: the nonlinear interconnected system model in step S1 is: Among them, σx i (t) Representing x in a discrete interconnected system i (t+1), when the nonlinear interconnected system model is a discrete interconnected system, define σx i (t) is expressed in a continuous interconnected system When the nonlinear interconnected system model is a continuous interconnected system, define In interconnected systems, and Respectively represent the state vectors of the ith and jth subsystems, the output vector and control input of the ith subsystem; g i [x i (t)] represents the uncertainty or disturbance term in the i-th subsystem; is the coefficient matrix with appropriate dimensions in the ith subsystem, is the interconnection matrix between the i-th subsystem and the j-th subsystem in the interconnected system.

[0006] The present invention is further configured as follows: the nonlinear term g in the nonlinear interconnected system model (Formula 1) is i [x i (t)] meets the following conditions:

[0007] The present invention is further configured as follows: the interval function observer in step S2 is: The system output is: In the formula, N i ,J i ,J ij ,H i ,E i , are the unknown coefficient matrices of the observer to be solved.

[0008] The present invention is further configured as follows: the interval function observer in step S2 performs interval estimation on the following state function: v i (t) = F i x i (t),i=1,2,…,N (Formula 5)

[0009] The present invention is further configured as follows: a sufficient condition for the existence of the interval function observer in step S3 is: Ni is the Nonnegative, Schur matrix (Equation 6) N i is the Metzler, Hurwitz matrix (Equation 7) G i A i -N i G i =J i C i (Formula 8) J ij C j -G i A ij =0 (Formula 9) F i -L i G i -E i C i =0 (Formula 10) H i =G i B i (Formula 11) In the formula: i, j = 1, 2,…, N, i ≠ j.

[0010] The present invention is further configured as follows: the process of solving the gain matrix in step S4 is as follows: S4-1: When (Equation 1) is a nonlinear discrete interconnected system, select Nonnegative and Schur matrix N i ; When (Equation 1) is a continuous interconnected system, select Metzler and Hurwitz matrix N i .; According to the nonlinear interconnected system model, the constant matrix F needs to be selected i ; S4-2: Based on the known matrix given in the nonlinear interconnected system (Equation 1), Construct the matrix; S4-3: Matrix obtained according to step S4-2 and Solving the matrix: To satisfy the conditions: S4-4: Assume unknown matrix According to the matrix N selected in step S4-1 i and the matrix polynomial obtained in step S4-3 and The following matrix G is expressed i and J i : S4-5: The unknown matrix Z selected in step S4-4 i and the unknown matrix L i , so that it satisfies the conditions: in: Description: Matrix and The second subscript of satisfies the following conditions: i≠j k =k,k=1,2,…,N and The N-1 second subscripts j1,…,j k ,…,j N Sort in ascending order, S4-6: According to step S2-5, a matrix G that satisfies the constraint conditions is obtained. i and Y i , S4-7: Solve based on the step S2-5 Finally, we get the matrix E i ,J ij ,i,j=1,2,…,N,i≠j; in: Represents X i The generalized inverse matrix of i represents any matrix, Description: Matrix The second subscript of satisfies the following conditions: (1)i≠j k =k, k = 1, 2, ..., N, (2) Matrix The N-1 second subscripts j1,…,j k ,…,j N Sort in ascending order, S4-8: According to step S4-5, the matrix G obtained by step S4-6 is obtained i and L i , we can get: And H i =G i B i , S4-9: Based on the gain matrix obtained from step S4-1 to step S4-8, the design of the interval function observer of the i-th subsystem is completed (i=1, 2, ..., N).

[0011] The beneficial effects of the present invention are as follows: a design method for interval function observation is proposed for nonlinear interconnected systems. In the design process of the observer, the two-way and continuous interconnected systems are unified into a model for representation. In addition to designing an interval function observer, the sufficient conditions for the existence of the observer are given using the monotone system theory. Based on the Sylvester equation, a solution method for the interval function observer's undetermined matrix is ​​given. Finally, specific steps for constructing an interval function observer are given. In actual complex physical systems, the order and complexity of traditional state observers can be effectively reduced, and the interval estimation of the state is more targeted. At the same time, the proposed interval function observer design method can be applied to two types of discrete and continuous interconnected systems, further broadening the design scope of the interval function observer, and has a certain reference role in the design of more complex Hualian large system interval function observers, enriching the design theory of interval function observers. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 A flow chart of the present invention is shown.

[0013] Figure 2 The interval estimation effect of the state function in the first subsystem of the nonlinear discrete interconnected system in Example 2 is shown.

[0014] Figure 3 The interval estimation effect of the state function in the second subsystem of the nonlinear discrete interconnected system in Example 2 is shown.

[0015] Figure 4 The interval estimation effect of the state function in the first subsystem of the nonlinear continuous interconnected system in Example 3 is shown.

[0016] Figure 5 The interval estimation effect of the state function in the second subsystem of the nonlinear continuous interconnected system in Example 3 is shown. DETAILED DESCRIPTION

[0017] Those skilled in the art can refer to the content of this article and appropriately improve the process parameters. It is particularly important to point out that all similar replacements and modifications are obvious to those skilled in the art, and they are all considered to be included in the present invention. The methods and applications of the present invention have been described through preferred embodiments, and relevant personnel can modify or appropriately change and combine the methods and applications described herein without departing from the content, spirit and scope of the present invention to implement and apply the technology of the present invention.

[0018] Example 1

[0019] The present invention proposes a method for designing an interval function observer for a nonlinear interconnected system, comprising the following steps: S1: Establish a nonlinear interconnected system model whose state changes discretely or continuously over time; S2: Designing an interval function observer for the nonlinear interconnected system model in step S1; S3: for the interval function observer in step S2, considering the nonlinear terms and interconnected terms existing in the nonlinear interconnected system, based on the stability of the designed interval function observer and the convergence of the estimation error of the state function of the nonlinear interconnected system, a set of constraints are designed as sufficient conditions for the existence of the interval function observer; S4: Based on the sufficient conditions for the existence of the interval function observer in step S3, a gain matrix of the interval function observer is calculated; S5: Based on the gain matrix in step S4, the design of the interval function observer for the nonlinear interconnected system is completed.

[0020] The nonlinear interconnected system model in step S1 is: Among them, σx i (t) Representing x in a discrete interconnected system i (t+1), when the nonlinear interconnected system model is a discrete interconnected system, define σx i (t) is expressed in a continuous interconnected system When the nonlinear interconnected system model is a continuous interconnected system, define

[0021] In interconnected systems, and Respectively represent the state vectors of the ith and jth subsystems, the output vector and control input of the ith subsystem; g i [x i (t)] represents the uncertainty or disturbance term in the i-th subsystem; is the coefficient matrix with appropriate dimensions in the ith subsystem, is the interconnection matrix between the i-th subsystem and the j-th subsystem in the interconnected system.

[0022] The nonlinear term g in the nonlinear interconnected system model (Equation 1) is i [x i (t)] meets the following conditions:

[0023] The interval function observer in step S2 is: The system output is:

[0024] In the formula, N i ,J i ,J ij ,H i ,E i , are the unknown coefficient matrices of the observer to be solved.

[0025] The interval function observer in step S2 performs interval estimation on the following state function: v i (t) = F i x i (t),i=1,2,…,N (Formula 5)

[0026] The sufficient condition for the existence of the interval function observer in step S3 is: N i is the Nonnegative, Schur matrix (Equation 6) N i is the Metzler, Hurwitz matrix (Equation 7) G i A i -N i G i =J i C i (Formula 8) J ij C j -G i A ij =0 (Formula 9) F i -L i G i -E i C i =0 (Formula 10) H i =G i B i (Formula 11) In the formula: i, j = 1, 2,…, N, i ≠ j.

[0027] Based on the conditions (Formula 6)-(Formula 11), the following conclusions can be obtained:

[0028] (a) When the nonlinear interconnected system (Equation 3) is a discrete interconnected system, that is, When (Equation 3)-(Equation 4) satisfies the conditions (Equation 6), (Equation 8)-(Equation 11), then (Equation 3) is called the interval function observer of the linear function (Equation 5) of the i-th subsystem in (Equation 1).

[0029] (b) When the nonlinear interconnected system (Equation 3) is a continuous interconnected system, that is, When (Equation 10) satisfies the following conditions (Equation 7)-(Equation 11), (Equation 3) is called the interval observer of the linear function (6) of the i-th subsystem in the system (Equation 1).

[0030] Proof: The following notation is given specifically:

[0031] Next, we will combine the three conditions in the definition of interval function observer to give the proof process. According to (Equation 1), (Equation 3) and (Equation 12), we can get:

[0032] According to (8)-(11), we can get

[0033] Combining the content, we can also conclude

[0034] When the nonlinear interconnected system model (Equation 1) is a discrete interconnected system, according to (Equation 6) and (Equation 16), when t ≥ 0, And the nonlinear interconnected system model (Equation 3) is asymptotically stable; when the nonlinear interconnected system model (Equation 1) is a continuous interconnected system, according to Equation (Equation 7) and (Equation 16), when t ≥ 0, Moreover, the nonlinear interconnected system model (Equation 3) is asymptotically stable, and the first condition of the interval function observer is proved.

[0035] According to (Equation 17), combined with the first condition of the above proof, we know Therefore The second condition for the interval observer is proved.

[0036] set up Based on the first condition of the above proof, according to (Equation 14), (Equation 15) and (Equation 16), we can get

[0037] in

[0038] When the nonlinear interconnected system model (Formula 1) is a nonlinear discrete interconnected system, due to N i is a Nonnegat ive and Schur matrix, so when When uniformly bounded, Uniformly bounded and convergent; when the nonlinear interconnected system model (Formula 1) is a continuous interconnected system, due to N i is a Metzler and Hurwitz matrix, so when When uniformly bounded, Uniformly bounded and convergent.

[0039] At the same time, Apparently

[0040] So, when When uniformly bounded converges to 0, e i,m (t) uniformly bounded converges to 0, that is, The uniformly bounded convergence to 0, the third condition of the interval observer is proved. And then the proof is completed.

[0041] When the nonlinear interconnected system model is a discrete interconnected system, let N i is the Nonnegat iv and Schur matrix; when the system is a continuous interconnected system, let N i It is the Metzler and Hurwitz matrix. Based on the conditions that each subsystem satisfies The nonlinear interconnected system (Eq. 1), function v i (t) = F i x i The undetermined matrix of the interval observer (Formula 3) with (t), i = 1, 2, ..., N can be solved as follows. in Z i and L i It satisfies the conditions Any matrix of .

[0042] Proof: Transposing both sides of the equal sign in (Equation 8) above, we can get Among them G i ,J i is the matrix to be solved.

[0043] Respectively replace I in (Equation 25) ni , Alternative solution method In According to (26), we can get

[0044] By transposing the matrices on both sides of the equal sign in (Equation 26), we can obtain (Equation 23) and (Equation 24) in the theorem.

[0045] Considering that the nonlinear interconnected system model (Formula 1) is an interconnected system, the subscripts of (Formula 9) and (Formula 10) in Theorem 1 are: i, j = 1, 2, ..., N, i ≠ j. The following notation is given for the correlation matrix in (Formula 9) and (Formula 10):

[0046] Description: Matrix and The second subscript i≠j k =k,k=1,2,…,N.

[0047] matrix and The N-1 second subscripts j1,…,j k ,…,j N Sort in ascending order.

[0048] According to (Equation 27)-(Equation 29), the matrix relationship in (Equation 9) and (Equation 10) can be described by the following equation: χ i X i =Y i (Formula 30)

[0049] The solution of the undetermined matrix in (Equation 9) and (Equation 10) is transformed into the solution of equation (Equation 30).

[0050] When the constant matrix X in (Equation 30) i and Y i Satisfy the conditions When , then (Equation 30) has a feasible solution.

[0051] By calculation The matrix E can be solved i ,J ij ,i,j=1,2,…,N,i≠j.

[0052] in, is the Moor-Penrose-inverse of X i .Φ i is an arbitrary matrix to be determined.

[0053] The process of solving the gain matrix in step S4 is as follows: S4-1: When (Equation 1) is a nonlinear discrete interconnected system, select Nonnegativ and Schur matrix N i ; When (Equation 1) is a continuous interconnected system, select Metzler and Hurwitz matrix N i According to the nonlinear interconnected system model, the constant matrix F needs to be selected i ; S4-2: Based on the known matrix given in the nonlinear interconnected system (Equation 1), Construct the matrix; S4-3: Matrix obtained according to step S4-2 and Solving the matrix: To satisfy the conditions: S4-4: Assume unknown matrix According to the matrix N selected in step S4-1 i and the matrix polynomial obtained in step S4-3 and The following matrix G is expressed i and J i : S4-5: The unknown matrix Z selected in step S4-4 i and the unknown matrix L i , so that it satisfies the conditions: in: Description: Matrix and The second subscript of satisfies the following conditions: i≠j k =k,k=1,2,…,N and The N-1 second subscripts j1,…,j k ,…,j N Sort in ascending order, S4-6: According to step S2-5, a matrix G that satisfies the constraint conditions is obtained. i and Y i , S4-7: Solve based on the step S2-5 Finally, we get the matrix E i ,J ij ,i,j=1,2,…,N,i≠j; in: Represents X i The generalized inverse matrix of i represents any matrix, Description: Matrix The second subscript of satisfies the following conditions: (1)i≠j k =k, k = 1, 2, ..., N, (2) Matrix The N-1 second subscripts j1,…,j k ,…,j N Sort in ascending order, S4-8: According to step S4-5, the matrix G obtained by step S4-6 is obtained i and L i , we can get: And H i =G i B i , S4-9: Based on the gain matrix obtained from step S4-1 to step S4-8, the design of the interval function observer of the i-th subsystem is completed (i=1, 2, ..., N).

[0054] Example 2

[0055] This embodiment considers a nonlinear discrete interconnected system model containing two subsystems according to the interval function observer design method for the nonlinear interconnected system disclosed in the embodiment.

[0056] The parameters of subsystem 1 are:

[0057] The parameters of subsystem 2 are:

[0058] For the interval function observer of nonlinear discrete interconnected systems, choose the constant matrix:

[0059] The initial state of the nonlinear discrete interconnected system is: x1(0)=[0.3 -0.3 0.4] T, x2(0)=[0.3 -0.2] T .

[0060] The gain matrix of the interval function observer of the first subsystem is as follows:

[0061] The gain matrix of the interval function observer of the second subsystem is as follows:

[0062] refer to Figure 2 , 3 , respectively show the interval function observer on the nonlinear discrete interconnected system state function interval estimation effect. It can be seen from the figure that the interval function observer can quickly implement interval estimation of the state function of the nonlinear discrete interconnected system.

[0063] Example 3

[0064] This embodiment considers a nonlinear discrete interconnected system model containing two subsystems according to the interval function observer design method for the nonlinear interconnected system disclosed in the embodiment.

[0065] The parameters of subsystem 1 are:

[0066] The parameters of subsystem 2 are:

[0067] For the interval function observer of nonlinear continuous interconnected systems, select the constant matrix:

[0068] The initial state of the nonlinear continuous interconnected system is: x1(0)=[0.1 -0.2 0.3] T , x2(0)=[0.2 -0.1 0.4] T .

[0069] The gain matrix of the interval function observer of the first subsystem is as follows:

[0070] refer to Figure 3 , 4 , respectively show the interval estimation structure of the interval function observer for the state function of the nonlinear continuous interconnected system. It can be seen from the figure that the interval function observer can quickly implement interval estimation of the state function of the nonlinear continuous interconnected system.

[0071] Explanation of symbols: 1. For any moment A, A T represents the transpose of matrix A, and rank(A) represents the rank of matrix A. 2. 0 represents a scalar or a zero matrix of appropriate dimension. 3. The relationship between vectors and matrices is understood based on elements. Vector x1 = [x 11 x 12 x 13 ]≤x2=[x 21 x 22 x 23] , which is x 11 ≤x 21 , x 12 ≤x 22 , and x 13 ≤x 23 ;Matrix A=[a ij ] m×n ≤B=[b ij ] m×n , which is a ij ≤ b ij ,i=1,2,…,m; j=1,2,…,n. 4. Vector x - and x + ∈R n Respectively represent x - ≤0, 0≤x + . 5. Matrix M∈R m×n It can be expressed as: M = M + -M - . Where M + =max(M,0),M - =max(-M,0).

[0072] Based on the above relationship, if there exists x,x - and x + ∈R n Satisfy x - ≤x≤x + , then the following inequality holds: M + x - -M - x + ≤Mx≤M + x + -M - x - .

[0073] In summary, the present invention proposes a design method of interval function observation for nonlinear interconnected systems. In the design process of the observer, the two-dimensional and continuous interconnected systems are unified into a model for representation, and the interval function observer is designed. The sufficient conditions for the existence of the observer are given using the monotone system theory, and the solution method of the interval function observer's undetermined matrix is ​​given based on the Sylvester equation. Finally, the specific steps for constructing the interval function observer are given. In actual complex physical systems, the order and complexity of traditional state observers can be effectively reduced, and the interval estimation of the state is more targeted. At the same time, the proposed interval function observer design method can be applied to both discrete and continuous interconnected systems, further broadening the design scope of the interval function observer, and has a certain reference role in the design of more complex Hualian large system interval function observers, enriching the design theory of interval function observers.

[0074] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be within the scope of protection of the present invention.

Claims

1. A method for designing an interval function observer for a nonlinear interconnected system, characterized in that: The following steps are involved: S1: Establish a nonlinear interconnected system model whose state changes discretely or continuously over time; S2: Designing an interval function observer for the nonlinear interconnected system model in step S1; S3: for the interval function observer in step S2, considering the nonlinear terms and interconnected terms existing in the nonlinear interconnected system, based on the stability of the designed interval function observer and the convergence of the estimation error of the state function of the nonlinear interconnected system, a set of constraints are designed as sufficient conditions for the existence of the interval function observer; S4: Based on the sufficient conditions for the existence of the interval function observer in step S3, a gain matrix of the interval function observer is calculated; S5: Based on the gain matrix in step S4, the design of the interval function observer for the nonlinear interconnected system is completed.

2. The interval function observer design method for nonlinear interconnected systems according to claim 1, characterized in that: The nonlinear interconnected system model in step S1 is: Among them, σx i (t) Representing x in a discrete interconnected system i (t+1), when the nonlinear interconnected system model is a discrete interconnected system, define σx i (t) is expressed in a continuous interconnected system When the nonlinear interconnected system model is a continuous interconnected system, we define In interconnected systems, and Respectively represent the state vectors of the ith and jth subsystems, the output vector and control input of the ith subsystem; g i [x i (t)] represents the uncertainty or disturbance term in the i-th subsystem; is the coefficient matrix with appropriate dimensions in the ith subsystem, is the interconnection matrix between the i-th subsystem and the j-th subsystem in the interconnected system.

3. The interval function observer design method for nonlinear interconnected systems according to claim 2, characterized in that: The nonlinear term g in the nonlinear interconnected system model (Equation 1) is i [x i (t)] meets the following conditions:

4. The interval function observer design method for nonlinear interconnected systems according to claim 3, characterized in that: The interval function observer in step S2 is: The system output is: In the formula, N i , J i , J ij , H i , E i , are the unknown coefficient matrices of the observer to be solved.

5. The interval function observer design method for nonlinear interconnected systems according to claim 4, characterized in that: The interval function observer in step S2 performs interval estimation on the following state function: v i v(t) = F i x i x(t), i = 1, 2, …, N (Equation 5).

6. The interval function observer design method for nonlinear interconnected systems according to claim 5, characterized in that: The sufficient condition for the existence of the interval function observer in step S3 is: N i is the Nonnegative, Schur matrix (Equation 6) N i is the Metzlera-Hurwitz matrix (Formula 7) G i A i -N i G i =J i C i (Formula 8) J ij C j -G i A ij =0 (Formula 9) F i -L i G i -E i c i =0 (Formula 10) H i =G i B i (Formula 11) In the formula: i, j = 1, 2,…, N, i≠j.

7. The interval function observer design method for nonlinear interconnected systems according to claim 6, characterized in that: The process of solving the gain matrix in step S4 is as follows: S4-1: When (Equation 1) is a nonlinear discrete interconnected system, select Nonnegative and Schur matrix N i ; When (Equation 1) is a continuous interconnected system, select the Metzler and Hurwitz matrices N i .; According to the nonlinear interconnected system model, the constant matrix F needs to be selected i ; S4-2: Based on the known matrix given in the nonlinear interconnected system (Equation 1), Construct the matrix; S4-3: Matrix obtained according to step S4-2 and Solve the matrix: To satisfy the conditions: S4-4: Assume unknown matrix According to the matrix N selected in step S4-1 i and the matrix polynomial obtained in step S4-3 and The following matrix G is expressed i and J i : S4-5: The unknown matrix Z selected in step S4-4 i and the unknown matrix L i , so that it satisfies the conditions: in: i=1,2,…,N Description: Matrix and The second subscript of satisfies the following conditions: i≠j k =k, k=1,2,…,N and The N-1 second subscripts j1, ..., j k , …, j N Sort in ascending order, S4-6: According to step S2-5, a matrix G that satisfies the constraint conditions is obtained. i and Y i , S4-7: Solve the problem based on step S2-5 Finally, we get the matrix E i , J ij ,i,j=1,2,…,N,i≠j; in: Represents X i The generalized inverse matrix of i represents any matrix, Description: Matrix The second subscript of satisfies the following conditions: (1)i≠j k =k,k=1,2,…,N, (2) Matrix The N-1 second subscripts j1, ..., j k , …, j N Sort in ascending order, S4-8: According to step S4-5, the matrix G obtained by step S4-6 is obtained i and L i , we can get: And H i =G i B i , S4-9: Based on the gain matrix obtained from step S4-1 to step S4-8, the design of the interval function observer of the i-th subsystem is completed (i=1, 2, ..., N).

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