Stability judgment method for cost control of polynomial fuzzy control system

Abstract

The present application relates to a kind of facing polynomial fuzzy control system cost control stability judging method, comprising the following steps: establishing polynomial fuzzy model;According to the first SOS condition based on Lyapunov function, try to solve the first polynomial matrix and the second polynomial matrix, omit non-convex term in solving process, judge whether it is successful, if yes, based on the first polynomial matrix and the second polynomial matrix, obtain feedback gain;Based on feedback gain, according to the second SOS condition, try to solve positive definite matrix, judge whether it is successful, if yes, based on positive definite matrix, construct polynomial Lyapunov function, realize cost control analysis based on polynomial Lyapunov function.Compared with prior art, the present application solves the non-convex optimization problem in cost control in two steps, avoids the influence of the constraint existing in input matrix itself on the flexibility of fuzzy control system design, and improves the performance of control system by obtaining the numerical value of cost J.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of system control, in particular to a stability judgment method for cost preserving control of a polynomial fuzzy control system. BACKGROUND

[0002] The research on polynomial fuzzy control system has made great progress, and it provides more superior performance compared with the previous T-S fuzzy control system. The traditional T-S fuzzy control system can be considered as a special case of the polynomial fuzzy control system, so the applicability of the polynomial fuzzy control system is more extensive.

[0003] Lyapunov stability theory is a theory for analyzing the stability of a system created by A.M. Lyapunov, a Russian mathematician and mechanician, in 1892. For a control system, stability is a basic problem to be studied. When studying linear constant systems, there are many criteria such as algebraic stability criterion and Nyquist stability criterion that can be used to determine the stability of the system. Lyapunov stability theory can be applied to analyzing the stability of linear systems and nonlinear systems, constant systems and time-varying systems, and is a more general stability analysis method. Lyapunov function is a function constructed for determining the stability of a system in nonlinear system control, and has wide application in the field of system control and mechanical engineering.

[0004] Polynomial fuzzy control system refers to a system capable of realizing fuzzy control. It mainly consists of a fuzzy controller, a controlled object, a detection module and a feedback part. Unlike ordinary fuzzy control systems, the system matrix and the feedback matrix of the polynomial fuzzy control system are composed of polynomials. Therefore, it can more accurately represent a nonlinear system with fewer fuzzy rules.

[0005] However, there are still some deficiencies in the processing method of the cost preserving control of the polynomial fuzzy control system. The mainstream processing method is to design a model of the system to eliminate the part that cannot be solved. This method not only has a big discount in effect, but also limits the flexibility of system design.

[0006] In summary, there is currently a lack of a cost preserving controller acquisition method to solve the problem of low flexibility of fuzzy control system design caused by the difficulty of solving non-convex terms and the difficulty of non-convex optimization in existing methods. SUMMARY

[0007] The purpose of the present application is to overcome the defects of the prior art and provide a stability judgment method for cost preserving control of a polynomial fuzzy control system. In the design process of the cost preserving control of the polynomial fuzzy control system, the non-convex optimization problem is solved in two steps, which overcomes or partially overcomes the influence of the constraints existing in the input matrix itself on the flexibility of the fuzzy control system design.

[0008] The objective of this invention can be achieved through the following technical solutions:

[0009] This invention provides a stability judgment method for cost-preservation control of polynomial fuzzy control systems, comprising the following steps:

[0010] Establish a polynomial fuzzy model;

[0011] Based on the polynomial fuzzy model, according to the preset first SOS (Sumof squares) condition set based on the Lyapunov function, attempt to solve the first polynomial matrix and the second polynomial matrix. Non-convex terms are omitted during the solution process. Determine whether the solution is successful. If not, end the cost-preserving control. If yes, obtain the feedback gain based on the first polynomial matrix and the second polynomial matrix.

[0012] Based on the feedback gain, according to the preset second SOS condition, attempt to solve the positive definite matrix, and determine whether the solution is successful. If not, end the cost-preserving control. If yes, construct a polynomial Lyapunov function based on the positive definite matrix, and determine the stability of the fuzzy control system based on the polynomial Lyapunov function.

[0013] Based on the feedback gain, the cost function of the polynomial fuzzy control system is calculated.

[0014] As a preferred technical solution, the process of establishing a polynomial fuzzy model includes the following steps:

[0015] Obtain the nonlinear system expression for the system state vector and system input vector;

[0016] Based on the nonlinear system expression, fuzzification processing is performed according to the preset fuzzy rule set and control rule set;

[0017] The polynomial fuzzy model is obtained through defuzzification.

[0018] As a preferred technical solution, for the expression being For nonlinear systems, fuzzification is performed based on a preset fuzzy rule set and a control rule set using the following formula:

[0019] if z1(t) is Q i1 and…and z p (t)is Q ip

[0020]

[0021] Where x(t)=[x1(t)x2(t)…x n (t)]T It is the state vector of the system, u(t)=[u1(t)u2(t)…u m (t)] T This is the system's input vector, where i = 1, 2, ..., r represents a total of r fuzzy rules, and z k (t) is a measurable time variable, k = 1, 2, ..., p represents a total of p control rules. It is an N×1 column vector. All elements are monomials of x(t) if and only if x(t) = 0. A i (x(t))∈R n×N B is the polynomial state matrix of the system. i (x(t))∈R n×m It is the polynomial input matrix of the system.

[0022] As a preferred technical solution, the polynomial fuzzy model is:

[0023]

[0024] in, And for any i, h i (z(t))≥0 and

[0025] As a preferred technical solution, the output of the polynomial fuzzy control system is:

[0026]

[0027] In order to simplify the symbols, and x(t) are respectively used When replacing x with x, note that although time t is removed, x remains a vector with respect to time t. (C) i (x) is a polynomial matrix.

[0028] As a preferred technical solution, the cost function of the polynomial fuzzy system is:

[0029]

[0030] Where Q and R are positive definite matrices.

[0031] As a preferred technical solution, the first SOS condition includes:

[0032] v1 T (X(x)-α(x))v1 is SOS (Condition 1)

[0033]

[0034]

[0035] is SOS (condition 3)

[0036] wherein X(x)R N×N is symmetric, N k (x) is the second polynomial matrix, a ii (x) is an arbitrary scalar greater than or equal to 0,

[0037] As a preferred technical solution, the second SOS condition comprises:

[0038] -v4 T (P(x)-a(x))v4 is SOS (condition 4)

[0039]

[0040]

[0041]

[0042] is SOS (condition 7)

[0043] wherein P(x)R N×N is the target matrix,

[0044] As a preferred technical solution, the feedback gain is F k (x)=N k (x)X(x) -1 .

[0045] As a preferred technical solution, SOSTOOLS is used to solve the first SOS condition and the second SOS condition.

[0046] Another aspect of the present application provides an application of the above-mentioned stability judgment method for cost-preserving control of a polynomial fuzzy control system in judging the stability of a fuzzy control system.

[0047] Compared with the prior art, the present application has the following advantages:

[0048] (1) In the cost preserving control design process of polynomial fuzzy control system, the non-convex optimization problem is solved in two steps. The first step is to solve the first SOS condition by ignoring the non-convex term, and in the second step, the obtained feedback gain is taken as a known quantity and brought into the second SOS condition. Thus, the non-convex optimization problem of cost preserving control is converted into a convex optimization problem, thereby overcoming or partially overcoming the influence of the constraints existing in the input matrix itself on the design flexibility of the fuzzy control system, and improving the design flexibility of the fuzzy control system.

[0049] (2) The method for obtaining the cost preserving controller of the polynomial fuzzy control system is given by the SOS design condition. A more general polynomial fuzzy model than the TS fuzzy model is given and the controller is designed, and then the cost preserving control design condition based on Lyapunov function is derived. Therefore, the method of the present application is more advanced than the existing LMI method based on Lyapunov function. BRIEF DESCRIPTION OF DRAWINGS

[0050] Figure 1 Flow chart for the Lyapunov function obtaining method in Example 1;

[0051] Figure 2 Schematic diagram of control system processing result;

[0052] Figure 3 Schematic diagram of cost preserving control result. DETAILED DESCRIPTION

[0053] The technical solutions in the embodiments of the present application will be described clearly and completely in combination with the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor should belong to the protection scope of the present application.

[0054] In the present application, the terms "first" and "second" are only used for description purposes, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Therefore, the features defined with "first" and "second" can explicitly or implicitly include one or more of the features. In the description of the present application, the meaning of "multiple" is two or more, unless otherwise specifically limited.

[0055] Example 1

[0056] As Figure 1 described, the present embodiment provides a stability judgment method for cost preserving control of polynomial fuzzy control system, comprising the following steps:

[0057] S1, establish a polynomial fuzzy model, the first step of the "two-step method", try to solve the symmetric polynomial matrix X(x) e R under the first SOS condition N×N and a polynomial matrix N k (x). The specific steps are as follows:

[0058] Step S101, establish a polynomial fuzzy model:

[0059] Consider the following nonlinear system:

[0060]

[0061] Where g is a nonlinear function, x(t) = [x1(t) x2(t) … x n (t)] T is the state vector of the system, u(t) = [u1(t) u2(t) … u m (t)] T is the input vector of the system.

[0062] The nonlinear system shown in (1) is accurately represented by the following polynomial fuzzy model:

[0063] The fuzzy rule i is as follows:

[0064] if z1(t) is Q i1 and…and z p (t) is Q ip

[0065]

[0066] Where i = 1, 2, …, r represents a total of r fuzzy rules. z k (t) is a measurable time variable, k = 1, 2, …, p represents a total of p control rules. is an N x 1 column vector, all of whose elements are monomials composed of x(t). In this paper, it is assumed that A i (x(t)) e R n×N is the polynomial state matrix of the system, B i (x(t)) e R n×m is the polynomial input matrix of the system.

[0067] De-fuzzification (2) can be expressed as the following accurate polynomial fuzzy model:

[0068]

[0069] Where

[0070]

[0071] According to the property of membership function, for any i, we have h i (z(t)) ≥ 0 and

[0072] Step S102, cost control:

[0073] This step gives the cost function of cost control. To simplify the notation, we replace and x(t) by and x, respectively. Note that although the time t is removed, x is always a vector with respect to time t.

[0074] The output model of the polynomial fuzzy control system is as follows:

[0075]

[0076] where C i (x) is a polynomial matrix, the cost function of the polynomial fuzzy system is represented as follows:

[0077]

[0078] where Q and R are positive definite matrices

[0079] Step S103, the first step of the "two-step method", derivation of the first SOS design condition

[0080] If there exists a symmetric polynomial matrix X(x)R N×N and a polynomial matrix N k (x) satisfying conditions 1, 2, and 3, then there exists a cost control controller such that the polynomial fuzzy control system is globally asymptotically stable.

[0081] v1 T (X(x)-α(x))v1 is SOS (condition 1)

[0082]

[0083]

[0084] is SOS (condition 3)

[0085] Proof: The candidate Lyapunov function is given as follows:

[0086]

[0087] The time derivative of the above candidate Lyapunov function is as follows:

[0088]

[0089]

[0090] where represents A i the s-th row of (x), T(x)∈R N×n is a polynomial matrix, whose element in the i-th row and the k-th column is

[0091] Note: According to the principle of two-step method, the non-convex is considered in the second step algorithm.

[0092] Let

[0093] If condition 2 and condition 3 are satisfied, there must be:

[0094]

[0095]

[0096] where

[0097]

[0098] where α ii (x) is defined as any scalar greater than or equal to 0, from (7) and (8) can be derived as follows inequality:

[0099]

[0100] Using Schur complement can get:

[0101]

[0102] From (10) can be derived:

[0103]

[0104] where After the latter part of (11) can be obtained:

[0105]

[0106] Because When x≠0 must be true. In summary, if conditions 1, 2, 3 are satisfied, this closed-loop system must be large range asymptotically stable.

[0107] According to (12), we have

[0108]

[0109] Integrating (13) from 0 to ∞, we have

[0110]

[0111] When t→∞, we have which can be transformed into

[0112]

[0113] The above is the first step of the two-step method to solve the cost constraint. B does not need to be considered i The term (x(t))u(t) avoids the emergence of non-convex terms. Using the N k (x) obtained by the first step, X(x) can express F k (x) as a known quantity and bring it into the second step to continue solving.

[0114] Step S2, determine whether there is a solution to step S1. If there is no solution, the algorithm is terminated. If there is a solution, continue.

[0115] Step S3, based on the N k (x) and X(x) obtained in step S1, calculate the feedback gain F k (x) = N k (x)X(x) -1 .

[0116] Step S4, design conditions for the second step of the two-step method. Under the second SOS condition, try to calculate the positive definite matrix P(x) ∈ R N ×N . The specific steps are as follows:

[0117] If there exists a positive definite matrix P(x) ∈ R N×N such that conditions 4, 5, 6, and 7 are satisfied, then the cost constraint controller of this system is F k (x) = N k (x)X(x) -1 . Solving conditions 4, 5, 6, and 7 can obtain an optimal cost.

[0118]

[0119] subject to

[0120] -v4 T (P(x)-α(x))v4 is SOS (condition 4)

[0121]

[0122]

[0123]

[0124] is SOS (condition 7)

[0125] where

[0126]

[0127] The following is a simulation example to verify the effectiveness of the method proposed in this article, the model of the polynomial fuzzy system is as follows:

[0128]

[0129]

[0130] Here

[0131]

[0132]

[0133]

[0134] C1(x) = [1 0] C2(x) = [1 0],

[0135] The membership function is given as follows:

[0136]

[0137]

[0138] According to the SOS design condition proposed in the present application, X(x) and F i (x) need to be solved, when there is a feasible solution, then the effectiveness of the cost-preserving method proposed in the present application is verified. It is worth noting that when the degree of X(x) and F i (x) is not 0, the theory of the present application has feasibility. When the degree of X(x) and F i (x) is 0, it indicates that X(x) and F i (x) become constant matrices, and the design condition of the fuzzy control system is changed from the SOS polynomial design condition to the LMI design condition.

[0139] As Figure 2The control result graph is shown in the figure. The six lines in the figure represent the trajectories of the selected six points over time, and the final trajectory converges at the origin, indicating that the Lyapunov function obtained can make the system large-scale asymptotically stable.

[0140] As Figure 3 The cost control result graph is shown in the figure. The two lines in the figure represent the trajectories of the state variables x1 and x2 over time t, which finally converge to 0, indicating that the Lyapunov function satisfying the solving condition can make the system stable. When Q=I, R=I, x(0)=[5 10] T , the cost control controller is obtained according to the cost control controller when the Lyapunov function is 0 or 1. When the Lyapunov function is 2, J=66.81 is obtained, when the Lyapunov function is 4, J=55.47 is obtained, when the Lyapunov function is 6, J=49.54 is obtained, and when the Lyapunov function is 6, J=45.77 is obtained. With the increase of the Lyapunov function, the value of J is gradually reduced, and the cost control effect is gradually improved.

[0141] The present application proposes a two-step method to solve the constraint condition of the cost control of the polynomial fuzzy control system, and the design condition is realized by minimizing the upper limit of the given performance function to ensure the cost control. In the process of stability analysis of the polynomial fuzzy control system, it is inevitable to solve the non-convex problem. Because the non-convex optimization problem is difficult to solve, it is often necessary to convert the non-convex optimization problem into a convex optimization problem for solving. For the polynomial fuzzy control system, the commonly used conversion method is to construct an input matrix with multiple non-zero rows. When the input matrix has multiple non-zero rows, the non-convex term in the polynomial fuzzy control system can be eliminated. In the process of designing the cost control of the polynomial fuzzy control system, the two-step method is used to solve the non-convex optimization problem in the control process. The two-step method is used to solve the non-convex optimization problem in two steps, which avoids the influence of the constraints of the input matrix itself on the design flexibility of the fuzzy control system. The method used in the present application not only makes the fuzzy control system large-scale asymptotically stable, but also obtains the minimum cost of making the system stable. The cost control is very important in industrial production, and reasonable design can greatly improve the efficiency of industrial production.

[0142] The present application provides a cost preserving control method for a polynomial fuzzy control system. A more general polynomial fuzzy model than the TS fuzzy model is given and a controller is designed. Then a cost preserving control design condition based on Lyapunov function is derived. Therefore, the present method is more advanced than the existing LMI method based on Lyapunov function. In addition, a two-step method is used to design the cost preserving control of the polynomial fuzzy control system, which avoids solving non-convex terms. The construction of an input matrix with all-zero rows also avoids solving non-convex terms. Obviously, the present method greatly enhances the flexibility of the cost preserving control design of the polynomial fuzzy control system. Finally, for the SOS design condition proposed in the present application, SOSTOOLS is used to solve the numerical value in two steps.

[0143] Embodiment 2

[0144] The present embodiment provides an application of the stability judgment method for the cost preserving control of the polynomial fuzzy control system in judging the stability of a fuzzy control system. First, a polynomial fuzzy model to be judged is obtained, and a Lyapunov function is obtained using the method described in Embodiment 1, so as to realize the stability judgment.

[0145] Preferably, in the application of the present embodiment in the field of variable frequency air conditioners, the fuzzy controller obtained in the process of obtaining the control Lyapunov function is applied to a household variable frequency air conditioner. The temperature, the frequency converter and the air conditioner compressor can be more closely associated. Using the method of the present application, the change of the temperature can be more quickly reflected on the frequency converter, while the air conditioner compressor is always kept at a reasonable speed. The application of the present application in the variable frequency air conditioner can prolong the service life of the air conditioner and save 30-40% of the electricity.

[0146] Embodiment 3

[0147] The present embodiment provides an electronic device, comprising one or more processors and a memory, wherein the memory stores one or more programs, and the one or more programs comprise instructions for executing the stability judgment method for the cost preserving control of the polynomial fuzzy control system as described in Embodiment 1.

[0148] Embodiment 4

[0149] The present embodiment provides a computer readable storage medium, comprising one or more programs for execution by one or more processors of an electronic device, and the one or more programs comprise instructions for executing the stability judgment method for the cost preserving control of the polynomial fuzzy control system as described in Embodiment 1.

[0150] The above merely illustrates the specific embodiments of the present application, but the protection scope of the present application is not limited thereto, and any skilled person in the art can easily think of various equivalent modifications or replacements within the technical range disclosed by the present application, and these modifications or replacements shall be covered within the protection scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.

Claims

1. A stability determination method for cost control of a polynomial fuzzy control system, characterized by, The method comprises the following steps: establishing a polynomial fuzzy model; based on the polynomial fuzzy model, according to a first SOS condition set based on a Lyapunov function, attempting to solve a first polynomial matrix and a second polynomial matrix, omitting non-convex terms in the solving process, judging whether the solving is successful, if not, ending the cost preserving control, if yes, obtaining feedback gain based on the first polynomial matrix and the second polynomial matrix; based on the feedback gain, according to a second SOS condition, attempting to solve a positive definite matrix, judging whether the solving is successful, if not, ending the cost preserving control, if yes, constructing a polynomial Lyapunov function based on the positive definite matrix, judging the stability of the fuzzy control system based on the polynomial Lyapunov function; based on the feedback gain, calculating a cost function of the polynomial fuzzy control system.

2. The method of claim 1, wherein the cost control system is a polynomial fuzzy control system. The process of establishing the polynomial fuzzy model comprises the following steps: obtaining a nonlinear system expression about a system state vector and a system input vector; based on the nonlinear system expression, performing fuzzy processing according to a preset fuzzy rule set and a control rule set; obtaining the polynomial fuzzy model through defuzzy processing.

3. The method of claim 2, wherein the cost control is a polynomial fuzzy control system. For the nonlinear system with expression The fuzzy processing according to the preset fuzzy rule set and control rule set is implemented by the following formula: where is the state vector of the system, is the input vector of the system, represents the total number of fuzzy rules, is a measurable time variable, represents the total number of control rules, is an N x 1 column vector, all elements of are monomials composed of if and only if , is the polynomial state matrix of the system, is the polynomial input matrix of the system.​ 4. The method of claim 3, wherein the cost control is a cost control of a polynomial fuzzy control system. The polynomial fuzzy model is: wherein , and for any have and .

5. The method of claim 4, wherein the cost control is a polynomial fuzzy control system. The output of the polynomial fuzzy control system is: where, for simplicity of notation, we have replaced and respectively by and respectively, noting that while the time has been removed, is always a vector with respect to time , is a polynomial matrix.

6. The method of claim 5, wherein the cost control is a polynomial fuzzy control system. The cost function of the polynomial fuzzy system is: wherein and are positive definite matrices, .

7. The method of claim 6, wherein the cost control is a polynomial fuzzy control system. The first SOS condition comprises: is SOS is SOS SOS wherein said first polynomial matrix is symmetric, said second polynomial matrix is symmetric, is an arbitrary scalar greater than or equal to 0, , , is a polynomial matrix, represents the first row.

8. The method of claim 7, wherein the cost control is a polynomial fuzzy control system. The second SOS condition comprises: is SOS is SOS is SOS SOS wherein is the target matrix, wherein is the feedback gain.

9. The method of claim 8, wherein the cost control is a polynomial fuzzy control system. The feedback gain is .

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