A single-input fuzzy control system stability judgment method and application

Abstract

This invention relates to a method and application for determining the stability of a single-input fuzzy control system, comprising the following steps: designing an optimization problem based on the Sanghta formula, solving the Lyapunov function and determining the stability of the single-input fuzzy control system; and, while ensuring the stability of the polynomial fuzzy control system, determining the stability of u. b Studying the case where (x) is not equal to 0 is more practical. This method uses the Sontag formula and then combines the properties of the Sontag formula with the Cauchy inequality to propose a new theory that can stabilize polynomial fuzzy control systems. Compared with previous design methods, this method is not constrained by the control Lyapunov function and therefore can better address the issue of u. b The case where (x) is not equal to 0 is studied. While ensuring the stability of the polynomial fuzzy control system, the case of u is investigated. b Studying the case where (x) is not equal to 0 is more relevant to reality.

Description

Technical Field

[0001] This invention relates to the field of system control technology, and in particular to a method and application for determining the stability of a single-input fuzzy control system. Background Technology

[0002] Significant progress has been made in the study of the Takagi-Sugeno (TS) fuzzy model to date, with substantial achievements in addressing the stability of systems under the TS fuzzy model within the framework of nonlinear systems. In particular, control design using linear matrix inequalities (LMIs) has brought about substantial advancements in the design of nonlinear control systems. The greatest advantage of LMI design is that it provides a simple, natural, and effective design procedure for nonlinear system control techniques that require specialized and relatively complex knowledge. Solutions to LMIs can be obtained using the interior-point method in convex optimization techniques.

[0003] A fuzzy control system is a system capable of implementing fuzzy control. It is an automatic control system mainly composed of a fuzzy controller, a controlled object, a detection module, and a feedback section. A single-input fuzzy control system has only one input variable.

[0004] While the linear matrix inequality approach has achieved great success, it still presents some unresolved design problems. These unresolved problems can lead to conservatism, and the use of Sontag's formulas in designing single-input fuzzy control systems has often been limited. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the existing technology by providing a method and application for judging the stability of a single-input fuzzy control system. Based on the Sanghta formula, an optimization problem is designed to solve the Lyapunov function and determine the stability of the single-input fuzzy control system. While ensuring the stability of the polynomial fuzzy control system, the stability of u is further assessed. b Studying the case where (x) is not equal to 0 is more relevant to reality.

[0006] The objective of this invention can be achieved through the following technical solutions:

[0007] This invention provides a method for determining the stability of a single-input fuzzy control system, comprising the following steps:

[0008] Step S1: Obtain the initial Lyapunov function of the single-input fuzzy control system;

[0009] Step S2: Based on the initial Lyapunov function and the preset first condition, solve the first optimization problem and determine whether the second condition is met. If yes, the objective solution is obtained and step S5 is executed; otherwise, step S3 is executed.

[0010] Step S3: Based on the initial Lyapunov function and the preset third condition, solve the second optimization problem, determine whether the fourth condition is met, if yes, obtain the target solution, and execute step S5; if no, execute step S4.

[0011] Step S4: Based on the initial Lyapunov function and the preset fifth condition, solve the third optimization problem, determine whether the sixth condition is met, if yes, obtain the objective solution, and execute step S5; if no, execute step S3.

[0012] Step S5: Based on the target solution, obtain the target Lyapunov function; based on the target Lyapunov function, obtain the stability judgment result.

[0013] The first, second, and third optimization problems are obtained based on the properties of the Sontag formula.

[0014] As a preferred technical solution, the first optimization problem is:

[0015]

[0016] As a preferred technical solution, the second condition is α < 0.

[0017] As a preferred technical solution, the second optimization problem is:

[0018]

[0019] Wherein, (18) is the condition of the first optimization problem, and (19) is:

[0020]

[0021] As a preferred technical solution, the third condition is s(x) = s (θ) (x).

[0022] As a preferred technical solution, the fourth condition is α. (θ) <0.

[0023] As a preferred technical solution, the third optimization problem is:

[0024]

[0025] As a preferred technical solution, the fifth condition is V(x) = V (θ) (x)+δV (θ) (x).

[0026] As a preferred technical solution, the sixth condition is α.(θ) <0.

[0027] As a preferred technical solution, obtaining the first optimization problem, the second optimization problem, and the third optimization problem includes the following steps:

[0028] The empty set condition is obtained based on the properties of the Sontag formula. The empty set condition is then transformed into the SOS condition. The first optimization problem, the second optimization problem, and the third optimization problem are obtained based on the SOS condition.

[0029] In another aspect, the present invention provides the application of the above-described method for judging the stability of a single-input fuzzy control system in judging the stability of a single-input fuzzy control system.

[0030] Compared with the prior art, the present invention has the following advantages:

[0031] High accuracy in stability assessment: Based on the Sanghta formula, an optimization problem is designed to solve the Lyapunov function and assess the stability of a single-input fuzzy control system. While ensuring the stability of the polynomial fuzzy control system, the stability of u is assessed. b Studying the case where (x) is not equal to 0 is more practical. Compared with previous design methods, this method is not constrained by the control Lyapunov function, and therefore can be applied to u. b Studying cases where (x) is not equal to 0 will lead to more accurate judgments. Attached Figure Description

[0032] Figure 1 This is a flowchart of the stability judgment method for a single-input fuzzy control system in Example 1;

[0033] Figure 2 This is a diagram showing the processing results of the control system.

[0034] Figure 3 This is a diagram showing the trajectory of the controller changing state. Detailed Implementation

[0035] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0036] Example 1

[0037] like Figure 1The present embodiment provides a method for judging the stability of a single-input fuzzy control system. It uses the Sontag formula and then combines the properties of the Sontag formula with the Cauchy inequality to propose a new theory that can stabilize a polynomial fuzzy control system.

[0038] First, establish a polynomial fuzzy model, considering the following polynomial nonlinear system:

[0039]

[0040] Here, x(t) = [x1(t) x2(t) … x n(t) ] T It is the state vector, and the input vector is u(t). This article studies the single-input case, so u(t) is a single input. The system shown in system (1) is a polynomial nonlinear system, where F(x(t)) and G(x(t)) are nonlinear functions and satisfy F(0) = 0. Then, a polynomial fuzzy model is constructed using the sector nonlinear definition. The polynomial nonlinear system shown in (1) can be accurately represented by the polynomial fuzzy model in the system.

[0041] Fuzzy rule i:

[0042] IF z1(t) is M i1 And...and z p (t) is M ip ,

[0043]

[0044] Here, r represents the existence of r fuzzy rules, z j (t)(j=1,2,...p) are known premise variables, M ij It is a fuzzy set related to the i-th fuzzy rule and the j-th premise variable. Where A i (x(t)) is an n×N real polynomial system matrix in x(t), B i (x(t)) is an n×q real polynomial input matrix with respect to x(t). Let x(t) be an N-row, one-column monomial vector, and let x(0) = 0. Therefore, defuzzification yields the following expression that accurately describes the dynamics of the nonlinear system:

[0045]

[0046] To simplify the expression of the formulas later, we will simplify the symbols. x(t) becomes x, meaning that the x that appears later is a function of t.

[0047] The definition is as follows:

[0048] Definition 1: A function with respect to x = [x1 x2 … x n The monomial form of ] is Where d1, d2, ..., d n It is a non-negative integer. Therefore, the degree of this monomial can be expressed as:

[0049] Definition 2: Define P as a polynomial universe. Suppose ω(x) is a polynomial with real coefficients and is itself a linear combination of a finite number of monomials. For any ω(x) ∈ P, when ω(x) is greater than or equal to 0, it is positive semi-definite. This set of positive semi-definite polynomials is called p. 0+ .

[0050] Definition 3: Define S as a SOS (Sum of squares) polynomial space contained in P. A polynomial ω(x) can be written as... In this form, we can consider ω(x) as an SOS polynomial, where (f i (x)) i＝1，…m ∈P. It can be seen that when ω(x)∈S, ω(x)∈P must hold. 0+ Therefore, when x ≠ 0 and ω(x) > 0, the SOS polynomial ω(x) can be considered positive definite. This set of positive definite SOS polynomials uses S... + express.

[0051] Definition 4: Given (G) i (x)) i＝1，…，z ∈P, multiplicative monoid M(G1(x), G2(x)…G v (x) is G i The set of finite products of (x). That is, M(G1(x), G2(x)...G v The elements in (x) satisfy the associative law, are closed, and have an identity element.

[0052] Definition 5: Given (F) γ (x)) γ＝1，…，l ∈P,F γ The cone of (x) can be represented in the following form:

[0053] Where ω is a positive integer, (S l (x)) l＝0，…ω ∈P,E l (x)∈M(F1(x), F2(x)…F n (x)).

[0054] Definition 6: Given (H) β (x)) β＝1，…，q∈P,H β The ideal case of (x) can be defined by the following expression:

[0055] Where p β (x)∈P and q is a positive integer.

[0056] Definition 7: General form of Cauchy's inequality

[0057] The P-satz lemma is as follows:

[0058] Polynomial (F) γ (x)) γ＝1，…，n , (G i (x)) i＝1，…，z , (H β (x)) i＝1，…，q If and only if there exist F∈C(F1(x), F2(x), ..., F n (x)), H∈Γ(H1(x), H2(x),…,H q (x)), G∈M(G1(x), G2(x),…,G v (x) makes:

[0059] F+G 2 +H=0 #(4)

[0060] Then the set {x∈R} n |F1(x)≥0,…,F n H(x)≥0, H1(x)=0, …, H q (x)=0, G1(x)≠0, …,G v (x)≠0 is an empty set.

[0061] The controller is designed as follows:

[0062]

[0063] The stability condition for a single-input system is derived as follows:

[0064] Under the controller design method mentioned in step four, if there exists a smooth and radially unbounded function V(x), an SOS polynomial s(x), and a positive definite SOS polynomial P(x) satisfying equations (6) and (7), where α is non-positive, then the closed-loop nonlinear polynomial fuzzy system shown in equation (3) will be asymptotically stable over a wide range.

[0065]

[0066] l1(x)∈S +The existence of l1(x) is mainly to ensure that V(x) is a radially unbounded positive definite function.

[0067] For the nonlinear system shown in (3), the derivative of the Lyapunov function is shown in (8):

[0068]

[0069] When V(x) is a radially unbounded positive definite function, it is only necessary to prove that Then the system can remain stable. Based on the controller shown in Part Four, we found the following characteristics:

[0070] (1) When u b When x > 0, u(x) must be < 0, that is,

[0071] (2) When u b When x < 0, u(x) > 0, that is, we have

[0072] In summary, when u b When (x)≠0, we can deduce the conclusion

[0073] Based on the above reasoning, we divide equation (8) into two parts. First, when... We just need to ensure It is also certain that its establishment will enable So the above ensures The condition can be rewritten in the following form:

[0074]

[0075] Next, the condition in equation (9) is transformed into an empty set condition, and then the P-satz lemma is used to process the empty set condition into an SOS constraint condition.

[0076] Here due to h i (z)>0, we can set h i (z) is transformed into a quadratic variable. Then the condition in equation (9) can be transformed into the following empty set condition.

[0077]

[0078]

[0079] Here, F1(x), H1(x), and G1(x) correspond to the values ​​in equation (10).

[0080] Applying the P-satz lemma to the empty set condition in equation (10), the empty set condition can be transformed into the following equation condition:

[0081]

[0082] Here, s0(x), s1(x)∈S, p(x)∈P, and α is a non-negative real number. Finally, the condition in equation (11) is transformed into the following SOS condition:

[0083]

[0084] Here Therefore, omit, The existence of this makes the computation too intensive. To reduce the computational burden, the Cauchy inequality will be used to transform it into a suitable polynomial:

[0085]

[0086] And because Therefore Therefore, equation (14) must hold true.

[0087]

[0088] If equation (14) is true, then by replacing the corresponding term in equation (12), we get equation (15).

[0089]

[0090] From equation (14), we can see that if there exists V(x) such that equation (15) holds, there must also exist V(x) such that equation (12) holds.

[0091] The specific steps of this method are as follows:

[0092] Step S1: Select a suitable initial Lyapunov function V (θ) (x)

[0093] Step S2: Let V(x) = V (θ) (x) and solve the following optimization problem:

[0094]

[0095] If the obtained α < 0, then the obtained V (θ) (x), S (θ) (x) is the solution to Theorem 1. If α (θ) If it is negative, then proceed to step 3.

[0096] Step S3: Let s(x) = s(θ) (x) and solve the following optimization problem:

[0097]

[0098] V(x)+δV(x)-l1(x) is SOS #(18)

[0099]

[0100] Solving for V yields (θ) (x), δV (θ) (x), if the solution yields α (θ) If V < 0, then the obtained V (θ) (x)+δV (θ (x), S (θ) (x) is the solution to Theorem 1. If α (θ) If it is negative, then proceed to step 4.

[0101] Step S4: Let V(x) = V (θ) (x)+δV (θ) (x) and solve the following optimization problem:

[0102]

[0103] V(x)-l1(x) is SOS #(21)

[0104]

[0105] Solving for δs (θ) (x), s (θ) (x), if the solution yields α (θ) If α < 0, then V(x) is the solution to Theorem 1. If α is non-negative, then proceed to step S3 again.

[0106] We will demonstrate the effectiveness of our proposed controller design method through a design example. We will take a fuzzy model with three rules, as shown in equation (2), as an example.

[0107] IF z1(t) is M i1 And...and z p (t) is M ip ,

[0108]

[0109] here

[0110]

[0111]

[0112]

[0113] The membership function of the fuzzy model is shown below:

[0114]

[0115]

[0116] M2(x) = 1 - M1(x) - M3(x)

[0117] Applying our proposed SOS constraint to this example improves system performance while ensuring system stability. Our stability constraint ensures system stability when a = 4 and b = 7. Compared to existing methods, our controller design is less restrictive and places lower demands on the system.

[0118] The Lyapunov function V(x) and polynomial s(x) obtained through the algorithm are as follows:

[0119] s(x)=0.08x1 2 +1.03x1x2+3.35x2 2 -1.67*e -11 x1-1.02*e -10 x1+0.0006

[0120] V(x) = 1.597 * e -8 x1 2 +6.166*e -8 x1x2+6.953*e -8 x2 2

[0121] The controller's processing result diagram and the turntable's change trajectory diagram are as follows: Figure 2 , Figure 3 As stated above.

[0122] Compared to previous design methods, our approach is not constrained by the control Lyapunov function, thus enabling control over u. b We investigate the case where (x) is not equal to 0. While ensuring the stability of the polynomial fuzzy control system, we examine u... b Studying the case where (x) is not equal to 0 is more relevant to reality.

[0123] Example 2

[0124] This embodiment provides an application of the stability judgment method for a single-input fuzzy control system as described in Embodiment 1 in judging the stability of a single-input fuzzy control system. First, an initial Lyapunov function is obtained, and then a target Lyapunov function is obtained using the method provided in Embodiment 1, thereby realizing the stability judgment of the fuzzy control system.

[0125] Preferably, the stability judgment method is applied to a single-input single-output closed-loop DC speed control system. The error speed input analog-to-digital converter converts the signal into a signal of a predetermined format and inputs it to the controller. The controller has built-in instructions to implement the stability judgment method for the single-input fuzzy control system as described in Example 1. After processing, the controller uses a digital-to-analog converter to transmit the signal to the DC motor, and the speed measuring device measures the speed and feeds it back to the input to achieve closed-loop control.

[0126] The Lyapunov acquisition method and controller design method used in this embodiment can enhance the stability of the DC speed control system and improve its control accuracy.

[0127] Example 3

[0128] This embodiment provides an electronic device, including: one or more processors and a memory, wherein the memory stores one or more programs, the one or more programs including instructions for executing the stability judgment method of the single-input fuzzy control system as described in Embodiment 1.

[0129] Example 4

[0130] This embodiment provides a computer-readable storage medium including one or more programs executable by one or more processors of an electronic device, the one or more programs including instructions for performing the stability determination method for a single-input fuzzy control system as described in Embodiment 1.

[0131] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and these modifications or substitutions should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for judging the stability of a single-input fuzzy control system, characterized in that, Includes the following steps: Step S1: Obtain the initial Lyapunov function of the single-input fuzzy control system; Step S2: Based on the initial Lyapunov function and the preset first condition, solve the first optimization problem and determine whether the second condition is met. If yes, the objective solution is obtained and step S5 is executed; otherwise, step S3 is executed. Step S3: Based on the initial Lyapunov function and the preset third condition, solve the second optimization problem, determine whether the fourth condition is met, if yes, obtain the target solution, and execute step S5; if no, execute step S4. Step S4: Based on the initial Lyapunov function and the preset fifth condition, solve the third optimization problem, determine whether the sixth condition is met, if yes, obtain the objective solution, and execute step S5; if no, execute step S3. Step S5: Based on the target solution, obtain the target Lyapunov function; based on the target Lyapunov function, obtain the stability judgment result. The first, second, and third optimization problems are derived based on the properties of the Sontag formula. The first optimization problem is: in, It is a smooth and radially unbounded function. For SOS polynomials, For non-positive values, , Let SOS be the set of positive definite polynomials. For state vectors, To determine the number of fuzzy rules, Represents N rows and one column about monomial vectors, It is a real polynomial system matrix. It is a real polynomial input matrix. The second optimization problem is: The third optimization problem is: 。 2. The stability judgment method for a single-input fuzzy control system according to claim 1, characterized in that, The second condition is .

3. The stability judgment method for a single-input fuzzy control system according to claim 1, characterized in that, The third condition is .

4. The stability judgment method for a single-input fuzzy control system according to claim 1, characterized in that, The fourth condition is .

5. The stability judgment method for a single-input fuzzy control system according to claim 1, characterized in that, The fifth condition is And the sixth condition is .

6. The stability judgment method for a single-input fuzzy control system according to claim 1, characterized in that, The acquisition of the first, second, and third optimization problems includes the following steps: The empty set condition is obtained based on the properties of the Sontag formula. The empty set condition is then transformed into the SOS condition. The first optimization problem, the second optimization problem, and the third optimization problem are obtained based on the SOS condition.

7. The application of the stability judgment method for a single-input fuzzy control system as described in any one of claims 1-6 in judging the stability of a single-input fuzzy control system.

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