Intermittent control method of type-2 switching fuzzy system

Abstract

The invention discloses an intermittent control method for a type-2 switching fuzzy system, and relates to the technical field of control engineering. The method comprises the following steps: firstly, in a switching nonlinear system, based on a dynamic state interval type-2 fuzzy set and under an If-Then rule, establishing a global fuzzy model of a discrete switching nonlinear system through an IT2 fuzzy set of a prerequisite variable and the emission intensity of a fuzzy rule; then, based on the global fuzzy model, proposing an IT2 fuzzy intermittent controller depending on a timer, and constructing a type 2 fuzzy intermittent controller structure framework; thirdly, inputting a type-2 fuzzy intermittent controller structure framework into the global fuzzy model, and defining a new switching signal to obtain a switching system with a 2N subsystem; and finally, controlling the new switching signal, and calculating the gain of the IT2 fuzzy intermittent controller, so that the IT2 fuzzy intermittent controller works intermittently, and meanwhile, ensuring the stability and H-infinity performance of the switching system of the 2N subsystem.

Description

Technical Field

[0001] The invention relates to the technical field of control engineering, and in particular to an intermittent control method for a type-two switching fuzzy system. Background Art

[0002] In complex practical control problems such as industrial automation, mechanical system control, and air traffic control, switched nonlinear systems have shown great application potential. Switched nonlinear systems are usually composed of multiple nonlinear subsystems and a switching rule that coordinates the switching between these subsystems. They can select the most appropriate control strategy according to different operating states of the system and improve the overall control performance of the system.

[0003] In the prior art, using the TS fuzzy model method, the switching nonlinear system is represented by a combination of several local linear modes and membership functions through If-Then rules, which greatly simplifies the difficulty of system analysis. So far, using the TS fuzzy model, many linear system analysis methods have been obtained, which effectively reduces the difficulty of analyzing the switching nonlinear system.

[0004] However, due to factors such as temperature changes and element wear, parameter uncertainty is widely present in actual nonlinear systems, and the TS fuzzy model does not take into account the parameter uncertainty of the membership function. To solve this problem, the researchers proposed a type-two (IT2) fuzzy model, which introduces a pair of upper and lower membership functions to describe parameter uncertainty, showing higher flexibility and robustness in dealing with the uncertainty of complex systems.

[0005] On the other hand, with the continuous development of control theory, intermittent control strategy has gradually attracted attention as an effective means to reduce control costs. Intermittent control means that the controller works alternately between busy and idle states, thereby minimizing the running time of the controller while ensuring system performance, reducing energy consumption and costs. Intermittent control strategy has been successfully applied to many fields such as communication control, industrial control, chemical reaction control, and transportation control.

[0006] Therefore, in view of the shortcomings of traditional control methods in dealing with the uncertainty of complex nonlinear systems, combining the IT2 fuzzy model with the intermittent control strategy and applying it to switching nonlinear systems is of great significance for improving system performance and reducing control costs. Summary of the invention

[0007] The problem to be solved by the present invention is to provide an intermittent control method for a type-two switching fuzzy system, combining the advantages of a type-two fuzzy model (IT2), switching system theory and intermittent control strategy to improve the performance of the switching system and reduce the control cost.

[0008] The present invention adopts the following technical solution: an intermittent control method for a type II switching fuzzy system, comprising the following steps:

[0009] S1. In the switched nonlinear system, based on the dynamic state interval type II fuzzy set, under the If-Then rule, the global fuzzy model of the discrete switched nonlinear system is established through the IT2 fuzzy set of the premise variables and the emission intensity of the fuzzy rules;

[0010] S2. Based on the global fuzzy model, a timer-dependent IT2 fuzzy intermittent controller is proposed, and the structural framework of the type II fuzzy intermittent controller is constructed;

[0011] S3, input the structure framework of the type II fuzzy intermittent controller into the global fuzzy model, define a new switching signal, and obtain a switching system with 2N subsystems;

[0012] S4, control the new switching signal, calculate the IT2 fuzzy intermittent controller gain, make the IT2 fuzzy controller work intermittently, and ensure the stability of the switching system of the 2N subsystem and H ∞ performance.

[0013] Specifically, in step S1, the dynamic state interval type II fuzzy set is DSIT2FSs, and under the If-Then rule, if θ1(x(k)) is θ2(x(k)) is yes represents the premise variable θ s (x(k)) fuzzy rule is m th IT2 fuzzy set, p represents the number of premise variables;

[0014] Then, the closed-loop DSIT2FSs is expressed as follows:

[0015]

[0016] Where x(k) is the state vector of the type-two switching fuzzy system, is a set of real numbers, n x is the vector dimension, and k is used to represent time. k ≥ 0, is an integer set; y(k) is the control output of the type-two switching fuzzy system; u(k) and ω(k) are the control input and disturbance input respectively, σ(k) is the switching signal; A im , B im , C im , D im 、E im 、F im is a known constant matrix, is a fuzzy set, r represents the number of fuzzy rules, σ(k)=i, A collection of subsystems.

[0017] m th The emission intensity interval set W of fuzzy rules im (x(k)), expressed as follows:

[0018]

[0019] in, and They are θ s The upper and lower membership functions LUMFs of (x(k)), ψ im (x(k)), are the lower and upper bounds of the membership function respectively;

[0020] Let σ(k)=i, Based on the fuzzy reasoning method, the global fuzzy model of discrete switching nonlinear system is obtained:

[0021]

[0022] Among them, ψ im (x(k)) is used to represent the membership function, and and is satisfied The nonlinear weighting function of .

[0023] Furthermore, in step S2, assuming that the switching signal σ(k) of the global fuzzy model of the discrete switching nonlinear system satisfies the minimum residence time constraint, a timer-dependent IT2 fuzzy intermittent controller is proposed. yes yes yes So:

[0024]

[0025] Among them, l represents the cycle number, τ is the defined timer, K in ,τ represents the controller gain;

[0026] [T 2l ,T 2l+1 ) is the controller busy interval, during which the controller is activated, [T 2l+1 ,T 2l+2) is the controller idle time interval, during which the controller stops running.

[0027] Represents premise variable n th Fuzzy sets, n th The emission intensity of fuzzy rules is expressed as an interval set:

[0028]

[0029] in, and They are The upper and lower membership functions LUMFs, φ in (x(k)), denote lower and upper membership, respectively;

[0030] In a similar way, the structure framework of the type-2 fuzzy intermittent controller is obtained, which is expressed as:

[0031]

[0032] Among them, φ in (x(k)) is used to represent the degree of membership, and Nonlinear weighting function and satisfy

[0033] Further, in step S3, using φ im and ψ im Represents φ im (x(k)) and ψ im (x(k)), so that:

[0034]

[0035]

[0036] Input the IT2 fuzzy intermittent controller structure framework into the global fuzzy model and obtain:

[0037]

[0038] Furthermore, in step S3, a new switching signal is defined

[0039]

[0040] Where N represents the number of subsystems, s(k) represents the working state of the controller, and is given by definition.

[0041] New switching signal It also depends on the switching signal σ(k) and the controller working state:

[0042]

[0043] in, A collection of subsystems representing the controller's work, Represents the collection of subsystems for which the controller is not functioning.

[0044] Then, the N-mode DSIT2FSs with intermittent control input in Equation (6), i.e., Equation (8), can be regarded as the following switching system with 2N subsystems:

[0045]

[0046] in, In the collection Take the value from.

[0047] Furthermore, we verify that the switching system with 2N subsystems in formula (11) has the switching signal Below is GUAS (globally uniform asymptotic stability) with l2-gain no greater than γ.

[0048] First, construct a class of non-increasing SPDLFs (Switching Path Dependent Lyapunov) functions:

[0049]

[0050] Among them, P c,τ >0, represents the timer, and c represents Switching paths within a controller operation cycle

[0051] Given an integer T b ≥1 and T f ≥1, if there exists a constant γ>0, a positive definite symmetric matrix X c,τ , matrix Ω iτ , U im,τ and the symmetric matrix Λ c,τ,m , with the right size The switching system of the 2N subsystem makes the switching signal Under this condition, it has globally uniform asymptotic stability with l2-gain no greater than γ, for any and Both have φ in -σ in ψ in ≥0 and satisfies the given integer Under the constraints of the conditions, the IT2 fuzzy intermittent controller gain is given by Calculated.

[0052] Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0053] The intermittent control method of the type II switching fuzzy system of the present invention can ensure the stability and H stability of the closed-loop system by designing the IT2 fuzzy controller to work intermittently by treating the switching system with intermittent control input as a switching system with two modes: a controller busy mode and a controller idle mode. ∞ performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a flow chart of the intermittent control method of the type II switching fuzzy system of the present invention;

[0055] Figure 2 is a state response diagram of a closed-loop system according to an embodiment of the present invention;

[0056] Figure 3 Schematic diagram of the control input of the closed-loop system according to the embodiment of the present invention. DETAILED DESCRIPTION

[0057] In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the application is further elaborated in detail below in conjunction with the accompanying drawings. The described embodiments are only a part of the embodiments involved in the present invention. All non-innovative embodiments of other researchers in the field on this embodiment belong to the protection scope of the present invention. At the same time, for the step numbering in the embodiment of the present invention, it is only set for the convenience of explanation, and the order between the steps is not limited in any way. The execution order of each step in the embodiment can be adaptively adjusted according to the understanding of those skilled in the art.

[0058] The present invention provides an intermittent control method for a type II switching fuzzy system, such as Figure 1 As shown, the following steps are included:

[0059] S1. In the switched nonlinear system, based on the dynamic state interval type II fuzzy set, under the If-Then rule, the global fuzzy model of the discrete switched nonlinear system is established through the IT2 fuzzy set of the premise variables and the emission intensity of the fuzzy rules;

[0060] S2. Based on the global fuzzy model, a timer-dependent IT2 fuzzy intermittent controller is proposed, and the structural framework of the type II fuzzy intermittent controller is constructed;

[0061] S3, input the structure framework of the type II fuzzy intermittent controller into the global fuzzy model, define a new switching signal, and obtain a switching system with 2N subsystems;

[0062] S4, control the new switching signal, calculate the IT2 fuzzy intermittent controller gain, make the IT2 fuzzy controller work intermittently, and ensure the stability of the switching system of the 2N subsystem and H ∞ performance.

[0063] The dynamic state interval type-2 fuzzy set is DSIT2FSs, which is a special form of type-2 fuzzy sets, which provides higher flexibility and expressiveness when dealing with uncertainty. Type-2 fuzzy sets not only consider the degree to which an element belongs to a set (i.e., membership), but also the uncertainty or imprecision of this degree itself. DSIT2FSs further models this uncertainty as a dynamically changing interval, allowing the system to more accurately reflect the complexity and dynamics of the real world.

[0064] In DSIT2FSs, the membership of each element is no longer a fixed value, but an interval that changes with time or other factors. This dynamic characteristic makes DSIT2FSs more effective in dealing with systems with time-varying characteristics.

[0065] In particular, in step S4, the following inference 1 proves that the switching signal The switched system with 2N subsystems under Equation (11) is a GUAS with l2-gain no greater than γ.

[0066] Consider the switched system with 2N subsystems in formula (11), given the integer T b ≥1 and T f ≥1. Assume that function kind Function κ 1,c,τ and κ 2,c,τ , and γ>0, then:

[0067]

[0068]

[0069]

[0070] Then the switching system with 2N subsystems in formula (11) has the switching signal Below is a GUAS with l2-gain no greater than γ.

[0071] First, let ω(k) = 0. From equations (14) and (15), we can get V c,τ+1 (k+1)-V c,τ (k)<-y T (k)y(k)<0, and Hengyou That is, SPDLF is constantly decreasing and meeting Combined with (13), it means that the switching system with 2N subsystems is a GUAS.

[0072] Next, consider the l2-gain of the switched system with 2N subsystems in Eq. (11): Let From equation (14) and equation (15), we can get Hengyou let:

[0073]

[0074] So, Under zero initial conditions, we have Hengyou

[0075] In summary, it is proved that when switching signal The switched system with 2N subsystems under Equation (11) is a GUAS with l2-gain no greater than γ.

[0076] Further, in step S4, it is derived that to ensure the stability of the switching system with 2N subsystems in the closed-loop formula (11) and H ∞ Performance, through the following reasoning 2, we get the sufficient conditions for the existence of periodic intermittent controller.

[0077] Reasoning 2: Consider the closed-loop switching system with 2N subsystems in formula (11), given the integer T b ≥ and T f ≥1, if there exists a constant constant γ>0, positive definite symmetric matrix X c,τ , matrix Ω iτ , U im,τ and the symmetric matrix Λ c,τ,m , with the right size A switching system with 2N subsystems switches the signal Next, it is a GUAS with l2-gain no greater than γ, so that for any and Both have φ in -σ in ψ in ≥0, and the following constraints apply:

[0078] S4.1. Given an integer For any The following conditions hold:

[0079] Ψ c,τ,mn -Λ c,τ,m <0,

[0080] σ im Ψ c,τ,mm -σ im Λ c,τ,m +Λ c,τ,m <0,

[0081] σ in Ψ c,τ,mn -σ in Λ c,τ,m +Λ c,τ,m +σ im Ψ c,τ,nm -σ im Λ c,τ,n +Λ c,τ,n <0, (17)

[0082] Where i = c τ ,and

[0083]

[0084] The superscript T indicates transposition.

[0085] S4.2. Given an integer For any The following conditions hold:

[0086]

[0087] Where j = c τ -N, for The following conditions hold:

[0088]

[0089] S4.3. Given an integer For any

[0090] The following conditions hold:

[0091]

[0092] in,

[0093] When there is a feasible solution for the constraints S4.1 to S4.3, the IT2 fuzzy intermittent controller gain is given by Calculated.

[0094] The proof is as follows: For the non-increasing SPDLFs in formula (12), first, consider , that is, the controller is activated at time k.

[0095] Let i = c τ , ξ T (k) = [x T (k)ω T (k)], for any:

[0096] According to formulas (8) and (12), we can get

[0097]

[0098] Because φ in -σ in ψ in ≥0 and According to formula (17), we can get:

[0099]

[0100] in,

[0101] Will c,τ <0 combined with Schur's complement lemma, we can get:

[0102]

[0103] This means:

[0104] V c,τ+1 (k+1)-V c,τ (k)+y T (k)y(k)-γ 2 ω T (k)ω(k)<0. (22)

[0105] Then, consider Another case, that is, the controller is idle at time k, let i = c τ -N,

[0106] By formula (11) and given two matrices and matrix A T PB+B T PA≤A T PA+B T PB.

[0107] Then, for any It can be deduced that:

[0108]

[0109] From formula (19) and Schur's complement theorem, we can get:

[0110]

[0111] Combining equations (23) and (24), we can obtain:

[0112]

[0113] for Switching path c will switch to the next switching path c + , similarly, from equations (20) and (21) we can get:

[0114]

[0115] Combining equations (22), (25) and (26), we can conclude that equations (14) and (15) are valid.

[0116] In conclusion, from Inference 1, we can know that the switching system with 2N subsystems in formula (11) has the switching signal The following is a GUAS with l2-gain no greater than γ. In Inference 2, the stability of the switching system and H in the closed-loop 2N subsystem are derived. ∞ Performance, sufficient conditions for the existence of periodic intermittent controllers.

[0117] In one embodiment of the present invention, the dynamic equation of the single-link robotic arm model is expressed as follows:

[0118]

[0119] Among them, the discrete parameter q k =(m k ,J k ,D k ) changes with the change of the angle parameter η(t), the above system can be regarded as a switching system with three modes, because the discrete parameter q k The following three sets of values ​​are used: q 1 =(1,1,2),q 2 =(5,5,2),q 3 =(10,10,2).

[0120] Through linearization, the IT2 fuzzy model of the single-link robot arm system is as follows:

[0121] If x1(k) is 0 radians, then:

[0122]

[0123] If x1(k) is radians, then:

[0124]

[0125] in, and

[0126]

[0127] The LUMFs of the controlled quantity are as follows:

[0128]

[0129] The LUMFs of the selected controller are:

[0130]

[0131] The nonlinear weighting parameters are chosen as:

[0132] a 11 (x(k))= a 21 (x(k))= a 31 (x(k))= a 12 (x(k))= a 22 (x(k))= a 32 (x(k))

[0133] =1-0.6sin 2 (0.1x(k)),

[0134]

[0135]

[0136]

[0137] First, the three-mode switching system is considered as a switching system with six modes (10) (i.e. ),

[0138] Then, set T b =2, T f =1,σ 11 =σ 21 =σ 31=0.7,σ 12 =σ 22 =σ 32 =0.8.

[0139] By solving the optimization problem, we can get γ min =2.5552, the controller gain is as follows:

[0140] K 11,1 =[-13.1483-7.2013],K 11,2 =[-14.1553-8.4895],

[0141] K 12,1 =[-13.1536-7.2463],K 12,2 =[-14.1534-8.4664],

[0142] K 21,1 =[-0.5935-5.1953],K 21,2 =[-1.8128-5.4478],

[0143] K 22,1 =[-0.5954-5.1926],K 22,2 =[-1.8256-5.4345],

[0144] K 31,1 =[-7.0988-5.3565],K 31,2 =[-7.2966-5.5249],

[0145] K 32,1 =[-7.0945-5.3525],K 32,2 =[-7.2843-5.5126]. (34)

[0146] Let the initial state be x T (0) = [0.5-0.5], the disturbance input is ω(k) = 0.5sin(k)e -0.1k , the results are as follows Figure 2 and Figure 3 As shown, Figure 2 The state response of the closed-loop system of Formula 30 is recorded, Figure 2 The control inputs to the closed-loop system of Equation 30 are recorded.

[0147] It can be seen from the figure that although the controller only works for two-thirds of the time, the state response gradually converges to zero, which shows that the designed periodic intermittent controller can effectively ensure the stability of the system and H ∞ performance.

[0148] 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 regarded as the scope of protection of the present invention.

Claims

1. An intermittent control method for a type II switching fuzzy system, characterized in that: The steps include: S1. In the switched nonlinear system, based on the dynamic state interval type II fuzzy set, under the If-Then rule, the global fuzzy model of the discrete switched nonlinear system is established through the IT2 fuzzy set of the premise variables and the emission intensity of the fuzzy rules; S2. Based on the global fuzzy model, a timer-dependent IT2 fuzzy intermittent controller is proposed, and the structural framework of the type II fuzzy intermittent controller is constructed; S3, input the structure framework of the type II fuzzy intermittent controller into the global fuzzy model, define a new switching signal, and obtain a switching system with 2N subsystems; S4, control the new switching signal, calculate the IT2 fuzzy intermittent controller gain, make the IT2 fuzzy controller work intermittently, and ensure the stability of the switching system of the 2N subsystem and H ∞ performance.

2. The intermittent control method of type II switching fuzzy system according to claim 1, characterized in that: In step S1, the dynamic state interval type II fuzzy set is DSIT2FSs, and under the If-Then rule, the closed-loop DSIT2FSs is expressed as follows: Where x(k) is the state vector of the type-two switching fuzzy system, is a set of real numbers, n x is the vector dimension, k is used to represent time, k ≥ 0, is an integer set; y(k) is the control output of the type-two switching fuzzy system; u(k) and ω(k) are the control input and disturbance input respectively, σ(k) is the switching signal; A im , B im , C im , D im 、E im 、F im is a known constant matrix, is a fuzzy set, r represents the number of fuzzy rules, σ(k)=i, A collection of subsystems.

3. The intermittent control method of type II switching fuzzy system according to claim 2, characterized in that: In step S1, a global fuzzy model of a discrete switching nonlinear system is established as follows: pass represents the premise variable θ s (x(k)) fuzzy rule is m th IT2 fuzzy set, p represents the number of premise variables; m th The emission intensity interval set W of fuzzy rules im (x(k)), expressed as follows: in, and They are θ s The upper and lower membership functions LUMFs of (x(k)), are the lower and upper bounds of the membership function respectively; Based on the fuzzy reasoning method, the global fuzzy model of the discrete switching nonlinear system is obtained, which is expressed as follows: Among them, ψ im (x(k)) is used to represent the membership function, and and To satisfy The nonlinear weighting function of .

4. The intermittent control method of type II switching fuzzy system according to claim 3, characterized in that: In step S2, assuming that the switching signal σ(k) of the global fuzzy model system satisfies the minimum residence time constraint, a timer-dependent IT2 fuzzy intermittent controller is proposed, which is expressed as follows: Among them, l represents the cycle number, τ is the defined timer, K in,τ represents the controller gain; [T 2l ,T 2l+1 ) is the controller busy interval, during which the controller is activated, [T 2l+1 ,T 2l+2 ) is the controller idle time interval, during which the controller stops running.

5. The intermittent control method of type II switching fuzzy system according to claim 4, characterized in that: In step S2, the structure framework of the type II fuzzy intermittent controller is constructed as follows: use Represents premise variable n th Fuzzy sets, n th The emission intensity interval set of fuzzy rules is expressed as follows: in, and They are The upper and lower membership functions LUMFs, denote lower and upper membership, respectively; The structure framework of the type-II fuzzy intermittent controller is obtained, which is expressed as: Among them, φ in (x(k)) is used to represent the degree of membership, and Nonlinear weighting function and satisfy 6. The intermittent control method of type II switching fuzzy system according to claim 4, characterized in that: In step S3, use φ im and ψ im Represents φ im (x(k)) and ψ im (x(k)), so that: Input the IT2 fuzzy intermittent controller structure framework into the global fuzzy model and obtain:

7. The intermittent control method of type II switching fuzzy system according to claim 4, characterized in that: In step S3, a new switching signal is defined Where N represents the number of subsystems, s(k) represents the working state of the controller, New switching signal It also depends on the switching signal σ(k) and the controller working state: in, A collection of subsystems representing the controller's work, Represents the collection of subsystems for which the controller is not functioning.

8. The intermittent control method of type II switching fuzzy system according to claim 7, characterized in that: In step S3, a switching system with 2N subsystems is constructed, which is expressed as follows: in, In the collection Take the value from.

9. The intermittent control method of type II switching fuzzy system according to claim 8, characterized in that: In step S4, based on the switching system of the closed-loop 2N subsystem, a class of non-increasing switching path-dependent Lyapunov functions SPDLFs is constructed, which is expressed as follows: Among them, P c,τ >0, represents the timer, and c represents Switching paths within a controller operation cycle Given an integer T b ≥1 and T f ≥1, if there exists a constant γ>0, a positive definite symmetric matrix X c,τ , matrix Ω iτ , U im,τ and the symmetric matrix Λ c,τ,m , with the right size The switching system of the 2N subsystem makes the switching signal Under this condition, it has globally uniform asymptotic stability with l2-gain no greater than γ, for any and Both have φ in -σ in ψ in ≥0 and satisfies the given integer The constraints under the conditions, represents the number of cycles in the feasible switching path, and the gain of IT2 fuzzy intermittent controller is given by Calculated.

10. The intermittent control method of type II switching fuzzy system according to claim 9, characterized in that: In step S4, given an integer The constraints include: S4.

1. For any The following conditions hold: Where i = c τ ,Ψ c,τ,mn is a symmetric matrix, and in, Indicates -Ω i,τ Transpose, γ represents l2l2H ∞ Gain coefficient, I represents the unit matrix, X c,τ+1 represents a positive definite matrix; S4.

2. For any T represents the controller working period, and the following conditions are met: Where j = c τ -N, for The following conditions hold: S4.

3. For any The following conditions hold: in, When there is a feasible solution for the constraints described in S4.1 to S4.3, the IT2 fuzzy intermittent controller gain is given by Calculated.

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