Improved T-S fuzzy stabilization criterion design method of nonlinear time delay system based on memory sampling strategy

Abstract

The invention discloses an improved T-S fuzzy stabilization criterion design method of a nonlinear time delay system based on a memory sampling strategy. It is assumed that the sampling period changes within an interval. In order to minimize the conservative property in an integral term, a new integral inequality is provided. In addition, an integral inequality (FMBII) based on a free matrix and an integral inequality (WII) based on a widinger are also adopted. An improved stabilization condition is obtained in a linear matrix inequality (LMI) by using an FMBII based on a ring functional method (LFA). In order to ensure asymptotic stability of a closed-loop system, a fuzzy memory sampling data controller is designed. Numerical research is carried out by using different methods, and the result shows that compared with the existing system, the obtained result has relatively small conservative property and superiority.

Description

Technical Field

[0001] The invention relates to the field of controller technology, and in particular to an improved TS fuzzy stabilization criterion design method for a nonlinear time-delay system based on a memory sampling strategy. Background Art

[0002] Studying nonlinear dynamic systems (NDS) has been considered as one of the largest research fields and has attracted numerous researchers over the past few decades. Due to the dynamic design of NDS and their numerous applications in the industrial sector, experts have recently focused on the stability and stabilization criteria of NDS. In terms of stability assessment, many researchers are interested in the control of NDS, as designing an effective controller can minimize the effects of es-estimation errors and external disturbances to achieve the desired results of the system.

[0003] Since it is challenging to analyze NDS directly, fuzzy control techniques can approximate NDS and are effective tools to stabilize NDS. Takagi-Sugeno (TS) fuzzy model is one of the most powerful techniques that mainly deals with stability analysis and synthesis of nonlinear systems. A set of fuzzy IF-THEN rules are introduced in the TS fuzzy model to represent the local linear input-output relationship of NDS. Based on the TS fuzzy model, in current research, there are several control methods such as sampled data control (SDC), state feedback control, impulse control, and sliding mode control to study some dynamic characteristics of the considered system. Among all the above control techniques, the TS fuzzy SDC strategy contains one or more constant or analog signals for the input or output signals. In addition, the SDC strategy only needs to understand the continuous-time system state at the sampling number (discrete), thereby reducing the amount of data transmitted.

[0004] SDC systems can use both continuous and discrete time signals. The main benefit of sampled data monitoring is that it only involves information about the state of the system during the sampling interval. This greatly reduces the amount of data transmitted, which improves the efficiency of supervision in practice. Therefore, a lot of attention has been paid to SDC systems to ensure that the sampling interval has a large upper limit, which is a characteristic of its stability.

[0005] With the increasing requirements of control system performance in industrial applications, the stability analysis and control design of nonlinear time-delay systems face greater challenges. The existing TS fuzzy control model has certain limitations when dealing with time-delay systems, and the sampling data control strategy needs to be further optimized to adapt to the changes in the sampling period and reduce the amount of data transmission. In addition, the existing integral inequality is too conservative, which affects the optimization of control performance. Therefore, the present invention proposes an improved TS fuzzy stabilization criterion design method to improve the control performance and computational efficiency of nonlinear time-delay systems while ensuring the asymptotic stability of the system. Summary of the invention

[0006] To solve the above problems, the present invention discloses an improved TS fuzzy stabilization criterion method for nonlinear time-delay systems based on a memory sampling strategy, which introduces new control strategies and mathematical tools to improve the stability and control performance of the system and meet the needs of modern industrial applications. Compared with the existing analysis strategies, the numerical complexity of the analysis strategy is lower and the results are improved.

[0007] To achieve the above object, the technical solution specifically adopted by the present invention is: an improved TS fuzzy stabilization criterion design method for nonlinear time-delay systems based on a memory sampling strategy, which mainly includes the following steps:

[0008] Step 1: Use TS fuzzy membership rules to transform the nonlinear system into a linear system:

[0009] Consider the following Delayed Nonlinear System (DNS):

[0010]

[0011] x(t)=ψ(t),

[0012] in is the state vector, f(.) is a smooth nonlinear continuous function (f(0,0,0) = 0); ψ(t) is the vector-valued initial condition on the interval [-θ, 0]. The function θ(t) is a time-varying delay that satisfies the following conditions:

[0013]

[0014] where θ, μ are positive scalars, and is the control input. By using TS fuzzy modeling, DNS can be expressed as:

[0015] Fuzzy rule i: if m 1 (t) is i1 And m 2 (t) is i2 And... and m p (t) is ip , then the fuzzy rule DNS is

[0016]

[0017] x(t)=ψ(t),0<θ(t)≤θ,

[0018] in is the state vector of the system, m 1 (t),m 2 (t),…,m p (t) is the premise variable,ij , (i=1,2,…,r,j=1,2,…,p) is the fuzzy membership function; A i , B i is a fuzzy system. The fuzzy rule DNS is written as:

[0019]

[0020] in

[0021]

[0022] The following relations are satisfied:

[0023]

[0024] Step 2: Considering the lower and upper bounds of the sampling period within the interval, the designed storage SDC solution is as follows: Considering the lower and upper bounds of the sampling period within the interval, the measured value x(t k ) Assume that only at sampling time t k Therefore, the control signal can be generated by applying a zero-order hold (ZOH) function, whose sampling instant sequence is:

[0025]

[0026] Assume that the sampling time belongs to an interval, that is

[0027]

[0028] where δ m and δ M (δ M ≥δ m ≥0) are given scalars, representing the lower and upper limits of the sampling interval respectively. Therefore, the memory-based SDC (sampled data controller) scheme is described as follows:

[0029] Controller rule j: If m 1 (t) is j1 , and m 2 (t) is j 2. And... and m p (t) is j p, then

[0030] u(t)=K 1j x(t k )+K 2j x(t k -χ), t k ≤t <t k+1, for \(j = 1, 2, \ldots, r\), the fuzzy memory sampled-data controller can be written as

[0031]

[0032] where is the memory-type SDC gain matrix and \(\chi\) is the constant delay. Using the controller in formula (6 - 8) and denoting and we get

[0033]

[0034] Step 3: Use the new integral inequalities FMBII and WII to approximate the upper bound of the integral term:

[0035] First, there is the following lemma for the integral term in solving the derivative of the considered cyclic functional.

[0036] Lemma 1. (FMBII) Let be a differentiable signal. For the vector matrix and the following relation holds:

[0037]

[0038] Lemma 2. For any constant matrix \(M>0\), given scalar values \(\alpha\) and \(b\) (\(\alpha < b\)), and a continuously differentiable function the following relation holds:

[0039]

[0040] Consider the following formula:

[0041]

[0042] where is the time derivative of the membership function, which can be positive or negative over time. Next, the switching idea can be used to confirm the following inequality:

[0043]

[0044] where there are 2 r-1 possible cases. Let \(H\) l : \(l = 1, 2, \ldots, 2\) r-1 be the set containing the possible permutations of , and \(C\) l be the set containing the constraints of \(P\) l . Then, \(f(n)\) can be expressed as:

[0045] If H l If established, then C l Established.

[0046] Based on the above analysis, the following lemma holds:

[0047] Lemma 3. For the matrix P i >0, is the membership function, and the switching rule is for l = 1, 2, …, 2 r-1 Established, then:

[0048]

[0049] The following symbols are defined:

[0050]

[0051] Lemma 4. (New integral inequality) Let is a differentiable function. For any given symmetric matrix R>0, any vector ω(a,b), any relaxation matrix The following inequality holds:

[0052]

[0053] in

[0054]

[0055] That is, the upper bound of the approximate integral term is obtained.

[0056] Step 4: Analyze the sampled data based on TS fuzzy memory. Establish a stabilization criterion based on memory sampled data for the following TS fuzzy system with time-varying delay:

[0057]

[0058] For a given scalar δ m >0,δ M >0 and gain k 1j ,k 2j ,(j=1,2,…,r), when there exists a matrix

[0059]

[0060] For i,j=1,2,…,r,

[0061]

[0062] in

[0063]

[0064] For a given scalar δ m >0,δ M >0,λ 2 >0,λ 3 >0, the condition for the asymptotic stability of the fuzzy system is that when there exists a matrix (for i,j=1,2,…,r), it satisfies:

[0065]

[0066]

[0067] in

[0068]

[0069] ξ 6 =z 1 -z 2 ,ξ 7 =z 1 -z 3 ,ξ 8 =z 1 +z 2 ,ξ 9 =z 1 +z 3 ,

[0070] Therefore, the memory sampling data controller can be calculated as and Compared with the prior art, the present invention has the following advantages:

[0071] 1. The present invention uses LFA to establish an effective TS fuzzy memory SDC stabilization criterion for time-delay nonlinear systems. Compared with traditional SDC, fuzzy SDC with stable signal transmission delay is a more important control type.

[0072] 2. The present invention proposes a new integral inequality and uses FMBII to estimate the derivative of the ring functional, obtaining a tighter upper bound for the integral term in the derivative of the ring functional. The proposed integral inequality contains fewer slack variables and is relatively simpler to calculate.

[0073] 3. Compared with the existing analysis strategies, the analysis strategy proposed by the present invention has lower numerical complexity and the results are improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 is the maximum allowed sampling interval δ m =δ M Histogram;

[0075] Figure 2 Take the state response waveform of the Lorentz system in Example 1 as an example;

[0076] Figure 3 Take the control input waveform in Example 1 as an example;

[0077] Figure 4 is the maximum allowed sampling interval δ m =δ M Histogram;

[0078] Figure 5 This is the state response waveform of the Rossler system in Example 1;

[0079] Figure 6 Take the control input waveform in Example 2 as an example;

[0080] Figure 7 is a schematic diagram of an inverted pendulum system;

[0081] Figure 8 The state response x(t) waveform in Example 2 is shown below;

[0082] Fig. 9 The control input u(t k ) waveform graph. DETAILED DESCRIPTION

[0083] The present invention is further explained below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. It should be noted that the words "front", "rear", "left", "right", "upper" and "lower" used in the following description refer to directions in the accompanying drawings, and the words "inner" and "outer" refer to directions toward or away from the geometric center of a specific component, respectively.

[0084] The present invention is verified by Matlab simulation. Three examples are given to illustrate the effectiveness of the proposed method. Examples 1 and 2 illustrate the superiority of this work by using the Lorentz system and the Roessler system respectively. The results are shown in Figure 1 and Figure 4 As shown, Figure 1 and Figure 4 A comparative study with recent existing works is shown. An inverted pendulum system is considered in Example 3, which shows the practicality of the proposed work.

[0085] Example 1. Consider the following Lorentz system with input terms:

[0086]

[0087] where u(t) is the control input and

[0088]

[0089]

[0090] For x 1 (t)∈[-d,d], the system can be described by the TS fuzzy model:

[0091]

[0092] in

[0093]

[0094] Where a = 10, b = 8 / 3, c = 28, d = 25. The membership function is expressed as In this example, in order to demonstrate the advantages of the method of the present invention over the existing methods, h k The boundary is set to (δ m =δ M ). Let λ 2 =0,λ 3 = 0.2, and the transmission delay is χ = 0.01. Then, by applying the LMI-based criterion through Theorem 2, δ M Calculated as δ M =0.0484. Detailed description as follows Figure 1 As shown. Figure 1 It can be seen that the results obtained by Theorem 1 are less conservative than the methods of similar studies.

[0095] When δ M =0.04, the controller gain matrix can be obtained as follows:

[0096] K 11 =[-31.1363-13.4209-5.5862],

[0097] K 12 =[-31.1363-13.4209-5.5862],

[0098] K 21 =[-0.0015-0.0007-0.0015],

[0099] K 22 =[-0.0015-0.0007-0.0015],

[0100] Figure 2 Indicates the system status response, Figure 3 Indicates control input.

[0101] Example 2. Consider the TS fuzzy model of the Rossler system [1] as follows:

[0102]

[0103] in

[0104]

[0105] Where a = 0.3, b = 0.5, c = 5, and d = 10. The membership function is expressed as Similar to Example 1, h k The boundary is set to (δ m =δ M ). Let λ 2 =0.52,λ 3 =3, and the transmission delay is χ = 0.01. Then, by applying the LMI-based criterion through Theorem 2, δ M Calculated as δ M =0.1723. Therefore, the MPSIδ can be found by using the method proposed in this chapter. M =0.1723. Detailed description as follows Figure 6 and Figure 5 As shown. Figure 4 It can be seen that the results obtained by Theorem 1 are less conservative than the methods in similar studies.

[0106] When δ M =0.1, the controller gain matrix can be obtained as follows:

[0107] K 11 =[6.57117.1441-12.0145],

[0108] K 12 =[6.57117.1441-12.0145],

[0109] K 21 =[0.04020.0470-0.0072],

[0110] K 22 =[0.04020.0470-0.0072].

[0111] Figure 5 Indicates the system status response, Figure 6 Indicates control input.

[0112] Example 3. Consider Figure 7The inverted pendulum system shown in Figure 1 is represented as a TS fuzzy model by applying the Eular first-order approximation technique. Then the fuzzy controller method proposed in this chapter is used to stabilize the inverted pendulum system.

[0113] The state space equation of the inverted pendulum is:

[0114]

[0115] Applying TS fuzzy modeling, the system can be written as follows:

[0116] Fuzzy rule 1: If x 2 (t), is approximately 0 radians,

[0117] Then x(t)=A 1 x(t)+A 1 x(t-θ)+B 1 u(t).

[0118] Fuzzy rule 2: If x 2 (t) is close to ρ radians (0 < ρ) < 1.57),

[0119] Then X(t)=A 2 x(t)+A 2 x(t-θ)+B 2 u(t).

[0120] in

[0121]

[0122] in β 2 = cosρ, |ρ| = 0.55, and is the membership function.

[0123] Let λ 2 =0.5,λ 3 = 0.25, μ = 0.2, transmission delay χ = 0.001, and delay upper bound θ = 2. Then, applying Theorem 1, the system under consideration can be stabilized as MPSIδ M =0.3892. M =0.1, the memory SDC fuzzy control gain matrix can be calculated as follows:

[0124] K 11 =[0.1400 56.9980 3.1689 32.3699],

[0125] K 12 =[0.1400 56.9980 3.1689 32.3699],

[0126] K 21 =[0.0028 0.1530 0.0112 0.0945],

[0127] K 22 =[0.0028 0.1530 0.0112 0.0945].

[0128] Status response such as Figure 8 As shown, the control input is Fig. 9 shown.

[0129] The technical means disclosed in the scheme of the present invention are not limited to the technical means disclosed in the above-mentioned implementation mode, but also include technical schemes composed of any combination of the above technical features.

Claims

1. An improved TS fuzzy stabilization criterion design method for nonlinear time-delay systems based on memory sampling strategy, characterized by: The following steps are involved: S1: Transform the nonlinear system into a linear system using TS fuzzy membership rules; S2: Considering the lower and upper bounds of the sampling period in the interval, a storage SDC scheme is designed; S3: Use the new integral inequalities FMBII and WII to approximate the upper bound of the integral term; S4: For nonlinear systems without time-varying delays, a stabilization criterion based on TS fuzzy memory sampling data is established, and the stability criterion in the context of LMI is improved and the proposed method is verified.

2. The improved TS fuzzy stabilization criterion design method for nonlinear time-delay systems based on memory sampling strategy according to claim 1 is characterized in that: In step S1, the following nonlinear system (DNS) is considered: x(t)=ψ(t), in is the state vector, f(.) is a smooth nonlinear continuous function (f(0,0,0) = 0); ψ(t) is the vector-valued initial condition on the interval [-θ,0]; the function θ(t) is a time-varying delay that satisfies the following conditions: where θ, μ are positive scalars, and is the control input; by using TS fuzzy modeling, DNS is expressed as: Fuzzy rule i: If m1(t) is Ξ i1 and m2(t) is i2 And... and m p (t) is ip , then the fuzzy rule DNS is x(t)=ψ(t),0<θ(t)≤θ, in is the state vector of the system, m1(t),m2(t),…,m p (t) is the premise variable, ij , (i=1,2,…,r,j=1,2,…,p) is the fuzzy membership function; A i , B i It is a fuzzy system; the fuzzy rule DNS can be written as; in The following relations are satisfied:

3. The improved TS fuzzy stabilization criterion design method for nonlinear time-delay systems based on memory sampling strategy according to claim 1 is characterized in that: The storage SDC scheme designed in step S2 is as follows: considering the lower and upper bounds of the sampling period within the interval, the measured value x(t k ) Assume that only at sampling time t k is available; therefore, the control signal can be generated by applying a zero-order hold (ZOH) function, whose sampling instant sequence is: Assume that the sampling time belongs to an interval, that is where δ m and δ M (δ M ≥δ m ≥0) are given scalars, representing the lower and upper limits of the sampling interval respectively; therefore, the memory-based SDC sampling data controller solution is described as follows: Controller rule j: If m1(t) is Ξ j1 , and m2(t) is j 2. And... and m p (t) is j p, then u(t)=K 1j x(t k )+K 2j x(t k -χ),t k ≤t<t k+1 ,j=1,2,…,r, The fuzzy memory sampling data controller is written Where K 1j , is the memory-type SDC gain matrix, and χ is the constant delay. Use the above fuzzy memory sampling data controller and remember and get 4. The improved TS fuzzy stabilization criterion design method for nonlinear time-delay systems based on memory sampling strategy according to claim 1 is characterized by: The step S3 uses the new integral inequalities FMBII and WII to approximate the upper bound of the integral term: first, there is the following lemma for solving the integral term in the derivative of the considered cyclic functional; Lemma 1. (FMBI I) Let is a differentiable signal. For the vector matrix and The following relationship holds: Lemma 2. For any constant matrix \(M>0\), given scalar values \(\alpha\) and \(b\) (\(\alpha < b\)), and a continuously differentiable function the following relation holds: Consider the following formula: in, is the time derivative of the membership function, which can be positive or negative over time. Next, we can use the switching idea to confirm the following inequality: There are 2 r-1 possible situations. Let H l :l=1,2,…,2 r-1 To include The set of possible permutations of l To include P l Then, f(n) can be expressed as: If H l If established, then C l Established. Based on the above analysis, the following lemma holds: Lemma 3. For the matrix P i >0, is the membership function, and the switching rule is for l = 1, 2, …, 2 r-1 Established, then: The following symbols are defined: Lemma 4. (New integral inequality) Let is a differentiable function. For any given symmetric matrix R>0, any vector ω(a,b), any relaxation matrix (i=0,1,2), the following inequality holds: in That is, the upper bound of the approximate integral term is obtained.

5. The improved TS fuzzy stabilization criterion design method for nonlinear time-delay systems based on memory sampling strategy according to claim 1 is characterized by: The step S4 analyzes the sampled data based on TS fuzzy memory; The stabilization criterion based on memory sampling data is established for the following TS fuzzy system with time-varying delay: x(t)=ψ(t),i=1,2,..,r. For a given scalar δ m >0,δ M >0 and gain K 1j ,K 2j ,(j=1,2,…,r), when there exists a matrix For i,j=1,2,…,r, it satisfies in For a given scalar δ m >0,δ M >0,λ2>0,λ3>0, the condition for the asymptotic stability of the fuzzy system is that when there exists a matrix (for i,j=1,2,…,r), it satisfies: in ξ6=z1-z2, ξ7=z1-z3, ξ8=z1+z2, ξ9=z1+z3, The memory sample data controller is calculated as and Therefore, a stabilization criterion based on TS fuzzy memory sampling data is established, and the stability criterion in the LMI context is improved. The proposed method will be verified later.

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