A Dissipative Sliding Mode Control Method under the Delta Operator Framework

Abstract

The present invention discloses a dissipative sliding mode control method under the Delta operator framework. The method is as follows: establish a dynamic model of a T-S fuzzy Delta operator system with time delay and external disturbance; design a fuzzy state observer for the dynamic model; obtain a corresponding observation error system according to the dynamic model and the fuzzy state observer; construct a fuzzy sliding mode surface and a fuzzy sliding mode controller; obtain the discriminant conditions for ensuring the asymptotic stability, strict dissipativity and reachability of the sliding mode surface of the fuzzy state observer system and the observation error system through the Lyapunov stability method, the Delta operator method and the convex optimization method; according to the discriminant conditions, obtain the fuzzy state observer gain matrix and the fuzzy sliding mode controller gain matrix and substitute them back into the fuzzy state observer and the fuzzy sliding mode controller. This method can achieve dissipative sliding mode control of a T-S fuzzy Delta operator system with time delay and external disturbance.

Description

Technical Field

[0001] The present invention relates to a sliding mode control method, and more particularly to a dissipative sliding mode control method for a T-S fuzzy Delta operator system with time delay and external disturbance in the case where system state information is not available. Background Art

[0002] In existing control strategies, sliding mode control is an effective method for dealing with model uncertainties, parameter variations, and external disturbances. It has the advantages of strong robustness, fast response speed, and easy implementation. Therefore, sliding mode control has been widely applied in fields such as vehicle control and robot control. With the rapid development of science and technology, actual control systems are becoming increasingly complex, and people's demand for high-speed sampling is increasing. If the mathematical model of a complex control system cannot be accurately given under high-speed sampling, the control effect will be unsatisfactory. In order to make the control system stable and achieve an ideal control effect, a sliding mode control method combining the T-S fuzzy model and the Delta operator framework has emerged. In addition, when a control system suffers from time delay and external disturbances, there will be a phenomenon that system state information cannot be fully obtained. Traditional sliding mode control methods are difficult to meet the performance requirements of the system in this case, while the dissipative sliding mode control method based on a state observer estimates the unknown state of the system by designing a state observer and ensures the good operation of the system's dissipative performance. Therefore, it has received extensive attention from scholars. Summary of the Invention

[0003] In order to solve the problem that the existing sliding mode control method cannot handle the influence of time delay and external disturbance on the system performance when the system state is not available, the present invention provides a dissipative sliding mode control method under the Delta operator framework. This method can achieve dissipative sliding mode control of a T-S fuzzy Delta operator system with time delay and external disturbance when the system state is not available.

[0004] The object of the present invention is achieved by the following technical solutions:

[0005] A dissipative sliding mode control method under the Delta operator framework includes the following steps:

[0006] Step 1: Establish a dynamic model of a T-S fuzzy Delta operator system with time delay and external disturbance, where:

[0007] The state space form of the dynamic model is:

[0008] System fuzzy rule i: If μ 1 (t) is μ 2 (t) is , …, μ B (t) is Then

[0009]

[0010] where \(i\) represents the fuzzy rule index; \(t\) represents the sampling time; \(\mu\) 1 (t) is the first antecedent variable at time \(t\), \(\mu\) 2 (t) is the second antecedent variable at time \(t\), \(\mu\) B (t) is the \(B\)th antecedent variable at time \(t\); is the first fuzzy set under the \(i\)th fuzzy rule, is the second fuzzy set under the \(i\)th fuzzy rule, is the \(B\)th fuzzy set under the \(i\)th fuzzy rule; \(x(t)\) is the state of the system at time \(t\), \(x(t - d)\) is the state of the system with time delay at time \(t\), \(d\) is the constant time delay; \(u(t)\) is the control input of the system at time \(t\); \(y(t)\) is the measured output of the system at time \(t\); \(\varphi(t)\) is the initial value of the system state; \(\nu(t)\) is the external disturbance; \(A\) i 、\(A\) id 、\(B\) i 、\(B\) iν 、\(C\) and \(D\) are all known matrices of appropriate dimensions; \(\delta(x(t))\) is the Delta operator of the system state \(x(t)\) at time \(t\);

[0011] Defuzzify the state - space form of the dynamic model of the T - S fuzzy Delta - operator system to obtain:

[0012]

[0013] where \(\tau\) i (\(\mu(t)\)) is the membership function under the \(i\)th fuzzy rule; represents the sum of the membership functions of \(F\) fuzzy rules; \(F\) is the total number of fuzzy rules;

[0014] Step 2: Design a fuzzy state observer for the dynamic model of the T - S fuzzy Delta - operator system with time delay and external disturbance established in Step 1, where:

[0015] The formula of the fuzzy state observer is:

[0016]

[0017] where is the estimated value of the system state \(x(t)\) at time \(t\), representing the state of the fuzzy state observer; is the estimated value of the system state \(x(t - d)\) with time delay at time \(t\); is the estimated value of the system output \(y(t)\) at time \(t\), representing the output of the fuzzy state observer; \(L\)i is the unknown gain matrix of the fuzzy state observer under the i-th observer fuzzy rule; ψ(t) is the initial value of

[0018] Defuzzify the fuzzy state observer to obtain:

[0019]

[0020] Step 3: According to the dynamic model established in Step 1 and the fuzzy state observer designed in Step 2, obtain the corresponding observation error system:

[0021]

[0022] where δ(e(t)) is the Delta operator of the state error of the fuzzy state observer; is the state error of the fuzzy state observer; e(t - d) is the state error of the fuzzy state observer with time delay; is the output error of the fuzzy state observer;

[0023] Step 4: Based on the fuzzy state observer designed in Step 2, construct a fuzzy sliding mode surface and a fuzzy sliding mode controller, where:

[0024] The specific form of the fuzzy sliding mode surface is:

[0025]

[0026] where s(t) is the sliding mode function at time t; is the transpose of matrix B i ; Q is the unknown matrix to be solved;

[0027] The specific form of the fuzzy sliding mode controller is:

[0028]

[0029] where K i is the unknown gain matrix of the fuzzy sliding mode controller; ε is a positive constant; is the norm of; sign(s(t)) is the sign function of the sliding mode function s(t);

[0030] Step 5: Using the fuzzy sliding mode surface and the fuzzy sliding mode controller constructed in Step 4, through the Lyapunov stability method, the Delta operator method, and the convex optimization method, obtain the discriminant conditions to ensure the asymptotic stability, strict dissipation, and reachability of the sliding mode surface of the fuzzy state observer system and the observation error system. The specific steps are as follows:

[0031] (1) If the external disturbance ν(t) and the output error z(t) of the fuzzy state observer satisfy:

[0032]

[0033] Then the fuzzy state observer system and the observation error system satisfy strict dissipativity; where ν T (t) is the transpose of the external disturbance ν(t); λ is the bound of the dissipative performance; h t is a number greater than 0; J(t) is the energy supply function;

[0034] (2) To ensure the establishment of strict dissipativity, the Lyapunov stability method is:

[0035]

[0036] Among them:

[0037] V(t) = V 1 (t) + V 2 (t) + V 3 (t) + V 4 (t) + V 5 (t)

[0038]

[0039]

[0040] In the formula, V(t) is the Lyapunov function at time t; δ(V(t)) is the Delta operator of V(t); is 's transpose, e T (t) is the transpose of e(t), is 's transpose, e T (t - ih) is the transpose of e(t - ih), w(t - jh) = e(t - jh) - e(t - (j - 1)h), w T (t - jh) is the transpose of w(t - jh); Q, R, S, and P are positive definite matrices to be solved;

[0041] From the above Lyapunov stability method, the discriminant conditions for the fuzzy state observer system and the observation error system to satisfy asymptotic stability and strict dissipativity are:

[0042] Λ ii <0, i = j

[0043] Λ ij + Λ ji <0, 1 ≤ i < j ≤ F

[0044] Wherein:

[0045]

[0046] In the formula, * represents the symmetric part of the block matrix; is the block matrix of the first row and first column of Λ ij ; is the block matrix of the first row and second column of Λ ij ; is the block matrix of the second row and second column of Λ ij ; is the block matrix of the first row and first column of; is the block matrix of the fourth row and fourth column of; is the block matrix of the first row and fourth column of; (A i - B i K j ) T is the transpose of the matrix (A i - B i K j ); (A i - L i C) T is the transpose of the matrix (A i - L i C); (B iν - L i D) T is the transpose of the matrix (B iν - L i D); C T is the transpose of the matrix C; D T is the transpose of the matrix D; is the transpose of the matrix L i ; is the matrix of the matrix M 2 ; α and β are known constants greater than ;

[0047] (3) To ensure that the sliding surface satisfies reachability, the Lyapunov stability method is:

[0048]

[0049] In the formula, W is the positive definite matrix to be solved;

[0050] From the above Lyapunov stability method, the discriminant condition for the sliding surface to satisfy reachability is:

[0051]

[0052] Ξ ii <0, i = j

[0053]

[0054] In the formula:

[0055]

[0056] where, γ ij is the scalar to be solved; is the block matrix of the first row and first column of Ξ ij ; is the block matrix of the second row and second column of Ξ ij ; is the block matrix of the third row and third column of Ξ ij ; is the block matrix of the fourth row and fourth column of Ξ ij ; is the block matrix of the fifth row and fifth column of Ξ ij ; is the transpose of matrix B i ; is the transpose of matrix B j ; is the transpose of matrix K j ; is the transpose of matrix A id ; is the transpose of matrix B iν ; is the transpose of matrix ;

[0057] Step Six: According to the discrimination condition in Step Five, obtain the fuzzy state observer gain matrix and the fuzzy sliding mode controller gain matrix. The specific steps are as follows:

[0058] Step Six - One: Initialize the parameters α > 0, β > 0, ε > 0, λ > 0, and the maximum number of iterations N, and let t = 0;

[0059] Step Six - Two: By solving the following convex optimization problem, obtain

[0060] min κ

[0061]

[0062] where:

[0063]

[0064]

[0065] Among them, is the block matrix at the first row and first column of is the block matrix at the fourth row and fourth column of is the block matrix at the fourth row and seventh column of is the block matrix at the seventh row and seventh column of is the block matrix at the first row and first column of is the block matrix at the first row and second column of is the block matrix at the second row and second column of is the block matrix at the first row and first column of is the block matrix at the fourth row and fourth column of, Γ 11 is the block matrix at the first row and first column of, Γ 12 is the block matrix at the first row and second column of, Γ 13 is the block matrix at the first row and third column of, Σ 11 is the block matrix at the first row and first column of, Σ 22 is the block matrix at the second row and second column of, Σ 33 is the block matrix at the third row and third column of is the transpose of matrix Y j is the transpose of matrix X i ;

[0066] Step Six Three: If t < N, then let t = t + 0.15 and execute Step Six Two; otherwise, execute the next step;

[0067] Step Six Four: If t ≥ N, then exit the loop, obtain the feasible solution of the convex optimization problem, and through L i = Q -1 X i and K j = Δ -1 Y j obtain the fuzzy state observer gain matrix and the fuzzy sliding mode controller gain matrix;

[0068] Step 7: Substitute the obtained fuzzy state observer gain matrix and fuzzy sliding mode controller gain matrix in Step 6 back into the fuzzy state observer and fuzzy sliding mode controller constructed in Step 4 to achieve the dissipative sliding mode control of the T-S fuzzy Delta operator system.

[0069] Compared with the prior art, the present invention has the following advantages:

[0070] 1. The present invention solves the dissipative sliding mode control problem that the system state cannot be obtained under the influence of time delay and external disturbance. Compared with the traditional sliding mode control problem, the present invention establishes a unified model for T-S fuzzy continuous systems and T-S fuzzy discrete systems, and the obtained results are applicable to both the dissipative sliding mode control problem of T-S fuzzy continuous systems and the dissipative sliding mode control problem of T-S fuzzy discrete systems; a fuzzy state observer is designed to estimate the unknown state of the controlled system, solving the problem that the system state cannot be obtained in practical problems; the present invention simultaneously considers the influence of time delay and external disturbance on the dissipative performance of the controlled system. The present invention is applicable to the dissipative sliding mode control of T-S fuzzy Detla operator systems.

[0071] 2. The present invention establishes a unified model for T-S fuzzy continuous systems and T-S fuzzy discrete systems under the Delta operator framework; solves the phenomenon of unknown system state by constructing a fuzzy state observer; designs an observer-dependent fuzzy sliding mode surface and a fuzzy sliding mode controller; uses the Lyapunov method, the Delta operator method, and the convex optimization method to obtain the discriminant conditions for ensuring the asymptotic stability and strict dissipation of the fuzzy state observer system and the observation error system and that the observer state can still be driven to the sliding mode surface under the influence of time delay and external disturbance. In the experiment of the present invention, when the sampling period h = 0, the obtained results are consistent with those of the T-S fuzzy continuous system; when the sampling period h = 1, the obtained results are consistent with those of the T-S fuzzy discrete system.

[0072] 3. The present invention can solve the dissipative sliding mode control problem of the T-S fuzzy Delta operator system with time delay and external disturbance in the case where the system state is not available or unknown. Description of the Drawings

[0073] Figure 1 is a flowchart of the dissipative sliding mode control method under the Delta operator framework of the present invention;

[0074] Figure 2 is a graph of the true state trajectory of the T-S fuzzy Delta operator system; where, "- -" represents the trajectory of the first component of the system state, "---" represents the trajectory of the second component of the system state, and "-·-·-" represents the trajectory of the third component of the system state;

[0075] Figure 3It is the state trajectory diagram of the fuzzy state observer (i.e., the state estimation trajectory diagram of the T-S fuzzy Delta operator system); where, "——" represents the trajectory of the first component of the fuzzy state observer state, "---" represents the trajectory of the second component of the fuzzy state observer state, and "-·-·-" represents the trajectory of the third component of the fuzzy state observer state;

[0076] Figure 4 It is the trajectory diagram of the observer error; where, "——" represents the trajectory of the first component of the observer error, "---" represents the trajectory of the second component of the observer error, and "-·-·-" represents the trajectory of the third component of the observer error;

[0077] Figure 5 It is the trajectory diagram of the sliding mode function; where, "-" represents the trajectory of the first component of the sliding mode function, and "---" represents the trajectory of the second component of the sliding mode function;

[0078] Figure 6 It is the trajectory diagram of the sliding mode controller; where, "-" represents the trajectory of the first component of the sliding mode controller, and "---" represents the trajectory of the second component of the sliding mode controller. Detailed implementation manner

[0079] The technical solution of the present invention will be further described below in conjunction with the accompanying drawings, but it is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.

[0080] The present invention provides a dissipative sliding mode control method under the Delta operator framework, as Figure 1 shown, the method includes the following steps:

[0081] Step 1: Establish a dynamic model of a T-S fuzzy Delta operator system with time delay and external disturbance.

[0082] In this step, the state space form of the dynamic model is:

[0083] System fuzzy rule i: If μ 1 (t) is μ 2 (t) is …, μ B (t) is Then

[0084]

[0085] In the formula, i represents the fuzzy rule index; t represents the sampling time, satisfying t = ch, where c is a positive integer and h represents the sampling period; μ1 (t) is the first antecedent variable at time t, μ 2 (t) is the second antecedent variable at time t, μ B (t) is the Bth antecedent variable at time t; is the first fuzzy set under the ith fuzzy rule, is the second fuzzy set under the ith fuzzy rule, is the Bth fuzzy set under the ith fuzzy rule; is the state of the system at time t, x(t - d) is the state of the system with time delay at time t, d is a constant time delay satisfying d = nh, and n is a positive integer; is the control input of the system at time t; is the measured output of the system at time t; φ(t) is the initial value of the system state; The external disturbance belongs to l 2 [0, +∞), and its upper bound is is a constant greater than 0; In addition, A i 、A id 、B i 、B iν 、C and D are all known matrices of appropriate dimensions; is n x - dimensional Euclidean space, is n u - dimensional Euclidean space, is n y - dimensional Euclidean space, is n ν - dimensional Euclidean space; δ(x(t)) is the Delta operator of the system state x(t) at time t, and its definition is:

[0086]

[0087] where, is the derivative of the system state x(t) at time t, x(t + h) is the state of the system at time t + h; When the sampling period h = 0, the system (1) represents a T - S fuzzy continuous system; When the sampling period h ≠ 0, the system (1) represents a T - S fuzzy discrete system; Thus, the system (1) is a unified framework for T - S fuzzy continuous systems and T - S fuzzy discrete systems.

[0088] Defuzzifying the state - space form of the dynamic model of the T - S fuzzy Delta - operator system, we can obtain:

[0089]

[0090] where, τ iμ(t) is the membership function under the i-th fuzzy rule, and it satisfies and 0 ≤ τ i μ(t) ≤ 1; denotes the sum of the membership functions of F fuzzy rules; F is the total number of fuzzy rules; the membership function τ i μ(t) is defined as follows:

[0091]

[0092] where, θ ig μ g (t) represents the membership of the g-th antecedent variable μ g (t) under the i-th fuzzy rule; is the product of b memberships under the i-th fuzzy rule.

[0093] Step 2: Design a fuzzy state observer for the dynamic model of the T-S fuzzy Delta operator system with time delay and external disturbance established in Step 1.

[0094] In this step, the formula of the fuzzy state observer is:

[0095] Observer fuzzy rule i: If μ 1 (t) is μ 2 (t) is …, μ B (t) is Then

[0096]

[0097] In the formula, is the estimated value of the system state x(t) at time t, representing the state of the fuzzy state observer; is the estimated value of the system state x(t - d) with time delay at time t; is the estimated value of the system output y(t) at time t, representing the output of the fuzzy state observer; L i is the unknown gain matrix of the fuzzy state observer under the i-th observer fuzzy rule; ψ(t) is the initial value of;

[0098] After defuzzifying the fuzzy state observer, we can obtain:

[0099]

[0100] Step 3: According to the dynamic model established in Step 1 and the fuzzy state observer designed in Step 2, obtain the corresponding observation error system.

[0101] In this step, the state error of the fuzzy state observer is defined as The output error of the fuzzy state observer is defined as According to the defuzzified T-S fuzzy Delta operator system (2) and the defuzzified fuzzy state observer (4), the corresponding observation error system can be obtained:

[0102]

[0103] where δ(e(t)) is the Delta operator of the state error of the fuzzy state observer; e(t - d) is the state error of the fuzzy state observer with time delay.

[0104] Step Four: Based on the fuzzy state observer designed in Step Two, construct a fuzzy sliding mode surface and a fuzzy sliding mode controller.

[0105] In this step, the specific form of the fuzzy sliding mode surface is:

[0106]

[0107] where s(t) is the sliding mode function at time t; is the transpose of matrix B i ; Q is an unknown matrix to be solved.

[0108] In this step, the specific form of the fuzzy sliding mode controller is:

[0109]

[0110] where K i is the unknown gain matrix of the fuzzy sliding mode controller; ε is a positive constant; is the norm of; sign(s(t)) is the sign function of the sliding mode function s(t).

[0111] Step Five: Using the fuzzy sliding mode surface and the fuzzy sliding mode controller constructed in Step Four, through the Lyapunov stability method, the Delta operator method, and the convex optimization method, obtain the discriminant conditions to ensure the asymptotic stability, strict dissipativity, and reachability of the sliding mode surface of the fuzzy state observer system and the observation error system. The specific steps are as follows:

[0112] (1) If the external disturbance ν(t) and the output error z(t) of the fuzzy state observer satisfy:

[0113]

[0114] Then the fuzzy state observer system and the observation error system satisfy strict dissipativity; where, ν T(t) is the transpose of the external disturbance ν(t); λ is the bound of the dissipation performance, and λ > 0; h t is a number greater than 0; J(t) is the energy supply function, and its definition is as follows:

[0115] J(t) = z T (t)M 1 z(t) + 2z T (t)M 2 ν(t) + ν T (t)M 3 ν(t)

[0116] wherein, M 1 is a semi - negative definite real matrix; M 2 is a real matrix; M 3 is a symmetric real matrix; z T (t) is the transpose of z(t).

[0117] (2) To ensure the strict dissipativity holds, the Lyapunov stability method is as follows:

[0118]

[0119] Among them:

[0120] V(t) = V 1 (t) + V 2 (t) + V 3 (t) + V 4 (t) + V 5 (t)

[0121]

[0122] wherein, V(t) is the Lyapunov function at time t; δ(V(t)) is the Delta operator of V(t); is the transpose of, e T (t) is the transpose of e(t), is the transpose of, e T (t - ih) is the transpose of e(t - ih), w(t - jh) = e(t - jh) - e(t - (j - 1)h), w T (t - jh) is the transpose of w(t - jh); Q, R, S, and P are positive definite matrices to be determined;

[0123] From the above Lyapunov stability method, the discriminant conditions for the fuzzy state observer system and the observation error system to satisfy asymptotic stability and strict dissipativity are:

[0124] Λ ii<0, i = j

[0125] Λ ij +Λ ji <0, 1 ≤ i < j ≤ F

[0126] where:

[0127]

[0128] In the formula, * represents the symmetric part of the block matrix; is the block matrix of the first row and first column of Λ ij ; is the block matrix of the first row and second column of Λ ij ; is the block matrix of the second row and second column of Λ ij ; is the block matrix of the first row and first column of is the block matrix of the fourth row and fourth column of is the block matrix of the first row and fourth column of; (A i -B i K j ) T is the transpose of the matrix (A i -B i K j ), (A i -L i ) T is the transpose of the matrix (A i -L i C), (B iν -L i ) T is the transpose of the matrix (B iν -L i D), C T is the transpose of the matrix C, D T is the transpose of the matrix D, is the transpose of the matrix L i ; is the matrix of the matrix M 2 ; Q, R, S, and P are positive definite matrices to be solved; α and β are known constants greater than .

[0129] (3) To ensure that the sliding surface satisfies reachability, the Lyapunov stability method is:

[0130]

[0131] In the formula, W is a positive definite matrix to be solved;

[0132] By the above Lyapunov stability method, the discriminant condition for the reachability of the sliding mode surface is obtained as follows:

[0133]

[0134] where:

[0135]

[0136] where γ ij is the scalar to be solved; is the block matrix of the first row and first column of Ξ ij , is the block matrix of the second row and second column of Ξ ij , is the block matrix of the third row and third column of Ξ ij , is the block matrix of the fourth row and fourth column of Ξ ij , is the block matrix of the fifth row and fifth column of Ξ ij ; is the transpose of matrix B i , is the transpose of matrix B j , is the transpose of matrix K j , is the transpose of matrix A id , is the transpose of matrix B iν , is the transpose of matrix .

[0137] Step 6: According to the discriminant condition in Step 5, the fuzzy state observer gain matrix and the fuzzy sliding mode controller gain matrix are obtained. The specific steps are as follows:

[0138] Step 6-1: Initialize the parameters α > 0, β > 0, ε > 0, λ > 0, and the maximum number of iterations N, and let t = 0;

[0139] Step 6-2: By solving the following convex optimization problem, obtain

[0140] min κ

[0141]

[0142] where:

[0143]

[0144]

[0145] Among them, is the block matrix of the first row and the first column of is the block matrix of the fourth row and the fourth column of is the block matrix of the fourth row and the seventh column of is the block matrix of the seventh row and the seventh column of is the block matrix of the first row and the first column of is the block matrix of the first row and the second column of is the block matrix of the second row and the second column of is the block matrix of the first row and the first column of is the block matrix of the fourth row and the fourth column of Γ 11 is the block matrix of the first row and the first column of 12 is the block matrix of the first row and the second column of 13 is the block matrix of the first row and the third column of Σ 11 is the block matrix of the first row and the first column of 22 is the block matrix of the second row and the second column of 33 is the block matrix of the third row and the third column of is the matrix Y j is the transpose of is the matrix X i is the transpose of;

[0146] Step Six Three: If t < N, then let t = t + 0.15 and execute Step Six Two; otherwise, execute the next step;

[0147] Step Six Four: If t ≥ N, then exit the loop, obtain the feasible solution of the convex optimization problem, and through L i = Q -1 X i and K j = Δ -1 Y j obtain the fuzzy state observer gain matrix and the fuzzy sliding mode controller gain matrix.

[0148] Step 7: Substitute the fuzzy state observer gain matrix and the fuzzy sliding mode controller gain matrix obtained in Step 6 back into the fuzzy state observer and the fuzzy sliding mode controller constructed in Step 4 to achieve the dissipative sliding mode control of the T-S fuzzy Delta operator system.

[0149] The following embodiments are used to verify the beneficial effects of the present invention:

[0150] Consider a T-S fuzzy Delta operator system with two fuzzy rules. The specific selection of all parameters involved in the system is as follows:

[0151] System parameters:

[0152]

[0153] d = 1, α = 0.2, β = 0.3, ε = 0.4

[0154] The external disturbance is expressed as ν(t) = 2sin(0.6t)exp(-0.8t); the strict dissipative performance bound is λ = 0.2, and the other dissipative performance parameters are:

[0155] M 3 = 0.5I

[0156] In addition, the sampling period is h = 0.15, the initial value of the system state is x(0) = [0 0 0] T (t ≤ 0), and the initial value of the fuzzy state observer is ; the membership function is selected as:

[0157]

[0158] τ 2 (μ 1 (t)) = 1 - τ 1 (μ 1 (t))

[0159] By solving the convex optimization problem, we can obtain:

[0160]

[0161] Using we can calculate the fuzzy state observer gain matrix as:

[0162]

[0163] Using we can calculate the fuzzy sliding mode controller gain matrix as:

[0164]

[0165] The effect of the constructed sliding mode controller is shown in Figures 2 - 6 . As can be seen from Figures 2 - 3 , the fuzzy state observer designed in the present invention can effectively estimate the true state of the T-S fuzzy Delta operator system. As can be seen from Figure 4 , the difference between the state estimated by the fuzzy state observer and the state of the true system is very small; Figure 5 and Figure 6 respectively describe the trajectory diagrams of the fuzzy sliding mode variable and the fuzzy sliding mode control variable. It can be seen that the curves of these two variables gradually tend to 0.

Claims

1. A dissipative sliding mode control method under the Delta operator framework, characterized in that The method comprises the following steps: Step 1: Establish a dynamic model of the TS fuzzy Delta operator system with time lag and external disturbance, where: The state space form of the dynamic model is: System fuzzy rule i: If μ1(t) is μ2(t) is μ B (t) Yes So In the formula, i represents the fuzzy rule index; t represents the sampling time; μ1(t) is the first antecedent variable at time t, μ2(t) is the second antecedent variable at time t, and μ B (t) is the Bth antecedent variable at time t; is the first fuzzy set under the i-th fuzzy rule, is the second fuzzy set under the i-th fuzzy rule, is the Bth fuzzy set under the i-th fuzzy rule; x(t) is the state of the system at time t, x(td) is the state of the system with time delay at time t, d is the steady-state time delay; u(t) is the control input of the system at time t; y(t) is the measured output of the system at time t; φ(t) is the initial value of the system state; ν(t) is the external disturbance; A i , A id , B i , B iν , C and D are all known matrices of appropriate dimensions; δ(x(t)) is the Delta operator of the system state x(t) at time t; Defuzzify the state space form of the dynamic model of the TS fuzzy Delta operator system and obtain: In the formula, τ i (μ(t)) is the membership function under the i-th fuzzy rule; represents the sum of the membership functions of F fuzzy rules; F is the total number of fuzzy rules; Step 2: Design a fuzzy state observer for the dynamic model of the TS fuzzy Delta operator system with time delay and external disturbance established in step 1, where: The formula of the fuzzy state observer is: In the formula, is the estimated value of the system state x(t) at time t, representing the state of the fuzzy state observer; is the estimated value of the system state x(td) with time lag at time t; is the estimated value of the system output y(t) at time t, representing the output of the fuzzy state observer; L i is the unknown gain matrix of the fuzzy state observer under the fuzzy rule of the i-th observer; ψ(t) is The initial value of Defuzzify the fuzzy state observer and get: Step 3: Based on the dynamic model established in step 1 and the fuzzy state observer designed in step 2, the corresponding observation error system is obtained: Where δ(e(t)) is the Delta operator of the state error of the fuzzy state observer; is the state error of the fuzzy state observer; e(td) is the state error of the fuzzy state observer with time lag; is the output error of the fuzzy state observer; Step 4: Based on the fuzzy state observer designed in step 2, construct the fuzzy sliding surface and the fuzzy sliding controller, where: The specific form of the fuzzy sliding surface is: Where s(t) is the sliding mode function at time t; is the matrix B i The transpose of ; Q is the unknown matrix to be determined; The specific form of the fuzzy sliding mode controller is: In the formula, K i is the unknown gain matrix of the fuzzy sliding mode controller; ε is a positive constant; yes The norm of ; sign(s(t)) is the sign function of the sliding mode function s(t); Step 5: Using the fuzzy sliding surface and fuzzy sliding mode controller constructed in step 4, through the Lyapunov stability method, Delta operator method and convex optimization method, obtain the judgment conditions that ensure the asymptotic stability and strict dissipation of the fuzzy state observer system and the observation error system and the reachability of the sliding surface. The specific steps are as follows: (1) If the external disturbance ν(t) and the output error z(t) of the fuzzy state observer satisfy: Then the fuzzy state observer system and the observation error system satisfy strict dissipativeness; where ν T (t) is the transpose of the external disturbance ν(t); λ is the limit of the dissipative performance; h t is a number greater than 0; J(t) is the energy supply function, defined as follows: J(t)=z T (t)M1z(t)+2z T (t)M2ν(t)+ν T (t)M3ν(t) Where M1 is a semi-negative definite real matrix; M2 is a real matrix; M3 is a symmetric real matrix; z T (t) is the transpose of z(t); (2) To ensure strict dissipativeness, the Lyapunov stability method is: in: V(t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t) Where V(t) is the Lyapunov function at time t; δ(V(t)) is the Delta operator of V(t); yes The transpose of T (t) is the transpose of e(t), yes The transpose of T (t-ih) is the transpose of e(t-ih), w(t-jh)=e(t-jh)-e(t-(j-1)h), w T (t-jh) is the transpose of w(t-jh); Q, R, S and P are the positive definite matrices to be determined; According to the above Lyapunov stability method, the judgment conditions for the fuzzy state observer system and the observation error system to satisfy asymptotic stability and strict dissipativeness are obtained as follows: L ii <0,i=j L ij +L ji <0.1≤i<j≤F in: In the formula, * represents the symmetric part of the block matrix; Yes ij The first row and first column of the block matrix, Yes ij The first row and second column of the block matrix, Yes ij The 2nd row and 2nd column block matrix, yes The first row and first column of the block matrix, yes The 4th row and 4th column of the block matrix, yes The 1st row and 4th column of the block matrix; (A i -B i K j ) T is the matrix (A i -B i K j ), (A i -L i C) T is the matrix (A i -L i C), (B iν -L i D) T is the matrix (B iν -L i D) is the transpose of C T is the transpose of matrix C, D T is the transpose of matrix D, is the matrix L i The transpose of is a matrix of matrix M2; α and β are greater than A known constant of (3) To ensure that the sliding surface meets the accessibility requirement, the Lyapunov stability method is: In the formula, W is the positive definite matrix to be determined; According to the above Lyapunov stability method, the judgment condition that the sliding surface satisfies the accessibility is obtained as follows: Where: Among them, γ ij is the scalar quantity to be sought; Yes ij The first row and first column of the block matrix, Yes ij The 2nd row and 2nd column block matrix, Yes ij The 3rd row and 3rd column of the block matrix, Yes ij The 4th row and 4th column of the block matrix, Yes ij The 5th row and 5th column of the block matrix; is the matrix B i The transpose of is the matrix B j The transpose of is the matrix K j The transpose of is the matrix A id The transpose of is the matrix B iν The transpose of is a matrix The transpose of Step 6: According to the judgment conditions of step 5, the fuzzy state observer gain matrix and the fuzzy sliding mode controller gain matrix are obtained. The specific steps are as follows: Step 61: Initialize parameters α>0, β>0, ε>0, λ>0, and the maximum number of iterations N, and set t=0; Step 62: By solving the following convex optimization problem, we get in: in, yes The first row and first column of the block matrix, yes The 4th row and 4th column of the block matrix, yes The 4th row and 7th column of the block matrix, yes The 7th row and 7th column of the block matrix, yes The first row and first column of the block matrix, yes The first row and second column of the block matrix, yes The 2nd row and 2nd column block matrix, yes The first row and first column of the block matrix, yes The 4th row and 4th column of the block matrix, Γ 11 yes The first row and first column of the block matrix, Γ 12 yes The first row and second column of the block matrix, Γ 13 yes The first row and third column of the block matrix, Σ 11 yes The first row and first column of the block matrix, Σ 22 yes The 2nd row and 2nd column block matrix, Σ 33 yes The 3rd row and 3rd column of the block matrix, is the matrix Y j The transpose of is the matrix X i The transpose of Step 63: If t<N, set t=t+0.15 and execute step 62; otherwise, execute the next step; Step 64: If t ≥ N, exit the loop, find a feasible solution to the convex optimization problem, and pass L i =Q -1 X i and K j =Δ - 1 Y j Obtain the fuzzy state observer gain matrix and the fuzzy sliding mode controller gain matrix; Step 7: Substitute the fuzzy state observer gain matrix and fuzzy sliding mode controller gain matrix obtained in step 6 back into the fuzzy state observer and fuzzy sliding mode controller constructed in step 4 to realize dissipative sliding mode control of the TS fuzzy Delta operator system.

2. The dissipative sliding mode control method under the Delta operator framework according to claim 1 is characterized in that In step 1, δ(x(t)) is defined as: In the formula, is the derivative of the system state x(t) at time t, and x(t+h) is the state of the system at time t+h.

3. The dissipative sliding mode control method under the Delta operator framework according to claim 1 is characterized in that In the step 1, τ i (μ(t)) is defined as follows: Among them, θ ig (μ g (t)) represents the g-th antecedent variable μ under the i-th fuzzy rule g The degree of membership of (t); It is the product of b membership degrees under the i-th fuzzy rule.

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