Fixed-time command filtering tracking control method for cascade continuous stirred reactor

Abstract

The present invention takes into account the requirements of the cascade continuous stirred reactor in the actual industrial system for the rapidity and stability of the convergence time, and proposes a fixed-time command filter tracking control method for the cascade continuous stirred reactor, which is used to solve the problem that the upper bound of the system's convergence time depends on the initial state, and realize the fixed-time convergence of the system under faster convergence speed and higher control accuracy. According to the working principle of the cascade continuous stirred reactor, the dynamic model of the system is established; based on the backstepping method combined with the fuzzy logic system, command filter control and fixed-time technology, a fixed-time command filter controller is designed; Lyapunov function analysis proves the fixed-time boundedness and stability of the closed-loop system; MATLAB simulation verifies the feasibility and effectiveness of the proposed fixed-time command filter tracking control for the cascade continuous stirred reactor system.

Description

Technical Field

[0001] The invention belongs to the field of automatic control, and in particular relates to a fixed time command filtering tracking control of a cascade continuous stirring reactor. Background Art

[0002] Continuously stirred tank reactor (CCSTR) is a comprehensive reaction vessel that occupies an important position in the reaction device. It is widely used in industrial processes such as biology, chemical industry, and pharmaceutical industry. However, it has the characteristics of high nonlinearity, strong coupling, uncertainty, time-varying and complexity, which makes the CCSTR system show obvious nonlinearity and instability. Therefore, it is more challenging to design an ideal controller to achieve high-precision tracking control of CCSTR system. In order to achieve better control performance of CCSTR, control strategies such as fuzzy adaptive control, backstepping method, dynamic surface, and command filtering are proposed. However, the above control methods can only achieve asymptotic stability of the system, so the stabilization time needs to be infinite, which is difficult to meet the control time limit requirements of actual industrial processes. Although the finite time control method has become a research hotspot in the field of continuous stirred tank reactor control because it can stabilize the system within a finite time, the convergence time of the finite time technology depends on the initial state. With the continuous change of the initial state variables of the system, the finite time technology will cause the system convergence time to gradually slow down or even tend to infinity, and still cannot make the system reach a stable state that meets the actual industrial requirements within a finite time. In actual systems, fast convergence speed can ensure the stability and safety of the system. Therefore, it is necessary to design a controller for the CCSTR system that can achieve fixed-time stability of the system under all initial conditions. Summary of the invention

[0003] The technical problem to be solved by the present invention is: with the continuous development of society, the requirements of industrial control systems for convergence time are getting higher and higher. The present invention takes into account the requirements of cascade continuous stirred reactors in actual industrial systems for rapidity and stability of convergence time, makes up for the shortcomings of the existing technology, and proposes a fixed time command filtering tracking control method.

[0004] In order to solve the above technical problems, the present invention provides a fixed time command filtering tracking control method for a cascade continuous stirred reactor, which adopts the following technical solutions:

[0005] Use fuzzy logic system to approximate unknown and complex nonlinear functions in the system;

[0006] Command filtering control is used to solve the problem of "computational explosion" in the backstepping method and an error compensation signal is introduced to eliminate static errors;

[0007] Fixed time technology is used to overcome the problem that the system's convergence time depends on the initial state, so that the system can accurately track the expected trajectory within a fixed time and ensure the overall fixed time stability of the system;

[0008] Lyapunov theorem is used to analyze and prove the fixed-time boundedness and stability of the closed-loop system;

[0009] Specifically, including

[0010] Step 1: construct a kinetic model of a cascade continuous stirred tank reactor system;

[0011] Step 2: Design a fixed-time command filter controller based on the backstepping method combined with fuzzy logic system, command filter control and fixed-time technology;

[0012] Step 3: Lyapunov function analysis proves the fixed-time boundedness and stability of the closed-loop system;

[0013] Step 4: Use the MATLAB / Simulink simulation platform to verify the feasibility and effectiveness of the proposed fixed-time command filter tracking control for the cascade continuous stirred reactor system.

[0014] Wherein, the kinetic model is:

[0015]

[0016]

[0017] make in and is a steady-state value, the kinetic model can be simplified to

[0018]

[0019] in,

[0020]

[0021]

[0022] The physical meaning of each symbol is as follows

[0023]

[0024] The steps of designing a fixed-time command filter controller based on the backstepping method combined with the fuzzy logic system, command filter control and fixed-time technology are specifically as follows:

[0025] Step 1: Define the tracking error as z 11 =x11 -x 1d , the error compensation tracking signal is v 11 = z 11 -ζ 11 , where x 1d = 0 is the reference signal.

[0026] Select the Lyapunov function as

[0027]

[0028] Select the virtual control function and the error compensation function as

[0029]

[0030] where k 11 , c 11a , c 11b , g 11a , g 11b > 0, p > 1, 0 < q < 1 are design parameters.

[0031] Then is

[0032]

[0033] Step 2: Define the tracking error as z 12 = x 12 - x 1,c , the error compensation tracking signal is

[0034] v 12 = z 12 -ζ 12 , where x 1,c is the output of the command filter.

[0035] Select the Lyapunov function as

[0036]

[0037] According to Young's inequality, we have

[0038]

[0039] Select the actual control function and the error compensation function as

[0040]

[0041] where k 12 , c 12a , c 12b , g 12a , g 12b , l 12>0, p > 1, 0 < q < 1 are design parameters.

[0042] Then is

[0043]

[0044] Step 3: Define the tracking error as z 21 = x 21 - x 2d , and the error compensation tracking signal as v 21 = z 21 - ζ 21 , where x 2d = 0 is the reference signal.

[0045] Select the Lyapunov function as

[0046]

[0047] According to Young's inequality, we have

[0048]

[0049] Select the virtual control function and the error compensation function as

[0050]

[0051]

[0052] where k 21 , c 21 a, c 21 b, g 21 a, g 21 b, l 21 >0, p > 1, 0 < q < 1 are design parameters.

[0053] Then is

[0054]

[0055] Step 4: Define the tracking error as z 22 = x 22 - x 2,c , and the error compensation tracking signal as v 22 = z 22 - ζ 22 , where x 2,c is the output of the command filter.

[0056] Select the Lyapunov function as

[0057]

[0058] It can be obtained from Young's inequality that

[0059]

[0060] Select the actual control function and error compensation function as

[0061]

[0062] where k 22 , c 22a , c 22b , g 22a , g 22b , l 22 > 0, p > 1, 0 < q < 1 are design parameters.

[0063] Then is

[0064]

[0065] Step 5: Define the tracking error as z 31 = x 31 - x 3d , and the error compensation tracking signal is v 31 = z 31 - ζ 31 , where x 3d = 0 is the reference signal.

[0066] Select the virtual control function and error compensation function as

[0067]

[0068] It can be obtained from Young's inequality that

[0069]

[0070] Select the virtual control function and error compensation function as

[0071]

[0072]

[0073] where k 31 , c 31a , c 31b , g 31a , g 31b , l 31 > 0, p > 1, 0 < q < 1 are design parameters.

[0074] Then is

[0075]

[0076] Step 6: Define the tracking error as z 32 = x 32 - x 3,c , and the error compensation tracking signal as v 32 = z 32 - ζ 32 , where x 3,c is the output of the command filter.

[0077] Select the Lyapunov function as

[0078]

[0079] From Young's inequality, we can get

[0080]

[0081] Select the actual control function and the error compensation function as

[0082]

[0083] where k 32 , c 32a , c 32b , g 32a , g 32b , l 32 > 0, p > 1, 0 < q < 1 are design parameters.

[0084] Then is

[0085]

[0086] Among them, the steps of analyzing and proving the fixed-time boundedness and stability of the closed-loop system by the Lyapunov function specifically include:

[0087] Define θ = max{||W 11 ||, ||W 21 ||, ||W 31 ||}, is the estimated value of θ, and the deviation between the estimated value and the actual value is That is:

[0088] Select the Lyapunov function as

[0089]

[0090] Construct the adaptive law as

[0091]

[0092] but for

[0093]

[0094] From Young's inequality we can get

[0095]

[0096] but for

[0097]

[0098] Among them, k ij >0,k ija >0,k ijb >0,l ij >0,l 12 >0,p>1,0 <q<1,m> 0,r>0 are design parameters.

[0099] From Young's inequality we can get

[0100]

[0101] but for

[0102]

[0103] Among them, k ij >0,k ija >0,k ijb >0,l ij >0,l 12 >0,p>1,0 <q<1,m> 0,r>0 are design parameters.

[0104] From Young's inequality we can get

[0105]

[0106] but for

[0107]

[0108] Where τ0=min{2k 11 ,2k 12 ,2k 21 ,2k 22 ,2k 31 ,2k 32 ,2 p k 11a ,2 p k 12a,2 p k 21a ,2 p k 22a ,2 p k 31a ,2 p k 32a ,2 q k 11b ,2 q k 12b ,2 q k 21b ,2 q k 22b ,2 q k 31b ,2 q k 32b ,m}, β0=τ0,

[0109] Among them, there exists a 0<φ0<1, so that the system eventually converges to

[0110]

[0111] The fixed time T is

[0112]

[0113] Among them, combined with the actual industrial requirements of CCSTR system, the given reference signal is y d =0;

[0114] Choose the initial condition of the state variable to be x 11 (0) = 2, x 12 (0) = -2; x 21 (0) = 5, x 22 (0) = -4; x 31 (0) = 2, x 32 (0) = -4;

[0115] The appropriate fuzzy membership function is chosen as

[0116] The controller design parameters are

[0117] k 11 =0.02,k 12 =3,k 21 =10,k 22 =380,k 31 =7,k 32 =1.41,

[0118] c 11a = 0.3, c 11b = 0.3, c 12a = 0.3, c 12b = 0.3, c 21a = 0.05, c 21b = 0.05,

[0119] c 22a = 0.05, c 22b = 0.05, c 31a = 0.01, c 31b = 0.01, c 32a = 0.01, c 32b = 0.01,

[0120] g 11a = 0.1, g 11b = 0.1, g 12a = 0.1, g 12b = 0.1, g 21a = 0.01, g 21b = 0.01,

[0121] g 22a = 0.01, g 22b = 0.01, g 31a = 0.01, g 31b = 0.01, g 32a = 0.01, g 32b = 0.01,

[0122] p = 2, q = 0.999.

[0123] Among them, the virtual control function α i is

[0124]

[0125] Among them, k 11 , c 11a , c 11b , k 21 , c 21a , c 21b , k 31 , c 31a , c 31b , l 21 , l 31 > 0, p > 1, 0 < q < 1 are design parameters.

[0126] Among them, the actual control function u i is

[0127]

[0128] Among them, k 12 , c 12a , c 12b , k 22 , c 22a , c 22b , k 32 , c 32a , c 32b , l 22 , l 32 > 0, p > 1, 0 < q < 1 are design parameters.

[0129] Among them, the error compensation function ζ ij is

[0130]

[0131] Among them, b 11 = 1, b 12 = 1, x i,c is the output of the command filter, α i is the virtual control function, k 11 , g 11a , g 11b , k 21 , g 21a , g 21b , k 31 , g 31a , g 31b , k 12 , g 12a , g 12b , k 22 , g 22a , g 22b , k 32 , g 32a , g 32b > 0, p > 1, 0 < q < 1 are design parameters.

[0132] The present invention has the following beneficial effects:

[0133] (1) The method of the present invention first introduces the fixed-time technology for the cascade continuous stirred tank reactor, and combines the command filter control to design a fixed-time command filter controller.

[0134] (2) The method of the present invention overcomes the problem that the upper bound of the convergence time of the system depends on the initial state, and realizes that the tracking error of the system can converge to a small neighborhood of the origin within a fixed time under any initial state.

[0135] (3) Compared with the existing related research, the present invention has a faster convergence speed and higher control accuracy, and better meets the general requirements of actual industrial systems. Description of the Drawings

[0136] Figure 1 The present invention is a flow chart of a fixed time command filtering and tracking control method for a cascade continuous stirred reactor.

[0137] Figure 2 This is a structural principle diagram of a CCSTR system of a fixed time command filtering tracking control method for a cascade continuous stirred reactor proposed by the present invention.

[0138] Figure 3 The tracking performance of the output signal y(t) of a cascade continuous stirred reactor system proposed in the present invention is shown in FIG. 1 and FIG. 2 : (a) fixed-time command filter control and (b) finite-time command filter control.

[0139] Figure 4 Tracking performance of input signal u(t) of a cascade continuous stirred reactor system proposed in the present invention for (a) fixed time command filter control and (b) finite time command filter control.

[0140] Figure 5 It is a closed-loop signal of a fixed-time command filtering and tracking control method for a cascade continuous stirred reactor proposed by the present invention.

[0141] Figure 6 The invention discloses an adaptive law for a fixed time command filtering tracking control method of a cascade continuous stirred reactor.

[0142] Among them, since the lines in the drawings provided by the present invention overlap, it is necessary to use colored drawings. DETAILED DESCRIPTION

[0143] The preferred embodiments of the present invention are described in detail below in conjunction with the accompanying drawings so that the advantages and features of the present invention can be more easily understood by those skilled in the art, thereby making a clearer and more definite definition of the protection scope of the present invention.

[0144] For CCSTR systems, the early traditional control strategies, such as position control devices, PID (or improved PID) control, input / input linearization and other methods, are difficult to meet the control accuracy and quality requirements of the system. In recent years, experts and scholars have proposed control strategies such as sliding mode control, predictive control, neural network control, fuzzy control and backstepping. Among them, backstepping is widely used in the control system design of CCSTR because it is suitable for high-order nonlinear systems composed of multiple integral links. However, the backstepping method has the problem of "computational explosion". To address this problem, control strategies such as dynamic surface and command filtering have been proposed one after another.

[0145] However, the above control method can only achieve asymptotic stability of the system, so the stabilization time needs to be infinite, which is difficult to meet the control time limit requirements in the actual industrial process. Therefore, finite time technology has received widespread attention. It has the advantages of high control accuracy, fast convergence speed, and good tracking performance, and can ensure that the state variables converge quickly to the equilibrium point. In 2021, finite time command filter control for CCSTR system was first proposed, and simulation verification achieved fast and accurate control of the system. Due to the lack of clear understanding of the internal mechanism of CCSTR, the unclear coupling effects between various environmental factors in the reactor and the random reaction dead corners in the reactor, the CCSTR system shows serious instability and nonlinearity in the actual industrial process. The convergence time of finite time technology depends on the initial state. If it is applied to the actual system, as the initial state variables of the system continue to change, the finite time technology will cause the system convergence time to gradually slow down or even tend to infinity, and the system still cannot reach a stable state that meets the actual industrial requirements within a limited time. In the actual system, the fast convergence speed can ensure the stability and safety of the system. Therefore, in view of the shortcomings of finite time technology, fixed time technology is proposed.

[0146] In 2012, Polyakov first proposed fixed-time stability based on finite-time control theory. Compared with finite-time technology, the convergence time of fixed-time technology is independent of the initial state, and has a faster convergence speed and higher control accuracy, so it is more suitable for practical engineering applications. In recent years, fixed-time technology has attracted much attention and has been continuously developed and applied in the fields of chemical processes, electronic circuits and mechatronics, biological systems, etc. At the same time, with the continuous development of society, industrial control systems have higher and higher requirements for convergence time. The present invention takes into account the requirements of cascade continuous stirred reactors in actual industrial systems for rapidity and stability of convergence time, makes up for the shortcomings of finite-time technology, applies fixed-time technology to CCSTR systems for the first time, and proposes a fixed-time command filter tracking control method for cascade continuous stirred reactors.

[0147] See also Figure 1 , the embodiment of the present invention includes:

[0148] A fixed-time command filter tracking control method for a cascade continuous stirred reactor is provided. According to the working principle of the cascade continuous stirred reactor, the dynamic model of the system is established; based on the backstepping method combined with fuzzy logic system, command filter control and fixed-time technology, a fixed-time command filter controller is designed; Lyapunov function analysis proves the fixed-time boundedness and stability of the closed-loop system; MATLAB simulation verifies the feasibility and effectiveness of the proposed fixed-time command filter tracking control for the cascade continuous stirred reactor system.

[0149] To achieve the control objectives, the following assumptions, definitions, and lemmas are provided:

[0150] Assumption 1: The volume and density of the materials in the CCSTR system remain unchanged before and after the reaction.

[0151] Assumption 2: During the reaction process, the CCSTR system parameters do not change.

[0152] Assumption 3: The expected output signal y d and its first-order derivative are known and continuously bounded.

[0153] Definition 1 Consider the system

[0154]

[0155] where x(t)∈R n is the state variable, and f(x) is a nonlinear function. Assuming that the system is stable in the Lyapunov sense, if there is a finite convergence time T(x0), then the system is finite-time stable; if for all times t≥T, x(t)=0 is satisfied, and the convergence time T has an upper bound that is independent of the state variable, then the system is fixed-time stable.

[0156] Lemma 1: The nonlinear continuous function f(x) is approximated by a fuzzy logic system and can be written as

[0157] f(x)=W T S(x)+ε.

[0158] in is the fuzzy basis function vector, W∈R N is the optimal weight vector, N>1 is the number of fuzzy rules, ε is the approximation error, for any Both

[0159] Basis function P i (x) is expressed as follows using the Gaussian function

[0160]

[0161] where μ i =[μ i1 ,μ i2 ,...,μ iN ] is the center vector, η i is the width of the Gaussian function.

[0162] Lemma 2 The command filter is defined as

[0163]

[0164] Among them, α 11 is the input signal; ωn , ζ is a design parameter.

[0165] If for any t ≥ 0, α 11 satisfies and where ρ1 and ρ2 are both positive constants, and then for any μ > 0, there must exist ω n > 0 and ζ ∈ (0, 1] such that and are both bounded.

[0166] Lemma 3 Assume χ1, χ2,..., χ n ≥ 0, 0 < k ≤ 1, b > 1, then there is

[0167]

[0168] Lemma 4 For any h > 0 and μ1 ≥ 0, μ2 > 0, the following inequality holds

[0169]

[0170] For any h > 0 and μ1 > 0, μ2 ≤ x, the following inequality holds

[0171] (μ1 - μ2) h ≥ μ2 h - μ1 h .

[0172] Lemma 5 For any given positive real numbers a, d, there is

[0173]

[0174] where ω, λ are arbitrary numbers, and k(ω, λ) is a real-valued function.

[0175] Lemma 6 If there exist design parameters α, β > 0, p > 1, 0 < q < 1, 0 < φ < 1, 0 < η < ∞ such that

[0176]

[0177] then the system is fixed-time stable, and its fixed time T satisfies

[0178]

[0179] Then the residual set of the system solution is

[0180]

[0181] The embodiment refers to Figure 1-6 , and specifically includes the following steps:

[0182] Step 1: Construct a kinetic model of the cascade continuous stirred tank reactor system.

[0183] According to the working principle of the cascade continuous stirred reactor, the schematic diagram is as follows Figure 2 The dynamic model of CCSTR system is constructed as follows:

[0184]

[0185] The physical meaning of each symbol is as follows

[0186]

[0187] make x 12 =f2,

[0188] in and is the steady-state value. Then the kinetic model can be simplified to

[0189]

[0190] where b 11 =1,b 12 =1,

[0191]

[0192] Note 1: This invention proposes a fixed-time command filter control for the CCSTR system for the first time. The control target is to construct a control input signal u for the CCSTR system. i (i=1,2,3), so that the output signal y i (i=1,2,3) Track and converge to the desired reference signal y faster in a fixed time d , and all signals of the closed-loop system are stable at a fixed time.

[0193] Step 2: Based on the backstepping method combined with fuzzy logic system, command filter control and fixed time technology, a fixed time command filter controller is designed. The design steps are as follows:

[0194] (1) Define the tracking error as z 11 =x 11 -x 1d , the error compensation tracking signal is v 11 =z 11 -ζ 11 , where x 1d =0 is the reference signal.

[0195] Select the Lyapunov function as

[0196]

[0197] Select the virtual control function and the error compensation function as

[0198]

[0199] where k 11 , c 11a , c 11b , g 11a , g 11b > 0, p > 1, 0 < q < 1 are design parameters.

[0200] Then is

[0201]

[0202] (2) Define the tracking error as z 12 = x 12 - x 1,c , and the error compensation tracking signal as v 12 = z 12 - ζ 12 , where x 1,c is the output of the command filter.

[0203] Select the Lyapunov function as

[0204]

[0205] According to the Young's inequality, we have

[0206]

[0207] Select the actual control function and the error compensation function as

[0208]

[0209] where k 12 , c 12a , c 12b , g 12a , g 12b , l 12 > 0, p > 1, 0 < q < 1 are design parameters.

[0210] Then is

[0211]

[0212] (3) Define the tracking error as z 21 = x 21 - x 2d , and the error compensation tracking signal is v 21 = z 21 - ζ 21 , where x 2d = 0 is the reference signal.

[0213] Select the Lyapunov function as

[0214]

[0215] It can be obtained from the Young's inequality that

[0216]

[0217] Select the virtual control function and the error compensation function as

[0218]

[0219] where k 21 , c 21a , c 21b , g 21a , g 21b , l 21 > 0, p > 1, 0 < q < 1 are design parameters.

[0220] Then is

[0221]

[0222] (4) Define the tracking error as z 22 = x 22 - x 2,c , and the error compensation tracking signal is v 22 = z 22 - ζ 22 , where x 2,c is the output of the command filter.

[0223] Select the Lyapunov function as

[0224]

[0225] It can be obtained from the Young's inequality that

[0226]

[0227] Select the actual control function and the error compensation function as

[0228]

[0229] where \(k\) 22 , \(c\) 22a , \(c\) 22b , \(g\) 22a , \(g\) 22b , \(l\) 22 >0, \(p>1\), \(0 < q < 1\) are design parameters.

[0230] Then is

[0231]

[0232] (5) Define the tracking error as \(z\) 31 \(= x\) 31 \(- x\) 3d , and the error compensation tracking signal as \(v\) 31 \(= z\) 31 \(- \zeta\) 31 , where \(x\) 3d \(= 0\) is the reference signal.

[0233] Select the Lyapunov function as

[0234]

[0235] According to Young's inequality, we have

[0236]

[0237] Select the virtual control function and the error compensation function as

[0238]

[0239] where \(k\) 31 , \(c\) 31a , \(c\) 31b , \(g\) 31a , \(g\) 31b , \(l\) 31 >0, \(p>1\), \(0 < q < 1\) are design parameters.

[0240] Then is

[0241]

[0242] (6) Define the tracking error as \(z\) 32 \(= x\) 32 \(- x\) 3,c , and the error compensation tracking signal as \(v\) 32 \(= z\) 32 \(- \zeta\) 32 , where \(x\) 3,c is the output of the command filter.

[0243] Select the Lyapunov function as

[0244]

[0245] It can be obtained from Young's inequality that

[0246]

[0247] Select the actual control function and error compensation function as

[0248]

[0249] where k 32 , c 32a , c 32b , g 32a , g 32b , l 32 > 0, p > 1, 0 < q < 1 are design parameters.

[0250] Then is

[0251]

[0252] Step 3: Analyze the Lyapunov function to prove the fixed-time boundedness and stability of the closed-loop system.

[0253] Define θ = max{||W 11 ||, ||W 21 ||, ||W 31 ||}, is the estimated value of θ, and the deviation between the estimated value and the actual value is That is:

[0254] Select the Lyapunov function as

[0255]

[0256] Construct the adaptive law as

[0257]

[0258] Then is

[0259]

[0260] It can be obtained from Young's inequality that

[0261]

[0262] Then is

[0263]

[0264] From Young's inequality we can get

[0265]

[0266] but for

[0267] where τ0=min{2k 11 ,2k 12 ,2k 21 ,2k 22 ,2k 31 ,2k 32 ,2 p k 11a ,2 p k 12a ,2 p k 21a ,2 p k 22a ,2 p k 31a ,2 p k 32a ,2 q k 11b ,2 q k 12b ,2 q k 21b ,2 q k 22b ,2 q k 31b ,2 q k 32b ,m}, β0=τ0,

[0268] According to Lemma 6, there exists a 0<φ0<1 such that the system eventually converges to

[0269]

[0270] The fixed time T is

[0271]

[0272] Note 2 For CCSTR systems that satisfy assumptions 1, 2, and 3, the fixed-time command filter controller, virtual control signal, error compensation signal, and adaptive law ensure that all signals in the closed-loop system are fixed-time bounded, and by properly adjusting the design parameters, the system tracking error can converge to a sufficiently small neighborhood near the origin within a fixed time. Therefore, the system ismax There will be stability within.

[0273] Step 4: Use the MATLAB / Simulink simulation platform to verify the feasibility and effectiveness of the proposed fixed-time command filter tracking control for the cascade continuous stirred reactor system.

[0274] The basic physical parameters of the selected CSTR model are as follows

[0275]

[0276]

[0277] Combined with the actual industrial requirements of CCSTR system, the reference signal is given as y d =0.

[0278] The initial conditions of the state variables are chosen to be

[0279] x 11 (0) = 2, x 12 (0) = -2; x 21 (0)=5,x 22 (0) = -4; x 31 (0) = 2, x 32 (0) = -4;

[0280] The appropriate fuzzy membership function is chosen as

[0281]

[0282] The controller design parameters are

[0283] k 11 =0.02,k 12 =3,k 21 =10,k 22 =380,k 31 =7,k 32 =1.41,

[0284] c 11a =0.3,c 11b =0.3,c 12a =0.3,c 12b =0.3,c 21a =0.05,c 21b =0.05,c 22a =0.05,c 22b =0.05,c 31a =0.01,c 31b =0.01,c 32a =0.01,c 32b=0.01,g 11a =0.1,g 11b =0.1,g 12a =0.1,g 12b =0.1,g 21a =0.01,g 21b =0.01,g 22a =0.01,g 22b =0.01,g 31a =0.01,g 31b =0.01,g 32a =0.01,g 32b =0.01,h 12 =1,h 21 =10,h 22 =10,h 31 =1,h 32 =1,p=2,q=0.999.

[0285] The fixed-time command filter control method and the finite-time command filter control method proposed in the present invention are respectively applied to the CCSTR system for comparative simulation to illustrate the superiority and feasibility of the method of the present invention. The simulation results are shown in Figures 3 to 6 shown. Figure 3 and Figure 4 They are the tracking performance curves of the system's output signal y(t) and input signal u(t), which clearly shows that the fixed-time command filter controller proposed in this paper is superior to the existing finite-time command filter controller. The output signal and input signal can track the expected signal within a fixed time, making the system have a faster convergence speed and better tracking performance. Figure 5 It shows that the fixed-time command filter controller can ensure the fixed-time stability of all closed-loop signals in the system; Figure 6 This shows that the adaptive law of the method of the present invention is smooth and reasonable.

[0286] The present invention improves the system convergence speed and control accuracy and the system tracking performance, while overcoming the problem that the system convergence time depends on the initial state, so that the closed-loop system has good stability and reliability.

[0287] The above descriptions are merely embodiments of the present invention and are not intended to limit the patent scope of the present invention. Any equivalent structure or equivalent process transformation made using the contents of the present invention specification and drawings, or directly or indirectly applied in other related technical fields, are also included in the patent protection scope of the present invention.

Claims

1. A fixed time command filtering tracking control method for a cascade continuous stirred reactor, characterized in that: include: Step 1: construct a kinetic model of a cascade continuous stirred tank reactor system; Step 2: Design a fixed-time command filter controller based on the backstepping method combined with fuzzy logic system, command filter control and fixed-time technology; Step 3: Lyapunov function analysis proves the fixed-time boundedness and stability of the closed-loop system; Step 4: Use the MATLAB / Simulink simulation platform to verify the feasibility and effectiveness of the proposed fixed-time command filter tracking control for the cascade continuous stirred tank reactor system; The kinetic model is: Assume that the volumes of reactor I and reactor II are V1=V2=V, where V1 is the volume of reactor I, V2 is the volume of reactor II, and the volume of the two cooling jackets is V j1 =V j2 =V j , where V j1 is the volume of the cooling layer of reactor I, V j2 is the volume of the cooling layer of reactor II, and the flow rate of the reactants is Q0=Q2=Q; wherein Q0 is the flow rate of reactant A flowing into reactor I, Q2 is the flow rate of reactant A flowing into reactor II, and Q R is the flow rate of reactant A fed back from reactor II to reactor I, c r is the specific heat capacity of the reactants, c j is the specific heat capacity of cooling water, T1 is the temperature of reactor I, T2 is the temperature in reactor II, T j1 is the temperature of the cooling water flowing out of reactor I, T j2 is the temperature of the cooling water flowing out of reactor II, Q j1 is the flow rate of cooling water flowing into and out of reactor I, Q j2 is the flow rate of cooling water flowing into and out of reactor II, ρ r is the reactant density, ρ j is the cooling water density, λ is the heat generation rate; C A0 is the concentration of reactant A flowing into reactor I, C A1 is the concentration of reactant A in reactor I, C A2 is the concentration of reactant A in reactor II, R is the universal gas constant, E is the activation energy, U is the heat transfer coefficient between the reactor and the jacket, A is the heat transfer area, and x 12 =f2, in T1 d , and is the steady-state value, α is the reaction rate constant, then the kinetic model can be simplified to in, b 11 =1,b 12 =1, in, is the value of T0 in a stable state, referred to as the steady-state value, For variable T j10 The steady-state value of For variable T j20 The steady-state value, T j10 is the temperature of the cooling water flowing into the reactor I, T j20 is the temperature of the cooling water flowing into the reactor II, and T0 is the temperature of the reactant A flowing into the reactor I; The steps of designing a fixed-time command filter controller based on the backstepping method combined with the fuzzy logic system, command filter control and fixed-time technology are specifically as follows: Step 1: Define the tracking error as z 11 =x 11 -x 1d , the error compensation tracking signal is v 11 =z 11 -ζ 11 , where x 1d =0 is the reference signal, ζ 11 is the error compensation function; Select the Lyapunov function as Among them, b 11 =1, where V0 is the Lyapunov function: The virtual control function and error compensation function are selected as where k 11 , c 11a , c 11b , g 11a , g 11b > 0, p> 1, 0 <q <1 are design parameters; ζ 12 is an error compensation function; but for Among them, k 11a ,k 11b is the design parameter; Step 2: Define the tracking error as z 12 =x 12 -x 1,c , the error compensation tracking signal is v 12 =z 12 -ζ 12 , where x 1,c is the output of the command filter; Select the Lyapunov function as Among them, b 12 =1, From Young's inequality we can get Among them, f 12 is a nonlinear approximation function based on fuzzy logic system, W 12 is the optimal weight vector, S 12 is the fuzzy basis function vector, is any positive number; The actual control function and error compensation function are selected as where k 12 , c 12a , c 12b , g 12a , g 12b , l 12 > 0, p > 1, 0 < q < 1 are design parameters, is the estimated value of the optimal parameter vector θ; but for: Among them, k 12a ,k 12b is the design parameter; Step 3: Define the tracking error as z 21 =x 21 -x 2d , the error compensation tracking signal is v 21 =z 21 -ζ 21 , where x 2d =0 is the reference signal, ζ 21 is the error compensation function; Select the Lyapunov function as From Young's inequality we can get Among them, f 21 is a nonlinear approximation function based on fuzzy logic system, W 21 is the optimal weight vector, S 21 is the fuzzy basis function vector, is any positive number; The virtual control function and error compensation function are selected as where k 21 , c 21a , c 21b , g 21a , g 21b , l 21 > 0, p > 1, 0 < q < 1 are design parameters; but for Step 4: Define the tracking error as z 22 =x 22 -x 2,c , the error compensation tracking signal is v 22 =z 22 -ζ 22 , where x 2,c is the output of the command filter, which is x 22 The expected value of 22 is the error compensation function; Select the Lyapunov function as From Young's inequality we can get Among them, f 22 is a nonlinear approximation function based on fuzzy logic system, W 22 is the optimal weight vector, S 22 is the fuzzy basis function vector, is any positive number; The actual control function and error compensation function are selected as where k 22 , c 22a , c 22b , g 22a , g 22b , l 22 > 0, p > 1, 0 < q < 1 are design parameters; but for Among them, k 21a ,k 21b ,k 22a ,k 22b is the design parameter; Step 5: Define the tracking error as z 31 =x 31 -x 3d , the error compensation tracking signal is v 31 =z 31 -ζ 31 , where x 3d =0 is the reference signal, ζ 31 is the error compensation function; Select the Lyapunov function as From Young's inequality we can get Among them, f 31 is a nonlinear approximation function based on fuzzy logic system, W 31 is the optimal weight vector, S 31 is the fuzzy basis function vector, is any positive number; The virtual control function and error compensation function are selected as where k 31 , c 31a , c 31b , g 31a , g 31b , l 31 > 0, p > 1, 0 < q < 1 are design parameters; but for Step 6: Define the tracking error as z 32 =x 32 -x 3,c , the error compensation tracking signal is v 32 =z 32 -ζ 32 , where x 3,c is the output of the command filter, ζ 32 is the error compensation function; Select the Lyapunov function as From Young's inequality we can get Among them, f 32 is a nonlinear approximation function based on fuzzy logic system, W 32 is the optimal weight vector, S 32 is the fuzzy basis function vector, is any positive number; The actual control function and error compensation function are selected as where k 32 , c 32a , c 32b , g 32a , g 32b , l 32 > 0, p > 1, 0 < q < 1 are design parameters but for 2. The fixed time command filtering tracking control method for a cascade continuous stirred tank reactor according to claim 1, characterized in that: The steps of the Lyapunov function analysis to prove the fixed-time boundedness and stability of the closed-loop system specifically include: Define θ=max{||W 11 ||,||W 21 ||,||W 31 ||}, is the estimated value of θ, and the deviation between the estimated value and the actual value is Right now: Select the Lyapunov function as Constructing adaptive laws for but for From Young's inequality we can get but for Among them, k ij >0,k ija >0,k ijb >0,l ij >0,l 12 >0,p>1,0 <q<1,m> 0,r>0 is the design parameter; From Young's inequality we can get but for where τ0 = min{2k 11 , 2k 12 , 2k 21 , 2k 22 , 2k 31 , 2k 32 , 2 p k 11a , 2 p k 12a , 2 p k 21a , 2 p k 22a , 2 p k 31a , 2 p k 32a , 2 q k 11b , 2 q k 12b , 2 q k 21b , 2 q k 22b , 2 q k 31b , 2 q k 32b , m}, β0 = τ0, Therefore, there exists a 0<φ0<1 such that the system eventually converges to The fixed time T is 3. The fixed time command filtering tracking control method for a cascade continuous stirred tank reactor according to claim 2, characterized in that: Combined with the actual industrial requirements of CCSTR system, the reference signal is given as y d =0; Choose the initial condition of the state variable to be x 11 (0) = 2, x 12 (0) = -2; x 21 (0)=5,x 22 (0) = -4; x 31 (0) = 2, x 32 (0) = -4; The appropriate fuzzy membership function is chosen as The controller design parameters are k 11 =0.02,k 12 =3,k 21 =10,k 22 =380,k 31 =7,k 32 =1.41, c 11a =0.3,c 11b =0.3,c 12a =0.3,c 12b =0.3,c 21a =0.05,c 21b =0.05, c 22a =0.05,c 22b =0.05,c 31a =0.01,c 31b =0.01,c 32a =0.01,c 32b =0.01, g 11a =0.1,g 11b =0.1,g 12a =0.1,g 12b =0.1,g 21a =0.01,g 21b =0.01, g 22a =0.01,g 22b =0.01,g 31a =0.01,g 31b =0.01,g 32a =0.01,g 32b =0.01,p=2,q=0.999。 4. The fixed time command filtering tracking control method for a cascade continuous stirred tank reactor according to claim 3, characterized in that: Virtual control function α i for where k 11 , c 11a , c 11b , k 21 , c 21a , c 21b , k 31 , c 31a , c 31b , l 21 , l 31 > 0, p > 1, 0 < q < 1 are design parameters.

5. The fixed time command filtering tracking control method for a cascade continuous stirred tank reactor according to claim 4, characterized in that: Actual control function u i for where k 12 , c 12a , c 12b , k 22 , c 22a , c 22b , k 32 , c 32a , c 32b , l 22 , l 32 > 0, p > 1, 0 < q < 1 are design parameters.

6. The fixed time command filtering tracking control method for a cascade continuous stirred tank reactor according to claim 5, characterized in that: Error compensation function ζ ij for where b 11 = 1, b 12 = 1, x i,c is the output of the command filter, α i is the virtual control function, k 11 , g 11a , g 11b , k 21 , g 21a , g 21b , k 31 , g 31a , g 31b , k 12 , g 12a , g 12b , k 22 , g 22a , g 22b , k 32 , g 32a , g 32b > 0, p > 1, 0 < q < 1 are design parameters.

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