A Robust Control Method for the Nonlinear Batch Process of a Stirred Tank Reactor

Abstract

The invention discloses a robust control method for the non-linear batch process of a stirred tank reactor, belonging to the technical field of iterative learning control. The robust control method for the non-linear batch process of the stirred tank reactor provided by the invention takes the reaction process of the stirred tank reactor as the research object in the stirred tank reactor. The sector non-linearity method is used to represent the non-linear batch process model as an uncertain T-S model with non-repetitive disturbances, and a two-dimensional composite ILC scheme is designed by using the two-dimensional and repetitive characteristics of the batch process. Based on the two-dimensional Lyapunov theory, sufficient conditions are derived to ensure that the obtained closed-loop system is asymptotically stable and has a specified H∞ attenuation level. The design problem is formulated as solving a set of LMIs. Finally, the verification results show that the developed method has a good attenuation effect and practical value for the non-repetitive disturbances of the fuzzy system.

Description

Technical Field

[0001] The present invention relates to the technical field of iterative learning control, and more specifically, to a robust control method for the nonlinear batch process of a stirred tank reactor. Background Art

[0002] In modern industrial production, as a common device, the performance optimization of a stirred tank reactor plays an important role in improving production efficiency and quality. In recent years, the use of iterative learning control methods to optimize it has been widely applied. However, the existing iterative learning control methods for optimizing the performance of stirred tank reactors are mainly based on two-dimensional composite iterative learning control algorithms, and most of these algorithms are based on linear models. However, the adaptability of these linear models is limited and they cannot fully characterize the nonlinear systems in actual situations. Some significant characteristics of the batch process, such as significant nonlinear characteristics, strong time-variation, and operating point instability, bring great difficulties to the comprehensive analysis of the system. Especially in the fields of special chemicals, polymers, pharmaceutical biochemistry, advanced alloys, modern agriculture, etc., these characteristics put forward higher requirements for the batch process, making the system control difficult and complex.

[0003] The existing such algorithms are difficult to fully meet the performance requirements in actual applications. Therefore, it is particularly important to develop a new type of nonlinear batch iterative learning control method suitable for the batch process. The new method needs to be able to more accurately establish the mathematical model of the batch process, understand its dynamic behavior and characteristics, and be able to design a control strategy with strong adaptability and high robustness to achieve precise control of the batch process with characteristics such as high nonlinearity, operating point instability, and unique repeatability, thereby improving the operation efficiency of the stirred tank reactor and the quality of production results.

[0004] Therefore, in order to meet the performance requirements in actual applications, it is necessary to develop a nonlinear batch iterative learning control scheme to more accurately describe the behavior of nonlinear systems and have the ability to adapt to the particularity of the batch process. Summary of the Invention

[0005] The present invention provides a robust control method for the nonlinear batch process of a stirred tank reactor to solve the control problem of the stirred tank reactor in the nonlinear and batch processes.

[0006] To solve the above technical problems, the technical solution of the present invention is as follows:

[0007] A robust control method for the nonlinear batch process of a stirred tank reactor, comprising the following steps:

[0008] S1. Establish the kinetic equation of the stirred tank reactor model;

[0009] S2. Establish a state - space model of the stirred - tank reactor based on the kinetic equation of the stirred - tank reactor model;

[0010] S3. According to the state - space model of the stirred - tank reactor, discretize it to obtain a discrete - linear model of the stirred - tank reactor;

[0011] S4. Based on the discrete - linear model of the stirred - tank reactor, construct a fuzzy system of the stirred - tank reactor;

[0012] S5. Based on the fuzzy system of the stirred - tank reactor, establish a closed - loop two - dimensional fuzzy system of the stirred - tank reactor;

[0013] S6. Analyze the performance of the closed - loop two - dimensional fuzzy system of the stirred - tank reactor to obtain the performance analysis results of the closed - loop two - dimensional fuzzy system of the stirred - tank reactor;

[0014] S7. Based on the performance analysis results of the closed - loop two - dimensional fuzzy system of the stirred - tank reactor, optimize the closed - loop two - dimensional fuzzy system of the stirred - tank reactor.

[0015] Furthermore, the kinetic equation of the stirred - tank reactor model in step S1 includes the rate of change of the concentration of substance A the rate of change of temperature Specifically:

[0016]

[0017]

[0018] Among them, is the rate of change of the concentration of substance A, is the rate of change of temperature, C A is the current concentration of substance A, C Af represents the concentration of substance A when feeding into the stirred - tank reactor, q is the feed flow rate, V is the recycle volume, k0 is the reaction rate constant, E0 / R0 is the ratio of the Arrhenius activation energy to the gas constant, T represents the reactor temperature, T c is the coolant temperature, T f is the temperature when feeding, -ΔH is the heat effect of the reaction, that is, the change in heat released or absorbed by the exothermic reaction, ρ is the density of the substance, C p is the constant - pressure heat capacity of the substance, U represents the product of the heat - transfer coefficient and the surface area of the reactor, V ρ is the product of the effective volume of the stirred - tank and the material density, A is the surface area of the stirred - tank reactor.

[0019] Furthermore, the expression of the state - space model of the stirred - tank reactor in step S2 is:

[0020]

[0021] where \(x(t,k)\in\mathbb{R}\) n , \(u(t,k)\in\mathbb{R}\) m and \(y(t,k)\in\mathbb{R}\) l represent the state, input, and output vectors at the instant \(t\) of the \(k\) - th batch, respectively; \(T\) d is the time period of a batch; \(f[x(t,k),u(t,k)]\) and \(g[x(t,k)]\) represent nonlinear functions with appropriate dimensions.

[0022] Furthermore, the expression of the discrete - time linear model in step S3 is as follows:

[0023]

[0024] In the expression, \(\theta(t,k)=[\theta_1(t,k),\theta_2(t,k),\cdots,\theta\) p (t,k)]\) represents the premise - variable vector, \(M_{ij}(i = 1,2,\cdots,r;j = 1,2,\cdots,p)\) are fuzzy sets, \(r\) is the number of fuzzy rules, and \(p\) is the number of premise variables;

[0025] With \(x(0,k)=x_0\) as the initial condition for each batch, \(w(t,k)\) represents a disturbance belonging to the \(L_2\) space, \(\{A\) i , B i , C i \} consists of system matrices of the same dimension;

[0026] Assume that the output matrices of all fuzzy subsystems are constant, i.e., \(C_1 = C_2=\cdots = C\) r ;

[0027] \{\(\Delta A\) i (t),\(\Delta B\) i (t)\} represents the form of the uncertainty terms as:

[0028] [\(\Delta A\) i (t)\(\Delta B\) i (t)] = E\(\Delta(t)[F\) Ai F Bi , \(\Delta(t)\) T \(\Delta(t)\leq I\)

[0029] where \(E\), \(F\) Ai and \(F\) Bi are known real - constant matrices of dimensions, and \(\Delta(t)\) represents an uncertain disturbance depending on time \(t\);

[0030] Denote \(\mu\) i (\theta(t,k))\) as the normalized fuzzy - basis function of the inferred fuzzy set and define it as:

[0031] ​​​​​​​​​​

[0032] Among them, M ij (θ j (t, k)) is the membership degree of θ ij in M j (t, k).

[0033] Furthermore, step S4 is specifically as follows: Represent the non-zero balance as and introduce and y = x2 as the system state, input, and output variables;

[0034] Set the sampling time to 0.05 min, and obtain the following fuzzy system by considering uncertainties and non-repetitive disturbances,

[0035] The fuzzy system matrix is:

[0036]

[0037] Furthermore, step S5 includes defining the tracking error for batch k:

[0038] e(t, k) = y d (t) - y(t, k)

[0039] Among them, y d (t) is the reference trajectory vector, and the tracking error e(t, k) is used to adjust the input vector of the next batch to make the actual output y(t, k) gradually approach the reference trajectory vector;

[0040] The expression of the closed-loop two-dimensional fuzzy system of the stirred tank reactor is:

[0041]

[0042] Furthermore, step S6 is specifically as follows:

[0043] S61. Set a set of conditions according to the given scalar and symmetric matrix for performance analysis of the two-dimensional closed-loop fuzzy system;

[0044] S62. Set the fuzzy Lyapunov function and calculate its value along the system trajectory;

[0045] S63. Review the conditions set in step S61 and determine a set of values that meet the set conditions;

[0046] S64. Sum up this set of values and confirm that the function decreases along the state trajectory by comparison;

[0047] S65. For any non-zero value, check the two-dimensional H-infinity performance under zero boundary conditions.

[0048] Compared with the prior art, the beneficial effects of the technical solution of the present invention are as follows:

[0049] In view of the non-linear motion characteristics of the stirred tank reactor, the present invention provides a robust control method for the non-linear batch process of the stirred tank reactor, that is, a robust fuzzy ILC scheme for the non-linear batch process based on the T-S fuzzy model. In the stirred tank reactor, the reaction process of the stirred tank reactor is taken as the research object. The non-linear batch process model is represented as an uncertain T-S model with non-repetitive disturbances by using the sector non-linearity method, and a two-dimensional composite ILC scheme is designed by using the two-dimensional and repetitive characteristics of the batch process. Based on the two-dimensional Lyapunov theory, sufficient conditions are derived to ensure that the resulting closed-loop system is asymptotically stable and has a specified H∞ attenuation level. The design problem is formulated as solving a set of LMIs. Finally, the verification results show that the developed method has good attenuation effect and practical value for the non-repetitive disturbances of the fuzzy system. Brief Description of the Drawings

[0050] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0051] Figure 1 is the flow chart of the present invention;

[0052] Figure 2 is the curve graph of RMS error vs. batch number;

[0053] Figure 3 is the curve graph of tracking error vs. time;

[0054] Figure 4 is the comparison graph of the output of the T-S fuzzy object and the desired trajectory. Detailed Description of the Preferred Embodiments

[0055] In order to better understand the purpose, structure and function of the present invention, the technical solution of the present invention will be further described in detail below with reference to the drawings and specific preferred embodiments.

[0056] In the description of the present invention, it should be understood that the orientation or positional relationships indicated by terms such as "left side", "right side", "upper part", "lower part", etc. are based on the orientation or positional relationships shown in the drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. "First", "second", etc. do not represent the importance of components, and therefore should not be construed as limitations on the present invention. The specific dimensions adopted in the embodiments are only for illustrating the technical solutions by way of example, and do not limit the protection scope of the present invention. For those skilled in the art, it is understandable that some well-known structures and their descriptions in the drawings may be omitted.

[0057] Unless otherwise clearly defined and limited, terms such as "installation", "setting", "connection", "fixation", etc. should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or integrated; it can be a mechanical connection or an electrical connection; it can be directly connected or indirectly connected through an intermediate medium, and it can be the communication inside two elements or the interaction relationship between two elements. For those of ordinary skill in the art, the specific meanings of the above terms in this application can be understood according to specific circumstances.

[0058] Example 1:

[0059] As Figure 1 shown, the present invention provides a technical solution: a robust control method for the nonlinear batch process of a stirred tank reactor, including the following steps:

[0060] S1. Establish the kinetic equation of the stirred tank reactor (CSTR) model;

[0061] Considering an uncertain T-S fuzzy system with non-repetitive disturbances, we demonstrate the effectiveness of the proposed method by applying the studied method to the highly nonlinear model of the CSTR. The kinetic description of the CSTR model is

[0062]

[0063] where C A is the current concentration of substance A, C Af represents the concentration of substance A when feeding into the stirred tank reactor, q is the feed flow rate, V is the recycle volume, k0 is the reaction rate constant, E0 / R0 is the ratio of the Arrhenius activation energy to the gas constant, T represents the reactor temperature, T c is the coolant temperature, T f is the temperature when feeding, -ΔH is the heat effect of the reaction, that is, the heat change released or absorbed by the exothermic reaction, ρ is the density of the substance, C p is the constant-pressure heat capacity of the substance, U represents the product of the heat transfer coefficient and the surface area of the reactor, Vρ The product of the effective volume of the stirred tank and the material density, the surface area of the A stirred tank reactor, is the rate of change of the concentration of substance A, represents the rate of change of temperature;

[0064] S2. Based on the kinetic equation of the stirred tank reactor model, establish the state space model of the stirred tank reactor;

[0065] The state space model of the said stirred tank reactor (CSTR), taking the water level in the reaction process of the stirred tank reactor (CSTR) as the research object, describes the relationship between the input and the output, and is designed according to the iterative learning control algorithm to establish the state space model as shown in (2)

[0066]

[0067] where x(t,k) ∈ R n , u(t,k) ∈ R m and y(t,k) ∈ R l respectively represent the state, input and output vectors at the instant t of the k-th batch; T d is the time period of one batch; f[x(t,k),u(t,k)] and g[x(t,k)] represent nonlinear functions with appropriate dimensions;

[0068] S3. According to the state space model of the stirred tank reactor, discretize it to obtain the discrete linear model of the stirred tank reactor;

[0069] Applying the local sector nonlinear method and using an appropriate average sampling time, a discrete linear model can be obtained. Therefore, the nonlinear object (2) can be transformed into a discrete uncertain two-dimensional T-S fuzzy model, which is represented by IF-THEN rules as follows

[0070] Rule: IF θ1(t,k) is M i1 , θ j (t,k) is M ij , θ p (t,k) is M ip , TURE

[0071]

[0072] where, θ(t,k) = [θ1(t,k), θ2(t,k),..., θ p (t,k)] represents the premise variable vector, and Mij (i = 1, 2,..., r; j = 1, 2,..., p) are fuzzy sets. r is the number of fuzzy rules, and p is the number of premise variables. It is considered that x(0,k) = x0 is the initial condition for each batch. w(t,k) represents the disturbance belonging to the L2 space. {Ai , B i , C i} are composed of system matrices with the same dimension. In addition, the output matrices of all fuzzy subsystems are assumed to be constant, i.e., C1 = C2 =... = C r . The constant C matrix condition can greatly reduce the complexity of design and calculation. In addition, {ΔA i (t), ΔB i (t)} represents the form of the uncertainty term as

[0073] [ΔA i (t) ΔB i (t)] = EΔ(t)[F Ai F Bi , Δ(t) T Δ(t) ≤ I (4)

[0074] where E, F Ai and F Bi are known real constant matrices with appropriate dimensions, and Δ(t) represents an uncertain perturbation depending on time t.

[0075] Denote μ i (θ(t,k)) as the normalized fuzzy basis function of the inferred fuzzy set , which is defined as:

[0076]

[0077] where M ij (θ j (t,k)) is regarded as the membership degree of θ ij in M j (t,k). In the following, for simplicity, μ i (θ(t,k)) is represented by μ i or μ i (t,k) (i = 1, 2,... r).

[0078] Applying the standard fuzzy inference method to step S2, the final uncertain T-S fuzzy system can be represented by the ILC setting as

[0079]

[0080] where,

[0081] S4. Based on the discrete linear model of the stirred tank reactor, construct the fuzzy system of the stirred tank reactor;

[0082] Represent the non-zero equilibrium as and introduce and \(y = x^2\) as the system state, input, and output variables. Set the sampling time to \(0.05\) min, then a following fuzzy system can be obtained by considering uncertainties and non-repetitive disturbances. The T-S fuzzy system matrices are given by

[0083]

[0084] S5. Based on the fuzzy system of the stirred tank reactor, a closed-loop two-dimensional fuzzy system of the stirred tank reactor is established;

[0085] Define the tracking error for batch \(k\)

[0086] \(e(t,k)=y\) d (t) - y(t,k) (8)

[0087] where \(y\) d (t) is the reference trajectory vector, and this tracking error is used to adjust the input vector of the next batch to make the actual output \(y(t,k)\) gradually approach the reference trajectory vector.

[0088] For step S2, in order to express the robust ILC design problem in the two-dimensional T-S fuzzy framework, the following classical ILC strategy can be considered. The form of the input for the current batch consists of the previous batch and a correction term

[0089] \(u(t,k)=u(t,k - 1)+r(t,k)\) (9)

[0090] where \(r(t,k)\) is the correction term, and \(u(t,0)\) is the initial value of the iterative algorithm, usually set to zero.

[0091] Introduce the intermediate vector

[0092] \(\delta x(t,k)=x(t,k)-x(t,k - 1)\) (10)

[0093] Let \(y\) d (0)=y(0,k)=Cx(0,k), so \(\delta x(0,k)=0\). From equations (2)-(9), we can obtain

[0094]

[0095] where

[0096]

[0097] \(w1(t,k)=(A(\delta\mu)+\Delta A(\delta\mu))x(t,k - 1)+(B(\delta\mu)+\Delta B(\delta\mu))u(t,k - 1)\),

[0098]

[0099]

[0100] Inspired by the parallel distributed compensation method, the present invention considers the following fuzzy ILC update law:

[0101] Rule: IF θ1(t,k) is M i1 , θ j (t,k) is M ij , θ p (t,k) is M ip , TURE

[0102]

[0103] where K i is the controller gain to be solved. Through fuzzy mixing, it can be expressed in the following form

[0104]

[0105] where

[0106] Let obtain From equations (11) to (13), the two-dimensional fuzzy system can be obtained as

[0107]

[0108] where

[0109]

[0110]

[0111] The boundary conditions are satisfied

[0112]

[0113] S6. Analyze the performance of the closed-loop two-dimensional fuzzy system of the stirred tank reactor to obtain the performance analysis results of the closed-loop two-dimensional fuzzy system of the stirred tank reactor;

[0114] Given a scalar r > 0, there exists a symmetric matrix P = diag{P h , P v} that satisfies the following conditions for all μ

[0115]

[0116] Perform performance analysis on the two-dimensional closed-loop fuzzy system (14) with the conditions set as Consider the fuzzy Lyapunov function as

[0117] V(t,k) = Vh (t, k) + V v (t, k)

[0118] V h (t, k) = x h (t, k) T P h x h (t, k)

[0119] V v (t, k) = x v (t, k) T P v x v (t, k) (16)

[0120] Then, along the trajectory of system (14), having

[0121]

[0122] Reviewing equation (15), A(μ) can be obtained T PA(μ) - P < 0. Therefore, for any The following condition holds

[0123] V h (t + 1, k) + V v (t, k + 1) ≤ V h (t, k) + V v (t, k) (18)

[0124] Summing both sides of inequality (18), with t from 0 to n and k from n to 0, it can be deduced that

[0125]

[0126] Obviously, the function decreases along the state trajectory, and it is easy to obtain That is

[0127]

[0128] From (20), it can be seen that system (14) satisfies the stability condition, that is, two-dimensional robust asymptotic stability.

[0129] Next, establish the two-dimensional H performance of system (14) under zero boundary conditions, that is, for any non-zero ∞ performance, that is, for any non-zero It can be obtained that

[0130]

[0131] From the establishment of (15), there is

[0132]

[0133] Summarizing both sides of (22), we can obtain

[0134]

[0135] It can be clearly observed that Combined with the zero boundary condition, we can obtain

[0136]

[0137] Therefore, the system (14) satisfies two-dimensional H ∞ The performance is satisfied.

[0138] So the ILC scheme of the obtained closed-loop fuzzy system (14) is asymptotically stable under the H ∞ performance r.

[0139] S7. Based on the performance analysis results of the closed-loop two-dimensional fuzzy system of the stirred tank reactor, optimize the closed-loop two-dimensional fuzzy system of the stirred tank reactor;

[0140] For step S6, in order to increase its applicability, by defining new variables and using matrix inequality transformation techniques, a new method is proposed to optimize the system.

[0141] Given a scalar r > 0, there exists a symmetric matrix P = diag{P h , P v}, and matrices w1, w2, w3, w4, w5, w6, w7, w8, w9 satisfy the following conditions for all μ

[0142]

[0143] Inequality (25) can be equivalently transformed into

[0144]

[0145] where

[0146]

[0147]

[0148] Based on the bilinear matrix inequality (LMI) theorem, the above inequality (25) establishes the following conditions

[0149]

[0150] Then, inequality (27) can be expressed as

[0151]

[0152] Among them

[0153]

[0154] Introduce the matrix Λ2 = [-I B T 0], W2 = [W7 W8 W9], It can be obtained that Apply the bilinear matrix inequality (LMI) theorem to (28) and deduce

[0155]

[0156] After matrix transformation, the inequality (29) can be reformulated as

[0157]

[0158] Equation (30) is the dual of equation (15) in step S5. It can be seen that the ILC scheme of the two-dimensional fuzzy system (14) is asymptotically stable under the H ∞ performance r.

[0159] Example 2:

[0160] On the basis of Example 1, add step S8, and the step S8 is to design a controller under the condition of an uncertain model;

[0161] In this step, the robust ILC scheme of the uncertain fuzzy object model is designed by widely using the results derived from the previous steps. When there is uncertainty in the model structure, that is, the matrices ΔA i (t) and ΔB i (t) exist in (3) and are in the form of (4). The results of steps S6 and S7 can be extended to the ILC scheme design of the uncertain fuzzy system. For this purpose, new LMI-based conditions for the solvability of the considered problem are designed.

[0162] Given a scalar r > 0, there exist symmetric matrices P = diag{P h , P v}, matrices w1, w2, w3, w4, w5, w6, w7, w8, w9, Y i and a scalar ε > 0 such that the following LMI holds

[0163] Ψ ii <0, i = 1, 2,... r (31)

[0164]

[0165] Among them

[0166]

[0167]

[0168] There is uncertainty in the state - space model, and the inequality (25) in step S7 is written as

[0169] R + sym{XΔ(t)Y}<0 (33)

[0170] where

[0171]

[0172] X=[0 E T 0 0 0], Y=[0 0 0 F A (μ)W1 + F B (μ)Y(μ) 0]

[0173] Applying the ε - Small Gain Theorem to handle the uncertainty in the inequality (33), we get

[0174]

[0175] where ε>0. Applying the Schur complement theorem, we deduce

[0176]

[0177] By left - multiplying and right - multiplying diag{I, I, I, I, I, εI, I} and its transpose respectively, we deduce

[0178]

[0179] According to the theorem: If the following inequalities hold

[0180] Φ ii <0, i=1, 2,..., r,

[0181]

[0182] then holds.

[0183] It can be deduced that equations (31) and (32) hold, and then (36) also holds.

[0184] In this case, the ILC law matrix required for (12) can be calculated

[0185] K i =Y i Y1 -1 .

[0186] Example 3:

[0187] Based on Example 2, step S9 is added. Step S9 is to achieve the tracking error of the stirred tank reactor under an uncertain model, specifically:

[0188] For the stirred tank reactor of an uncertain T-S fuzzy system with non-repetitive disturbances, as the number of batches increases, the actual output of the T-S fuzzy object gradually approaches the desired trajectory. The controlled system exhibits good robust stability and convergence, confirming that the method proposed in the present invention has a good attenuation effect on the non-repetitive disturbances of the fuzzy system.

[0189] As Figure 2 shown, Figure 2 is a graph of the RMS error versus the number of batches, used to depict the RMS response of the tracking error reduced to a bounded value with respect to the batches;

[0190] As Figure 3 shown, Figure 3 is a graph of the tracking error varying with time, used to show the process of the tracking error e(t,k);

[0191] As Figure 4 shown, Figure 4 is a comparison graph of the output of the T-S fuzzy object and the desired trajectory, used to show that as the batches progress, the actual output of the T-S fuzzy object gradually approaches the desired trajectory.

[0192] Example 4:

[0193] Based on Example 1, for the fuzzy system (7) of the stirred tank reactor (CSTR), x2 is selected as the premise variable, and the membership function is defined as

[0194]

[0195] where

[0196]

[0197] and

[0198]

[0199] The constraint on x2 is imposed as |x2(t,k)| ≤ β (β = 10), and the model parameters are given as follows

[0200]

[0201] Let the non-repetitive disturbance be w(t,k) = [0.01 0.02] Tsin(0.1tδ3 + 0.15kδ3), where δ3 varies randomly within the interval [0, 1]. For k ≥ 1, the initial state vector x(0, k) and the input vector u(0, k) are set to zero. The control objective is to adjust the reactor temperature to the equilibrium point. Therefore, the reference trajectory is selected as y d (t) = 0.95(5t - 0.5t 2 )sin(0.2πt), t ∈ [0, 10] min.

[0202] Applying step S7 with the H∞ performance bound γ = 2.5, the parameters of the fuzzy iterative learning controller can be obtained as

[0203] K1 = [-60.2317 -9.1955 9.1477];

[0204] K2 = [6.6190 -9.4438 9.1757];

[0205] K3 = [-28.3136 -9.4766 8.8549];

[0206] K4 = [-28.3535 -9.1338 9.4011].

[0207] To evaluate the tracking performance of the batch, the root mean square (RMS) is introduced

[0208]

[0209] where H is the number of sampled data during the test time, and H is taken as 200, and e(t, k) represents the error in the k-th batch.

[0210] Obviously, the above embodiments of the present invention are merely examples for clearly explaining the present invention, and are not intended to limit the implementation manners of the present invention. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to enumerate all the implementation manners here. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the claims of the present invention.

Claims

1. A robust control method for the non-linear batch process of a stirred tank reactor, characterized in that, It includes the following steps: S1. Establish the kinetic equation of the stirred tank reactor model; The kinetic equation of the stirred tank reactor model includes the rate of change of the concentration of substance A The rate of change of temperature Specifically: Among them, is the rate of change of the concentration of substance A, is the rate of change of temperature, C A is the current concentration of substance A, C Af represents the concentration of substance A when feeding into the stirred tank reactor, q is the feed flow rate, V is the recycle volume, k0 is the reaction rate constant, E0 / R0 is the ratio of the Arrhenius activation energy to the gas constant, T represents the reactor temperature, T c is the coolant temperature, T f is the temperature at feeding, -ΔH is the heat effect of the reaction, that is, the change in heat released or absorbed by the exothermic reaction, ρ is the density of the substance, C p is the specific heat capacity at constant pressure of the substance, U represents the product of the heat transfer coefficient and the surface area of the reactor, V ρ is the product of the effective volume of the stirred tank and the material density, A is the surface area of the stirred tank reactor; S2. Based on the kinetic equation of the stirred tank reactor model, establish the state - space model of the stirred tank reactor; The expression of the state - space model of the stirred tank reactor is: where \(x(t,k)\in\mathbb{R}\) n , \(u(t,k)\in\mathbb{R}\) m and \(y(t,k)\in\mathbb{R}\) l represent the state, input, and output vectors at the instant \(t\) of the \(k\)-th batch, respectively; \(T\) d is the time period of a batch; \(f[x(t,k),u(t,k)]\) and \(g[x(t,k)]\) represent nonlinear functions with appropriate dimensions; S3. According to the state - space model of the stirred tank reactor, discretize it to obtain the discrete - linear model of the stirred tank reactor; The expression of the discrete - linear model is as follows: In the expression, θ(t,k) = [θ1(t,k), θ2(t,k),..., θ p (t,k)] represents the premise variable vector, Mij (i = 1, 2,..., r; j = 1, 2,..., p) are fuzzy sets, r is the number of fuzzy rules, and p is the number of premise variables; With \(x(0,k) = x_0\) as the initial condition for each batch, \(w(t,k)\) represents the perturbation belonging to the \(L_2\) space, and \(\{A i ,B i ,C i \}\) consists of system matrices with the same dimension; Assume that the output matrices of all fuzzy subsystems are constant, i.e., C1 = C2 =... = C r ; {ΔA i (t), ΔB i (t)} represents the form of the uncertainty term as: [ΔA i (t)ΔB i (t)] = EΔ(t)[F Ai F Bi , Δ(t) T Δ(t) ≤ I where E, F Ai and F Bi are known real constant matrices of dimensions, and Δ(t) represents an uncertain perturbation depending on time t; Express μ i (θ(t,k)) as the normalized fuzzy basis function of the inference fuzzy set which is defined as: Among them, M ij (θ j (t, k)) is the membership degree of θ ij in M j (t, k); S4. Based on the discrete - linear model of the stirred tank reactor, construct the fuzzy system of the stirred tank reactor; Step S4 specifically is: Represent the non-zero balance as and introduce and y = x2 as system state, input, and output variables; Set the sampling time to 0.05 min, and obtain a following fuzzy system by considering uncertainties and non-repetitive disturbances. The fuzzy system matrix is: S5. Based on the fuzzy system of the stirred tank reactor, establish the closed - loop two - dimensional fuzzy system of the stirred tank reactor; Step S5 includes: Define the tracking error of batch k: e(t,k) = y d (t) - y(t,k) Among them, y d (t) is the reference trajectory vector, and the tracking error e(t, k) is used to adjust the input vector of the next batch, so that the actual output y(t, k) gradually approaches the reference trajectory vector; The expression of the closed - loop two - dimensional fuzzy system of the stirred tank reactor is: S6. Analyze the performance of the closed - loop two - dimensional fuzzy system of the stirred tank reactor to obtain the performance analysis result of the closed - loop two - dimensional fuzzy system of the stirred tank reactor; S7. Based on the performance analysis result of the closed - loop two - dimensional fuzzy system of the stirred tank reactor, optimize the closed - loop two - dimensional fuzzy system of the stirred tank reactor.

2. The robust control method for the non-linear batch process of the stirred tank reactor according to claim 1, characterized in that Step S6 is specifically: S61. Set a set of conditions according to the given scalar and symmetric matrix for the performance analysis of the two - dimensional closed - loop fuzzy system; S62. Set the fuzzy Lyapunov function and calculate its value along the system trajectory; S63. Review the conditions set in step S61 and determine a set of values that meet the set conditions; S64. Sum up this set of values and confirm that the function decreases along the state trajectory by comparison; S65. For any non - zero value, check the two - dimensional H - infinity performance under the zero - boundary condition.

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