CN108227494B - Nonlinear batch process 2D optimal constraint fuzzy fault-tolerant control method - Google Patents

Abstract

The purpose of the invention is to improve the control performance and tracking performance of the control method in the nonlinear batch process, and propose a 2D optimal constraint fuzzy fault-tolerant controller design method for the nonlinear batch process. The invention establishes a 2D T-S fuzzy state space model through the nonlinear and two-dimensional characteristics of the batch process, further combines the system state error and output error, and uses the Roesser model to transform the dynamic model of the original system into a prediction form. The closed-loop fault fuzzy system model transforms the design constraint fuzzy iterative learning fault-tolerant control law into a deterministic constraint update law; according to the designed infinite optimization performance index and the Lyapunov stability theory of the 2D system, the guarantee is given in the form of linear matrix inequality (LMI) constraints Real-time online design of robust asymptotically stable fuzzy fault-tolerant update law for closed-loop systems. The invention solves the problem that the system model is difficult to handle under nonlinear conditions, and ensures that the system can still run smoothly under the worst condition and has the best tracking performance.

Description

2D Optimal Constrained Fuzzy Fault Tolerant Control Method for Nonlinear Batch Processes

technical field

The invention belongs to the advanced control field of industrial processes, and relates to a 2D optimal constraint fuzzy fault-tolerant control method for a nonlinear batch process.

Background technique

As one of the production methods, there are roughly two types of system descriptions for the batch process, one is linear and the other is nonlinear. In the early stage, most of the control of batch process was directly aimed at the linear model. However, in the actual industrial process, the batch process itself has strong nonlinear characteristics, and there is a large mismatch between the linear model and the actual process. It is difficult to achieve the best control effect in practical application. There are certain difficulties in directly dealing with nonlinear systems. For this reason, a new model is needed to approximate the nonlinear system.

With the increase of production scale and the complexity of production steps, the uncertainty in actual production has become increasingly prominent, which not only affects the efficient and stable operation of the system, but even threatens the quality of products. Moreover, these complex operating conditions correspondingly increase the probability of system failure. Among them, the actuator failure is a common failure, which will affect the operation of the process and reduce the control performance, and even endanger personal safety. Although control methods such as iterative learning reliable fault-tolerant control appear in the batch processing process, it can solve the control problem that the system still runs stably when the actuator fails. However, for high-precision equipment, the probability of failure is extremely low. If there is no failure, using reliable control will result in a waste of resources. If things go on like this, the cost will also increase, which is obviously not in line with energy conservation and emission reduction. Environmental concept. In the event of a serious failure, the reliable control law may completely lose its control function. In this case, it is very likely that the system will collapse, resulting in heavy property damage and casualties.

In addition, although the robust iterative learning and reliable control strategy adopted at this stage can effectively resist the influence of uncertainties and faults in the production process, ensure the stability of the system, and maintain the control performance of the system, the control law is Based on the solution of the whole production process, it is obtained that the control effect belongs to the optimal control covering the whole world, that is, the same control law is used from beginning to end. However, in actual operation, under the influence of disturbance and fault, the system state cannot completely change according to the obtained control law; if the system state at the current moment deviates from the set value to a certain extent, the same Control law, with the passage of time, the deviation of the system state will increase, and the current robust iterative learning and reliable control method cannot solve the problem of system state deviation, which will inevitably lead to poor system stability and control performance. Impact. In addition, for control law design and system output, the existing literature does not consider constraints, but in the actual production process, constraints must be considered.

Model Predictive Control (MPC) can well meet the needs of real-time update and correction of the control law, and obtain the optimal control law at each moment by means of "rolling optimization" and "feedback correction" to ensure that the system state can follow as much as possible. The set trajectory runs. However, most of the existing technologies use a one-dimensional control law in an infinite time domain. There is a lack of "learning" process between batches, and the control effect is not improved with the increment of batches; there is another method that only considers batches The process of "learning" between times, this method cannot realize the control problem of intermittent process with uncertain initial value. Obviously, the discussion of infinite time-domain constrained optimization problems for systems with uncertainties and faults needs to be further discussed. Therefore, it is urgent to propose a new control method to make up for the deficiencies of the existing methods, so as to achieve the goals of energy saving, cost reduction, and even the occurrence of accidents that endanger personal safety in the batch production process.

Most of the current predictive control technologies design control laws in one-dimensional direction, only consider the time direction or batch direction, and only consider the time direction, so that each batch is only a simple repetition, and the control performance cannot be improved with the increase of batches. ; Only considering the batch direction can not realize the control problem of batch process with uncertain initial value. Although there are a few results considering time and batch direction, there are no good research results for nonlinear and actuator failures.

Therefore, in order to solve the above problems, respond to the call for energy saving and emission reduction in the production process, and ensure the control performance of the system, it is extremely necessary to propose a 2D fuzzy constraint fault-tolerant control method for nonlinear batch process optimization in infinite time domain.

SUMMARY OF THE INVENTION

In order to solve the above existing technical problems, the present invention provides a 2D optimal fuzzy constraint fault-tolerant control method for a nonlinear batch process. A 2D fuzzy iterative learning control law for nonlinear infinite time domain optimization is designed for batch process models with nonlinear disturbances and actuator failures. Using this design method to design the control law can not only ensure the smooth operation of the system when a fault occurs, so as to achieve the goals of energy saving, consumption reduction and cost reduction, and even reduce the occurrence of accidents that endanger personal safety.

The purpose of the invention is to improve the control performance and tracking performance of the control method in the nonlinear batch process, and propose a 2D optimal constraint fuzzy fault-tolerant controller design method for the nonlinear batch process. The invention establishes a 2D T-S fuzzy state space model through the nonlinear and two-dimensional characteristics of the batch process, further combines the system state error and output error, and uses the Roesser model to convert the dynamic model of the original system into a closed-loop fault expressed in the form of prediction The system model transforms the design constraint iterative learning fault-tolerant control law into a deterministic constraint update law; according to the designed infinite optimization performance index and the Lyapunov stability theory of the 2D system, it is given in the form of Linear Matrix Inequality (LMI) constraints to ensure the robustness of the closed-loop system Asymptotically stable fuzzy fault-tolerant update law real-time online design. The present invention is devoted to the design of a fuzzy optimal fault-tolerant controller in the event of a failure of a nonlinear batch process executor. Firstly, it solves the problem that the nonlinear system model is difficult to deal with, and secondly, it solves the design of the constrained fault-tolerant control law in the event of failure.

The present invention is achieved through the following technical solutions:

A 2D optimal constrained fuzzy fault-tolerant control method for nonlinear batch processes. The specific steps of the method are:

Step 1. Establish a nonlinear batch process equivalent 2D-Rosser error augmentation model:

Step 1.1 Considering the actuator gain fault, according to the nonlinear and two-dimensional characteristics of the batch process, a 2D T-S fuzzy fault state space model is established, which is expressed by equation (1):

And its input and output constraints satisfy:

分别是输入、实际输出的上界约束值，t，k分别表示在批次内的运行时刻与批次；T_p表示一个批次运行的总时间；p为前提变量数目；r为模糊规则数目；A_i,B_i,C_i为相应模糊规则i下的系统状态矩阵、系统输入矩阵、系统输出矩阵；x(0,k)为第k个批次的初始状态；M_ij为模糊集，M_ij(x_j(t,k))为x_j(t,k)属于M_ij的隶属度；

由

可得

Among them, x(t,k), y(t,k), u(t,k), ω(t,k) respectively represent the state of the system, the output of the system, the control input of the system and the unknown disturbance;

are the upper bound constraint values of input and actual output, respectively, t, k represent the running time and batch in the batch, respectively; T _p represents the total running time of a batch; p is the number of prerequisite variables; r is the number of fuzzy rules ; A _i , B _i , C _i are the system state matrix, system input matrix, and system output matrix under the corresponding fuzzy rule i; x(0,k) is the initial state of the kth batch; M _ij is the fuzzy set, M _ij (x _j (t, k)) is the degree of membership that x _j (t, k) belongs to M _ij ;

Depend on

Available

Different α values are defined to represent different fault types of the actuator. When α>0, it means a partial failure; when α=0, it means a complete failure, which does not involve the problem of the optimal controller;

For the partial failure of the actuator, α>0 must satisfy the following form:

和

是已知的常数；In the formula,

and

is a known constant;

Step 1.2 Design a 2D iterative learning controller u(t,k), as shown in equation (3):

It can be seen that, to design u(t,k), we only need to design the update law r(t,k) at time t in k batches, so that the system output y(t,k) can track the given expected output y _d (t ,k);

Step 1.3 Define the state error and output error in the batch direction as follows:

δ(x(t,k))=x(t,k)-x(t,k-1) (4a)

则(1)式转化为等价误差模型为式(5)：make

Then formula (1) is transformed into the equivalent error model as formula (5):

δ(ω(t,k))＝ω(t,k)-ω(t,k-1)，in,

δ(ω(t,k))=ω(t,k)-ω(t,k-1),

I为适维的单位矩阵；并设

则上述模型表示为：δ(hi(x(t, _k )))=hi(x(t, _k ))-hi(x(t, _k -1)),

I is a suitable dimensional identity matrix; and set

Then the above model is expressed as:

分别为适维向量的水平和垂直状态分量，Z(t,k)是系统的被控输出；in,

are the horizontal and vertical state components of the dimensional vector, respectively, and Z(t,k) is the controlled output of the system;

Step 2. Design an iterative learning control law for the batch process model with disturbance and actuator failure:

Step 2.1 Design a 2D predictive fault-tolerant controller for the above model (5) to achieve the minimum optimal control under the maximum disturbance and maximum fault, even if the model (5) reaches a steady state and satisfies the following robust performance indicators at each moment:

limit:

代表输入增量；And Q(Q>0) and R(R>0) are weighted matrices of appropriate dimensions, r(t+i|t,k) is the predicted value of the input at time t for time t+i, and r(t, k)=r(t|t,k),

represents the input increment;

Step 2.2 Define the state feedback control law to make the system achieve quadratic stability. The selected update law is:

Then the closed-loop model of (5) is expressed as:

则其闭环预测模型表示为：in,

Then its closed-loop prediction model is expressed as:

Step 2.3 Use the 2D Lyapunov function to prove the stability of the system, and define the Lyapunov function as:

Among them, M>0

Step 2.4 Model (8c) can still run smoothly within the allowable range of faults, and must meet:

(1) 2D Lyapunov function inequality constraints:

矩阵Y_i,Y_j(i＝1,2,...r,)，标量ε_i,ε_j,γ,θ＞0，0＜α＜1,0＜μ＜1，可使得下面的矩阵不等式成立：(2) For a given positive semi-definite symmetric matrix R, Q, there is a positive definite symmetric matrix M=diag{M ^h , M ^v }, and the semi-positive definite symmetric matrix

Matrices Y _i , Y _j (i=1, 2,...r,), scalars ε _i , ε _j , γ, θ>0, 0<α<1, 0<μ<1, can make the following matrix The inequality holds:

且

and

且

and

in,

The robust update law gain is:

将其带入u(t,k)＝u(t,k-1)+r(t,k)，便可得到2D约束迭代学习控制律设计u(t,k)，在下一时刻，不断重复步骤2.4，继续求解新的控制量u(t,k)，并依次循环。Therefore, the further update law is expressed as:

Bring it into u(t,k)=u(t,k-1)+r(t,k), the 2D constrained iterative learning control law design u(t,k) can be obtained, and at the next moment, repeating Step 2.4, continue to solve the new control variable u(t,k), and cycle in turn.

Compared with the prior art, the beneficial effects of the present invention are:

This method designs a fuzzy fault-tolerant iterative learning control law on the basis of the control system model with nonlinearity, disturbance and fault, introduces state error and output error, and uses the Roesser model to transform the dynamic model of the original system into a predictive model. The closed-loop system model transforms the designed fuzzy fault-tolerant iterative learning control law into a deterministic update law; according to the designed infinite optimization performance index and the Lyapunov stability theory of the 2D system, it is given in the form of Linear Matrix Inequality (LMI) constraints to ensure the robustness of the closed-loop system The asymptotically stable update law is designed online in real time, which can effectively solve the problem that the nonlinear system model is difficult to handle and the design of the constrained fuzzy optimal fault-tolerant control law in the event of a fault. It effectively solves the problem that the control performance of the nonlinear batch process cannot be improved with the increment of the batch, and realizes that the system can be optimized in real time within the range of variable constraints regardless of whether there is a fault. Worst-case smooth operation and optimal tracking performance. Ultimately, it can save energy and reduce consumption, reduce costs, and reduce the occurrence of accidents that endanger personal safety.

Detailed ways

The present invention will be further described below with reference to specific embodiments.

A 2D optimal constrained fuzzy fault-tolerant control method for nonlinear batch processes. The specific steps of the method are:

Step 1. Establish a nonlinear batch process equivalent 2D-Rosser error augmentation model:

Step 1.1 Considering the actuator gain fault, according to the nonlinear and two-dimensional characteristics of the batch process, a 2D T-S fuzzy fault state space model is established, which is expressed by equation (1):

And its input and output constraints satisfy:

分别是输入、实际输出的上界约束值，t，k分别表示在批次内的运行时刻与批次；T_p表示一个批次运行的总时间；p为前提变量数目；r为模糊规则数目；A_i,B_i,C_i为相应模糊规则i下的系统状态矩阵、系统输入矩阵、系统输出矩阵；x(0,k)为第k个批次的初始状态；M_ij为模糊集，M_ij(x_j(t,k))为x_j(t,k)属于M_ij的隶属度；

由

可得

Among them, x(t,k), y(t,k), u(t,k), ω(t,k) respectively represent the state of the system, the output of the system, the control input of the system and the unknown disturbance;

are the upper bound constraint values of input and actual output, respectively, t, k represent the running time and batch in the batch, respectively; T _p represents the total running time of a batch; p is the number of prerequisite variables; r is the number of fuzzy rules ; A _i , B _i , C _i are the system state matrix, system input matrix, and system output matrix under the corresponding fuzzy rule i; x(0,k) is the initial state of the kth batch; M _ij is the fuzzy set, M _ij (x _j (t,k)) is the degree of membership that x _j (t, k) belongs to M _ij ;

Depend on

Available

Different α values are defined to represent different fault types of the actuator. When α>0, it means a partial failure; when α=0, it means a complete failure, which does not involve the problem of the optimal controller;

For the partial failure of the actuator, α>0 must satisfy the following form:

是已知的常数；where α ( α ≤ 1) and

is a known constant;

Step 1.2 Design a 2D iterative learning controller u(t,k), as shown in equation (3):

It can be seen that, to design u(t,k), we only need to design the update law r(t,k) at time t in k batches, so that the system output y(t,k) can track the given expected output y _d (t ,k);

Step 1.3 Define the state error and output error in the batch direction as follows:

δ(x(t,k))=x(t,k)-x(t,k-1) (4a)

则(1)式转化为等价误差模型为式(5)：make

Then formula (1) is transformed into the equivalent error model as formula (5):

δ(ω(t,k))＝ω(t,k)-ω(t,k-1)，in,

δ(ω(t,k))=ω(t,k)-ω(t,k-1),

为适维的单位矩阵；并设

则上述模型表示为：δ(hi(x(t, _k )))=hi(x(t, _k ))-hi(x(t, _k -1)),

is a suitable dimensional identity matrix; and set

Then the above model is expressed as:

分别为适维向量的水平和垂直状态分量，Z(t,k)是系统的被控输出；in,

are the horizontal and vertical state components of the dimensional vector, respectively, and Z(t,k) is the controlled output of the system;

Step 2. Design an iterative learning control law for the batch process model with disturbance and actuator failure:

Step 2.1 Design a 2D predictive fault-tolerant controller for the above model (5) to achieve the minimum optimal control under the maximum disturbance and maximum fault, even if the model (5) reaches a steady state and satisfies the following robust performance indicators at each moment:

limit:

代表输入增量；And Q(Q>0) and R(R>0) are weighted matrices of appropriate dimensions, r(t+i|t,k) is the predicted value of the input at time t for time t+i, and r(t, k)=r(t|t,k),

represents the input increment;

Step 2.2 Define the state feedback control law to make the system achieve quadratic stability. The selected update law is:

Then the closed-loop model of (5) is expressed as:

则其闭环预测模型表示为：in,

Then its closed-loop prediction model is expressed as:

Step 2.3 Use the 2D Lyapunov function to prove the stability of the system, and define the Lyapunov function as:

Among them, M>0

Step 2.4 Model (8c) can still run smoothly within the allowable range of faults, and must meet:

(1) 2D Lyapunov function inequality constraints:

矩阵Y_i,Y_j(i＝1,2,...r,)，标量ε_i,ε_j,γ,θ＞0，0＜α＜1,0＜μ＜1，可使得下面的矩阵不等式成立：(2) For a given positive semi-definite symmetric matrix R, Q, there is a positive definite symmetric matrix M=diag{M ^h , M ^v }, and the semi-positive definite symmetric matrix

Matrices Y _i , Y _j (i=1, 2,...r,), scalars ε _i , ε _j , γ, θ>0, 0<α<1, 0<μ<1, can make the following matrix The inequality holds:

且

and

且

and

in,

The robust update law gain is:

将其带入u(t,k)＝u(t,k-1)+r(t,k)，便可得到2D约束迭代学习控制律设计u(t,k)，在下一时刻，不断重复步骤2.4，继续求解新的控制量u(t,k)，并依次循环。Therefore, the further update law is expressed as:

Bring it into u(t,k)=u(t,k-1)+r(t,k), the 2D constrained iterative learning control law design u(t,k) can be obtained, and at the next moment, repeating Step 2.4, continue to solve the new control variable u(t,k), and cycle in turn.

Example

Consider a nonlinear continuous stirred tank:

Among them, C _A is the concentration of A in the irreversible reaction (A→B) process; T is the temperature of the reactor; T _C is the temperature of the cooling flow, as the manipulated variable q=100 (L/min), V=100 (L) , C _Af =1(mol/L), T _f =400(K), ρ=1000(g/L), C _P =1(J/gK), k ₀ =4.71×10 ⁸ (min ⁻¹ ) , E/R=8000 (K), ΔH=-2×10 ⁵ (J/mol), UA=1×10 ⁵ (J/minK). The variable range is limited to 200≤T _C ≤450(K), 0.01≤C _A ≤1(mol/L), 250≤T≤500(K); y(t,k)=Cx(t,k) is the output . The above nonlinear model transforms into:

in,

C=[1 0]

The control objective is to make the reactor temperature follow a given curve:

The simulation was run in 50 batches, each running 600 steps. The evaluation index uses the sum of square root error (RSSE) to evaluate the control effect.

The calculated initial stage controller gains are:

K1=[-0.0905 0.0041 0.5031];

K2=[0.1120 0.0021 0.5799];

K3 = [0.1344 -0.0078 0.2622];

K4=[0.0260 0.0042 0.4630].

This method designs a fuzzy iterative learning control law for the nonlinear batch process with disturbances and faults, and effectively solves the problem that the nonlinear system model is difficult to handle and the design of the constrained fuzzy optimal fault-tolerant control method in the event of a fault. It effectively solves the problem that the control performance of the nonlinear batch process cannot be improved with the increment of the batch, realizes that the system can be optimized in real time within the range of variable constraints regardless of whether there is a fault or not, improves the control performance of the system, and ensures that the system can Worst-case smooth operation and optimal tracking performance. Ultimately, it can save energy and reduce consumption, reduce costs, and reduce the occurrence of accidents that endanger personal safety.

Claims (1)

1. The nonlinear batch process 2D optimal constraint fuzzy fault-tolerant control method is characterized by comprising the following steps:

consider a non-linear continuous stirred tank:

wherein, C_AThe concentration of A in the irreversible reaction A → B process; t is the temperature of the reaction kettle; t is_CFor the cooling flow temperature, the manipulated variables q are 100L/min, V100L, C_AF＝1mol/L，T_f＝400K，ρ＝1000g/L，C_p＝1J/gK，k₀＝4.71×10⁸min^-1，E/R＝8000K，ΔH＝-2×10⁵J/mol，UA＝1×10⁵J/minK; the variable range is limited to 200 ≦ T_C≤450K，0.01≤C_AT is not less than 1mol/L, not less than 250 and not more than 500K; y (T, k) is output, x (T, k) is system state, represented by T and C_AComposition, representing temperature and concentration; the above nonlinear model translates into:

wherein,

C＝[1 0]

the control objective is to let the reactor temperature follow a given curve:

the simulation is carried out for 50 batches, each batch is operated for 600 steps, and the evaluation index uses the square sum root error RSSE for evaluating the control effect;

the initial phase controller gain calculated is:

K1＝[-0.0905 0.0041 0.5031]；

K2＝[0.1120 0.0021 0.5799]；

K3＝[0.1344 -0.0078 0.2622]；

K4＝[0.0260 0.0042 0.4630]；

the method comprises the following specific steps:

step 1, establishing an equivalent 2D-Rosser error augmentation model of a nonlinear batch process:

step 1.1, considering actuator gain faults, and establishing a 2D T-S fuzzy fault state space model according to the nonlinear and two-dimensional characteristics of the batch process, wherein the model is represented by formula (1):

and the input and output constraints thereof meet:

wherein x (t, k), y (t, k), u (t, k), ω (t, k) respectively represent the state of the system, the output of the systemControl inputs to the system and unknown disturbances;

the upper bound constraint values of input and actual output are respectively, and t and k respectively represent the running time and the batch in the batch; t is_pRepresents the total time of a batch run; p is the number of preconditions; r is the number of fuzzy rules; a. the_i,B_i,C_iA system state matrix, a system input matrix and a system output matrix under the corresponding fuzzy rule i are obtained; x (0, k) is the initial state of the kth batch; m_ijFor fuzzy sets, M_ij(x_j(t, k)) is x_j(t, k) is M_ijDegree of membership of;

by

Can obtain the product

Defining different alpha values to indicate different fault types of the actuator, and indicating partial failure fault when alpha is more than 0; when alpha is 0, the failure is completely failed, and the problem of an optimal controller is not involved;

for partial actuator failure, α > 0 should satisfy the following form:

wherein α is not more than 1 and

is a known constant;

step 1.2, designing a 2D iterative learning controller u (t, k), as shown in formula (3):

therefore, u (t, k) is designed, and only k batches of the updating law r (t, k) at t moment are designed to realize that the system output y (t, k) tracks the given expected output y_d(t,k)；

Step 1.3 defines the state error and output error in the batch direction as follows:

δ(x(t,k))＝x(t,k)-x(t,k-1) (4a)

order to

Then the equation (1) is converted into an equivalent error model which is equation (5):

wherein,

δ(ω(t,k))＝ω(t,k)-ω(t,k-1)，

δ(h_i(x(t,k)))＝h_i(x(t,k))-h_i(x(t,k-1))，

i is an identity matrix; and is provided with

The above model is then expressed as:

wherein,

the horizontal and vertical state components of the adaptive vector, respectively, and Z (t, k) is the controlled output of the system;

step 2, designing an iterative learning control law for batch process models with interference and actuator faults:

step 2.1, a 2D predictive fault-tolerant controller is designed for the model (5) to achieve minimum optimal control under the maximum interference and the maximum fault, even if the model (5) achieves a steady state and meets the following robust performance indexes at each moment:

and (3) limiting:

and Q > 0 and R > 0 are weighting matrices, R (t + i | t, k) is a predicted value input at time t to time t + i, and R (t, k) ═ R (t | t, k),

represents an input increment;

step 2.2, defining a state feedback control law to enable the system to achieve secondary stability, wherein the selected updating law is as follows:

the closed-loop model of (5) is expressed as:

wherein,

its closed-loop prediction model is represented as:

step 2.3, the stability of the system is proved by using a 2D Lyapunov function, wherein the Lyapunov function is defined as follows:

wherein M > 0

t≥0；

Step 2.4 the model (8c) can still run stably within the fault tolerance range, and the following requirements must be met:

(1) the 2D lyapunov function is inequality constrained:

(2) for a given semi-positive definite symmetric matrix R, Q, there is a positive definite symmetric matrix M ═ diag { M^h,M^v}, semi-positive definite symmetric matrix

Matrix Y_i，Y_jWhere i 1,2, r, a scalar e_i,ε_jγ, θ > 0, 0 < α < 1,0 < μ < 1, such that the following matrix inequality holds:

and is

And is

Wherein,

the robust update law gain is:

therefore, the further update law is represented as:

and (3) the value is substituted into u (t, k) ═ u (t, k-1) + r (t, k), so that a 2D constraint iterative learning control law design u (t, k) can be obtained, the step 2.4 is continuously repeated at the next moment, the new controlled variable u (t, k) is continuously solved, and the steps are sequentially circulated.