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

Nonlinear batch process 2D optimal constraint fuzzy fault-tolerant control method Download PDF

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CN108227494B
CN108227494B CN201810009893.4A CN201810009893A CN108227494B CN 108227494 B CN108227494 B CN 108227494B CN 201810009893 A CN201810009893 A CN 201810009893A CN 108227494 B CN108227494 B CN 108227494B
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罗卫平
王立敏
余维燕
王鹏
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Hainan Normal University
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Abstract

The invention aims to improve the control performance and the tracking performance of a control method in a nonlinear batch process, and provides a design method of a 2D optimal constraint fuzzy fault-tolerant controller in the nonlinear batch process. According to the method, a 2D T-S fuzzy state space model is established through the nonlinear and two-dimensional characteristics of a batch process, a system state error and an output error are further combined, a dynamic model of an original system is converted into a closed-loop fault fuzzy system model represented in a prediction mode through a Roesser model, and a design constraint fuzzy iterative learning fault-tolerant control law is converted into a determination constraint updating law; according to designed infinite optimization performance indexes and a 2D system Lyapunov stability theory, a fuzzy fault-tolerant update law real-time online design for ensuring the stability of the robustness asymptotic of a closed-loop system is given in a Linear Matrix Inequality (LMI) constraint form. The invention well solves the problem that a system model is difficult to process under the nonlinearity, ensures that the system can still run stably under the worst condition and has the optimal tracking performance.

Description

Nonlinear batch process 2D optimal constraint fuzzy fault-tolerant control method
Technical Field
The invention belongs to the field of advanced control of industrial processes, and relates to a nonlinear batch process 2D optimal constraint fuzzy fault-tolerant control method.
Background
As a batch process, which is one of the production methods, there are roughly two types of system descriptions, one is linear and the other is nonlinear. The early control of the intermittent process is mostly directly aimed at the linear model, however, in the actual industrial process, the intermittent process has strong nonlinear characteristics, and the linear model and the actual process have a large mismatch problem. Making it difficult to achieve optimal control in practical applications. Direct processing of non-linear systems presents certain difficulties. For this purpose, a new model is used to approximate the nonlinear system.
Along with the increase of the production scale and the increase of the complexity of the production steps, the uncertainty existing in the actual production is increasingly prominent, so that the high-efficiency and smooth operation of the system is influenced, and even the quality of the product is threatened. These complex operating conditions, in turn, increase the probability of system failure. Among them, actuator failure is a common failure, which affects the operation of the process and reduces the control performance, even endangers the personal safety. Although control methods such as iterative learning reliable fault-tolerant control and the like appear in the batch processing process, the control problem that the system still stably runs when an actuator fails can be well solved. However, for equipment with high precision, the possibility of failure occurrence is extremely low, if no failure occurs, the equipment is reliably controlled to use, so that resource waste is caused, the cost is increased in the past, and the environment-friendly concept of energy conservation and emission reduction is obviously not met. In the event of a serious fault, the reliable control law may completely lose control, which may lead to system breakdown, resulting in significant property loss and casualties.
In addition, although the robust iterative learning reliable control strategy adopted at the present stage can effectively resist the influence caused by uncertainty and faults in the production link, ensure the stability of the system and maintain the control performance of the system, the control law is obtained by solving based on the whole production process and belongs to global-covering optimization control in the control effect, namely the same control law is used all the time. However, in actual operation, under the influence of interference and faults, the system state cannot change completely according to the obtained control law action; if the system state at the current moment deviates from the set value to a certain extent, the same control law is still continuously adopted, the deviation of the system state is increased gradually along with the lapse of time, and the existing robust iterative learning reliable control method cannot solve the problem of the deviation of the system state, which inevitably has adverse effects on the stable operation and the control performance of the system. In addition, the existing literature does not consider the constraint problem for control law design and system output, and the constraint must be considered in the actual production process.
The Model Predictive Control (MPC) can well meet the requirement of real-time update and correction of the control law, and the optimal control law at each moment is obtained through rolling optimization and feedback correction, so that the system state can be ensured to run along the set track as much as possible. However, in the prior art, a one-dimensional infinite time domain control law is mostly adopted, a learning process is lacked among batches, and the control effect is not improved along with the increment of the batches; there is also a process that only considers batch-to-batch "learning" and this approach does not achieve the control problem of an intermittent process where the initial value is uncertain. It is clear that the discussion of the infinite time domain constraint optimization problem for systems with uncertainty and faults is left to be further. Therefore, a new control method is urgently needed to make up the defects of the existing method so as to achieve the aims of saving energy, reducing consumption, reducing cost, even reducing the occurrence of accidents which harm human safety and the like in the batch production process.
Most of the existing prediction control technologies design a control law in a one-dimensional direction, only the time direction or the batch direction is considered, only the time direction is considered, so that each batch is only simply repeated, and the control performance cannot be improved along with the increment of the batch; the control problem of the intermittent process with uncertain initial values cannot be realized only by considering the batch direction. Although there are few results considering time and batch direction, there is no good research result for the situations of nonlinearity, actuator failure and the like.
Therefore, in order to solve the problems, respond to calls for energy conservation, emission reduction and the like in the production process and ensure the control performance of the system, it is necessary to provide a 2D fuzzy constraint fault-tolerant control method for infinite time domain optimization in a nonlinear batch process.
Disclosure of Invention
In order to solve the technical problems, the invention provides a nonlinear batch process 2D optimal fuzzy constraint fault-tolerant control method. And designing a nonlinear infinite time domain optimized 2D fuzzy iterative learning control law for batch process models with nonlinear interference and actuator faults. The control law is designed by the design method, so that the system can be ensured to stably run when a fault occurs, the aims of saving energy, reducing consumption, reducing cost and the like are fulfilled, and the aims of reducing the occurrence of personal safety hazard and the like can be fulfilled.
The invention aims to improve the control performance and the tracking performance of a control method in a nonlinear batch process, and provides a 2D optimal constraint fuzzy fault-tolerant controller design method for the nonlinear batch process. According to the method, a 2D T-S fuzzy state space model is established through the nonlinear and two-dimensional characteristics of a batch process, a system state error and an output error are further combined, a dynamic model of an original system is converted into a closed-loop fault system model represented in a prediction mode through a Roesser model, and a design constraint iterative learning fault-tolerant control law is converted into a determination constraint updating law; according to designed infinite optimization performance indexes and a 2D system Lyapunov stability theory, a fuzzy fault-tolerant update law real-time online design for ensuring the stability of the robustness asymptotic of a closed-loop system is given in a Linear Matrix Inequality (LMI) constraint form. The invention is directed to the design of a fuzzy optimal fault-tolerant controller under the condition that an actuator of a nonlinear batch process fails. The control algorithm can achieve the aims of saving energy, reducing consumption, reducing cost, reducing the occurrence of personal safety hazards and the like.
The invention is realized by the following technical scheme:
the nonlinear batch process 2D optimal constraint fuzzy fault-tolerant control 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):
Figure BDA0001539937320000041
and the input and output constraints thereof meet:
Figure BDA0001539937320000042
wherein 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;
Figure BDA0001539937320000043
upper bound of input, actual output, respectivelyConstraint values, t, k, respectively represent run time and batch within a batch; t ispRepresents the total time of a batch run; p is the number of preconditions; r is the number of fuzzy rules; a. thei,Bi,CiA 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; mijFor fuzzy sets, Mij(xj(t, k)) is xj(t, k) is MijDegree of membership of;
Figure BDA0001539937320000044
by
Figure BDA0001539937320000045
Can obtain the product
Figure BDA0001539937320000046
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:
Figure BDA0001539937320000047
in the formula (I), the compound is shown in the specification,
Figure BDA0001539937320000048
and
Figure BDA0001539937320000049
is a known constant;
step 1.2, designing a 2D iterative learning controller u (t, k), as shown in formula (3):
Figure BDA00015399373200000410
from this, u (t, k) is designed onlyThe k-batch time-t update law r (t, k) needs to be designed to achieve that the system output y (t, k) tracks the given desired output yd(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)
Figure BDA0001539937320000051
order to
Figure BDA0001539937320000052
Then the equation (1) is converted into an equivalent error model which is equation (5):
Figure BDA0001539937320000053
wherein the content of the first and second substances,
Figure BDA0001539937320000054
δ(ω(t,k))＝ω(t,k)-ω(t,k-1)，
Figure BDA0001539937320000055
δ(hi(x(t,k)))＝hi(x(t,k))-hi(x(t,k-1))，
Figure BDA0001539937320000056
Figure BDA0001539937320000057
i is an adaptive identity matrix; and is provided with
Figure BDA0001539937320000058
Figure BDA0001539937320000059
The above model is then expressed as:
Figure BDA00015399373200000510
wherein the content of the first and second substances,
Figure BDA00015399373200000511
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:
Figure BDA0001539937320000061
Figure BDA0001539937320000062
and (3) limiting:
Figure BDA0001539937320000063
and Q (Q > 0) and R (R > 0) are weighting matrices of appropriate dimensions, R (t + i | t, k) is the predicted value input at time t to t + i, and R (t, k) ═ R (t | t, k),
Figure BDA0001539937320000064
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:
Figure BDA0001539937320000065
the closed-loop model of (5) is expressed as:
Figure BDA0001539937320000066
wherein the content of the first and second substances,
Figure BDA0001539937320000067
its closed-loop prediction model is represented as:
Figure BDA0001539937320000068
step 2.3, the stability of the system is proved by using a 2D Lyapunov function, wherein the Lyapunov function is defined as follows:
Figure BDA0001539937320000071
wherein M > 0
Figure BDA0001539937320000072
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:
Figure BDA0001539937320000073
(2) for a given semi-positive definite symmetric matrix R, Q, there is a positive definite symmetric matrix M ═ diag { Mh,Mv}, semi-positive definite symmetric matrix
Figure BDA0001539937320000074
Matrix Yi,Yj(i ═ 1, 2.. r.,), scalar epsiloni,εjγ, θ > 0, 0 < α < 1,0 < μ < 1, such that the following matrix inequality holds:
Figure BDA0001539937320000075
Figure BDA0001539937320000076
Figure BDA0001539937320000077
Figure BDA0001539937320000078
and is
Figure BDA0001539937320000079
Figure BDA00015399373200000710
And is
Figure BDA00015399373200000711
Wherein the content of the first and second substances,
Figure BDA00015399373200000712
the robust update law gain is:
Figure BDA0001539937320000081
therefore, the further update law is represented as:
Figure BDA0001539937320000082
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.
Compared with the prior art, the invention has the beneficial effects that:
designing a fuzzy fault-tolerant iterative learning control law on the basis of a control system model with nonlinearity, interference and faults, introducing a state error and an output error, converting a dynamic model of an original system into a closed-loop system model represented in a prediction form by using a Roesser model, and converting the designed fuzzy fault-tolerant iterative learning control law into a determined updating law; according to designed infinite optimization performance indexes and a 2D system Lyapunov stability theory, an update law real-time online design for ensuring asymptotic stability of a closed-loop system robustness is given in a Linear Matrix Inequality (LMI) constraint form, and the problems that a system model is difficult to process under nonlinearity and a fuzzy optimal fault-tolerant control law is constrained under a fault condition are effectively solved. The method effectively solves the problem that the control performance of the nonlinear batch process cannot be improved along with the increment of the batch, realizes the real-time optimization of the system in a variable constraint range regardless of the existence of faults, improves the control performance of the system, ensures that the system can still run stably under the worst condition and has the optimal tracking performance. Finally, the purposes of saving energy, reducing consumption, reducing cost and reducing the occurrence of accidents damaging personal safety are achieved.
Detailed Description
The present invention will be further described with reference to the following specific examples.
The nonlinear batch process 2D optimal constraint fuzzy fault-tolerant control 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):
Figure BDA0001539937320000091
and the input and output constraints thereof meet:
Figure BDA0001539937320000092
wherein 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;
Figure BDA0001539937320000093
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 ispRepresents the total time of a batch run; p is the number of preconditions; r is the number of fuzzy rules; a. thei,Bi,CiA 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; mijFor fuzzy sets, Mij(xj(t, k)) is xj(t, k) is MijDegree of membership of;
Figure BDA0001539937320000094
by
Figure BDA0001539937320000095
Can obtain the product
Figure BDA0001539937320000096
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:
Figure BDA0001539937320000097
in the formula (I), the compound is shown in the specification,α(α1) and
Figure BDA0001539937320000098
is a known constant;
step 1.2, designing a 2D iterative learning controller u (t, k), as shown in formula (3):
Figure BDA0001539937320000099
thus, the design is madeu (t, k), the k batches are only required to be designed to update the law r (t, k) at time t so as to realize that the system output y (t, k) tracks the given expected output yd(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)
Figure BDA0001539937320000101
order to
Figure BDA0001539937320000102
Then the equation (1) is converted into an equivalent error model which is equation (5):
Figure BDA0001539937320000103
wherein the content of the first and second substances,
Figure BDA0001539937320000104
δ(ω(t,k))＝ω(t,k)-ω(t,k-1)，
Figure BDA0001539937320000105
δ(hi(x(t,k)))＝hi(x(t,k))-hi(x(t,k-1))，
Figure BDA0001539937320000106
is an adaptive identity matrix; and is provided with
Figure BDA0001539937320000107
The above model is then expressed as:
Figure BDA0001539937320000108
wherein the content of the first and second substances,
Figure BDA0001539937320000109
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:
Figure BDA0001539937320000111
Figure BDA0001539937320000112
and (3) limiting:
Figure BDA0001539937320000113
and Q (Q > 0) and R (R > 0) are weighting matrices of appropriate dimensions, R (t + i | t, k) is the predicted value input at time t to t + i, and R (t, k) ═ R (t | t, k),
Figure BDA0001539937320000114
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:
Figure BDA0001539937320000115
the closed-loop model of (5) is expressed as:
Figure BDA0001539937320000116
wherein the content of the first and second substances,
Figure BDA0001539937320000117
its closed-loop prediction model is represented as:
Figure BDA0001539937320000118
step 2.3, the stability of the system is proved by using a 2D Lyapunov function, wherein the Lyapunov function is defined as follows:
Figure BDA0001539937320000121
wherein M > 0
Figure BDA0001539937320000122
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:
Figure BDA0001539937320000123
(2) for a given semi-positive definite symmetric matrix R, Q, there is a positive definite symmetric matrix M ═ diag { Mh,Mv}, semi-positive definite symmetric matrix
Figure BDA0001539937320000124
Matrix Yi,Yj(i ═ 1, 2.. r.,), scalar epsiloni,εjγ, θ > 0, 0 < α < 1,0 < μ < 1, such that the following matrix inequality holds:
Figure BDA0001539937320000125
Figure BDA0001539937320000126
Figure BDA0001539937320000127
Figure BDA0001539937320000128
and is
Figure BDA0001539937320000129
Figure BDA00015399373200001210
And is
Figure BDA00015399373200001212
Wherein the content of the first and second substances,
Figure BDA00015399373200001211
the robust update law gain is:
Figure BDA0001539937320000131
therefore, the further update law is represented as:
Figure BDA0001539937320000132
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.
Examples
Consider a non-linear continuous stirred tank:
Figure BDA0001539937320000133
Figure BDA0001539937320000134
wherein, CAConcentration of A during irreversible reaction (A → B); t is the temperature of the reaction kettle; t isCFor the cooling stream temperature, the manipulated variables q are 100(L/min), V100 (L), CAf＝1(mol/L)，Tf＝400(K)，ρ＝1000(g/L)，CP＝1(J/gK)，k0＝4.71×108(min-1)，E/R＝8000(K)，ΔH＝-2×105(J/mol)，UA＝1×105(J/minK). The variable range is limited to 200 ≦ TC≤450(K)，0.01≤CAT is more than or equal to 1(mol/L) and more than or equal to 250 and less than or equal to 500 (K); y (t, k) ═ Cx (t, k) is the output. The above nonlinear model translates into:
Figure BDA0001539937320000135
Figure BDA0001539937320000136
wherein the content of the first and second substances,
Figure BDA0001539937320000137
Figure BDA0001539937320000141
Figure BDA0001539937320000142
Figure BDA0001539937320000143
Figure BDA0001539937320000144
C＝[1 0]
the control objective is to let the reactor temperature follow a given curve:
Figure BDA0001539937320000145
the simulation was performed for 50 batches, each run for 600 steps. The evaluation index uses a sum of squares root error (RSSE) for evaluating the control effect.
Figure BDA0001539937320000146
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 designs a fuzzy iterative learning control law under the condition of interference and faults in the nonlinear batch process, and effectively solves the problems that a system model is difficult to process under the nonlinear condition and the design problem of a constraint fuzzy optimal fault-tolerant control method under the condition of faults. The method effectively solves the problem that the control performance of the nonlinear batch process cannot be improved along with the increment of the batch, realizes the real-time optimization of the system in a variable constraint range regardless of the existence of faults, improves the control performance of the system, ensures that the system can still run stably under the worst condition and has the optimal tracking performance. Finally, the purposes of saving energy, reducing consumption, reducing cost and reducing the occurrence of accidents damaging personal safety are achieved.

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:
Figure FDA0003157763450000011
Figure FDA0003157763450000012
wherein, CAThe concentration of A in the irreversible reaction A → B process; t is the temperature of the reaction kettle; t isCFor the cooling flow temperature, the manipulated variables q are 100L/min, V100L, CAF＝1mol/L，Tf＝400K，ρ＝1000g/L，Cp＝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 ≦ TC≤450K，0.01≤CAT 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 CAComposition, representing temperature and concentration; the above nonlinear model translates into:
Figure FDA0003157763450000013
Figure FDA0003157763450000014
wherein the content of the first and second substances,
Figure FDA0003157763450000015
Figure FDA0003157763450000021
Figure FDA0003157763450000022
Figure FDA0003157763450000023
Figure FDA0003157763450000024
C＝[1 0]
the control objective is to let the reactor temperature follow a given curve:
Figure FDA0003157763450000025
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;
Figure FDA0003157763450000026
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):
Figure FDA0003157763450000031
and the input and output constraints thereof meet:
Figure FDA0003157763450000032
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;
Figure FDA0003157763450000033
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 ispRepresents the total time of a batch run; p is the number of preconditions; r is the number of fuzzy rules; a. thei,Bi,CiA 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; mijFor fuzzy sets, Mij(xj(t, k)) is xj(t, k) is MijDegree of membership of;
Figure FDA0003157763450000034
by
Figure FDA0003157763450000035
Can obtain the product
Figure FDA0003157763450000036
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:
Figure FDA0003157763450000037
wherein α is not more than 1 and
Figure FDA0003157763450000038
is a known constant;
step 1.2, designing a 2D iterative learning controller u (t, k), as shown in formula (3):
Figure FDA0003157763450000041
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 yd(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)
Figure FDA0003157763450000042
order to
Figure FDA0003157763450000043
Then the equation (1) is converted into an equivalent error model which is equation (5):
Figure FDA0003157763450000044
wherein the content of the first and second substances,
Figure FDA0003157763450000045
δ(ω(t,k))＝ω(t,k)-ω(t,k-1)，
Figure FDA0003157763450000046
δ(hi(x(t,k)))＝hi(x(t,k))-hi(x(t,k-1))，
Figure FDA0003157763450000047
Figure FDA0003157763450000048
i is an identity matrix; and is provided with
Figure FDA0003157763450000049
The above model is then expressed as:
Figure FDA00031577634500000410
wherein the content of the first and second substances,
Figure FDA0003157763450000051
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:
Figure FDA0003157763450000052
Figure FDA0003157763450000053
and (3) limiting:
Figure FDA0003157763450000054
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),
Figure FDA0003157763450000055
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:
Figure FDA0003157763450000056
the closed-loop model of (5) is expressed as:
Figure FDA0003157763450000057
wherein the content of the first and second substances,
Figure FDA0003157763450000061
its closed-loop prediction model is represented as:
Figure FDA0003157763450000062
step 2.3, the stability of the system is proved by using a 2D Lyapunov function, wherein the Lyapunov function is defined as follows:
Figure FDA0003157763450000063
wherein M > 0
Figure FDA0003157763450000067
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:
Figure FDA0003157763450000064
(2) for a given semi-positive definite symmetric matrix R, Q, there is a positive definite symmetric matrix M ═ diag { Mh,Mv}, semi-positive definite symmetric matrix
Figure FDA0003157763450000065
Matrix Yi，YjWhere i 1,2, r, a scalar ei,εjγ, θ > 0, 0 < α < 1,0 < μ < 1, such that the following matrix inequality holds:
Figure FDA0003157763450000066
Figure FDA0003157763450000071
Figure FDA0003157763450000072
Figure FDA0003157763450000073
and is
Figure FDA0003157763450000074
Figure FDA0003157763450000075
And is
Figure FDA0003157763450000076
Wherein the content of the first and second substances,
Figure FDA0003157763450000077
Figure FDA0003157763450000078
the robust update law gain is:
Figure FDA0003157763450000079
therefore, the further update law is represented as:
Figure FDA00031577634500000710
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.
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