CN111505937A - Industrial process improved model prediction fault-tolerant control method under multiple modes - Google Patents

Industrial process improved model prediction fault-tolerant control method under multiple modes Download PDF

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CN111505937A
CN111505937A CN202010141674.9A CN202010141674A CN111505937A CN 111505937 A CN111505937 A CN 111505937A CN 202010141674 A CN202010141674 A CN 202010141674A CN 111505937 A CN111505937 A CN 111505937A
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model
switching
time
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industrial process
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王立敏
罗卫平
张日东
王心如
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Hangzhou Dianzi University
Hainan Normal University
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Hangzhou Dianzi University
Hainan Normal University
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Abstract

A multi-modal industrial process improved model prediction fault-tolerant control method belongs to the technical field of industrial process advanced control. The method comprises the following steps: step 1, establishing a switching system model of a controlled object based on a state space model; and 2, designing a model prediction tracking controller and a switching law. The invention introduces a performance function, designs a controller capable of resisting faults by adjusting variables in the function, and simultaneously meets the requirement of optimal control performance. The method effectively solves the control problem of model mismatching caused by system faults and the switching problem of each mode, effectively improves the tracking performance and robustness of the industrial process, shortens the running time of each mode of the system, realizes good control effect under the condition of model mismatching caused by the system faults, and improves the production efficiency.

Description

Industrial process improved model prediction fault-tolerant control method under multiple modes
Technical Field
The invention relates to the technical field of advanced control of industrial processes, in particular to a multi-mode industrial process improved model prediction fault-tolerant control method.
Background
In modern industrial production, industrial processes are widely applied, especially in the food industry, the pharmaceutical industry, the chemical industry and the like, and the research on the control theory of the industrial processes also makes a great breakthrough. But still presents a challenge in terms of high-precision control of modern industrial processes. The main reasons are the high quality production level requirements and the complex and variable process conditions. Thus, the probability of system failure also increases. Among these faults, actuator faults are the most common one. Due to the characteristics of friction, dead zones, saturation, etc., the actuator inevitably experiences some malfunction during its execution, which makes it difficult to reach a specified or desired position. If the fault is not detected and corrected in a timely manner, the production performance inevitably deteriorates, and even safety problems of equipment and personnel are caused.
In order to solve the problems, the fault-tolerant control problem of the industrial process is widely applied, and attention is paid to a fault-tolerant control method of repetitive iterative learning, but when the fault of an actuator becomes serious or external disturbance exists, the existing reliable control method of robust iterative learning cannot solve the problem of system state deviation, namely the same control law is adopted from beginning to end, and the deviation of the system is increased along with the increase of time. This can have a negative effect on the continuous stable operation and control performance of the system, even compromising the quality of the product. In recent years, Model Predictive Control (MPC) has the potential for performance improvement and real-time update of control laws for optimization. However, in situations where the models, processes do not match, and actuator failure is severe, there is still a problem of improving the performance of the MPC to achieve the desired product quality.
In addition, the industrial process has a multi-modal characteristic, the variables controlled by two different modes are different, the control targets are different, when the mode is switched from one mode to the other mode, and the operation time of each mode directly influences the production efficiency and the product quality. Although there is some research effort for multi-modality, the controller gain cannot be adjusted throughout the process. In actual industrial control, due to factors such as drift, process nonlinearity and system external interference existing in actual working conditions, the control performance of the control system may be reduced after the control system operates for a period of time, and the operation time of each mode may be prolonged. If the switching signal and the repair controller are not designed in time to improve the control quality, the economic benefit obtained by the control system is reduced.
Disclosure of Invention
In order to solve the problem that a system in the prior art cannot update a control law in real time under multiple modes and track the performance of the control law, so that the control performance of the system is influenced, the invention provides an industrial process improved model prediction fault-tolerant control method under multiple modes, effectively solves the control problem that a model is not matched and the switching problem of each mode caused by system faults, effectively improves the tracking performance and robustness of the industrial process, shortens the running time of each mode of the system, achieves good control effect under the condition of model mismatch caused by system faults, and improves the production efficiency.
In order to achieve the above object, the present invention provides a multi-modal industrial process improved model prediction fault-tolerant control method, which comprises the following steps:
step 1, establishing a switching system model of a controlled object based on a state space model;
step 1.1, constructing a system model with faults in the novel multi-mode industrial process;
the system formula for partial actuator failure is as follows:
Figure BDA0002399285020000021
the actuator failure is represented as:
uiF(k)＝αiui(k) (2)
wherein, yi(z),ui(z) and wi(z) are each output yi(k) Input ui(k) And interference wi(k) Z transformation of αi,(0＜αi1) is expressed as the failure coefficient, Ai(z-1)、Bi(z-1) And
Figure BDA0002399285020000022
is a dimensional polynomial;
setting an industrial process with uncertainty, which can be represented by a multiple-input multiple-output discrete transfer function model introducing a difference operator Δ, specifically:
Figure BDA0002399285020000031
thus define:
Δxi(k)＝[Δyi(k)Δyi(k-1)…Δyi(k-m+1)Δui(k-1)Δui(k-2)…Δui(k-n+1)]T(4)
wherein, yi(k) And ui(k) Respectively an output and a control input at time k,
Figure BDA0002399285020000032
and
Figure BDA0002399285020000033
is the corresponding coefficient in the formula;
thus, equation (1) can be transformed into the following state space form:
Figure BDA0002399285020000034
step 1.2, defining an output tracking error, wherein the output tracking error is specifically expressed as:
ei(k)＝yi(k)-ri(k) (6)
the dynamic relationship of the output tracking error obtained is:
Figure BDA0002399285020000035
wherein, yi(t)、
Figure BDA0002399285020000036
Actual output values of the k-time and i-mode and tracking set points of the k-time and i-mode, respectively, ei(k) Is the output error of the i mode at time k, Δ ri(k +1) is the difference value of the ith modal setting value of the industrial process at the moment of k + 1;
step 1.3, introducing a new state variable, specifically:
Figure BDA0002399285020000037
wherein the content of the first and second substances,
Figure BDA0002399285020000038
is composed of state-based extension information ei(k) Determining;
step 1.4, setting a new state variable
Figure BDA0002399285020000039
The method specifically comprises the following steps:
Figure BDA00023992850200000310
step 1.5, converting the space model into an equivalent error model containing extended information through the steps:
Figure BDA0002399285020000041
the ith modal prediction control model is:
Figure BDA0002399285020000042
shown with the switching system model as:
Figure BDA0002399285020000043
wherein σ (k) is Z+→N1,2, …, N represents the switching signal, N is sub-system,
Figure BDA0002399285020000044
and
Figure BDA0002399285020000045
representing the above-mentioned model (11) for different modalities;
step 1.6, when the system is switched from the current mode to the next mode, the state transition matrix J is usediAssociating the two modal states, specifically as follows:
xi+1(Ti)＝Jixi(Ti) (13a)
wherein, when Ji＝IiWhen the system is in a normal state, two adjacent modes have the same system state;
and when the system is in the i mode of any batch at any time, if G is satisfiedi(x (k)) less than 0, the system switches the i mode at the previous moment to the i +1 mode at the next moment, and the definition of the switching time is in a specific form:
Ti＝min{k＞Ti-1|Mi(x(k))＜0},T0＝0 (13b)
wherein, TiSwitching time points;
at the runtime of the system as a whole, the switching sequence of each modality for each batch can be expressed as:
∑＝{(T1,ρ(T1)),(T2,ρ(T2)),...,(Ti,ρ(Ti)),...} (13c)
wherein, taupThe average residence time of each mode is taken as the time interval between two adjacent modes in the same batch to satisfy Ti-Ti-1≥τi，TiRepresents the switching time point, T, of the last moment of the same batchi+1Is the switching time point of the next moment of the same batch;
step 1.7, constructing a novel closed-loop predictive control system;
step 1.7.1, aiming at the ith mode, designing the following prediction updating law:
Figure BDA0002399285020000051
Figure BDA0002399285020000052
and
Figure BDA0002399285020000053
respectively representing the predicted value of the state of the kth batch at the time t and the predicted value of the output of the kth batch at the time t, Delauui(k + j | k) represents the prediction update law of the kth batch at the time t, wherein,
Figure BDA0002399285020000054
Δui(k|k)＝Δui(k)；
step 1.7.2, designing the optimal performance index based on the system (11) to ensure that the performance index
Figure BDA0002399285020000055
Minimization under constraints is specifically expressed as follows:
Figure BDA0002399285020000056
Figure BDA0002399285020000057
the constraint condition is
Figure BDA0002399285020000058
Wherein the content of the first and second substances,
Figure BDA0002399285020000059
and
Figure BDA00023992850200000510
each represents a matrix of related weights,
Figure BDA00023992850200000511
and
Figure BDA00023992850200000512
are respectively a variable Δ ui(k + j | k) and
Figure BDA00023992850200000513
Δ r (k + j | k) is an indeterminate set;
step 2, designing a model prediction tracking controller and a switching law;
step 2.1, designing a prediction updating law aiming at the switching system model, researching the robustness stability of the system, and according to the switching system model, specifically, an ith modal closed-loop prediction model is as follows:
Figure BDA00023992850200000514
step 2.2, design controller gain
Figure BDA00023992850200000515
Such that:
Figure BDA00023992850200000516
step 2.2.1, defining L yapunov functions, specifically:
Figure BDA0002399285020000061
wherein, P1 i,
Figure BDA0002399285020000062
Are all undetermined positive definite matrixes;
step 2.2.2, the following Lyapunov inequality constraints are satisfied:
Figure BDA0002399285020000063
step 2.2.3, setting a series of initial conditions of the i-th mode closed-loop prediction model, so that:
Figure BDA0002399285020000064
wherein j is a positive integer, l1Infinity is a positive integer, corresponding
Figure BDA0002399285020000065
A boundary in a time direction;
step 2.2.4, superposing the Lyapunov inequality constraints (18) to obtain the following inequalities:
Figure BDA0002399285020000066
wherein, thetaiIs composed of
Figure BDA0002399285020000067
An upper boundary of (d);
step 2.2.5, the following inequalities are used to make the above expressions (18) to (20) true, and the inequalities are specifically:
Figure BDA0002399285020000068
Figure BDA0002399285020000069
Figure BDA00023992850200000610
Figure BDA0002399285020000071
and the control law gain matrix that is obtained can be expressed as follows:
Figure BDA0002399285020000072
wherein, Δ ri(k+1)≠0，
Figure BDA0002399285020000073
Positive definite matrix
Figure BDA0002399285020000074
At this time
Figure BDA0002399285020000075
Given of P1 i，
Figure BDA0002399285020000076
Exists in a positive definite symmetric matrix, and matrix Y1 i,
Figure BDA0002399285020000077
And positive numberi＞0，θiIf the result is more than 0, the result is to be determined;
step 2.3, designing a switching law;
designing a switching point according to the switching signal;
delta V is known from the inequality (21a)iIs < 0, i.e.
Figure BDA0002399285020000078
Wherein t is0K < t, the following inequality is obtained:
Figure BDA0002399285020000079
wherein the content of the first and second substances,
Figure BDA00023992850200000710
a switching time for the ith modality;
from Vi＜μiVi-1The following can be obtained:
Figure BDA00023992850200000711
suppose that
Figure BDA00023992850200000712
Then:
Figure BDA0002399285020000081
from the above condition, when the switching signal is satisfied
Figure BDA0002399285020000082
When, Vσ(t)And (t) is convergence, i.e. the system is asymptotically stable.
Preferably, A in step 1.1i(z-1)、Bi(z-1) And
Figure BDA0002399285020000083
the dimensional polynomial expressed is specifically:
Figure BDA0002399285020000084
Figure BDA0002399285020000085
Figure BDA0002399285020000086
preferably, in the expression (5) in the form of state space represented in step 1.1, wherein:
Figure BDA0002399285020000087
preferably, the step 1.5 includes an equivalent error model of the extension information, wherein:
Figure BDA0002399285020000088
preferably, in the step 2.2.4, when the lyapunov inequality constraints (18) are superimposed, j ═ 0 to j ∞.
Has the advantages that: according to the method, firstly, a state error, an output tracking error and a new state variable are introduced according to a given system model with faults and based on the repeatability of an industrial process, and the state error, the output tracking error and the new state variable are expanded into an equivalent model containing the state error, the output tracking error and expansion information, so that a corresponding switching system model is obtained. In order to research the optimal control performance, the invention introduces a performance function, and designs a controller capable of resisting faults by adjusting variables in the function, and simultaneously meets the requirement of optimal control performance. And the minimum operation of each mode is designed by using an average residence time method. The design process has the advantages of simple design, small operand, short system operation time and good tracking performance. The method effectively solves the control problem of model mismatching caused by system faults and the switching problem of each mode, effectively improves the tracking performance and robustness of the industrial process, shortens the running time of each mode of the system, realizes good control effect under the condition of model mismatching caused by the system faults, and improves the production efficiency.
Drawings
FIG. 1 is a tracking performance diagram of an improved model predictive fault-tolerant control method for an industrial process under multiple modes under normal faults;
FIG. 2 is a comparison diagram of state concentrations of frequent faults of the industrial process improved model predictive fault-tolerant control method under multiple modes provided by the invention;
FIG. 3 is a comparison graph of inputs of the improved model predictive fault-tolerant control method for industrial processes in multiple modes according to the present invention;
FIG. 4 is a tracking performance diagram of the time-varying fault of the industrial process improved model predictive fault-tolerant control method under multiple modes, provided by the invention;
FIG. 5 is a comparison graph of state concentrations under time-varying faults of the industrial process improved model predictive fault-tolerant control method under multiple modes provided by the invention;
FIG. 6 is an input comparison diagram under time-varying fault of the industrial process improved model predictive fault-tolerant control method under multiple modes.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention.
The invention provides a multi-modal industrial process improved model prediction fault-tolerant control method, which comprises the following steps:
step 1, establishing a switching system model of a controlled object aiming at different modes in an industrial process on the basis of a state space model;
step 1.1, constructing a system model with faults in the novel multi-mode industrial process;
the multi-input multi-output process comprises the following steps:
Figure BDA0002399285020000101
and actuator failure is represented as:
uiF(k)＝αiui(k) 2
thus, the system equation with partial actuator failure is as follows:
Figure BDA0002399285020000102
wherein, yi(z),ui(z) and wi(z) are each output yi(k) Input ui(k) And interference wi(k) Z transformation of αi,(0＜αi1) is expressed as the failure coefficient, Ai(z-1)、Bi(z-1) And
Figure BDA0002399285020000103
is a dimensional polynomial;
setting an industrial process with uncertainty can be described by the following multiple-input multiple-output discrete transfer function model:
Figure BDA0002399285020000104
after the difference operator delta is introduced, the multiple-input multiple-output discrete transfer function model is specifically expressed as follows:
Figure BDA0002399285020000105
thus define:
Δxi(k)＝[Δyi(k)Δyi(k-1)…Δyi(k-m+1)Δui(k-1)Δui(k-2)…Δui(k-n+1)]T4
wherein, yi(k) And ui(k) Respectively an output and a control input at time k,
Figure BDA0002399285020000111
and
Figure BDA0002399285020000112
is the corresponding coefficient in the formula;
thus, equation 1 can be transformed into the following state space form:
Figure BDA0002399285020000113
step 1.2, in order to enable the system to keep a stable running state and have better tracking performance, defining an output tracking error, wherein the output tracking error is specifically represented as:
ei(k)＝yi(k)-ri(k) 6
the dynamic relationship of the output tracking error obtained is:
Figure BDA0002399285020000114
wherein, yi(t)、
Figure BDA0002399285020000115
Actual output values of the k-time and i-mode and tracking set points of the k-time and i-mode, respectively, ei(k) Is the output error of the i mode at time k, Δ ri(k +1) is the difference value of the ith modal setting value of the industrial process at the moment of k + 1;
step 1.3, introducing a new state variable, specifically:
Figure BDA0002399285020000116
wherein the content of the first and second substances,
Figure BDA0002399285020000117
is composed of state-based extension information ei(k) Determining;
step 1.4, setting a new state variable
Figure BDA0002399285020000118
The method specifically comprises the following steps:
Figure BDA0002399285020000119
step 1.5, converting the space model into an equivalent error model containing extended information through the steps:
Figure BDA00023992850200001110
the ith modal prediction control model is:
Figure BDA00023992850200001111
shown with the switching system model as:
Figure BDA0002399285020000121
wherein σ (k) is Z+→N1,2, …, N represents switching signal, switching signal and timeOr state-dependent, N is sub-system,
Figure BDA0002399285020000122
and
Figure BDA0002399285020000123
representation of the above-described model 11 for different modalities;
step 1.6, in the actual production process, because the model dimensions of two adjacent modes may be different, when the system is switched from the current mode to the next mode, the state transition matrix J is usediAssociating the two modal states, specifically as follows:
xi+1(Ti)＝Jixi(Ti) 13a
wherein, when Ji＝IiWhen the system is in a normal state, two adjacent modes have the same system state;
the switching time of the system becomes critical when the system state is known, and when the system is in the i mode of any batch at any time, if G is satisfiedi(x (k)) less than 0, the system switches the i mode at the previous moment to the i +1 mode at the next moment, and the definition of the switching time is in a specific form:
Ti＝min{k＞Ti-1|Mi(x(k))＜0},T0＝0 13b
wherein, TiSwitching time points;
at the runtime of the system as a whole, the switching sequence of each modality for each batch can be expressed as:
∑＝{(T1,ρ(T1)),(T2,ρ(T2)),...,(Ti,ρ(Ti)),...} 13c
wherein, taupThe average residence time of each mode is taken as the time interval between two adjacent modes in the same batch to satisfy Ti-Ti-1≥τi，TiRepresents the switching time point, T, of the last moment of the same batchi+1Is the switching time point of the next moment of the same batch;
step 1.7, constructing a novel closed-loop predictive control system;
step 1.7.1, aiming at the ith mode, designing the following prediction updating law:
Figure BDA0002399285020000124
Figure BDA0002399285020000125
and
Figure BDA0002399285020000126
respectively representing the predicted value of the state of the kth batch at the time t and the predicted value of the output of the kth batch at the time t, Delauui(k + j | k) represents the prediction update law of the kth batch at the time t, wherein,
Figure BDA0002399285020000131
Δui(k|k)＝Δui(k)；
step 1.7.2, design the best performance index based on the system 11, so that the performance index
Figure BDA0002399285020000132
Minimization under constraints is specifically expressed as follows:
Figure BDA0002399285020000133
Figure BDA0002399285020000134
the constraint condition is
Figure BDA0002399285020000135
Wherein the content of the first and second substances,
Figure BDA0002399285020000136
and
Figure BDA0002399285020000137
each represents a matrix of related weights,
Figure BDA0002399285020000138
and
Figure BDA0002399285020000139
are respectively a variable Δ ui(k + j | k) and
Figure BDA00023992850200001310
Δ r (k + j | k) is an indeterminate set;
step 2, designing a model prediction tracking controller and a switching law;
step 2.1, aiming at the theory of predictive control adopted by the switching system model, designing a predictive updating law, researching the robustness stability of the system, and according to the switching system model, the ith modal closed-loop predictive model is specifically as follows:
Figure BDA00023992850200001311
step 2.2, design controller gain
Figure BDA00023992850200001312
Such that:
Figure BDA00023992850200001313
step 2.2.1, the stability of the system is proved by using L yapunov stability theorem to define L yapunov function, which is specifically as follows:
Figure BDA00023992850200001314
wherein, P1 i,
Figure BDA00023992850200001315
Are all undetermined positive definite matrixes;
step 2.2.2, in order to ensure the robustness stability of the system and optimize the problem, the following Lyapunov inequality can be established, so that the constraint is established:
Figure BDA0002399285020000141
step 2.2.3, setting a series of initial conditions of the i-th mode closed-loop prediction model, so that:
Figure BDA0002399285020000142
wherein j is a positive integer, l1Infinity is a positive integer, corresponding
Figure BDA0002399285020000143
A boundary in a time direction;
step 2.2.4, superposing the Lyapunov inequality constraints 18 to obtain the following inequalities:
Figure BDA0002399285020000144
wherein, thetaiIs composed of
Figure BDA0002399285020000145
An upper boundary of (d);
step 2.2.5, the following inequalities are used to make the above expressions 18 to 20 true, where the inequalities are specifically:
Figure BDA0002399285020000146
Figure BDA0002399285020000147
Figure BDA0002399285020000148
Figure BDA0002399285020000149
and the control law gain matrix that is obtained can be expressed as follows:
Figure BDA0002399285020000151
wherein, Δ ri(k+1)≠0，
Figure BDA0002399285020000152
Positive definite matrix
Figure BDA0002399285020000153
At this time
Figure BDA0002399285020000154
Given of P1 i，
Figure BDA0002399285020000155
Exists in a positive definite symmetric matrix, and matrix Y1 i,
Figure BDA0002399285020000156
And positive numberi＞0，θiIf the result is more than 0, the result is to be determined;
step 2.3, designing a switching law;
designing a switching point according to the switching signal;
from the 21a inequality, Δ V can be knowniIs < 0, i.e.
Figure BDA0002399285020000157
Wherein t is0K < t, the following inequality is obtained:
Figure BDA0002399285020000158
wherein the content of the first and second substances,
Figure BDA0002399285020000159
a switching time for the ith modality;
from Vi＜μiVi-1The following can be obtained:
Figure BDA00023992850200001510
suppose that
Figure BDA00023992850200001511
Then:
Figure BDA00023992850200001512
from the above condition, when the switching signal is satisfied
Figure BDA00023992850200001513
When, Vσ(t)And (t) is convergence, i.e. the system is asymptotically stable.
As a further development of the invention, in step 1.1Ai(z-1)、Bi(z-1) And
Figure BDA00023992850200001514
the dimensional polynomial expressed is specifically:
Figure BDA0002399285020000161
Figure BDA0002399285020000162
Figure BDA0002399285020000163
as a further refinement of the invention, in said step 1.1 expression 5 in the form of a state space is represented, wherein:
Figure BDA0002399285020000164
as a further improvement of the present invention, said step 1.5 contains an equivalent error model of the extended information, wherein:
Figure BDA0002399285020000165
as a further improvement of the present invention, in the step 2.2.4, when the lyapunov inequality constraints 18 are added, j ═ 0 to j ∞ are added.
The first embodiment is as follows:
when in a highly nonlinear Continuous Stirred Tank Reactor (CSTR):
Figure BDA0002399285020000166
wherein, CAExpressed as a reversible reaction, the concentration in (A → B) is A, T is the temperature of the reactor, T isCIs the coolant temperature as a controlled variable, q-100 (L/min), V-100 (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) with an operating constraint of 200. ltoreq.TC≤450(K)、0.01≤CAT is less than or equal to 1 (mol/L) and 250 is less than or equal to 500 (K).
The goal of the operation is to move the system from an undesirable equilibrium point (361.141,0.8986) to the desired equilibrium point (398.972,0.52), where the system response moves to the undesirable equilibrium point in a cyclic operation.
To achieve the above batch control objective, 3 balance points 1# (361.141,0.8986, 302), 2# (370, 0.8391, 307.8121), and 3# (380, 0.7469, 309.3844) are selected, respectively, for the piecewise affine control design of the loop operation. 1# to 3# transition from (361.141,0.8986) (1#) to (398.972,0.52) (0 #). Accordingly, three piecewise affine operation regions are divided for the control design. The sampling time is truncated to 1.2(s), and a first order euler approximation is used for simulation testing of discrete continuous time systems.
For the model predictive fault-tolerant control design, a linearized state space model of an operating region in the balance points 1# to 3# can be obtained simply through the Jacobian matrix of the original system relative to the balance points 1# to 3 #. As shown in table 1, it proves that these linear models are very coarse by representing the dynamic response characteristics in these regions, and therefore, are considered as the "worst case" of the control design.
TABLE 1 piecewise-linear model obtained for CSTR
Figure BDA0002399285020000171
Wherein the control objective is to let the reactor temperature follow a given curve, in particular:
Figure BDA0002399285020000181
in order to show that the method for predicting fault-tolerant control of the industrial process under multiple modes has better effect, the invention uses MAT L AB to perform comparison experiments on the proposed method with extended information (SSMPC4y) and the proposed method without extended information (SSMPC5y), such as a tracking performance graph in FIG. 1, a state concentration comparison graph in FIG. 2, and an input comparison graph in FIG. 3, and particularly illustrates that the color of the schematic line of the method without extended information (SSMPC5y) is slightly darker than that of the schematic line of the method with extended information (SSMPC4 y).
In industrial processes, faults are common and not negligible, and are roughly classified into constant faults and time-varying faults.
The type one is as follows: frequent failure
The failure occurrence time k is 350(s), the failure value α is 0.6, and the corresponding simulation graphs are 1-3.
Type two: time varying fault
The failure occurrence time is 350(s), the failure value is α is 0.5+0.1sin (k), and the corresponding simulation diagram is 4-6.
According to the data waveform diagrams shown in fig. 1-6, it can be seen that although the two methods have slight fluctuation at the three turning points of 120(s), 300(s) and 480(s), the tracking performance of the control method with extended information provided by the present invention is better than that of the control method without extended information, which further illustrates that the control effect of the industrial process improved model predictive fault-tolerant control method under multiple modes provided by the present invention is better.

Claims (5)

1. A multi-modal industrial process improved model prediction fault-tolerant control method is characterized by comprising the following steps: the method comprises the following steps:
step 1, establishing a switching system model of a controlled object based on a state space model;
step 1.1, constructing a system model with faults in the novel multi-mode industrial process;
the system formula for partial actuator failure is as follows:
Figure FDA0002399285010000011
the actuator failure is represented as:
uiF(k)＝αiui(k) (2)
wherein, yi(z),ui(z) and wi(z) are each output yi(k) Input ui(k) And interference wi(k) Z transformation of αi,(0＜αi1) is expressed as the failure coefficient, Ai(z-1)、Bi(z-1) And
Figure FDA0002399285010000012
is a dimensional polynomial;
setting an industrial process with uncertainty, which can be represented by a multiple-input multiple-output discrete transfer function model introducing a difference operator Δ, specifically:
Figure FDA0002399285010000013
thus define:
Δxi(k)＝[Δyi(k) Δyi(k-1)…Δyi(k-m+1) Δui(k-1) Δui(k-2)…Δui(k-n+1)]T(4)
wherein, yi(k) And ui(k) Respectively an output and a control input at time k,
Figure FDA0002399285010000014
and
Figure FDA0002399285010000015
is the corresponding coefficient in the formula;
thus, equation (1) can be transformed into the following state space form:
Figure FDA0002399285010000016
step 1.2, defining an output tracking error, wherein the output tracking error is specifically expressed as:
ei(k)＝yi(k)-ri(k) (6)
the dynamic relationship of the output tracking error obtained is:
Figure FDA0002399285010000021
wherein, yi(t)、
Figure FDA0002399285010000022
Actual output values of the k-time and i-mode and tracking set points of the k-time and i-mode, respectively, ei(k) Is the output error of the i mode at time k, Δ ri(k +1) is the difference value of the ith modal setting value of the industrial process at the moment of k + 1;
step 1.3, introducing a new state variable, specifically:
Figure FDA0002399285010000023
wherein the content of the first and second substances,
Figure FDA0002399285010000024
is composed of state-based extension information ei(k) Determining;
step 1.4, setting a new state variable
Figure FDA0002399285010000025
The method specifically comprises the following steps:
Figure FDA0002399285010000026
step 1.5, converting the space model into an equivalent error model containing extended information through the steps:
Figure FDA0002399285010000027
the ith modal prediction control model is:
Figure FDA0002399285010000028
shown with the switching system model as:
Figure FDA0002399285010000029
wherein σ (k) is Z+→N1,2, …, N represents the switching signal, N is sub-system,
Figure FDA00023992850100000210
and
Figure FDA00023992850100000211
representing the above-mentioned model (11) for different modalities;
step 1.6, when the system is switched from the current mode to the next mode, the state transition matrix J is usediTwo are combinedThe modal states are correlated as follows:
xi+1(Ti)＝Jixi(Ti) (13a)
wherein, when Ji＝IiWhen the system is in a normal state, two adjacent modes have the same system state;
and when the system is in the i mode of any batch at any time, if G is satisfiedi(x (k)) less than 0, the system switches the i mode at the previous moment to the i +1 mode at the next moment, and the definition of the switching time is in a specific form:
Ti＝min{k＞Ti-1|Mi(x(k))＜0},T0＝0 (13b)
wherein, TiSwitching time points;
at the runtime of the system as a whole, the switching sequence of each modality for each batch can be expressed as:
∑＝{(T1,ρ(T1)),(T2,ρ(T2)),...,(Ti,ρ(Ti)),...} (13c)
wherein, taupThe average residence time of each mode is taken as the time interval between two adjacent modes in the same batch to satisfy Ti-Ti-1≥τi，TiRepresents the switching time point, T, of the last moment of the same batchi+1Is the switching time point of the next moment of the same batch;
step 1.7, constructing a novel closed-loop predictive control system;
step 1.7.1, aiming at the ith mode, designing the following prediction updating law:
Figure FDA0002399285010000031
Figure FDA0002399285010000032
and
Figure FDA0002399285010000033
respectively representing the predicted value of the state of the kth batch at the time t and the predicted value of the output of the kth batch at the time t, Delauui(k + j | k) represents the prediction update law of the kth batch at the time t, wherein,
Figure FDA0002399285010000034
Δui(k|k)＝Δui(k)；
step 1.7.2, designing the optimal performance index based on the system (11) to ensure that the performance index
Figure FDA0002399285010000035
Minimization under constraints is specifically expressed as follows:
Figure FDA0002399285010000036
Figure FDA0002399285010000037
the constraint condition is
Figure FDA0002399285010000041
Wherein the content of the first and second substances,
Figure FDA0002399285010000042
and
Figure FDA0002399285010000043
each represents a matrix of related weights,
Figure FDA0002399285010000044
and
Figure FDA0002399285010000045
are respectively a variable Δ ui(k + j | k) and
Figure FDA0002399285010000046
Δ r (k + j | k) is an indeterminate set;
step 2, designing a model prediction tracking controller and a switching law;
step 2.1, designing a prediction updating law aiming at the switching system model, researching the robustness stability of the system, and according to the switching system model, specifically, an ith modal closed-loop prediction model is as follows:
Figure FDA0002399285010000047
step 2.2, design controller gain
Figure FDA0002399285010000048
Such that:
Figure FDA0002399285010000049
step 2.2.1, defining L yapunov functions, specifically:
Figure FDA00023992850100000410
wherein, P1 i,
Figure FDA00023992850100000411
Are all undetermined positive definite matrixes;
step 2.2.2, the following Lyapunov inequality constraints are satisfied:
Figure FDA00023992850100000412
step 2.2.3, setting a series of initial conditions of the i-th mode closed-loop prediction model, so that:
Figure FDA00023992850100000413
wherein j is a positive integer, l1Infinity is a positive integer, corresponding
Figure FDA00023992850100000414
A boundary in a time direction;
step 2.2.4, superposing the Lyapunov inequality constraints (18) to obtain the following inequalities:
Figure FDA00023992850100000415
wherein, thetaiIs composed of
Figure FDA00023992850100000416
An upper boundary of (d);
step 2.2.5, the following inequalities are used to make the above expressions (18) to (20) true, and the inequalities are specifically:
Figure FDA0002399285010000051
Figure FDA0002399285010000052
Figure FDA0002399285010000053
Figure FDA0002399285010000054
and the control law gain matrix that is obtained can be expressed as follows:
Figure FDA0002399285010000055
wherein, Δ ri(k+1)≠0，
Figure FDA0002399285010000056
Positive definite matrix
Figure FDA0002399285010000057
At this time
Figure FDA0002399285010000058
Given that the number of the first and second sets of data,
Figure FDA0002399285010000059
exists in a positive definite symmetric matrix, and the matrix
Figure FDA00023992850100000510
And positive numberi＞0，θiIf the result is more than 0, the result is to be determined;
step 2.3, designing a switching law;
designing a switching point according to the switching signal;
delta V is known from the inequality (21a)iIs < 0, i.e.
Figure FDA00023992850100000511
Wherein t is0K < t, the following inequality is obtained:
Figure FDA00023992850100000512
wherein the content of the first and second substances,
Figure FDA0002399285010000061
a switching time for the ith modality;
from Vi＜μiVi-1The following can be obtained:
Figure FDA0002399285010000062
suppose that
Figure FDA0002399285010000063
Then:
Figure FDA0002399285010000064
from the above condition, when the switching signal is satisfied
Figure FDA0002399285010000065
When, Vσ(t)And (t) is convergence, i.e. the system is asymptotically stable.
2. The method according to claim 1, wherein A in step 1.1 is a fault-tolerant control method of industrial process improvement model prediction under multiple modesi(z-1)、Bi(z-1) And
Figure FDA0002399285010000066
the dimensional polynomial expressed is specifically:
Figure FDA0002399285010000067
Figure FDA0002399285010000068
Figure FDA0002399285010000069
3. the method for predictive fault-tolerant control of an industrial process improved model under multiple modes according to claim 2, wherein the step 1.1 is expressed in an expression (5) in a state space form, wherein:
Figure FDA0002399285010000071
4. the method according to claim 1, wherein the step 1.5 comprises an equivalent error model with extended information, wherein:
Figure FDA0002399285010000072
5. the method according to claim 1, wherein in step 2.2.4, when the lyapunov inequality constraints (18) are added, j ═ 0 is added to j ∞.
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