CN113515106A - Industrial process multi-dimensional fault-tolerant predictive control method - Google Patents

Industrial process multi-dimensional fault-tolerant predictive control method Download PDF

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CN113515106A
CN113515106A CN202110438773.8A CN202110438773A CN113515106A CN 113515106 A CN113515106 A CN 113515106A CN 202110438773 A CN202110438773 A CN 202110438773A CN 113515106 A CN113515106 A CN 113515106A
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李恺如
赵东辉
杨俊友
张日东
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Shenyang University of Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B23/00—Testing or monitoring of control systems or parts thereof
    • G05B23/02—Electric testing or monitoring
    • G05B23/0205—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults
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    • G05B23/0243—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults characterised by the fault detection method dealing with either existing or incipient faults model based detection method, e.g. first-principles knowledge model
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract

The invention discloses a multidimensional fault-tolerant predictive control method for an industrial process, which comprises the following steps of 1, establishing an equivalent multi-time-lag novel error model aiming at an intermittent process with time lag; step 2, on the basis of the novel error model, establishing a 2D-Roesser state space model based on a multi-step prediction idea and giving performance index representation; step 3, establishing a closed-loop control system of the 2D-Roesser state space model, and establishing a sufficient condition that the system has invariant set characteristics; step 4, designing a prediction controller and selecting a performance index function which is resistant to external interference and has terminal constraint based on the prediction models in the steps 1 and 3 along the time and batch directions, and providing sufficient conditions that the terminal constraint set of the prediction model is an invariant set; and 5, constructing an optimization algorithm aiming at the selected performance index function to obtain expected control performance along the time and batch directions. The invention can well make up the defects of the traditional fault-tolerant control and has advantages in the aspects of controller design and calculation.

Description

Industrial process multi-dimensional fault-tolerant predictive control method
Technical Field
The invention relates to the technical field of control of industrial processes, in particular to a multidimensional fault-tolerant predictive control method for an industrial process.
Background
With the rapid development of science and technology, industrial production gradually presents the characteristics of small scale, multiple varieties, high added value and the like, and the intermittent process draws attention of people again. At present, batch production technology has been widely applied in various fields of manufacturing industry, pharmacy, metal synthesis and the like. As the operating processes and flows of industrial production become more and more complex, the probability of system failure increases. Meanwhile, a time lag phenomenon is ubiquitous in industrial processes. The existence of factors such as faults and time lag gradually becomes an obstacle to the stable and efficient operation of the intermittent process. Faults are classified as sensor faults, actuator faults, and other component faults of the system. Of all failures, actuator failures are most common in industrial production. The existence of actuator faults can reduce the operation precision of the system, damage the control performance of the system and even influence the production efficiency. The existence of the time lag can cause the response speed of the system to be delayed and the tracking performance to be deteriorated, and even influence the stability of the system. Therefore, under the dual effects of faults and time lag, an effective and feasible control method is found to ensure stable and efficient operation of the control process, and the method has important significance for industrial production.
In order to solve the problem of faults, the fault-tolerant control technology of the intermittent process is widely applied, but the current technical level mainly adopts one dimension, and the one-dimensional method only considers the influence of time and specific industrial production. In addition, in actual production, there are factors such as actuator failure, drift and system external interference, and the control performance of the system is greatly affected. The intermittent process has two-dimensional characteristics, and the fault of the current batch has high possibility to influence the next batch and even a plurality of future batches. Coupled with the existence of time lags during the batch process, it is clear that the difficulty of controller design is increased. It becomes necessary to find new optimal control methods for batch processes under the dual influence of faults and time lags.
Disclosure of Invention
The invention aims to provide a multidimensional fault-tolerant predictive control method for an industrial process, aiming at an intermittent process with multiple time lags, interference and actuator faults, the control law can be updated in real time, the tracking performance and the anti-interference performance of the control method in the batch process are improved, the control performance of the system is ensured to be optimal, and the efficient production is realized.
In order to achieve the purpose, the invention provides the following technical scheme: a multi-dimensional fault-tolerant predictive control method for industrial process includes
Step 1, aiming at an intermittent process with time lag, establishing an equivalent multi-time-lag novel error model;
step 2, on the basis of the novel error model, establishing a 2D-Roesser state space model based on a multi-step prediction idea and giving performance index representation;
step 3, establishing a closed-loop control system of the 2D-Roesser state space model, and establishing a sufficient condition that the system has invariant set characteristics;
step 4, designing a prediction controller and selecting a performance index function which is resistant to external interference and has terminal constraint based on the prediction models in the steps 1 and 3 along the time and batch directions, and providing sufficient conditions that the terminal constraint set of the prediction model is an invariant set;
and 5, constructing an optimization algorithm aiming at the selected performance index function to obtain expected control performance along the time and batch directions.
Compared with the prior art, the invention has the beneficial effects that: by applying the concept of dimension expansion, a V function containing corresponding time lag is designed aiming at each time lag, so that the design can well make up the defects of the traditional fault-tolerant control, and the method has advantages in the aspects of design and calculation of the controller. Especially, the system has the characteristics of simple design, small calculated amount and the like for a small time lag system.
Aiming at the intermittent process with multiple time lags, interference and actuator faults, the invention combines the iterative learning control law, selects the Lyapunov-Razumikhin function (LRF), and provides the intermittent process 2D prediction fault-tolerant control method with time lags and disturbance by utilizing the model prediction fault-tolerant control method, so that the control law can be updated in real time, the tracking performance and the anti-interference performance of the control method in the batch process are improved, the control performance of the system is ensured to be optimal, and the efficient production is realized.
Drawings
FIG. 1 is a graph of the performance of different R-traces under repeated disturbances in accordance with the present invention.
FIG. 2 is a graph of the input trace of different batches under repeated perturbation according to the present invention.
FIG. 3 is a graph of the output traces of different batches under repeated perturbation in accordance with the present invention.
FIG. 4 is a graph of the update law of different batches under repeated perturbation according to the present invention.
FIG. 5 is a graph of tracking error for different batches under repeated perturbations in accordance with the present invention.
FIG. 6 is a graph of different R tracking performance under non-repetitive disturbances in accordance with the present invention.
FIG. 7 is a graph of the input trace of different batches under the non-repetitive perturbation of the present invention.
FIG. 8 is a graph of the output traces of different batches under the non-repetitive disturbance of the present invention.
FIG. 9 is a graph of the update law trajectories for different batches under non-repetitive disturbance according to the present invention.
FIG. 10 is a plot of tracking error traces for different batches under non-repetitive perturbations in accordance with the present invention.
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, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention provides a technical scheme that: a multi-dimensional fault-tolerant predictive control method for industrial process includes
Step 1, aiming at an intermittent process with time lag, establishing an equivalent multi-time-lag novel error model;
step 2, on the basis of the novel error model, establishing a 2D-Roesser state space model based on a multi-step prediction idea and giving performance index representation;
step 3, establishing a closed-loop control system of the 2D-Roesser state space model, and establishing a sufficient condition that the system has invariant set characteristics;
step 4, designing a prediction controller and selecting a performance index function which is resistant to external interference and has terminal constraint based on the prediction models in the steps 1 and 3 along the time and batch directions, and providing sufficient conditions that the terminal constraint set of the prediction model is an invariant set;
and 5, constructing an optimization algorithm aiming at the selected performance index function to obtain expected control performance along the time and batch directions.
Step 1 is concretely
1.1 construct the following model of an intermittent process with time lag
Figure BDA0003033500440000041
Wherein t and k represent the time of run and the batch, respectively; x (t + s, k) is belonged to Rn，y(t，k)∈Rl，uF(t，k)∈RmRespectively representing state variables with time lag, output variables and input variables of the system at the kth batch at the time t;
Figure BDA0003033500440000042
representing an adaptive constant matrix x0，kDenotes the initial state of the k-th batch, dmRepresents a maximum value of the state skew; wherein I represents an adaptive identity matrix, and ω (t, k) represents an external unknown disturbance; considering a partial failure fault α, the system input signal is u (t, k), so this fault type can be expressed as follows:
uF(t，k)＝αu(t，k) (2)
Figure BDA0003033500440000043
wherein the content of the first and second substances,α(α1) and
Figure BDA0003033500440000044
is a known variable;
definition of
Figure BDA0003033500440000045
Figure BDA0003033500440000046
α＝diag[α 1，α 2，...，α m] (6)
α＝diag[a1，α2，...，αm] (7)
Thus, an intermittent process with time lag and actuator failure can be described as follows:
Figure BDA0003033500440000051
1.2 building an equivalent 2D closed-loop state space model
1.2.1 introduction of an iterative learning control strategy
Aiming at the intermittent process of the formula (1), an iterative learning control law can be designed by using an iterative learning control strategy:
Figure BDA0003033500440000052
wherein u (t, 0) represents the initial value of the iterative process, typically set to zero; r (t, k) is belonged to RmRepresenting an iterative learning updating law to be designed; obviously, the design of the iterative learning controller u (t, k) can be converted into the design of the update law r (t, k) so that the control output y (t, k) can track the upper reference output y as much as possibler(t)；
1.2.2 defining tracking error variables
e(t，k)＝y(t，k)-yr(t) (10)
Defining an error function along the batch direction
δf(t，k)＝f(t，k)-f(t，k-1) (11)
Wherein, f can represent system state variable, output variable and external disturbance;
according to the formulae (9) to (11), can be obtained
Figure BDA0003033500440000053
Figure BDA0003033500440000054
Order to
Figure BDA0003033500440000061
1.2.3 extended equivalent 2D-Roesser model the following was obtained
Figure BDA0003033500440000062
Wherein the content of the first and second substances,
Figure BDA0003033500440000063
order to
Figure BDA0003033500440000064
Figure BDA0003033500440000065
2D closed-loop state space model based on 2D-Roesser model can be obtained
Figure BDA0003033500440000066
Wherein the content of the first and second substances,
Figure BDA0003033500440000067
1.2.4 design update law as follows:
Figure BDA0003033500440000068
step 2 is concretely
2.1 establishing 2D state space model based on 2D-Roesser model under prediction mode as follows
Figure BDA0003033500440000071
2.2 selecting MPC as the limited optimization performance index
Figure BDA0003033500440000072
Where l (t + i | t, k + j | k) and VT(x (t + N | t, k + N | k) is referred to as the phase cost and the terminal cost, respectively.
Figure BDA0003033500440000073
Wherein Q and R are weight matrices, and τ is a positive scalar;
2.3 optimization problem, which can be described in the form
Figure BDA0003033500440000074
Wherein the content of the first and second substances,
Figure BDA0003033500440000075
Figure BDA0003033500440000076
is a terminal constraint set, interference and control input satisfied
Figure BDA0003033500440000077
Figure BDA0003033500440000078
Where eta is a known constant, rkIs the kth element of the update law,
Figure BDA0003033500440000079
step 3 is specifically
3.1 set Ωπ，tIs a set of RPIs, if there is a positive scalar
Figure BDA00030335004400000710
So that
Figure BDA0003033500440000081
Wherein
Figure BDA0003033500440000082
Figure BDA0003033500440000083
Representing that the RPI set omega is used at any time of t and r is used as a corresponding updating law; let LRF:
Figure BDA0003033500440000084
definition of
Figure BDA0003033500440000085
Wherein
Figure BDA0003033500440000086
The corresponding update law is
Figure BDA0003033500440000087
3.2 set Ωπ，tIs a set of RPIs, consider the system (16), for a given matrix
Figure BDA0003033500440000088
If a symmetric positive definite matrix exists
Figure BDA0003033500440000089
The matrix Y belongs to R(n+l)×(n+l)，G∈R(n+l)×(n+l)，Z∈R(n+l)×(n+l)Positive scalar quantity
Figure BDA00030335004400000810
So that the following matrix inequality is resolvable:
Figure BDA00030335004400000811
Figure BDA00030335004400000812
wherein the content of the first and second substances,
Figure BDA00030335004400000813
Figure BDA00030335004400000814
P＝ξX-1，K＝YG-1then Ωπ，tIs a set of RPIs;
3.2.1 according to the dilation theory-GTX-1G≤X-GT-G, left multiplying (23) by diag { G-T，G-T...，G-TMultiplying the transpose of the I, I, I, I } right to obtain
Figure BDA0003033500440000091
Wherein the content of the first and second substances,
Figure BDA0003033500440000092
3.2.2 formula (25) can be converted to the following form:
Figure BDA0003033500440000093
wherein
Figure BDA0003033500440000094
Using schur theorem on (26) and left-multiplying the resulting inequality
Figure BDA0003033500440000095
And right-multiplying its transpose, the following equation can be obtained:
Figure BDA0003033500440000096
wherein the content of the first and second substances,
Figure BDA0003033500440000101
3.2.3 order
Figure BDA0003033500440000102
Then (27) may be equivalent to
Figure BDA0003033500440000103
Because of the fact that
Figure BDA0003033500440000104
Then
Figure BDA0003033500440000105
Set omegaπ，tIs a set of RPIs;
3.3 proof for equation (24):
because of the fact that
Figure BDA0003033500440000106
Is provided with
Figure BDA0003033500440000107
Then
Figure BDA0003033500440000108
By applying the shur complement theorem, the control input constraint condition (24) can be obtained.
The specific method of the step 4 is
4.1 sufficient conditions for the prediction model to be a invariant set are given as follows
Terminal constraint set of any batch at time t
Figure BDA0003033500440000109
Two conditions should be met, first omegaπ，tIs a set of RPIs, followed by the presence of alpha1，α2∈κ∞And positive definite function
Figure BDA00030335004400001010
So that
Figure BDA00030335004400001011
Figure BDA00030335004400001012
4.2 equation (30) can be obtained by solving eigenvalues of a positive definite matrix,
Figure BDA0003033500440000111
wherein λ ismin：＝min{ρmin(P)}，λmax：＝min{ρmax(P)}，ρmin(. and ρ)max(. cndot.) represents the minimum and maximum eigenvalues, respectively, and is thus certified;
4.3 the main work focuses on the demonstration of conditional equation (31), as follows:
considering the system (18), if (23), (24) hold, the following matrix inequality can be solved
Figure BDA0003033500440000112
Figure BDA0003033500440000113
Is a set RPI, wherein
Figure BDA0003033500440000114
For (32) left-multiplying diag { G-T G-T … G-TRight-hand multiplying the transpose to obtain a matrix inequality equivalent to
Figure BDA0003033500440000121
Wherein the content of the first and second substances,
Figure BDA0003033500440000125
let X-1＝ξ-1P, formula (33) can be written as
Figure BDA0003033500440000122
Wherein σ ═ g + KTRK-γ0P，
Figure BDA0003033500440000123
Apply schur complement theory to (34), then take the left
Figure BDA0003033500440000124
Right-multiplying by its transpose to obtain
Figure BDA0003033500440000131
Because of the fact that
Figure BDA0003033500440000132
Then
Figure BDA0003033500440000133
Then the following equation holds
Figure BDA0003033500440000134
Therefore, the temperature of the molten metal is controlled,
Figure BDA0003033500440000135
is a terminal constraint set.
The concrete method of the step 5 is
5.1 terminal constraint set due to the aforementioned RPI Properties
Figure BDA0003033500440000136
The following conditions should be satisfied
Figure BDA0003033500440000137
Thus, it is possible to provide
Figure BDA0003033500440000138
The following optimization problem is considered,
min xi, such that
Figure BDA0003033500440000139
Figure BDA00030335004400001310
Is that
Figure BDA00030335004400001311
τ can be optimized taking into account (32) the conditions of the terminal constraint set; taking the new variable η τ ξ, so η is minimized; then (32) can be written as
Figure BDA0003033500440000141
5.2 the terminal constraint set based on the prediction model is a sufficient condition of an invariant set, and the steps of designing an optimization control algorithm are as follows:
a: all the states of the system (including time lag) are obtained
Figure BDA0003033500440000142
b: if t is 0, for any batch k, solve for
Figure BDA0003033500440000143
So that the (23), (24), (38), (39),
Figure BDA0003033500440000144
the optimum lambda is obtained and is recorded as lambda*Continuing to step c;
otherwise t ≠ 0, for any batch k, using lambda*Instead of λ, thereby solving
Figure BDA0003033500440000145
So that (23), (24), (38), (39), continues to step c;
c: if obtained, is
Figure BDA0003033500440000146
Then t +1, k, and then return to step a.
The invention is explained in more detail below with reference to the figures and examples:
the invention uses a nonlinear continuous stirred tank as a control object to carry out simulation, and comprises the following two differential equations
Figure BDA0003033500440000151
Wherein, CAIs the concentration of A during the irreversible reaction (A → B); t is the temperature of the reactor; t isjIs the temperature of the cooling stream. As the variable to be operated on,
Figure BDA0003033500440000152
k0＝2.53×1019(1/mol min)，E/R＝13，500(K)，T(0)＝25(℃)，CA(0)＝0.9(mol/L)。
for system discrimination, a 26 ℃ transfer test was performed with a sampling interval of 1. From this, a transfer model can be derived
Figure BDA0003033500440000153
Assuming the system is second order, a least squares method with a transfer input and a transfer response is used.
x1(t，k)＝y(t，k)，x2(t, k) — 0.0013y (t-1, k) +0.0425u (t-1, k). The transfer function can be converted into the following state space model:
Figure BDA0003033500440000154
after discretization, the time lag extension model corresponding to the state space model is as follows:
Figure BDA0003033500440000155
wherein the content of the first and second substances,
Figure BDA0003033500440000156
C＝[1 0]，α＝0.8。
in this simulation example, the actuator fault we consider is a partial failure fault (α ═ 0.8). Through simulation experiments, the tracking performance, input, output, updating law and tracking error control effect of the system under the control method are obtained, and the effectiveness of the proposed two-dimensional iterative learning prediction fault-tolerant controller is verified.
In a practical industrial process, interference is inevitable. The invention respectively considers the robustness of repeated interference and non-repeated interference and carries out simulation.
Referring to FIGS. 1-5, the repetitive interference ω (t, k) e R2，ω(t，k)＝cos(t)×[0.001 0.002]T. In this case, ω (t, k) depends only on t, i.e., ω (t, k) ═ ω (t).
Referring to fig. 6-10, simulation studies were conducted on the robustness of non-repetitive interference. Wherein the non-repetitive interference omega (t, k) epsilon R2，ω(t，k)＝(0.4Δ1 0.4Δ2)T，Δ1∈[-1 1]，Δ2∈[-1 1]ω (t, k) depends on t and k.

Claims (6)

1. A multi-dimensional fault-tolerant predictive control method for an industrial process is characterized by comprising the following steps: comprises that
Step 1, aiming at an intermittent process with time lag, establishing an equivalent multi-time-lag novel error model;
step 2, on the basis of the novel error model, establishing a 2D-Roesser state space model based on a multi-step prediction idea and giving performance index representation;
step 3, establishing a closed-loop control system of the 2D-Roesser state space model, and establishing a sufficient condition that the system has invariant set characteristics;
step 4, designing a prediction controller and selecting a performance index function which is resistant to external interference and has terminal constraint based on the prediction models in the steps 1 and 3 along the time and batch directions, and providing sufficient conditions that the terminal constraint set of the prediction model is an invariant set;
and 5, constructing an optimization algorithm aiming at the selected performance index function to obtain expected control performance along the time and batch directions.
2. The industrial process multi-dimensional fault-tolerant predictive control method of claim 1, further comprising: step 1 is concretely
1.1 construct the following model of an intermittent process with time lag
Figure FDA0003033500430000011
Wherein t and k represent the time of run and the batch, respectively; x (t + s, k) is belonged to Rn，y(t，k)∈Rl，uF(t，k)∈RmRespectively representing state variables with time lag, output variables and input variables of the system at the kth batch at the time t;
Figure FDA0003033500430000012
representing an adaptive constant matrix x0，kDenotes the initial state of the k-th batch, dmRepresents a maximum value of the state skew; wherein I represents an adaptive identity matrix, and ω (t, k) represents an external unknown disturbance; considering a partial failure fault α, the system input signal is u (t, k), so this fault type can be expressed as follows:
uF(t，k)＝αu(t，k) (2)
Figure FDA0003033500430000021
wherein the content of the first and second substances,α(α1) and
Figure FDA0003033500430000022
is a known variable;
definition of
Figure FDA0003033500430000023
Figure FDA0003033500430000024
α＝diag[α 1，α 2，...，α m] (6)
α＝diag[α1，α2，...，αm] (7)
Thus, an intermittent process with time lag and actuator failure can be described as follows:
Figure FDA0003033500430000025
1.2 building an equivalent 2D closed-loop state space model
1.2.1 introduction of an iterative learning control strategy
Aiming at the intermittent process of the formula (1), an iterative learning control law can be designed by using an iterative learning control strategy:
Figure FDA0003033500430000026
wherein u (t, 0) represents the initial value of the iterative process, typically set to zero; r (t, k) is belonged to RmRepresenting an iterative learning updating law to be designed; obviously, the design of the iterative learning controller u (t, k) can be converted into the design of the update law r (t, k) so that the control output y (t, k) can track the upper reference output y as much as possibler(t)；
1.2.2 defining tracking error variables
e(t，k)＝y(t，k)-yr(t) (10)
Defining an error function along the batch direction
δf(t，k)＝f(t，k)-f(t，k-1) (11)
Wherein, f can represent system state variable, output variable and external disturbance;
according to the formulae (9) to (11), can be obtained
Figure FDA0003033500430000031
Figure FDA0003033500430000032
Order to
Figure FDA0003033500430000033
1.2.3 extended equivalent 2D-Roesser model the following was obtained
Figure FDA0003033500430000034
Wherein the content of the first and second substances,
Figure FDA0003033500430000035
order to
Figure FDA0003033500430000036
Figure FDA0003033500430000037
2D closed-loop state space model based on 2D-Roesser model can be obtained
Figure FDA0003033500430000038
Wherein the content of the first and second substances,
Figure FDA0003033500430000039
1.2.4 design update law as follows:
Figure FDA0003033500430000041
3. the industrial process multi-dimensional fault-tolerant predictive control method of claim 2, characterized in that: step 2 is concretely
2.1 establishing 2D state space model based on 2D-Roesser model under prediction mode as follows
Figure FDA0003033500430000042
2.2 selecting MPC as the limited optimization performance index
Figure FDA0003033500430000043
Where l (t + i | t, k + j | k) and VT(x (t + N | t, k + N | k) is called the phase cost and the terminal cost, respectively
Figure FDA0003033500430000044
Wherein Q and R are weight matrices, and τ is a positive scalar;
2.3 optimization problem, which can be described in the form
Figure FDA0003033500430000045
Wherein the content of the first and second substances,
Figure FDA0003033500430000046
Figure FDA0003033500430000047
is a terminal constraint set, interference and control input satisfied
Figure FDA0003033500430000051
Figure FDA0003033500430000052
Where eta is a known constant, rkIs the kth element of the update law,
Figure FDA0003033500430000053
4. the method of claim 3, wherein: step 3 is specifically
3.1 set Ωπ，tIs a set of RPIs, if there is a positive scalar
Figure FDA0003033500430000054
So that
Figure FDA0003033500430000055
Wherein
Figure FDA0003033500430000056
Figure FDA0003033500430000057
Representing that the RPI set omega is used at any time of t and r is used as a corresponding updating law; let LRF:
Figure FDA0003033500430000058
definition of
Figure FDA0003033500430000059
Wherein
Figure FDA00030335004300000510
The corresponding update law is
Figure FDA00030335004300000511
3.2 set Ωπ，tIs a set of RPIs, consider the system (16), for a given matrix
Figure FDA00030335004300000512
If a symmetric positive definite matrix exists
Figure FDA00030335004300000513
The matrix Y belongs to R(n+l)×(n+l)，G∈R(n+l)×(n+l)，Z∈R(n +l)×(n+l)Positive scalar quantity
Figure FDA00030335004300000514
So that the following matrix inequality is resolvable:
Figure FDA0003033500430000061
Figure FDA0003033500430000062
wherein the content of the first and second substances,
Figure FDA0003033500430000063
Figure FDA0003033500430000064
P＝ξX-1，K＝YG-1then Ωπ，tIs a set of RPIs;
3.2.1 according to the dilation theory-GTX-1G≤X-GT-G, left multiplying (23) by diag { G-T，G-T，...，G-TMultiplying the transpose of the I, I, I, I } right to obtain
Figure FDA0003033500430000065
Wherein the content of the first and second substances,
Figure FDA0003033500430000066
3.2.2 formula (25) can be converted to the following form:
Figure FDA0003033500430000071
wherein
Figure FDA0003033500430000072
Using schur theorem on (26) and left-multiplying the resulting inequality
Figure FDA0003033500430000073
And right-multiplying its transpose, the following equation can be obtained:
Figure FDA0003033500430000074
wherein the content of the first and second substances,
Figure FDA0003033500430000075
3.2.3 order
Figure FDA0003033500430000076
Then (27) may be equivalent to
Figure FDA0003033500430000077
Because of the fact that
Figure FDA0003033500430000078
Then
Figure FDA0003033500430000079
Set omegaπ，tIs a set of RPIs;
3.3 proof for equation (24):
because of the fact that
Figure FDA00030335004300000710
Is provided with
Figure FDA0003033500430000081
Then
Figure FDA0003033500430000082
By applying the shur complement theorem, the control input constraint condition (24) can be obtained.
5. The method of claim 4, wherein: the specific method of the step 4 is
4.1, giving sufficient conditions that the terminal constraint set of the prediction model is an invariant set, which are as follows:
terminal constraint set of any batch at time t
Figure FDA0003033500430000083
Two conditions should be met, first omegaπ，tIs a set of RPIs, followed by the presence of alpha1，α2∈κ∞And positive definite function
Figure FDA0003033500430000084
So that
Figure FDA0003033500430000085
Figure FDA0003033500430000086
4.2 equation (30) can be obtained by solving eigenvalues of a positive definite matrix,
Figure FDA0003033500430000087
wherein λ ismin：＝min{ρmin(P)}，λmax：＝min{ρmax(P)}，ρmin(. and ρ)max(. cndot.) represents the minimum and maximum eigenvalues, respectively, and is thus certified;
4.3 the main work focuses on the demonstration of conditional equation (31), as follows:
considering the system (18), if (23), (24) hold, the following matrix inequality can be solved
Figure FDA0003033500430000091
Figure FDA0003033500430000092
Is a set RPI, wherein
Figure FDA0003033500430000093
For (32) left-multiplying diag { G-T G-T … G-TRight-hand multiplying the transpose to obtain a matrix inequality equivalent to
Figure FDA0003033500430000094
Wherein the content of the first and second substances,
Figure FDA0003033500430000095
let X-1＝ξ-1P, formula (33) can be written as
Figure FDA0003033500430000101
Wherein, σ ═ Q + KTRK-γ0P，
Figure FDA0003033500430000102
Apply schur complement theory to (34), then take the left
Figure FDA0003033500430000103
Right-multiplying by its transpose to obtain
Figure FDA0003033500430000104
Because of the fact that
Figure FDA0003033500430000105
Then the following equation holds
Figure FDA0003033500430000106
Therefore, the temperature of the molten metal is controlled,
Figure FDA0003033500430000107
is a terminal constraint set.
6. The industrial process multi-dimensional fault-tolerant predictive control method of claim 5, wherein: the concrete method of the step 5 is
5.1 terminal constraint set due to the aforementioned RPI Properties
Figure FDA0003033500430000111
The following conditions should be satisfied
Figure FDA0003033500430000112
Thus, it is possible to provide
Figure FDA0003033500430000113
The following optimization problem is considered,
min xi, such that
Figure FDA0003033500430000114
Figure FDA0003033500430000115
Is that
Figure FDA0003033500430000116
τ can be optimized taking into account (32) the conditions of the terminal constraint set; taking the new variable η τ ξ, so η is minimized; then (32) can be written as
Figure FDA0003033500430000117
5.2 the terminal constraint set based on the prediction model is a sufficient condition of an invariant set, and the steps of designing an optimization control algorithm are as follows:
a: all the states of the system (including time lag) are obtained
Figure FDA0003033500430000118
b: if t is 0, for any batch k, solve for
Figure FDA0003033500430000121
So that the (23), (24), (38), (39),
Figure FDA0003033500430000122
the optimum lambda is obtained and is recorded as lambda*Continuing to step c;
otherwise t ≠ 0, for any batch k, using lambda*Instead of λ, thereby solving
Figure FDA0003033500430000123
So that (23), (24), (38), (39), continues to step c;
c: if obtained, is
Figure FDA0003033500430000124
Then t +1, k, and then return to step a.
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