CN107976942A - The batch process 2D constraint fault tolerant control methods of Infinite horizon optimization - Google Patents
The batch process 2D constraint fault tolerant control methods of Infinite horizon optimization Download PDFInfo
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Abstract
The invention belongs to the Dynamic matrix control field of automatic technology, and fault tolerant control method is constrained more particularly, to the batch process 2D of Infinite horizon optimization.For the constraint fault control system modelling iterative learning control law of interference, introduce state error and output error, the dynamic model of original system is converted into a closed-loop model represented in the form of prediction with Roesser models, design constraint iterative learning control law is converted into and determines constraint more new law；According to designed infinite optimality criterion and 2D system Lyapunov Theory of Stability, with linear matrix inequality（LMI）Constraint type provides the more new law real-time online design for ensuring closed-loop system robust asymptotically stabilization.The present invention solves control performance and can not improve with being incremented by for batch, and the control problem of the uncertain batch process of initial value, is finally reached energy-saving consumption-reducing, reduces the generation of cost, harm reduction personal safety accident.
Description
Technical field
The invention belongs to the Dynamic matrix control field of automatic technology, more particularly, to a kind of interval mistake of Infinite horizon optimization
Journey 2D constrains fault tolerant control method.
Background technology
Batch process becomes one of mostly important mode of production of modern manufacturing industry, with production-scale increase, with
And the increase of production stage complexity, it is uncertain increasingly prominent present in actual production, not only influence system
It is efficiently and smoothly operated, or even threatened the quality of product.And these complicated operating conditions, accordingly increase system event
Hinder the probability occurred.Wherein, actuator failures are a kind of common failures, can influence the operation of technical process and reduce controlling
Can, or even endanger personal safety.Although occur the controlling parties such as the reliable faults-tolerant control of iterative learning during batch processed
Method, can solve the control problem of system still stable operation when actuator failures occur well.But for high-accuracy
For the equipment of degree, the possibility that failure occurs is extremely low, if no matter either with or without failure, will cause resource using reliable control
Waste, if things go on like this, cost can also increase, it is clear that and not meet the environmental protection concept of energy-saving and emission-reduction.When catastrophe failure occurs
The reliable possible effect out of hand completely of control law, most likely results in system crash, causes great property in this case
Loss and casualties.
In addition, although the reliable control strategy of the Robust Iterative Learning used at this stage can be resisted effectively in production link
Uncertainty and failure caused by influence, ensure the stability of system, maintain the control performance of system, but the control law is
Solve and draw based on whole production process, the global optimal control of covering is belonged in control effect, i.e., is used from beginning to end
Same control law.
However, in actual motion, especially (so-called constrained system refers to its performance variable and controlled variable all to constrained system
Need to meet physical constraint) under the influence of interference and failure, system mode can not possibly fully according to obtained control law act on and
Change；If certain deviation occurs for the system mode at current time and setting value, continue to use same control law, with when
Between passage, the deviation of system mode can more increase, and existing Robust Iterative Learning reliable control method can not solve be
The problem of system state deviates, stable operation and control performance that this will certainly be to system exert an adverse impact.
Model Predictive Control (MPC) can meet the modified needs of control law real-time update well, by " rolling excellent
Change " and the mode of " feedback compensation " obtain the optimal control law at each moment, it is ensured that system mode can be as much as possible along setting
Fixed track operation.However, the prior art mostly using one-dimensional form infinite horizon control law, lack between batch
The process of " study ", control effect are not improved with being incremented by for batch；Also a kind of " study " between only considering batch
Process, this method can not achieve the control problem of the uncertain batch process of initial value.It is it will be apparent that uncertain for having
And the discussion of the constrained system Infinite horizon optimization problem of failure is up for continuing deeper into.Thus it is badly in need of proposing a kind of new control
Method makes up the deficiency of existing method, to realize energy-saving consumption-reducing in batch production process, reduce cost even harm reduction people
The targets such as body security incident generation.
The existing Prediction and Control Technology design control law in one-dimensional square mostly, only considers time orientation or batch side
To only consideration time orientation make it that each batch is simple repetition, and control performance can not be obtained with being incremented by for batch
It is perfect；Only consider that batch direction can not achieve the control problem of the uncertain batch process of initial value.
The content of the invention
In order to solve above-mentioned technical problem, the present invention provides a kind of Infinite horizon optimization batch process 2D about
Beam fault tolerant control method, efficiently solving control performance can not improve with being incremented by for batch, and initial value is not true
No matter the control problem of fixed batch process, realizes system either with or without failure, in the range of variable bound can real-time optimization,
It is finally reached energy-saving consumption-reducing, reduces the targets such as cost, the generation of harm reduction personal safety accident.
The technical solution adopted by the present invention is as follows：
The batch process 2D constraint fault tolerant control methods of Infinite horizon optimization, comprise the following steps：
A, batch process model of the structure with interference and actuator failures, it is described to have between interference and actuator failures
Process model (1) of having a rest is represented by (1a) and (1b)：
And its input, output constraint meet：
Wherein, t represents the time, and k represents batch,In be system state,It is the defeated of system
Enter,It is the reality output of system,It is input, the upper bound binding occurrence of reality output respectively,It is its exterior unknown disturbances, andΔ A is unknown uncertain system perturbation matrices, Δ
A (t, k)=D Δs (t, k) E, Δ (t, k) ΔT(t, k)≤I, { A, B2,C2Be appropriate dimension sytem matrix, { D, E } is appropriate
The constant matrices of dimension, I are the unit matrixs of appropriate dimension；Define different α values and represent the different fault type of actuator, when
During α ＞ 0, part failure of removal is represented；As α=0, represent entirely ineffective failure, be not involved in the problems, such as optimal controller；
For actuator partial failure, α ＞ 0 need to meet following form：
α (α≤1) in formula, andIt is known constant；
B, the batch process dynamic model with interference and actuator failures is converted into one in the form of predicted value to represent
Closed-loop model, for model (1), introduce following iterative learning control law：
U (t, k)=u (t, k-1)+r (t, k), u (0, k)=0, t=0,1,2 ... T (3)
Wherein, u (0, k) is the initial value of iteration, is normally provided as 0,Iterative learning more new law；
State error and output error on definition batch direction is as follows：
δk(f (t, k))=f (t, k)-f (t, k-1) (4a)
Obtained by model (1) and iterative learning control law (3)：
Wherein,
The error model being then augmented is write as following form with Roesser models：
Wherein： C3=[0 0 I],
And assumeDivide the level of suitable dimensional vector
With plumbness component, Z (t, k) is the controlled output of system；
C, iterative learning control law is gone out to the batch process modelling with interference and actuator failures,
For above-mentioned model (6) design 2D prediction fault-tolerant controllers, reach the minimum under maximum interference and maximum failure
Optimal control, even if model (6) reaches stable state and meets following robust performance index at each moment：
Limitation：
And Q (Q ＞ 0) and R (R ＞ 0) are the weighting matrixs of appropriate dimension, and r (t+i | t, k) it is moment t defeated to the t+i moment
The predicted value entered, and r (t, k)=r (t | t, k),Represent input increment；
Definition status Feedback Control Laws, make system reach Quadratic Stability, and the more new law of selection is：
Then the closed low predictions model of (6) is expressed as
Using the stabilization of 2D Lyapunov function proof systems, defining Lyapunov functions is：
Wherein
Model (6) still can even running in fault tolerance, it is necessary to meets：
(1) 2D Liapunov functions inequality constraints：
(2) there are the matrix M of appropriate dimensionj, the nonsingular matrix G, any scalar ε ＞ 0, θ ＞ 0 of H, Y and appropriate dimension,
γj＞ 0 may be such that following MATRIX INEQUALITIES is set up：
And
And
Wherein, T1=- (G+GT-Mj),
Optimal performance index meets J at this time∞(t,k)≤θ；
Robust more new law gain is K (t, k)=YG-1；
Therefore, further more new law is expressed as：
Carry it into u (t, k)=u (t, k-1)+r (t, k), can obtain 2D constraint iterations study design of control law u (t,
K), in subsequent time, constantly repeat (11a)-(11b) and continue to solve new controlled quentity controlled variable u (t, k), and circulate successively.
Matrix Q (Q ＞ 0) and R (R ＞ 0) is adjusted, initial value x (t | t, k) is given, solves formula (11a)-(11d), it is minimum to find θ
When Y, G, x (t | t, k) can be different at different moments, and K (t, k) also can constantly change with the time；If system does not break down, by profit
With normal system controller；Failure is different, and K (t, k) value also can be different, if α=0, there is inequality (11a) β=β0=0, this
When (11a) formula may be unsatisfactory for zero condition of being less than, be embodied in real process be exactly controller not in action, system is unstable,
State deviates original track quickly.Equipment, or even shutdown processing should be just checked at this time.For this point, harm reduction people
Body security incident occurs.
Compared with prior art, beneficial effects of the present invention are：The advantages of the present invention are in interference event
Iterative learning control law is designed on the basis of barrier control system model, state error and output error are introduced, with Roesser models
The dynamic model of original system is converted into a closed-loop model represented in the form of prediction, design iteration is learnt into control law
It is converted into and determines more new law；According to designed infinite optimality criterion and 2D system Lyapunov Theory of Stability, with line
Property MATRIX INEQUALITIES (LMI) constraint type provide ensure closed-loop system robust asymptotically stabilization more new law real-time online design, have
Solving to effect control performance can not improve with being incremented by for batch, and the control of the uncertain batch process of initial value
No matter problem, realizes system either with or without failure, can real-time optimization in the range of variable bound.It is finally reached energy-saving consumption-reducing, drop
Low cost, the generation of harm reduction personal safety accident.On the whole, can not only using this design method design control law
The even running in fault tolerance of guarantee system, to realize energy-saving consumption-reducing, reduce the targets such as cost, or even can also realize
The targets such as harm reduction personal safety accident generation.
Brief description of the drawings
Fig. 1 is that the batch process 2D of Infinite horizon of the present invention optimization constrains fault tolerant control method flow chart.
Fig. 2 is tracking performance figure caused by R differences in the quadratic form of one embodiment of the present invention.
Fig. 3 is the tracking performance comparison diagram of one embodiment of the present invention two kinds of distinct methods in R=200000.
Fig. 4 is the output tracking comparison diagram of one embodiment of the present invention two kinds of distinct methods in R=200000.
Fig. 5 is the tracking performance comparison diagram of one embodiment of the present invention two kinds of distinct methods in α=0.
Fig. 6 is the output tracking comparison diagram of one embodiment of the present invention two kinds of distinct methods in α=0.
Fig. 7 is the more new law comparison diagram of two kinds of distinct methods of one embodiment of the present invention.
Embodiment
The present invention will be described in detail with specific embodiment below in conjunction with the accompanying drawings.
As shown in Figure 1, the batch process 2D constraint fault tolerant control methods of Infinite horizon optimization, comprise the following steps：
A, batch process model of the structure with interference and actuator failures, it is described to have between interference and actuator failures
Process model (1) of having a rest is represented by (1a) and (1b)：
And its input, output constraint meet：
Wherein, t represents the time, and k represents batch,In be system state,It is the defeated of system
Enter,It is the reality output of system,It is input, the upper bound binding occurrence of reality output respectively,
It is its exterior unknown disturbances, andΔ A is unknown uncertain system perturbation matrices, Δ A (t, k)=D
Δ (t, k) E, Δ (t, k) ΔT(t, k)≤I, { A, B2,C2Be appropriate dimension sytem matrix, { D, E } is the normal of appropriate dimension
Matrix number, I are the unit matrixs of appropriate dimension；Define different α values and represent the different fault type of actuator, as α ＞ 0,
Represent part failure of removal；As α=0, represent entirely ineffective failure, be not involved in the problems, such as optimal controller；
For actuator partial failure, α ＞ 0 need to meet following form：
In formulaα(α≤ 1), andIt is known constant；
B, the batch process dynamic model with interference and actuator failures is converted into one in the form of predicted value to represent
Closed-loop model, for model (1), introduce following iterative learning control law：
U (t, k)=u (t, k-1)+r (t, k), u (0, k)=0, t=0,1,2 ... T (3)
Wherein, u (0, k) is the initial value of iteration, is normally provided as 0,Iterative learning more new law；
State error and output error on definition batch direction is as follows：
δk(f (t, k))=f (t, k)-f (t, k-1) (4a)
Obtained by model (1) and iterative learning control law (3)：
Wherein,
The error model being then augmented is write as following form with Roesser models：
Wherein： C3=[0 0 I],
And assumeDivide the level of suitable dimensional vector
With plumbness component, Z (t, k) is the controlled output of system；
C, iterative learning control law is gone out to the batch process modelling with interference and actuator failures,
For above-mentioned model (6) design 2D prediction fault-tolerant controllers, reach the minimum under maximum interference and maximum failure
Optimal control, even if model (6) reaches stable state and meets following robust performance index at each moment：
Limitation：
And Q (Q ＞ 0) and R (R ＞ 0) are the weighting matrixs of appropriate dimension, and r (t+i | t, k) it is moment t defeated to the t+i moment
The predicted value entered, and r (t, k)=r (t | t, k),Represent input increment；
Definition status Feedback Control Laws, make system reach Quadratic Stability, and the more new law of selection is：
Then the closed low predictions model of (6) is expressed as
Using the stabilization of 2D Lyapunov function proof systems, defining Lyapunov functions is：
Wherein
Model (6) still can even running in fault tolerance, it is necessary to meets：
(1) 2D Liapunov functions inequality constraints：
(2) there are the matrix M of appropriate dimensionj, the nonsingular matrix G, any scalar ε ＞ 0, θ ＞ 0 of H, Y and appropriate dimension,
γj＞ 0 may be such that following MATRIX INEQUALITIES is set up：
And
And
Wherein, T1=- (G+GT-Mj),
Optimal performance index meets J at this time∞(t,k)≤θ；Robust more new law gain is K (t, k)=YG-1；
Therefore, further more new law is expressed as：
Carry it into u (t, k)=u (t, k-1)+r (t, k), can obtain 2D constraint iterations study design of control law u (t,
K), in subsequent time, constantly repeat (11a)-(11b) and continue to solve new controlled quentity controlled variable u (t, k), and circulate successively.
Matrix Q (Q ＞ 0) and R (R ＞ 0) is adjusted, gives initial valueSolution formula (11a)-(11d), finds θ minimums
When Y, G, at different momentsCan be different, K (t, k) also can constantly change with the time；If system does not break down, by profit
With normal system controller；Failure is different, and K (t, k) value also can be different, if α=0, there is inequality (11a) β=β0=0, this
When (11a) formula may be unsatisfactory for zero condition of being less than, be embodied in real process be exactly controller not in action, system is unstable,
State deviates original track quickly.
Embodiment
Injection molding process is a complicated industrial manufacturing process, and the quality of injecting products depends on material parameter, machine
The reciprocation of device parameter, procedure parameter and these parameters.The quality of injecting products includes many aspects, such as appearance matter
Amount, accuracy to size and machinery (optics, electricity) performance etc..Different users is different to the focus of quality.These matter
Figureofmerit is together decided on by the control accuracy of the material, mould and the procedure parameter that are used in process.Meanwhile note
All there is various disturbing factors for different links during modeling.
Injection moulding process is substantially a kind of process of multistage Batch Process product, have in each Main Stage one or
Multiple key parameters play end product quality conclusive.The injection speed of injection stage, the pressurize in packing stage
Pressure, the melt temperature in plastic phase is the critical process variables in these stages, so must stablize to these parameters
Controlled with accurate, so that it is guaranteed that the product quality of production.
Packing stage is the important stage of decision product quality, in this stage since low temperature mould has cooling effect, is
Prevent melt in die cavity is inverse from overwhelming stream and melt cooling and cause product to shrink, injection nozzle there remains certain pressure.Cause
This, nozzle exit pressure is this stage most important controlled variable, this pressure is also referred to as dwell pressure.
The control of dwell pressure causes plastics industry circle and the attention of related researcher already.Although largely study work
Have been proven that the importance of dwell pressure, the research for packing stage is still relatively fewer, and reason is on the one hand to protect
Pressure analysis needs the result of mold filling analysis, and as primary condition, being on the other hand then must because being furtherd investigate to packing stage
The compressibility of melt must be considered, it is necessary to consider more physical parameters, problem is become more complicated.
In addition, in injection moulding process, control valve opening is larger, although the possibility of blocking can be reduced to a certain extent
Property, effectively prevent failure, but for the system with high-accuracy degree, failure possibility occurrence is inherently very low,
Larger valve opening can cause the increase of cost during the waste and control of raw material.Therefore, solve the problems, such as this to pass
It is important.
The failsafe valve and existing disturbance that the present invention may occur for system, in the case where there is constrained variable, design science
Practise control and feedback control organically combines and 2D prediction faults-tolerant control rules are designed for uniformity, closed-loop control is carried out to dwell pressure.
Algorithm above is solved, it is (this moment is considered as normal condition) to obtain initial time controller gain：
KNormally=[- 0.0065-0.0040 0.0041]；
Failure occurs in 30 batches, is worth and is：α=0.6, controller gain are：
KFailure=[- 0.0085-0.0068 0.0045]；
More new law constraint satisfaction | r (t+i | t, k) |≤0.1.
It is the control effect of its controller below：If as shown in Fig. 2, want to obtain minimum performance indicator, it is necessary to adjust two
Set matrix is given in secondary type, obtain so that tracking performance preferably when, controller at this time is obtained, with this controller and conventional method
Contrasted, four figures are comparative result figures below.The tracking performance that Fig. 3 can clearly give expression to context of methods is more preferable, although
Initial and several batch tracking performances are poor after breaking down, but it can even realize that zero error tracks afterwards.Fig. 4 is that system is defeated
Going out the tracing figure of track, a few batch control effects are preferable not to the utmost after still having initial batches and failure in the diagram, but once
Reach stable, tracking effect is extremely well, is especially become apparent at step point；It is the control after system thoroughly fails to see Fig. 5, Fig. 6
Effect processed, it is clear that either that control method, desired control effect of all arriving, should just shut down in this case
Check.Fig. 7 is more new law expression figure, is found out from this figure, and the change of context of methods more new law is extremely gentle, it is seen that its control performance is more
It is good.
The above is only the preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art
For member, various improvements and modifications may be made without departing from the principle of the present invention, these improvements and modifications also should
It is considered as protection scope of the present invention.
Claims (1)
1. the batch process 2D constraint fault tolerant control methods of Infinite horizon optimization, it is characterised in that：Comprise the following steps：
A, constraint batch process model of the structure with interference and actuator failures, it is described to have between interference and actuator failures
Process model (1) of having a rest is represented by (1a) and (1b)：
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And its input, output constraint meet：
Wherein, t represents the time, and k represents batch,In be system state,It is the input of system,It is the reality output of system,It is input, the upper bound binding occurrence of reality output respectively,It is
Its exterior unknown disturbances, andΔ A is unknown uncertain system perturbation matrices, Δ A (t, k)=D Δs
(t, k) E, Δ (t, k) ΔT(t, k)≤I, { A, B2,C2Be appropriate dimension sytem matrix, { D, E } is the constant of appropriate dimension
Matrix, I are the unit matrixs of appropriate dimension；Define different α values and represent the different fault type of actuator, as α ＞ 0, table
Show partial failure failure；As α=0, represent entirely ineffective failure, be not involved in the problems, such as optimal controller；
For actuator partial failure, α ＞ 0 need to meet following form：
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In formulaWithIt is known constant；
B, by with interference and actuator failures batch process dynamic model be converted into one represented in the form of predicted value close
Loop system model, for model (1), introduces following iterative learning control law：
U (t, k)=u (t, k-1)+r (t, k), u (0, k)=0, t=0,1,2 ... T (3)
Wherein, u (0, k) is the initial value of iteration, is normally provided as 0,Iterative learning more new law；
State error and output error on definition batch direction is as follows：
δk(f (t, k))=f (t, k)-f (t, k-1) (4a)
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<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<msub>
<mi>&delta;</mi>
<mi>k</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
<mi>&alpha;</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
<mover>
<mi>w</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mi>b</mi>
<mo>)</mo>
</mrow>
</mrow>
Wherein,
The error model being then augmented is write as following form with Roesser models：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>B</mi>
<mi>&alpha;</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>C</mi>
<mn>1</mn>
</msub>
<mover>
<mi>w</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>y</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mover>
<mo>=</mo>
<mi>&Delta;</mi>
</mover>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mover>
<mi>C</mi>
<mo>&OverBar;</mo>
</mover>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>Z</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mover>
<mo>=</mo>
<mi>&Delta;</mi>
</mover>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein：
<mrow>
<msub>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>=</mo>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mi>I</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>+</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>&Delta;</mi>
<mi>A</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
<mi>&Delta;</mi>
<mi>A</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mi>B</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<msub>
<mi>C</mi>
<mn>1</mn>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>I</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>-</mo>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
</mrow>
C3=[0 0I],
And assumeDivide the horizontal and vertical of suitable dimensional vector
State component, Z (t, k) are the controlled output of system；
C, iterative learning control law is gone out to the batch process modelling with interference and actuator failures,
For above-mentioned model (6) design 2D prediction fault-tolerant controllers, reach the minimum optimization under maximum interference and maximum failure
Control, even if model (6) reaches stable state and meets following robust performance index at each moment：
<mrow>
<munder>
<mi>min</mi>
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<mi>&infin;</mi>
</mrow>
</munder>
<munder>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
<mrow>
<msup>
<mi>&gamma;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>|</mo>
<mo>|</mo>
<mi>w</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>&le;</mo>
<mi>&gamma;</mi>
<mo>|</mo>
<mo>|</mo>
<mi>Z</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
</mrow>
</munder>
<msub>
<mi>J</mi>
<mi>&infin;</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mi>a</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>J</mi>
<mi>&infin;</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>:</mo>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mi>T</mi>
</msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>Q</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>R</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>r</mi>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mi>b</mi>
<mo>)</mo>
</mrow>
</mrow>
Limitation：
And Q (Q ＞ 0) and R (R ＞ 0) are the weighting matrixs of appropriate dimension, and r (t+i | t, k) it is that moment t inputs the t+i moment
Predicted value, and r (t, k)=r (t | t, k),Represent input increment；
Definition status Feedback Control Laws, make system reach Quadratic Stability, and the more new law of selection is：
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>K</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>&infin;</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mi>a</mi>
<mo>)</mo>
</mrow>
</mrow>
Then the closed low predictions model of (6) is expressed as
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>B</mi>
<mi>&alpha;</mi>
<mi>K</mi>
<mo>)</mo>
</mrow>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>C</mi>
<mn>1</mn>
</msub>
<mover>
<mi>w</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>y</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mover>
<mo>=</mo>
<mi>&Delta;</mi>
</mover>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>e</mi>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>e</mi>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mover>
<mi>C</mi>
<mo>&OverBar;</mo>
</mover>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>Z</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mover>
<mo>=</mo>
<mi>&Delta;</mi>
</mover>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mi>b</mi>
<mo>)</mo>
</mrow>
</mrow>
Using the stabilization of 2D Lyapunov function proof systems, defining Lyapunov functions is：
<mrow>
<mi>V</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>&theta;</mi>
<mo>&CenterDot;</mo>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<msup>
<mi>M</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>&infin;</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein M ＞ 0
Model (6) still can even running in fault tolerance, it is necessary to meets：
(1) 2D Liapunov functions inequality constraints：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mo>&lsqb;</mo>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
<mo>,</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mi>Q</mi>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>i</mi>
<mo>|</mo>
<mi>t</mi>
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<mo>+</mo>
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<mi>R</mi>
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</mrow>
</mrow>
</mtd>
</mtr>
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<mo>&le;</mo>
<mi>V</mi>
<mrow>
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</mover>
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</mtable>
<mo>-</mo>
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<mrow>
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<mn>10</mn>
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</mrow>
</mrow>
(2) there are the matrix M of appropriate dimensionj, the nonsingular matrix G, any scalar ε ＞ 0, θ ＞ 0, γ of H, Y and appropriate dimensionj＞
0 may be such that following MATRIX INEQUALITIES is set up：
<mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>T</mi>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<msubsup>
<mi>GA</mi>
<mn>1</mn>
<mi>T</mi>
</msubsup>
<mo>+</mo>
<msup>
<mi>Y</mi>
<mi>T</mi>
</msup>
<msup>
<mi>&beta;B</mi>
<mi>T</mi>
</msup>
</mrow>
</mtd>
<mtd>
<mrow>
<msubsup>
<mi>GC</mi>
<mn>3</mn>
<mi>T</mi>
</msubsup>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>G</mi>
<msup>
<mover>
<mi>E</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>Y</mi>
<mi>T</mi>
</msup>
<mi>&beta;</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>Y</mi>
<mi>T</mi>
</msup>
<msup>
<mi>R</mi>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</msup>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>GQ</mi>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
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<mo>*</mo>
</mtd>
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<mi>T</mi>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<msubsup>
<mi>HC</mi>
<mn>1</mn>
<mi>T</mi>
</msubsup>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
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<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
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<mtr>
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</mtd>
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<mtd>
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<mo>-</mo>
<msub>
<mi>M</mi>
<mi>j</mi>
</msub>
<mo>+</mo>
<mi>&epsiv;</mi>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<msup>
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<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<mo>+</mo>
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<mi>&epsiv;B&beta;</mi>
<mn>0</mn>
<mn>2</mn>
</msubsup>
<msup>
<mi>B</mi>
<mi>T</mi>
</msup>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
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<mtd>
<mn>0</mn>
</mtd>
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<mn>0</mn>
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<mi>I</mi>
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</mtable>
</mfenced>
<mo><</mo>
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<mi>a</mi>
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</mrow>
</mrow>
<mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
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<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
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<mi>t</mi>
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<mi>k</mi>
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<mi>T</mi>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
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<mtd>
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<mo>-</mo>
<msub>
<mi>M</mi>
<mi>j</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&le;</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mi>b</mi>
<mo>)</mo>
</mrow>
</mrow>
And And
Wherein,
Optimal performance index meets J at this time∞(t,k)≤θ；
Robust more new law gain is K (t, k)=YG-1；
Therefore, further more new law is expressed as：
U (t, k)=u (t, k-1)+r (t, k) is carried it into, 2D constraint iterations study design of control law u (t, k) can be obtained,
Subsequent time, constantly repeats (11a)-(11b) and continues to solve new controlled quentity controlled variable u (t, k), and circulates successively.
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| CN109212971A (en) * | 2018-10-11 | 2019-01-15 | 海南师范大学 | Multistage batch process 2D linear quadratic tracks fault tolerant control method |
| CN109212971B (en) * | 2018-10-11 | 2022-02-18 | 海南师范大学 | Multi-stage intermittent process 2D linear secondary tracking fault-tolerant control method |
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| CN110045611A (en) * | 2019-04-24 | 2019-07-23 | 华北电力大学 | A kind of Robust Iterative Learning model predictive control method applied to batch stirred tank reactor |
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| CN110568763A (en) * | 2019-10-11 | 2019-12-13 | 海南师范大学 | model prediction H-infinity fault-tolerant control method for resisting intermittent process disturbance and time lag |
| CN110568763B (en) * | 2019-10-11 | 2022-03-29 | 海南师范大学 | Model prediction H-infinity fault-tolerant control method for resisting intermittent process disturbance and time lag |
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