CN109254531B - Method for optimal cost control of a multi-stage batch process with time lag and disturbances - Google Patents

Method for optimal cost control of a multi-stage batch process with time lag and disturbances Download PDF

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CN109254531B
CN109254531B CN201711221845.3A CN201711221845A CN109254531B CN 109254531 B CN109254531 B CN 109254531B CN 201711221845 A CN201711221845 A CN 201711221845A CN 109254531 B CN109254531 B CN 109254531B
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王立敏
刘冰
李平
张日东
于晶贤
施惠元
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Liaoning Shihua University
Hangzhou Dianzi University
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Abstract

The invention belongs to the field of advanced control of industrial processes, and relates to an optimal cost control method of a multi-stage intermittent process with time lag and interference. Firstly, a two-dimensional switching state space model is constructed, the constructed two-dimensional multi-stage state space model is converted into a two-dimensional switching system model, then a hybrid controller is designed according to the constructed two-dimensional state space model and an optimal cost control algorithm, and finally the gain of the controller is solved in a linear matrix inequality mode. The invention designs the optimal cost-guaranteed controllers corresponding to different stages aiming at the multi-stage intermittent process with interval time-varying time lag and interference, thereby not only ensuring the stable operation of the intermittent process, but also reducing the operation time of each stage and minimizing the production cost, realizing energy conservation and emission reduction for enterprise production and maximizing the economic benefit.

Description

Method for optimal cost control of a multi-stage batch process with time lag and disturbances
Technical Field
The invention belongs to the field of advanced control of industrial processes, and relates to an optimal cost control method of a multi-stage intermittent process with time lag and interference.
Background
In industrial production, due to the influence of factors such as nonlinearity of an actual process and external disturbance of a system, the control performance of a control system is reduced, and thus the production efficiency is reduced. If the quality is not controlled in time, the yield is reduced and energy waste is caused. Along with the increasingly high requirements on the refinement and functions of products in production, the high-precision control problem of the intermittent process is emphasized, the intermittent process has a hysteresis characteristic, the problem which generally exists in actual production after information is wrong is one of important reasons for causing the performance reduction of a system controller, and a corresponding controller is designed aiming at the situation to ensure that the intermittent process has better control performance and becomes the current hotspot control problem. A great deal of research results are already available for the high-precision control of the single-stage intermittent process with time lag, and corresponding patents are also embodied. However, the intermittent process in the actual production process has a multi-stage characteristic, different stages affect each other, and when switching from one stage to another stage not only affects the system stability but also affects the final quality of the product. This has led to interest in the control studies of multi-stage batch processes, with a small amount of research effort now occurring. But most of them are only aimed at the no-time-lag situation, and moreover, the study of the stability of the batch process is only the most basic requirement of the system production. Studying its optimal cost control on a stable basis would be a matter of great concern to researchers and even producers. Unfortunately, this study does not exist for the presence of time lags.
Aiming at the multi-stage intermittent process under the conditions of time lag and interference, in order to ensure that the multi-stage intermittent process has optimal cost control while running efficiently, a more effective optimal cost control method is necessary.
Disclosure of Invention
In order to solve the above problems, the present invention provides an optimal cost control method for a multi-stage intermittent process with time lag and interference, which aims to improve the anti-interference capability of the multi-stage intermittent process and the control performance of the system.
The method comprises the steps of designing an iterative learning control law aiming at each stage of an intermittent process, establishing a 2D-FM (two-dimensional-frequency modulation) different-dimensional switching system model with interval time-varying time lag by defining system output and state errors and introducing expansion information, selecting a 2D Lyapunov functional and combining a residence time method by considering a performance index function, providing an optimal cost control law design scheme when an index of the system depending on the interval time lag is stable in a matrix inequality constraint form, solving a controller gain meeting the upper bound of the performance index by introducing a convex optimization algorithm, and simultaneously solving the minimum operation time of each stage depending on the time lag.
On the basis of newly designing an equivalent model, the invention ensures that the system stably runs and simultaneously improves the tracking performance of the system through the design of the optimal cost control law, shortens the running time and aims to improve the production efficiency and the like.
The technical scheme adopted by the invention is as follows:
a method for optimal cost control of a multi-stage batch process with time-lag and disturbances, comprising the steps of:
step 1, constructing an equivalent two-dimensional switching system model
Step 1.1, constructing a two-dimensional switching state space model
Establishing an equivalent switching error state space model with expansion information according to the discrete state space model of each stage in each batch of system operation;
for intermittent processes with time lag and disturbances, the model is described below
Figure GDA0003232582930000031
Wherein k and t respectively represent the batch of the intermittent process and the running time of the batch, and x (t, k +1), y (t, k +1) and u (t, k +1) respectively represent the system state, system output and system input of the batch of k +1 at the time of t; d (t) represents a state time lag in the direction of time t and satisfies dm≤d(t)≤dM；ρ(·,·):Z+×Z+→qWhere {1,2, …, q } represents the switching signal, q represents the total number of phases per batch of the batch process, x0,k+1Initial state of the k +1 th duty cycle, ωρ(t,k)(t, k +1) is an unknown external disturbance;
for a multi-stage batch process, each stage corresponds to a subsystem, and when the corresponding subsystem is activated during the operation of the multi-stage batch process, the formula (1) can be rewritten into the formula (2):
Figure GDA0003232582930000032
wherein i represents the stage of the batch process,
Figure GDA0003232582930000033
Figure GDA0003232582930000034
wherein the content of the first and second substances,
Figure GDA0003232582930000039
in the form of a matrix of constants of appropriate dimensions,
Figure GDA0003232582930000035
perturbation matrix for unknown uncertain parameters and satisfy
Figure GDA0003232582930000036
Wherein, FiT(t,k)Fi(t,k)≤Ii 0≤t≤T；k＝1,2,…，
Figure GDA0003232582930000037
Is a known dimension-adaptive constant matrix;
step 1.2, converting the constructed two-dimensional multi-stage state space model into a two-dimensional switching system model
Aiming at the system (2), a multi-stage intermittent process two-dimensional augmentation model is constructed; introduction of
Figure GDA0003232582930000038
Figure GDA0003232582930000041
Figure GDA0003232582930000042
Wherein u isi(t,0) denotes an initial value of the iterative algorithm, ri(t, k +1) represents the iterative learning update law for phase i, δ (f)i(t, k +1)) represents the variable fi(t, k +1) error in the k +1 direction; e.g. of the typei(t, k +1) represents the actual output value y of the systemi(t, k +1) and the system output set value
Figure GDA0003232582930000043
An error of (2);
Figure GDA0003232582930000044
is in an expanded state; substituting the expressions (3) and (4) into the expression (1) to obtain a two-dimensional state error space model and a two-dimensional output error space model of the intermittent process stage i represented by the expressions (6) and (7);
Figure GDA0003232582930000045
Figure GDA0003232582930000046
wherein
Figure GDA0003232582930000047
The intermittent process equivalent two-dimensional augmentation model represented by the formula (8a) can be obtained by combining and representing the formulas (5), (6) and (7) in a matrix form
Figure GDA0003232582930000048
Wherein the content of the first and second substances,
Figure GDA0003232582930000049
Figure GDA00032325829300000410
Figure GDA00032325829300000411
Iiis an adaptive identity matrix;
the equation (8a) model is equivalently reproduced as a switching system model
Figure GDA00032325829300000412
Step 2, designing a controller according to the constructed multi-stage two-dimensional state space model and the optimal cost control algorithm
From the above description, the iterative learning update law of phase i can be expressed as follows:
Figure GDA0003232582930000051
by substituting equation (9) for equation (8a) and reproducing it in the form of a switching model as in equation (8b), a two-dimensional closed-loop switching state space model of an intermittent process can be obtained, which is represented by equation (10):
Figure GDA0003232582930000052
wherein the content of the first and second substances,
Figure GDA0003232582930000053
aiming at the two-dimensional closed-loop switching state space model of the multi-stage intermittent process, an updating law r is designedi(t, k +1) and which satisfies the following cost function
Figure GDA0003232582930000054
System state for different phases simultaneously
Figure GDA0003232582930000055
The initial condition of which is satisfied
Figure GDA0003232582930000056
Wherein
Figure GDA0003232582930000057
Is a positive real number and is,
Figure GDA0003232582930000058
and
Figure GDA0003232582930000059
is a constant that is given to the user,
Figure GDA00032325829300000510
called zero initial condition, selecting segmented Lyapunov function for each stage of multi-stage intermittent process with interval time-varying time lag
Figure GDA00032325829300000511
Figure GDA00032325829300000512
Figure GDA00032325829300000513
Figure GDA00032325829300000514
Figure GDA00032325829300000515
In increments of
Figure GDA0003232582930000061
Figure GDA0003232582930000062
Figure GDA0003232582930000063
Figure GDA0003232582930000064
Figure GDA0003232582930000065
Wherein the content of the first and second substances,
Figure GDA0003232582930000066
Pi，Qi，Wiand RiA positive definite matrix corresponding to the ith stage is to be solved; alpha is alphaiIs a positive number less than 1; t represents matrix transposition;
and step 3: solving for controller and runtime dependent on upper and lower time lag bounds with minimal control cost
Obtained according to the above
Figure GDA0003232582930000067
Has the following formula
Figure GDA0003232582930000068
Wherein the content of the first and second substances,
Figure GDA0003232582930000069
Figure GDA00032325829300000610
obtaining controller parameter, i.e. state feedback gain
Figure GDA00032325829300000611
In fact, (16) holds, only the following holds
Figure GDA0003232582930000071
(17) The essential condition for the establishment of the formula is that the following formula is established
Figure GDA0003232582930000072
Wherein
Figure GDA0003232582930000073
Figure GDA0003232582930000074
Figure GDA0003232582930000075
Figure GDA0003232582930000076
Figure GDA0003232582930000077
Figure GDA0003232582930000078
Figure GDA0003232582930000079
Figure GDA0003232582930000081
Figure GDA0003232582930000082
d2＝(dM-dm-1)-1。
While the cost function (11) satisfies the following constraint:
Figure GDA0003232582930000083
limiting
Figure GDA0003232582930000084
0＜λi，αi＜1，
Wherein
Figure GDA0003232582930000085
γ3＝λmax(Xi)-1,Ωi＝(Pi)-1,Xi＝(Ri)-1,
Figure GDA0003232582930000086
Figure GDA0003232582930000087
Figure GDA0003232582930000088
Representative of the system initial conditions (12) in a transformable form xiiIs a given matrix, by solving (20) - (21) that satisfy the above constraints,
Figure GDA0003232582930000089
can be obtained by substituting the formula (3) intoi(t,k+1)；
It is clear that each phase ui(t, k +1) obtaining a time lag value and a cost index function; for each phase of the dwell time (the dwell time refers to the run time of the system in the corresponding phase) satisfied formula
Figure GDA00032325829300000810
It is clear thatiIs obtained depending on
Figure GDA00032325829300000811
While
Figure GDA00032325829300000812
With time lag, i.e. mu, iniThe magnitude is affected by the time lag, while alpha is the value of the inequality being solvediThe value size is also affected by a time lag, and obviously, the running time of each stage is affected by the time lag. However, the method provided by the invention not only ensures the stable performance of the system, but also shortens the running time of the system, and simultaneously ensures that the system has the minimum cost.
The invention has the advantages of
The invention designs the optimal cost-guaranteed controllers corresponding to different stages aiming at the multi-stage intermittent process with interval time-varying time lag and interference, thereby not only ensuring the stable operation of the intermittent process, but also reducing the operation time of each stage and minimizing the production cost, realizing energy conservation and emission reduction for enterprise production and maximizing the economic benefit.
Drawings
Fig. 1 is a tracking performance graph.
Fig. 2 is a switching time diagram.
FIG. 3 is a graph of batch 6 output response.
Fig. 4 is a 50 th batch output response graph.
FIG. 5 is a flow chart of an optimal cost control method for a multi-stage batch process with time-lag and disturbance in accordance with the present invention.
Detailed Description
The method for controlling the optimal cost of the multi-stage intermittent process with time lag and interference in the invention is further explained by combining the attached drawings:
example 1
As shown in fig. 1, a method for optimal cost control of a multi-stage batch process with time-lag and disturbances, comprising the steps of:
step 1, constructing an equivalent two-dimensional switching system model
Step 1.1, constructing a two-dimensional switching state space model
And establishing an equivalent switching error state space model with expansion information according to the discrete state space model of each stage in each batch of system operation.
For intermittent processes with time lag and disturbances, the model is described below
Figure GDA0003232582930000101
Wherein k and t respectively represent the batch of the batch process and the running time of the batch, and x (t, k +1), y (t, k +1) and u (t, k +1) respectively represent the system state, system output and system input of the batch of k +1 at the time of t. d (t) represents a state time lag in the direction of time t and satisfies dm≤d(t)≤dM。ρ(·,·):Z+×Z+→qWhere {1,2, …, q } represents the switching signal, q represents the total number of phases per batch of the batch process, x0,k+1Initial state of the k +1 th duty cycle, ωρ(t,k)(t, k +1) is an unknown external disturbance.
For a multi-stage batch process, each stage corresponds to a subsystem, and when the corresponding subsystem is activated during the operation of the multi-stage batch process, the formula (1) can be rewritten into the formula (2):
Figure GDA0003232582930000102
wherein i represents the stage of the batch process,
Figure GDA0003232582930000103
Figure GDA0003232582930000104
wherein the content of the first and second substances,
Figure GDA0003232582930000105
in the form of a matrix of constants of appropriate dimensions,
Figure GDA0003232582930000106
perturbation matrix for unknown uncertain parameters and satisfy
Figure GDA0003232582930000107
Wherein, FiT(t,k)Fi(t,k)≤Ii 0≤t≤T；k＝1,2,…，
Figure GDA0003232582930000108
Is a known dimension-adaptive constant matrix;
step 1.2, converting the constructed two-dimensional multi-stage state space model into a two-dimensional switching system model
Aiming at the system (2), a multi-stage intermittent process two-dimensional augmentation model is constructed; introduction of
Figure GDA0003232582930000109
Figure GDA0003232582930000111
Figure GDA0003232582930000112
Wherein u isi(t,0) denotes an initial value of the iterative algorithm, ri(t, k +1) represents the iterative learning update law for phase i, δ (f)i(t, k +1)) represents the variable fi(t, k +1) error in the k +1 direction; e.g. of the typei(t, k +1) represents the actual output value y of the systemi(t, k +1) and the system output set value
Figure GDA0003232582930000113
An error of (2);
Figure GDA0003232582930000114
in the extended state. Substituting equations (3) and (4) into equation (1) to obtain a two-dimensional state error space model and a two-dimensional output error space of the intermittent process stage i represented by equations (6) and (7)A model;
Figure GDA0003232582930000115
Figure GDA0003232582930000116
wherein
Figure GDA0003232582930000117
The intermittent process equivalent two-dimensional augmentation model represented by the formula (8a) can be obtained by combining and representing the formulas (5), (6) and (7) in a matrix form
Figure GDA0003232582930000118
Wherein the content of the first and second substances,
Figure GDA0003232582930000119
Figure GDA00032325829300001110
Figure GDA00032325829300001111
Iiis an adaptive identity matrix;
the equation (8a) model is equivalently reproduced as a switching system model
Figure GDA00032325829300001112
Step 2, designing a controller according to the constructed multi-stage two-dimensional state space model and the optimal cost control algorithm
From the above description, the iterative learning update law of phase i can be expressed as follows:
Figure GDA0003232582930000121
by substituting equation (9) for equation (8a) and reproducing it in the form of a switching model as in equation (8b), a two-dimensional closed-loop switching state space model of an intermittent process can be obtained, which is represented by equation (10):
Figure GDA0003232582930000122
wherein the content of the first and second substances,
Figure GDA0003232582930000123
aiming at the two-dimensional closed-loop switching state space model of the multi-stage intermittent process, an updating law r is designedi(t, k +1) and which satisfies the following cost function
Figure GDA0003232582930000124
System state for different phases simultaneously
Figure GDA0003232582930000125
The initial condition of which is satisfied
Figure GDA0003232582930000126
Wherein
Figure GDA0003232582930000127
Is a positive real number and is,
Figure GDA0003232582930000128
and
Figure GDA0003232582930000129
is a constant that is given to the user,
Figure GDA00032325829300001210
called zero initial condition, selecting segmented Lyapunov function for each stage of multi-stage intermittent process with interval time-varying time lag
Figure GDA00032325829300001211
Figure GDA00032325829300001212
Figure GDA00032325829300001213
Figure GDA00032325829300001214
Figure GDA00032325829300001215
In increments of
Figure GDA0003232582930000131
Figure GDA0003232582930000132
Figure GDA0003232582930000133
Figure GDA0003232582930000134
Figure GDA0003232582930000135
Wherein the content of the first and second substances,
Figure GDA0003232582930000136
Pi，Qi，Wiand RiA positive definite matrix corresponding to the ith stage is to be solved; alpha is alphaiIs a positive number less than 1; t represents matrix transposition;
and step 3: solving for controller and runtime dependent on upper and lower time lag bounds with minimal control cost
Obtained according to the above
Figure GDA0003232582930000137
Has the following formula
Figure GDA0003232582930000138
Wherein the content of the first and second substances,
Figure GDA0003232582930000139
Figure GDA00032325829300001310
obtaining controller parameter, i.e. state feedback gain
Figure GDA00032325829300001311
In fact, (16) holds, only the following holds
Figure GDA0003232582930000141
(17) The essential condition for the establishment of the formula is that the following formula is established
Figure GDA0003232582930000142
Wherein
Figure GDA0003232582930000143
Figure GDA0003232582930000144
Figure GDA0003232582930000145
Figure GDA0003232582930000146
Figure GDA0003232582930000147
Figure GDA0003232582930000148
Figure GDA0003232582930000149
Figure GDA0003232582930000151
Figure GDA0003232582930000152
d2＝(dM-dm-1)-1。
While the cost function (11) satisfies the following constraint:
Figure GDA0003232582930000153
limiting
Figure GDA0003232582930000154
0＜λi，αi＜1， (21)
Wherein
Figure GDA0003232582930000155
γ3＝λmax(Xi)-1,Ωi＝(Pi)-1,Xi＝(Ri)-1,
Figure GDA0003232582930000156
Figure GDA0003232582930000157
Figure GDA0003232582930000158
Representative of the system initial conditions (12) in a transformable form xiiIs a given matrix, by solving (20) - (21) that satisfy the above constraints,
Figure GDA0003232582930000159
can be obtained by substituting the formula (3) intoi(t,k+1)。
It is clear that each phase uiThe (t, k +1) gains depend not only on the magnitude of the time lag, but also on the cost index function. For each phase of the dwell time (the dwell time refers to the run time of the system in the corresponding phase) satisfied formula
Figure GDA00032325829300001510
It is clear thatiIs obtained depending on
Figure GDA00032325829300001511
While
Figure GDA00032325829300001512
With time lag, i.e. mu, iniThe magnitude is affected by the time lag, while alpha is the value of the inequality being solvediThe value size is also affected by a time lag, and obviously, the running time of each stage is affected by the time lag. However, the method provided by the invention not only ensures the stable performance of the system, but also shortens the running time of the system, and simultaneously ensures that the system has the minimum cost.
Example 2
The injection molding process is a typical multi-stage batch process, and each batch mainly includes an injection section → a pressure holding section → a cooling section.
The injection molding cycle generally begins with the mold closed. The injection section has the function of uniformly plasticizing the plastic in the barrel, and then injecting the molten material into the mold cavity through the screw at high speed and high pressure until the mold cavity is completely filled with the melt. After the injection phase is complete, the system enters a dwell phase in order to continue the polymer into the mold cavity to compensate for the volumetric shrinkage due to cooling and solidification. There is a speed/pressure switch (V/P switch) between the injection phase and the dwell phase, this switch point indicating the end of the injection phase and the start of the dwell phase. And after the pressure maintaining stage is finished, the injection molding process enters a cooling plasticizing stage. The screw is driven by a motor to rotate, and molten materials in the charging barrel are conveyed forwards; and the pressure of the storage chamber is increased due to the continuous accumulation of the melt at the head of the screw rod, the screw rod is pushed to move backwards, the screw rod stops rotating until the screw rod retreats to the preset position, and the plasticizing process is finished. After the plasticizing process is finished, the polymer in the mold cavity is continuously cooled until the polymer is completely solidified, and the product is ejected. This is the cooling and mold opening stage. The above process is a complete injection molding process.
In order to ensure the product quality and the production efficiency, key variables need to be considered in the production process of each batch so as to achieve high-precision control of the whole production process. For example, to reasonably control the injection speed, too slow or too fast will affect the pressure in the mold cavity to reduce the product quality, and then the pressure is maintained, and the sudden change of pressure will also affect the injection speed, so as to increase the operation time. Generally, sudden changes in these variables during production can affect subsequent production, and more seriously, the batch of product can become waste if overshooting or shaking occurs during production. In addition, in the injection molding process, the control valve is opened, the set controller does not drive the screw immediately, information can have a certain lag, the lag information is too long, the stability of the system is obviously reduced and even unstable, and meanwhile, the pressure maintaining stage is further influenced. The influence is reflected in the practical production, and the product is a poor product. Therefore, the problem of system stability caused by time lag in a multi-stage intermittent process is very important for actual production, and not only is the stable operation only the basic requirement of the production process at present, but also the problem of ensuring the lowest cost is the most concerned by producers under the condition. The injection molding process is widely applied to plastic processing and other related fields, so that attention of researchers is attracted.
Different control laws are designed for the injection section and the pressure maintaining section in the injection molding process
Solving through the inequality to obtain the gain of the controller as follows:
Figure GDA0003232582930000171
then according to
Figure GDA0003232582930000172
Is calculated to obtain tau1＝90，τ2When the cost performance index of the two stages is 91, J is obtained1*＝1280.7，J2*＝69.4231。
As shown in fig. 1 to 4, the control effects obtained in the simulation of the present embodiment are respectively shown. Fig. 1 shows that the tracking performance of the method of the invention is superior to that of an intermittent process with interval time-varying time lag and disturbance. As can be seen from fig. 2, after several batches, the switching time is stabilized at 90 steps, i.e. the operation time of phase 1 is 90 steps, and the second phase is solved to 91 steps, so that the minimum operation time of the switching system is 181 steps. Therefore, the system introducing the expansion information has higher efficiency in practical application. As can be seen from fig. 3 and 4, although the previous batches may experience an unstable period, the tracking error becomes smaller and smaller after a period of operation, and almost reaches zero-error tracking. For injection molding processes, the run time of each stage has an effect on the total time of the run of the batch, and in addition to relying on process improvements to reduce run time, it is also feasible to design efficient controllers to reduce time. The controller designed by the optimal insurance cost control method is adopted for the multi-stage intermittent process with interval time-varying time lag and interference, so that the system running time is greatly shortened while the product quality is ensured, and a feasible control method is provided for the efficient running of the intermittent process.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.

Claims (1)

1. A method for optimal cost control of a multi-stage batch process with time-lag and disturbances, characterized by: the method comprises the following steps:
step 1, constructing an equivalent two-dimensional switching system model
Step 1.1, constructing a two-dimensional switching state space model
Establishing an equivalent switching error state space model with expansion information according to the discrete state space model of each stage in each batch of system operation;
for intermittent processes with time lag and disturbances, the model is described below
Figure FDA0003232582920000011
Wherein k and t respectively represent the batch of the intermittent process and the running time of the batch, and x (t, k +1), y (t, k +1) and u (t, k +1) respectively represent the system state, system output and system input of the batch of k +1 at the time of t; d (t) represents a state time lag in the direction of time t and satisfies dm≤d(t)≤dM；ρ(·,·):Z+×Z+→qWhere {1,2, …, q } represents the switching signal, q represents the total number of phases per batch of the batch process, x0,k+1Initial state of the k +1 th duty cycle, ωρ(t,k)(t, k +1) is an unknown external disturbance;
for a multi-stage batch process, each stage corresponds to a subsystem, and when the corresponding subsystem is activated during the operation of the multi-stage batch process, the formula (1) can be rewritten into the formula (2):
Figure FDA0003232582920000012
wherein i represents the stage of the batch process,
Figure FDA0003232582920000013
Figure FDA0003232582920000014
wherein the content of the first and second substances,
Figure FDA0003232582920000015
in the form of a matrix of constants of appropriate dimensions,
Figure FDA0003232582920000016
perturbation matrix for unknown uncertain parameters and satisfy
Figure FDA0003232582920000017
Wherein, FiT(t,k)Fi(t,k)≤Ii 0≤t≤T；k＝1,2,… ，
Figure FDA0003232582920000021
Is a known dimension-adaptive constant matrix;
step 1.2, converting the constructed two-dimensional multi-stage state space model into a two-dimensional switching system model
Aiming at the system (2), a multi-stage intermittent process two-dimensional augmentation model is constructed; introduction of
Figure FDA0003232582920000022
Figure FDA0003232582920000023
Figure FDA0003232582920000024
Wherein u isi(t,0) denotes an initial value of the iterative algorithm, ri(t, k +1) represents the iterative learning update law for phase i, δ (f)i(t, k +1)) represents the variable fi(t, k +1) error in the k +1 direction; e.g. of the typei(t, k +1) represents the actual output value y of the systemi(t, k +1) and the system output set value
Figure FDA00032325829200000211
An error of (2);
Figure FDA00032325829200000212
is in an expanded state; substituting the expressions (3) and (4) into the expression (1) to obtain a two-dimensional state error space model and a two-dimensional output error space model of the intermittent process stage i represented by the expressions (6) and (7);
Figure FDA0003232582920000025
Figure FDA0003232582920000026
wherein
Figure FDA0003232582920000027
The intermittent process equivalent two-dimensional augmentation model represented by the formula (8a) can be obtained by combining and representing the formulas (5), (6) and (7) in a matrix form
Figure FDA0003232582920000028
Wherein the content of the first and second substances,
Figure FDA0003232582920000029
Figure FDA00032325829200000210
Figure FDA0003232582920000031
Iiis an adaptive identity matrix;
the equation (8a) model is equivalently reproduced as a switching system model
Figure FDA0003232582920000032
Step 2, designing a controller according to the constructed multi-stage two-dimensional state space model and the optimal cost control algorithm
From the above description, the iterative learning update law of phase i can be expressed as follows:
Figure FDA0003232582920000033
by substituting equation (9) for equation (8a) and reproducing it in the form of a switching model as in equation (8b), a two-dimensional closed-loop switching state space model of an intermittent process can be obtained, which is represented by equation (10):
Figure FDA0003232582920000034
wherein the content of the first and second substances,
Figure FDA0003232582920000035
aiming at the two-dimensional closed-loop switching state space model of the multi-stage intermittent process, an updating law r is designedi(t, k +1) and which satisfies the following cost function
Figure FDA0003232582920000036
System state for different phases simultaneously
Figure FDA0003232582920000037
The initial condition of which is satisfied
Figure FDA0003232582920000038
Wherein
Figure FDA0003232582920000039
Is a positive real number and is,
Figure FDA00032325829200000310
and
Figure FDA00032325829200000311
is a given constant.
Figure FDA00032325829200000312
Referred to as zero initial condition, for multiple orders with interval time varying skewSelecting segmented Lyapunov functions at each stage of the segment intermittent process
Figure FDA0003232582920000041
Figure FDA0003232582920000042
Figure FDA0003232582920000043
Figure FDA0003232582920000044
Figure FDA0003232582920000045
In increments of
Figure FDA0003232582920000046
Figure FDA0003232582920000047
Figure FDA0003232582920000048
Figure FDA0003232582920000049
Figure FDA00032325829200000410
Wherein the content of the first and second substances,
Figure FDA00032325829200000411
Pi，Qi，Wiand RiA positive definite matrix corresponding to the ith stage is to be solved; alpha is alphaiIs a positive number less than 1; t represents matrix transposition;
and step 3: solving for controllers dependent on upper and lower time-lag bounds with minimal control cost and runtime derived from the above
Figure FDA00032325829200000412
Has the following formula
Figure FDA00032325829200000413
Wherein the content of the first and second substances,
Figure FDA0003232582920000051
Figure FDA0003232582920000052
obtaining controller parameter, i.e. state feedback gain
Figure FDA0003232582920000053
In fact, (16) holds, only the following holds
Figure FDA0003232582920000054
(17) The essential condition for the establishment of the formula is that the following formula is established
Figure FDA0003232582920000055
Wherein
Figure FDA0003232582920000056
Figure FDA0003232582920000057
Figure FDA0003232582920000058
Figure FDA0003232582920000059
Figure FDA0003232582920000061
Figure FDA0003232582920000062
Figure FDA0003232582920000063
Figure FDA0003232582920000064
Figure FDA0003232582920000065
While the cost function (11) satisfies the following constraint:
Figure FDA0003232582920000066
limiting
Figure FDA0003232582920000067
0＜λiαi＜1， (21)
Wherein
Figure FDA0003232582920000068
Figure FDA0003232582920000069
Figure FDA00032325829200000610
Figure FDA00032325829200000611
Representative of the system initial conditions (12) in a transformable form xiiIs a given matrix, by solving (20) - (21) that satisfy the above constraints,
Figure FDA00032325829200000612
can be obtained by substituting the formula (3) intoi(t, k + 1); it is clear that each phase ui(t, k +1) obtaining a time lag value and a cost index function; for each phase of the dwell time (the dwell time refers to the run time of the system in the corresponding phase) satisfied formula
Figure FDA0003232582920000071
It is clear thatiIs obtained depending on
Figure FDA0003232582920000072
While
Figure FDA0003232582920000073
With time lag, i.e. mu, iniThe magnitude is affected by the time lag, while alpha is the value of the inequality being solvediThe value size is also affected by a time lag, and obviously, the running time of each stage is affected by the time lag.
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