CN109212972B - Limited rolling time domain hybrid 2D tracking control method for intermittent process - Google Patents

Abstract

The invention relates to an advanced 2D tracking control design method, in particular to a limited rolling time domain hybrid 2D tracking control method of an intermittent process, which comprises the following steps: step 1, establishing a state space model of an equivalent 2D intermittent process; step 2, designing a limited rolling time domain hybrid 2D tracking controller; step 3, designing optimal controllers and switching signals in different stages according to performance indexes in different stages to obtain switching conditions and operating time in different stages; and 4, analyzing stability, obtaining a system stability condition and designing a switching signal meeting the system stability according to a Lyapunov function stability theory and an average residence time method of the 2D system. The method provided by the invention is simple and easy to implement, the system operation time is obviously shortened, so that the production efficiency is improved, and the method has the characteristics of real-time updating, quick tracking and good control performance.

Description

Limited rolling time domain hybrid 2D tracking control method for intermittent process

Technical Field

The invention relates to an advanced 2D tracking control design method, in particular to a limited rolling time domain hybrid 2D tracking control method of an intermittent process, which is a design method for an injection molding process controller.

Background

At this stage, our country is the largest manufacturing country in the world, but most of the product processing is concentrated in the middle and low end markets. One of the main reasons for this phenomenon is the low technical content and high energy consumption of the production and processing equipment. Therefore, it is important to design a more efficient energy-saving controller without changing the original equipment, and the most important design of the controller is an algorithm, so that it is critical to design a new algorithm to realize multi-stage high-precision control in the intermittent process. The reason why the currently used system is high precision realized by aiming at a single stage, the intermittent process is a multi-stage process, the realization of the high precision of a certain stage does not represent that the whole process is high precision, is because the intermittent process is a process produced according to a certain processing procedure and a multi-stage production process, and adjacent stages are mutually associated and mutually influenced, so that corresponding algorithms must be designed aiming at different stages, thereby ensuring the quality of the produced products.

In addition, the control technology of the intermittent process is mostly one-dimensional, only the influence of time on the production process is considered, and the control performance of the control system is reduced after the control system runs for a period of time due to the fact that the actual working condition has a complex environment. At present, aiming at the repeatability and the two-dimensional characteristic of the intermittent process, attention is paid to a feedback and iterative learning control method, but under the influence of interference and other factors, the problem of system state deviation cannot be solved by the existing robust iterative learning reliable control, namely, the same control law is adopted from beginning to end, and the system deviation is increased along with the time. This can have a negative effect on the continuous stable operation and control performance of the system, even compromising the quality of the product. Although some of the controllers can adjust, the switching time and the operation time cannot be accurately adjusted to achieve the expected effect.

Therefore, it is necessary to provide a more effective control method for seeking suitable switching conditions and operation time at different stages of the intermittent process, and for solving the problems of model mismatch and interference in control, unchanged controller gain, and the like.

Disclosure of Invention

For the above-mentioned cases that occur with batch processes: the invention designs a multi-stage intermittent process limited rolling time domain hybrid 2D tracking controller, so that a system still stably operates under the condition of model mismatch and interference, and good tracking performance is realized. The invention solves the defect that the gain of the controller is not adjustable in the prior method by adding the controller with more flexible adjustment of the adjustable weighting coefficient, improves the control quality and realizes better control performance.

The invention aims to seek the appropriate switching condition and running time at different stages of the intermittent process; and secondly, providing a limited rolling time domain hybrid 2D tracking control method for the intermittent process to improve the control performance and the anti-interference performance of the control method in the intermittent process. The method comprises the steps of firstly selecting a proper state variable to establish a multi-stage state space model through an i-th stage input and output correlation model of an intermittent process, further converting the state space model into a 2D-Roesser model containing the state variable and an output tracking error, representing the model by using a switching system, and then obtaining an optimal control law by combining Riccati equations and boundary conditions according to performance indexes of different stages. And finally designing a residence time method depending on the Lyapunov function aiming at different stages. The method provided by the invention is simple and easy to implement, the system operation time is obviously shortened, so that the production efficiency is improved, and the method has the characteristics of real-time updating, quick tracking and good control performance.

The invention is realized by the following technical scheme:

the limited rolling time domain hybrid 2D tracking control method of the intermittent process is characterized by comprising the following steps: the method comprises the following specific steps:

step 1, establishing a state space model of an equivalent 2D intermittent process:

the correlation model of the input and output at the ith stage of the batch process is as follows:

F_i(z^-1)yⁱ(t,k)＝H_i(z^-1)uⁱ(t,k) (1)

wherein, yⁱ(t, k) and uⁱ(t, k) are the output and input variables of the input-output model, respectively, F_i(z^-1) And H_i(z^-1) Are each yⁱ(t, k) and uⁱ(t, k) coefficient polynomials, and the following equations exist:

F_i(z^-1)＝1+f₁ ⁱz^-1+f₂ ⁱz^-2+···+f_n ⁱz^-n (2)

H_i(z^-1)＝h₁ ⁱz^-1+h₂ ⁱz^-2+···+h_m ⁱz^-m (3)

wherein f is_l ⁱ,h_s ⁱAre the corresponding coefficients, l 1., n, s 1., m; z is a radical of^-1A backward shift operator; m, n is the input-output model order;

introducing a non-minimum state space variable:

x_m ⁱ(t,k)^T＝[yⁱ(t,k)^T,yⁱ(t-1,k)^T,…,yⁱ(t-n+1,k)^T,uⁱ(t-1,k)^T,uⁱ(t-2,k)^T,···,uⁱ(t-m+1,k)^T] (4)

combining (2) to (4), the input-output model (1) is transformed into a state space model:

x_o ⁱ(t+1,k)＝A_o ⁱx_o ⁱ(t,k)+B_o ⁱuⁱ(t,k) (5)

yⁱ(t+1,k)＝C_o ⁱx_o ⁱ(t+1,k) (6)

wherein x is_o ⁱ(t+1,k),yⁱ(t +1, k) are the value of the state variable and the value of the output variable at time t +1, respectively, uⁱ(t, k) is the value of the input delta variable at time t, yⁱ(t-l,k),uⁱ(t-s, k) are the output delta and input delta values, respectively, at time t-i, where l is 0, 1. A. the_o ⁱ，B_o ⁱ，C_o ⁱRespectively corresponding state matrix, input matrix and output matrix; t is the number of the transposed symbols taken,

and:

B_o ⁱ＝[h ₁ ⁱ 0…0 I ⁱ 0…0 0]^T，C_o ⁱ＝[Iⁱ 0 0…0 0 0 0](ii) a Wherein IⁱIs an identity matrix;

step 2, designing a limited rolling time domain hybrid 2D tracking controller of the intermittent process;

step 3, designing optimal controllers and switching signals sigma (t, k) at different stages according to performance indexes at different stages to obtain switching conditions and operating time at different stages;

step 4, stability analysis, namely obtaining a system stability condition and designing a switching signal meeting the system stability according to a Lyapunov function stability theory and an average residence time method of the 2D system;

the step 2 specifically comprises the following steps:

establishing an equivalent 2D model:

if the operation result has an error with the expected value, namely: when inputting uⁱ(t, k) it is difficult to reach the expected value after the system operation, and for better tracking performance, the tracking error is defined according to the designed state space model for the ith stage:

is given a desired trajectory;

aiming at the state space model of the ith stage, the following iterative learning control law is introduced:

uⁱ(t,k)＝uⁱ(t,k-1)+rⁱ(t,k),uⁱ(t,0)＝0，t＝0，1，2 (7)

wherein u isⁱ(t,0) is an initial value for the iteration, typically set to 0, rⁱ(t, k) belongs to R and is an iterative learning updating law, and an iterative learning updating law R is designedⁱThe purpose of (t, k) is to determine that it makes yⁱ(t, k) tracking

Defining:

the state space model from step 1.2 can be extended to a 2D model containing state variables and output tracking error, of the form:

models (9) - (10) were combined and extended to the following form using the Roesser model:

wherein the content of the first and second substances,

0 in the matrix represents a zero matrix, then the system (11) is reproduced as a switching system model:

wherein σ (t, k) is Z⁺→ N (1, 2) indicates a switching signal, which is related to time or system state, N is the number of sub-phases, and the switching sequence is defined as S: ═ T₀,T₁...T_t... }; all the time intervals of the successive interruptions satisfy T_t+1-T_t≥τⁱ,t＝0，1，2，...,T_tRepresents the T-th switching time, T₀Is the initial time, τⁱIs the dwell time of the different phases, and its value depends on the lyapunov function,

represented by a switching system model for the different phases;

the update law is designed for each phase i as:

wherein，KⁱIn order for the gain matrix to be determined,

the step 3 specifically comprises the following steps:

considering a non-minimum realization expansion state space model containing a free terminal state, selecting corresponding performance indexes:

wherein M isⁱ,Nⁱ,

Weight matrixes of the ith state variable, the controlled input and the tail end state respectively,

for switching in system model

In order to optimize the time domain for the rolling,

respectively the starting end time and the tail end time;

according to the model performance indexes, the optimal control laws of the controllers in different stages are obtained as follows:

will update law rⁱ(t, k) using an iterative learning control law to obtain optimal control:

uⁱ(t,k)＝rⁱ(t,k)+uⁱ(t,k-1) (18)

at the next moment, new control quantity is continuously solved according to the same method steps, and the steps are circulated in sequence,

the step 4 specifically comprises the following steps:

for the switching system, controllers at different stages are designed:

wherein the content of the first and second substances,

for each phase, let:

the switching system may become:

for a given

Has a horizontal convergence index of not more than

Vertical convergence index of not more than

Existence of diagonal matrix

So that the following inequality holds:

wherein the content of the first and second substances,

the system is stable if the switching signal with the average dwell time satisfies the following inequality:

wherein the content of the first and second substances,

minimum run time of

And rounding up.

The invention has the beneficial effects that:

compared with the prior art, the method has the advantages and beneficial effects that a limited rolling time domain hybrid 2D tracking control law is designed on the basis of an i-th stage input and output correlation model of an intermittent process, a proper state variable is selected to establish a multi-stage state space model, a state error and an output error are introduced, the model is expanded into a 2D-Roesser model and is represented by a switching system model, according to performance indexes including terminal states designed at different stages, a 2D system Lyapunov function stability theory and an average residence time method, an optimal hybrid control law is given, the defect that the gain of a controller in the existing method is not adjustable is effectively overcome, the control quality is improved, better control performance is realized, sufficient conditions for stabilizing robust indexes of the system along the time and batch directions and the minimum running time of each stage are given, resulting in a significant reduction in system runtime. Finally, the aims of energy conservation, consumption reduction, cost reduction and high-precision control are achieved.

Drawings

FIG. 1 is a graph of tracking errors for lots 5, 30, and 60 of the present invention.

FIG. 2 is a system input diagram for lots 5, 30, and 60 of the present invention.

FIG. 3 is a graph of tracking error for all lots in accordance with the present invention.

FIG. 4 is a timing chart of the first phase switching of all lots in accordance with the present invention.

In fig. 1, the horizontal axis represents the number of steps, and the vertical axis represents the tracking error value; in fig. 2, the horizontal axis represents the number of steps, and the vertical axis represents the input value; in fig. 3, the horizontal axis represents the lot, and the vertical axis represents the tracking error value; in fig. 4, the horizontal axis represents the lot and the vertical axis represents the number of steps.

Detailed Description

The invention will be further described with reference to the following detailed description of the embodiments.

As shown in fig. 1 to 4, the limited rolling time domain hybrid 2D tracking control method of the intermittent process includes the following specific steps:

step 1, establishing a state space model of an equivalent 2D intermittent process:

1.1 the correlation model of the input and output at the ith stage of the batch process is as follows:

F_i(z^-1)yⁱ(t,k)＝H_i(z^-1)uⁱ(t,k) (1)

wherein, yⁱ(t, k) and uⁱ(t, k) are the output and input variables of the input-output model, respectively, F_i(z^-1) And H_i(z^-1) Are each yⁱ(t, k) and uⁱ(t, k) coefficient polynomials, and the following equations exist:

F_i(z^-1)＝1+f₁ ⁱz^-1+f₂ ⁱz^-2+···+f_n ⁱz^-n (2)

H_i(z^-1)＝h₁ ⁱz^-1+h₂ ⁱz^-2+···+h_m ⁱz^-m (3)

wherein f is_l ⁱ,h_s ⁱAre the corresponding coefficients, l 1., n, s 1., m; z is a radical of^-1A backward shift operator; m, n are input inputsDemolding the order of the model;

1.2 introduce non-minimum state space variables:

x_m ⁱ(t,k)^T＝[yⁱ(t,k)^T,yⁱ(t-1,k)^T,…,yⁱ(t-n+1,k)^T,uⁱ(t-1,k)^T,uⁱ(t-2,k)^T,···,uⁱ(t-m+1,k)^T] (4)

combining (2) to (4), the input-output model (1) is transformed into a state space model:

x_o ⁱ(t+1,k)＝A_o ⁱx_o ⁱ(t,k)+B_o ⁱuⁱ(t,k) (5)

yⁱ(t+1,k)＝C_o ⁱx_o ⁱ(t+1,k) (6)

wherein x is_o ⁱ(t+1,k),yⁱ(t +1, k) are the value of the state variable and the value of the output variable at time t +1, respectively, uⁱ(t, k) is the value of the input delta variable at time t, yⁱ(t-l,k),uⁱ(t-s, k) are the output delta and input delta values, respectively, at time t-i, where l is 0, 1. A. the_o ⁱ，B_o ⁱ，C_o ⁱRespectively corresponding state matrix, input matrix and output matrix; t is the number of the transposed symbols taken,

and:

B_o ⁱ＝[h ₁ ⁱ 0…0 Iⁱ 0…0 0]^T，C_o ⁱ＝[Iⁱ 0 0…0 0 0 0](ii) a Wherein IⁱIs an identity matrix;

step 2, designing a limited rolling time domain hybrid 2D tracking controller; the method comprises the following steps:

2.1 establishing an equivalent 2D model:

if the operation result has an error from the expected value,namely; when inputting uⁱ(t, k) it is difficult to reach the expected value after the system operation, and for better tracking performance, the tracking error is defined according to the designed state space model for the ith stage:

is given a desired trajectory;

aiming at the state space model of the ith stage, the following iterative learning control law is introduced:

uⁱ(t,k)＝uⁱ(t,k-1)+rⁱ(t,k),uⁱ(t,0)＝0，t＝0，1，2 (7)

wherein u isⁱ(t,0) is an initial value for the iteration, typically set to 0, rⁱ(t, k) epsilon R is an iterative learning updating law, and the purpose of designing the iterative learning updating law is to determine the updating law Rⁱ(t, k) is such that yⁱ(t, k) tracking

Defining:

△x_o ⁱ(t,k)＝x_o ⁱ(t,k)-x_o ⁱ(t,k-1) (8)

the state space model from step 1.2 can be extended to a 2D model containing state variables and output tracking error, of the form:

models (9) - (10) were combined and extended to the following form using the Roesser model:

wherein the content of the first and second substances,

0 in the matrix represents a zero matrix, then the system (11) is reproduced as a switching system model:

wherein σ (t, k) is Z⁺→ N (1, 2) indicates a switching signal, which is related to time or system state, N is the number of sub-phases, and the switching sequence is defined as S: ═ T₀,T₁...T_t... }; all the time intervals of the successive interruptions satisfy T_t+1-T_t≥τⁱ,t＝0，1，2，...,T_tRepresents the T-th switching time, T₀Is the initial time, τⁱIs the dwell time of the different phases, and its value depends on the lyapunov function,

represented by a switching system model for the different phases;

2.2 design the update law for each phase i as:

wherein, KⁱIn order for the gain matrix to be determined,

step 3, designing optimal controllers and switching signals sigma (t, k) at different stages according to performance indexes at different stages to obtain switching conditions and operating time at different stages; the method comprises the following steps:

3.1 considering the non-minimum realization extension state space model containing the free terminal state, selecting the corresponding performance indexes:

wherein M isⁱ,Nⁱ,

Weight matrixes of the ith state variable, the controlled input and the tail end state respectively,

for switching in system model

In order to optimize the time domain for the rolling,

respectively the starting end time and the tail end time;

3.2 according to the model performance index, obtaining the optimal control law of the controllers in different stages, as follows:

3.3 will update law rⁱ(t, k) using an iterative learning control law to obtain optimal control:

uⁱ(t,k)＝rⁱ(t,k)+uⁱ(t,k-1) (18)

3.4 at the next moment, continuously solving new control quantity according to the same method steps, and sequentially circulating;

step 4, stability analysis, namely obtaining a system stability condition and designing a switching signal meeting the system stability according to a Lyapunov function stability theory and an average residence time method of the 2D system; the method comprises the following steps:

4.1 for the switching system, the controllers at different stages are designed:

wherein the content of the first and second substances,

for each phase, let:

the switching system may become:

for a given

Has a horizontal convergence index of not more than

Vertical convergence index of not more than

Presence of diagonal matrices

So that the following inequality holds:

wherein the content of the first and second substances,

the system is stable if the switching signal with the average dwell time satisfies the following inequality:

wherein the content of the first and second substances,

minimum run time of

And rounding up.

Examples

The control effects of the injection and pressure maintaining stages in the injection molding process directly affect the quality and production efficiency of products, wherein the injection speed and the pressure of a mold cavity in the pressure maintaining stage have the greatest influence on the corresponding stage in the injection section, and a fixed value for tracking control needs to be given.

And (3) an injection stage: (1-0.9291 z)^-1-0.03191z^-1)IV＝(8.687z^-1-5.617z^-2)VO

And (3) pressure maintaining stage: (1-1.317 z)^-1-0.3259z^-2)NP＝(171.8z^-1-156.8z^-2)VO

Nozzle pressure model of injection velocity: (1-z)^-1)NP＝(0.1054z^-1)IV

For the injection speed, let:

it is possible to obtain:

the state model of the injection segment can be found as:

for the dwell phase, let:

it is possible to obtain:

the state model for the dwell phase can be obtained as:

wherein IV is the injection speed of the injection section, and the set value is 40 mm/s; and the pressure NP of the die cavity of the pressure maintaining section is the pressure of the die cavity of the pressure maintaining section, the set value is 300bar, and the VO represents the opening degree of the valve. The design switching condition is [ 0010 ]]xⁱ(t, k) ≧ 350, i.e., once the mold cavity pressure is greater than 350, the system will switch from the injection section to the hold pressure section. We can get a run time of 87 for the first phase and 88 for the second phase. And taking batches 5, 30 and 60 as examples, the experimental results are as follows: as can be seen from the figure, the batch error of the method provided by the invention is increased along with the increase of the batch, the error is almost a straight line which tends to 0, the tracking performance is good, and the system input tracking effect is initially realizedThe phase is poor, but as the batch increases, the tracking effect becomes better and better, and the input value is a smooth curve. The two-dimensional switching time tends to be a constant value as the batch increases. The feasibility and the superiority of the method are verified.

Claims (1)

1. The limited rolling time domain hybrid 2D tracking control method of the intermittent process is characterized by comprising the following steps: the method comprises the following specific steps:

step 1, establishing a state space model of an equivalent 2D intermittent process:

the correlation model of the input and output at the ith stage of the batch process is as follows:

F_i(z^-1)yⁱ(t,k)＝H_i(z^-1)uⁱ(t,k) (1)

wherein, yⁱ(t, k) and uⁱ(t, k) are the output and input variables of the input-output model, respectively, F_i(z^-1) And H_i(z^-1) Are each yⁱ(t, k) and uⁱ(t, k) coefficient polynomials, and the following equations exist:

F_i(z^-1)＝1+f₁ ⁱz^-1+f₂ ⁱz^-2+···+f_n ⁱz^-n (2)

H_i(z^-1)＝h₁ ⁱz^-1+h₂ ⁱz^-2+···+h_m ⁱz^-m (3)

wherein f is_l ⁱ,h_s ⁱAre the corresponding coefficients, l 1., n, s 1., m; z is a radical of^-1A backward shift operator; m, n is the input-output model order;

introducing a non-minimum state space variable:

x_m ⁱ(t,k)^T＝[yⁱ(t,k)^T,yⁱ(t-1,k)^T,…,yⁱ(t-n+1,k)^T,uⁱ(t-1,k)^T,uⁱ(t-2,k)^T,···,uⁱ(t-m+1,k)^T] (4)

combining (2) to (4), the input-output model (1) is transformed into a state space model:

x_o ⁱ(t+1,k)＝A_o ⁱx_o ⁱ(t,k)+B_o ⁱuⁱ(t,k) (5)

yⁱ(t+1,k)＝C_o ⁱx_o ⁱ(t+1,k) (6)

wherein x is_o ⁱ(t+1,k),yⁱ(t +1, k) are the value of the state variable and the value of the output variable at time t +1, respectively, uⁱ(t, k) is the value of the input delta variable at time t, yⁱ(t-l,k),uⁱ(t-s, k) are the output delta and input delta values, respectively, at time t-i, where l is 0, 1. A. the_o ⁱ，B_o ⁱ，C_o ⁱRespectively corresponding state matrix, input matrix and output matrix; t is the number of the transposed symbols taken,

and:

B_o ⁱ＝[h₁ ⁱ 0…0 Iⁱ 0…0 0]^T，C_o ⁱ＝[Iⁱ 0 0…0 0 0 0](ii) a Wherein IⁱIs an identity matrix;

step 2, designing a limited rolling time domain hybrid 2D tracking controller of the intermittent process;

step 3, designing optimal controllers and switching signals sigma (t, k) at different stages according to performance indexes at different stages to obtain switching conditions and operating time at different stages;

step 4, stability analysis, namely obtaining a system stability condition and designing a switching signal meeting the system stability according to a Lyapunov function stability theory and an average residence time method of the 2D system;

the step 2 specifically comprises the following steps:

establishing an equivalent 2D model:

if the operation result has an error with the expected value, namely: when inputting uⁱ(t, k) it is difficult to reach the expected value after the system operation, and for better tracking performance, the tracking error is defined according to the designed state space model for the ith stage:

is given a desired trajectory;

aiming at the state space model of the ith stage, the following iterative learning control law is introduced:

uⁱ(t,k)＝uⁱ(t,k-1)+rⁱ(t,k),uⁱ(t,0)＝0，t＝0，1，2 (7)

wherein u isⁱ(t,0) is an initial value for the iteration, typically set to 0, rⁱ(t, k) belongs to R and is an iterative learning updating law, and an iterative learning updating law R is designedⁱThe purpose of (t, k) is to determine that it makes yⁱ(t, k) tracking

Defining:

the state space model from step 1.2 can be extended to a 2D model containing state variables and output tracking error, of the form:

models (9) - (10) were combined and extended to the following form using the Roesser model:

wherein the content of the first and second substances,

0 in the matrix represents a zero matrix, then the system (11) is reproduced as a switching system model:

wherein σ (t, k) is Z⁺→ N (1, 2) indicates a switching signal, which is related to time or system state, N is the number of sub-phases, and the switching sequence is defined as S: ═ T₀,T₁...T_t... }; all the time intervals of the successive interruptions satisfy T_t+1-T_t≥τⁱ,t＝0，1，2，...,T_tRepresents the T-th switching time, T₀Is the initial time, τⁱIs the dwell time of the different phases, and its value depends on the lyapunov function,

represented by a switching system model for the different phases;

the update law is designed for each phase i as:

wherein, KⁱIn order for the gain matrix to be determined,

the step 3 specifically comprises the following steps:

considering a non-minimum realization expansion state space model containing a free terminal state, selecting corresponding performance indexes:

wherein M isⁱ,Nⁱ,

Weight matrixes of the ith state variable, the controlled input and the tail end state respectively,

for switching in system model

In order to optimize the time domain for the rolling,

respectively the starting end time and the tail end time;

according to the model performance indexes, the optimal control laws of the controllers in different stages are obtained as follows:

will update law rⁱ(t, k) using an iterative learning control law to obtain optimal control:

uⁱ(t,k)＝rⁱ(t,k)+uⁱ(t,k-1) (18)

at the next moment, new control quantity is continuously solved according to the same method steps, and the steps are circulated in sequence,

the step 4 specifically comprises the following steps:

for the switching system, controllers at different stages are designed:

wherein the content of the first and second substances,

for each phase, let:

the switching system may become:

for a given

Has a horizontal convergence index of not more than

Vertical convergence index of not more than

Presence of diagonal matrices

So that the following inequality holds:

wherein the content of the first and second substances,

the system is stable if the switching signal with the average dwell time satisfies the following inequality:

wherein the content of the first and second substances,

minimum run time of

And rounding up.

Images