CN107942669A - The limited rolling time horizon of batch injection moulding process mixes tracking and controlling method - Google Patents

The limited rolling time horizon of batch injection moulding process mixes tracking and controlling method Download PDF

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CN107942669A
CN107942669A CN201711232764.3A CN201711232764A CN107942669A CN 107942669 A CN107942669 A CN 107942669A CN 201711232764 A CN201711232764 A CN 201711232764A CN 107942669 A CN107942669 A CN 107942669A
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王立敏
罗卫平
余维燕
王鹏
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Hainan Normal University
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Abstract

The invention belongs to the Dynamic matrix control field of industrial process, is related to a kind of limited rolling time horizon of batch injection moulding process and mixes tracking and controlling method.This method establishes different phase input/output model by gathering inputoutput data first, suitable state variable is chosen again establishes multistage state-space model, state-space model is further converted to the Extended state space model comprising state variable and output tracking error, and represented with switching system model, then chosen for different phase and include the performance indicator of the SOT state of termination, tried to achieve with reference to Riccati equations and boundary condition and optimal mix control law.For this control law by increasing adjustable weighting coefficient, adjusting is more flexible, and ensures that system obtains more preferable control performance.Finally it is directed to different phase, residence time method of the design dependent on Lyapunov functions, the result that the method is drawn is not required to quote any other variable, simple and practicable and can shorten system operation time, that is, improves production efficiency.

Description

The limited rolling time horizon of batch injection moulding process mixes tracking and controlling method
Technical field
The invention belongs to the Dynamic matrix control field of industrial process, is related to a kind of limited rolling time horizon of batch injection moulding process and mixes Miscellaneous tracking and controlling method.
Background technology
Injection molding process is widely used in the association areas such as plastic processing, although having part for injection molding process Research, but be still a challenge in the high-precision control aspect of modern plastics processing.Main reason is that its complicated dynamic is special Property, and changeable process conditions.Injection molding process is typical multistage batch process, and each batch mainly includes injection It is injection speed and dwell pressure respectively in the variable that injection portion and pressurize section need to control, two are not with two stages of pressurize Different with the variable of stage control, control targe is different, when from a stage switches to another stage, and each stage running The length of time, directly affects production efficiency and product quality.Obviously, for such production process design high-precision controller and The switching condition of adjacent phases and the run time in each stage, will be most important.
High-precision control currently for the single stage is ripe, but single process is not related to switching condition, will not relate to And run time.Although also there is certain achievement in research for the multistage, controller gain cannot be adjusted in whole process Section.And in actual industrial control, since there is drift in actual condition, process is non-linear and the factor such as its exterior interference, control System processed its control performance after a period of time is run may decline.If repair controller is to improve Control platform not in time, The economic benefit that reduction control system is obtained.
Therefore, to seek the suitable switching condition of batch injection moulding process different phase, run time, to solve mould in controlling The problems such as type mismatch and interference, propose that a kind of significantly more efficient method of controlling is extremely necessary.
The content of the invention
The object of the invention one is to seek the suitable switching condition of batch injection moulding process different phase, run time；Second, for Improve the tracking performance and anti-interference of control method in batch process, propose the limited rolling time horizon tracking of batch injection moulding process Control method.This method establishes different phase input/output model by gathering inputoutput data first, then chooses suitably State variable establishes multistage state-space model, further by state-space model be converted to comprising state variable and output with The Extended state space model of track error, and represented with switching system model, then chosen for different phase and include terminal shape The performance indicator of state, tries to achieve with reference to Riccati equations and boundary condition and optimal mixes control law.This control law can by increase The weighting coefficient of adjusting, adjusting is more flexible, and ensures that system obtains more preferable control performance.Different phase is finally directed to, Residence time method of the design dependent on Lyapunov functions, the result that the method is drawn are not required to quote any other variable, letter List is easy and can shorten system operation time, that is, improves production efficiency.
The limited rolling time horizon of batch injection moulding process mixes tracking and controlling method, and this method comprises the concrete steps that：
Step 1, for different phase in batch process, establish the switching based on state-space model of controlled device System model, is specifically：
Step 1, for different phase in batch injection moulding process, establish controlled device based on state-space model Switching system model, is specifically：
1.1 gather the inputoutput data of batch injection moulding process first, and batch process respective stage is established using the data Model, form are as follows：
Ai(z-1)yi(z)=Bi(z-1)ui(z)
Wherein, yi(z), ui(z) be respectively i-th stage output and input z-transform, ai,biIt is multinomial A respectivelyi (z-1), Bi(z-1) in corresponding coefficient, m, n are A respectivelyi(z-1), Bi(z-1) maximum order；
1.2 the model in step 1.1 is further handled, Δ difference operator, y are introducedi(k)∈R,ui(k) ∈ R are respectively k The output in i-th stage of moment batch process and input variable；Obtain error model：
And choose non-minimum state space variable Δ x0 i(k)T, form is as follows：
Δx0 i(k)T=[Δ yi(k)T,Δyi(k-1)T,…,Δyi(k-n+1)T,Δui(k-1)T,Δui(k-2)T,…, Δui(k-m+1)T]
Wherein, Δ x0 i(k) dimension is (m-1) × p+n × q, and p is the dimension of input variable, and q is the dimension of output variable Number；
I-th new of stage condition spatial model is obtained from above：
Wherein,
Wherein,It is the unit matrix of a p dimension,It is the unit matrix of a q dimension；
1.3 in order to there is preferable tracking performance, for stage i, define output tracking error ei(k)=yi(k)-ri(k), With reference to step 1.2, it is as follows to try to achieve tracking error form：
Wherein, ri(k) it is the k moment, the desired output in i stages；
1.4 were directed to for the i-th stage, chose new state variable again, further expanded model and obtained new extended mode sky Between model, make it includes state variable and output tracking error, its form is as follows：
zi(k+1)=Aizi(k)+BiΔui(k)
Wherein,Matrix AiIn 0 represent null matrix；
Said system is reproduced as switching system model by 1.5 is：
Z (k+1)=Aσ(k)z(k)+Bσ(k)Δu(k).
Wherein, σ (k):Z+→N:What={ 1,2 ..., N } was represented is switching signal, it may be with time or system mode phase Close, N is the number of stages of subsystem, and switching sequence is defined as S:={ T0,T1,T2,...,Tt,...}；It is all be continuously interrupted when Between interval meet Tt+1-Tt≥τi, t=0,1,2 ..., TtRepresent t-th of switching instant, T0It is initial time, τiFor not same order The residence time and its value of section depend on liapunov function, Aσ(k),Bσ(k)For different phase above formula model 1.4 Represent；
Step 2, design batch injection moulding process different phase controller and switching signal σ (k) are switched with obtaining different phase Condition and run time, are specifically：
2.1 consideration the non-minimum of the state containing free terminal realize Extended state space model, choose corresponding performance indicator Form is as follows：
Wherein, Qi, Ri,The weight matrix of the i-th stage condition variable, controlled input and the SOT state of termination is represented respectively,For rolling optimization time domain；Respectively beginning and terminal juncture；
2.2 ask for the optimal control law of different phase controller according to the performance indicator in step 2.1, and form is as follows：
The controlled quentity controlled variable obtained in step 2.2 is acted on controlled device by 2.3：
ui(k)=Δ ui(k)+ui(k-1)
2.4 in subsequent time, repeat step 2.1 to 2.3 continues to solve new controlled quentity controlled variable, and circulates successively；
2.5 for different phase design switching signal be σ (k)；
2.5.1 the switching system being directed in step 1.5, design different phase controller are：
Δui(k)=- Kizi(k)
Wherein,
Then each stage i, switching system can be changed to：
Z (k+1)=(Ai-BiKi)z(k)
2.5.2 for i-th of subsystem, following liapunov function is selected：
Vi(k)=zT(k)Pi(k)z(k)
Wherein, Pi(k), i ∈N,N:=1,2 ..., and N } it is to rely on residence time τiMatrix；Then：
If switching system is stablized, there must be Δ Vi(k) ＜ 0, it is equivalent to：
With reference to step 2.2, above-mentioned inequality is solved, the τ of different phase can be obtainedi。
Compared with prior art, beneficial effects of the present invention are：Noted for two important stages that injection moulding process need to control Section and pressurize section are penetrated, its switching condition is designed with run time, realizes that it is efficiently produced.Deposited for it in model mismatch and interference In the case of, design by the more flexible controller of the adjusting for increasing reconcilable weighting coefficient, improve its Control platform, it is real Now more preferable control performance.The method advantage is that the setting of other specification is not required, and direct to be worth, simple and practicable, this is obvious Other methods are superior to, such as average residence time method, so-called average residence time method, refers to system in each stage There is the average value of residence time.Average residence time method usually assumes that a certain variable in its condition gives, this can undoubtedly increase The run time in big a certain stage, so as to extend the run time of whole production process.As it can be seen that the method for the present invention is ensuring System stable operation and while have optimum control performance, in the case where ensureing product quality, improves production efficiency.
Embodiment
With reference to specific embodiment, the present invention is described further.
The limited rolling time horizon of batch injection moulding process mixes tracking and controlling method, and this method comprises the concrete steps that：
Step 1, for different phase in batch process, establish the switching based on state-space model of controlled device System model, is specifically：
Step 1, for different phase in batch injection moulding process, establish controlled device based on state-space model Switching system model, is specifically：
1.1 gather the inputoutput data of batch injection moulding process first, and batch process respective stage is established using the data Model, form are as follows：
Ai(z-1)yi(z)=Bi(z-1)ui(z)
Wherein, yi(z), ui(z) be respectively i-th stage output and input z-transform, ai,biIt is multinomial A respectivelyi (z-1), Bi(z-1) in corresponding coefficient, m, n are A respectivelyi(z-1), Bi(z-1) maximum order；
1.2 further handle the model in step 1.1, introduce Δ difference operator, yi(k)∈R,ui(k) ∈ R are respectively k The output in i-th stage of moment batch process and input variable；Obtain error model：
And choose non-minimum state space variable Δ x0 i(k)T, form is as follows：
Δx0 i(k)T=[Δ yi(k)T,Δyi(k-1)T,…,Δyi(k-n+1)T,Δui(k-1)T,Δui(k-2)T,…, Δui(k-m+1)T]
Wherein, Δ x0 i(k) dimension is (m-1) × p+n × q, and p is the dimension of input variable, and q is the dimension of output variable Number；
I-th new of stage condition spatial model is obtained from above：
Wherein,
Wherein,It is the unit matrix of a p dimension,It is the unit matrix of a q dimension；
1.3 in order to there is preferable tracking performance, for stage i, define output tracking error ei(k)=yi(k)-ri(k), With reference to step 1.2, it is as follows to try to achieve tracking error form：
Wherein, ri(k) it is the k moment, the desired output in i stages；
1.4 were directed to for the i-th stage, chose new state variable again, further expanded model and obtained new extended mode sky Between model, make it includes state variable and output tracking error, its form is as follows：
zi(k+1)=Aizi(k)+BiΔui(k)
Wherein,Matrix AiIn 0 represent null matrix；
Said system is reproduced as switching system model by 1.5 is：
Z (k+1)=Aσ(k)z(k)+Bσ(k)Δu(k).
Wherein, σ (k):Z+→N:What={ 1,2 ..., N } was represented is switching signal, it may be with time or system mode phase Close, N is the number of stages of subsystem, and switching sequence is defined as S:={ T0,T1,T2,...,Tt,...}；It is all be continuously interrupted when Between interval meet Tt+1-Tt≥τi, t=0,1,2 ..., TtRepresent t-th of switching instant, T0It is initial time, τiFor not same order The residence time and its value of section depend on liapunov function, Aσ(k),Bσ(k)For different phase above formula model 1.4 Represent；
Step 2, design batch injection moulding process different phase controller and switching signal σ (k) are switched with obtaining different phase Condition and run time, are specifically：
2.1 consideration the non-minimum of the state containing free terminal realize Extended state space model, choose corresponding performance indicator Form is as follows：
Wherein, Qi, Ri,The weight matrix of the i-th stage condition variable, controlled input and the SOT state of termination is represented respectively,For rolling optimization time domain；Respectively beginning and terminal juncture；
2.2 ask for the optimal control law of different phase controller according to the performance indicator in step 2.1, and form is as follows：
The controlled quentity controlled variable obtained in step 2.2 is acted on controlled device by 2.3：
ui(k)=Δ ui(k)+ui(k-1)
2.4 in subsequent time, and repeat step 2.1 to 2.3 continues to solve new controlled quentity controlled variable, and circulates successively；
2.5 for different phase design switching signal be σ (k)；
2.5.1 the switching system being directed in step 1.5, design different phase controller are：
Δui(k)=- Kizi(k)
Wherein,
Then each stage i, switching system can be changed to：
Z (k+1)=(Ai-BiKi)z(k)
2.5.2 for i-th of subsystem, following liapunov function is selected：
Vi(k)=zT(k)Pi(k)z(k)
Wherein, Pi(k), i ∈N,N:=1,2 ..., and N } it is to rely on residence time τiMatrix；Then：
If switching system is stablized, there must be Δ Vi(k) ＜ 0, it is equivalent to：
With reference to step 2.2, above-mentioned inequality is solved, the τ of different phase can be obtainedi。
Embodiment
Injection moulding process is typical batch production process, and each batch mainly includes three steps, i.e. injection portion → pressurize Section → cooling section.In injection portion, screw rod, which travels forward, to be stored in melt (the heated shape after enclosing heating of raw material of machine barrel front end Into) extrude forward, flow through running channel, runner, cast gate, in the closed mold cavity of entrance (die cavity).When die cavity is completely filled with Afterwards, forming process switches to pressurize section by injection portion.In pressurize section, screw rod is pushed ahead with very low speed, to keep Certain nozzle exit pressure.A small amount of melt goes successively to die cavity, compensates due to volume contraction caused by material cooling and curing.One Product minimum cast gate in denier mould middle section cures substantially, and pressurize section stops, and process enters cooling section, ideally melt at this time Flowing should stop.Injection mechanism is plasticized in cooling section, is ready for next circulation；At the same time, in die cavity Material continues cooling until being fully cured.Finally, mould is opened, and thimble ejects product, completes a circulation.
Therefore, injection molding process is mainly comprising injection portion, pressurize section, cooling section three phases.Injection portion, pressurize section Control effect, which has product final mass, to be directly affected, and wherein injection portion injection speed, pressurize section cavity pressure are to corresponding rank Section control effect influences maximum, it is necessary to control tracking set-point.The two parameters are controlled by corresponding valve, valve Aperture affecting parameters.In addition, in injection portion, when cavity pressure reaches certain value, process enters pressurize section, thus in injection portion mould Cavity pressure is needed to be detected but need not be directly controlled.Only high temperature manufactured goods are cooled down in cooling section, are not taken Control measure；Thus need to establish the hybrid state spatial model of injection molding process injection portion and pressurize section.
The Frequency Domain Mathematical Model of existing injection molding process injection portion and pressurize section is as follows：
Injection portion Frequency Domain Mathematical Model is：(1-0.9291z-1-0.03191z-2) IV=(8.687z-1-5.617z-2) VO,
Pressurize section Frequency Domain Mathematical Model is：(1-1.317z-1+0.3259z-2) NP=(171.8Z-1-156.8Z-2)VO；
Wherein, IV represents injection portion injection speed, setting value 40mm/s；NP represents cavity pressure, is set in pressurize section It is worth for 300bar；VO represents valve opening.
Two stage input/output models of injection molding process are rewritten as switching system augmentation mould of equal value using step 1 Type is as follows：
Z (k+1)=Aσ(k)z(k)+Bσ(k)Δ u (k), σ (k)={ 1,2 }
Definition injection portion is the stage 1, and pressurize section is the stage 2, i.e. σ (t, k)=1, σ (t, k)=2 represent stage 1, rank respectively Section 2.
Using step 2, according to different phase design accordingly can flexible modulation in real time controller, to improve its control product Matter, solves the drawbacks of controller gain cannot be adjusted in whole process in existing method.Different phase is finally directed to, is designed Go out to depend on the residence time method of Lyapunov functions, the result that the method is drawn is not required to quote any other variable, simply It is easy and cause system operation time shorten, that is, improve production efficiency.
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 limited rolling time horizon of batch injection moulding process mixes tracking and controlling method, it is characterised in that：The specific steps of this method It is：
Step 1, for different phase in batch injection moulding process, establish the switching based on state-space model of controlled device System model, is specifically：
1.1 gather the inputoutput data of batch injection moulding process first, and batch process respective stage model is established using the data, Form is as follows：
Ai(z-1)yi(z)=Bi(z-1)ui(z)
<mrow> <msup> <mi>A</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mi>i</mi> </msubsup> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>a</mi> <mn>2</mn> <mi>i</mi> </msubsup> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>...</mo> <msubsup> <mi>a</mi> <mi>n</mi> <mi>i</mi> </msubsup> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mi>n</mi> </mrow> </msup> </mrow>
<mrow> <msup> <mi>B</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>b</mi> <mn>1</mn> <mi>i</mi> </msubsup> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>b</mi> <mn>2</mn> <mi>i</mi> </msubsup> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>b</mi> <mi>m</mi> <mi>i</mi> </msubsup> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mi>m</mi> </mrow> </msup> </mrow>
Wherein, yi(z), ui(z) be respectively i-th stage output and input z-transform, ai,biIt is multinomial A respectivelyi(z-1), Bi(z-1) in corresponding coefficient, m, n are A respectivelyi(z-1), Bi(z-1) maximum order；
1.2 further handle the model in step 1.1, introduce Δ difference operator, yi(k)∈R,ui(k) ∈ R are respectively the k moment The output in i-th stage of batch process and input variable；Obtain error model：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mi>&amp;Delta;y</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mi>i</mi> </msubsup> <msup> <mi>&amp;Delta;y</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>a</mi> <mn>2</mn> <mi>i</mi> </msubsup> <msup> <mi>&amp;Delta;y</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>...</mn> <mo>+</mo> <msubsup> <mi>a</mi> <mi>n</mi> <mi>i</mi> </msubsup> <msup> <mi>&amp;Delta;y</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>b</mi> <mn>1</mn> <mi>i</mi> </msubsup> <msup> <mi>&amp;Delta;u</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>b</mi> <mn>2</mn> <mi>i</mi> </msubsup> <msup> <mi>&amp;Delta;u</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>...</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msubsup> <mi>b</mi> <mi>m</mi> <mi>i</mi> </msubsup> <msup> <mi>&amp;Delta;u</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
And choose non-minimum state space variableForm is as follows：
Δx0 i(k)T=[Δ yi(k)T,Δyi(k-1)T,…,Δyi(k-n+1)T,Δui(k-1)T,Δui(k-2)T,…,Δui (k-m+1)T]
Wherein, Δ x0 i(k) dimension is (m-1) × p+n × q, and p is the dimension of input variable, and q is the dimension of output variable；
I-th new of stage condition spatial model is obtained from above：
<mrow> <msup> <msub> <mi>&amp;Delta;x</mi> <mn>0</mn> </msub> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>A</mi> <mn>0</mn> <mi>i</mi> </msubsup> <msubsup> <mi>&amp;Delta;x</mi> <mn>0</mn> <mi>i</mi> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>B</mi> <mn>0</mn> <mi>i</mi> </msubsup> <msup> <mi>&amp;Delta;u</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow>
<mrow> <msup> <mi>&amp;Delta;y</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>C</mi> <mn>0</mn> <mi>i</mi> </msubsup> <msubsup> <mi>&amp;Delta;x</mi> <mn>0</mn> <mi>i</mi> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Wherein,
<mrow> <msubsup> <mi>A</mi> <mn>0</mn> <mi>i</mi> </msubsup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mi>i</mi> </msubsup> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msubsup> <mi>a</mi> <mn>2</mn> <mi>i</mi> </msubsup> </mrow> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mrow> <mo>-</mo> <msubsup> <mi>a</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>i</mi> </msubsup> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msubsup> <mi>a</mi> <mi>n</mi> <mi>i</mi> </msubsup> </mrow> </mtd> <mtd> <msubsup> <mi>b</mi> <mn>1</mn> <mi>i</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>b</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>i</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>b</mi> <mi>m</mi> <mi>i</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>I</mi> <mi>q</mi> <mi>i</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>I</mi> <mi>q</mi> <mi>i</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mo>...</mo> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mo>...</mo> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>I</mi> <mi>q</mi> <mi>i</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>I</mi> <mi>p</mi> <mi>i</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mo>...</mo> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mo>...</mo> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>I</mi> <mi>p</mi> <mi>i</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>
<mrow> <msubsup> <mi>B</mi> <mn>0</mn> <mi>i</mi> </msubsup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>b</mi> <mn>1</mn> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>I</mi> <mi>p</mi> <mi>i</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>
<mrow> <msubsup> <mi>C</mi> <mn>0</mn> <mi>i</mi> </msubsup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>I</mi> <mi>q</mi> <mi>i</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>
Wherein,It is the unit matrix of a p dimension,It is the unit matrix of a q dimension；
1.3 in order to there is preferable tracking performance, for stage i, define output tracking error ei(k)=yi(k)-ri(k), with reference to Step 1.2, it is as follows to try to achieve tracking error form：
<mrow> <msup> <mi>e</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>C</mi> <mn>0</mn> <mi>i</mi> </msubsup> <msubsup> <mi>A</mi> <mn>0</mn> <mi>i</mi> </msubsup> <msubsup> <mi>&amp;Delta;x</mi> <mn>0</mn> <mi>i</mi> </msubsup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>C</mi> <mn>0</mn> <mi>i</mi> </msubsup> <msubsup> <mi>B</mi> <mn>0</mn> <mi>i</mi> </msubsup> <msup> <mi>&amp;Delta;u</mi> <mi>i</mi> </msup> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow>
Wherein, ri(k) it is the k moment, the desired output in i stages；
1.4 were directed to for the i-th stage, chose new state variable again, further expanded model and obtained new extended mode spatial mode Type, makes it includes state variable and output tracking error, its form is as follows：
zi(k+1)=Aizi(k)+BiΔui(k)
Wherein,Matrix AiIn 0 represent null matrix；
Said system is reproduced as switching system model by 1.5 is：
Z (k+1)=Aσ(k)z(k)+Bσ(k)Δu(k).
Wherein, σ (k):Z+→N:What={ 1,2 ..., N } was represented is switching signal, it may be related to time or system mode, N It is the number of stages of subsystem, switching sequence is defined as S:={ T0,T1,T2,...,Tt,...}；Between all times being continuously interrupted Every meeting Tt+1-Tt≥τi, t=0,1,2 ..., TtRepresent t-th of switching instant, T0It is initial time, τiFor different phase Residence time and its value depends on liapunov function, Aσ(k),Bσ(k)Represented for different phase above formula model 1.4；
Step 2, design batch injection moulding process different phase controller and switching signal σ (k) to obtain different phase switching condition And run time, it is specifically：
2.1 consideration the non-minimum of the state containing free terminal realize Extended state space model, choose corresponding performance indicator form It is as follows：
<mrow> <msup> <mi>J</mi> <mi>i</mi> </msup> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <msubsup> <mi>k</mi> <mn>0</mn> <mi>i</mi> </msubsup> </mrow> <mrow> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mo>&amp;lsqb;</mo> <msup> <mi>z</mi> <mi>i</mi> </msup> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msup> <mi>Q</mi> <mi>i</mi> </msup> <msup> <mi>z</mi> <mi>i</mi> </msup> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msup> <mi>&amp;Delta;u</mi> <mi>i</mi> </msup> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msup> <mi>R</mi> <mi>i</mi> </msup> <msup> <mi>&amp;Delta;u</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>+</mo> <msup> <mi>z</mi> <mi>i</mi> </msup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>Q</mi> <mi>f</mi> <mi>i</mi> </msubsup> <msup> <mi>z</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> </mrow>
Wherein, Qi, Ri,The weight matrix of the i-th stage condition variable, controlled input and the SOT state of termination is represented respectively, For rolling optimization time domain；Respectively beginning and terminal juncture；
2.2 ask for the optimal control law of different phase controller according to the performance indicator in step 2.1, and form is as follows：
<mrow> <msup> <mi>&amp;Delta;u</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msup> <mi>R</mi> <mrow> <mo>-</mo> <mi>i</mi> </mrow> </msup> <msup> <mi>B</mi> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msup> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mi>I</mi> <mi>i</mi> </msup> <mo>+</mo> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>R</mi> <mrow> <mo>-</mo> <mi>i</mi> </mrow> </msup> <msup> <mi>B</mi> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <msup> <mi>A</mi> <mi>i</mi> </msup> <msup> <mi>z</mi> <mi>i</mi> </msup> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <mo>=</mo> <msup> <mi>A</mi> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msup> <msup> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msup> <mi>I</mi> <mi>i</mi> </msup> <mo>+</mo> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>R</mi> <mrow> <mo>-</mo> <mi>i</mi> </mrow> </msup> <msup> <mi>B</mi> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>+</mo> <msup> <mi>Q</mi> <mi>i</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msup> <mi>A</mi> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msup> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>A</mi> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msup> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>R</mi> <mi>i</mi> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msup> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <msup> <mi>B</mi> <mi>i</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>B</mi> <mrow> <mi>i</mi> <mi>T</mi> </mrow> </msup> <msubsup> <mi>K</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>+</mo> <msup> <mi>Q</mi> <mi>i</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <msubsup> <mi>K</mi> <mrow> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> <mo>,</mo> <msubsup> <mi>k</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow> <mi>i</mi> </msubsup> <mo>=</mo> <msubsup> <mi>Q</mi> <mi>f</mi> <mi>i</mi> </msubsup> </mrow>
The controlled quentity controlled variable obtained in step 2.2 is acted on controlled device by 2.3：
ui(k)=Δ ui(k)+ui(k-1)
2.4 in subsequent time, and repeat step 2.1 to 2.3 continues to solve new controlled quentity controlled variable, and circulates successively；
2.5 for different phase design switching signal be σ (k)；
2.5.1 the switching system being directed in step 1.5, design different phase controller are：
Δui(k)=- Kizi(k)
Wherein,
Then each stage i, switching system can be changed to：
Z (k+1)=(Ai-BiKi)z(k)
2.5.2 for i-th of subsystem, following liapunov function is selected：
Vi(k)=zT(k)Pi(k)z(k)
Wherein, Pi(k), i ∈N,N:=1,2 ..., and N } it is to rely on residence time τiMatrix；Then：
<mrow> <msup> <mi>&amp;Delta;V</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mi>z</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>K</mi> <mi>i</mi> </msup> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>P</mi> <mrow> <mi>h</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>i</mi> </msubsup> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>K</mi> <mi>i</mi> </msup> </mrow> <mo>)</mo> <mo>-</mo> <msubsup> <mi>P</mi> <mi>h</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <mi>z</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>h</mi> <mo>=</mo> <mi>k</mi> <mo>-</mo> <msub> <mi>k</mi> <mrow> <mi>t</mi> <mo>,</mo> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mo>&amp;lsqb;</mo> <msub> <mi>k</mi> <mi>t</mi> </msub> <mo>,</mo> <msub> <mi>k</mi> <mi>t</mi> </msub> <mo>+</mo> <msup> <mi>&amp;tau;</mi> <mi>i</mi> </msup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>z</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>K</mi> <mi>i</mi> </msup> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>P</mi> <msup> <mi>&amp;tau;</mi> <mi>i</mi> </msup> <mi>i</mi> </msubsup> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>K</mi> <mi>i</mi> </msup> </mrow> <mo>)</mo> <mo>-</mo> <msubsup> <mi>P</mi> <msup> <mi>&amp;tau;</mi> <mi>i</mi> </msup> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <mi>z</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mo>&amp;lsqb;</mo> <msub> <mi>k</mi> <mi>t</mi> </msub> <mo>+</mo> <msup> <mi>&amp;tau;</mi> <mi>i</mi> </msup> <mo>,</mo> <msub> <mi>k</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
If switching system is stablized, there must be Δ Vi(k) ＜ 0, it is equivalent to：
<mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>K</mi> <mi>i</mi> </msup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>P</mi> <mrow> <mi>h</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>i</mi> </msubsup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>K</mi> <mi>i</mi> </msup> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>P</mi> <mi>h</mi> <mi>i</mi> </msubsup> <mo>&lt;</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>&amp;le;</mo> <mi>h</mi> <mo>&amp;le;</mo> <msup> <mi>&amp;tau;</mi> <mi>i</mi> </msup> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>i</mi> <mo>&amp;Element;</mo> <munder> <mi>N</mi> <mo>&amp;OverBar;</mo> </munder> <mo>,</mo> </mrow>
<mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>K</mi> <mi>i</mi> </msup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>P</mi> <msup> <mi>&amp;tau;</mi> <mi>i</mi> </msup> <mi>i</mi> </msubsup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mi>i</mi> </msup> <mo>-</mo> <msup> <mi>B</mi> <mi>i</mi> </msup> <msup> <mi>K</mi> <mi>i</mi> </msup> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>P</mi> <msup> <mi>&amp;tau;</mi> <mi>i</mi> </msup> <mi>i</mi> </msubsup> <mo>&lt;</mo> <mn>0</mn> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>i</mi> <mo>&amp;Element;</mo> <munder> <mi>N</mi> <mo>&amp;OverBar;</mo> </munder> </mrow>
<mrow> <msubsup> <mi>P</mi> <mn>0</mn> <mi>i</mi> </msubsup> <mo>-</mo> <msubsup> <mi>P</mi> <msup> <mi>&amp;tau;</mi> <mi>i</mi> </msup> <mi>j</mi> </msubsup> <mo>&amp;le;</mo> <mn>0</mn> <mo>,</mo> <mi>i</mi> <mo>&amp;NotEqual;</mo> <mi>j</mi> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>&amp;Element;</mo> <munder> <mi>N</mi> <mo>&amp;OverBar;</mo> </munder> </mrow>
With reference to step 2.2, above-mentioned inequality is solved, the τ of different phase can be obtainedi。
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