CN104657549B - A kind of iterative learning forecast Control Algorithm based on Orthogonal Parameter LTV models - Google Patents
A kind of iterative learning forecast Control Algorithm based on Orthogonal Parameter LTV models Download PDFInfo
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Abstract
Iterative learning forecast Control Algorithm of the injection moulding process pressurize section based on Orthogonal Parameter LTV models is directed to the invention discloses a kind of.Implementation steps are as follows：(1) Orthogonal Parameter LTV models are established；(2) model parameter estimation；(3) order selects；(4) ILC MPC control laws are derived.The present invention makes full use of injection moulding process to rerun, process characteristic known to pursuit path, by introducing Orthogonal Parameter LTV models, the characteristics of changing traditional modeling method complexity high, extrapolation difference.In control process, the control strategy being combined with Model Predictive Control is controlled using iterative learning, history batch information can be both made full use of, rolling optimization can be carried out online according to forecast model again, so as to more more stably complete demand for control rapidly.
Description
Technical field
The present invention relates to for injection moulding process field, and in particular to one kind is directed to injection moulding process pressurize section Orthogonal Parameter
The iterative learning forecast Control Algorithm of LTV (linear time-varying, LTV) model.
Background technology
Plastic products are widely used among people's daily life, and injection molding is to process the main method of plastic products,
80% plastic products are produced using the technique in the world at present.The capital equipment of injection moulding process is injection moulding machine, letter
Claim injection machine.Injection molding develops according to metal die casting principle, by using injection machine and injection mold by plastics
Raw material is changed into moulded products, is a kind of typical batch production process.It is the thermophysical property using plastics, material from
Hopper is added in barrel, and barrel is outer to be heated by heating collar, makes material melts, is driven in barrel built with the effect of outer power motor
The screw rod of dynamic rotation, material are fed forward and are compacted along screw channel in the presence of screw rod, and material is in external heat and screw rod shearing
Double action under little by little plastify, melt and be homogenized, when screw rod rotates, effect of the material in screw channel frictional force and shearing force
Under, molten material is shifted onto the head of screw rod, at the same time, screw rod retreats under the reaction of material, makes screw head
Storing space is formed, completes plasticizing process, then, screw rod will be stored up in the presence of injection cylinder piston thrust with high speed, high pressure
Expect in die cavity of the indoor melting charge by nozzle injection to mould, the melt in die cavity is by pressurize, cooling, solidifying and setting
Afterwards, mould is in the presence of clamping, opening mold, and stereotyped product is fallen from mould ejection by liftout attachment
Under.Therefore once completely injection flow mainly includes closing mould, injection, pressurize, cooling, five parts of die sinking, injection machine is made
Industry based on this, continuous circulation process.
Pressurize section is to determine an important stage of final product quality, and its key variables are dwell pressures, but due to pressurize
Typically no steady operation point, process variable can big ups and downs in very large range in process operation section.Further, since melt
Flow velocity, pressure distribution inhomogeneities, and the factor such as material, process conditions influence, and process can be caused to show strong non-thread
Property and time-varying characteristics, therefore, LTI model can not fully describe injection moulding process, and the control based on linear model
Strategy can not play a role well, and these features determine that pressure maintaining period control is more complicated than continuous processing industry,
It is therefore desirable to study that the nonlinear model of pressure maintaining period can be better described, and it is based on this model cootrol process variable.
Mainly there are mechanism model, fuzzy model, neutral net mould currently for the nonlinear model of injection moulding process pressurize section
Type, multi-model etc., but for mechanism model, it is relatively difficult to establish a mechanism model that can fully describe process；
The method shortcoming of fuzzy model is that the design of fuzzy model lacks systematicness, and the simple Fuzzy Processing of information will cause system
Control accuracy reduces and dynamic quality is deteriorated；The main inconvenience of neural network model is to model difficulty and extrapolation is poor；Multimode
Type method shortcoming is that model selection and design waste time and energy.
However, there is injection moulding process process to rerun, process characteristic known to pursuit path, above-mentioned modeling method is not
This feature is made full use of to reduce modeling difficulty.
The content of the invention
To overcome the shortcomings of above-mentioned modeling method, injection moulding process is made full use of to rerun, technique known to pursuit path
Feature, the invention provides a kind of iterative learning forecast Control Algorithm based on Orthogonal Parameter LTV models, has both met that precision will
Ask, reduce complexity again, and optimal iterative learning predictive control algorithm is derived on the basis of Orthogonal Parameter LTV models.
To achieve the above object, the present invention, which adopts the following technical scheme that, is achieved：
Step (1), establish Orthogonal Parameter LTV models：
Reruned using injection moulding process, process characteristic known to pursuit path, establish Orthogonal Parameter LTV models, used
To characterize process nonlinear problem；This model reduces complexity, when time-varying coefficient is again on the basis of required precision is met
Between axial coordinate t nonlinear function；
Step (2), model parameter estimation：
The purpose for determining model parameter is that model accuracy can be issued to highest in relevant control criterion, utilizes
Levenberg-Marquardt methods ask lsqnonlin to determine model parameter.
Step (3), order selection；
The purpose for determining model order/structure is that obtained model can meet that control requires with full accuracy, utilizes Chi Chi
(Akaike information criterion, AIC) information criterion determines model order.
Step (4), derive ILC-MPC control laws；
The solving-optimizing proposition on the basis of obtained LTV models, obtain optimal iterative learning Predictive control law.The present invention's has
Beneficial effect is：
The present invention makes full use of injection moulding process to rerun, process characteristic known to pursuit path, by introducing orthogonal ginseng
Numberization LTV models, the characteristics of changing traditional modeling method complexity high, extrapolation difference.In control process, using iteration
The control strategy that control is combined with Model Predictive Control is practised, can both make full use of history batch information, again can be according to pre-
Survey model and carry out rolling optimization online, so as to more more stably complete demand for control rapidly.
Brief description of the drawings
Fig. 1 is to be based on LTV model ILC-MPC control system block diagrams；
Fig. 2 is ILC-MPC algorithm flow charts；
Fig. 3 is control effect figure.
Embodiment
The present invention is further analyzed below in conjunction with the accompanying drawings.
Iterative learning Predictive Control System block diagram based on Orthogonal Parameter LTV models is as shown in Figure 1.
It is as follows based on Orthogonal Parameter LTV model iterative learning control method implementation steps：
Step (1), establish Orthogonal Parameter LTV models：
For injection moulding process pressurize section, represented using following LTV models：
yk(t)=G (q, t) uk(t)+vk(t) t=1 ..., N, k=1,2 ... (1)
Wherein t and k represents time axial coordinate and batch axial coordinate respectively；N is each batch time length；yk(t),uk
(t),vk(t) dwell pressure of k-th of batch t, valve opening and disturbance are represented respectively；G (q, t) is from uk(t) y is arrivedk
(t) linear time-varying transmission function；
For injection moulding process, because disturbance is there is very strong batch correlation, therefore, measurable disturbance is not available as follows
Described along the integration white noise of batch axle：vk(t)=vk-1(t)+wk(t), wherein wk(t) it is zero mean Gaussian white noise；
It is formula (1) to above-mentioned LTV models, adjacent two lot data is done difference, then had
Wherein
A (q, t)=1+a1(t)q-1+...+an(t)q-n (4)
B (q, t)=1+b1(t)q-1+...+bn(t)q-n (5)
Above-mentioned model is a kind of LTV-OE model structures (see formula 2~7), and wherein n is OE model orders, aiAnd b (t)i
(t) it is time-varying model coefficient, wherein i=1,2 ..., n.
Although injection moulding process has obvious non-linear and time variation, there is process and rerun, track in its process
Distinguishing feature known to track, therefore aiAnd b (t)i(t) time axial coordinate n nonlinear function can be expressed as, so as to both full
Sufficient accuracy requirement, complexity is reduced again, i.e.,
Wherein,It is one group of polynomial basis function, m is multinomial order, ai jAnd bi jFor weight coefficient.Due to orthogonal
Multinomial has good numerical stability, and in orthogonal polynomial various forms, the Legendre that leading coefficient is 1 is multinomial
Formula form is simple, recursion is convenient, high order approximation error is smaller, therefore, selects it as orthogonal basis function.
Obtaining the Legendre multinomials that leading coefficient is 1 needs to convert by two steps.First, Legendre multinomials are determined
On [- 1,1], its general type is justice：
Due to Pn(x) x innCoefficient beOrderThen it can obtain
Leading coefficient is 1 Legendre multinomials.Secondly as Legendre polynomials domain is [- 1,1], and aiAnd b (t)i
(t) defined on [1, N], such as down conversion is passed throughT ∈ [1, N] can be then converted to μ ∈ [-
1,1]。
Make basic function from the Legendre multinomials that leading coefficient is 1, bring formula (8) into, in (9)：
Step (2), model parameter estimation
See formula (2) for LTV-OE models, the model parameter for needing to estimate is：
Then optimum predictor is a step forward：
Therefore, by minimizing prediction error criterion function VKN(θ), to determine parameter vector θ
Wherein,
Due to εk(t | θ) and A (q, t) are in non-linear relation, then analytic solutions are not present in optimal problem (15), need to use numerical value
Method is solved.For above-mentioned non-linear least square problem, existing Newton-Raphson, Gauss-Newton,
The numerical optimizations such as Levenberg-Marquardt, because Levenberg-Marquardt methods have fast convergence rate, number
The advantages of value stabilization is good, therefore use Levenberg-Marquardt methods to carry out numerical solution, and iterated to calculate as follows
Journey：
WhereinFor the estimate of the r times iteration, μ is step factor, and I is unit battle array,
When carrying out the numerical optimization of model parameter in aforementioned manners, need to knowGradient ψk(t,θ).For
Formula (14), both sides are respectively to ai jAnd bi j：
A (q, t) is substituted into above formula, gradient ψ can be obtainedkThe calculation formula of (t, θ).
Step (3), order selection
The mould of LTV-OE models is determined using red pond information criterion (Akaike information criterion, AIC)
Type order n and orthogonal polynomial order m.
Red pond information criterion is：
Wherein, NkFor Identification Data number, Nk=KN；D is model parameter number, d=2nm.
Step (4), derive ILC-MPC control laws
It is that the adjacent batch of formula (1) is done difference and had for LTV models：
I.e.
Being write as state space form is：
Wherein,
Forecast model：
OrderThen
yk(t)-yk-1(t)=Cxk(t)+wk(t) therefore have：
P is prediction time domain.
Due to xk(t) it is known state, then is had according to recurrence formula：
·
·
·
·
·
·
It is M due to controlling time domain, then has：
To sum up derive, can obtain,
Order
Then it is represented by：
Bringing (27) into has：
Arrange,
Wherein,
The object function is selected to be：
Unconstrained problem analytic solutions are：
Make dT=[10 ... 0] (GTQG+R)-1GTQ, then
Wherein,
Solving-optimizing propositional formula (33), obtain finally parsing solution formula (35), be so as to obtain control law：
uk(t)=rk(t), k=1；uk(t)=rk(t)+uk-1(t), k ＞ 1 (37)
Iterative learning forecast Control Algorithm flow chart based on Orthogonal Parameter LTV models is as shown in Figure 2.
As shown in figure 3, establishing Orthogonal Parameter LTV models, model order and parameter are determined with the method for System Discrimination,
Emulation is controlled to pressurize section dwell pressure on the basis of this model, obtains Fig. 3 curves, it can be seen that by about 2-3 batches
After secondary study, curve is set in output (dwell pressure) curve tracking.
It is that the present invention is not limited only to above-described embodiment, as long as meeting for limitation of the invention that above-described embodiment, which is not,
Application claims, belong to protection scope of the present invention.
Claims (1)
- A kind of 1. iterative learning forecast Control Algorithm based on Orthogonal Parameter LTV models, it is characterised in that this method include with Lower step：Step (1), establish Orthogonal Parameter LTV models：Reruned using injection moulding process, process characteristic known to pursuit path, Orthogonal Parameter LTV models are established, to table Levy injection moulding process nonlinear problem；Wherein time-varying coefficient is time axial coordinate t nonlinear function；Step (2), model parameter estimation：Because model accuracy can be issued to highest in relevant control criterion, solved using Levenberg-Marquardtt methods non- Linear least squares minimization problem determines model parameter；Step (3), order selection：In order that model can meet that control requires with full accuracy, model order is determined using red pond information criterion；Step (4), derive ILC-MPC control laws：The solving-optimizing proposition on the basis of step (1) obtains LTV models, optimal iterative learning Predictive control law is obtained, realize note The control of molding machine pressurize section dwell pressure；Step (1) is represented particularly directed to injection moulding process pressurize section using following LTV models：yk(t)=G (q, t) uk(t)+vk(t), t=1 ..., N, k=1,2 ... (1)Wherein t and k represents time axial coordinate and batch axial coordinate respectively；N is each batch time length；yk(t),uk(t),vk (t) dwell pressure of k-th of batch t, valve opening and disturbance are represented respectively；G (q, t) is from uk(t) y is arrivedk(t) Linear time-varying transmission function；For injection moulding process, due to disturbing there is very strong batch correlation, therefore measurable disturbance is not available such as lower edge batch The integration white noises of secondary axes describes：vk(t)=vk-1(t)+wk(t), wherein wk(t) it is zero mean Gaussian white noise；It is formula (1) to above-mentioned LTV models, adjacent two lot data is done difference, then had<mrow> <mover> <msub> <mi>y</mi> <mi>k</mi> </msub> <mo>-</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mover> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>-</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>WhereinA (q, t)=1+a1(t)q-1+...+an(t)q-n (4)B (q, t)=1+b1(t)q-1+...+bn(t)q-n (5)<mrow> <msub> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>y</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow><mrow> <msub> <mover> <mi>u</mi> <mo>&OverBar;</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>Above-mentioned formula (2)~(7) are LTV-OE model structures, and wherein n is OE model orders, aiAnd b (t)i(t) it is time-varying model Coefficient；Although injection moulding process has an obvious non-linear and time variation, its process exist process rerun, pursuit path Known distinguishing feature, therefore aiAnd b (t)i(t) time axial coordinate n nonlinear function can be expressed as, i.e.,Wherein,It is one group of polynomial basis function, m is multinomial order,WithFor weight coefficient；If leading coefficient is 1 Legendre multinomials are as orthogonal basis function；Obtaining the Legendre multinomials that leading coefficient is 1 needs to convert by two steps；First, Legendre polynomials exist On [- 1,1], its general type is：<mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> <mfrac> <msup> <mi>d</mi> <mi>n</mi> </msup> <mrow> <msup> <mi>dx</mi> <mi>n</mi> </msup> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>Due to Pn(x) x innCoefficient beOrderIt then can obtain leading coefficient Number is 1 Legendre multinomials；Secondly as Legendre polynomials domain is [- 1,1], and aiAnd b (t)i(t) it is fixed in Justice passes through such as down conversion on [1, N]Then t ∈ [1, N] can be converted to μ ∈ [- 1,1]；Make basic function from the Legendre multinomials that leading coefficient is 1, bring formula (8) into, in (9)：<mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msup> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>j</mi> </msup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mrow> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>t</mi> <mo>-</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow><mrow> <msub> <mi>b</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msup> <msub> <mi>b</mi> <mi>i</mi> </msub> <mi>j</mi> </msup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mrow> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>t</mi> <mo>-</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>Step (2) is specifically that the model parameter for needing to estimate is for LTV-OE models：<mrow> <mi>&theta;</mi> <mo>=</mo> <msup> <mrow> <mo>&lsqb;</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mn>1</mn> </msubsup> <mo>,</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mi>m</mi> </msubsup> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>a</mi> <mi>n</mi> <mn>1</mn> </msubsup> <mo>,</mo> <msubsup> <mi>a</mi> <mi>n</mi> <mn>2</mn> </msubsup> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>a</mi> <mi>n</mi> <mi>m</mi> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mn>1</mn> <mn>1</mn> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>b</mi> <mn>1</mn> <mi>m</mi> </msubsup> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>b</mi> <mi>n</mi> <mn>1</mn> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mi>n</mi> <mn>2</mn> </msubsup> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>b</mi> <mi>n</mi> <mi>m</mi> </msubsup> <mo>&rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>Then optimum predictor is a step forward：<mrow> <msub> <mover> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <msub> <mover> <mi>u</mi> <mo>&OverBar;</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>Therefore, by minimizing prediction error criterion function VKN(θ), to determine parameter vector θ：<mrow> <msub> <mi>V</mi> <mrow> <mi>K</mi> <mi>N</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>K</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <mo>&lsqb;</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>&epsiv;</mi> <mi>k</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>Wherein,Due to εk(t | θ) and A (q, t) are in non-linear relation, then analytic solutions are not present in optimal problem (15), need to use numerical method Solved；For above-mentioned non-linear least square problem, numerical solution is carried out using L-M methods, and iterated to calculate as follows Process：<mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>r</mi> </msub> <mo>+</mo> <msup> <mrow> <mo>&lsqb;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>&psi;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>r</mi> </msub> <mo>)</mo> </mrow> <msup> <msub> <mi>&psi;</mi> <mi>k</mi> </msub> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>&mu;</mi> <mi>I</mi> <mo>&rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&lsqb;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>&psi;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>r</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>&epsiv;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>WhereinFor the estimate of the r times iteration, μ is step factor, and I is unit battle array,<mrow> <msub> <mi>&psi;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msup> <mrow> <mo>&lsqb;</mo> <mfrac> <mo>&part;</mo> <mrow> <mo>&part;</mo> <mi>&theta;</mi> </mrow> </mfrac> <msub> <mi>&epsiv;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>=</mo> <msup> <mrow> <mo>&lsqb;</mo> <mfrac> <mo>&part;</mo> <mrow> <mo>&part;</mo> <mi>&theta;</mi> </mrow> </mfrac> <msub> <mover> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>When carrying out the numerical optimization of model parameter in aforementioned manners, need to knowGradient ψk(t,θ)；For formula (14), both sides are right respectivelyWithDerivation obtains：<mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <mfrac> <mrow> <mo>&part;</mo> <msub> <mover> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <msup> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>j</mi> </msup> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mrow> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>t</mi> <mo>-</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msub> <mover> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>i</mi> <mo>|</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow><mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <mfrac> <mrow> <mo>&part;</mo> <msub> <mover> <mover> <mi>y</mi> <mo>&OverBar;</mo> </mover> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <msup> <msub> <mi>b</mi> <mi>i</mi> </msub> <mi>j</mi> </msup> </mrow> </mfrac> <mo>=</mo> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mrow> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>t</mi> <mo>-</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msub> <mover> <mi>u</mi> <mo>&OverBar;</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>A (q, t) is substituted into above formula, gradient ψ can be obtainedkThe calculation formula of (t, θ)；Step (3) is specifically that the model order n and orthogonal polynomial order of LTV-OE models are determined using red pond information criterion m；Described red pond information criterion is：<mrow> <mi>A</mi> <mi>I</mi> <mi>C</mi> <mo>=</mo> <msub> <mi>logV</mi> <mrow> <mi>K</mi> <mi>N</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mi>d</mi> </mrow> <msub> <mi>N</mi> <mi>k</mi> </msub> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>Wherein, NkFor Identification Data number, Nk=KN；D is model parameter number, d=2nm；Step (4) is specifically to do difference for the adjacent batch of LTV models to have：<mrow> <msub> <mi>y</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&lsqb;</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>+</mo> <msub> <mi>v</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>v</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>I.e.<mrow> <msub> <mi>y</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <msub> <mi>r</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>Being write as state space form is：<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>r</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>Cx</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>Wherein,Forecast model：<mrow> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>j</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>Cx</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>j</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>w</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>j</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>OrderThen<mrow> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>j</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>Cx</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>j</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>yk(t)-yk-1(t)=Cxk(t)+wk(t) therefore have：<mrow> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mrow> <mi>p</mi> <mi>x</mi> <mn>1</mn> </mrow> </msub> <mo>&CenterDot;</mo> <mi>C</mi> <mo>&CenterDot;</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mrow> <mi>p</mi> <mi>x</mi> <mn>1</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>P is prediction time domain；Due to xk(t) it is known state, then is had according to recurrence formula：<mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mover> <mi>r</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow><mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msub> <mi>r</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> <mn>...</mn> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> <mn>...</mn> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mn>...</mn> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>...</mn> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>...</mn> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mn>...</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>P</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>P</mi> <mo>)</mo> </mrow> <mn>...</mn> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>P</mi> <mo>)</mo> </mrow> <mn>...</mn> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mn>...</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>P</mi> <mo>)</mo> </mrow> <mn>...</mn> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow>It is M due to controlling time domain, then has：To sum up derive, can obtain,<mrow> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>P</mi> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>M</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>|</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>Order<mrow> <mi>h</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>g</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>M</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>Then it is represented by：<mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>h</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>t</mi> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>Bringing (27) into has：<mrow> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mrow> <mi>p</mi> <mi>x</mi> <mn>1</mn> </mrow> </msub> <mo>&CenterDot;</mo> <mi>C</mi> <mo>&CenterDot;</mo> <mo>&lsqb;</mo> <mi>h</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>t</mi> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>+</mo> <msub> <mi>I</mi> <mrow> <mi>P</mi> <mi>x</mi> <mn>1</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow>Arrange,<mrow> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>G</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>t</mi> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mi>k</mi> </msub> <msubsup> <mo>(</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow>Wherein,<mrow> <mi>G</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>C</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>M</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>C</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>C</mi> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow><mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> <msubsup> <mo>(</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mrow> <mi>P</mi> <mi>x</mi> <mn>1</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mrow>The object function is selected to be：<mrow> <mi>J</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>M</mi> <mo>,</mo> <mi>P</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>y</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mi>Q</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>r</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>t</mi> </msubsup> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mi>R</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> </mrow>Unconstrained problem analytic solutions are：<mrow> <msub> <mi>r</mi> <mi>k</mi> </msub> <mo>*</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>G</mi> <mi>T</mi> </msup> <mi>Q</mi> <mi>G</mi> <mo>+</mo> <mi>R</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>G</mi> <mi>T</mi> </msup> <mi>Q</mi> <mo>&CenterDot;</mo> <mo>&lsqb;</mo> <msub> <mi>y</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mi>f</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>Make dT=[1 0 ... 0] (GTQG+R)-1GTQ, then<mrow> <msub> <mi>r</mi> <mi>k</mi> </msub> <mo>=</mo> <msup> <mi>d</mi> <mi>T</mi> </msup> <mo>&lsqb;</mo> <msub> <mi>y</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mi>f</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow><mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> <msubsup> <mo>(</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>I</mi> <mrow> <mi>P</mi> <mi>x</mi> <mn>1</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>Wherein,<mrow> <mi>H</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>C</mi> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>G</mi> <mrow> <mo>(</mo> <msubsup> <mo>|</mo> <mrow> <mi>t</mi> <mo>+</mo> <mi>P</mi> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>C</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>M</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>C</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>C</mi> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>C</mi> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>P</mi> </munderover> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>i</mi> <mo>)</mo> </mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mi>M</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>Solving-optimizing propositional formula (33), obtain finally parsing solution formula (35), be so as to obtain control law：uk(t)=rk(t), k=1；uk(t)=rk(t)+uk-1(t), k ＞ 1 (37).
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