CN106991075A - Multimodal extremum search method based on Newton method extremum search - Google Patents
Multimodal extremum search method based on Newton method extremum search Download PDFInfo
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Abstract
The invention discloses a kind of multimodal extremum search method based on Newton method extremum search, for single-input single-output system, given input initial point carries out the extremum search of preset time using Newton method extremum search, determine whether to search local extremum by judging the difference of search two similar time points of processing completion time used for them, if searching local extremum, the initial next local extremum of point search of input is then updated, otherwise extending search time continues search for the corresponding local extremum of the initial point.The present invention is extended based on Newton method extremum seeking algorithm, add the outer circulation for changing input initial point, by the change to initial point, the extreme value near different initial points is searched out using Newton method extremum seeking algorithm, so as to complete the extremum search process to Solving Multimodal Function.
Description
Technical field
The invention belongs to extremum search technical field, more specifically, it is related to a kind of based on Newton method extremum search
Multimodal extremum search method.
Background technology
For single-input single-output system, it is inputted and output can meet certain functional relation.In actual applications, lead to
Often need input being adjusted to suitable value, so that output reaches extreme value, be now accomplished by using extremum search method.Pass at present
The extremum seeking algorithm of system can only search local peaking, if system function is Solving Multimodal Function, then can only just search just
A peak value near initial point, the search for other peak values is helpless, and local peaking's not necessarily global peak.It is aobvious
So, want to search for the extreme value for obtaining Solving Multimodal Function system, depend heavily on the setting of initial point, and initial point is usually artificial
Set, it is difficult to break away from the influence of subjective factor, therefore need a kind of method that can be scanned for the extreme value of Solving Multimodal Function badly.
The content of the invention
It is an object of the invention to overcome the deficiencies of the prior art and provide a kind of multimodal pole based on Newton method extremum search
Be worth searching method, by constantly update initial point carry out extremum search, using realize to system function as Solving Multimodal Function single input
The multimodal extremum search of single output system.
For achieving the above object, the multimodal extremum search method of the invention based on Newton method extremum search includes following
Step：
S1：Set the input initial point x of extremum search0；
S2：With initial point x0Inputted as system, a length of t+m extreme value is searched when being carried out using Newton method extremum seeking algorithm
Rope, the system output y at record moment ttY is exported with moment t+m systemt+m, t and m are the time set according to actual conditions
Parameter；The specific steps of Newton method extremum seeking algorithm include：
S2.1：System is made to input
S2.2：System is inputtedIt is added, is inputted with disturbance S (t)
S2.3：System is inputted into x input systems, corresponding system output y is obtained, unstable ripple is filtered off by wave filter
Shape, the output stablized
S2.4：Will outputIt is multiplied respectively with disturbance M (t) and disturbance N (t), obtains signal The estimation of the inverse matrix of Hessian (Hai Sen) matrix H is constructed by Riccati (Li Kadi) equation
Matrix Γ, is then multiplied by output by estimated matrix ΓResult after being multiplied with disturbance M (t), produces variable
S2.5：It is integrated after Δ y is multiplied by into a constant coefficient-k, wherein k ＞ 0, in estimation pointPlace searches out next
Estimate iteration pointI.e.OrderReturn to step S2.2；
S3：Judge whether | yt+m-yt| ＜ ε, ε represent predetermined threshold value, ε ＞ 0, if it is, by yt+mAs current initial
Point x0Corresponding extreme value, into step S6, otherwise into step S4；
S4：Extend the search time of Newton method extremum seeking algorithm, the search time of extension represents the extension time for τ+m, τ
Parameter, records the system output y of time instant τ in this extension search timeτY is exported with time instant τ+m systemτ+m；
S5：Judge whether | yτ+m-yτ| ＜ ε, ε represent predetermined threshold value, if it is, by yτ+mIt is used as current initial point x0It is right
The extreme value answered, into step S6, otherwise return to step S4；
S6：Make initial point x0=x0+ h, return to step S2, h represent that initial point updates step-length；
During step S2 to step S6 circulations are performed, parameter is terminated to search and is monitored, once reach that search terminates
Condition, then multimodal extremum search terminate.
Multimodal extremum search method of the invention based on Newton method extremum search, for single-input single-output system, gives
The extremum search that initial point carries out preset time using Newton method extremum search is inputted, by judging search two phases of processing completion time used for them
The difference at nearly time point determines whether to search local extremum, if searching local extremum, updates input initial point and search
The next local extremum of rope, otherwise extend search time continue search for the corresponding local extremum of the initial point.
The present invention can not be to single-input single-output system of the system function for Solving Multimodal Function for traditional extremum seeking algorithm
The situation of multimodal extremum search is carried out, is extended based on Newton method extremum seeking algorithm, change input initial point is added
Outer circulation, by the change to initial point, the extreme value near different initial points is searched out using Newton method extremum seeking algorithm,
So as to complete the extremum search process to Solving Multimodal Function.By simulating, verifying it can be found that the present invention is a double loop system,
The corresponding extreme value of different input initial points can be accurately searched, search speed is fast, and stability is good.
Brief description of the drawings
Fig. 1 is the embodiment flow chart of the multimodal extremum search method of the invention based on Newton method extremum search；
Fig. 2 is Newton method extremum seeking algorithm schematic diagram in the present invention；
Fig. 3 is the curve map of system function in the present embodiment；
Fig. 4 is the crest value and the variation diagram of time obtained using multimodal extremum search method of the present invention in the present embodiment；
Fig. 5 is correspondence extreme point and the variation diagram of time in Fig. 4；
Fig. 6 is the crest value and the variation diagram of time for inputting Newton method extremum seeking algorithm when initial point is 3；
Fig. 7 is the crest value and the variation diagram of time for inputting Newton method extremum seeking algorithm when initial point is 4；
Fig. 8 is the crest value and the variation diagram of time for inputting Newton method extremum seeking algorithm when initial point is 5；
Fig. 9 is the valley value and the variation diagram of time obtained using multimodal extremum search method of the present invention in the present embodiment；
Figure 10 is correspondence extreme point and the variation diagram of time in Fig. 9；
Figure 11 is the valley value and the variation diagram of time of Newton method extremum seeking algorithm when initial point is 3；
Figure 12 is the valley value and the variation diagram of time of Newton method extremum seeking algorithm when initial point is 4；
Figure 13 is the valley value and the variation diagram of time of Newton method extremum seeking algorithm when initial point is 5.
Embodiment
The embodiment to the present invention is described below in conjunction with the accompanying drawings, so as to those skilled in the art preferably
Understand the present invention.Requiring particular attention is that, in the following description, when known function and design detailed description perhaps
When can desalinate the main contents of the present invention, these descriptions will be ignored herein.
Embodiment
Fig. 1 is the embodiment flow chart of the multimodal extremum search method of the invention based on Newton method extremum search.
As shown in figure 1, the specific steps of the multimodal extremum search method of the invention based on Newton method extremum search include：
S101：Set initial parameter：
Set the input initial point x of extremum search0。
S102：Newton method extremum search：
With initial point x0Inputted as system, a length of t+m extremum search when being carried out using Newton method extremum seeking algorithm,
Record the system output y at moment ttY is exported with moment t+m systemt+m, t and m are the time ginsengs set according to actual conditions
Number.
Fig. 2 is Newton method extremum seeking algorithm schematic diagram in the present invention.As shown in Fig. 2 Newton method extremum seeking algorithm
Detailed process comprises the following steps：
S201：System is made to input
S202：Input disturbance is handled：
System is inputtedIt is added, is inputted with disturbance S (t)
The disturbance expression used in the present embodiment for：
Wherein, η (t)=[cos (W (t)), sin (W (t))]TIt is the motion of Blang's unit circle,It is one
Constant vector, wherein, h1＞ 0, h2＞ 0 and a ＞ 0 is design parameter.
S203：Obtain correspondence system output：
System is inputted into x input systems, corresponding system output y is obtained, unstable waveform is filtered off by wave filter, is obtained
To stable output
S204：Output disturbance processing：
Will outputIt is multiplied respectively with disturbance M (t) and disturbance N (t), obtains signal
M (t) and disturbance N (t) are disturbed in the present embodiment to be expressed as：
OutputBe multiplied with M (t) be in order to estimate outputFirst derivative.OutputObtained after being multiplied with disturbance N (t)
As a resultThe estimated matrix Γ of the inverse matrix of Hessian matrix Hs is constructed by Riccati equations, then by estimated matrix Γ
It is multiplied by outputResult after being multiplied with disturbance M (t), produces variable
S205：Find next estimation iteration point：
It is integrated after Δ y is multiplied by into a constant coefficient-k, wherein k ＞ 0, in estimation pointPlace searches out next estimation
Iteration pointI.e.OrderReturn to step S202.
The iterative formula of Newton method is：WhereinAs
Hessian matrix Hs,Estimation be Γ,Exported after as filteringAfter being multiplied with disturbance M (t)
Estimation,Make bearing the difference of Newton direction, here two estimation points：
So next estimation iteration point can be searched out after integration
S103：Judge whether that search obtains local extremum：
Y is exported by the system for comparing moment ttY is exported with moment t+m systemt+mDifference whether be less than predetermined threshold value
Can be determined whether that search obtains local extremum, that is, judge whether | yt+m-yt| ＜ ε, ε represent predetermined threshold value, ε ＞ 0, if it is,
Then by yt+mIt is used as current initial point x0Corresponding extreme value, into step S107, otherwise into step S104.Obviously, m is one
Shorter time interval.
S104：Extend search time：
If not searching current initial point x also0Corresponding extreme value, then need to extend searching for Newton method extremum seeking algorithm
Rope time, search time of extension is τ+m, and τ represents to extend time parameter, and record time instant τ in this extension search time is
System output yτY is exported with time instant τ+m systemτ+m。
S105：Judge whether extension search searches for and obtain local extremum：
It is similar with step S103, y is exported by the system for comparing time instant τ in extension search timeτWith time instant τ+m system
Export yτ+mDifference whether be that can be determined whether that search obtains local extremum less than predetermined threshold value, that is, judge whether | yτ+m-yτ|
＜ ε, ε represent predetermined threshold value, if it is, by yτ+mIt is used as current initial point x0Corresponding extreme value, into step S106, otherwise
Return to step S104.
It can be seen that, if from initial point x0Start timing as the initial input of Newton method extremum search, can carry out first
Shi Changwei t+m extremum search, if search also may proceed to carry out duration τ+m extremum search less than extreme value, by that analogy,
A length of τ+m search when extending every time, always search is gone down.
S106：Update initial point：
Make initial point x0=x0+ h, return to step S102, h represent that initial point updates step-length, are set according to actual conditions
's.
It can be seen that, step S102 to step S106 forms a circulation, and lasting search obtains each extreme value of system.In step
During rapid S102 to step S106 circulations are performed, parameter is terminated to search and is monitored, once reach search termination condition, then it is many
Peak extremum search terminates.In general, search end parameter could be arranged to two kinds, and a kind of is the total of multimodal extremum search method
The running body time, i.e., the total runtime of multimodal extremum search method is recorded using a timer, when overall operation
Between reach predetermined threshold value, then multimodal extremum search terminates；A kind of is the value of current initial point, if i.e. initial point x0Beyond default
Scope, then multimodal extremum search terminate.User can be selected according to the situation of real system using which kind of search termination condition
Judged.
In order to illustrate the feasibility and correctness of the present invention, pass through theory deduction and simulating, verifying two ways separately below
To carry out analytic explanation to technical scheme.
The present invention uses Newton method extremum seeking algorithm when carrying out local peaking's search to each initial value, should
Method can be carried out from closed loop cycle, so that it is attached to search given initial point for a given initial value by continuous iteration
Near peak value, the detailed process that it is proved is as follows.
For any nonlinear system：
Y=f (x), x ∈ R (4)
Wherein f () is unknown system function, it is known that f () is in x=x*There is local extremum y at place*=f (x*), it is assumed that f
() is that Second Order Continuous can be micro-, and there is a constant value x*∈ R, have
It can be seen from the extremum search method based on Newton method shown in Fig. 2：
Subscript " " represents first derivative, h ＞ 0.
Define evaluated errorx*Represent extreme point,Subscript " -1 " represents inverse matrix, can
By assuming to understand f (x) in x=x*The Taylor expansion at place is
Bring formula (5) and formula (10) into formula (8) and formula (9), error system can be obtained：
It is theoretical using average to error system, its equal valve system can be obtained：
Equilibrium point can be obtained by equal valve system：
Its corresponding Jacobian matrix：
By h ＞ 0 and k ＞ 0, it is known that Jacobian matrix is Hull dimension hereby matrix, then equal valve system is in equilibrium point (0,0)
Place is exponentially stable, is so far demonstrated to any one given initial point, can be searched by the way that Newton method extremum seeking algorithm is stable
Rope is to the extreme value near it.
Multimodal extremum search method of the present invention is by comparing whether the difference of two similar time point output valves is less than a valve
Value judges whether to search the extreme value near the initial point, if searching, initial point is increased every time after h as it is new just
Initial point.By the value for updating initial point, you can realize the multimodal extremum search to single-input single-output system.
In order to prove the technique effect of the present invention, simulating, verifying has been carried out using a specific system.Used system
System function it is as follows：
Fig. 3 is the curve map of system function in the present embodiment.As shown in figure 3, system function is a week in the present embodiment
Phase function, there is infinite multiple peak values.Simulating, verifying only is carried out to wherein several peak values in the present embodiment.
Then the carry out peak value searching of the system is entered using Newton method extremum seeking algorithm using the inventive method first
The local unimodal search of row is with as a comparison, because Newton method extremum seeking algorithm can only search a peak value every time, therefore
Repeatedly searched for using the input initial value of the inventive method.
When carrying out peak value searching to the system using the inventive method, each parameter setting is as follows：Initial value x0=3, just
Initial value updates step-length h=1, search time parameter t=20, m=1, extension time parameter τ=4, the judgment threshold ε of local extremum
=0.1, search termination condition is using the total runtime t searched forAlways=50.In order to preferably be contrasted, this emulation is tested
Crest and trough are scanned for respectively in card, i.e., crest judges the condition of Local Extremum for 0≤yt+m-yt＜ ε and 0≤
yτ+m-yτ＜ ε, trough judges the condition of Local Extremum for 0≤yt-yt+m＜ ε and 0≤yτ-yτ+m＜ ε.
Fig. 4 is the crest value and the variation diagram of time obtained using multimodal extremum search method of the present invention in the present embodiment.
Fig. 5 is correspondence extreme point and the variation diagram of time in Fig. 4.As shown in Figure 4 and Figure 5, it is known that searched 3 different crests altogether
Value, corresponding initial point is respectively 3,4 and 5, as initial point x0Y when=30=3.6136, work as x0Y when=41=6.1658, work as x0
Y when=52=14.4970.The crest value searched in Fig. 3 and Fig. 4 and correspondence extreme point are compared with Fig. 3, it can be seen that be
Match.
Initial value x is set respectively0Unimodal search is carried out using Newton method extremum seeking algorithm for 3,4,5.Fig. 6 is that input is first
The crest value and the variation diagram of time of Newton method extremum seeking algorithm when initial point is 3.Fig. 7 is Newton method when input initial point is 4
The crest value of extremum seeking algorithm and the variation diagram of time.Fig. 8 is the ripple for inputting Newton method extremum seeking algorithm when initial point is 5
Peak value and the variation diagram of time.As shown in Figure 6 to 8, for Newton method extremum seeking algorithm, as given initial value x0=3
When y=3.6140, as given initial value x0Y=6.1650 when=4, as given initial value x0Y=14.4950 when=5, will
Knowable to the search result of this three width figure and Fig. 4 search result are contrasted, error is respectively less than 0.1, it is possible thereby to illustrate searching
The correctness of the present invention in terms of rope crest value.
Fig. 9 is the valley value and the variation diagram of time obtained using multimodal extremum search method of the present invention in the present embodiment.
Figure 10 is correspondence extreme point and the variation diagram of time in Fig. 9.As shown in Figure 9 and Figure 10, it is known that searched 3 different ripples altogether
Valley, corresponding initial point is respectively 3,4 and 5, as initial point x0Y when=30=-2.6362, work as x0Y when=41=-3.5853,
Work as x0Y when=52=-12.8680.The valley value searched in Fig. 9 and Figure 10 and correspondence extreme point are compared with Fig. 3, can be with
It is seen to be what is matched.
Similarly, initial point x is set respectively0Valley value is carried out using Newton method extremum seeking algorithm for 3,4,5 unimodal
Search.Figure 11 is the valley value and the variation diagram of time of Newton method extremum seeking algorithm when initial point is 3.Figure 12 is that initial point is
The valley value of Newton method extremum seeking algorithm and the variation diagram of time when 4.Figure 13 is that Newton method extremum search is calculated when initial point is 5
The valley value of method and the variation diagram of time.As shown in Figure 11 to Figure 13, it is initial when what is given for Newton method extremum seeking algorithm
Point x0Y=-2.6390 when=3, as given initial point x0Y=-3.5820 when=4, as given initial point x0Y=- when=5
12.8650, the search result of the search result of this three width figure and Fig. 9 is contrasted knowable to, error is respectively less than 0.1, thus may be used
To illustrate that the inventive method is also correct in terms of trough is searched for.
In summary, it is known that the peak value for searching Solving Multimodal Function that algorithm is capable of accurate stable is carried herein, for crest
Search with trough is all suitable for.
Although illustrative embodiment of the invention is described above, in order to the technology of the art
Personnel understand the present invention, it should be apparent that the invention is not restricted to the scope of embodiment, to the common skill of the art
For art personnel, as long as various change is in the spirit and scope of the present invention that appended claim is limited and is determined, these
Change is it will be apparent that all utilize the innovation and creation of present inventive concept in the row of protection.
Claims (3)
1. a kind of multimodal extremum search method based on Newton method extremum search, it is characterised in that comprise the following steps：
S1：Set the input initial point x of extremum search0；
S2：With initial point x0Input, a length of t+m extremum search when being carried out using Newton method extremum seeking algorithm, remember as system
Record the system output y at moment ttY is exported with moment t+m systemt+m, t and m are the time parameters set according to actual conditions；
The specific steps of Newton method extremum seeking algorithm include：
S2.1：System is made to input
S2.2：System is inputtedIt is added, is inputted with disturbance S (t)
S2.3：System is inputted into x input systems, corresponding system output y is obtained, unstable waveform is filtered off by wave filter,
The output stablized
S2.4：Will outputIt is multiplied respectively with disturbance M (t) and disturbance N (t), obtains signal The estimated matrix Γ of the inverse matrix of Hessian matrix Hs is constructed by Riccati equations, then will be estimated
Meter matrix Γ is multiplied by outputResult after being multiplied with disturbance M (t), produces variable
S2.5：It is integrated after Δ y is multiplied by into a constant coefficient-k, wherein k ＞ 0, in estimation pointPlace searches out next estimation
Iteration pointI.e.OrderReturn to step S2.2；
S3：Judge whether | yt+m-yt| ＜ ε, ε represent predetermined threshold value, ε ＞ 0, if it is, by yt+mIt is used as current initial point x0It is right
The extreme value answered, into step S6, otherwise into step S4；
S4：Extend the search time of Newton method extremum seeking algorithm, the search time of extension is τ+m, record this extension search
The system output y of time instant τ in timeτY is exported with time instant τ+m systemτ+m；
S5：Judge whether | yτ+m-yτ| ＜ ε, ε represent predetermined threshold value, if it is, by yk+mIt is used as current initial point x0It is corresponding
Extreme value, into step S6, otherwise return to step S4；
S6：Make initial point x0=x0+ h, return to step S2, h represent that initial point updates step-length；
During step S2 to step S6 circulations are performed, parameter is terminated to search and is monitored, once search termination condition is reached,
Then multimodal extremum search terminates.
2. multimodal extremum search method according to claim 1, it is characterised in that the search terminates parameter and uses multimodal
The total runtime of extremum search method, i.e., when total runtime reaches predetermined threshold value, then multimodal extremum search terminates.
3. multimodal extremum search method according to claim 1, it is characterised in that the search terminates parameter using initial
The value of point, if i.e. initial point x0Beyond preset range, then multimodal extremum search terminates.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109347214A (en) * | 2018-07-16 | 2019-02-15 | 福州大学 | The searching method of multimodal extreme value in radio energy transmission system frequency modulation control |
| CN110262618A (en) * | 2019-06-19 | 2019-09-20 | 电子科技大学 | A kind of photovoltaic peak power tracking method based on no steady oscillation extremum search |
| CN112612029A (en) * | 2020-12-24 | 2021-04-06 | 哈尔滨工业大学芜湖机器人产业技术研究院 | Grid map positioning method fusing NDT and ICP |
-
2017
- 2017-03-01 CN CN201710117344.4A patent/CN106991075A/en active Pending
Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109347214A (en) * | 2018-07-16 | 2019-02-15 | 福州大学 | The searching method of multimodal extreme value in radio energy transmission system frequency modulation control |
| CN109347214B (en) * | 2018-07-16 | 2021-12-28 | 福州大学 | Method for searching multi-peak extreme value in frequency modulation control of wireless power transmission system |
| CN110262618A (en) * | 2019-06-19 | 2019-09-20 | 电子科技大学 | A kind of photovoltaic peak power tracking method based on no steady oscillation extremum search |
| CN110262618B (en) * | 2019-06-19 | 2020-07-31 | 电子科技大学 | Photovoltaic peak power tracking method based on unsteady state oscillation extremum search |
| CN112612029A (en) * | 2020-12-24 | 2021-04-06 | 哈尔滨工业大学芜湖机器人产业技术研究院 | Grid map positioning method fusing NDT and ICP |
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