CN113359482A - Output tracking control of high-order under-actuated mechanical system disturbed by second moment process - Google Patents

Output tracking control of high-order under-actuated mechanical system disturbed by second moment process Download PDF

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CN113359482A
CN113359482A CN202110838853.2A CN202110838853A CN113359482A CN 113359482 A CN113359482 A CN 113359482A CN 202110838853 A CN202110838853 A CN 202110838853A CN 113359482 A CN113359482 A CN 113359482A
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李武全
王瑞桃
徐晓宇
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Ludong University
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Abstract

The invention discloses a method for researching the output tracking problem of an under-actuated high-order mechanical system disturbed by a second moment process. Unlike prior studies on the interference of systems by wiener processes, a more practical noise, the second moment process, is considered herein. A new controller is designed by a reverse-push design method, the tracking error can be adjusted to be arbitrarily small, a closed-loop system has a unique solution, and all states are bounded in probability. Finally, the feasibility of the controller design is demonstrated by simulations. The method has strong practicability and is suitable for the output tracking control problem of a random high-order system.

Description

Output tracking control of high-order under-actuated mechanical system disturbed by second moment process
Technical Field
The invention belongs to the field of stability control research of random high-order nonlinear systems. Specifically, a high-order under-actuated mechanical system interfered by a second moment process is considered, and a controller is designed and stability analysis is carried out mainly by adopting a reverse-thrust design method.
Background
The widespread use of stochastic systems in life has led to increased attention being paid to the associated control problems. At present, all results of the output tracking control problem of the random high-order nonlinear system consider that the system is interfered by the wiener process, but the second moment process is more common in real life. For example, in some electrical systems, a second moment process is used to describe the disturbance of an electrical component, or the second moment process may be used in engineering applications to simulate the effect of road irregularities on a mechanical system. The present invention takes into account that in many practical engineering systems, white noise cannot simulate some practical interference noise. Therefore, compared with the previous research on the tracking control problem of a random high-order nonlinear system, the method has more theoretical significance and practical significance in researching the tracking control problem of the high-order under-actuated mechanical system disturbed by the second-order moment process.
Disclosure of Invention
The embodiment of the invention provides an output tracking control method for a random mechanical system interfered by a secondary moment process, which is used for solving the problem of poor practicability caused by the fact that the interference of the secondary moment process is not considered in the existing control method.
In a first aspect, the present invention provides a method of designing a controller, comprising: performing variable replacement on a kinematic equation of a mechanical system, and converting the kinematic equation into a state space model; and introducing a group of coordinate transformation and virtual controllers into the obtained state space model, and finally designing a proper controller.
And in the second aspect, the controller designed in the first aspect is used as input to an original system, derivative is carried out on a correspondingly designed Lyapunov function, and stability analysis is carried out by combining a Lyapunov second discrimination method.
The embodiment of the invention provides a method for designing a controller, which carries out corresponding processing on the secondary moment process interference on a system through the reasonably designed controller, and overcomes the problem that the existing control algorithm ignores the influence of the secondary moment process, so that the system can still effectively track a reference signal under the condition of the secondary moment process interference.
Drawings
In order to clearly and accurately illustrate the embodiments of the present invention, a brief description of the drawings needed to describe the operation steps of the embodiments is provided below, and it is obvious to those skilled in the art that other drawings can be obtained according to the drawings without inventive labor.
FIG. 1 is a diagram of an under-actuated mechanical system perturbed by a second moment process for the present case;
FIG. 2 is a schematic illustration of the 2-th moment of the tracking error provided by the present invention;
fig. 3 is a schematic diagram of u of the controller according to the embodiment of the present invention.
Detailed Description
1 example System description of the invention
The invention concerns an under-actuated mechanical system consisting of a mass on a horizontal smooth surface
Figure 396833DEST_PATH_IMAGE001
Inverted pendulum without mass bar support
Figure 275927DEST_PATH_IMAGE002
Composition, as shown in figure 1. Mass block
Figure 186115DEST_PATH_IMAGE001
Is connected with the wall surface through a linear spring, and simultaneously uses a nonlinear spring with a three-time force-deformation relation and an inverted pendulum
Figure 393236DEST_PATH_IMAGE002
And (4) connecting. Is provided with
Figure 758490DEST_PATH_IMAGE003
Is a mass block
Figure 616724DEST_PATH_IMAGE001
Is detected by the displacement of (a) a,
Figure 979704DEST_PATH_IMAGE004
for inverted pendulum
Figure 533176DEST_PATH_IMAGE002
At an angle to the vertical line of
Figure 623492DEST_PATH_IMAGE005
And
Figure 809667DEST_PATH_IMAGE006
. The spring is not stretched. A control force
Figure 327236DEST_PATH_IMAGE007
Act on
Figure 305687DEST_PATH_IMAGE001
. The system has two degrees of freedom that are under-actuated. The units of these variables are: the unit of the mass block and the inverted pendulum is kilogram (kg); displacement of
Figure 12743DEST_PATH_IMAGE003
，
Figure 111149DEST_PATH_IMAGE008
Length of
Figure 19194DEST_PATH_IMAGE009
The unit of (d) is meter (m); angle of rotation
Figure 734209DEST_PATH_IMAGE004
Unit of (d) is radian (rad); force of
Figure 979377DEST_PATH_IMAGE007
，
Figure 214180DEST_PATH_IMAGE010
The unit of (a) is newton (N); acceleration of gravity
Figure 276814DEST_PATH_IMAGE011
Has the unit of
Figure 964278DEST_PATH_IMAGE012
(ii) a The time unit is seconds(s). The motion equation is as follows:
Figure 278716DEST_PATH_IMAGE013
wherein the content of the first and second substances,
Figure 617294DEST_PATH_IMAGE014
is the spring constant in
Figure 601561DEST_PATH_IMAGE015
，
Figure 760010DEST_PATH_IMAGE016
Spring constant of a non-linear spring in units of
Figure 284663DEST_PATH_IMAGE017
. Assume parameters in the above system
Figure 556376DEST_PATH_IMAGE001
，
Figure 23129DEST_PATH_IMAGE002
，
Figure 622869DEST_PATH_IMAGE009
，
Figure 934902DEST_PATH_IMAGE016
Are all unknown constants but belong to known intervals
Figure 998804DEST_PATH_IMAGE018
。
Subjecting the system to coordinate transformation
Figure 449508DEST_PATH_IMAGE019
Given domain of definition
Figure 51390DEST_PATH_IMAGE020
So as to obtain a state space model of the system
Figure 183426DEST_PATH_IMAGE021
Wherein the function
Figure 164151DEST_PATH_IMAGE022
Is defined in
Figure 972707DEST_PATH_IMAGE023
On
Figure 293179DEST_PATH_IMAGE024
Function and initial value
Figure 478172DEST_PATH_IMAGE025
。
Figure 313404DEST_PATH_IMAGE026
Is a standard second moment process defined on the complete probability space,
Figure 43594DEST_PATH_IMAGE027
is that
Figure 354490DEST_PATH_IMAGE028
Adaptive and continuous in segments, meet
Figure 562748DEST_PATH_IMAGE029
Figure 439437DEST_PATH_IMAGE030
Is a constant. For the
Figure 668424DEST_PATH_IMAGE031
Selecting a reference signal
Figure 14086DEST_PATH_IMAGE032
Suppose that
Figure 212986DEST_PATH_IMAGE033
And
Figure 694915DEST_PATH_IMAGE034
all have a constant bound, i.e. there is a normal number
Figure 281754DEST_PATH_IMAGE035
So that
Figure 911449DEST_PATH_IMAGE036
According to the system (3) have
Figure 992669DEST_PATH_IMAGE037
，
Figure 578371DEST_PATH_IMAGE038
. Definition of
Figure 758948DEST_PATH_IMAGE039
. Selecting a series of Lyapunov functions
Figure 328469DEST_PATH_IMAGE040
And a series of virtual controllers and a final real controller
Figure 275697DEST_PATH_IMAGE041
2 detailed implementation steps of the invention
Nonlinear systems perturbed by second-order moment processes are generally described as
Figure 669900DEST_PATH_IMAGE042
Wherein is
Figure 598542DEST_PATH_IMAGE043
The status of the system is such that,
Figure 468409DEST_PATH_IMAGE044
is the system state
Figure 94693DEST_PATH_IMAGE045
The derivative of (a) of (b),
Figure 999196DEST_PATH_IMAGE046
is a second moment process, and is applied to all
Figure 770842DEST_PATH_IMAGE047
Satisfy the requirement of
Figure 65689DEST_PATH_IMAGE048
Description of the invention
Figure 77421DEST_PATH_IMAGE049
Is a standard second moment process.
Before the controller design process, some relevant reasoning is given.
Introduction 1 for any
Figure 757801DEST_PATH_IMAGE050
Definition of time of rest
Figure 451081DEST_PATH_IMAGE051
Wherein it is assumed that
Figure 951333DEST_PATH_IMAGE052
. Stochastic process for the above system
Figure 44054DEST_PATH_IMAGE046
Is segmented and continuous, and
Figure 329673DEST_PATH_IMAGE029
，
Figure 708701DEST_PATH_IMAGE030
is a constant. If there is a positive definite function
Figure 446981DEST_PATH_IMAGE053
Sum constant
Figure 999186DEST_PATH_IMAGE054
For all
Figure 467207DEST_PATH_IMAGE055
Is provided with
Figure 830186DEST_PATH_IMAGE056
Figure 508292DEST_PATH_IMAGE057
The system is in
Figure 677237DEST_PATH_IMAGE058
There is a unique solution.
Lemma 2. assume that the above system has a unique solution, domain, in an almost everywhere sense
Figure 140710DEST_PATH_IMAGE059
There is a positive definite function
Figure 923859DEST_PATH_IMAGE060
And parameters
Figure 964627DEST_PATH_IMAGE061
So that
Figure 609366DEST_PATH_IMAGE062
Is true for all
Figure 911034DEST_PATH_IMAGE063
，
Figure 943712DEST_PATH_IMAGE064
This is true. Therefore, for any initial value
Figure 143881DEST_PATH_IMAGE065
The solution probability of the system is bounded.
3.1 Lei
Figure 779261DEST_PATH_IMAGE066
Is a random process with a regular sample path. If it is not
Figure 341961DEST_PATH_IMAGE067
Is limited, then
Figure 217644DEST_PATH_IMAGE068
。
Lei 4. for all
Figure 357638DEST_PATH_IMAGE069
And any positive real number
Figure 78601DEST_PATH_IMAGE070
The following inequality holds
Figure 507526DEST_PATH_IMAGE071
Introduction 5. set
Figure 6640DEST_PATH_IMAGE072
Is a finite constant that is set to be constant,
Figure 243717DEST_PATH_IMAGE073
if for
Figure 33950DEST_PATH_IMAGE074
，
Figure 695876DEST_PATH_IMAGE075
Is established then by
Figure 975678DEST_PATH_IMAGE076
2.1 adopting a recursion design method to construct a distributed controller for the system (3).
The first step is as follows: designing distributed virtual controllers
Figure 575418DEST_PATH_IMAGE077
。
Introducing transformations
Figure 418609DEST_PATH_IMAGE078
The result of the calculation is obtained,
Figure 810407DEST_PATH_IMAGE079
selecting the Lyapunov function
Figure 136478DEST_PATH_IMAGE080
Therefore, it is required to have
Figure 269519DEST_PATH_IMAGE081
By Young inequality and tracking signal
Figure 463871DEST_PATH_IMAGE082
By the nature of (1), we obtain
Figure 585542DEST_PATH_IMAGE083
Wherein the content of the first and second substances,
Figure 394098DEST_PATH_IMAGE084
is an arbitrary constant which is a constant number,
Figure 358643DEST_PATH_IMAGE085
a non-negative smooth function.
It is apparent that the virtual controller is in the form of
Figure 232052DEST_PATH_IMAGE086
Wherein
Figure 332863DEST_PATH_IMAGE087
Is a function of the positive smoothness of the image,
Figure 515583DEST_PATH_IMAGE088
is a freely designed parameter.
Substituting (10) - (11) into (9) to obtain
Figure 701845DEST_PATH_IMAGE089
The second step is that: designing distributed virtual controlDevice for cleaning the skin
Figure 175682DEST_PATH_IMAGE090
。
Defining transformations
Figure 521213DEST_PATH_IMAGE091
Due to the fact that
Figure 484621DEST_PATH_IMAGE077
Is smooth, we have
Figure 33545DEST_PATH_IMAGE092
Considering the Lyapunov function
Figure 627251DEST_PATH_IMAGE093
The method according to (13), wherein,
Figure 624026DEST_PATH_IMAGE094
for the
Figure 899281DEST_PATH_IMAGE095
According to the theorem 4, the sum of the inequalities Young
Figure 794555DEST_PATH_IMAGE096
And has an unequal relationship of the above-mentioned values,
Figure 125043DEST_PATH_IMAGE097
wherein the content of the first and second substances,
Figure 664740DEST_PATH_IMAGE098
is an arbitrary normal number which is a constant number,
Figure 156901DEST_PATH_IMAGE099
is a positive smooth function.
From the bounded nature of the tracking signal, it is apparent thatA smooth function
Figure 805051DEST_PATH_IMAGE100
So that
Figure 362065DEST_PATH_IMAGE101
Similarly, the Young inequality and the nature of the tracking signal illustrate that there is a smooth function
Figure 67853DEST_PATH_IMAGE102
And an arbitrary constant
Figure 809544DEST_PATH_IMAGE103
There is a relationship that,
Figure 554777DEST_PATH_IMAGE104
substitution of (15) and (17) into (14) gives
Figure 227067DEST_PATH_IMAGE105
Therefore, the virtual control law is selected
Figure 865990DEST_PATH_IMAGE106
So that
Figure 653949DEST_PATH_IMAGE107
Wherein
Figure 932483DEST_PATH_IMAGE108
Is a parameter that is freely designed and is,
Figure 221513DEST_PATH_IMAGE109
is a positive definite smooth function.
The third step: design distributionVirtual controller
Figure 590309DEST_PATH_IMAGE110
。
Defining transformations
Figure 860753DEST_PATH_IMAGE111
Because of
Figure 705212DEST_PATH_IMAGE112
Is smooth and can be obtained
Figure 407720DEST_PATH_IMAGE113
Wherein each function is
Figure 942607DEST_PATH_IMAGE114
Selecting a Lyapunov function
Figure 462581DEST_PATH_IMAGE115
Figure 669703DEST_PATH_IMAGE116
Derivation of this can yield:
Figure 46675DEST_PATH_IMAGE117
similar to the previous two steps, some items are processed
Figure 701647DEST_PATH_IMAGE118
Figure 736730DEST_PATH_IMAGE119
Figure 555782DEST_PATH_IMAGE120
Substituting (24) and (25) into (23) to obtain
Figure 911677DEST_PATH_IMAGE121
Thus a virtual controller
Figure 109571DEST_PATH_IMAGE122
So that
Figure 627140DEST_PATH_IMAGE123
Wherein the content of the first and second substances,
Figure 667908DEST_PATH_IMAGE124
is a parameter that is freely designed and is,
Figure 312647DEST_PATH_IMAGE125
is a positive definite smooth function.
The fourth step: design true controller
Figure 879895DEST_PATH_IMAGE126
Defining transformations
Figure 584677DEST_PATH_IMAGE127
Can be obtained by the same way
Figure 112741DEST_PATH_IMAGE128
Wherein
Figure 748122DEST_PATH_IMAGE129
Figure 982925DEST_PATH_IMAGE130
Selecting a Lyapunov function
Figure 186505DEST_PATH_IMAGE131
Combined with the above transformation to obtain
Figure 326499DEST_PATH_IMAGE132
Similar to the previous proof of procedure, we can obtain the following relationship:
Figure 375358DEST_PATH_IMAGE133
Figure 464667DEST_PATH_IMAGE134
Figure 963782DEST_PATH_IMAGE135
wherein the content of the first and second substances,
Figure 200859DEST_PATH_IMAGE136
，
Figure 194354DEST_PATH_IMAGE137
and
Figure 387438DEST_PATH_IMAGE138
are all constant in number, and are,
Figure 667241DEST_PATH_IMAGE139
are positively smooth functions. Therefore, we design the controller
Figure 532560DEST_PATH_IMAGE140
The derivative of the final Lyapunov function satisfies
Figure 375751DEST_PATH_IMAGE141
Wherein
Figure 490251DEST_PATH_IMAGE142
Is a parameter that is freely designed and is,
Figure 816321DEST_PATH_IMAGE143
is a positive definite smooth function.
2.2 stability analysis
In the present invention we reach the following stability conclusions.
Theorem 1 for a controller having a shape like (37) for a system (3), the following conclusion holds:
1) closed loop system in
Figure 214942DEST_PATH_IMAGE144
There is a unique solution;
2) all states of the closed-loop system are bounded probabilistically;
3) tracking error satisfaction
Figure 674873DEST_PATH_IMAGE145
The right side can be made small enough by a reasonable choice of design parameters.
And (3) proving that: first, the closed-loop system is proved
Figure 530965DEST_PATH_IMAGE144
There is a unique solution.
According to the Young inequality, for arbitrary constants
Figure 418149DEST_PATH_IMAGE146
We can get
Figure 304066DEST_PATH_IMAGE147
That is to say
Figure 911896DEST_PATH_IMAGE148
Wherein
Figure 934078DEST_PATH_IMAGE149
。
Combining (39) and (40), we obtain
Figure 257743DEST_PATH_IMAGE150
Wherein
Figure 319372DEST_PATH_IMAGE151
。
We rewrite (41) to
Figure 855526DEST_PATH_IMAGE152
Wherein
Figure 607582DEST_PATH_IMAGE153
。
Is provided with
Figure 711935DEST_PATH_IMAGE154
For any one
Figure 306864DEST_PATH_IMAGE155
Definition of time of rest
Figure 646710DEST_PATH_IMAGE156
According to (42), can be obtained
Figure 597480DEST_PATH_IMAGE157
According to
Figure 918739DEST_PATH_IMAGE158
By definition and second moment process properties, we obtain
Figure 814014DEST_PATH_IMAGE159
In conjunction with lemma 1, a closed loop system is described
Figure 832917DEST_PATH_IMAGE160
The only syndrome of syndrome is solved.
Next, we demonstrate that all states of the closed-loop system are bounded with probability.
Is provided with
Figure 762827DEST_PATH_IMAGE161
To obtain the formula
Figure 520567DEST_PATH_IMAGE162
According to
Figure 790223DEST_PATH_IMAGE163
And the second moment process property, we get
Figure 393243DEST_PATH_IMAGE164
According to (47) and Fubini's theorem 3
Figure 443238DEST_PATH_IMAGE165
Means that
Figure 60296DEST_PATH_IMAGE166
The above formula illustrates that the derivation and expectation may be switched in order of operations.
Without being provided with
Figure 930163DEST_PATH_IMAGE167
To obtain
Figure 336873DEST_PATH_IMAGE168
Overwrite (50) according to the theorem 5
Figure 116742DEST_PATH_IMAGE169
That is to say
Figure 29334DEST_PATH_IMAGE170
Therefore, we have
Figure 307869DEST_PATH_IMAGE171
According to lemma 2, it was concluded that all states of a closed-loop system are probabilistically bounded proven.
The third conclusion is demonstrated below. From the formula (52), it can be derived
Figure 206686DEST_PATH_IMAGE172
In view of
Figure 965694DEST_PATH_IMAGE173
By definition of (1), we obtain
Figure 236138DEST_PATH_IMAGE174
Wherein
Figure 955964DEST_PATH_IMAGE175
By
Figure 438898DEST_PATH_IMAGE176
And
Figure 317992DEST_PATH_IMAGE177
are independent of each other. We can choose appropriate parameters to make the right side of (55) small enough. This is done for certification.
2.3 simulation
The invention simulates the output tracking control of a system under the interference of a second moment process through Matlab simulation, and embodies the effectiveness of the algorithm of the invention. For such actual mechanical systems, we select appropriate parameters according to actual conditions. To avoid loss of generality, we assume
Figure 447753DEST_PATH_IMAGE178
. According to the calculation steps of the algorithm, we can obtain
Figure 700880DEST_PATH_IMAGE179
Wherein
Figure 331713DEST_PATH_IMAGE180
Figure 675101DEST_PATH_IMAGE181
Figure 100397DEST_PATH_IMAGE182
In the simulation, we select the reference signal
Figure 44082DEST_PATH_IMAGE183
In the unit of
Figure 150709DEST_PATH_IMAGE184
. Giving an initial value
Figure 473237DEST_PATH_IMAGE185
In the unit of
Figure 194069DEST_PATH_IMAGE184
，
Figure 488698DEST_PATH_IMAGE186
，
Figure 133437DEST_PATH_IMAGE187
，
Figure 700684DEST_PATH_IMAGE188
. The adjustable parameter is taken as
Figure 139887DEST_PATH_IMAGE189
，
Figure 667951DEST_PATH_IMAGE190
，
Figure 568911DEST_PATH_IMAGE191
，
Figure 538135DEST_PATH_IMAGE192
，
Figure 476136DEST_PATH_IMAGE193
，
Figure 616130DEST_PATH_IMAGE194
，
Figure 868251DEST_PATH_IMAGE195
，
Figure 285457DEST_PATH_IMAGE196
，
Figure 518992DEST_PATH_IMAGE197
，
Figure 693753DEST_PATH_IMAGE198
，
Figure 546302DEST_PATH_IMAGE199
，
Figure 942648DEST_PATH_IMAGE200
，
Figure 160134DEST_PATH_IMAGE201
，
Figure 884508DEST_PATH_IMAGE202
，
Figure 462120DEST_PATH_IMAGE203
. From fig. 2 it can be seen that the tracking error squared order moment satisfies
Figure 463705DEST_PATH_IMAGE204
Thus, the feasibility of the above invention was verified.

Claims (5)

1. A research method aiming at output tracking of an under-actuated mechanical system interfered by a second moment process is characterized in that: the controller is designed and the stability of the system after control is analyzed.
2. The design control section according to claim 1, wherein the design control section is realized by a reverse-thrust design method including: performing variable replacement on a kinematic equation of a mechanical system, and converting the kinematic equation into a state space model; and introducing a group of coordinate transformation and virtual controllers into the obtained state space model, and finally designing a proper controller.
3. The method of claim 2, wherein a state space model of the system under study is constructed as follows:
Figure 698443DEST_PATH_IMAGE001
wherein the content of the first and second substances,
Figure 736806DEST_PATH_IMAGE002
it is the state of the system that is,
Figure 824979DEST_PATH_IMAGE003
is that
Figure 140554DEST_PATH_IMAGE002
The derivative of (a) of (b),
Figure 272458DEST_PATH_IMAGE004
is a control input to the control unit,
Figure 283270DEST_PATH_IMAGE005
is a standard second-order moment process,
Figure 768609DEST_PATH_IMAGE006
and
Figure 63324DEST_PATH_IMAGE007
is a known non-linear function;
designing a set of transformations for the system state, as follows:
Figure 710338DEST_PATH_IMAGE008
wherein the content of the first and second substances,
Figure 474025DEST_PATH_IMAGE009
is a set of virtual controllers, given at the beginning of design
Figure 153268DEST_PATH_IMAGE010
Designing a Lyapunov function on the basis of the transformation, wherein the Lyapunov function is as follows:
Figure 646698DEST_PATH_IMAGE011
based on the Lyapunov function and the state transformation, a set of controllers is obtained, which is as follows:
Figure 995771DEST_PATH_IMAGE012
wherein the content of the first and second substances,
Figure 981175DEST_PATH_IMAGE013
is a real controller and is finally used as an input to control the system;
under the action of the controller, the Lyapunov function
Figure 198530DEST_PATH_IMAGE014
Satisfies the following equation:
Figure 218570DEST_PATH_IMAGE015
wherein
Figure 472965DEST_PATH_IMAGE016
Is a normal number that is designed to be,
Figure 194933DEST_PATH_IMAGE017
is a parameter that is freely designed and is,
Figure 954992DEST_PATH_IMAGE018
and
Figure 688593DEST_PATH_IMAGE019
is an arbitrary constant.
4. The stability analysis process of claim 1, wherein the stability analysis using several lemmas proves, comprising: and inputting the controller designed in the first aspect into an original system as input, performing derivation on the correspondingly designed Lyapunov function, and performing stability analysis by combining a Lyapunov second judgment method.
5. The method of claim 4, wherein the stability analysis is performed based on the Lyapunov second method, and wherein the specific demonstration procedure is described as follows: first, a stopping time is introduced on the basis of the following formula
Figure 504102DEST_PATH_IMAGE020
Figure 588733DEST_PATH_IMAGE021
Wherein the parameters
Figure 898623DEST_PATH_IMAGE022
Are all larger than 0; according to
Figure 876943DEST_PATH_IMAGE014
The definition of (1) and the second moment process property, the following holds:
Figure 738720DEST_PATH_IMAGE023
closed loop system in
Figure 920434DEST_PATH_IMAGE024
The only syndrome of solution exists;
second, the following relationship holds based on the described system and theorem
Figure 283282DEST_PATH_IMAGE025
Figure 991475DEST_PATH_IMAGE026
Obtaining the conclusion that all states of the closed-loop system are proved according to probability;
thirdly, based on the relational expression, obtaining
Figure 899519DEST_PATH_IMAGE027
Wherein the content of the first and second substances,
Figure 693163DEST_PATH_IMAGE028
wherein the content of the first and second substances,
Figure 594123DEST_PATH_IMAGE029
is an arbitrary constant, and the right side of the above equation is chosen to be small enough.
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