CN113655715A - Performance optimization method of multi-channel discrete network control system - Google Patents

Performance optimization method of multi-channel discrete network control system Download PDF

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CN113655715A
CN113655715A CN202110849018.9A CN202110849018A CN113655715A CN 113655715 A CN113655715 A CN 113655715A CN 202110849018 A CN202110849018 A CN 202110849018A CN 113655715 A CN113655715 A CN 113655715A
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expression
channel
control system
decomposition
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CN113655715B (en
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张斌
姜晓伟
李刚
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China University of Geosciences
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China University of Geosciences
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    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
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Abstract

The invention provides a performance optimization method of a multi-channel discrete network control system, which establishes a multi-channel discrete network control system model, simulates data packet loss by utilizing a binary random process, assumes that channel noise is additive white Gaussian noise, and takes network-induced delay as constant delay through all-pass decomposition, inside-outside decomposition and H-pass decomposition2And the control system model is deduced by using tools such as a spatial decomposition technology, Youla parameterization of a controller and the like, so that the optimal tracking performance of the control system is obtained.

Description

Performance optimization method of multi-channel discrete network control system
Technical Field
The invention relates to the technical field of network system control, in particular to a performance optimization method of a multi-channel discrete network control system.
Background
A system model is introduced in the document "Performance limitation of network control systems with network delay and channel noises constraints", and the limit of the tracking Performance of a network control system with dual-channel noise constraints and network-induced delay constraints is researched. The network parameters mainly consider network-induced delay and additive white gaussian noise in the forward channel and additive white gaussian noise constraints in the feedback channel. And selecting an optimal single-parameter structure by using a spectrum decomposition technology to obtain a display expression of the tracking performance limit of the system. Although the system considers the time delay and the additive white gaussian noise constraint in the forward channel and the feedback channel, in the actual network communication channel, the constraints of packet loss, coding and decoding and the like exist, the network constraint considered by the model is not comprehensive enough, and the research on the tracking performance limit of the model on the network control system needs to be further deepened.
The literature "Optimal Tracking Performance of NCSs with Time-delay and Encoding-decoding Constraints" introduces a more complex research model, and researches the Optimal Performance of a network control system with network-induced delay Constraints, two-channel additive white Gaussian noise Constraints and Encoding and decoding Constraints. The network parameters mainly consider the coding and decoding constraint and the additive white Gaussian noise constraint in a forward channel and the network-induced delay constraint and the additive white Gaussian noise constraint in a feedback channel, and utilize H2Norm and spectrum decomposition technology is used for obtaining a display expression of the tracking performance limit of the system based on an optimal single-parameter structure. For this model, the network constraints to be considered are more complex, but still further studies can be made, for example, to study the influence of packet loss on system tracking performance on this basis.
Disclosure of Invention
One of the main problems solved by the present invention is the problem of how to further optimize the tracking performance of a multiple-input multiple-output discrete network control system.
The invention provides a performance optimization method of a multi-channel discrete network control system, which comprises the following steps: establishing a multi-channel discrete network control system model, wherein system input of the multi-channel discrete network control system model is expressed as a first expression:
Figure BDA0003181769100000021
wherein the content of the first and second substances,
Figure BDA0003181769100000022
for the input of a model of a multi-channel discrete network control system, n1、n2Separately, additive white Gaussian noise in the feedforward path and in the feedback path, A, A-1Representing transfer functions of encoding and decoding, respectively, z-τRepresenting time delay, K being a single degree of freedom controller, parameter drRepresenting packet loss, r-is the reference input
Figure BDA0003181769100000023
To be aOutputting the system;
the output of the multi-channel discrete network control system model is represented as a second expression:
Figure BDA0003181769100000024
wherein the content of the first and second substances,
Figure BDA0003181769100000025
g is a controlled object;
based on the error signal
Figure BDA0003181769100000026
Is a third expression:
Figure BDA0003181769100000027
wherein the content of the first and second substances,
Figure BDA0003181769100000028
is a reference input;
and a tracking performance index J, J being a fourth expression:
Figure BDA0003181769100000029
where λ is 0 ≦ λ ≦ 1, λ is a trade-off between system tracking error and channel input constraints, Γ is a predefined constraint value for the channel input energy,
Figure BDA00031817691000000210
represents the energy of the system output signal, an
Figure BDA00031817691000000211
Figure BDA00031817691000000212
Energy of error signal is expressed to obtainA first optimal expression of the channel discrete network control system model:
Figure BDA00031817691000000213
where V is the direction vector of the reference input, z is the transfer function argument, V ═ diag (β)1 2,...,βm 2),W＝diag(γ1 2,...,γm 2)，βi 2、γi 2Respectively, additive white Gaussian noise n in channel i1、n2M is a natural number, TryIs a reference input
Figure BDA0003181769100000031
To the system output
Figure BDA0003181769100000032
The transfer function of (a) is selected,
Figure BDA0003181769100000033
additive white Gaussian noise for forward channel
Figure BDA0003181769100000034
To the system output
Figure BDA0003181769100000035
The transfer function of (a) is selected,
Figure BDA0003181769100000036
additive white Gaussian noise for feedback channel
Figure BDA0003181769100000037
To the system output
Figure BDA0003181769100000038
The transfer function of (a) is selected,
Figure BDA0003181769100000039
is Q ∈ RH∞Representing a stable, regular, real rational transfer function (matrix) set, inf representing an infimum bound;
calculating to obtain reference input based on co-prime decomposition, all-pass decomposition and Youla parameterized form of single-degree-of-freedom controller of rational transfer function matrix
Figure BDA00031817691000000310
To the system output
Figure BDA00031817691000000311
Transfer function T ofryForward channel additive white gaussian noise
Figure BDA00031817691000000312
To the system output
Figure BDA00031817691000000313
Transfer function of
Figure BDA00031817691000000314
Additive white Gaussian noise of sum feedback channel
Figure BDA00031817691000000315
To the system output
Figure BDA00031817691000000316
Transfer function of
Figure BDA00031817691000000317
And, TryExpressed as a fifth expression:
Figure BDA00031817691000000318
Tn1yexpressed as a sixth expression:
Figure BDA00031817691000000319
Figure BDA00031817691000000320
expressed as a seventh expression:
Figure BDA00031817691000000321
where q is the packet loss probability, I is the identity matrix, z-τTau is a time delay coefficient for network time delay;
converting the obtained fifth expression and sixth expression based on the co-prime decomposition of the rational transfer function matrix, the double-Bezout equation and the Youla parameterized form of the single-degree-of-freedom controller to obtain a converted fifth expression:
Figure BDA00031817691000000322
and the converted sixth expression:
Figure BDA0003181769100000041
and the converted seventh expression:
Figure BDA0003181769100000042
and calculating the first optimal expression by utilizing a spatial decomposition technology, and selecting an optimal controller to enable the decomposed expression related to the controller parameters to be 0, so that the optimal tracking performance of the multi-channel discrete network control system model is obtained.
Further, calculating the first optimal expression using a spatial decomposition technique includes:
definition of
Figure BDA0003181769100000043
Is an eighth tableThe expression is as follows:
Figure BDA0003181769100000044
definition of
Figure BDA0003181769100000045
Is a ninth expression:
Figure BDA0003181769100000046
definition of
Figure BDA0003181769100000047
Is a tenth expression:
Figure BDA0003181769100000048
wherein the content of the first and second substances,
Figure BDA0003181769100000049
for the first part of the first optimal expression,
Figure BDA00031817691000000410
as a second part of the first optimal expression
Figure BDA00031817691000000411
In a third part of the first optimal expression, Q is a single degree of freedom controller parameter,
Figure BDA00031817691000000412
to conform to the double Bezout equation
Figure BDA00031817691000000413
And belong to RH∞Is determined by the matrix of the first and second matrices,
Figure BDA00031817691000000414
is the factor of the controlled object obtained by left co-prime decomposition, N is the factor of the controlled object obtained by right co-prime decomposition, and q is a constant.
Further, the computing the first optimal expression using a spatial decomposition technique further includes computing J1 *：
N is a factor obtained by right cross-prime decomposition of the controlled object and comprises all zero points of the controlled object, and the expression of N is an eleventh expression:
N＝LzNm，
wherein L iszThe non-minimum phase zero point z of the controlled object is included as an all-pass factori,i＝1,2,...,Nz，NmThe non-minimum phase factor contains all minimum phase zeros of the controlled object;
Lzdecomposed into a twelfth expression:
Figure BDA0003181769100000051
wherein s isiIs a non-minimum phase zero point and,
Figure BDA0003181769100000052
for its conjugate zero, z is the transfer function argument,
according to the eleventh expression and the twelfth expression, simplifying the eighth expression to obtain a first simplified expression:
Figure BDA0003181769100000053
further, for the first simplified expression, defining f expression as a thirteenth expression:
Figure BDA0003181769100000054
wherein f is a self-defined function about a non-minimum phase zero;
then the first simplified equation is converted to a second simplified equation according to the thirteenth expression:
Figure BDA0003181769100000055
further, due to
Figure BDA0003181769100000056
Then there is a third simplified expression based on the spatial decomposition technique:
Figure BDA0003181769100000061
definition of
Figure BDA0003181769100000062
And
Figure BDA0003181769100000063
is provided with
Figure BDA0003181769100000064
Expressed as a fourteenth expression:
Figure BDA0003181769100000065
wherein f is-1Is the inverse of the above-mentioned self-defined function;
Figure BDA0003181769100000066
expressed as a fifteenth expression:
Figure BDA0003181769100000067
computing
Figure BDA0003181769100000068
According to CauchiThe theorem has a sixteenth expression:
Figure BDA0003181769100000069
wherein s isjIs another non-minimum phase zero, dz is the calculus sign;
substituting the sixteenth expression into the fourteenth expression to obtain a seventeenth expression:
Figure BDA00031817691000000610
wherein H is a conjugate transpose;
then calculate
Figure BDA00031817691000000611
From the all-pass decomposition formula:
M＝BpMm，
wherein B ispThe all-pass factor includes all unstable poles p of the controlled objecti,i＝1,2,...,Np；
BpDecomposed into an eighteenth expression:
Figure BDA00031817691000000612
wherein M ismThe minimum phase factor includes all stable poles of the controlled object, NpNumber of unstable poles, pjFor the jth unstable pole, the number,
Figure BDA0003181769100000071
is the conjugation thereof;
the fifteenth expression is thus simplified to:
Figure BDA0003181769100000072
wherein the content of the first and second substances,
Figure BDA0003181769100000073
is the whole flux factor BpThe inverse of (a) is,
Figure BDA0003181769100000074
is composed of
Figure BDA0003181769100000075
A minimum phase part obtained by all-pass decomposition;
there is a nineteenth expression based on the partial fraction decomposition:
Figure BDA0003181769100000076
wherein, aiIs an expression for the pole of instability, and
Figure BDA0003181769100000077
substituting the nineteenth expression into the simplified fifteenth expression to obtain a twentieth expression:
Figure BDA0003181769100000078
wherein R is1(s)、R2(s) are all RH∞，
Figure BDA0003181769100000079
Is an unstable pole piConjugation of (1);
because:
Figure BDA00031817691000000710
then based on the spatial decomposition technique to obtain
Figure BDA00031817691000000711
The twenty-first expression of (1):
Figure BDA0003181769100000081
further, the selecting the optimal controller so that the decomposed expression related to the controller parameter is 0, thereby obtaining the optimal tracking performance of the multi-channel discrete network control system model includes:
selecting an appropriate controller parameter Q such that
Figure BDA0003181769100000082
Then it is possible to obtain:
Figure BDA0003181769100000083
and because of
Figure BDA0003181769100000084
And the double Bezout equation yields:
Figure BDA0003181769100000085
therefore, it is not only easy to use
Figure BDA0003181769100000086
Further simplified to a twenty-second expression:
Figure BDA0003181769100000087
calculating to obtain a twenty-third expression according to the twenty-second expression:
Figure BDA0003181769100000088
further obtain
Figure BDA0003181769100000089
A twenty-fourth expression:
Figure BDA00031817691000000810
further, calculating
Figure BDA00031817691000000811
And
Figure BDA00031817691000000812
method and calculation of
Figure BDA00031817691000000813
The method of (1), wherein, after the calculation
Figure BDA00031817691000000814
Expressed as a twenty-fifth expression:
Figure BDA0003181769100000091
wherein, t(s)i)＝(si)τNm(si)M-1(si)，t(si)HIs t(s)i) Conjugate transpose of(s)j)＝(sj)τNm(sj)M-1(sj)，
Figure BDA0003181769100000092
For the variance of additive white gaussian noise in the forward channel i,
Figure BDA0003181769100000093
wiis zero point siIn the direction of (a) of (b),
Figure BDA0003181769100000094
is a conjugate transpose thereof, wherein ejIs a unit vector with the jth element being 1;
and after calculation
Figure BDA0003181769100000095
Expressed as a twenty-sixth expression:
Figure BDA0003181769100000096
wherein the content of the first and second substances,
Figure BDA0003181769100000097
in order to be a conjugate thereof,
Figure BDA0003181769100000098
l(pi)Hin order to be a conjugate transpose thereof,
Figure BDA0003181769100000099
Om(pj) Substituting the minimum phase part obtained by the encoder through all-pass decomposition into the unstable pole pjAs a result of (a) the process of (b),
Figure BDA00031817691000000910
is its inverse, L-1(pj) Substituting the instability pole p for the twelfth expressionjInverse of the result of (1), γi 2For the variance of additive white gaussian noise in the feedback channel i,
Figure BDA00031817691000000911
ηiis an unstable pole piIn the direction of (a) of (b),
Figure BDA00031817691000000912
is a conjugate transpose thereof, wherein ejIs a unit vector with the jth element being 1.
Further, obtaining an optimal performance expression of the multi-channel discrete network control system model according to the twenty-fourth expression, the twenty-fifth expression and the twenty-sixth expression is as follows:
Figure BDA0003181769100000101
the invention establishes a multi-channel discrete network control system model, simulates data packet loss by utilizing a binary random process, assumes that channel noise is additive white Gaussian noise, and network-induced delay is constant time delay and is realized by all-pass decomposition, inside and outside decomposition and H2And the spatial decomposition technology and tools such as Youla parameterization of the controller are used for deducing the multi-channel discrete network control system model to obtain the optimal tracking performance of the control system.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description, serve to explain the principles of the invention.
Fig. 1 is a schematic diagram of a model of a mimo discrete network control system according to an embodiment of the present invention.
Fig. 2 is a schematic diagram of tracking performance limits under different time delays in the embodiment of the present invention.
Fig. 3 is a schematic diagram of the tracking performance limit under different packet loss probabilities in the embodiment of the present invention.
Detailed Description
Various exemplary embodiments of the present invention will be described in detail below with reference to the accompanying drawings. It should be noted that: the relative arrangement of the components and steps, the numerical expressions and numerical values set forth in these embodiments do not limit the scope of the present invention unless specifically stated otherwise.
Meanwhile, it should be understood that the sizes of the respective portions shown in the drawings are not drawn in an actual proportional relationship for the convenience of description.
The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the invention, its application, or uses.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to specific embodiments and the accompanying drawings.
Techniques, methods, and apparatus known to those of ordinary skill in the relevant art may not be discussed in detail but are intended to be part of the specification where appropriate.
In all examples shown and discussed herein, any particular value should be construed as merely illustrative, and not limiting. Thus, other examples of the exemplary embodiments may have different values.
It should be noted that: like reference numbers and letters refer to like items in the following figures, and thus, once an item is defined in one figure, further discussion thereof is not required in subsequent figures.
In a first embodiment, as shown in fig. 1, a multiple-input multiple-output discrete network control system is provided, and for the network system, an optimization method of a multiple-channel discrete network control system is provided:
firstly, establishing a multi-input multi-output discrete network control system model, wherein the input of the multi-channel discrete network control system model is expressed as formula (1):
Figure BDA0003181769100000111
wherein the content of the first and second substances,
Figure BDA0003181769100000112
for the input of a model of a multi-channel discrete network control system, n1、n2Additive white Gaussian noise in and in the feedforward path, A, A, respectively-1Representing transfer functions of encoding and decoding, respectively, z-τRepresenting time delay, K being a single degree of freedom controller, parameter drWhich represents a loss of a data packet,
Figure BDA0003181769100000113
is a reference input
Figure BDA0003181769100000114
Outputting for the system;
the output of the multi-channel discrete network control system model is expressed as a formula:
Figure BDA0003181769100000115
wherein the content of the first and second substances,
Figure BDA0003181769100000116
g is a controlled object;
based on the error signal
Figure BDA0003181769100000117
The expression of (a) is:
Figure BDA0003181769100000118
wherein the content of the first and second substances,
Figure BDA0003181769100000119
is a reference input;
and tracking performance index J, the expression of J is:
Figure BDA0003181769100000121
where λ is 0 ≦ λ ≦ 1, λ is a trade-off between system tracking error and channel input constraints, Γ is a predefined constraint value for the channel input energy,
Figure BDA0003181769100000122
represents the energy of the system output signal, an
Figure BDA0003181769100000123
Figure BDA0003181769100000124
Representing the energy of the error signal to obtain an optimal expression of the multi-channel discrete network control system model:
Figure BDA0003181769100000125
where V is the direction vector of the reference input, z is the transfer function argument, V ═ diag (β)1 2,...,βm 2),W＝diag(γ1 2,...,γm 2)，βi 2、γi 2Respectively, additive white Gaussian noise n in channel i1、n2M is a natural number, TryIs a reference input
Figure BDA0003181769100000126
To the system output
Figure BDA0003181769100000127
Transfer function of, Tn1yAdditive white Gaussian noise for forward channel
Figure BDA0003181769100000128
To the system output
Figure BDA0003181769100000129
The transfer function of (a) is selected,
Figure BDA00031817691000001210
additive white Gaussian noise for feedback channel
Figure BDA00031817691000001211
To the system output
Figure BDA00031817691000001212
The transfer function of (a) is selected,
Figure BDA00031817691000001213
is Q ∈ RH∞Representing a stable, regular, real rational transfer function (matrix) set, inf representing an infimum bound;
calculation of a reference form based on co-prime decomposition, all-pass decomposition and Youla parameterization of a single degree of freedom controller of a rational transfer function matrixInput device
Figure BDA00031817691000001214
To the system output
Figure BDA00031817691000001215
Transfer function T ofryForward channel additive white gaussian noise
Figure BDA00031817691000001216
To the system output
Figure BDA00031817691000001217
Transfer function T ofn1yAdditive white Gaussian noise of sum feedback channel
Figure BDA00031817691000001218
To the system output
Figure BDA00031817691000001219
Transfer function of
Figure BDA00031817691000001220
And, TryThe expression of (a) is:
Figure BDA00031817691000001221
Tn1ythe expression of (a) is:
Figure BDA00031817691000001222
Tn2ythe expression of (a) is:
Figure BDA0003181769100000131
where q is the packet loss probability, I is the identity matrix, z-τFor network delay, τ is the delay systemCounting;
converting the obtained formulas (6) - (7) based on the co-prime decomposition of the rational transfer function matrix, the double Bezout equation and the Youla parameterized form of the single degree of freedom controller to obtain a converted expression:
Figure BDA0003181769100000132
and the converted sixth expression:
Figure BDA0003181769100000133
and the converted seventh expression:
Figure BDA0003181769100000134
then, the optimal expression (5) is calculated by using a spatial decomposition technology:
definition of
Figure BDA0003181769100000135
Is expressed as:
Figure BDA0003181769100000136
definition of
Figure BDA0003181769100000137
Is expressed as:
Figure BDA0003181769100000138
definition of
Figure BDA0003181769100000139
Is expressed as:
Figure BDA00031817691000001310
wherein the content of the first and second substances,
Figure BDA00031817691000001311
for the first part of the optimal expression,
Figure BDA00031817691000001312
for the second part of the optimal expression,
Figure BDA00031817691000001313
is a third part of the optimal expression, the optimal expression being a combination of the three parts, and wherein Q is a single degree of freedom controller parameter,
Figure BDA0003181769100000141
to conform to the double Bezout equation
Figure BDA0003181769100000142
And belong to RH∞Is determined by the matrix of the first and second matrices,
Figure BDA0003181769100000143
is the factor of the controlled object obtained by left co-prime decomposition, N is the factor of the controlled object obtained by right co-prime decomposition, and q is a constant.
Decomposing the optimal expression into three parts, respectively calculating the values of the three parts, firstly calculating
Figure BDA0003181769100000144
N is a factor obtained by right-side co-prime decomposition of the controlled object and comprises all zeros of the controlled object, and the expression of N is as follows:
N＝LzNm (15)，
wherein L iszThe non-minimum phase zero point z of the controlled object is included as an all-pass factori,i＝1,2,...,Nz，NmIs not the minimum phase factor and includes all the controlled objectsMinimum phase zero of;
Lzthe decomposition is expressed as:
Figure BDA0003181769100000145
wherein s isiIs a non-minimum phase zero point and,
Figure BDA0003181769100000146
for its conjugate zero, z is the transfer function argument,
according to the formulas (15) - (16), simplifying the optimal expression to obtain a first simplified expression:
Figure BDA0003181769100000147
for the first simplified form, the expression of the function f defining the non-minimum phase zero is:
Figure BDA0003181769100000148
wherein f is a self-defined function about a non-minimum phase zero;
then according to said (18), the first reduction is convertible to a second reduction:
Figure BDA0003181769100000151
due to the fact that
Figure BDA0003181769100000152
Then equation (19) is further simplified based on the spatial decomposition technique:
Figure BDA0003181769100000153
for ease of calculation, define
Figure BDA0003181769100000154
And
Figure BDA0003181769100000155
is provided with
Figure BDA0003181769100000156
Expressed as:
Figure BDA0003181769100000157
wherein f is-1Is the inverse of the above-mentioned self-defined function;
Figure BDA0003181769100000158
expressed as:
Figure BDA0003181769100000159
computing
Figure BDA00031817691000001510
According to the cauchy theorem, the method comprises the following steps:
Figure BDA00031817691000001511
wherein s isjIs another non-minimum phase zero, dz is the calculus sign;
substituting the (23) into the (21) to obtain:
Figure BDA00031817691000001512
wherein H is a conjugate transpose.
Then calculate
Figure BDA00031817691000001513
From the all-pass decomposition formula:
M＝BpMm (25)，
wherein B ispThe all-pass factor includes all unstable poles p of the controlled objecti,i＝1,2,…,Np；
BpThe decomposition is as follows:
Figure BDA0003181769100000161
wherein M ismThe minimum phase factor includes all unstable poles of the controlled object, NpFor unstable pole bits, pjIs the jth unstable pole, pjIs the conjugation thereof;
thus simplifying to obtain:
Figure BDA0003181769100000162
wherein the content of the first and second substances,
Figure BDA0003181769100000163
is the whole flux factor BpThe inverse of (a) is,
Figure BDA0003181769100000164
is composed of
Figure BDA0003181769100000165
A minimum phase part obtained by all-pass decomposition;
based on partial fraction decomposition:
Figure BDA0003181769100000166
wherein, aiIs an expression for the pole of instability, and
Figure BDA0003181769100000167
substituting the (28) into the (27) after the simplification to obtain:
Figure BDA0003181769100000168
wherein R is1(s)、R2(s) are all RH∞，
Figure BDA0003181769100000169
Is an unstable pole piConjugation of (1);
and because:
Figure BDA0003181769100000171
then based on the spatial decomposition technique to obtain
Figure BDA0003181769100000172
Expression (c):
Figure BDA0003181769100000173
finally, selecting the optimal controller to enable a part of expressions related to the controller parameters in the decomposed formula to be 0, thereby obtaining the optimal tracking performance of the multi-channel discrete network control system model, wherein the calculation step comprises the following steps:
selecting an appropriate controller parameter Q such that:
Figure BDA0003181769100000174
then it is possible to obtain:
Figure BDA0003181769100000175
and because of
Figure BDA0003181769100000176
And the double Bezout equation yields:
Figure BDA0003181769100000177
therefore, it is not only easy to use
Figure BDA0003181769100000178
Further simplifying as follows:
Figure BDA0003181769100000179
the following are obtained through simple calculation:
Figure BDA0003181769100000181
according to the calculated above
Figure BDA0003181769100000182
And
Figure BDA0003181769100000183
expressions (24) and (35), to obtain
Figure BDA0003181769100000184
Comprises the following steps:
Figure BDA0003181769100000185
computing
Figure BDA0003181769100000186
And
Figure BDA0003181769100000187
method and calculation of
Figure BDA0003181769100000188
The method of (1), wherein, after the calculation
Figure BDA0003181769100000189
Expressed as:
Figure BDA00031817691000001810
wherein, t(s)i)＝(si)τNm(si)M-1(si)，t(si)HIs t(s)i) Conjugate transpose of (1), t(s)j)＝(sj)τNm(sj)M-1(sj)，
Figure BDA00031817691000001811
For the variance of additive white gaussian noise in the forward channel i,
Figure BDA00031817691000001812
wiis zero point siIn the direction of (a) of (b),
Figure BDA00031817691000001813
is a conjugate transpose thereof, wherein ejIs a unit vector with the jth element being 1;
and after calculation
Figure BDA00031817691000001814
Expressed as:
Figure BDA00031817691000001815
wherein the content of the first and second substances,
Figure BDA00031817691000001816
in order to be a conjugate thereof,
Figure BDA00031817691000001817
l(pi)Hin order to be a conjugate transpose thereof,
Figure BDA00031817691000001818
Om(pj) Substituting the minimum phase part obtained by the encoder through all-pass decomposition into the unstable pole pjAs a result of (a) the process of (b),
Figure BDA00031817691000001819
is its inverse, L-1(pj) Substituting the instability pole p for the twelfth expressionjInverse of the result of (1), γi 2For the variance of additive white gaussian noise in the feedback channel i,
Figure BDA00031817691000001820
ηiis an unstable pole piIn the direction of (a) of (b),
Figure BDA00031817691000001821
is a conjugate transpose thereof, wherein ejIs a unit vector with the jth element being 1.
The optimal performance expression of the multi-channel discrete network control system model obtained according to the formulas (36) to (38) is as follows:
Figure BDA0003181769100000191
the invention utilizes a binary random process to simulate the data packet loss, assumes that the channel noise is additive white Gaussian noise, and the network induced delay is a constant delay which is decomposed through full-pass decomposition, internal and external decomposition and H2And the model is deduced by using tools such as a spatial decomposition technology, a Youla parameterization of a controller and the like, so that the optimal tracking performance of the system is obtained.
Compared with the prior art, the invention has the advantages that: 1. comprehensively considering multiple communication constraints of double-channel additive white Gaussian noise, data packet loss, communication time delay and coding and decoding, and establishing a network control system model under the multiple communication constraints; 2. an optimal controller is designed by utilizing tools such as cross-prime decomposition, Youla parameterization and the like, and the optimal controller is ensuredOn the premise of system stability, the tracking performance of the multi-input multi-output discrete network control system is greatly improved; 3. through the frequency domain H2The optimal control method obtains the infimum boundary of the tracking performance of the multi-input multi-output discrete network control system, and deeply reveals the internal relation between the performance of the network control system and various communication constraints on the basis of the prior art.
The following experimental data demonstrate the outstanding optimization effect that this embodiment can produce:
considering a discrete multi-input multi-output controlled object, a transfer function matrix model of the controlled object is as follows:
Figure BDA0003181769100000192
the transfer function matrix includes a non-minimum phase zero z ═ k, and an output zero direction η ═ 1,0TThe device comprises an unstable pole p which is 2, and the pole direction is omega (0,1)TDefining the input vector as v ═ (1,0)TSelecting:
Figure BDA0003181769100000201
then:
Figure BDA0003181769100000202
selecting:
Figure BDA0003181769100000203
then there are:
Figure BDA0003181769100000204
we can get from the controlled object model:
Figure BDA0003181769100000205
when in use
Figure BDA0003181769100000206
0.2, 0.5 and 0.8, respectively, there is a performance limit expression:
Figure BDA0003181769100000207
the limit of the tracking performance of the multiple-input multiple-output discrete network control system under different time delay constraints is shown in fig. 2, and by comparing the tracking performance when T is 0.2, T is 0.5, and T is 0.8, it can be seen that the larger the time delay parameter in the feedback channel of the multiple-input multiple-output network control system is, the worse the performance of the discrete multiple-input multiple-output network control system is. And as can also be seen from fig. 2, when the unstable pole of the controlled object is sufficiently close to the non-minimum phase zero, the tracking performance of the discrete multiple-input multiple-output network control system may be deteriorated sharply. As can be seen from fig. 3, the tracking performance becomes worse as the packet loss probability increases.
The above description is only exemplary of the present invention and should not be taken as limiting the invention, as any modification, equivalent replacement, or improvement made within the spirit and scope of the present invention should be included in the present invention.
It should also be noted that the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other identical elements in the process, method, article, or apparatus that comprises the element.

Claims (8)

1. A performance optimization method for a multi-channel discrete network control system is characterized by comprising the following steps:
establishing a multi-channel discrete network control system model, wherein system input of the multi-channel discrete network control system model is expressed as a first expression:
Figure FDA0003181769090000011
wherein the content of the first and second substances,
Figure FDA0003181769090000012
system input, n, for a model of a multi-channel discrete network control system1、n2Separately, additive white Gaussian noise in the feedforward path and in the feedback path, A, A-1Representing transfer functions of encoding and decoding, respectively, z-τRepresenting time delay, K being a single degree of freedom controller, parameter drWhich represents a loss of a data packet,
Figure FDA0003181769090000013
for the purpose of reference input, the system is,
Figure FDA0003181769090000014
Figure FDA0003181769090000015
outputting for the system;
the output of the multi-channel discrete network control system model is represented as a second expression:
Figure FDA0003181769090000016
wherein the content of the first and second substances,
Figure FDA0003181769090000017
g is a controlled object;
based on the error signal
Figure FDA0003181769090000018
Is a third expression:
Figure FDA0003181769090000019
wherein the content of the first and second substances,
Figure FDA00031817690900000110
is a reference input;
and a tracking performance index J, J being a fourth expression:
Figure FDA00031817690900000111
where λ is 0 ≦ λ ≦ 1, λ is a trade-off between system tracking error and channel input constraints, Γ is a predefined constraint value for the channel input energy,
Figure FDA00031817690900000112
represents the energy of the system output signal, an
Figure FDA00031817690900000113
Figure FDA00031817690900000114
Representing the energy of the error signal to obtain a first optimal expression of the multi-channel discrete network control system model:
Figure FDA00031817690900000115
where V is the direction vector of the reference input, z is the transfer function argument, V ═ diag (β)1 2,...,βm 2),W＝diag(γ1 2,...,γm 2)，βi 2、γi 2Respectively, additive white Gaussian noise n in channel i1、n2Is 1,2, m is a positive integer, TryIs a reference input
Figure FDA0003181769090000021
To the system output
Figure FDA0003181769090000022
Transfer function of, Tn1yAdditive white Gaussian noise for forward channel
Figure FDA0003181769090000023
To the system output
Figure FDA0003181769090000024
Transfer function of, Tn2,yAdditive white Gaussian noise for feedback channel
Figure FDA0003181769090000025
To the system output
Figure FDA0003181769090000026
The transfer function of (a) is selected,
Figure FDA0003181769090000027
Q∈RH∞representing a stable, regular, real rational transfer function (matrix) set, inf representing an infimum bound;
calculating to obtain reference input based on co-prime decomposition, all-pass decomposition and Youla parameterized form of single-degree-of-freedom controller of rational transfer function matrix
Figure FDA0003181769090000028
To the system output
Figure FDA0003181769090000029
Transfer function T ofryForward channel additive white gaussian noise
Figure FDA00031817690900000210
To the system output
Figure FDA00031817690900000211
Transfer function of
Figure FDA00031817690900000218
Additive white Gaussian noise of sum feedback channel
Figure FDA00031817690900000212
To the system output
Figure FDA00031817690900000213
Transfer function of
Figure FDA00031817690900000219
And, TryExpressed as a fifth expression:
Figure FDA00031817690900000214
Figure FDA00031817690900000220
expressed as a sixth expression:
Figure FDA00031817690900000215
Figure FDA00031817690900000221
expressed as a seventh expression:
Figure FDA00031817690900000216
where q is the packet loss probability, I is the identity matrix, z-τTau is a time delay coefficient for network time delay;
converting the obtained fifth expression and sixth expression based on the co-prime decomposition of the rational transfer function matrix, the double-Bezout equation and the Youla parameterized form of the single-degree-of-freedom controller to obtain a converted fifth expression:
Figure FDA00031817690900000217
and the converted sixth expression:
Figure FDA0003181769090000031
and the converted seventh expression:
Figure FDA0003181769090000032
and calculating the first optimal expression by utilizing a spatial decomposition technology, and selecting an optimal controller to enable the expression related to the controller parameters after decomposition to be 0, thereby obtaining the optimal tracking performance of the multi-channel discrete network control system model.
2. The method of claim 1, wherein computing the first optimal expression using a spatial decomposition technique comprises:
definition of
Figure FDA0003181769090000033
Is an eighth expression:
Figure FDA0003181769090000034
definition of
Figure FDA0003181769090000035
Is a ninth expression:
Figure FDA0003181769090000036
definition of
Figure FDA0003181769090000037
Is a tenth expression:
Figure FDA0003181769090000038
wherein the content of the first and second substances,
Figure FDA0003181769090000039
for the first part of the first optimal expression,
Figure FDA00031817690900000310
as a second part of the first optimal expression
Figure FDA00031817690900000311
In a third part of the first optimal expression, Q is a single degree of freedom controller parameter,
Figure FDA00031817690900000312
to conform to the double Bezout equation
Figure FDA00031817690900000313
And belong to RH∞Is determined by the matrix of the first and second matrices,
Figure FDA00031817690900000314
is the factor of the controlled object obtained by left co-prime decomposition, N is the factor of the controlled object obtained by right co-prime decomposition, and q is a constant.
3. The method of claim 2, wherein said computing said first optimal expression using a spatial decomposition technique further comprises computing a first optimal expression for a multi-channel discrete network control system
Figure FDA0003181769090000041
N is a factor obtained by right cross-prime decomposition of the controlled object and comprises all zero points of the controlled object, and the expression of N is an eleventh expression:
N＝LzNm，
wherein L iszThe non-minimum phase zero point z of the controlled object is included as an all-pass factori,i＝1,2,...,Nz，NmThe non-minimum phase factor contains all minimum phase zeros of the controlled object;
Lzdecomposed into a twelfth expression:
Figure FDA0003181769090000042
wherein s isiIs a non-minimum phase zero point and,
Figure FDA0003181769090000043
for its conjugate zero, z is the transfer function argument,
according to the eleventh expression and the twelfth expression, simplifying the eighth expression to obtain a first simplified expression:
Figure FDA0003181769090000044
4. the method as claimed in claim 3, wherein for the first simplified expression, f is defined as a thirteenth expression:
Figure FDA0003181769090000045
wherein f is a self-defined function about a non-minimum phase zero;
then the first simplified equation is converted to a second simplified equation according to the thirteenth expression:
Figure FDA0003181769090000051
5. the method as claimed in claim 4, wherein the performance optimization method is due to the fact that
Figure FDA0003181769090000052
Then there is a third simplified expression based on the spatial decomposition technique:
Figure FDA0003181769090000053
definition of
Figure FDA0003181769090000054
And
Figure FDA0003181769090000055
is provided with
Figure FDA0003181769090000056
Figure FDA0003181769090000057
Expressed as a fourteenth expression:
Figure FDA0003181769090000058
wherein f is-1Is the inverse of the above-mentioned self-defined function;
Figure FDA0003181769090000059
expressed as a fifteenth expression:
Figure FDA00031817690900000510
computing
Figure FDA00031817690900000511
There is a sixteenth expression according to cauchy theorem:
Figure FDA00031817690900000512
wherein s isjIs another non-minimum phase zero, dz is the calculus sign;
substituting the sixteenth expression into the fourteenth expression to obtain a seventeenth expression:
Figure FDA00031817690900000513
wherein H is a conjugate transpose;
then calculate
Figure FDA00031817690900000514
From the all-pass decomposition formula:
M＝BpMm，
wherein B ispThe all-pass factor includes all unstable poles p of the controlled objecti,i＝1,2,...,Np；BpDecomposed into an eighteenth expression:
Figure FDA0003181769090000061
wherein M ismThe minimum phase factor includes all unstable poles of the controlled object, NpNumber of unstable poles, pjFor the jth unstable pole, the number,
Figure FDA00031817690900000612
is the conjugation thereof;
the fifteenth expression is thus simplified to:
Figure FDA0003181769090000062
wherein the content of the first and second substances,
Figure FDA0003181769090000063
is the whole flux factor BpThe inverse of (a) is,
Figure FDA0003181769090000064
is composed of
Figure FDA0003181769090000065
A minimum phase part obtained by all-pass decomposition;
there is a nineteenth expression based on the partial fraction decomposition:
Figure FDA0003181769090000066
wherein, aiIs an expression for the pole of instability, and
Figure FDA0003181769090000067
substituting the nineteenth expression into the simplified fifteenth expression to obtain a twentieth expression:
Figure FDA0003181769090000068
wherein R is1(s)、R2(s) are all RH∞，
Figure FDA0003181769090000069
Figure FDA00031817690900000610
Is an unstable pole piConjugation of (1);
because:
Figure FDA0003181769090000071
then based on the spatial decomposition technique to obtain
Figure FDA0003181769090000072
The twenty-first expression of (1):
Figure FDA0003181769090000073
6. the method as claimed in claim 2, wherein the selecting the optimal controller to make the decomposed expression related to the controller parameter 0 to obtain the optimal tracking performance of the model of the multi-channel discrete network control system comprises:
selecting a controller parameter Q such that
Figure FDA0003181769090000074
Then it is possible to obtain:
Figure FDA0003181769090000075
and because of
Figure FDA0003181769090000076
And the double Bezout equation yields:
Figure FDA0003181769090000077
therefore, it is not only easy to use
Figure FDA0003181769090000078
Further simplified to a twenty-second expression:
Figure FDA0003181769090000079
calculating to obtain a twenty-third expression according to the twenty-second expression:
Figure FDA00031817690900000710
further obtain
Figure FDA0003181769090000081
A twenty-fourth expression:
Figure FDA0003181769090000082
7. the method of claim 6, wherein the calculating comprises calculating a performance optimization function for the multi-channel discrete network control system
Figure FDA0003181769090000083
And
Figure FDA0003181769090000084
method and calculation of
Figure FDA0003181769090000085
The method of (1), wherein, after the calculation
Figure FDA0003181769090000086
Expressed as a twenty-fifth expression:
Figure FDA0003181769090000087
wherein, t(s)i)HIs t(s)i) Conjugate transpose of (1), t(s)i)＝(si)τNm(si)M-1(si)，βi 2For the variance of additive white gaussian noise in the forward channel i,
Figure FDA0003181769090000088
wiis zero point siIn the direction of (a) of (b),
Figure FDA0003181769090000089
is a conjugate transpose thereof, wherein ejIs a unit vector with the jth element being 1;
and after calculation
Figure FDA00031817690900000810
Expressed as a twenty-sixth expression:
Figure FDA00031817690900000811
wherein the content of the first and second substances,
Figure FDA00031817690900000812
Figure FDA00031817690900000813
in order to be a conjugate thereof,
Figure FDA00031817690900000814
Figure FDA00031817690900000815
l(pi)Hin order to be a conjugate transpose thereof,
Figure FDA00031817690900000820
Om(pj) Substituting the minimum phase part obtained by the encoder through all-pass decomposition into the unstable pole pjAs a result of (a) the process of (b),
Figure FDA00031817690900000816
is its inverse, L-1(pj) Substituting the instability pole p for the twelfth expressionjThe inverse of the result of the calculation of (c),
Figure FDA00031817690900000817
for the variance of additive white gaussian noise in the feedback channel i,
Figure FDA00031817690900000818
ηiis an unstable pole piIn the direction of (a) of (b),
Figure FDA00031817690900000819
is a conjugate transpose thereof, wherein ejIs a unit vector with the jth element being 1.
8. The method for optimizing the performance of the multi-channel discrete network control system according to claim 7, wherein the obtaining of the optimal performance expression of the multi-channel discrete network control system model according to the twenty-fourth expression, the twenty-fifth expression and the twenty-sixth expression is as follows:
Figure FDA0003181769090000091
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114895645A (en) * 2022-03-31 2022-08-12 中国地质大学（武汉） Network control system performance limit analysis method considering non-zero mean noise

Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6161209A (en) * 1997-03-28 2000-12-12 Her Majesty The Queen In Right Of Canada, As Represented By The Minister Of Industry Through The Communications Research Centre Joint detector for multiple coded digital signals
CN101155083A (en) * 2006-09-27 2008-04-02 中兴通讯股份有限公司 Network state estimation method based on packet loss rate
CN101527674A (en) * 2008-03-04 2009-09-09 中国移动通信集团公司 Method and device for processing data
CN107168053A (en) * 2017-05-04 2017-09-15 南京理工大学 A kind of finite field filter design method with stochastic filtering change in gain
CN109039508A (en) * 2018-09-30 2018-12-18 上海科梁信息工程股份有限公司 Wireless multipath fading channel simulator system and method
CN112904271A (en) * 2021-03-03 2021-06-04 西北大学 Fourth-order cumulant DOA estimation method based on co-prime array and augmented extended array

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6161209A (en) * 1997-03-28 2000-12-12 Her Majesty The Queen In Right Of Canada, As Represented By The Minister Of Industry Through The Communications Research Centre Joint detector for multiple coded digital signals
CN101155083A (en) * 2006-09-27 2008-04-02 中兴通讯股份有限公司 Network state estimation method based on packet loss rate
CN101527674A (en) * 2008-03-04 2009-09-09 中国移动通信集团公司 Method and device for processing data
CN107168053A (en) * 2017-05-04 2017-09-15 南京理工大学 A kind of finite field filter design method with stochastic filtering change in gain
CN109039508A (en) * 2018-09-30 2018-12-18 上海科梁信息工程股份有限公司 Wireless multipath fading channel simulator system and method
CN112904271A (en) * 2021-03-03 2021-06-04 西北大学 Fourth-order cumulant DOA estimation method based on co-prime array and augmented extended array

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
FENG LIN YU等: "Tunable bandpass filters with constant absolute bandwidth and high linearity", 《2012 INTERNATIONAL CONFERENCE ON MICROWAVE AND MILLIMETER WAVE TECHNOLOGY (ICMMT)》 *
李刚等: "GIS技术在电信固网拓展应用中的研究", 《四川大学学报（工程科学版）》 *

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114895645A (en) * 2022-03-31 2022-08-12 中国地质大学（武汉） Network control system performance limit analysis method considering non-zero mean noise
CN114895645B (en) * 2022-03-31 2024-04-16 中国地质大学（武汉） Network control system performance limit analysis method considering non-zero mean noise

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