CN110764409A - Optimal linear quadratic Gaussian control method for discrete system - Google Patents

Optimal linear quadratic Gaussian control method for discrete system Download PDF

Info

Publication number
CN110764409A
CN110764409A CN201910965583.4A CN201910965583A CN110764409A CN 110764409 A CN110764409 A CN 110764409A CN 201910965583 A CN201910965583 A CN 201910965583A CN 110764409 A CN110764409 A CN 110764409A
Authority
CN
China
Prior art keywords
matrix
linear quadratic
control
optimal
formula
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Granted
Application number
CN201910965583.4A
Other languages
Chinese (zh)
Other versions
CN110764409B (en
Inventor
梁笑
张琦妍
卢晓
王海霞
张桂林
牟宗磊
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Shandong University of Science and Technology
Original Assignee
Shandong University of Science and Technology
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Shandong University of Science and Technology filed Critical Shandong University of Science and Technology
Priority to CN201910965583.4A priority Critical patent/CN110764409B/en
Publication of CN110764409A publication Critical patent/CN110764409A/en
Application granted granted Critical
Publication of CN110764409B publication Critical patent/CN110764409B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • 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
    • G05B13/0205—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric not using a model or a simulator of the controlled system
    • G05B13/024—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric not using a model or a simulator of the controlled system in which a parameter or coefficient is automatically adjusted to optimise the performance

Landscapes

  • Engineering & Computer Science (AREA)
  • Health & Medical Sciences (AREA)
  • Artificial Intelligence (AREA)
  • Computer Vision & Pattern Recognition (AREA)
  • Evolutionary Computation (AREA)
  • Medical Informatics (AREA)
  • Software Systems (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
  • Feedback Control In General (AREA)

Abstract

The invention discloses an optimal linear quadratic Gaussian control method for a discrete system, and particularly relates to a linear quadratic Gaussian control technology. The method is based on the discrete time optimal Linear Quadratic Gaussian (LQG) control problem of multi-time lag and state/control correlated noise, firstly, a maximum value principle of a linear quadratic Gaussian system with multi-time lag and state/control correlated noise is derived by using a variational method, a solution of forward and backward random difference equations (FBSDEs) consisting of an adjoint equation and a balance condition is obtained by a mathematical induction method, sufficient necessary conditions existing in the optimal linear quadratic Gaussian controller are given based on the solution of the FBSDEs, and an explicit solution and a minimum quadratic performance index of the optimal controller are obtained.

Description

Optimal linear quadratic Gaussian control method for discrete system
Technical Field
The invention relates to a linear quadratic Gaussian control technology, in particular to an optimal linear quadratic Gaussian control method for a discrete system.
Background
The Linear Quadratic Gaussian (LQG) control problem is an optimal stochastic control problem with additive white gaussian noise and state/control dependent noise systems, which combines a linear quadratic regulator for full state feedback and a kalman filter for state estimation. In recent years, optimal LQG control has been applied in various fields such as wind turbines, drones and industrial machine tools, and particularly in Network Control Systems (NCSs). With the rapid development of scientific technology, the former LQG control technology has gradually not kept pace with industrial applications, which has prompted us to study more complex LQG control systems with multiple input lags and state/control related noise.
As is known, time delay and data loss inevitably occur in the data transmission process of NCSs, and there are many LQG control results related to time delay and data packet loss. By using the dual principle, an optimal controller for controlling an input delay system is given. An explicit solution to the dual sensor distributed LQG problem is proposed for the linear system LQG problem with single input delay and state feedback.
On the other hand, since the state/control related noise can be applied to characterize the data packet loss, many documents have focused on the LQG system having the state/control related noise. An optimal encoder-decoder is designed by introducing a standard LQR state feedback controller, and the optimal LQG problem with any data packet loss is solved. Aiming at a packet loss network control system with a remote controller and a local controller, sufficient necessary conditions of a finite time domain problem and infinite time domain mean square stability are respectively given. For a discrete LQG system with single input delay and multiplicative noise, an explicit solution of an optimal controller and a suboptimal controller is given by introducing a maximum value principle.
However, the previous literature only studied systems of single input delay and multiplicative noise, and for noise systems where there are multiple input delays and state/control dependent noise, to our knowledge the optimal LQG control problem hardly makes any progress, and when the state variables are not fully available, the separation principle does not hold, so the optimal LQG control problem for this system becomes even more challenging.
Disclosure of Invention
The invention aims to solve the defects, provides a maximum value principle of an LQG system with multiple time lags and state/control correlated noise, obtains the solution of a forward and backward random difference equation by a mathematical induction method, and provides a method for solving an optimal LQG controller on the basis of the solution of the forward and backward random difference equation.
The invention specifically adopts the following technical scheme:
the optimal linear quadratic Gaussian control method for the discrete system adopts a formula (2.1) to express the discrete linear quadratic Gaussian system with multiple input time lags and state/control related noise,
Figure RE-GDA0002305343190000021
wherein x isk∈RnRepresentative of the state, uk∈RnRepresenting control variables, d > 0 representing input time lag, vkIs an arbitrary mean value of zero and a variance of phi2White noise scalar of omegak∈RnIs zero mean value with pen variation and satisfies
Figure RE-GDA00023053431900000213
The matrix C is a matrix of a number of,
Figure RE-GDA0002305343190000022
D0，
Figure RE-GDA0002305343190000023
a constant matrix of appropriate dimensions, vkAnd wkUncorrelated, initial states of x0With a mean value of
Figure RE-GDA00023053431900000212
And the initial control variable usIs known at s ═ d., 1;
Rnrepresenting an n-dimensional real euclidean space, I representing an identity matrix of appropriate dimensions, superscript' representing the transpose of the matrix,
Figure RE-GDA0002305343190000024
representing a random variable wkComplete probability space of, random variable wkIs formed by
Figure RE-GDA0002305343190000025
Is generated, i.e.
Figure RE-GDA0002305343190000026
By
Figure RE-GDA0002305343190000027
All of
Figure RE-GDA0002305343190000028
The symmetric matrix A is more than 0 (more than or equal to 0) and represents that the matrix A is a positive definite (semi-positive definite) matrix, and Tr (A) represents the trace of the matrix A;
the cost function associated with the system of equation (2.1) is shown in equation (2.2),
wherein the matrix Q, R,is a semi-positive constant matrix with appropriate dimensions, N being the length;
definition of
Figure RE-GDA00023053431900000211
When i is 0, d, formula (2.1) is represented by formula (3.1)
xk+1＝C(k)xk+D0(k)uk+Dd(k)uk-d+ωk(3.1)；
Maximum principle of discrete linear quadratic gaussian system with multiple input time lag and state/control correlated noise when k is 0
The expansion is as follows:
Figure RE-GDA0002305343190000031
Figure RE-GDA0002305343190000033
therein, ζkIs a covariate, and k > N, ζk＝0；
Further introduces a coupled Riccati difference equation shown in formula (3.5)
Figure RE-GDA0002305343190000034
Wherein the content of the first and second substances,
Figure RE-GDA0002305343190000036
wherein the content of the first and second substances,
Figure RE-GDA0002305343190000037
Figure RE-GDA0002305343190000038
Figure RE-GDA0002305343190000039
Figure RE-GDA00023053431900000310
terminal value of
Figure RE-GDA0002305343190000041
Figure RE-GDA0002305343190000042
Let us assume that when k is 0k＞0
In a discrete linear quadratic gaussian system with multiple input dead time and state/control dependent noise,
if and only if ΩkWhen N is a positive timing matrix, the optimal controller u is set to 0kIs shown in a formula (3.14),
Figure RE-GDA0002305343190000044
the optimum performance index is shown in formula (3.15),
Figure RE-GDA0002305343190000045
wherein omegak，Mk，
Figure RE-GDA0002305343190000046
Satisfy coupling equations (3.5) - (3.11) with a final value of (3.12);
preferably, in formula (3.15), use is made ofAnd
Figure RE-GDA0002305343190000048
when, j < 0
Figure RE-GDA0002305343190000051
Preferably, when there is no time delay in a discrete linear quadratic gaussian system with multiple input dead time and state/control related noise, i.e. d is 0, the coefficients are described as
Figure RE-GDA0002305343190000052
The optimal controller degradation is:
Figure RE-GDA0002305343190000053
the optimum performance index at this time becomes:
Figure RE-GDA0002305343190000054
preferably, when a more complex system with multiple time lags, a discrete linear quadratic Gaussian system with multiple input lags and state/control related noise is represented by equation (3.16),
Figure RE-GDA0002305343190000055
wherein the content of the first and second substances,
Figure RE-GDA0002305343190000056
the cost function is given by the equation (2.2),
if and only if ΩkWhen k is 0.. and both are positive timing matrices, the optimal controller u corresponding to equation (3.16)kComprises the following steps:
Figure RE-GDA0002305343190000058
the optimal cost function is (3.26),
Figure RE-GDA0002305343190000061
wherein D isiDefined as 0 when i > d, optimal common mode ζk-1And state xkThe relational expression (2) is the following expression (3.13).
The invention has the following beneficial effects:
the maximum principle of a linear quadratic Gaussian system with multiple time lags and state/control related noise is introduced, the solution of forward and backward random difference equations (FBSDEs) is obtained through piecewise induction, sufficient necessary conditions of optimal linear quadratic Gaussian control are provided on the basis of the solution of the FBSDEs, and an optimal controller and a minimized performance index are obtained.
Drawings
FIG. 1 is ΩkWhen the linear quadratic Gaussian solution (LQG) is only optimal in the formula (2.1) when the linear quadratic Gaussian solution is more than 0, the optimal controller is shown in a schematic diagram.
Detailed Description
The following description of the embodiments of the present invention will be made with reference to the accompanying drawings:
the optimal linear quadratic Gaussian control method for the discrete system adopts a formula (2.1) to express the discrete linear quadratic Gaussian system with multiple input time lags and state/control related noise,
Figure RE-GDA0002305343190000062
wherein x isk∈RnRepresentative of the state, uk∈RmRepresenting a control variable with a delay d > 0, vkIs a mean of zero and a variance of phi2Scalar white noise of wk∈RnIs a zero mean random variable, and satisfies
Figure RE-GDA0002305343190000066
The matrix C is a matrix of a number of,
Figure RE-GDA0002305343190000063
D0，
Figure RE-GDA0002305343190000064
being a constant matrix of appropriate dimensions, vkAnd wkUncorrelated, initial states x0With a mean value ofAnd controlling the variable usIs known at S ═ d., 1;
Rnrepresenting an n-dimensional real euclidean space, I representing an identity matrix of appropriate dimensions, superscript' representing the transpose of the matrix,
Figure RE-GDA0002305343190000071
representing a random variable wkComplete probability space of, random variable wkIs formed by
Figure RE-GDA0002305343190000072
Is generated, i.e.
Figure RE-GDA0002305343190000073
The symmetric matrix A is more than 0 (more than or equal to 0) to represent that the matrix A is a positive definite (semi-positive definite) matrix, Tr (A) represents the trace of the matrix A;
the performance index related to the system shown in the formula (2.1) is shown in the formula (2.2),
Figure RE-GDA0002305343190000074
wherein the matrix Q, R,
Figure RE-GDA0002305343190000075
is a semi-positive constant matrix with proper dimensionality, and N is the time length;
definition of
When i is 0, d, formula (2.1) is represented by formula (3.1)
xk+1＝C(k)xk+D0(k)uk+Dd(k)uk-d+wk(3.1)；
On the basis of equation (3.1), the maximum principle of a discrete linear quadratic gaussian system with multiple input dead time and state/control correlated noise when k is 0.
Figure RE-GDA0002305343190000077
Figure RE-GDA0002305343190000078
Figure RE-GDA0002305343190000079
Therein, ζkIs a covariate, and k > N, ζk＝0；
Further introduces a coupled Riccati difference equation shown in formula (3.5)
Figure RE-GDA0002305343190000081
Wherein the content of the first and second substances,
Figure RE-GDA0002305343190000082
Figure RE-GDA0002305343190000083
wherein the content of the first and second substances,
Figure RE-GDA0002305343190000084
Figure RE-GDA0002305343190000086
Figure RE-GDA0002305343190000087
terminal value of
Figure RE-GDA0002305343190000088
Figure RE-GDA0002305343190000089
It is emphasized that the key to solving the problem of discrete linear quadratic gaussian system control with multiple input dead time and state/control related noise is to obtain the solutions of FBSDEs (3.1) and (3.2) - (3.4), and the relationship between the optimal co-state ζ k-1 and state xk is as follows.
Let us assume that when k is 0k> 0, equation
Figure RE-GDA00023053431900000810
Are solutions of FBSDEs (3.1) and (3.2) - (3.4) at the same time
Figure RE-GDA0002305343190000091
Satisfying equations (3.5), (3.10), (3.11) with the termination condition (3.12), the optimal control theory in equation (2.1) can be obtained on the basis of the known maximum value principle and the FBSDEs solution (3.13).
In a discrete linear quadratic gaussian system with multiple input dead time and state/control dependent noise,
if and only if ΩkWhen N is a positive timing matrix, the optimal controller u is set to 0kIs shown in a formula (3.14),
Figure RE-GDA0002305343190000092
the optimum performance index is shown in formula (3.15),
Figure RE-GDA0002305343190000093
wherein omegak，Mk，The coupling equations (3.5) - (3.11) with the final value of (3.12) are satisfied.
When there is no time delay in a discrete linear quadratic gaussian system with multiple input dead time and state/control related noise, i.e., d is 0, the coefficients are described as
Figure RE-GDA0002305343190000095
It is readily apparent that equations (3.8) and (3.10) can be rewritten as:
Figure RE-GDA0002305343190000096
will be provided with
Figure RE-GDA0002305343190000097
And
Figure RE-GDA0002305343190000098
instead of equations (3.9) and (3.11), it can be deduced that j is 0At this time, the difference equations (3.5) and (3.7) satisfy
Figure RE-GDA0002305343190000102
Wherein the content of the first and second substances,
Figure RE-GDA0002305343190000104
the optimal controller degradation is:
the optimum performance index at this time becomes:
Figure RE-GDA0002305343190000106
for complex systems with multiple time lags, the system equations are described as,
wherein the content of the first and second substances,
Figure RE-GDA0002305343190000108
i ═ 0., d, the performance index is formula (2.2),
on the basis of the known maximum principle shown in equation (2.1), the maximum principle of more complex systems with multiple time lags is expanded to:
Figure RE-GDA0002305343190000111
Figure RE-GDA0002305343190000112
Figure RE-GDA0002305343190000113
where i is 0,.. ang.d, and k is 0,.. ang.n, with k > N, ζk＝0。
Equations (3.5) - (3.11) are extended to the following equations:
Figure RE-GDA0002305343190000114
Figure RE-GDA0002305343190000115
Figure RE-GDA0002305343190000116
Figure RE-GDA0002305343190000117
Figure RE-GDA0002305343190000118
the termination value is shown as equation (3.12), which gives the main results of a discrete random linear quadratic gaussian system with state/control dependent noise and multiple input lags.
If and only if ΩkWhen N is a positive timing matrix, the optimal controller u corresponding to equation (3.16)kComprises the following steps:
Figure RE-GDA0002305343190000121
the optimum performance index is (3.26),
Figure RE-GDA0002305343190000122
wherein D isiDefined as 0 when i > d, optimal synergy ζk-1And state xkThe relational expression (2) is the following expression (3.13).
Considering the scalar case of a time-invariant LQG control system with state/control-related noise and multiple-input skew, the coefficients of equation (3.2) are as follows:
C＝2，
Figure RE-GDA0002305343190000123
D0＝1，
Figure RE-GDA0002305343190000124
Dd＝1.2
Figure RE-GDA0002305343190000127
u-1＝-1，Q＝1，R＝1，
Figure RE-GDA0002305343190000126
according to the theorem and equations (3.3) - (3.10), it can be calculated that:
Figure RE-RE-GDA0002305343190000131
obviously, Ωk> 0, therefore, there is a unique optimal linear quadratic Gaussian solution (LQG) for the available equation (2.1) and the optimal controller is as shown in FIG. 1.
According to the theorem, the optimal performance index at this time is
With the system (2.1) with cost function (2.2), we can get, by its maximum principle (3.1) - (3.4), that when k is equal to N,
Figure RE-GDA0002305343190000133
substituting (3.7) - (3.9), the optimal controller uNCan obtain the product
Figure RE-GDA0002305343190000134
Meanwhile, according to (3.2) to (3.4), there are
Figure RE-GDA0002305343190000141
Substituting into (3.6) - (3.12), ξN-1To obtain
Figure RE-GDA0002305343190000142
At this time, (3.14) was confirmed to be established when k is equal to N. Assuming that k is ≧ N +1 and N > N-d, ζk-1Like (3.14), it turns out that (3.14) is also true when k is equal to n. U additionallykAt all k ≧ n +1 are optimum ξnCan be calculated to obtain
Figure RE-GDA0002305343190000143
Insertion of ζ into (3.3)n(3.3) into
Figure RE-GDA0002305343190000151
Thus, N ═ N., N-d +1, the optimum controller is
Figure RE-GDA0002305343190000152
ζ Using equations (2.1), (3.3) and (6.5)n-1Obtaining:
Figure RE-GDA0002305343190000161
the above formula (3.14) is satisfied when k is N, and N-d < N ≦ N.
At this time, continue to obtain
Figure RE-GDA0002305343190000162
Figure RE-GDA0002305343190000163
Analogy to the above approach, assume that for all k > N +1, N-0k-1Expression ofAll the formulae (3.14) will be confirmed to be true when k is equal to n, as will be described later. Due to zetanThe calculation yields (6.8), then for all N ═ 0
Figure RE-GDA0002305343190000171
Figure RE-GDA0002305343190000182
In the formula (I), the compound is shown in the specification,
the substitution of (2.1) and (6.8) can be summarized as
In this case, the optimum controller is designed to be N-d, i.e., 0
Figure RE-GDA0002305343190000191
In the same way, unSubstitution (3.3) also proves
Figure RE-GDA0002305343190000192
"necessity": assuming that there is a uniqueness
Figure RE-GDA0002305343190000193
Measurable ukTo minimize the cost function (2.2).
By induction, omega is obtainedkN is reversible, while the optimum controller can be designed to be (3.14). Definition of
Figure RE-GDA0002305343190000194
Wherein k is 0, N, and when k is N, the above formula becomes
Figure RE-GDA0002305343190000195
The formula (2.1) is used. It can be appreciated that the uniqueness of the optimal controller depends only on uNIf > 0 is true. Another xN0 and uN-d0, J (N) may be represented by
Figure RE-GDA0002305343190000196
J (N) is known to be uNAnd because the system (2.1) has a unique solution, J (N) > 0 should satisfy ΩN> 0, i.e. omegakPositive when k is N. To accomplish the proof, assume all k ≧ n +1 times ΩkIs greater than 0. Omega will be demonstrated belown＞0。
Using (2.1), (3.3) and (3.4) gives
Figure RE-GDA0002305343190000201
To obtain the form j (N), on both sides of the above equation, k is accumulated from k N +1 to k N,
Figure RE-GDA0002305343190000202
then
Substitution of formula (2.2) to give
Figure RE-GDA0002305343190000211
Let x be the same as in the case where k is equal to Nn＝0，un-iWhen the value is 0 and (3.14) is inserted into formula (6.4), the compound is obtained
Figure RE-GDA0002305343190000212
Analogy omegaNThe case of > 0, and easily yields 0, for all kn> 0 is true.
"sufficiency": suppose k is equal to or greater than 0 and is equal to ΩNIf > 0 is true, it is necessary to prove that the cost function (2.2) is minimal
Figure RE-GDA0002305343190000224
Measurable uniqueness.
Definition of
Figure RE-GDA0002305343190000221
First, toEquivalent substitutions j ═ j +1, i ═ i +1, and m ═ m +1 can be obtained
Figure RE-GDA0002305343190000231
Structure of the device
Figure RE-GDA0002305343190000232
Form, then calculated
Figure RE-GDA0002305343190000252
Introduction of
The simultaneous substitution of (7.8) and (3.5) - (3.11), (7.7) becomes
Figure RE-GDA0002305343190000253
On both sides (7.9) from k-0 to k-N, gives
Figure RE-GDA0002305343190000261
In this case, the cost function (2.2) is shaped as
Figure RE-GDA0002305343190000262
Because of omegak> 0, then the only optimal controller must satisfy the condition Δk0. In this case, the cost function is minimal, i.e. the optimal controller should be
Figure RE-GDA0002305343190000263
And the optimal performance index is
Figure RE-GDA0002305343190000264
It is to be understood that the above description is not intended to limit the present invention, and the present invention is not limited to the above examples, and those skilled in the art may make modifications, alterations, additions or substitutions within the spirit and scope of the present invention.

Claims (3)

1. The optimal linear quadratic Gaussian control method of the discrete system is characterized in that the discrete linear quadratic Gaussian system with multiple input time lags and state/control related noise is expressed by a formula (2.1),
wherein x isk∈RnRepresentative of the state, uk∈RmRepresenting control variables, d > 0 representing input delay, vkIs a mean of zero and a variance of phi2Scalar white noise of wk∈RnIs a zero mean random variable and satisfies
Figure FDA0002230370640000012
The matrix C is a matrix of a number of,
Figure FDA0002230370640000013
D0，
Figure FDA0002230370640000014
constant matrix of appropriate dimensions, vkAnd ωkUncorrelated, initial states x0With a mean value ofAnd the initial control variable usIs known at s ═ d., 1;
Rnrepresenting an n-dimensional real euclidean space, I representing an identity matrix of appropriate dimensions, superscript' representing the transpose of the matrix,
Figure FDA0002230370640000016
representing a random variable wkComplete probability space of, random variable wkIs formed by
Figure FDA0002230370640000017
Is generated, i.e.
Figure FDA0002230370640000018
Symmetric matrix A > 0(≧ 0) means that matrix A is a positive (semi-positive) matrix, Tr(A) A trace representing matrix A;
the performance index related to the system shown in the formula (2.1) is shown in the formula (2.2),
Figure FDA0002230370640000019
wherein the matrix Q, R,
Figure FDA00022303706400000110
is a semi-positive constant matrix with proper dimensionality, and N is a time length;
definition of
When i is 0, d, formula (2.1) is represented by formula (3.1)
xk+1＝C(k)xk+D0(k)uk+Dd(k)uk-d+wk(3.1)；
The maximum principle of the discrete linear quadratic gaussian system with multiple input dead time and state/control correlated noise when k is 0
Figure FDA0002230370640000021
Figure FDA0002230370640000022
Figure FDA0002230370640000023
Therein, ζkIs a covariate, and k > N, ζk＝0；
Further introduces a coupled Riccati difference equation shown in formula (3.5)
Figure FDA0002230370640000024
Wherein the content of the first and second substances,
Figure FDA0002230370640000025
Figure FDA0002230370640000026
wherein the content of the first and second substances,
Figure FDA0002230370640000027
Figure FDA0002230370640000028
Figure FDA0002230370640000029
Figure FDA00022303706400000210
terminal value of
Figure FDA0002230370640000031
Figure FDA0002230370640000032
Let us assume that when k is 0kIf more than 0, the solution of the forward and backward random difference equation can be obtained as
Figure FDA0002230370640000033
In a discrete linear quadratic gaussian system with multiple input dead time and state/control dependent noise,
if and only if ΩkWhen N is a positive timing matrix, the optimal controller u is set to 0kIs shown in a formula (3.14),
Figure FDA0002230370640000034
the optimum performance index is shown in formula (3.15),
wherein omegak，Mk，
Figure FDA0002230370640000036
Satisfy coupled Riccati equations (3.5) - (3.11) with a final value of (3.12);
2. the discrete system optimal linear quadratic gaussian control method according to claim 1, wherein when there is no time delay in the discrete linear quadratic gaussian system with multiple input skew and state/control dependent noise, i.e. d is 0, the coefficients are described as
Figure FDA0002230370640000041
The optimal controller degradation is:
the optimum performance index at this time becomes:
3. the method of discrete system optimal linear quadratic Gaussian control according to claim 1, wherein when a more complex system with multiple time lags, the discrete linear quadratic Gaussian system with multiple input time lags and state/control dependent noise is represented by equation (3.16),
Figure FDA0002230370640000044
wherein the content of the first and second substances,
i ═ 0., d, the performance index is formula (2.2),
if and only if ΩkWhen k is 0.. and both are positive timing matrices, the optimal controller u corresponding to equation (3.16)kComprises the following steps:
Figure FDA0002230370640000046
the optimum performance index is (3.26),
Figure FDA0002230370640000051
wherein D isiDefined as 0 when i > d, optimal synergy ζk-1And state xkThe relational expression (2) is the following expression (3.13).
CN201910965583.4A 2019-10-12 2019-10-12 Optimal linear quadratic Gaussian control method for discrete system Active CN110764409B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201910965583.4A CN110764409B (en) 2019-10-12 2019-10-12 Optimal linear quadratic Gaussian control method for discrete system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201910965583.4A CN110764409B (en) 2019-10-12 2019-10-12 Optimal linear quadratic Gaussian control method for discrete system

Publications (2)

Publication Number Publication Date
CN110764409A true CN110764409A (en) 2020-02-07
CN110764409B CN110764409B (en) 2022-04-29

Family

ID=69331739

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201910965583.4A Active CN110764409B (en) 2019-10-12 2019-10-12 Optimal linear quadratic Gaussian control method for discrete system

Country Status (1)

Country Link
CN (1) CN110764409B (en)

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112130451A (en) * 2020-09-23 2020-12-25 兰州理工大学 High-precision control method for mine filling slurry concentration
CN112182484A (en) * 2020-09-22 2021-01-05 深圳职业技术学院 Approximate solving method for optimal linear quadratic Gaussian control of UDP-like system
CN112512110A (en) * 2020-11-19 2021-03-16 合肥工业大学 Wireless transmitting power prediction control method for intelligent substation reliability demand constraint

Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20070112445A1 (en) * 2005-09-28 2007-05-17 Aulac Technologies Inc. Quadratic Performance, Infinite Steps, Set Point Model Tracking Controllers
CN101004592A (en) * 2007-01-25 2007-07-25 上海交通大学 Control method of feed forward, feedback control system for interferential and time delayed stable system
CN102301290A (en) * 2009-02-02 2011-12-28 费希尔-罗斯蒙特***公司 Model predictive controller with tunable integral component to compensate for model mismatch
US20120277888A1 (en) * 2011-04-29 2012-11-01 Georgia Tech Research Corporation Systems and methods for parameter dependent riccati equation approaches to adaptive control
CN109933888A (en) * 2019-03-11 2019-06-25 济南大学 A kind of stochastic system Tracking Control Design method with multiplicative noise and time lag

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20070112445A1 (en) * 2005-09-28 2007-05-17 Aulac Technologies Inc. Quadratic Performance, Infinite Steps, Set Point Model Tracking Controllers
CN101004592A (en) * 2007-01-25 2007-07-25 上海交通大学 Control method of feed forward, feedback control system for interferential and time delayed stable system
CN102301290A (en) * 2009-02-02 2011-12-28 费希尔-罗斯蒙特***公司 Model predictive controller with tunable integral component to compensate for model mismatch
US20120277888A1 (en) * 2011-04-29 2012-11-01 Georgia Tech Research Corporation Systems and methods for parameter dependent riccati equation approaches to adaptive control
CN109933888A (en) * 2019-03-11 2019-06-25 济南大学 A kind of stochastic system Tracking Control Design method with multiplicative noise and time lag

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
LIANG X等: "Optimal Control for Remote and Local Controllers with Packet Dropout and Input Delay", 《INTERNATIONAL CONFERENCE ON CONTROL AUTOMATION ROBOTICS AND VISION》 *
崔鹏等: "线性离散奇异***的最优估计新方法", 《山东大学学报(工学版)》 *

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112182484A (en) * 2020-09-22 2021-01-05 深圳职业技术学院 Approximate solving method for optimal linear quadratic Gaussian control of UDP-like system
CN112130451A (en) * 2020-09-23 2020-12-25 兰州理工大学 High-precision control method for mine filling slurry concentration
CN112130451B (en) * 2020-09-23 2021-07-23 兰州理工大学 High-precision control method for mine filling slurry concentration
CN112512110A (en) * 2020-11-19 2021-03-16 合肥工业大学 Wireless transmitting power prediction control method for intelligent substation reliability demand constraint
CN112512110B (en) * 2020-11-19 2022-03-15 合肥工业大学 Wireless transmitting power prediction control method for intelligent substation reliability demand constraint

Also Published As

Publication number Publication date
CN110764409B (en) 2022-04-29

Similar Documents

Publication Publication Date Title
CN110764409B (en) Optimal linear quadratic Gaussian control method for discrete system
Li et al. Robust H∞ filtering for nonlinear stochastic systems with uncertainties and Markov delays
Yang et al. Hybrid adaptive and impulsive synchronization of uncertain complex networks with delays and general uncertain perturbations
Zhang et al. New approach on designing stochastic sampled-data controller for exponential synchronization of chaotic Lur’e systems
Xu et al. Controlling the ultimate state of projective synchronization in chaotic systems of arbitrary dimension
Wu et al. Saturated impulsive control for synchronization of coupled delayed neural networks
Park et al. H∞ sampled-state feedback control for synchronization of chaotic Lur’e systems with time delays
Peng et al. Finite-time synchronization of quaternion-valued neural networks with delays: A switching control method without decomposition
Hilhorst et al. Control of linear parameter-varying systems using B-splines
Ren et al. Prescribed-time leader-following consensus for stochastic second-order multi-agent systems subject to actuator failures via sliding mode control strategy
Hwang Decentralized Fuzzy Control of Nonlinear Interconnected Dynamic Delay Systems via Mixed $ H_2/\! H_\infty $ Optimization With Smith Predictor
Li et al. Explicit Milstein schemes with truncation for nonlinear stochastic differential equations: Convergence and its rate
Wang et al. Optimal index and averaging principle for Itô–Doob stochastic fractional differential equations
Kaldmäe et al. Integrability for nonlinear time-delay systems
Wu et al. Maximum principle for partially-observed optimal control problems of stochastic delay systems
Chen et al. Robust adaptive neural control for a class of perturbed nonlinear systems with unmodeled dynamics and output disturbances
Lin et al. New delay-range-dependent exponential estimates for singular systems with time-varying delay
Sivaganesh Master stability functions for a class of coupled simple nonlinear electronic circuits
Pinchas Two blind adaptive equalizers connected in series for equalization performance improvement
Yu et al. Decentralized partial-variable periodic intermittent control for a class of interconnected fractional-order systems
Zhong et al. Decentralized control and synchronization of time-varying complex dynamical network
An Improved delay-derivative-dependent stability criteria for linear systems using new bounding techniques
Ghamgui et al. Stability and stabilization of 2D continuous state-delayed systems
Hewitt et al. Algebraically stable general linear methods and the G-matrix
US20050216542A1 (en) Transform domain adaptive filter, equalizer and wireless communication device using the same, and decision feedback equalization method

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant