CN111007721B - Method for dynamically selecting sampling period in linear steady system - Google Patents
Method for dynamically selecting sampling period in linear steady system Download PDFInfo
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- CN111007721B CN111007721B CN201911268105.4A CN201911268105A CN111007721B CN 111007721 B CN111007721 B CN 111007721B CN 201911268105 A CN201911268105 A CN 201911268105A CN 111007721 B CN111007721 B CN 111007721B
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- 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
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Abstract
The invention belongs to the technical field of analysis and design of a general control system and the like, and discloses a method for dynamically selecting a sampling period in a linear steady system. Different from the traditional research idea of recursion of a discrete control design from a continuous control design, the method directly develops the control design in a discrete time domain, and avoids the problem that the conservative form of the discrete control law method is obtained by approaching the continuous control law in the prior art; in the design process of the discrete control law, the scale transformation of the system state is adopted, and the additional degree of freedom of the adjustable scale factor is introduced, so that the effective suppression of uncertain behaviors in the system is ensured, the integrated design of the scale factor and the sampling period is realized, the selection range of the sampling period in the sampling control design is further expanded, and the Zeno phenomenon in the sampling design process is avoided.
Description
Technical Field
The invention belongs to the technical field of analysis and design of a general control system and the like, and particularly relates to a dynamic selection method for a sampling period in a linear steady system.
Background
Due to the development of digital computers and microelectronic technology, the application of digital controllers in control systems is on the rapid growth trend, and is generally applied to various aspects of aviation, aerospace, production and life. Compared with an analog controller, the digital controller not only has the advantages of high sensitivity, low cost and the like, but also has the advantages of flexible programming, insensitivity to noise, capability of obtaining better control performance and the like.
The selection of the sampling period is always a core problem in the analysis and design of the sampling control system, and the research of the problem has important influence on the performance analysis of the closed-loop digital control system and the computer implementation of the sampling controller. Useful information in a controlled system or a dynamic process is easily lost due to an overlarge sampling period, and higher requirements are provided for the arithmetic capability of a processor and the hardware realization of a controller due to an undersize sampling period; meanwhile, with the popularization and wide application of networks, the access network devices have great differences in data acquisition and information processing, so that a control scheme capable of realizing dynamic adjustment of a sampling period is urgently needed, so that the problem caused by the fact that the relationship between the dynamic performance and the steady-state performance of the system is considered in a compromise mode, and even the problem of heterogeneity among different systems of the access network is solved.
Disclosure of Invention
In view of this, the present invention provides a method for dynamically selecting a sampling period in a linear steady system, where the method utilizes the coupling effect between an adjustable control gain and a sampling period selection to implement an integrated design of the control gain selection and the sampling period selection of the linear system, and can dynamically adjust the sampling period in the linear control system without affecting the stability of a controlled system.
In order to solve the technical problem, the invention provides a dynamic selection method of a sampling period in a linear steady system, which comprises the following steps:
s1, performing coordinate transformation on a continuous state space model of a controlled system by adopting an adjustable scale factor, and then performing discretization to obtain a discrete state space model, wherein the discrete state space model comprises a nominal part and a disturbance part;
s2, aiming at the nominal part of the discrete state space model, firstly designing a stabilizing control law based on sampling state feedback, and then estimating the influence of the disturbance part;
s3, finding an integrated parameter for ensuring the stability of the nominal part of the discrete state space model; calculating a range of control gain parameters based on the perturbation component; and then dynamically selecting a sampling period according to the range of the control gain parameter, thereby realizing the integrated design of the control gain parameter and the sampling period, wherein the integrated parameter is the product of the control gain parameter and the sampling period.
Preferably, the step S1 specifically includes:
s11, carrying out coordinate transformation on the continuous state space model of the controlled system by adopting an adjustable scale factor according to the following formula to obtain a continuous state space model after coordinate transformation:
wherein, x = [ x ] 1 ,x 2 ,…,x n ] T ∈R n Is a state variable of the continuous state space model; u is an element of R p Input variables for a continuous state space model; l is a control gain parameter; n is the dimension of the continuous state space model; z = [ z ] 1 ,z 2 ,…,z n ] T ∈R n Is the state variable of the continuous state space model after coordinate transformation; v is an input variable of the continuous state space model after coordinate transformation;
s12, discretizing the continuous state space model obtained after the coordinate transformation according to the following formula to obtain a discrete state space model comprising a nominal part and a disturbance part;
wherein, the first and the second end of the pipe are connected with each other,
t is the sampling period, k is the sampling time,
as a disturbance term, a i I =0,1, \8230, n-1 is a continuous state space model system matrix
The last row of (2).
Preferably, the step S2 specifically includes:
s21, designing a stabilization control law based on sampling state feedback, wherein the stabilization control law is shown as the following formula;
v(k)＝-k 1 z 1 (k)-k 2 z 2 (k)-…-k n z n (k)；
wherein k is 1 ,k 2 ,…k n Control law coefficients for sampling;
s22, constructing a disturbance-free nominal closed-loop system matrix shown as the following formula according to the stabilization control law;
s23, estimating the influence of the disturbance part by using a constant variation method according to the following formula;
where Λ is an upper bound constant for the perturbation portion.
Preferably, the step S3 specifically includes:
s31, determining an integration parameter based on the requirement of the matrix stability of the undisturbed nominal closed-loop system, wherein the integration parameter is the product of a control gain parameter and a sampling period;
s32, calculating the range of the control gain parameter based on the disturbance part:
and S33, dynamically selecting a sampling period according to the range of the control gain parameter obtained by calculation in the step S32.
Compared with the prior art, the invention adopts an integrated design method of controlling gain parameters and sampling periods, can realize dynamic selection of the sampling periods on the basis of covering the existing sampling period adjustment interval, further enlarges the selection range of the sampling periods of a linear system, and provides greater freedom for optimizing the dynamic performance of the system; meanwhile, the integrated design scheme directly provided in the discrete time domain avoids the conservatism brought by the fact that the continuous time control law is discretized to deduce the discrete time control law, and avoids the Zeno phenomenon in the traditional discrete control design.
Drawings
Fig. 1 is a flowchart of a method for dynamically selecting a sampling period in a linear constancy system according to the present invention;
FIG. 2 is a schematic diagram of an integrated design structure of a controlled system according to an embodiment of the present invention;
fig. 3 is a schematic diagram illustrating the state adjustment of the controlled system when the control gain L =0.7 and the sampling period T =1 according to the embodiment of the present invention;
fig. 4 is a schematic diagram of control inputs of the controlled system when the control gain L =0.7 and the sampling period T =1 according to the embodiment of the present invention;
fig. 5 is a schematic diagram illustrating state adjustment when the controlled system control gain L =7 and the sampling period T =0.1 according to the embodiment of the present invention;
fig. 6 is a schematic diagram of control inputs of the controlled system when the control gain L =7 and the sampling period T =0.1 in the embodiment of the present invention.
Detailed Description
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.
As shown in fig. 1, the present invention provides a dynamic selection method for sampling period in a linear steady system, which comprises the following steps:
s1, performing coordinate transformation on a continuous state space model of a controlled system by adopting an adjustable scale factor, and then performing discretization to obtain a discrete state space model, wherein the discrete state space model comprises a nominal part and a disturbance part;
linear constancy system described for a transfer function as shown in equation (1)
Let u be an element of R 1 For an input variable, y ∈ R 1 For the output variable and x ∈ R n For the state variables, the above linear steady system is described as a continuous state space model of controllable standard type as shown in equation (2):
wherein A ∈ R n×n ，b∈R n×1 ，C∈R 1×n Respectively a system matrix, an input matrix and an output matrix of the continuous state space model shown in the formula (2),
the continuous state space model shown in the formula (2) is rewritten into the form shown in the formula (3)
Wherein, the first and the second end of the pipe are connected with each other,
based on the viewpoint of the disturbance system, the stabilized control problem of the continuous state space model shown in the formula (3) after rewriting is regarded as the stabilized control problem of the nominal system shown in the formula (4) under the action of the disturbance part f (x),
for the nominal system shown in the formula (4), coordinate transformation is carried out by adopting an adjustable scale factor based on the formula (5)
If the control gain parameter L is an undetermined constant, the continuous state space model corresponding to the nominal system shown in equation (4) can be described again as
Meanwhile, the disturbance part in the rewritten continuous state space model shown in formula (3) can also be described as
Further, the rewritten continuous state space model shown in the formula (3) can be described again as a continuous state space model shown in the formula (7) under the coordinate transformation:
3) Discretizing the continuous state space model shown in the formula (7) to obtain a discrete state space model containing a nominal part and a disturbance part shown in the formula (8):
wherein the content of the first and second substances,
s2: aiming at the nominal part of the discrete state space model, firstly, a stabilizing control law based on sampling state feedback is designed, and then the influence of the disturbance part is estimated.
1) And (3) designing a calm control law: nominal part for the discrete state space model shown in equation (8)
Z(k+1)＝ΦZ(k)+Γv(k), (9)
Design of the stabilization control law shown in the formula (10)
v(k)＝-k 1 z 1 (k)-k 2 z 2 (k)-…-k n z n (k) (10)
Wherein k is 1 ,k 2 ,…k n Is a calm control law coefficient.
Then, a closed loop system as shown in the formula (11) is constructed
Z(k+1)＝ΩZ(k) (11)
Wherein the coefficient k of the calm control law 1 ,k 2 ,…k n Designed such that the polynomial shown in equation (12) is a constant of the Hurwitz polynomial,
λ n +k n λ n-1 +…+k 2 λ+k 1 ＝0. (12)
based on the stationary control law coefficient k selected according to equation (12) i Find all the characteristics of the closed loop system matrix shown in the formula (13)The integral parameter LT of which the eigenvalue falls on the left half plane of the complex plane,
2) Estimation of disturbance part influence: as can be seen from the specific form of the disturbance part f (x) in the continuous state space model shown in equation (3), the existence of the constant c makes the estimation of the disturbance part shown in equation (14) true,
|f(x 1 (t),…,x n (t))|≤c(|x 1 (t)|+…+|x n (t)|) (14)
further, the existence constant can be known
The disturbance portion in the continuous state space model shown in equation (7) satisfies the condition shown in equation (15):
the disturbance component in the discrete state space model represented by the formula (8) is obtained by a constant variational method, and it is known that the influence of the disturbance component is estimated as follows by the upper bound constant Λ of the disturbance component
3) Selection of the lower bound of the control gain parameter: since the nominal part of the discrete state space model is fully controllable, there is a calm control law that makes the corresponding closed-loop system as shown in equation (11) globally asymptotically stable. From the conclusion of the linear stationary system stability, it can be known that there is a positive stationary symmetric matrix P such that in the Lyapunov function V (Z (k)) = Z T (k) The solution of PZ (k) along the nominal part shown in equation (9) satisfies
V(Z(k+1))-V(Z(k))＝-||Z(k)|| 2 . (17)
Further, along with the solution of the discrete state space model shown in equation (8), the Lyapunov function V (Z (k)) satisfies
Selecting
Can obtain
The influence of the disturbance part in the discrete state space model shown in the formula (8) is suppressed, and meanwhile, the asymptotic stability of the discrete state space model shown in the formula (8) is ensured.
S3, finding an integrated parameter for ensuring the stability of the nominal part of the discrete state space model; calculating a range of control gain parameters based on the perturbation portion; and then dynamically selecting a sampling period according to the range of the control gain parameter, thereby realizing the integrated design of the control gain parameter and the sampling period, wherein the integrated parameter is the product of the control gain parameter and the sampling period.
1) Determining a calm control law coefficient k 1 ,k 2 ,…k n : based on the expected characteristic value of the closed-loop system shown in the formula (11), an expected characteristic polynomial (12) is determined, and a calm control law coefficient k is obtained 1 ,k 2 ,…k n ；
2) Determining an integration parameter LT: based on the fact that (phi, gamma) is a controllable pair, the nominal part of the discrete state space model shown in the formula (9) is completely controllable, the closed-loop poles of the discrete state space model can be randomly configured by a stabilized control law sampled as shown in the formula (10), namely, the eigenvalue of the closed-loop system matrix shown in the formula (13) depends on the coefficient k of the stabilized control law 1 ,k 2 ,…k n With the integration parameter LT, based on the selected calm control law coefficient k 1 ,k 2 ,…k n Determining an integration parameter LT, wherein the integration parameter LT enables the characteristic value of the closed-loop system matrix to be in a left half plane;
3) Controlling the linkage selection of the gain parameter L and the sampling period T: based on the selected integration parameter LT, satisfy at the control gain parameter L
To further understand the present invention, a third order system is incorporated below
The embodiment of the present invention is simulated, fig. 2 is a schematic view of an integrated design structure of the controlled system, fig. 3 is a schematic view of a state adjustment of the controlled system when a control gain L =0.7 and a sampling period T =1, fig. 4 is a graph of a change of a control input of the controlled system when the control gain L =0.7 and the sampling period T =1, fig. 5 is a schematic view of a state adjustment of the controlled system when the control gain L =7 and the sampling period T =0.1, and fig. 6 is a graph of a change of a control input of the controlled system when the control gain L =7 and the sampling period T = 0.1. It can be seen from the above simulation results that, based on the integrated design thought provided by the invention, the sampling period can be realized to be [0.1,1]Due to the linkage design of the control gain and the sampling period, the stability of the whole closed-loop system is not influenced by the dynamic adjustment of the sampling period.
While there have been shown and described what are at present considered the fundamental principles and essential features of the invention and its advantages, it will be apparent to those skilled in the art that the invention is not limited to the details of the foregoing exemplary embodiments, but is capable of other specific forms without departing from the spirit or essential characteristics thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that changes, modifications, substitutions and alterations can be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.
Claims (1)
1. A method for dynamically selecting a sampling period in a linear steady system is characterized by comprising the following steps:
s1, performing coordinate transformation on a continuous state space model of a controlled system by adopting an adjustable scale factor, and then performing discretization to obtain a discrete state space model, wherein the discrete state space model comprises a nominal part and a disturbance part;
the step S1 specifically comprises the following steps:
s11, carrying out coordinate transformation on a continuous state space model of the controlled system by adopting an adjustable scale factor according to the following formula to obtain a continuous state space after coordinate transformation:
wherein, x = [ x ] 1 ,x 2 ,…,x n ] T ∈R n Is a state variable in a continuous state space model; u is an element of R 1 Input variables for a continuous state space model; l is a control gain parameter; n is the dimension of the continuous state space model; z = [ z ] 1 ,z 2 ,…,z n ] T ∈R n Is the state variable of the continuous state space model after coordinate transformation; v is an input variable of the continuous state space model after coordinate transformation;
s12, discretizing the continuous state space model after coordinate transformation according to the following formula to obtain a discrete state space model comprising a nominal part and a disturbance part;
wherein，
T is the sampling period, k is the sampling time,
as a disturbance term, a i I =0,1, \ 8230, n-1 is a continuous state space model system matrix
The last row of (2);
s2, aiming at the nominal part of the discrete state space model, firstly designing a stabilization control law based on sampling state feedback, and then estimating the influence of the disturbance part;
the step S2 specifically comprises the following steps:
s21, designing a stabilization control law based on sampling state feedback, wherein the stabilization control law is shown as the following formula:
v(k)＝-k 1 z 1 (k)-k 2 z 2 (k)-…-k n z n (k)；
wherein k is 1 ,k 2 ,…k n To calm the control law coefficients;
s22, constructing a disturbance-free nominal closed-loop system matrix shown as the following formula according to the calm control law;
s23, estimating the influence of the disturbance part by using a constant variation method according to the following formula;
wherein Λ is an upper bound constant of the disturbance part;
s3, finding an integrated parameter for ensuring the stability of the nominal part of the discrete state space model; calculating a range of control gain parameters based on the perturbation component; then, dynamically selecting a sampling period according to the range of the control gain parameter, and further realizing the integrated design of the control gain parameter and the sampling period, wherein the integrated parameter is the product of the control gain parameter and the sampling period;
the step S3 specifically includes:
s31, determining an integration parameter based on the requirement of the matrix stability of the undisturbed nominal closed-loop system, wherein the integration parameter is the product of a control gain parameter and a sampling period;
s32, calculating the range of the control gain parameter based on the disturbance part:
and S33, dynamically selecting a sampling period according to the range of the control gain parameters obtained by calculation in the step S32.
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| US6539290B1 (en) * | 1995-06-07 | 2003-03-25 | Dabulamanzi Holdings, Llc | Method, apparatus and design procedure for controlling multi-input, multi-output (MIMO) parameter dependent systems using feedback LTI'zation |
| CN101578584A (en) * | 2005-09-19 | 2009-11-11 | 克利夫兰州立大学 | Controllers, observers, and applications thereof |
| CN107463095A (en) * | 2017-07-20 | 2017-12-12 | 南京邮电大学 | A kind of output feedback controller design method with time-varying sampling period |
| CN108205259A (en) * | 2016-12-19 | 2018-06-26 | 中国航天科工飞航技术研究院 | Multiplex control system and its design method based on linear extended state observer |
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| US6539290B1 (en) * | 1995-06-07 | 2003-03-25 | Dabulamanzi Holdings, Llc | Method, apparatus and design procedure for controlling multi-input, multi-output (MIMO) parameter dependent systems using feedback LTI'zation |
| CN101578584A (en) * | 2005-09-19 | 2009-11-11 | 克利夫兰州立大学 | Controllers, observers, and applications thereof |
| CN108205259A (en) * | 2016-12-19 | 2018-06-26 | 中国航天科工飞航技术研究院 | Multiplex control system and its design method based on linear extended state observer |
| CN107463095A (en) * | 2017-07-20 | 2017-12-12 | 南京邮电大学 | A kind of output feedback controller design method with time-varying sampling period |
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