CN106970526A - A kind of Design of Feedback Controller method of the time-delay/time-varying system based on convex combination method - Google Patents
A kind of Design of Feedback Controller method of the time-delay/time-varying system based on convex combination method Download PDFInfo
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Abstract
The present invention proposes a kind of Design of Feedback Controller method of the time-delay/time-varying system based on convex combination method, based on Lyapunov Theory of Stability, using time lag dividing method, by decomposing delay interval for multiple equidistant minizones, the Lyapunov krasovskii functionals itemized with triple product are constructed, the smaller stability criteria of conservative is obtained based on convex combination method；Design of feedback controller, by handling nonlinear terms with Linearization Method, provides the existence consition and concrete form of controller on this basis.Its advantage is：The present invention is on the basis of time lag segmentation, and construction triple product itemizes, makes full use of time lag lower bound information, the integral inequality using popularization, and these characteristics all cause the reduction of system conservative；Compared with other time lag dividing methods, present invention introduces less matrix variables, the complexity of calculating is reduced, and is easy to obtain the controller existence consition based on LMI.
Description
Technical field
The present invention relates to controller design field, a kind of feedback of the time-delay/time-varying system based on convex combination method is particularly related to
Controller design method.
Background technology
Time delay is widely present in communication system, Chemical Engineering, biosystem, nuclear reactor, process control and network
Networked control systems etc..
The generation of time lag may deteriorate systematic function or even make system loss of stability, if ignoring time lag in analysis
Influence can cause incorrect design；Therefore, in recent years, the uncertainty, stability and controller on time lag system
Extensive concern of the relevant issues such as design by many experts and scholars.
Currently, the stability study of time-delay/time-varying system is general first by constructing suitable Lyapunov-
Krasovskii functionals, the stability of derivation system is carried out with reference to methods such as free-form curve and surface, integral inequality, time lag cutting techniques
Criterion.In document Robust stability and control for uncertain neutral time delay
Discretization thought is based in systems [J], by Discrete-Delay and the interval non-uniformly distributed load of distributed delay into several pieces, and right
The suitable Lyapunov-krasovskii functionals of interval structure are answered, the stability criteria of system is derived；In document Delay-
partitioning approach to robust stability of uncertain neutral system with
Mixed delays [J] are based on time lag cutting techniques and Jensen inequality methods, derive that band edible organic acid mixture does not know neutral type
The stable adequate condition of system；Delay interval is divided into several pieces by document above, although the less result of conservative can be obtained,
But add the complexity of calculating；In order to reduce due to the computational complexity that Concourse Division is caused, document Improved
robust stabilization method for linear systems with interval time-varying
Input delays by using Wirtinger inequality [J] are directed to the robust stabilizing of uncertain time-delay/time-varying system
Problem, is divided into two subintervals by delay interval, constructs suitable Lyapunov-krasovskii functionals, in Wirtinger not
On the basis of equation, interactive convex combination method is utilized[11]Obtain improved On Delay-Dependent Stability conclusion；Document A delay
decomposition approach to robust stability analysis of uncertain systems with
Time-varying delay are based on time lag split plot design and free-form curve and surface method, and delay interval is divided into two subintervals, right
System carries out stability analysis；Document Novel delaydependent robust stability criteria for
neutral systems with mixed time-varying delaysand nonlinear perturbations[J]
Based on time lag cutting techniques and free-form curve and surface method, the delay interval of the time_varying delay Neutral Differential Equations with disturbance is divided into two
Individual subinterval, obtains the robust stability criterion of system；Although document above has obtained the simple conclusion of form, but due to construction
Functional is relatively easy, so as to cause the conservative of conclusion.
The content of the invention
The problem of present invention is solves existing, proposes a kind of feedback controller of the time-delay/time-varying system based on convex combination method
Design method.
The technical proposal of the invention is realized in this way：
A kind of Design of Feedback Controller method of the time-delay/time-varying system based on convex combination method, it is characterised in that：The control
Device design process is as follows：
A. varying-delay system model is set up,
Wherein：x(t)∈RnFor the state vector of system；A、A1It is known suitable dimension matrix；It is continuous initial vector
Function；Time_varying delay h (t) is met：0≤hd≤h(t)≤hD,Wherein, 0≤hd≤hDIt is constant with μ >=0；
B. delay interval is split, delay interval is divided into N parts of equidistant minizones, N>0, and be integer, then hi(i=
1,2 ..., N+1) meet：hd=h1<h2<…<hN<hN+1=hD, wherein the length of each minizone is hα=hi+1-hi=(hD-
hd)/N；
C. stability analysis is carried out to system, during analysis, introduces lemma 1, lemma 2, lemma 3, lemma 4；
D. Lyapunov-Krasovskii functionals are constructed, derivation, lemma 1, lemma in step b are carried out to functional
2nd, lemma 3, lemma 4 finally give stability of a system condition,
As h (t) ∈ [hi,hi+1] when, present invention construction Lyapunov-Krasovskii functionals are：
Wherein,
Vi1(t)=ξT(t) P ξ (t),
Wherein,
The derivative that can obtain functional V (t) along the track of system (1) is：
Following formula can be obtained with lemma 1 and lemma 2 to set up：
Wherein
Obtained with lemma 3
Wherein, δT(t)=[xT(t-hi) xT(t-h(t)) xT(t-hi+1)],
Order
Then
Wherein,
Therefore obtained with reference to inequality (9)~(13)
For h (t) ∈ [hi,hi+1], according to lemma 4,
If following inequality is set up：
Then have：
Then haveFrom Lyapunov Theory of Stability, system (1) is asymptotically stable；
E. condition-theorem 1 of the stability of a system is drawn：For giving scalar 0<hd<hDWith 0≤μ<1, system is asymptotic steady
Fixed, if there is positive definite matrixRi>0 (i=1,2) and Ui>0 (i=1,2)；With
And the matrix T of any appropriate dimensionj, Yj(j=1,2,3), set up following MATRIX INEQUALITIES：
Wherein,
Ξ12=-P12+P13,Ξ13=P11A1,Ξ14=-P13,
Ξ15=ATP12+P22+hiU1,Ξ16=ATP13+P23+hαU2,Ξ22=-Q1-2R1+Y1+Y1 T, Ξ26=-P23+P33, Ξ46=-P33,Ξ56=0, Ξ66
=-U2,
Ac=[the A of A 01 0 0 0],
And meet hα=hi+1-hi=(hD-hd)/N,
F. the design of subduer, considers the Stabilization of the controller of system, order control in step e on the basis of theorem 1
The form of device processed is：
U (t)=Kx (t) (14)
Its closed-loop system is：
(14) are brought into system (15), can be obtained
J. on the basis of theorem 1, provide existence consition-theorem 2 of subduer, replace A with A+BK, and formula (2),
(3) the premultiplication right side in two ends multiplies diag { X, X, X, X, X, X, X, X } and its transposition, wherein makingDefinition
I.e.It can obtain Wherein
It is known for any inequality J>0, inequality (21) is set up：
Due to J>0, have to Arbitrary Matrix X
(X-J)J-1(X-J)>0
For any given ε
(ε-1X-J)(ε-2J)-1(ε-1X-J)
=XJ-1X-2εX+ε2J>0
I.e.：
XJ-1X>2εX-ε2J (22)
By formula 19)~(22), formula (17), (18) can be obtained by mending lemma using Schur；
Theorem 2 is for giving scalar 0<hd<hD,0≤μ<1 and ε, if in the presence of matrix L=L of appropriate dimensionT>0, if in the presence of
Positive definite matrixAndAnd the matrix of any appropriate dimension Set up following LMI：
Ξ13=A1X,
Ξ35=0, Ξ36=0,
Ξ56=0,
And meet hα=hi+1-hi=(hD-hd)/N,
SoIt is asymptotically stable and controls the feedback oscillator to be：K=VX-1。
The Design of Feedback Controller method of time-delay/time-varying system based on convex combination method, due to existing in formula (18)~(20)
Nonlinear terms XJ-1X, it is impossible to solve controller gain using convex optimized algorithm, therefore using Linearization Method to non-linear
Item is handled.
The beneficial effects of the invention are as follows：In the functional construction of the present invention, the lower bound information of time lag is not only taken full advantage of,
And triple product subitem is with the addition of, this two parts characteristic can all make the reduction of system conservative；By constructing triple integral functional, knot
The integral inequality promoted is closed, the stability conclusion based on convex combination mode has been obtained, is with time lag split plot design with other
System is compared, invention introduces less matrix variables, reduces the complexity of calculating, and obtain based on LMI for next step
Controller existence consition provide convenience.
Brief description of the drawings
In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing
There is the accompanying drawing used required in technology description to be briefly described, it should be apparent that, drawings in the following description are only this
Some embodiments of invention, for those of ordinary skill in the art, on the premise of not paying creative work, can be with
Other accompanying drawings are obtained according to these accompanying drawings.
H when Fig. 1 is μ=0.5dTake the time lag upper bound h of different valueDThe contrast of value；
H when Fig. 2 is μ unknowndTake the time lag upper bound h of different valueDThe contrast of value；
Fig. 3 is hdμ takes the time lag upper bound h of different value when=0D(N=2)；
Fig. 4 is the different corresponding closed-loop system zero input state response curve of ε values；
Fig. 5 is the time that system mode response curve tends towards stability when ε takes different value.
Embodiment
Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete
Site preparation is described, it is clear that described embodiment is only a part of embodiment of the invention, rather than whole embodiments.It is based on
Embodiment in the present invention, it is every other that those of ordinary skill in the art are obtained under the premise of creative work is not paid
Embodiment, belongs to the scope of protection of the invention.
A kind of Design of Feedback Controller method of the time-delay/time-varying system based on convex combination method, the controller design process is such as
Under：
A. varying-delay system model is set up,
Wherein：x(t)∈RnFor the state vector of system；A、A1It is known suitable dimension matrix；It is continuous initial vector
Function；Time_varying delay h (t) is met：0≤hd≤h(t)≤hD,Wherein, 0≤hd≤hDIt is constant with μ >=0；
B. delay interval is split, delay interval is divided into N parts of equidistant minizones, N>0, and be integer, then hi(i=
1,2 ..., N+1) meet：hd=h1<h2<…<hN<hN+1=hD, wherein the length of each minizone is hα=hi+1-hi=(hD-
hd)/N；
C. stability analysis is carried out to system, during analysis, introduces following lemma：
Lemma 1 is for x (t) ∈ RnWith single order continuous derivative, work as Rn×n>0, h>0 and vector valued functionFollowing integral inequality is set up：
Wherein,
Matrix W ∈ R of the lemma 2 for any appropriate dimensionn×n, W=WT>0 constant h>0 and vector valued functionFollowing integral inequality is set up：
Wherein,
Lemma 3 sets r1≤r(t)≤r2, then to Arbitrary Matrix R ∈ Rn×n, R=RT>0, and any appropriate dimension matrix
Ti, Yi, (i=1,2,3) following integral inequality is set up：
Wherein,xT(t-r(t))xT(t-r2)],
Lemma 4H1,H2,H3For the constant matrices with appropriate dimension, η (t) is the time-varying function with bound, η1≤η
(t)≤η2, then inequality H1+(η2-η(t))H2+(η(t)-η1)H3<0 necessary and sufficient condition set up is that following formula is set up
D. Lyapunov-Krasovskii functionals are constructed, derivation, lemma 1, lemma in step b are carried out to functional
2nd, lemma 3, lemma 4 finally give stability of a system condition,
As h (t) ∈ [hi,hi+1] when, present invention construction Lyapunov-Krasovskii functionals are：
Wherein,
Vi1(t)=ξT(t) P ξ (t),
Wherein,
The derivative that can obtain functional V (t) along the track of system (1) is：
With
Lemma 1 and lemma 2 can obtain following formula and set up：
Wherein
Obtained with lemma 3
Wherein, δT(t)=[xT(t-hi) xT(t-h(t)) xT(t-hi+1)],
Order
Then
Wherein,
Therefore obtained with reference to inequality (9)~(13)
For h (t) ∈ [hi,hi+1], according to lemma
4,
If following inequality is set up：
Then have：
Then haveFrom Lyapunov Theory of Stability, system (1) is asymptotically stable；
E. condition-theorem 1 of the stability of a system is drawn：For giving scalar 0<hd<hDWith 0≤μ<1, system is asymptotic steady
Fixed, if there is positive definite matrixRi>0 (i=1,2) and Ui>0 (i=1,2)；With
And the matrix T of any appropriate dimensionj, Yj(j=1,2,3), set up following MATRIX INEQUALITIES：
Wherein,
Ξ12=-P12+P13,Ξ13=P11A1,Ξ14=-P13,Ξ 15=ATP12+P22+hiU1,Ξ16=ATP13+P23+hαU2,Ξ22=-Q1-2R1+Y1+Y1 T, Ξ26=-P23+P33, Ξ46=-P33,Ξ56=0, Ξ66
=-U2,
Ac=[the A of A 01 0 0 0],
And meet hα=hi+1-hi=(hD-hd)/N,
F. the design of subduer, considers the Stabilization of the controller of system, order control in step e on the basis of theorem 1
The form of device processed is：
U (t)=Kx (t) (14)
Its closed-loop system is：
(14) are brought into system (15), can be obtained
J. on the basis of theorem 1, provide existence consition-theorem 2 of subduer, replace A with A+BK, and formula (2),
(3) the premultiplication right side in two ends multiplies diag { X, X, X, X, X, X, X, X } and its transposition, wherein makingDefinition
I.e.It can obtain Wherein
It is known for any inequality J>0, inequality (21) is set up：
Due to J>0, have to Arbitrary Matrix X
(X-J)J-1(X-J)>0
For any given ε
(ε-1X-J)(ε-2J)-1(ε-1X-J)
=XJ-1X-2εX+ε2J>0
I.e.：
XJ-1X>2εX-ε2J (22)
By formula 19)~(22), formula (17), (18) can be obtained by mending lemma using Schur；
Theorem 2 is for giving scalar 0<hd<hD,0≤μ<1 and ε, if in the presence of matrix L=L of appropriate dimensionT>0, if in the presence of
Positive definite matrix AndAnd the matrix of any appropriate dimension Set up following LMI：
Ξ13=A1X,
Ξ35=0, Ξ36=0,
Ξ56=0,
And meet hα=hi+1-hi=(hD-hd)/N,
SoIt is asymptotically stable and controls the feedback oscillator to be：K=VX-1。
Due to there is nonlinear terms XJ in formula (18)~(20)-1X, it is impossible to solve controller gain using convex optimized algorithm,
Therefore nonlinear terms are handled using Linearization Method.
To verify the validity and smaller conservative of result above, simulation analysis are carried out to following 2 examples：
Example 1. considers time-delay/time-varying system (1), and its coefficient matrix is：
As μ=0.5, time lag lower bound h is givend, and as the i=N in theorem 1, ask for can guarantee that system Asymptotic Stability
The maximum time lag upper bound, it is credible for increase, by theorem 1 of the present invention and document a：Improved delay-range-
Dependent stability criteria for linear systems with time-varying delays and text
Offer b：Robust stability criteria for uncertain linear systems with interval
Time-varying delay are compared, and Fig. 1 gives hdWhen taking different value, the value in the time lag upper bound can from Fig. 1
Go out, compared with other systems with time lag split plot design, the variable number of theorem 1 is less, calculate simple；By with document a with text
B is offered to be compared, as μ=0.5, system time lags lower bound hdWhen=1, the time lag Greatest lower bound drawn, its Literature b is
2.3912, and apply theorem 1, what is obtained as N=2 is 2.4526, and what is obtained as N=4 is 2.9413, is when what N=9 was obtained
3.5353；The conservative of theorem 1 is smaller as shown in Figure 1.
Example 2. considers time-delay/time-varying system (1), and its coefficient matrix is：
When μ is unknown, time lag lower bound h is givend, and as the i=N in theorem 1, ask for can guarantee that system is asymptotically stable
The maximum time lag upper bound, by this theorem and document c：Further improvement on delay-range-dependent
Stability results for linear systems with interval time-varying delays and document
d：A novel approach to delay–fractional–dependent stability criterion for
Linear systems with interval delay are contrasted, such as Fig. 2；Work as hdWhen=0, μ values are given, and when theorem 1
In i=N when, ask for can guarantee that the asymptotically stable maximum time lag upper bound of system, by this theorem and document e：New
stability criteria for continuous time systems with interval time-varying
Delay and document f：Improved delay-range-dependent stability criteria for linear
Systems with interval time-varying delays are contrasted, such as Fig. 3；
Understand that the conservative of the system is smaller by above simulation analysis, this is due to be constructed in the present invention
Lyapunov-krasovskii functionals are also not only multiple etc. using delay interval is decomposed fully using triple product subitem is added
Away from minizone method to time-delay/time-varying system carry out stability analysis.
Next B=[0 is set；1], the system initial state of example 1 is x0=[1-1]T, example 1 is obtained into system applied to theorem 2
Feedback control gain be K=[- 0.6731-1.6609]；And emulation draws closed-loop system zero input state response curve such as Fig. 4
It is shown：
When ε takes different value, the time that system mode response tends towards stability is listed in Fig. 4, it can be seen that using present invention control
Device design method, takes different ε values, and the time tended towards stability is also different, and as ε=0.1, system mode response curve becomes in 4.9s
In stable；As ε=1, system mode response curve tends towards stability in 3.9s；During ε=4, system mode response curve is in 2.8s
Tend towards stability；And work as ε>System mode response curve still tends towards stability in 2.8s when 4；And document Improved robust
stabilization method for linear systems with interval time-varying input
Delays by using Wirtinger inequality [J] and document Novel delay-partitioning
stabilization approach for networked control system via Wirtinger-based
The time that system mode response curve tends towards stability in inequalities [J] is respectively 7s and 28s or so, this demonstrate that this hair
Bright institute's extracting method is correct and effective.
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the invention, all essences in the present invention
God is with principle, and any modification, equivalent substitution and improvements made etc. should be included in the scope of the protection.
Claims (2)
1. a kind of Design of Feedback Controller method of the time-delay/time-varying system based on convex combination method, it is characterised in that：The controller
Design process is as follows：
A. varying-delay system model is set up,
Wherein：x(t)∈RnFor the state vector of system；A、A1It is known suitable dimension matrix；It is continuous initial function；
Time_varying delay h (t) is met：0≤hd≤h(t)≤hD,Wherein, 0≤hd≤hDIt is constant with μ >=0；
B. delay interval is split, delay interval is divided into N parts of equidistant minizones, N>0, and be integer, then hi(i=1,
2 ..., N+1) meet：hd=h1<h2<…<hN<hN+1=hD, wherein the length of each minizone is hα=hi+1-hi=(hD-
hd)/N；
C. stability analysis is carried out to system, during analysis, introduces lemma 1, lemma 2, lemma 3, lemma 4；
D. Lyapunov-Krasovskii functionals are constructed, derivation is carried out to functional, lemma 1, lemma 2 in step b, drawn
Reason 3, lemma 4 finally give stability of a system condition,
As h (t) ∈ [hi,hi+1] when, present invention construction Lyapunov-Krasovskii functionals are：
Wherein,
Vi1(t)=ξT(t) P ξ (t),
Wherein,
The derivative that can obtain functional V (t) along the track of system (1) is：
Following formula can be obtained with lemma 1 and lemma 2 to set up：
Wherein
Obtained with lemma 3
Wherein, δT(t)=[xT(t-hi) xT(t-h(t)) xT(t-hi+1)],
Order
Then
Wherein,
Therefore obtained with reference to inequality (9)~(13)
For h (t) ∈ [hi,hi+1], according to lemma 4, if following inequality is set up：
Then have：
Then haveFrom Lyapunov Theory of Stability, system (1) is asymptotically stable；
E. condition-theorem 1 of the stability of a system is drawn：For giving scalar 0<hd<hDWith 0≤μ<1, system be it is asymptotically stable,
If there is positive definite matrixQi>0 (i=1,2,3), Ri>0 (i=1,2) and Ui>0 (i=1,2)；And
The matrix T of any appropriate dimensionj, Yj(j=1,2,3), set up following MATRIX INEQUALITIES：
Wherein,
Ξ12=-P12+P13,Ξ13=P11A1,Ξ14=-P13,
Ξ15=ATP12+P22+hiU1,Ξ16=ATP13+P23+hαU2,Ξ22=-Q1-2R1+Y1+Y1 T, Ξ26=-P23+P33, Ξ46=-P33,Ξ56=0, Ξ66=-U2,
Ac=[the A of A 01 0 0 0],
And meet hα=hi+1-hi=(hD-hd)/N,
F. the design of subduer, the Stabilization of the controller of system is considered in step e, controller is made on the basis of theorem 1
Form be：
U (t)=Kx (t) (14)
Its closed-loop system is：
(14) are brought into system (15), can be obtained
J. on the basis of theorem 1, existence consition-theorem 2 of subduer is provided, A is replaced with A+BK, and in formula (2), (3)
The two ends premultiplication right side multiplies diag { X, X, X, X, X, X, X, X } and its transposition, wherein makingDefinition
I.e.XP11A1X=A1X, can be obtained
Wherein
It is known for any inequality J>0, inequality (21) is set up：
Due to J>0, have to Arbitrary Matrix X
(X-J)J-1(X-J)>0
For any given ε
(ε-1X-J)(ε-2J)-1(ε-1X-J)
=XJ-1X-2εX+ε2J>0
I.e.：
XJ-1X>2εX-ε2J (22)
By formula (19)~(22), formula (17), (18) can be obtained by mending lemma using Schur；
Theorem 2 is for giving scalar 0<hd<hD,0≤μ<1 and ε, if in the presence of matrix L=L of appropriate dimensionT>0, if there is positive definite
MatrixAndAnd the matrix of any appropriate dimension Set up following LMI：
Ξ13=A1X,
Ξ35=0, Ξ36=0,
Ξ56=0,
And meet hα=hi+1-hi=(hD-hd)/N,
SoIt is asymptotically stable and controls the feedback oscillator to be：K=VX-1。
2. the Design of Feedback Controller method of the time-delay/time-varying system according to claim 1 based on convex combination method, it is special
Levy and be：Due to there is nonlinear terms XJ in formula (18)~(20)-1X, it is impossible to solve controller gain using convex optimized algorithm, because
This is handled nonlinear terms using Linearization Method.
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| CN108646566A (en) * | 2018-06-12 | 2018-10-12 | 南京理工大学 | A kind of slave torque feedback control method of the bilateral teleoperation system based on sampling |
| CN109591804A (en) * | 2018-11-22 | 2019-04-09 | 湖南大学 | Consider the vehicle platoon stability control method of communication delay |
| CN112186780A (en) * | 2020-09-24 | 2021-01-05 | 河南理工大学 | Method for designing ballast of uncertain power system based on interval time lag influence |
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