CN105589340B - A kind of stability judging method of uncertain network Systems with Multiple Time-Delays - Google Patents
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Abstract
The invention discloses a kind of stability judging methods of uncertain network Systems with Multiple Time-Delays, establish closed network multiple time delay control system model, the uncertainty of closed network multiple time delay control system is mapped in convex polyhedron parameter space, convex polyhedron uncertain network multiple time delay control system model is obtained；Lyapunov functions based on construction, using free-form curve and surface method, the Robust Delay-Dependence Stability of the convex polyhedron uncertain network multiple time delay control system judged obtains time lag correlation robust stability adequate condition.The invention avoids identical Lyapunov functions are used to all convex polyhedron vertex in Instable Space, reduce the conservative of robust stability adequate condition.
Description
Technical field
The invention belongs to automatic control technology fields, are related to a kind of judgement of stability of uncertain network Systems with Multiple Time-Delays
Method.
Background technology
Network control system needs that data volume that is shared and exchanging is increasing, type becomes increasingly complex, in practical work
In journey field, due to the complexity of the limitation and objective things itself of human cognitive ability and method, system is hardly resulted in
It determines or accurately describes, so as to cause a large amount of uncertain presence；Simultaneously with the continuous expansion of modern control system scale
Greatly, complexity increases sharply, system structure ambiguity, non-modeling parameters ambiguity, external environment unpredictability, external disturbance
Randomness etc. further increase complexity and the source of system uncertain factor.Peace of the modern scientist field to complication system
The requirement of full property and reliability is higher and higher, therefore considers the uncertain factor of system, studies the robust control of uncertain system
Strategy, to remain to keep preferable performance that there is important theoretical meaning when ensureing that system dynamic characteristic changes in certain perturbation range
Justice and more practical value.
In addition, with the development of mechanics of communication and complex network, large scale network networked control systems are at low cost with its, connect
Flexibly, it is easily installed extension, safeguards that the advantages that simple, function is complicated has become complex large system objective demand.But due to net
The characteristics of network is to communication media time-sharing multiplex, when multiple nodes by network carry out data interaction when, usually occur data collision,
Phenomena such as information occlusion, disconnecting, multiframe transmission, thus inevitably there is the non real-time transmission of information, therefore in addition to
Except the uncertainty of system, Time Delay is the another main problem faced in network control system research, it is often
Lead to the major reason of system performance degradation.
In the uncertainty of the system of description frequently with norm-bounded ambiguous model and convex polyhedron；Wherein, norm has
Boundary's uncertainty description method is to limit probabilistic Maximum tolerance based on small gain theorem, has certain office
It is sex-limited；The stability analysis of convex polyhedron uncertain system and robust Controller Design are mostly based on Lyapunov Quadratic Stabilities
It is theoretical.But since Lyapunov Quadratic Stability concepts are identical to being used all convex polyhedron vertex in Instable Space
Lyapunov functions, cause the conservative of result larger.With parameter rely on Lyapunov stability thoughts proposition and
The development of LMI methods is perfect, and gradually this thinking is used for in the analysis and design of convex polyhedron uncertain control systems.
At present for the STUDY ON ROBUST CONTROL of the discrete system with crowned design, the side of being mainly the following
Method：(1) announcement effects method constructs N number of linear matrix inequality to N convex polyhedrons vertex and obtains discrete convex polyhedron
The criterion for robust stability of uncertain system converts Infinite-dimensional problem to finite dimension problem.The method requires convex polyhedron
Each vertex uses a public Lypaunov matrix, therefore has larger conservative.(2) added martix constructs
Lyapunov function methods release matrix P (a) in Lyapunov functions by introducing the added martix variable such as right of freedom tactical deployment of troops
With sytem matrix A (a) product terms, to reduce conservative existing for system Robust Stability.(3) Algebraic Structure constructs
Then Lyapunov function methods are closed by exploring the Algebraic Structure for the Lyapunov stable conditions that uncertain system parameter relies on
And similar terms, ensure the robust stability of uncertain system to require all positive definite of each single item in multinomial.Based on this thinking institute
The result of acquirement is than the result conservative smaller based on Lyapunov Quadratic Stability theories, but most of achievements in research at present
Both for continuous system, the research of discrete system is also seldom related to.
Invention content
For the above-mentioned prior art the problem of or defect, the object of the present invention is to provide a kind of uncertain nets
The stability judging method of network Systems with Multiple Time-Delays.
To achieve the goals above, the present invention adopts the following technical scheme that：
A kind of stability judging method of uncertain network Systems with Multiple Time-Delays, specifically includes following steps：
Step 1, closed network multiple time delay control system model is established；
Step 2, the uncertainty of closed network multiple time delay control system is mapped in convex polyhedron parameter space, is obtained
To convex polyhedron uncertain network multiple time delay control system model；
Step 3, construction includes the Lyapunov functions of multiple time delay information；
Step 4, using free-form curve and surface method, the convex polyhedron uncertain network multiple time delay that judgment step 2 obtains controls
The Robust Delay-Dependence Stability of system obtains time lag correlation robust stability adequate condition；If meeting time lag correlation robust stability
Adequate condition, then uncertain network Systems with Multiple Time-Delays is stable, if not satisfied, then uncertain network Systems with Multiple Time-Delays is
Unstable.
Specifically, the closed network multiple time delay control system model in the step 1 is：
Wherein,
Positive integerFor Time-varying time-delays；Ap,Bp,CpIt is the real constant coefficient matrix of suitable dimension, k is current sampling
Moment, x (k) ∈ RnFor the system state variables of augmentation, Ac,Bc,Cc,DcIt is the real constant coefficient matrix of suitable dimension.
Specifically, the convex polyhedron uncertain network multiple time delay control system model in the step 2 is：
Wherein, A0α,A1α…,A3αIt is sytem matrix group.
Specifically, constructed in the step 3 include multiple time delay information Lyapunov functions it is as follows：
V (x (k))=V1+V2+V3+V4
Wherein,
V1=xT(k)Pαx(k)
In formula, Pα,R1α,R2α,R3α∈Rn×nTo rely on parameter alphai(t) symmetric positive definite matrix, 0≤αi(t)≤1, i=1,
2 ..., n is Bounded Real scalar function.
Specifically, the time lag correlation robust of the convex polyhedron uncertain network multiple time delay control system in the step 4
Stable adequate condition is：
For Time-varying time-delaysThere are symmetric positive definite matrix Pα=Pα T>0, Qα=Qα T
>0,And the matrix N of arbitrary appropriate dimensioniα,Miα,Siα(i=1,2), Xα,Yα,Zα>=0 energy
Enough set up following one group of linear matrix inequality：
Λ23α=dmaxΞ23α
Λ24α=dmaxΞ24α
Λ34α=dmaxΞ34α
Ξijα=Xijα+Yijα+Zijα, i, j=1,2,3
Compared with prior art, the present invention has the following technical effects：
1, the present invention is directed to data/address bus extensive use in the controls, introduces unreliable logical caused by transmission time lag
Letter factor and from internal system and external uncertain factor, completes system Multi Time Lag and uncertain from physical space
To the mapping of mathematical space, establishes closed loop and do not know network multiple time delay control system model.
2, the present invention is described and models in three dimensions to the uncertainty of control system, by systematic uncertainty
It is mapped in convex polyhedron Instable Space, is compared with uncertain two-dimensional space description, it is dynamic that description method is more in line with system
Step response can flexibly describe the uncertainty from exterior and system self structure, characterising parameter uncertainty institute
Obtained system has less conservative to the description of probabilistic Maximum tolerance ratio two-dimensional space.
3, the present invention uses Lyapunov function methods, and dynamical system Multi Time Lag information is included into Lyapunov functions
In.Right of freedom battle array is based on according to Lyapunov Theory of Stability by constructing an aobvious Lyapunov function containing time lag
Method analyzes the stability of network control system, avoids and is used all convex polyhedron vertex in Instable Space
Identical Lyapunov functions, reduce the conservative of robust stability adequate condition.
4, in Robust Delay-Dependence Stability determination method, the Lyapunov functions containing time lag are shown by construction, and draw
Enter added martix variable, counteracted the quadratic form integral term occurred in Lyapunov function difference, reduces time lag correlation Shandong
Stick stablizes the conservative of adequate condition.
Description of the drawings
Fig. 1 is closed network networked control systems Time-Delay model；
Fig. 2 is to work as dmaxConvex polyhedron uncertain network networked control systems condition responsive when=9；
Fig. 3 is to work as dmaxConvex polyhedron uncertain network networked control systems condition responsive when=4；
Fig. 4 is to work as dmaxCondition responsive of the deterministic network networked control systems on convex polyhedron vertex 1 when=9；
Fig. 5 is to work as dmaxCondition responsive of the deterministic network networked control systems on convex polyhedron vertex 2 when=9；
Fig. 6 is to work as dmaxCondition responsive of the deterministic network networked control systems on convex polyhedron vertex 3 when=9；
Explanation and illustration in further detail is done to the solution of the present invention with reference to the accompanying drawings and detailed description.
Specific implementation mode
The time lag characteristic of closed network networked control systems is analyzed, is divided according to the position of time lag distribution, can be incited somebody to action
Time lag is divided into three parts：Sampling transmission time lag τSC, control calculating time lag τCAnd control action time lag τCA, generation position
Distribution is as shown in Figure 1, existing analyze above-mentioned three parts time lag：
(1) sampling transmission time lag τSC, i.e. the transmission time lag of sensor to controller.In network control system, it will pass
This period that sensor samples sampled data arrival controller is known as " sensor-controller time lag ", is denoted asK is to work as
Preceding sampling instant, thenIt is defined asWhereinWithIndicate that controller starts operation control signal respectively
At the time of moment and sensor start sampling system output.
(2) control calculates time lag τC, the calculating time lag of controller execution operation generation.It, will in network control system
Controller starts calculating to calculating completion and obtains control signal this period to be known as " controller calculating time lag ", is denoted asK is
Current sampling instant, thenIt is defined asWhereinWithIt is that controller calculating is completed to obtain control letter respectively
Number at the time of and start calculate at the time of.
(3) control action time lag τCA, the transmission time lag of controller to actuator.In network control system, it will control
This period for being performed device reception from the time of device sends control signal to the signal is known as " controller-actuator time lag ",
It is denoted asK is current sampling instant, thenIt is defined asWhereinWithIt is that actuator receives respectively
At the time of control signal starts action and at the time of controller operation completes to obtain control signal.
Typically time-varying, withWithIt compares, its value is negligible, and change rate is almost
Zero.When carrying out the analysis and synthesis of network control system, can start with from hardware aspect is influenced to be reduced to minimum, institute
It is exactly with what the Internet-Game Addiction of ordinary circumstance controlling bus introducing was often referred toWithBoth time lags.
The stability judging method of the uncertain network Systems with Multiple Time-Delays of the present invention, specifically includes following steps：
Step 1, closed network multiple time delay control system model is established, does not consider that system is not true during this model foundation
It is qualitative.
Based on the structure chart of the closed network networked control systems in Fig. 1, the discrete state equations for obtaining controlled device are：
Wherein xp∈RnIndicate the state vector of controlled device, up∈RmIndicate input vector, yp∈RpIndicate output vector,
K is current sampling instant；N, m, p indicate the dimension of controlled device in control system, actuator and sensor respectively；Ap,Bp,
CpIt is the real constant coefficient matrix of suitable dimension.
The discrete state equations of controller can be indicated by following equation：
Wherein xc∈RnIndicate the state vector of controller, uc∈RpIndicate the input vector of controller, yc∈RmIndicate control
The output vector of device processed；N, p, m indicate the dimension of controller, sensor and actuator； Ac,Bc,Cc,DcIt is the real often system of suitable dimension
Matrix number.
As shown in Figure 1, the collected controlled device output vector y by analog-to-digital conversion of sensorpBy transmission channel
After S-C, sampling transmission time lag τ attachedSCInformation, as input vector ucInto controller；And pass through controller operation
Dominant vector ycAs output, after transmission channel C-A, control action time lag τ attachedCAInformation, as input quantity, and
Enter actuator after digital-to-analogue conversion.
DefinitionIndicate the sampling transmission time lag τ between k-th of sampling period S-CSC,Indicate k-th of sampling period C-A
Between control action time lag τCA, to which the time lag in network can be described with following relationships：
Wherein,dmin,dmaxBe respectively minimum transfer time lag between node and
Maximum transmitted time lag.
Introduce the system state variables vector of augmentationNet then based on time lag system theory
Network control system state equation can be expressed as：
Definition
Then closed network multiple time delay control system can be write as：
Wherein
Can be seen that the closed network multiple time delay control system that formula (6) indicates from system above model is a band
Constant autonomous system, can study it using time lag system theory when having a LINEAR CONTINUOUS of multiple time delay.
Step 2, the closed network multiple time delay control system model obtained based on step 1 further considers that system is not true
It is qualitative, the uncertainty of closed network multiple time delay control system is mapped in convex polyhedron parameter space, convex multi-panel is obtained
Body uncertain network multiple time delay control system model.
Crowned design can be used following model to indicate：
Wherein, Ai∈Rn×mAnd Adi∈Rn×mFor known real matrix, uncertain parameter A and AdIt is bounded, belongs to limited
The convex combination of known matrix, A and AdIt can be expressed as：
S:={ A1,A2,…,Am,Ad1,Ad2,…,Adn}∈Θ (8)
Wherein0≤αi(t)≤1, i=1,2 ..., n is the real scalar function of bounded, and
And meet：
Due to uncertain parameter αi(t) it may be time-varying, it is also necessary to assuming that its change rate is bounded, that is, meet：
In formulaIndicate uncertain parameter αi(t) change rate.
From geometrically seeing uncertain parameter matrix A and AdIt is with A respectivelyiAnd AdiFor the convex polyhedron on vertex；θi, i=
1 ..., the known scalar that m is to determine.
Consider the convex polyhedron uncertain network multiple time delay control system model with multiple-state delay as follows：
Wherein x (k) ∈ RnFor the system state variables of augmentation, positive integerFor Time-varying time-delays.Sytem matrix
Group (A0α,A1α,…,A3α) description system uncertainty, be bounded and belong to the convex combination collection of limited known matrix：
Wherein A0i,A1i,…,A3i∈Rn×mFor known real matrix.As can be seen from the above equation, any matrix for belonging to set Ψ
(A0,A1,…,A3)αIt can be by N number of vertex matrix (A in set Ψ0i,A1i,…,A3i), the convex combination table of i=1 ..., N
Show.0≤α in above formulai(t)≤1, i=1,2 ..., n is Bounded Real scalar function, is met simultaneously：
Due to uncertain parameter αi(t) it may be time-varying, it is also necessary to assuming that its change rate is bounded, that is, meet：
From geometrically seeing that uncertain parameter space is with A0,A1,A2,A3For the convex polyhedron on vertex；θi, i=1 ..., m are
Determining known scalar.
Crowned design describe method can by define the arbitrary convex polyhedron of uncertain parameter spatial description,
In practical modeling process, internal system and external most uncertainties can use convex polyhedron uncertain parameter model convenient
Ground describes.Any norm-bounded uncertain parameter can use the convex polyhedron uncertain parameter model on respective number vertex in fact
Arbitrarily approached.
Step 3, the convex polyhedron uncertain network multiple time delay control system model obtained based on step 2, construction include
There are the Lyapunov functions of multiple time delay information.
The Lyapunov functions of convex polyhedron uncertain network multiple time delay control system shown in constructive formula (11)：
V (x (k))=V1+V2+V3+V4 (15)
Wherein,
In formula, Pα,R1α,R2α,R3α∈Rn×nTo rely on parameter alphai(t) symmetric positive definite matrix.
For arbitrary primary condition, along the arbitrary trajectory of system, the single order forward difference of Lyapunov functions is：
Wherein,
Formula (18) is managed using Schur complement fixed, it is known that Λ<0, it is equivalent to：
Step 4, the Lyapunov functions constructed based on step 3, using free-form curve and surface method, judgment step 2 obtains convex
The Robust Delay-Dependence Stability of polyhedron uncertain network multiple time delay control system, it is abundant to obtain time lag correlation robust stability
Condition.If meeting time lag correlation robust stability adequate condition, uncertain network Systems with Multiple Time-Delays is stable, if discontented
Foot, then uncertain network Systems with Multiple Time-Delays is unstable.
According to whether related to time lag information, it is based on the obtained determination of stability side of time lag system method for analyzing stability
Method and adequate condition can be divided into two classes：One kind is On Delay-Dependent Stability condition, and one kind is Delay-Independent Stability condition.Its
Middle Delay-Independent Stability condition does not have any restrictions to time lag, does not consider the size of time lag.Time lag don't-care condition is for arbitrary
Time lag is all set up.Due to requiring no knowledge about the relevant information of system time lags, Delay-Independent Stability method can be analyzed and be handled
The unknown time lag of system.Under normal conditions, the unrelated conclusion of time lag is fairly simple and is easily verified that.However for Small Time Lag or
The case where person's time lag bounded, the unrelated stable condition of time lag necessarily bring larger conservative.Correspondingly, On Delay-Dependent Stability
Method is then by time lag information in view of in the analysis of system stability, embodying influence of the time lag size to system stability.It is logical
Often, in the case of time lag bounded or smaller time lag, On Delay-Dependent Stability condition has more than Delay-Independent Stability condition
Low conservative.Further, according to the information for whether including time lag derivative in stability condition, On Delay-Dependent Stability condition
It can be divided into that time lag is related and time lag derivative correlation is related to time lag and unrelated two kinds of time lag derivative again.Since time lag is related and time lag
Derivative correlated condition contains more time lag informations, thus it has smaller conservative compared with the latter.
In order to further decrease conservative, the convex polyhedron obtained the present invention is based on right of freedom tactical deployment of troops judgment step 2 is not true
The Robust Delay-Dependence Stability for determining networking Systems with Multiple Time-Delays model obtains time lag correlation robust stability adequate condition.
Definition status x (l) forward differences first are：
Y (l)=x (l+1)-x (l) (20)
Then there is the establishment of following formula equation：
The Lyapunov functions of convex polyhedron uncertain network Systems with Multiple Time-Delays shown in constructive formula (11)：
V (k)=V1(k)+V2(k)+V3(k)+V4(k)+V5(k) (23)
Wherein Pα=Pα T>0, Qα=Qα T>0,For symmetric positive definite matrix undetermined, definition
Lyapunov function single order forward difference Δ V (k)=V (k+1)-V (k), then have：
ΔV1(k)=2xT(k)Pαy(k)+yT(k)Pαy(k) (25)
Meanwhile using formula (22), for Arbitrary Matrix Niα,Miα,Siα(i=1,2,3) has following zero equation to set up：
On the other hand, for the matrix X of any appropriate dimension, Y, Z >=0 has following zero equation to set up：
WhereinBy zero equation (30) and
(31) the left side is added to Δ V (k), then the single order forward difference Δ V (k) of Lyapunov functions is further transformed to：
Wherein,According to Lyapunov
Theory of Stability, the time lag correlation robust stability of convex polyhedron uncertain network multiple time delay control system shown in formula (11)
Adequate condition be Δ V (k)<0 sets up.
For Time-varying time-delays If there is symmetric positive definite matrix Pα=Pα T>0, Qα
=Qα T>0,And the matrix N of arbitrary appropriate dimensioniα,Miα,Siα(i=1,2), Xα,Yα,Zα≥
0 makes following linear matrix inequality set up, then Δ V (k)<0 sets up：
Wherein,
Λ23α=dmaxΞ23α
Λ24α=dmaxΞ24α
Λ34α=dmaxΞ34α
Ξijα=Xija+Yijα+Zijα, i, j=1,2,3
In other words, the time lag correlation robust of convex polyhedron uncertain network multiple time delay control system shown in formula (11)
Stable adequate condition is：For Time-varying time-delays There are symmetric positive definite matrix Pα=Pα T＞
0, Qα=Qα T＞ 0,And the matrix N of arbitrary appropriate dimensioniα, Miα, Siα(i=1,2), Xα, Yα,
Zα>=0 can be such that above-mentioned one group of linear matrix inequality sets up.
Embodiment：
Using the stability judging method of uncertain network Systems with Multiple Time-Delays proposed by the present invention, in given network time service
Minimum time lag boundary dminWhen, find the maximum time lag boundary of the network time service of convex polyhedron uncertain network networked control systems
dmaxSo that whenWhen, closed network networked control systems are robust asymptotic stabilities.And it is directed to
Network control system three vertex of convex polyhedron determinacy case, when providing minimal network time lag boundary dminWhen, it obtains
The maximum network time lag boundary d on three vertex of convex polyhedronmax, and the result of several situations is compared.Concrete methods of realizing
It is as follows：
Step 1：Controlled device is with convex polyhedron uncertain network multiple time delay control system, state-space model
For：
Wherein, x (k) ∈ RnFor the system state variables of augmentation, positive integerFor Time-varying time-delays.Matrix (A0,
A1,…,A3) it is uncertain matrix group, belong to the uncertain set of convex polyhedron：
Here α=3 are set, i.e. set Ψ is 3 vertex matrix (A0i,A1i,…,A3i), i=1,2,3 convex combination.Wherein：
Step 2, it is solved using the tool boxes Matlab LMI, works as dminIt, can be in the hope of convex polyhedron uncertain network when=6
Change the maximum time lag boundary d of Systems with Multiple Time-Delaysmax=9, and the solution of LMIs is：
Step 3, according to different minimum time lag boundary dmin, convex polyhedron uncertain network multiple time delay control system and
It is in the determinacy case on three vertex of convex polyhedron, maximum time lag boundary dmaxSimulation result it is as shown in table 1：
Table 1 gives dminIn the case of, calculate dmax
From table 1 it follows that more based on the time-varying with crowned design that free-form curve and surface method obtains
Lagging network networked control systems time lag correlation asymptotically stability condition is with deterministic network networked control systems on three tops of convex polyhedron
The time lag correlation asymptotically stability condition obtained based on inequality method when point has higher conservative, ensures uncertain network control
The maximum allowable time lag that system processed is stablized is smaller than the maximum allowable time lag of deterministic network networked control systems.
Step 4, it is x (0)=[5 8] to give dynamical system original stateTWhen and step 2 in Matlab LMI tools
Case solve as a result, simulate the closed network networked control systems condition responsive of Unequal time lag situation with Matlab, if Fig. 2 is to scheming
Shown in 6.
Fig. 2 and Fig. 3 is shape of the convex polyhedron uncertain network multiple time delay control system under different maximum time lag boundary
State response curve, it can be seen that when maximum allowable delay bound increases, the regulating time that system is stablized is consequently increased, it is seen that
For the convex polyhedron uncertain network networked control systems, the time-delay response time of system.Fig. 2 is the maximum acquired
Time lag boundary is dmaxCondition responsive curve in the case of=9, when time lag meetsWhen, system is still
It is asymptotically stability, shows the validity of this method.
Being to determine property of Fig. 4, Fig. 5 and Fig. 6 network control system is respectively at three vertex of convex polyhedron, maximum time lag side
Boundary dmaxCondition responsive curve when=9, with convex polyhedron uncertain network networked control systems in maximum time lag boundary dmaxWhen=9
Condition responsive curve graph 2 compare, it can be seen that in identical time lag, convex polyhedron uncertain network networked control systems
The system stable regulation time will be grown, and this also illustrates the uncertainties of system to affect response time and the stability of system
Energy.
Claims (1)
1. a kind of stability judging method of uncertain network Systems with Multiple Time-Delays, which is characterized in that specifically include following steps：
Step 1, closed network multiple time delay control system model is established；
Step 2, the uncertainty of closed network multiple time delay control system is mapped in convex polyhedron parameter space, is obtained convex
Polyhedron uncertain network multiple time delay control system model；
Step 3, construction includes the Lyapunov functions of multiple time delay information；
Step 4, using free-form curve and surface method, convex polyhedron uncertain network multiple time delay control system that judgment step 2 obtains
Robust Delay-Dependence Stability, obtain time lag correlation robust stability adequate condition；If it is abundant to meet time lag correlation robust stability
Condition, then uncertain network Systems with Multiple Time-Delays is stable, if not satisfied, then uncertain network Systems with Multiple Time-Delays is unstable
Fixed；
Closed network multiple time delay control system model in the step 1 is：
Wherein,
Positive integerFor Time-varying time-delays；Ap,Bp,CpIt is the real constant coefficient matrix of suitable dimension, k is current sampling instant,For the system state variables of augmentation, Ac,Bc,Cc,DcIt is the real constant coefficient matrix of suitable dimension；
Convex polyhedron uncertain network multiple time delay control system model in the step 2 is：
Wherein, A0α,A1α…,A3αIt is sytem matrix group；
Constructed in the step 3 include multiple time delay information Lyapunov functions it is as follows：
V (x (k))=V1+V2+V3+V4
Wherein,
V1=xT(k)Pαx(k)
In formula, Pα,R1α,R2α,R3α∈Rn×nTo rely on parameter alphai(t) symmetric positive definite matrix, 0≤αi(t)≤1, i=1,2 ...,
N is Bounded Real scalar function；T is the time variable of uncertain network Systems with Multiple Time-Delays, αi(t) it is time dependent uncertain
Parameter；
The abundant item of the time lag correlation robust stability of convex polyhedron uncertain network multiple time delay control system in the step 4
Part is：
For Time-varying time-delays There are symmetric positive definite matrix Pα=Pα T>0, Qα=Qα T>0,And the matrix N of arbitrary appropriate dimensioniα,Miα,Siα(i=1,2), Xα,Yα,Zα>=0 can make
Following one group of linear matrix inequality are set up：
Λ23α=dmaxΞ23α
Λ24α=dmaxΞ24a
Λ34α=dmaxΞ34α
Ξijα=Xijα+Yijα+Zijα, i, j=1,2,3
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