CN108732926A - Networked system method for estimating state based on insufficient information - Google Patents

Networked system method for estimating state based on insufficient information Download PDF

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CN108732926A
CN108732926A CN201810570473.3A CN201810570473A CN108732926A CN 108732926 A CN108732926 A CN 108732926A CN 201810570473 A CN201810570473 A CN 201810570473A CN 108732926 A CN108732926 A CN 108732926A
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matrix
state
estimator
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networked system
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董宏丽
侯男
李雪融
韩非
张勇
路阳
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Northeast Petroleum University
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    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/0265—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion
    • G05B13/027—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion using neural networks only
    • 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
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

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Abstract

The networked system method for estimating state based on insufficient information of the present invention, the networked system being applicable under different situations, it considers wherein non-linear, time lag, variation topology, coupling in random, estimator executes influence of phenomena such as change in gain and incomplete metrical information to state estimation performance, the dynamic model and state estimator model of structure system, it calculates evaluated error and carries out augmentation with original system state, provide the performance constraints that estimator to be designed need to meet, then obtaining can be so that augmented system meets the adequate condition of performance requirement, the LMI forms of the unknown gain parameter of state estimator can be solved by again converting the condition to.Compared to existing state estimator design method, the present invention can have generality suitable for the pervasive network system reality.

Description

Networked system method for estimating state based on insufficient information
Technical field
The present invention relates to network control technology fields, and in particular to a kind of networked system state based on insufficient information Method of estimation.
Background technology
Network control system is made of the spatially distributed control system of closed loop communication network.With network technology Fast development, the application field of networked system is more and more extensive, and control object is increasingly complicated, and control system is towards more bonus point Dispersion, intelligentized direction are developed.The performance of network is continuously improved, network technology also relative maturity, network can make user into Row long-range data transmission and interactive operation, and its cost, routing complexity, maintenance difficulties etc. substantially reduce.Due to network The various advantages of technology, largely research and application are begun to focus on is used for long-range Industry Control and industry certainly by data network In the fields such as dynamicization, and advanced aircraft, spaceship, automobile, robot, building intelligent, tele-medicine, remote teaching and Prodigious achievement is achieved in the applications of complex controls process such as experiment, military commanding and the manufacturing.But the intervention of network makes The scale and complexity of control system dramatically increase, and network inducing phenomena also inevitably occurs, such as communication delay, parameter Uncertain, sensor saturation etc., these phenomenons are often common in practical engineering application.So the research of network control system Have great importance.
State estimation problem is always one of the hot research problem of control theory and field of signal processing, it is therefore an objective to be utilized The measurement for being estimated system exports the internal state of reasonably estimating system.This is because in practical applications, about due to physics The reasons such as the measurement cost of beam, technology restriction or costliness, what system mode can not always directly obtain.Therefore, networking The state estimation problem of system receives extensive research and pays attention in recent years.It is proposed can be suitable for it is various in the case of networking The state estimator design method of system is just provided with important practical significance.
Invention content
For the defects in the prior art, the present invention provides a kind of networked system state estimation based on insufficient information Method, this method can be suitably used for pervasive network system, have generality.
Networked system method for estimating state provided by the invention based on insufficient information, specifically includes following steps：
Step 1：The dynamic model of networked system to be studied is built, includes networked system in the dynamic model State sets the condition that function meets in dynamic model；
Step 2：The state estimator model containing unknown parameter is built, according to state estimator model to the dynamic Model carries out state estimation, obtains state estimation error, carries out augmentation to the system mode and evaluated error, the state of obtaining is estimated Count augmented system model；
Step 3：The constraints that setting evaluated error need to meet obtains ensuring that state estimation augmented system model meets The adequate condition of performance requirement；
Step 4：The unknown parameter in state estimator is calculated according to adequate condition existing for state estimator.
Beneficial effects of the present invention：
The networked system method for estimating state based on insufficient information of the present invention, the net being applicable under different situations Network system, wherein consider non-linear, time lag, variation it is topological, it is random in coupling, estimator execute change in gain and endless Influence of the phenomena such as full metrical information (output nonlinear, quantization, the random sensor that occurs are saturated) to state estimation performance, structure The dynamic model and state estimator model of system calculate evaluated error and carry out augmentation with original system state, provide to be designed The performance constraints that estimator need to meet, then obtaining can be so that augmented system meets the adequate condition of performance requirement, then will The condition is converted into the LMI forms that can solve the unknown gain parameter of state estimator.Compared to existing state estimator design Method, the present invention can have generality suitable for the pervasive network system reality.
Description of the drawings
It, below will be to specific in order to illustrate more clearly of the specific embodiment of the invention or technical solution in the prior art Embodiment or attached drawing needed to be used in the description of the prior art are briefly described.In all the appended drawings, similar element Or part is generally identified by similar reference numeral.In attached drawing, each element or part might not be drawn according to actual ratio.
Fig. 1 shows a kind of stream of networked system method for estimating state based on insufficient information provided by the present invention Cheng Tu；
Fig. 2 shows the system modes of the embodiment of the present invention 1 to switch figure；
Fig. 3 shows that the embodiment of the present invention 1 emulates obtained evaluated error figure；
Fig. 4 shows that the embodiment of the present invention 1 emulates obtained practical neural network state trajectory x1(k) and its state is estimated Count track
Fig. 5 shows that the embodiment of the present invention 1 emulates obtained practical neural network state trajectory x2(k) and its state is estimated Count track
Fig. 6 shows that the embodiment of the present invention 1 emulates obtained practical neural network state trajectory x3(k) and its state is estimated Count track
Fig. 7 shows that the mode of the network change topology in the embodiment of the present invention 2 develops figure；
Fig. 8 is shown in the embodiment of the present invention 2 to the output signal z of 3 nodesi(k) estimation of (i=1,2,3) misses Poor curve；
Fig. 9 shows the state estimation error (e of 2 interior joint 1 of the embodiment of the present invention1(k)) the setting upper limit of covariance It is worth curve and actual value curve；
Figure 10 shows the state estimation error (e of 2 interior joint 2 of the embodiment of the present invention2(k)) the setting upper limit of covariance It is worth curve and actual value curve；
Figure 11 shows the state estimation error (e of 2 interior joint 3 of the embodiment of the present invention3(k)) the setting upper limit of covariance It is worth curve and actual value curve；
Figure 12 is shown between the event triggering moment and adjacent events triggering moment of 3 interior joint 1 of the embodiment of the present invention Time interval figure；
Figure 13 is shown between the event triggering moment and adjacent events triggering moment of 3 interior joint 2 of the embodiment of the present invention Time interval figure；
Figure 14 shows the augmentation evaluated error under mean square meaning in the embodiment of the present invention 3Figure Shape.
Specific implementation mode
The embodiment of technical solution of the present invention is described in detail below in conjunction with attached drawing.Following embodiment is only used for Clearly illustrate technical scheme of the present invention, therefore be intended only as example, and the protection of the present invention cannot be limited with this Range.
It should be noted that unless otherwise indicated, technical term or scientific terminology used in this application should be this hair The ordinary meaning that bright one of ordinary skill in the art are understood.
Fig. 1 shows the flow of the networked system method for estimating state provided by the present invention based on insufficient information Figure, this method specifically include following steps：
Step 1：The dynamic model of networked system to be studied is built, includes networked system in the dynamic model State sets the condition that function meets in dynamic model；
Step 2：The state estimator model containing unknown parameter is built, according to state estimator model to the dynamic Model carries out state estimation, obtains state estimation error, carries out augmentation to the system mode and evaluated error, the state of obtaining is estimated Count augmented system model；
Step 3：The constraints that setting evaluated error need to meet obtains ensuring that state estimation augmented system model meets The adequate condition of performance requirement；
Step 4：The unknown parameter in state estimator is calculated according to adequate condition existing for state estimator.
The symbol description used in embodiment：
RnIndicate Euclidean n-space, Rn×mIndicate the set of all n × m ranks real matrixes.Z-Indicate non-positive integer collection It closes.Symbol P > 0 indicate that P is real symmetric tridiagonal matrices.I and 0 indicates the unit matrix of suitable dimension, null matrix respectively.||x|| The Euclid norm of representation vector x.| | A | | be defined as | | A | |=(trace (ATA))1/2Matrix A norm.MTIt indicates The transposition of matrix M.For real symmetric matrix X, Y, symbol X >=Y (X > Y) indicate that X-Y is positive semidefinite (positive definite).We establish one A probability spaceWherein Prob is that the probability that summation is 1 measures.E { x } and E { x | y } respectively represent the phase of x Hope the expectation with x under condition y.diag{A1,A2,...,AnExpression diagonal blocks be matrix A1,A2,...,AnBlock diagonal matrix. In symmetrical matrix, symbol * indicates the omission to symmetrical item.If M is a symmetrical matrix, λmax(M) and λmin(M) it indicates The maximum eigenvalue and minimal eigenvalue of matrix M.SymbolIndicate Kronecker product.If there is no clear specified matrix in specification Dimension then assumes that its dimension is suitble to the algebraic operation of matrix.
Embodiment 1
Networked system to be studied is the Discrete-Delay nerve network system that parameter is jumped containing markov, and structure has The specific method of Discrete-Delay nerve network system dynamic model that markov jumps parameter includes：Markov chain θ (k) (k >= 0) value, transition probability matrix Λ=[λ in finite state space S={ 1,2 ..., s }ij]s×sFor
Wherein, λij>=0 (i, j ∈ S) is the transition probability from i to j, andIt is refreshing using containing n Discrete-time Markovian through member jumps neural network, is described using following dynamical equation：
Y (k)=D (θ (k)) x (k)+E (θ (k)) h (x (k)), (1-2)
Wherein, x (k)=[x1(k),x2(k),…,xn(k)]TFor neural state vector；G (x (k))=[g1(x1(k)),g2 (x2(k)),…,gn(xn(k))]TExpression primary condition is the nonlinear activation function of g (0)=0；d1(θ (k)) and d2(θ(k)) Indicate Discrete-Delay；A (θ (k))=diag { a1(θ(k)),a2(θ(k)),...,an(θ (k)) } it describes to work as certain neuron and network When being disconnected with external input, current potential is reset to the rate of isolated quiescent condition；Ad(θ (k))=diag { ad1(θ (k)),ad2(θ(k)),...,adn(θ (k)) } be states with time-delay parameter matrix；W (θ (k))=[wij(θ(k))]n×nIt is connection Weight matrix；For Discrete-Delay connection weight matrix；Y (k) is output；H (x (k)) is output In Nonlinear perturbations, ψ (k) is given initiation sequence, it is assumed that Nonlinear Vector value function h: To be continuous, and for all x,Meet following fan-shaped Bounded Conditions
[h(x)-h(y)-Φ(x-y)]T[h(x)-h(y)-Ω(x-y)]≤0 (1-3)
Wherein Φ and Ω is the real matrix of suitable dimension,
Nonlinear Vector value function g (x (k)) meets：
‖g(x(k)+δ(k))-g(x(k))‖≤‖Bδ(k)‖ (1-4)
For all system modes, B=diag { b1,b2,...,bn> 0 be a known matrix, δ (k) be one to Amount.
The state estimator model containing unknown parameter is built, the dynamic model is carried out according to state estimator model State estimation obtains state estimation error, carries out augmentation to the system mode and evaluated error, obtains state estimation augmentation system System model specific method include：The formula of the discrete-time state estimator of foundation is：
WhereinFor the state of estimator, KiFor gain matrix to be designed, Δ KiIt indicates change in gain, has Norm-bounded multiplication form be：
ΔKi=KiHkF(k)Ek (1-6)
Wherein, Hk,EkFor the known matrix of appropriate dimension, F (k) is to meet FT(k) unknown matrix of F (k)≤I；
It enablesFormula (1-1), (1-2) and (1-5) is substituted into respectively Obtaining error dynamics formula is：
Define η (k)=[xT(k) eT(k)]T,WithIn conjunction with evaluated error formula (1-7) and neural network formula (1-1) and formula (1-2), using augmented state variable method, obtained augmented system model is as follows：
Wherein,
It provides evaluated error and needs the constraints met, it is special based on Lyapunov Theory of Stability, matrix operation, matrix Value indicative property and inequality transformation, obtain ensuring that state estimation augmented system model meets the adequate condition of particular characteristic requirement Specific method includes：Error dynamics formula (1-8) is meansquare exponential stability, full if there is normal number μ > 0 and 0 < α < 1 Foot
Definition d 1=min { d1i| i ∈ S }, d 2=min { d2i | i ∈ S } andλ=min { λii| i ∈ S }, enable parameterFor it is known that augmented system (1-8) if being exponentially stable There are one group of matrixesTwo matrix Q > 0, R > 0 and normal number κ1i, κ2i, κ3iMeet
Wherein
Obtain exponentially stable adequate condition.
Using MATRIX INEQUALITIES technology and by matrix operation, shape is calculated according to adequate condition existing for state estimator The method of unknown parameter in state estimator includes：There may be index state estimators to make augmented system formula (1-8) index Stablize, if there is two groups of matrixesNormal numberTwo matrix Q > 0, R > 0 and normal number κ1i, κ2i, κ3i, meet
Wherein
The gain of estimator is
Below by way of writing Matlab program solutions linear matrix inequality (1-11) and drawing simulation curve, in fact with emulation The networked system method for estimating state based on insufficient information of the bright present invention of illustration jumps the discrete of parameter for markov The validity of nerve network system state estimation：
Using tool there are three neural network formula (1-1), (1-2) of neuron, parameter is as follows：
d1(1)=2, d1(2)=6, d2(1)=1, d2(2)=5, A (1)=diag { 0.4,0.3,0.3 },
A (2)=diag { 0.5,0.2, -0.3 }, Ad(1)=diag { 0.05,0.01,0.04 },
Ad(2)=diag { 0.03,0.02, -0.01 }, B=diag { 0.2,0.3,0.1 },
Take following excitation function：
g1(x1(k))=- 0.2tanh (x1(k)),g2(x2(k))=0.3tanh (x2(k)),g3(x3(k))=0.1tanh (x3(k)) wherein xl(k) (l=1,2,3) indicates first of element of neural network state x (k).Meanwhile Nonlinear Vector value letter Number h (x (k)) is taken as
Based on the above parameter, with the tool boxes LMI of Matlab, solution formula (1-11) obtains following feasible solution：
κ11=4.7167, κ21=7.4847, κ31=5.4295,
κ12=4.1119, κ22=7.5078, κ32=2.7628.
Since there are index state estimators so that augmented system formula (1-8) Exponential Stability, formula (1-5) are nerve net The index state estimator of network formula (1-1), formula (1-2), then calculated estimator gain matrix be
Being emulated with simulation software further proves the stability and estimator formula of neural network formula (1-1) and (1-2) The performance of (1-5).Fig. 2 describes system mode switching figure.Fig. 3 describes primary condition x (k)=[0.26-0.2 0.1]T(k∈ [- 5,0]) under evaluated error curve.Fig. 4 is practical neural network state trajectory x1(k) and its state estimation trackFigure 5 be practical neural network state trajectory x2(k) and its state estimation trackFig. 6 is practical neural network state trajectory x3 (k) and its state estimation trackFrom fig. 4 to fig. 6 as it can be seen that jumping the Discrete Neural Network system of parameter for markov The state estimator design method of the present invention can effectively estimate the dbjective state of system.
Embodiment 2
Networked system to be studied is the complex networks system with random variation topology.Build complex web to be studied The dynamic model of network system includes networked system state in the dynamic model, sets the item that function meets in dynamic model The specific method of part includes：Using the Random Discrete time-varying complex network being made of M switching node,
Wherein,For the state vector of i-th of node,For the output of i-th of node, interference InputFor probability spaceOn zero mean Gaussian white noise sequence, variance V1> 0, A (k),And Mi(k) (i=1,2 ..., M) is the known matrix with suitable dimension, Random Discrete time-varying network it is outer Coupled configuration matrixFor non-zero matrix, whereinDue to this Coupled configuration matrix is symmetrical, i.e. W(l)=W(l)T, and meet
Γ=diag { r1I,r2I,...,rnI } it is interior coupling matrix,For probability spaceOn Gaussian sequence, meetδ () is Kronecker that delta function, i.e.,：
τ (k) is the stochastic variable for describing complex network and changing topology at random, when considering that sequence { τ (k) } satisfaction one is discrete Between similar Markov chain, value in finite state space below
S=1,2 ..., s } (2-3)
Wherein Ξ=[λmn]s×sFor transition probability matrix, element definition is
λmn=Prob τ (k+1)=n | τ (k)=m } (2-4)
Nonlinear function f (k, xi(k)) it is a Nonlinear Stochastic function for meeting f (k, 0)=0, there is following system Count characteristic：
Wherein, i=1,2 ..., M, q are a known nonnegative integer, Θr(k) andFor The matrix of known appropriate dimension, wherein the quantization measure equation of i-th of sensor is
WhereinFor the measurement output of i-th of node；For the exogenous disturbances of i-th of node, It is probability spaceOn a zero mean Gaussian white noise sequence, variance V2> 0, it is assumed that τ (k), ω (k), v1(k), v2(k) and f (k, xi(k)) (i=1,2 ..., M；L=1,2 ..., s) is independent of each other, Ci(k) andIt is the known matrix with suitable dimension, quantizer q () is expressed as
Wherein quantizer q () is logarithm type, for each qj() (1≤j≤m), quantization level collection is described For
Quantizer entirely will be divided partly according to quantization level, logarithmic quantization device qj() can be expressed as
Wherein κj=(1- ρj)/(1+ρj),
Wherein
The state estimator model containing unknown parameter is built, the dynamic model is carried out according to state estimator model State estimation obtains state estimation error, carries out augmentation to the system mode and evaluated error, obtains state estimation augmentation system System model specific method include：DefinitionQuantify the following formula of measure equation (2-6) Expression：
For network (2-1), based on measurement yi(k) (i=1,2 ..., M), builds following state estimator：
Wherein,For state xi(k) estimation,For the estimator output of i-th of node；Gi (k, τ (k)) and Ki(k, τ (k)) is estimator gain to be calculated；It is defined as follows new variables：
M (k)=diag { M1(k),M2(k),...,MM(k)},
C (k)=diag { C1(k),C2(k),...,CM(k)},
G (k, τ (k))=diag { G1(k,τ(k)),...,GM(k,τ(k))},
K (k, τ (k))=diag { K1(k,τ(k)),...,KM(k,τ(k))},
The state estimation error of i-th of node is enabled to beThe output estimation error of i-th of node ForBy formula (2-1), (2-8), (2-9) substitutes into expression formulaThe evaluated error dynamic of complex network obtains：
Wherein,
In conjunction with (2-5),Statistical property be：
DefinitionWithF (k) is uncertain real value time-varying matrix, is met FT(k) F (k)≤I enables η (k)=[xT(k) eT(k)]TWithUtilize augmented state variable side Original system state kinetics equation and error equation are integrated into a new Augmentation approach equation, obtained augmented system by method Model：
Wherein,
The error co-variance matrix of dynamical system (2-14) is defined as
It provides evaluated error and needs the constraints met, arranged using stochastic analysis technology, matrix operation method, matrix Technology and MATRIX INEQUALITIES technology, obtaining can be so that state augmented system meets H∞The adequate condition of performance requirement and Variance Constraints Specific method include：In order to find state estimator parameter Gi(k, τ (k)) and Ki(k, τ (k)) (i=1,2 ..., M, k=0, 1 ..., N-1) so that following two requires to be met simultaneously：
1) for given interference attenuation horizontal γ > 0, positive definite matrix Uv(τ(k)),Uη(τ (k)) and original state η (0), Filtering errorMeet following H∞Performance constraints：
Wherein
2) evaluated error covariance meets following constraint：
Wherein γ (k) (0≤k < N) is given matrix sequence,
In conjunction with complex network formula (2-1), the state estimator gain G in formula (2-9) is enabledi(k, m) and Ki(k, m) is It is known that for a positive scalar γ > 0, positive definite matrix Uv(m) > 0 and Uη(m) 0 >, for all non-zero v (k), formula (2-16) The H of definition∞Performance constraints are satisfied, if in primary condition
P (0, m)≤γ2Uη(m), m ∈ S
Under, there are a series of positive definite matrixes { P (k, m) }1≤k≤N+1, m ∈ SMeet following recursion matrix inequality：
Wherein,
In conjunction with complex network formula (2-1), state estimator gain G is enabledi(k, m) and Ki(k, m) is it is known that obtainingIf in primary conditionUnder, There are a series of positive definite matrixes { Q (k, m) }1≤k≤N+1, m ∈ SMeet following recursion matrix inequality：
Wherein,
In conjunction with complex network formula (2-1), it is assumed that the parameter G in formula (2-9)i(k, m) and Ki(k, m) be it is known, for One positive scalar γ > 0, positive definite matrix Uv(m) > 0 and Uη(m) 0 >, if there is groups of positive definite matrix P (k, m)}1≤k≤N+1, m ∈ S, { Q (k, m) }1≤k≤N+1, m ∈ S{ ηr(k)}0≤k≤N(r=1,2 ..., q) meets following recursion matrix and differs Formula：
In primary condition
Wherein,
∑5(k, m)=diag {-Q (k, m) ,-Q (k, m) ,-V-1, Ω33(k, m) },
Ω33(k, m)=diag {-σ1(k, m) I ,-σ2(k, m) I ... ,-σq(k, m) I },
So for evaluated error system-computed formula (2-14), J is obtained10 Hes of <
The specific side of the unknown gain parameter in state estimator is calculated according to adequate condition existing for state estimator Method includes：Using MATRIX INEQUALITIES technology, matrix variables replacement and matrix operation, adequate condition existing for state estimator is write Specific method at the LMI forms that can solve state estimator gain includes：For the horizontal γ > of a given interference attenuation 0, positive definite matrix Uv(m) 0 >,With a series of scheduled variance upper limit { Υ (k)}0≤k≤N+1, obtain the Variance Constraints H of complex network formula (2-1)∞Estimator, if, in primary condition
Under, there are groups of positive definite matrixes
(Q1(k, m) }0≤k≤N+1, m ∈ S, { Q2(k, m) }0≤k≤N+1, m ∈ S,
Positive scalar
{ρ1(k, m) }0≤k≤N, m ∈ S, { ρ2(k, m) }0≤k≤N, m ∈ S, { ηr(k)}0≤k≤N(r=1,2 ..., q)
Groups of real value matrix
{Q3(k, m) }0≤k≤N+1, m ∈ S, { Gi(k, m) }0≤k≤N, m ∈ S, { Ki(k, m) }0≤k≤N, m ∈ S
Meet following Recursive Linear MATRIX INEQUALITIES：
Parameter updates formula
Wherein
∑6(k, m)=diag { Ξ1111(k, m) ,-Y (k, m) },
Ξ21(k, m)=diag { ∑s9(k, m), ∑10(k, m) },
Λ21(k, m)=diag { Σ14(k,m),Σ15(k,m)},
Using Matlab software programming program solution Recursive Linear MATRIX INEQUALITIESs (2-25)-(2-28) and draw emulation song Line, for the non-linear complex network that coupling and quantization measure in topological, random is changed at random, an emulation is given below Example illustrates validity of the networked system method for estimating state for the random topological complex networks system of variation of the present invention.
Consider a non-linear complex network formula (2-1) being made of 3 nodes, there is following outer coupled configuration matrix
Select nonlinear function f (k, xi(k))：
F (k, xi(k))=[0.1 0.3]T×(0.2xi1(k)ξ1(k)+0.3xi2(k)ξ2(k))
Wherein xir(k) (r=1,2) is xi(k) r-th of element, ξr(k) (r=1,2) is orthogonal Gauss white noise Sound sequence, is desired for 0, has unit variance.In addition, ξr(k) also uncorrelated to ω (k).It can be seen that above-mentioned random Non-linear satisfaction
The other parameters of complex network formula (2-1) are
Γ=diag { 0.5,0.5 }, M3(k)=[0.2-0.1sin (3k)], C1(k)=[0.5-sin (4k)],
C2(k)=[0.3 sin (2k)], C3(k)=[0.4-sin (3k)],
Select quantizer parameters for κ1=κ2=κ3=0.25 HeEnable H∞Performance indicator is γ=0.9, Uv(1)=Uv(2)=diag { 1,1 },And V1=V2=1, according to formula (2- 24) initial matrix is selected.Based on the variance restrained condition algorithm for estimating, formula (2-25), (2-26), (2- can be solved 27), the solution of (2-28), as shown in table 1.
1 Variance Constraints state estimator parameter of table
It is assumed that original state is x1(0)=[1.5 1.5]T, x2(0)=[1.2-1.6]T, x3(0)=[- 2.0 0.8]T,WithSimulation result is shown in Fig. 7 to Figure 11, Middle Fig. 7 indicates that the mode of network change topology develops figure, and Fig. 8 shows the output signal z of 3 nodesi(k) (i=1,2,3) Evaluated error curve；Fig. 9 indicates the state estimation error (e of node 11(k)) the setting upper limit value curve and actual value of covariance Curve；Figure 10 indicates the state estimation error (e of node 22(k)) the setting upper limit value curve and actual value curve of covariance；Figure 11 indicate the state estimation error (e of node 33(k)) the setting upper limit value curve and actual value curve of covariance.According to analogous diagram Shape can show that the networked system method for estimating state using the present invention based on insufficient information, which is directed to, has random variation The validity of the complex networks system state estimation of topology.
Embodiment 3
Networked system to be studied is complicated with the random time lag that sensor saturation and estimator change in gain occurs Network system builds the dynamic analog with the random time lag complex networks system that sensor saturation and estimator change in gain occurs Type, includes complex network node state in dynamic model, and the specific method for setting the condition that function meets in dynamic model includes： The Discrete-Delay complex networks system formula used for：
Wherein,Indicate the state vector of i-th of node； D (k) meets dm≤d(k)≤dMTime-varying state time lag is described；F () is Nonlinear Vector value function； It is wijThe coupled configuration matrix of >=0 (i ≠ j) but not all network for being zero；Work as γjWhen ≠ 0, Γ=diag { γ1, γ2,...,γn} >=0 is the inherent coupling matrix for connecting j-th of state variable；In probability space On be scalar zero mean Gaussian white noise sequence, as i ≠ j, Ai, Adi, BiWith EiIt is the known constant real matrix of suitable dimension,
Nonlinear Vector value function f () (f (0)=0) is continuous and meets
Wherein, φ and ψ is known constant matrix；
Assuming that preceding q0The output of a node is available：
Wherein,It is with random hair The measurement output of i-th of node of raw sensor saturation；Stochastic variableIt is that a probability distribution isBernoulli Jacob be distributed white sequence and be used for indicating to occur at random Sensor saturated phenomenon, whereinIt is a known constant；CiIt is a known real constant square with suitable dimension Battle array；Saturation functionIt is expressed as
WhereinSign () is sign function, ulDescription Saturated level；According to saturation functionDefinition, there are a diagonal matrix Λ meet 0≤Λ < I and
In order to avoid the transmission of inessential information between sensor and estimator, network is saved with an event trigger method Resource, wherein the event triggering function formula of i-th of node is：
Wherein,It is previous event triggering moment of i-th of node based on current sample time k；ΩiIt is one symmetrical Positive definite weight matrix；θiIt is a positive adjustable threshold for determining triggering frequency, function formula (3-6) is triggered based on event, Only when event triggering function meetsWhen, the measuring signal of node i just can be for transmission to corresponding Estimator,
The triggering moment sequence for defining i-th of node isAnd work asWhen,Then the new triggering moment of i-th of node can be determined as with iteration：
Establish following uncatalyzed coking state estimator：
Wherein,It is xi(k) estimation,WithIt is estimator gain square to be designed Battle array, stochastic variable α (k) and β (k) with Gaussian Profile are used for adjusting the random generation of estimator change in gain, α (k) and β (k) expectation is respectivelyThe variance of α (k) and β (k) is respectivelyα (k), β (k) and μ (k) are mutually solely Vertical,
Norm-bounded does not know gain matrix Δ Ki(k) and Δ Mi(k) it provides as follows：
ΔKi(k)=KiH1iF1(k)N1i,ΔMi(k)=MiH2iF2(k)N2i (3-9)
Wherein, H1i, H2i, N1iAnd N2iIt is the known constant matrix of suitable dimension, uncertain matrix F1(k) and F2(k) meetWithDefinitionIndicate i-th of node at upper one The difference of sensor measurement output between triggering moment and present sampling instant；DefinitionIt is that state is estimated Error is counted, by substituting into variable expression and matrix operation, then ei(k) there is following expression：
Wherein,
It is easy for expression, definition
F (x (k))=[fT(x1(k)) fT(x2(k)) … fT(xN(k))]T,
A=diag { A1, A2..., AN, Ad=diag { Ad1, Ad2..., AdN, B=diag { B1, B2..., BN,
By matrix operation, the compact formula of evaluated error formula (3-10) is obtained：
Define η (k)=[xT(k) eT(k)]T, it is based on formula (3-1) and (3-11), passes through augmented state variable method and square Battle array operation, the then formula for obtaining augmented system are：
Wherein,
It provides evaluated error and needs the constraints met, based on Lyapunov Theory of Stability, inequality technology, matrix Eigenvalue and matrix analysis technology obtain ensuring that state estimation augmented system model meets the abundant item of particular characteristic requirement The specific method of part includes：If there are scalar a > 0,Make with r > 0
It sets up, then the differentiation of augmentation evaluated error dynamical system (3-12) is index ultimate boundness under mean square meaning, r isA upper bound.
Known K, M and 0 c≤1 < of scalar, if there is matrix P=diag { P1,P20 (P of >1=diag { P11, P12,...,P1N,), Q=diag { Q1,Q20 (Q of >1=diag { Q11,Q12,...,Q1N,), Ξ1=diag { ε11,ε12,...,ε1N> 0, With a positive scalar ε3Make following inequality：
It sets up, then augmentation evaluated error dynamical system formula (3-12) is index ultimate boundness under mean square meaning,
Wherein,
By MATRIX INEQUALITIES technology and matrix variables replacement technology, adequate condition existing for state estimator is write as can Solve the LMI forms of state estimator parameter；
0 c≤1 < of known scalar, if there is matrix P=diag { P1,P20 (P of >1=diag { P11,P12,...,P1N,), Q=diag { Q1,Q20 (Q of >1=diag { Q11,Q12,...,Q1N,), Ξ1=diag { ε11,ε12,...,ε1N> 0,With positive scalar ε3, κ makes following linear matrix inequality set up：
Wherein,
Using Matlab software programming program solution linear matrix inequality (3-15) and simulation curve is drawn, using emulation Example proves that the networked system method for estimating state based on insufficient information of the present invention is directed to have and sensor occurs at random The validity of the time lag complex networks system state estimation of saturation and estimator change in gain：
For a time lag complex network comprising 4 nodes, it is assumed that have the measurement of 2 nodes that can utilize, i.e., q0=2, other network parameters are as follows：
H11=H12=H13=H14=diag { 1,1,1 }, H21=H22=diag { 1,1 },
SelectionIt can obtain dM=3 and dm=2.
Select following Nonlinear Vector value function f (xi(k)) (i=1,2,3,4)：
Wherein xir(k) (r=1,2,3) is xi(k) r-th of element.Then, Wo Menyou With
The event triggering threshold of preceding 2 nodes is θ1=θ2=0.4, weight matrix Ω1=Ω2=diag { 1,1 }.
By solving the feasible solution of linear matrix inequality (3-15), the increasing of state estimator formula (3-8) can be obtained Beneficial matrix is：
The upper bound of each moment augmentation evaluated error is calculated, as shown in table 2.
k 1 2 3 4 5 ... 77 78 79 80
γ(k) 1860 3710 5530 7340 9130 ... 100350 101210 102060 102900
The upper bound of 2 augmentation evaluated error of table
Choose the uncertain matrix F in formula (3-9)1(k)=diag { cos (0.01k), cos (0.01k), cos (0.01k) }, F2(k)=diag { cos (0.01k), cos (0.01k) } selects primary condition
Simulation result is shown in Figure 12, Figure 13 and Figure 14.When Figure 12 indicates that the event triggering moment of node 1 and adjacent events trigger Time interval figure between quarter, Figure 13 indicate the time interval between the event triggering moment and adjacent events triggering moment of node 2 Figure.Figure 14 indicates the augmentation evaluated error under mean square meaningFigure.According to the value of the γ (k) in table 2, It can obtainI.e. augmentation evaluated error dynamical system is index ultimate boundness under mean square meaning.Cause This, for the random time lag complex networks system shape using the present invention that sensor saturation and estimator change in gain occurs State estimator design method can effectively estimate dbjective state.
The first embodiment is estimated mainly for the state for the Discrete-Delay nerve network system for jumping parameter containing markov Meter, second of embodiment are directed to the state estimation of the complex networks system with random variation topology, the third embodiment is directed to State estimation with the random time lag complex networks system that sensor saturation and estimator change in gain occurs.It needs further Illustrate, the step in the method for the present invention gives the method for implementing the state estimation, and those skilled in the art also may be used Further to improve the step method in details or realization according to the prior art respectively according to understanding.
Finally it should be noted that：The above embodiments are only used to illustrate the technical solution of the present invention., rather than its limitations；To the greatest extent Present invention has been described in detail with reference to the aforementioned embodiments for pipe, it will be understood by those of ordinary skill in the art that：Its according to So can with technical scheme described in the above embodiments is modified, either to which part or all technical features into Row equivalent replacement；And these modifications or replacements, various embodiments of the present invention technology that it does not separate the essence of the corresponding technical solution The range of scheme should all cover in the claim of the present invention and the range of specification.

Claims (10)

1. a kind of networked system method for estimating state based on insufficient information, which is characterized in that specifically include following steps：
Step 1：The dynamic model of networked system to be studied is built, includes networked system state in the dynamic model, Set the condition that function meets in dynamic model；
Step 2：The state estimator model containing unknown parameter is built, according to state estimator model to the dynamic model State estimation is carried out, state estimation error is obtained, augmentation is carried out to the system mode and evaluated error, obtains state estimation increasing Wide system model；
Step 3：The constraints that setting evaluated error need to meet, obtains state estimation augmented system model and meets performance requirement Adequate condition；
Step 4：The unknown parameter in state estimator is calculated according to adequate condition existing for state estimator.
2. the networked system method for estimating state based on insufficient information as described in claim 1, which is characterized in that described Networked system to be studied is the Discrete-Delay nerve network system that parameter is jumped containing markov.
3. the networked system method for estimating state based on insufficient information as claimed in claim 2, which is characterized in that structure The specific method of Discrete-Delay nerve network system dynamic model that parameter is jumped with markov includes：Markov chain θ (k) (k >=0) value, transition probability matrix Λ=[λ in finite state space S={ 1,2 ..., s }ij]s×sFor
Wherein, λij>=0 (i, j ∈ S) is the transition probability from mode i to mode j, andConsideration contains n The discrete-time Markovian of a neuron jumps neural network, is described using following dynamical equation：
Y (k)=D (θ (k)) x (k)+E (θ (k)) h (x (k)), (1-2)
Wherein, x (k)=[x1(k),x2(k),…,xn(k)]TFor neural state vector；G (x (k))=[g1(x1(k)),g2(x2 (k)),…,gn(xn(k))]TExpression primary condition is the nonlinear activation function of g (0)=0；d1(θ (k)) and d2(θ (k)) is indicated Discrete-Delay；A (θ (k))=diag { a1(θ(k)),a2(θ(k)),…,an(θ (k)) } description is when certain neuron and network and outside When portion's input disconnects, current potential is reset to the rate of isolated quiescent condition；Ad(θ (k))=diag { ad1(θ(k)),ad2 (θ(k)),…,adn(θ (k)) } be states with time-delay parameter matrix；W (θ (k))=[wij(θ(k))]n×nIt is connection weight matrix；For Discrete-Delay connection weight matrix；Y (k) is output；H (x (k)) is non-in output Linear disturbance, ψ (k) are given initiation sequence, it is assumed that Nonlinear Vector value functionFor even It is continuous, and for all x,Meet following fan-shaped Bounded Conditions
[h(x)-h(y)-Φ(x-y)]T[h(x)-h(y)-Ω(x-y)]≤0 (1-3)
Wherein Φ and Ω is the real matrix of suitable dimension；Nonlinear Vector value function g (x (k)) meets：
‖g(x(k)+δ(k))-g(x(k))‖≤‖Bδ(k)‖ (1-4)
For all system modes, B=diag { b1,b2,…,bn}>0 is a known matrix, and δ (k) is a vector.
4. the networked system method for estimating state based on insufficient information as claimed in claim 3, which is characterized in that structure Discrete-time state estimator model containing unknown parameter is：
Wherein,For the state of estimator, KiFor matrix to be designed, Δ KiIt indicates change in gain, there is following model Number bounded multiplicative form：
ΔKi=KiHkF(k)Ek (1-6)
Wherein, Hk,EkFor the known matrix of appropriate dimension, F (k) is to meet FT(k) unknown matrix of F (k)≤I.
5. the networked system method for estimating state based on insufficient information as described in claim 1, which is characterized in that described Networked system to be studied is the complex networks system with random variation topology.
6. the networked system method for estimating state based on insufficient information as claimed in claim 5, which is characterized in that use The model for the Random Discrete time-varying complex network being made of M switching node is,
Wherein,For the state vector of i-th of node,For the output of i-th of node, exogenous disturbancesFor probability spaceOn zero mean Gaussian white noise sequence, variance V1>0, A (k),And Mi(k) (i=1,2 ..., M) is the known matrix with suitable dimension, Random Discrete time-varying complex network it is outer Coupled configuration matrixFor non-zero matrix, whereinW(l)=W(l )T, and meet
Γ=diag { r1I,r2I,…,rnI } it is interior coupling matrix,For probability spaceOn height This white noise sequence meetsδ () is Kronecker that delta function, i.e.,：
τ (k) is the stochastic variable for describing complex network and changing topology at random, and it is same to consider that sequence { τ (k) } meets a discrete time Class Markov chain, value in finite state space below
S=1,2 ..., s } (2-3)
Wherein Ξ=[λmn]s×sFor transition probability matrix, element definition is
λmn=Prob τ (k+1)=n | τ (k)=m } (2-4)
Nonlinear function f (k, xi(k)) it is a Nonlinear Stochastic function for meeting f (k, 0)=0, has following statistics special Property：
Wherein, i=1,2 ..., M, q are a known nonnegative integer, Θr(k) andIt is known The matrix of appropriate dimension, wherein the quantization measure equation of i-th of sensor is
Wherein,For the measurement output of i-th of node；For the exogenous disturbances of i-th of node, it is Probability spaceOn a zero mean Gaussian white noise sequence, variance V2>0, it is assumed that τ (k), ω (k), v1 (k), v2(k) and f (k, xi(k)) (i=1,2 ..., M；L=1,2 ..., s) it is independent of each other, Ci(k) andIt is the known matrix with suitable dimension, quantizer q () is expressed as
Wherein quantizer q () is logarithm type, for each qj() (1≤j≤m), quantization level collection is described as
Quantizer divides entire part, logarithmic quantization device q according to quantization levelj() can be expressed as
Wherein, κj=(1- ρj)/(1+ρj),
Wherein,
7. the networked system method for estimating state based on insufficient information as claimed in claim 6, which is characterized in that be based on Measure yi(k) (i=1,2 ..., M), the model for building complex network state estimator are：
WhereinFor state xi(k) estimation,For the estimator output of i-th of node；Gi(k,τ And K (k))i(k, τ (k)) is estimator gain to be calculated.
8. the networked system method for estimating state based on insufficient information as described in claim 1, which is characterized in that described Networked system to be studied is with the random time lag complex networks system that sensor saturation and estimator change in gain occurs.
9. the networked system method for estimating state based on insufficient information as claimed in claim 8, which is characterized in that structure Time lag complex networks system dynamic model be：
Wherein,Indicate the state vector of i-th of node, d (k) (dm≤d(k)≤dM) description time-varying state time lag, f () is Nonlinear Vector value function,It is wij≥0 (i ≠ j) but not all network coupled configuration matrix for being zero, works as γjWhen ≠ 0, Γ=diag { γ1,γ2,…,γn}≥ 0 is the interior coupling matrix for connecting j-th of state variable,In probability spaceOn be that scalar zero is equal It is worth Gaussian sequence, as i ≠ j, Ai, Adi, BiAnd EiIt is properly to tie up Several known constant real matrixes, Nonlinear Vector value function f () (f (0)=0) are continuous and meet
Wherein, φ and ψ is known constant matrix；
Assuming that preceding q0The output of a node is available：
Wherein,It is with random The measurement output of i-th of node of sensor saturation occurs；Stochastic variableIt is that a probability distribution isBernoulli Jacob be distributed white sequence, be used for indicating to send out at random Raw sensor saturated phenomenon, whereinIt is a known constant；CiIt is that a known constant with suitable dimension is real Matrix；Saturation functionIt is expressed as
WhereinSign () is sign function, ulDescription saturation It is horizontal；According to saturation functionDefinition, there are a diagonal matrix Λ to meet 0≤Λ<I and
In order to avoid the transmission of inessential information between sensor and estimator, network money is saved with an event trigger method Source, wherein the event triggering function formula of i-th of node is：
Wherein,It is previous event triggering moment of i-th of node based on current sample time k；ΩiIt is a symmetric positive definite Weight matrix；θiIt is a positive adjustable threshold for determining triggering frequency, based on event triggering function formula (3-6), only When event triggering function meetsWhen, the measuring signal of node i just can be for transmission to estimating accordingly Gauge,
The triggering moment sequence for defining i-th of node isAnd work asWhen,Then the new triggering moment of i-th of node can be determined as with iteration：
10. the networked system method for estimating state based on insufficient information as claimed in claim 9, which is characterized in that institute The method of stating further includes establishing following uncatalyzed coking state estimator：
Wherein,It is xi(k) estimation,WithIt is estimator gain matrix to be designed, tool There are the stochastic variable α (k) and β (k) of Gaussian Profile to be used for adjusting the random generation of estimator change in gain, the phase of α (k) and β (k) Prestige is respectivelyThe variance of α (k) and β (k) is respectivelyα (k), β (k) and μ (k) are independent from each other,
Norm-bounded does not know gain matrix Δ Ki(k) and Δ Mi(k) it provides as follows：
ΔKi(k)=KiH1iF1(k)N1i,ΔMi(k)=MiH2iF2(k)N2i (3-9)
Wherein, H1i, H2i, N1iAnd N2iIt is the known constant matrix of suitable dimension, uncertain matrix F1(k) and F2(k) meet F1 T (k)F1(k)≤I and
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CN113922413B (en) * 2021-09-22 2023-08-22 东北石油大学 Island micro-grid distributed secondary control method based on dynamic event triggering protocol
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