| CPC H04L 63/1475 (2013.01) [H04L 63/04 (2013.01)] | 8 Claims |
|
1. An anti-eavesdropping distributed fusion filtering method for a multi-rate nonlinear system, comprising the following steps:
Step 1: establishing a dynamic model for the multi-rate nonlinear system over sensor networks:
 wherein, tk is a state update instant of the multi-rate nonlinear system; x(tk) is a state vector of the multi-rate nonlinear system at time tk; x(tk+1) is a state vector of the multi-rate nonlinear system at time tk+1; f(x(tk)) is a continuous and differentiable nonlinear function with a bounded second-order derivative of the multi-rate nonlinear system at time tk; B(tk) is a coefficient matrix of process noise at time tk; ω(tk) is process noise with zero mean and covariance Q(tk) at time tk; i is a label of sensor nodes, i=1,2, . . . , N, N represents a number of sensor nodes; sk is a measurement sampling instant of sensor; x(sk) is a state vector of the multi-rate nonlinear system at time sk; yi(sk) is a measurement output signal of an i-th sensor node in the multi-rate nonlinear system at time sk; Ci(sk) is a measurement matrix of the i-th sensor node based on the multi-rate nonlinear system at time sk; vi(sk) is measurement noise of the i-th sensor node in the multi-rate nonlinear system at time sk; Λi(sk) is used to describe a phenomenon of fading measurements;
Step 2: transforming the dynamic model for the multi-rate nonlinear system over sensor networks in Step 1 into a single-rate nonlinear system dynamic model through a prediction compensation strategy:
 wherein, yi(tk) is a measurement output signal of the i-th sensor node in a single-rate nonlinear system at time tk; β(tk) is an auxiliary variable; Ci(tk) is a measurement matrix of the i-th sensor node based on the single-rate nonlinear system at time tk; xi(tk|tk−1) is a one-step prediction of the i-th sensor node at time tk−1; Λi=diag{λi1,λi2, . . . , λiny}; ny is a dimension of yi(sk); λiu is an expectation of a random variable λiu(sk), u=1,2, . . . , ny; yi(tk) is a measurement output signal of the i-th sensor node in the multi-rate nonlinear system at time tk; st is the measurement sampling instant of sensor node;
Step 3: designing an anti-eavesdropping distributed fusion filter for the single-rate nonlinear system dynamic model in Step 2; wherein
Step 3a: when a sensor node exchanges information, in order to prevent transmitted data from being eavesdropped by an eavesdropper and ensure a security of information transmission, adding artificial noise to the one-step prediction xj(tk|tk−1) of a sensor node j before xj(tk|tk−1) being sent to the sensor node i:
 wherein, j∈Ni, Ni is a set of neighboring nodes of the i-th sensor node; xj(tk|tk−1) represents a one-step prediction of the j-th sensor node at time tk−1;
 is a transmitted message from the sensor node j to the sensor node i at time tk; I is an nx-dimensional identity matrix; nx is a dimension of the state vector x(tk); aij(tk) is the artificial noise with zero mean and covariance Qij(tk) at time tk; Lij(tk) is a selection matrix at time tk;
Step 3b: when the sensor node i receives the information
 transmitted by the sensor node j, obtaining a compensated one-step prediction at time tk according to a zero-order holder compensation rule:
 wherein,
 is the compensated one-step prediction at time tk;
 is a compensated one-step prediction at time tk−1;
Step 3c: designing a local distributed filter:
 wherein, xi(tk+1|tk) represents a one-step prediction of the i-th sensor node at time tk;
 represents a compensated one-step prediction at time tk+1; xi(tk+1|tk+1) represents a filter of the i-th sensor node at time tk+1; xi(tk|tk) represents a filter of the i-th sensor node at time tk; f(xi(tk|tk)) represents a nonlinear function filtering form based on the single-rate nonlinear system of the i-th sensor node at time tk; Ki(tk+1) represents a local distributed filter parameter of the i-th sensor node at time tk+1; yi(tk+1) represents a measurement output signal of the i-th sensor node in the single-rate nonlinear system at time tk+1; Ci(tk+1) is a measurement matrix of the i-th sensor node based on the single-rate nonlinear system at time tk+1; εi represents a predefined consensus parameter of the i-th sensor node; hij represents a connection coefficient between the i-th sensor node and the j-th sensor node; and
Step 3d: obtaining an anti-eavesdropping distributed fusion filter based on a local filter xi(tk|tk) and a covariance intersection fusion criterion:
 wherein, a superscript “−1” represents an inverse of a matrix; xCI(tk|tk) is a fusion filter at time tk;
 CI(tk|tk) is fusion filtering error covariance at time tk;  (tk|tk) is an upper bound on a local filtering error covariance of the i-th sensor node at time tk;  (tk|tk) is an inverse of a matrix is an inverse of a matrix
 ωi is a scalar;
Step 4: calculating an upper bound on the one-step prediction error covariance
 (tk+1|tk) of the i-th sensor node at time tk by solving a matrix difference equation: wherein, a superscript “T” represents the transpose of a matrix; ò1 is a known scaling parameter; ò1−1 is an inverse of ò1; Ai(tk) is a partial derivative of the continuous and differentiable nonlinear function f(x(tk)) corresponding to a system state at the local filter xi(tk|tk) at time tk; Mi(tk) and Di(tk) are known error matrices obtained by Taylor series based on f(x(tk));
 represent transposes of Ai(tk), B(tk), Mi(tk) and Di(tk), respectively;
Step 5: according to
 i(tk+1|tk) obtained in Step 4, deriving the local distributed filter parameter Ki(tk+1) of the i-th sensor node at time tk+1 by minimizing a trace of the upper bound on the local filtering error covariance:Ki(tk+1)=δ(β(tk+1),1)(1+ò2)
 (tk+1|tk)CiT(tk+1)ΛiΠi−1(tk+1)wherein,
 wherein, δ(a,b) is a Kronecker function; β(tk+1) is an auxiliary variable at time tk+1; ò2 and ò3 are known scaling parameters; ò3−1 is an inverse of ò3;
 is a transpose of Ci(tk+1);
 is a transpose of xi(tk+1|tk); Πi−1(tk+1) is an inverse of Πi(tk+1); Ri(tk+1) is a covariance matrix of measurement noise vi(tk+1) of the i-th sensor node at time tk+1;
Step 6: by maximizing an estimation error covariance of the eavesdropper, deriving the selection matrix Lij(tk+1) at time tk+1 from the following optimization problem:
 wherein,
 wherein,
 are diagonal matrices with elements of 0 or 1 and a sum of the diagonal elements are ñi; xj(tk+1|tk) represents a one-step prediction of the j-th sensor node at time tk;
 is a transpose of xj(tk+1|tk); Qij(tk+1) is a covariance matrix of artificial noise aij(tk+1) at time tk+1;
 represents that an objective function f(x) is maximized by selecting a decision variable x;
Step 7: substituting Ki(tk+1) obtained in Step 5 and Lij(tk+1) obtained in Step 6 into Step 3 to obtain the fusion filter xCI(tk+1|tk+1) at time tk+1; determining whether tk+1 reaches a total duration M, if tk+1<M, performing Step 8, otherwise, ending;
Step 8: based on Ki(tk+1) obtained in Step 5 and Lij(tk+1) obtained in Step 6, solving for the upper bound on the local filtering error covariance
 (tk+1|tk+1) of the i-th sensor node at time tk+1: (tk+1|tk+1)=(1−β(tk+1))Δi(tk+1)+β(tk+1)Θi(tk+1)wherein,
 wherein, ò2−1 is an inverse of ò2;
 is a square of εi;
 (tk+1|tk+1) is the upper bound on the local filtering error covariance of the i-th sensor node at time tk−1; is a transpose of Ki(tk+1);
 is a transpose of Xij(tk+1); (I−Ki(tk+1)ΛiCi(tk+1))T is a transpose of I−Ki(tk+1)ΛiCo(tk+1); øi represents a penetration of the i-th sensor node;
let tk=tk+1 and performing Step 3 until tk+1=M is satisfied; and
Step 9: estimating the multi-rate nonlinear system when simultaneously considering eavesdroppers and fading measurements for transmitted data via the sensor network, wherein when a fading probability rises from 0.3 to 0.7, an average mean square error is reduced by approximately 37%; and when the fading probability increases from 0.7 to 0.8, a reduction in the average mean square error is approximately 88%, thus improving an accuracy of a filtering performance of such problems and wherein the updated eavesdroppers and fading measurements presents a complexity of transmitted measurements of the nonlinear system to an environment monitoring computer display indicating accuracy of filtering performance.
|