CN109728796B - Filtering method based on event trigger mechanism - Google Patents
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Abstract
The invention discloses a filtering method based on an event trigger mechanism, and relates to a filtering method based on an event trigger mechanism. The invention solves the problem of large estimation error of the existing filtering method. The process is as follows: 1. establishing a dynamic model of a nonlinear stochastic system; 2. carrying out filter design on a dynamic model of the nonlinear stochastic system under an event trigger mechanism; 3. calculating the one-step prediction error covariance matrix upper bound of the filter; 4. calculating a filter gain matrix; 5. substituting the filter gain matrix into two to obtain state estimation at the k +1 th moment; judging whether k +1 reaches the total network duration M, if k +1 is less than M, executing step six, and if k +1= M, ending; 6. calculating an upper bound of a filtering error covariance matrix; another k = k +1, two are performed until k +1= m is satisfied. The invention is used in the field of filtering of event trigger mechanisms.
Description
Technical Field
The invention relates to a filtering method based on an event trigger mechanism.
Background
The filtering problem of the nonlinear stochastic system is an important research part in a networked control system, and the nonlinear stochastic system is widely applied to signal estimation tasks in the fields of system engineering, global positioning systems, target tracking systems and the like.
Nonlinearity and randomness are ubiquitous in real life, but the existing filtering method cannot simultaneously process a nonlinear random system with multiple measurement loss phenomena with uncertain loss probability, usually deteriorates the estimation performance of a filter, and in addition, the uncertain loss probability can cause the distortion behavior of signals;
in conclusion, the existing filtering method has the problem of large estimation error.
Disclosure of Invention
The invention solves the problem of large estimation error of the existing filtering method and provides a filtering method based on an event trigger mechanism.
A multi-time-varying filtering method based on an event trigger mechanism comprises the following specific processes:
step one, establishing a dynamic model of a nonlinear stochastic system;
step two, carrying out filter design on a dynamic model of the nonlinear stochastic system under an event trigger mechanism;
step three, calculating the upper bound omega of the covariance matrix of the one-step prediction error of the filter k+1|k ；
Step four, predicting the upper bound omega of the error covariance matrix according to one step k+1|k Calculating a filter gain matrix K k+1 ；
Step five, the filter gain matrix K calculated in the step four k+1 Substituting the state estimation formula 9 in the step two to obtain the state estimation at the k +1 th moment
Judging whether k +1 reaches the total network duration M, if k +1 is less than M, executing the step six, and if k +1 is not greater than M, ending the step;
step six, according to the filter gain matrix K in the step four k+1 And calculating the upper bound omega of the covariance matrix of the filtering error k+1|k+1 ；
And k = k +1, and step two is executed until k +1= m is satisfied.
Effects of the invention
The filtering method of the nonlinear stochastic system considers the influence of the multiple measurement loss phenomenon with the uncertain loss probability on the filtering performance, utilizes the extended Kalman filtering method to comprehensively consider the effective information of the covariance matrix of the filtering error, and compared with the filtering method of the existing nonlinear stochastic system with multiple measurement loss, the filtering method of the nonlinear stochastic system processes the multiple measurement loss phenomenon with the uncertain loss probability, obtains the filtering method of the nonlinear stochastic system based on the extended Kalman filtering, achieves the purpose of disturbance resistance, has the advantages of easy solution and realization, and reduces the estimation error of the existing filtering method. The invention realizes good estimation effect and can ensure that the relative error at each moment is less than 0.5.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is the actual state trajectory x of a nonlinear stochastic system 1,k And its corresponding filtered trace contrast map, x k ＝[x 1,k x 2,k ] T Is a system state trajectory;
is x 1,k (ii) is estimated;
FIG. 3 is a diagram of the actual state trajectory x of a nonlinear stochastic system 2,k And its corresponding filtered trajectory contrast map, x k ＝[x 1,k x 2,k ] T Is a system state trajectory;
is x 2,k (ii) is estimated;
FIG. 4 is a non-linear stochastic system trajectory x 1,k The logarithm of the filtered mean square error (log (MSE 1)) of (a) and its corresponding upper bound; upper bound 1 is Ω k+1|k+1 The logarithmic value of the first component of (a);
FIG. 5 is a trace x of a nonlinear stochastic system 2,k A plot of the logarithm of the filtered mean square error (log (MSE 2)) and its corresponding upper bound; upper bound 2 is Ω k+1|k+1 The log value of the second component of (a).
Detailed Description
In a first specific embodiment, the first embodiment is described with reference to the first drawing, and a filtering method based on an event trigger mechanism in the first embodiment is, as shown in fig. 1, specifically includes the following steps:
step one, establishing a dynamic model of a nonlinear stochastic system of a multiple measurement loss phenomenon with uncertain loss probability;
step two, carrying out filter design on a dynamic model of a nonlinear random system of a multiple measurement loss phenomenon with uncertain loss probability under an event trigger mechanism;
step three, calculating the upper bound omega of the covariance matrix of the one-step prediction error of the filter k+1|k ；
Step four, predicting the upper bound omega of the error covariance matrix according to one step k+1|k Calculating a filter gain matrix K k+1 ；
Step five, the filter gain matrix K calculated in the step four k+1 Substituting the state estimation formula 9 in the step two to obtain the state estimation at the k +1 th moment
Realizing the filter design of a dynamic model of a nonlinear stochastic system with multiple measurement loss phenomena with uncertain loss probability;
judging whether k +1 reaches the total network duration M, if k +1 is less than M, executing the step six, and if k +1= M, ending;
step six, according to the filter gain matrix K in the step four k+1 And calculating the upper bound omega of the covariance matrix of the filtering error k+1|k+1 (ii) a And k = k +1, and step two is executed until k +1= m is met.
Many times the embodiment becomes common general knowledge of those skilled in the art, referring to different times;
the second embodiment is as follows: the difference between the first embodiment and the first embodiment is that, in the first step, a dynamic model of a nonlinear stochastic system of a multiple measurement loss phenomenon with uncertain loss probability is established; the specific process is as follows:
the dynamic model state space form of the nonlinear stochastic system of the multiple measurement loss phenomenon with uncertain loss probability is as follows:
y k ＝Ξ k C k x k +h(x k ,ζ k )+ν k (2)
in the formula (I), the compound is shown in the specification,
state variables at the time k and the time k +1 respectively; initial value x 0 Has a mean value of
Variance is P 0|0 ；
A real number domain of states of a dynamic model of the nonlinear stochastic system, n being a dimension;
is the output of the measurement at the k-th time,
of states of dynamic models for non-linear stochastic systemsA real number domain, m being the dimension; eta k And ζ k Is gaussian white noise with a mean value of zero and a variance of 1;
is that the mean is zero and the variance is Q k Process noise of (Q) k ＞0，
Is the real number domain of the states of the dynamic model of the nonlinear stochastic system, p is the dimension; f (x) k ) For non-linear perturbations, g (x) k ,η k )、h(x k ,ζ k ) Is a non-linear function;
is a mean of zero and a variance of R k Measurement noise of (2), R k ＞0；C k And D k Respectively, a measurement matrix at the time k and a noise driving matrix at the time k.
Other steps and parameters are the same as those in the first embodiment.
The third concrete implementation mode: the difference between this embodiment and one or two of the embodiments is that the data loss matrix xi k ＝diag{ξ 1,k ,ξ 2,k ,...,ξ m,k The diag { · } is a diagonal matrix; xi i,k For m variables that are independent of each other with respect to i and k, i =1, 2.
Where i is the location of the data loss, k is the time,
is a definite mathematical expectation, Δ ξ i Uncertainty of delineation probability, i =1,2, \8230M, prob { } is a probability,
is a desire of {. Cndot.);
non-linear function g (x) k ,η k ) And h (x) k ,ζ k ) Satisfies g (0, eta) k )＝0，h(0,ζ k ) =0 and the following conditions:
wherein s > 0 is a known constant,
and
are all non-linear parameter matrices, r =1, 2.., s;
is composed of
Transpose of g (x) j ,η j )、h(x j ,ζ j ) Is a nonlinear function, j is time, k is not equal to j;
is composed of
The method (2) is implemented by the following steps,
represents the expectation of {. Cndot. },
is x k Transposing;
the following assumption xi i,k ，η k ，ζ k ，ω k ，ν k And
are independent of each other.
Other steps and parameters are the same as those in the first or second embodiment.
The fourth concrete implementation mode is as follows: the difference between this embodiment and the first to third embodiments is that, in the second step, a filter design is performed on a dynamic model of a nonlinear stochastic system with multiple measurement loss phenomena having uncertain loss probability under an event trigger mechanism; the specific process is as follows:
firstly, aiming at the dynamic model of the nonlinear stochastic system established in the step one, the following event trigger formula is selected
(y k+j -y k ) T (y k+j -y k )＞σ (7)
Where σ > 0 is a known adjustment threshold; y is k+j Is the measurement output at the k + j time, i.e. when equation (7) is satisfied, y k+j Allowed to transmit, j being time;
the actual output after event triggering from equation (7) is:
wherein k is a In order to trigger the moment of time,
is the actual output after the event is triggered,
to trigger the time k a A is the number of the current trigger sequences (i.e., the trigger sequences are triggered a times at this time),
the actual output value after the event at the moment k is triggered;
aiming at the dynamic model of the nonlinear stochastic system established in the step one, the following filter structure is designed:
in the formula (I), the compound is shown in the specification,
an estimation function that is non-linear;
is x k One-step prediction at time k;
and with
Are each x k The estimates at time k and time k +1,
is an initial value of the estimation;
actual output at time k + 1;
in order to be a mathematical expectation of the data loss matrix,
diag {. Is a diagonal matrix;
for actual output after event triggering, C k+1 For the measurement matrix at the time K +1, K k+1 A filter gain matrix at the moment k +1 to be designed;
other steps and parameters are the same as those in one of the first to third embodiments.
The fifth concrete implementation mode: the difference between this embodiment and one of the first to the fourth embodiments is that the upper bound Ω of the covariance matrix of the one-step prediction error of the filter is calculated in the third step k+1|k (ii) a The specific process is as follows:
calculating the upper bound omega of the covariance matrix of the one-step prediction error of the filter according to the following formula k+1|k ：
In the formula, omega k+1|k An upper bound of a covariance matrix of the one-step prediction error at the moment k; omega k|k The upper bound of the filtering error covariance matrix at the moment k; epsilon 1 A known constant greater than zero; gamma ray k > 0 is satisfied
I is an identity matrix;
is f (x) k ) In that
Point about x k Solving a matrix after the partial derivation; m is a group of k And L k Is f (x) k ) In that
Point Taylor series expansion type rear high-order term partA corresponding known matrix; d k Driving the matrix for noise at time k;
are respectively A k ，M k ，D k ，
L k Transposing; tr {. Cndot } represents a trace of {. Cndot }.
Other steps and parameters are the same as in one of the first to fourth embodiments.
The sixth specific implementation mode is as follows: the difference between this embodiment and the first to the fifth embodiments is that the upper bound Ω of the covariance matrix of the one-step prediction error obtained in the fourth step is obtained from the third step k+1|k Calculating a filter gain matrix K k+1 (ii) a The specific process is as follows:
calculating a filter gain matrix K according to the following formula k+1 ：
In the formula, K k+1 A filter gain matrix at time k + 1;
representing a Hadamard product (whose elements are defined as the product of corresponding elements of the two matrices); i is the identity matrix, Ψ k+1|k In order to estimate the term for the non-linearity,
ε l is a constant greater than zero, l =1,2,3,4,5;
is epsilon l The inverse of (c);
is a constant number of times, and is,
are all nonlinear parameter matrices, R =1,2 k+1 Tr {. Is the covariance of the measurement noise at time k + 1, and represents the trace of {. Cndot.);
and
are respectively as
C k+1 And
the transposing of (1).
Other steps and parameters are the same as in one of the first to fifth embodiments.
The seventh embodiment: the difference between this embodiment and one of the first to sixth embodiments is that the gain matrix K is filtered according to the fourth step in the sixth step k+1 And calculating the upper bound omega of the covariance matrix of the filtering error k+1|k+1 (ii) a The specific process is as follows:
filtering error covariance matrix upper bound omega k+1|k+1 Is composed of
Wherein omega k+1|k+1 The covariance matrix upper bound of the filtering error at the moment of k + 1;
are respectively as
The transposing of (1).
Other steps and parameters are the same as those in one of the first to sixth embodiments.
Eighth embodiment, the difference between this embodiment and one of the first to seventh embodiments, is that f (x) in the first step k ) In that
A point Taylor series is expanded by
Wherein
Is the filtering error at the time instant k,
the high-order error term after Taylor series expansion is obtained; delta k Is an unknown time-varying matrix, satisfies
I is an identity matrix and is a matrix of the identity,
is Δ k Transposing;
the upper bound of the filtering error covariance matrix, namely omega, is solved k+1|k+1 So that P is k+1|k+1 ≤Ω k+1|k+1 In which
Is the filter error covariance matrix at time k + 1,
is the filtering error at the time instant k +1,
is the expectation of the element { · },
is composed of
The transposing of (1).
Because the filter error covariance matrix has uncertain items, the true value of the filter error covariance matrix cannot be obtained. Optimization of filtering error covariance matrix upper bound omega k+1|k+1 Can obtain the filter gain matrix K at the moment K + 1 k+1 。
Other steps and parameters are the same as those in one of the first to seventh embodiments.
The following examples were employed to demonstrate the beneficial effects of the present invention:
the method of the invention is adopted for simulation:
system parameters:
from g (x) k ,ξ k ) And h (x) k ,ζ k ) The expression form of (A) is:
other simulation initial values are selected as follows:
x 0 ＝[1.35 -0.55] T ，x 0|0 ＝[1.35 -0.55] T ，Ω 0|0 ＝10I，
M k ＝diag{0.1,0.2}，L k ＝0.1I，γ k ＝0.05，ε i ＝0.05(i＝1,2,…,5)，σ＝0.004，Q k ＝0.05，R k ＝0.01I。
the filter effect is as follows:
as shown in fig. 2,3,4, 5:
FIG. 2 is a diagram of the actual state trajectory x of a nonlinear stochastic system 1,k And its corresponding filtered trajectory contrast map, x k ＝[x 1,k x 2,k ] T The system state track is shown, wherein a solid line is a real track graph of the system, and a dotted line is a corresponding filtering track graph;
FIG. 3 is a diagram of the actual state trajectory x of a nonlinear stochastic system 2,k And its corresponding filtered trace contrast map, x k ＝[x 1,k x 2,k ] T The system state track is shown, wherein a solid line is a real track graph of the system, and a dotted line is a corresponding filtering track graph;
FIG. 4 is a trace x of a non-linear stochastic system 1,k The logarithm of the filtered mean square error (log (MSE 1)) of (a) and the corresponding logarithm of the upper bound, x k ＝[x 1,k x 2,k ] T The system state trajectory, wherein the solid line is the trajectory graph of the logarithm of the filtered mean square error and the dashed line is the trajectory graph of the corresponding upper bound;
FIG. 5 is a trace x of a non-linear stochastic system 2,k The logarithm of the filtered mean square error (log (MSE 2)) of (a) and the corresponding logarithm of the upper bound, x k ＝[x 1,k x 2,k ] T The system state trajectory, where the solid line is the trajectory plot of the logarithm of the filtered mean square error and the dashed line is the trajectory plot of the corresponding upper bound.
As can be seen from fig. 2 to 3, the filter of the present invention can effectively estimate the target state for the nonlinear stochastic system of the multiple measurement loss phenomenon with uncertain probability under the event trigger mechanism.
Claims (2)
1. A filtering method based on an event trigger mechanism is characterized in that: the method comprises the following specific processes:
step one, establishing a dynamic model of a nonlinear stochastic system;
secondly, designing a filter for a dynamic model of the nonlinear stochastic system under an event trigger mechanism;
step three, calculating the upper bound omega of the covariance matrix of the one-step prediction error of the filter k+1|k ；
Step four, predicting the upper bound omega of the error covariance matrix according to one step k+1|k Calculating a filter gain matrix K k+1 ；
Step five, the filter gain matrix K calculated in the step four k+1 Substituting step two to obtain the state estimation at the k +1 th time
Judging whether k +1 reaches the total network duration M, if k +1 is less than M, executing the step six, and if k +1= M, ending;
step six, according to the filter gain matrix K in the step four k+1 And calculating the upper bound omega of the covariance matrix of the filtering error k+1|k+1 ；
If k = k +1, executing step two until k +1= m is satisfied;
establishing a dynamic model of a nonlinear stochastic system in the first step; the specific process is as follows:
the dynamic model state space form of the nonlinear stochastic system is as follows:
y k ＝Ξ k C k x k +h(x k ,ζ k )+ν k (2)
in the formula (I), the compound is shown in the specification,
state variables at the time k and the time k +1 respectively; initial value x 0 Has a mean value of
Variance of P 0|0 ；
A real number domain of states of a dynamic model of the nonlinear stochastic system, n being a dimension;
is the measurement output at the time of the k-th time,
a real number domain of states of a dynamic model of the nonlinear stochastic system, m being a dimension; eta k And ζ k Is gaussian white noise with a mean value of zero and a variance of 1;
mean value is zero and variance is Q k Process noise, Q k ＞0，
Is the real number domain of the state of the dynamic model of the nonlinear stochastic system, p is the dimension; f (x) k ) For non-linear perturbations, g (x) k ,η k )、h(x k ,ζ k ) Is a non-linear function;
is a mean of zero and a variance of R k Measurement noise of (2), R k ＞0；C k And D k Respectively a measuring matrix at the k moment and a noise driving matrix at the k moment;
data loss matrix xi k ＝diag{ξ 1,k ,ξ 2,k ,...,ξ m,k -diag { · } is a diagonal matrix; xi i,k For m variables that are independent of each other with respect to i and k, i =1, 2.. Said., m, obeys a bernoulli distribution that takes a value of 1 or 0, and satisfies the following condition:
where i is the location of the data loss, k is the time,
is a definite mathematical expectation, Δ ξ i Uncertainty of delineation probability, i =1,2, \8230 { }, m, prob { } is probability,
is a desire of {. Cndot.);
non-linear function g (x) k ,η k ) And h (x) k ,ζ k ) Satisfies g (0, eta) k )＝0，h(0,ζ k ) =0 and the following conditions:
wherein s > 0 is a constant,
and
are all non-linear parameter matrices, r =1, 2.., s;
is composed of
Transpose of (g), g (x) j ,η j )、h(x j ,ζ j ) Is a nonlinear function, j is a time, k is not equal to j;
is composed of
The transpose of (a) is performed,
represents the expectation of {. Cndot. },
is x k Transposing;
in the second step, filter design is carried out on the dynamic model of the nonlinear stochastic system under the event trigger mechanism; the specific process is as follows:
firstly, aiming at the dynamic model of the nonlinear stochastic system established in the step one, the following event trigger formula is selected
(y k+j -y k ) T (y k+j -y k )＞σ (7)
Where σ > 0 is the adjustment threshold; y is k+j Is the measurement output at the k + j time, i.e. when equation (7) is satisfied, y k+j Allowed to transmit, j is time;
the actual output after event triggering from equation (7) is:
wherein k is a In order to trigger the moment of time,
is the actual output after the event is triggered,
for triggering a time k a A is the number of the current trigger sequence,
the actual output value after the event at the moment k is triggered;
aiming at the dynamic model of the nonlinear stochastic system established in the step one, the following filter structure is designed:
in the formula (I), the compound is shown in the specification,
an estimation function that is non-linear;
is x k One-step prediction at time k;
and
are each x k At the k-th time and k +1 timeIs estimated by the estimation of (a) a,
is an initial value of the estimation;
actual output at time k + 1;
in order to be a mathematical expectation of the data loss matrix,
diag {. Is a diagonal matrix;
for actual output after event triggering, C k+1 For the measurement matrix at time K +1, K k+1 A filter gain matrix at the k +1 moment to be designed;
calculating the one-step prediction error covariance matrix upper bound omega of the filter in the step three k+1|k (ii) a The specific process is as follows:
calculating the upper bound omega of the covariance matrix of the one-step prediction error of the filter according to the following formula k+1|k ：
In the formula, omega k+1|k An upper bound of a covariance matrix of the one-step prediction error at the moment k; omega k|k The upper bound of the filtering error covariance matrix at the moment k; epsilon 1 Is a constant greater than zero; gamma ray k ＞0；A k Is f (x) k ) In that
Point about x k Solving a matrix after the partial derivation; m k And L k Is f (x) k ) In that
A known matrix corresponding to the high-order term part after the point Taylor series expansion; d k Driving the matrix for noise at time k;
are respectively A k ，M k ，D k ，
L k Transposing; tr {. Is } represents the trace of {. Is };
in the fourth step, the upper bound omega of the one-step prediction error covariance matrix obtained in the third step k+1|k Calculating a filter gain matrix K k+1 (ii) a The specific process is as follows:
calculating a filter gain matrix K according to the following formula k+1 ：
In the formula, K k+1 A filter gain matrix at time k + 1;
represents a Hadamard product; i is the identity matrix, Ψ k+1|k In order to estimate the term for the non-linearity,
ε l is a constant greater than zero, l =1,2,3,4,5;
is epsilon l The inverse of (1);
is a constant;
are all nonlinear parameter matrices, R =1,2 k+1 For the covariance of the measurement noise at time k +1, tr {. Cndot } represents the trace of {. Cndot.;
and
are respectively as
C k+1 And
transposing;
in the sixth step, the gain matrix K is filtered according to the fourth step k+1 And calculating the upper bound omega of the covariance matrix of the filtering error k+1|k+1 (ii) a The specific process is as follows:
filtering error covariance matrix upper bound omega k+1|k+1 Is composed of
Wherein omega k+1|k+1 The covariance matrix upper bound of the filtering error at the moment of k + 1;
are respectively as
C k+1 ,K k+1 The transposing of (1).
2. The filtering method based on the event trigger mechanism as claimed in claim 1, wherein: the non-linear perturbation f (x) k ) In that
A point Taylor series expansion has
Wherein
Is the filtering error at the time instant k,
the high-order error term after Taylor series expansion is adopted; delta k Is an unknown time-varying matrix, satisfies
I is a unit matrix, and the unit matrix is,
is Δ k The transposing of (1).
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