Design of unbiased reduced order filter for stochastic systems with sampled measurements

The problem of synthesizing reduced order filter, that forms an unbiased estimate of linear functional from an unknown phase vector of a continuous time stochastic system with sampled measurements and additive noises, is considered in this work. The H 2 norm of the weighted transfer function from input noises to estimation error is used as a performance criterion. The method for reduced order filter design is based on the expression of the explicit dependence of performance criterion on unknown parameters in the canonical basis and the application of the non-convex optimization technique. Numerical simulations are performed to illustrate the proposed approach.


Introduction
The object of the work is the development of methods for constructing reduced order filters for continuous time stochastic systems with sampled measurements and additive noises.
The well-known [1] full order Kalman, Kalman-Bucy filters and Kolmogorov, Wiener filters provide the minimum variance and minimum steady-state variance of the estimation error for the full phase vector of discrete and continuous linear systems, respectively.For continuous systems with discrete measurements, there is also a well-known [1] full order continuous-discrete Kalman filter.Instead of these full order filters, to decrease the computational effort of calculating the state estimation of large-scale or high-speed systems, one uses reduced order filters or functional filters to estimate a linear functional from the system state vector.
There are various methods for constructing reduced order discrete and continuous filters or observers using the technique of generalized inverse matrices [2], linear matrix inequalities [2,3], linear algebraic operations [4], Oja flow [5], matrix decomposition [6].For the synthesis of sampled-data control systems, including continuous-discrete full order filters, a new technique of differential linear matrix inequalities [7] has been developed.
In this work, one uses an approach integral quadratic performance measures [8][9][10] that allows to determine the explicit dependence of the optimality criterion on unknown parameters in the canonical basis and apply the non-convex optimization technique.
The following standard notations are used.C, R, R + , Z + and N denote the sets of complex, real, non-negative real, non-negative integer and natural numbers, respectively.E [•] denotes the mathematical expectation operator.(•) T indicates transposition of a vector or matrix.I n and 0 represent the identity matrix of size n, and the zero or zero matrix of the appropriate size.j denotes an integer or √ −1.tr [X] indicates the trace of the square matrix X. span {S} represents set of all linear combinations of the vectors in S.
The work is organized as follows.In the next section, a problem statement is given and the dynamics of the original system and the desired filter are described.The unbiased conditions of the desired filter are studied in Section 3. In Section 4 an approach for criterion calculation is given by the filter design algorithm through the integral quadratic performance measures.Section 5 contains a numerical example to illustrate the obtained results.Finally, Section 6 concludes the work.

Problem Statement
Consider an n-dimensional linear continuous-discrete system: where x(•) ∈ R n is the state vector; y(t i ) ∈ R l is the measured output available only at the sampling times defined as is the vector to be estimated.The matrices A, C and F are constant.Zero mean white noise processes w(t), v(t i ) of dimensions n and l, respectively, are uncorrelated with each other and with the initial state x(0) is a random variable with known probabilistic characteristics: where i, j ∈ Z + , t, τ ∈ R + ; Q, P 0 are a priori known positive semidefinite matrices and R is a priori known positive matrix; δ(t − τ ) is Dirac delta function; δ ij is Kronecker delta function.
The desired reduced order filter has the following state-space implementation where q(•) ∈ R k is the state vector of the filter and the estimation of the vector q(•) = T x(•), k < n; q − (t i ) is the state vector of the filter before measurement y(t i ) is processed; σ(t) ∈ R p is the unbiased estimation formed by the filter.
The problem is to design the reduced order filter ( 5)-( 7) for the continuous-discrete system (1)-(3) with probabilistic characteristics (4) such that the sum of the steady-state mean values of the squared continuous and discrete estimation error e(

Main Result
The following Lemma is used to obtain the main result about the unbiasedness of the designed filter.Lemma 1.The mean value of the state vector of the system (1) is Proof.The solution of the system (1) with initial value x(0) is given by Determining the mathematical expectation of both parts of the previous equation and using the probabilistic characteristics (4) we obtain that E [x(t)] = e At x0 .From the definition of the matrix exponential in terms of a power series and the Cayley-Hamilton theorem, we get E [x(t)] ∈ span x0 , Ax 0 , . . ., A n−1 x0 .
Conditions for the unbiased estimate of the linear functional of the state vector can be formulated as the following Theorem.Theorem 1. Necessary and sufficient conditions for the unbiasedness of the estimate formed by the reduced order filter ( 5)- (7) for the system (1)-( 3) with probabilistic characteristics (4) are as follows where C A,x 0 is controllability matrix for pair {A, x0 }, i.e.

Proof.
Denoting the estimation error ε(•) = q(•) − q(•) and the estimation error ε − (t i ) = q(t i ) − q − (t i ) before processing y(t i ) from equations (1), ( 2), ( 5), (6), we get with initial state ε(0) = T (x(0) − x0 ).The solution of the estimation error system (12) on the [t j , t j+1 ) is given by It follows from Lemma 1 and ( 4), ( 13) that Expressions for the estimation error e(t) are derived similarly from equations ( 3), (7) and Lemma 1 From these relations and Lemma 1 can be deduced that the estimates q(•) and σ(•) are unbiased if and only if the conditions ( 9), ( 10) and ( 11) are satisfied.Remark 1.In the case of the full order filter ( 5)- (7) where k = n, T = I n and rank C A,x 0 = n, the conditions of Theorem 1 give the matrices of the stationary continuous-discrete Kalman filter [1,7]: where K ∈ R n×l is optimal gain matrix.

Criterion Calculation
To calculate the optimality criterion (8), one can interpret the steady-state value of the squared observation error as the H 2 norm of the weighted transfer function from the noises w(t) and v(t i ) to the estimation error e(•) For continuous time noise w(t) transfer function is determined from the equations ( 1), ( 12), (14), as Introducing with respect to discrete time noise v(t i ) a discrete time system equivalent to the estimation error system (12)-( 14) by the Further calculation of the optimality criterion is possible using integral quadratic performance measures and non-convex optimization technique, as was described in previous works [8][9][10].3) defined in the canonical Luenberger basis with h = 0.1, following matrices and probabilistic characteristics (4) From the Theorem 1 we obtain following matrices of the first order unbiased filter ( 5)-( 7) where the parameters a, b, c ∈ R are such that e Nch N d = e h a ∈ (−1, 1) for convergence of the estimation error.
From the Section 4 we get following transfer matrices from the noises w(t) and v(t i ) to the estimation error e(•) W ew (s) = 0 Hence, the value of the optimality criterion for the continuous component according to the formula (15) is The explicit dependence on the parameters of the optimality criterion for the discrete component by the formula (16) has the form The global minimum point and the global minimum value of the function (18) over the feasible set are From the Remark 1 we obtain the following matrices of the full order stationary continuousdiscrete Kalman filter ( 5)-( 7

Conclusion
In this work, a new method for the synthesis of unbiased reduced order filters was proposed.Necessary and sufficient conditions for the unbiasedness of the estimate formed by a reduced order filter for a continuous time stochastic system with sampled measurements and additive noises were obtained.Calculation of the optimality criterion is based on the use of integral quadratic performance measures that made it possible to determine the explicit dependence of the optimality criterion on unknown parameters.The computational example was carried out to illustrate the results.
c t P T P e Nct dt; we can obtain the transfer function for discrete time noise v(t i ) W ev (z) = −P d N d (zI k − e Nch N d ) −1 e Nch M − P d M, z ∈ C.

Figure 1 .
Figure 1.Dynamics of root-mean-square errors in continuous and discrete time