Nonlinear spatial-time-varying filtering and identification algorithms for correlation-extremum dynamic systems with random structure and their linearization

1.1. The estimation problem statement

The problem under consideration is adaptive estimation for the dynamic state process model described by a stochastic nonlinear differential equation (Eq. (1)) [12-15]:

where $\mathbf{\Lambda }\left(t\right)$ is the state vector (of dimension $n$, in general case), $\mathbf{\Lambda }\in R^n$, which includes the random, unknown, and time-varying parameters vector ${\mathit{a}}^{\mathrm{T}}=(a_1,\dots ,a_q)$, with initial Gaussian value $\mathbf{\Lambda }{(t}_0)$, ${\mathbf{F}}^{\left(l\right)}(\mathbf{\Lambda },\mathbf{u},\mathrm{}t)$ is the nonlinear deterministic Lipschitz continuous vector function ${\mathbf{F}}^{\left(l\right)}\left(\mathbf{\Lambda },\mathbf{u},\mathrm{}t\right)=‖f_i^{\left(l\right)}(\mathbf{\Lambda },\mathbf{u},\mathrm{}t)‖$, $(i=\overline{1,n})$, $\mathbf{u}\left(t\right)$ is the known control vector, which may be a function of the state vector estimates components, $l\left(t\right)$ is a stationary Markov process taking values in the set $\left\{\mathrm{1,2},\dots ,p\right\}$ (system mode index or number of the state). Here ${\mathbf{W}}^{\left(l\right)}\left(t\right)$ is a vector process, $\mathit{W}\in R^d$, of the state Gaussian white noise with a zero-mean $E\left[{\mathbf{W}}^{\left(l\right)}\left(t\right)\right]=0$ and correlation matrix $K_w\left(t,\tau \right)=E\left[{\mathbf{W}}^{\left(l\right)}\left(t\right){{\mathbf{W}}^{\left(l\right)}}^T\left(\tau \right)\right]={\mathbf{Q}}^{\left(l\right)}\left(t\right)\mathrm{}\delta (t-\tau )$, where ${\mathbf{Q}}^{\left(l\right)}\left(t\right)$ is the diagonal intensity matrix ${\mathbf{Q}}^{\left(l\right)}\left(t\right)=‖{\mathrm{q}}_j^{\left(l\right)}\left(t\right)‖$, $(j=\overline{1,d},d\le n)$, $\delta$ is the delta function. The following notations are used: $E[\bullet ]$ denotes the expectation operator for stochastic processes; ${{\mathbf{W}}^{\left(l\right)}}^T$ is a transposed vector for ${\mathbf{W}}^{\left(l\right)}$.

The following measurement equation (Eq. (2)):

describes the observable signal $\mathbf{r}\left(x,\mathrm{}y,\mathrm{}t\right)$ as the spatial-time-varying process (of dimension $m$), $\mathit{r}\in R^m,(m\le n)$,where $x$, $y$are the space variables – space coordinates at any point – $x\in X=[x_0,x_X]$, $y\in \mathrm{Y}=[y_0,y_Y]$ , $t$ is the time variable $t\in T=[t_0,t_T]$, ${\mathbf{S}}^{\left(l\right)}\left(x,\mathrm{}y,\mathrm{}\mathbf{\Lambda },t\right)$ is the vector of STV signals of different physical nature, ${\mathbf{N}}^{\left(l\right)}\left(x,\mathrm{}y,\mathrm{}t\right)$ is the measurements STVGWN with diagonal intensity matrix ${\mathbf{C}}_0^{\left(l\right)}$ and correlation function $h^{\left(l\right)}(\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }t)={\mathbf{C}}_0^{\left(l\right)}\delta (\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }t)$. The state system noise and the measurements noise are assumed independent and temporally uncorrelated.

The Markov process describes the random changes of the structure with p -finite states and the transition intensities ${\mathrm{\nu }}_{jl}\left(t\right)$ and ${\mathrm{\nu }}_j\left(t\right)$, where $j,\mathrm{}l=\overline{1,p}$. The system behavior and the features of the system may be explicated as follows. The system begins in a particular mode of operation, say $l\left(t\right)=1$, then at a random time the system jumps to one of the other $\left(p-1\right)$ possible modes of operation and may or may not remain in this state, and the dynamics differential equations corresponding to the different switch positions form the description of the system dynamics for each state. The transitions may occur from one state (or location) to another under varying and uncertain external conditions (e.g., when the process dynamics at a certain state would take the associated continuous state outside a distinct region of the state space), with Markov process models describing the continuous stochastic process of system (e.g., target) dynamics and digital process of the mode or structure changes.

The nonlinear filtering or estimation problem described above by the system and measurement equations (Eq. (1-2)) is to determine the finite-dimensional dynamical system whose output is the best minimum variance estimate of the joint Markov process ${\left(\mathbf{\Lambda }\right(t),\mathrm{}l(t\left)\right)}^T$, for $t\ge 0$ given the STV observed data $\mathit{r}(x,y,t)$.

1.2. The algorithms synthesis and linearization problem solution

The STV signal processing algorithms synthesis and analysis problem solution [12-15] was built upon an integration of two theories – the correlation-extremum systems theory and the theory of stochastic systems with random structure, and was based on the generalized Fokker-Plank-Kolmogorov-Stratonovich differential equation for the evolution of joint conditional probability density function of the state dynamics $\mathbf{\Lambda }\left(t\right)$ and the system structure $l\left(t\right)$ given the STV measurement data $\mathit{r}\left(x,y,t\right)$, $\omega \left(\mathbf{\Lambda },l,t|\mathit{r}\left(x,y,\tau \right),t_0\le \tau \le t\right)=\widehat{\omega }\left(\mathbf{\Lambda },\mathrm{}l,t\right)={\widehat{\omega }}^{\left(l\right)}\left(\mathbf{\Lambda },\mathrm{}t\right)$ (with the initial value $\omega \left({\mathbf{\Lambda }}_0,t_0\right)$ of probability density of the state dynamics $\mathbf{\Lambda }\left(t_0\right)$) (the sign ^ means the a posteriori function value).

The a posteriori probability density $\widehat{\mathrm{\omega }}\left(\mathbf{\Lambda },t\right)$ for the whole dynamics process is determined by the following expression (Eq. (3)):

where ${\widehat{P}}_l\left(t\right)$ is the a posteriori probability of the $l\mathrm{t}\mathrm{h}$ state, whose evolution is defined by the state probability estimate differential equations (Eq. (4)) (or discrete (for a discrete problem statement)), the presence of which in the filtering and identification algorithms and the relation between them form the main distinguishing properties of signal processing in systems with the random structure [12, 13, 15]:

where $\mathrm{}l=\overline{1,p}$, ${\mathcal{F}}^{\left(l\right)}(\mathbf{\Lambda },\mathbf{r},t)$ is the derivative of the likelihood function logarithm in the $l\mathrm{t}\mathrm{h}$ state ($z$ is the variable, $z\in \mathbf{\Lambda })$. The a priori state probabilities $P_l\left(t\right)$ can be found using the known Kolmogorov equations.

All processes are defined on the probability space ($\mathrm{\Omega }$, $\mathcal{F}$, $\tilde{P}$), where $\left\{{\tilde{P}}_{\lambda .},\lambda \in \mathbf{\Lambda }\right\}$ is a family of probability measures on ($\mathrm{\Omega }$, $\mathcal{F}$) which are absolutely continuous with respect to a fixed probability measure ${\tilde{P}}_0$, with the corresponding $\sigma$-algebra $\mathcal{F}$. The likelihood function $L_f\left(\mathbf{\Lambda }\right)$ for obtaining the estimate $\widehat{\mathbf{\Lambda }\mathbf{}}$ of the state process $\mathbf{\Lambda }$ (including parameters) is based on the information contained in STV measurements $\mathit{r}\left(x,y,\tau \right),t_0\le \tau \le t$, and both are defined by the expressions:

(the relationship between the likelihood function and the Radon-Nikodym derivative), and $\widehat{\mathbf{\Lambda }}\in \underset{\mathrm{\lambda }\in \mathbf{\Lambda }}{\mathrm{argmax}}{L}_f\left(\mathbf{\Lambda }\right).$ To determine the function ${\mathcal{F}}^{\left(l\right)}(\mathbf{\Lambda },\mathbf{r},t)$ it is necessary to know the likelihood functional for the measurements additive STVGWN ${\mathbf{N}}^{\left(l\right)}\left(x,\mathrm{}y,\mathrm{}t\right)$. The conditional probability density $\omega \left(\mathit{r}\left(x,y,\tau \right)|\mathbf{\Lambda },\mathrm{}l,\tau ,t_0\le \tau \le t\right)$ called likelihood function $L_f\left(\mathbf{\Lambda },\mathrm{}l\right)=L_f^{\left(l\right)}\left(\mathbf{\Lambda }\right)$ (as a function of $\mathbf{\Lambda }$ and $l$ (for systems with random structure)) is indubitably normal for a linear system. In the case under consideration the measurement $\mathit{r}\left(x,y,\tau \right)$ is the sum of the normal random process ${\mathbf{N}}^{\left(l\right)}\left(x,\mathrm{}y,\mathrm{}\tau \right)$ and the deterministic signal function ${\mathbf{S}}^{\left(l\right)}\left(x,\mathrm{}y,\mathrm{}\mathbf{\Lambda },\tau \right)$ (or may be random signal function depending on Gaussian value $\mathbf{\Lambda }\left(\tau \right)$) (for example, when the target intensity pattern is modeled by a Gaussian function).

Denote the limit of the conditional probability density $\omega \left(\mathit{r}\left(x,y,\tau \right)|\mathbf{\Lambda },\mathrm{}l,\tau ,t_0\le \tau \le t\right)$ by $L^{\left(l\right)}\left(\mathbf{\Lambda }\right):$

The principle of the likelihood functional $L^{\left(l\right)}\left(\mathbf{\Lambda }\right)$ maximum for the STVGWN ${\mathbf{N}}^{\left(l\right)}\left(x,\mathrm{}y,\mathrm{}t\right)$ with the spectral density matrix ${\mathbf{C}}_0^{\left(l\right)}$ is first extended in this research to the systems with random structure in the form (Eq. (5)):

where $c^{\left(l\right)}$ is a value depending on ${\mathbf{C}}_0^{\left(l\right)}$; $X$, $Y$ and $T$ are the spatial and time limits of integration (or spatial and time domains of observation).

The suboptimal estimate $\widehat{\mathbf{\Lambda }}\left(t\right)$ of the state dynamics Markov process $\mathbf{\Lambda }\left(t\right)$ on the assumption of mean square loss function is the conditional mathematical expectation. The optimal estimate of discrete process $l\left(t\right)$ by the a posteriori probability criterion will be such a value of $l$ that makes the value of the a posteriori probability ${\widehat{P}}_l\left(t\right)$ maximum.

The suboptimal (due to nonlinearities) estimate of the state is $\widehat{\mathbf{\Lambda }}\left(t\right)=\sum _{l=1}^p{\widehat{P}}_l\left(t\right){\widehat{\mathbf{\Lambda }}}^{\left(l\right)}\left(t\right)$.

The signal position on the image plane $XOY$ (for example, the FLIR image plane) can be determined by parameters vector: ${\mathrm{\lambda }}_x\left(t\right)={\phi }_x(\mathbf{\Lambda },t)$, ${\lambda }_y\left(t\right)={\phi }_y(\mathbf{\Lambda },t)$ (whose dynamics defines the target intensity pattern on image plane). Then the signal ${\mathbf{S}}^{\left(l\right)}\left(x,\mathrm{}y,\mathrm{}\mathbf{\Lambda },t\right)$ may be represented as a function ${\mathbf{S}}^{\left(l\right)}\left(x,\mathrm{}y,\mathrm{}\mathbf{\Lambda },t\right)={\mathbf{S}}^{\left(l\right)}\left(x-{\lambda }_x,\mathrm{}y-{\lambda }_y,t\right)$. In this case the suboptimal estimator is defined as a tracker system. Assuming that the signal or image position along one of the axes (e.g., in $y$ direction) is known, to simplify the derivation, and denoting the STV signal ${\mathbf{S}}^{\left(l\right)}\left(x,\mathbf{\Lambda },t\right)={\mathbf{S}}^{\left(l\right)}\left(x-{\lambda }_x,t\right)$, and the state parameter ${\lambda }_x$ without index ${\lambda }_x=\lambda .$, the measurement equation (Eq. (2)) becomes: $\mathbf{r}\left(x,t\right)={\mathbf{S}}^{\left(l\right)}\left(x-\lambda .,t\right)+{\mathbf{N}}^{\left(l\right)}\left(x,\mathrm{}t\right).$

For the case when the changes of the structure and corresponding parameters form Markov process with two states ($l=\overline{1,2}$) and the transition intensity $\mathrm{\nu }\left(t\right)$ the nonlinear filtering problem solution for correlation-extremum systems with random structure derived in [12, 13] is presented by the following correlation-extremum algorithms (Eq. (6-9)).

The differential Eq. (6) for the a posteriori probabilities of state ${\widehat{\mathrm{P}}}_l\left(t\right)$ is presented below:

${\widehat{P}}_2\left(t\right)=1-{\widehat{P}}_1\left(t\right)$, where ${\widehat{P}}_2\left(t\right)$ is the a posteriori probability of the second state, ${\mathrm{\Delta }\mathrm{\lambda }}_{\left(l\right)}\left(t\right)$ is the state estimate error ${\Delta \lambda }_{\left(l\right)}\left(t\right)=\lambda \left(t\right)-{\widehat{\lambda }}^{\left(l\right)}\left(t\right)$, $l=\overline{\mathrm{1,2}}$; ${\sigma }_{\left(l\right)}^2\left(t\right)$ is the variance of the a posteriori probability density function ${\sigma }_{\left(l\right)}^2\left(t\right)=〈{\left[\right(\lambda \left(t\right)-{\widehat{\lambda }}^{\left(l\right)}\left(t\right)]}^2〉$, $l=\overline{1,2}$; $B^{\left(l\right)}\left({\Delta \lambda }_{\left(l\right)},t\right)$ is the spatial correlation function in the $l\mathrm{t}\mathrm{h}$ state $B^{\left(l\right)}\left({\Delta \lambda }_{\left(l\right)},t\right)=\langle {{\mathbf{S}}^{\left(l\right)}}^T\left(x-{\widehat{\lambda }}^{\left(l\right)},t\right){\mathbf{S}}^{\left(l\right)}(x-\lambda ,t)\rangle$, (or for the scalar measurement: $B^{\left(l\right)}\left({\Delta \lambda }_{\left(l\right)},t\right)=\langle S^{\left(l\right)}\left(x-{\widehat{\lambda }}^{\left(l\right)},t\right)S^{\left(l\right)}(x-\lambda ,t)\rangle$); ${\mathrm{C}}_X^{\left(l\right)}$ is the specific spectral intensity of the STVGWN ${\mathbf{N}}^{\left(l\right)}\left(x,\mathrm{}t\right)$ in the $l$th state: $C_X^{\left(l\right)}=C_0^{\left(l\right)}/X$.

The derivation of the equations Eq. (6) assumes that the parameters are considered as “unpowered” (the term well known in the signal processing theory, first applied in radar signal processing), which is to say that the integrals $\underset{–X}{\overset{X}{\int }}[{S^{\left(l\right)}\left(x-{\widehat{\lambda }}^{\left(l\right)},t\right)]}^{2}dx$ (representing the signal power) and $\underset{-X}{\overset{X}{\int }}{r}^2\left(x,t\right)dx$ (which are explicitly independent of the estimated parameter), may be taken into account in the coefficients $k_1$ and $k_2$. A further assumption in the algorithms synthesis is that the integral limits X and Y are vastly larger than the signal correlation intervals $(X\gg {{∆}_X}_{cor},Y\gg {{∆}_Y}_{cor})$.

The state estimate equation (Eq. (7)) has been derived in the form:

where $N_X^{\left(l\right)}={\int }_{-X}^X\frac{\partial {{\mathbf{S}}^{\left(l\right)}}^T\left(x–{\widehat{\lambda }}^{\left(l\right)},t\right)}{\partial {\widehat{\lambda }}^{\left(l\right)}}{\mathbf{N}}^{\left(l\right)}\left(x,t\right)dx$ (As it is known from the signal theory, the measurements signals and noises in many cases are (or are assumed to be) uncorrelated). The suboptimal state estimate of the whole process (for two states) can be obtained by using a weighted sum: $\widehat{\lambda }\left(t\right)={\widehat{P}}_1\left(t\right){\widehat{\lambda }}^{\left(1\right)}\left(t\right)+{\widehat{P}}_2\left(t\right){\widehat{\lambda }}^{\left(2\right)}\left(t\right)$.

The variance differential equation is presented below:

and the estimate error variance for the whole process is: ${\sigma }^2\left(t\right)={\widehat{P}}_1\left(t\right){\sigma }_{\left(1\right)}^2\left(t\right)+{\widehat{P}}_2\left(t\right){\sigma }_{\left(2\right)}^2\left(t\right)$.

The suboptimal (due to nonlinearities) estimate $\widehat{\lambda }\left(t\right)$ and covariance ${\sigma }^2\left(t\right)$ for the whole process represent weighted sums, where the $l$th weighting factor ${\widehat{P}}_l\left(t\right)$ is the a posteriori $l$th hypothesis conditioned probability.

In this paper the spatial-time-varying filtering in systems with random structure with changing noise statistics (intensities) (as well as other characteristics) for each state is proposed. For this noise intensity identification problem the suboptimal estimate (in the minimum mean square error sense) of the measurement noise intensity $C_0$ (considering as a time-varying parameter) at a given time $t_i$ can be obtained from the following conditional mean value Eq. (9) (originally presented in this paper for spatial-time varying signal processing):

where $P_l\left(t_i\right)|{\mathit{r}}_i$ is the conditional probability of the state or mode $l$, conditioned on the observed measurements to time $t_i$, (e.g., for two states (or structures) two STVGWN models can be used with different intensity levels corresponding to the lower and upper statistics bounds). The spectral densities may be treated for the case of stationary measurements noise ${\mathbf{C}}_0^{\left(l\right)}$, and for the case of nonstationary noise ${\mathbf{C}}_0^{\left(l\right)}\left(t\right)$ depending on some of the state vector components or parameters (e.g., [13]).

Spatial-time-varying filtering in STVGMCN, proposed in [15] may also be useful to discriminate between target (or another object) and background intensity patterns.

The filtering algorithm (Eq. (6-8)) is composed of two separate estimators processing the STV signals or images in parallel and exchanging information according to the state estimate equation (Eq. (7)). The signals of different nature fields are used to compute the a posteriori probabilities ${\widehat{P}}_l\left(t\right)$ via Eq. (6). The suboptimal parameter estimate $\widehat{\mathrm{\lambda }}\left(t\right)$ corresponds to the maximum value of the cross-correlation function of the reference and received STV signals, when $\mathrm{\Delta }{\mathrm{\lambda }}_{\left(l\right)}\left(t\right)=0$.

The proposed algorithms (Eq. (6-9)) can be applied to the solution of the linearization problem in the case of great estimation errors. The fundamental difference between the approaches taken in [1] and in the present work lies in the first application of the systems with random structure theory here to STV signal processing.

In this paper the following new linearized correlation-extremum algorithms for systems with random structure have been derived (Eqs. (10-12)) using the Taylor series expansion 1) of the cross-correlation function $B^{\left(l\right)}\left(\mathrm{\Delta }{\mathrm{\lambda }}_{\left(l\right)}\right)$, and 2) of the cross-correlation function second derivative $\frac{{\partial }^2{B}^{\left(l\right)}\left(\mathrm{\Delta }{\lambda }_{\left(l\right)}\right)}{\partial \mathrm{\Delta }{\lambda }_{\left(l\right)}^2}$, taking into account the features of the singular processes (${B^{\text{'}}}^{\left(l\right)}\left(0\right)=0$, and ${B^{\text{'}\text{'}}}^{\left(l\right)}\left(0\right)<0$), to simplify the study of the a posteriori probabilities functions ${\widehat{P}}_l\left(t\right)$ behavior.

The equations for the a posteriori probabilities of state may be defined as:

The linearization of the state estimate equation (Eq. (7)) has been obtained by 1) applying the series approximation to $f\left({\widehat{\lambda }}^{\left(l\right)},u,t\right)$, and 2) the series approximation of the spatial correlation function derivatives $\frac{\partial B^{\left(l\right)}\left(\mathrm{\Delta }{\lambda }_{\left(l\right)},t\right)}{\partial \mathrm{\Delta }{\lambda }_{\left(l\right)}}$, in the form:

For the whole process the suboptimal state estimate is: $\widehat{\lambda }\left(t\right)={\widehat{P}}_1\left(t\right){\widehat{\lambda }}^{\left(1\right)}\left(t\right)+{\widehat{P}}_2\left(t\right){\widehat{\lambda }}^{\left(2\right)}\left(t\right)$. The variance equation (the Riccati-type one) with linearized functions is:

and for the whole process the estimate error variance: ${\sigma }^2\left(t\right)={\widehat{P}}_1\left(t\right){\sigma }_{\left(1\right)}^2\left(t\right)+{\widehat{P}}_2\left(t\right){\sigma }_{\left(2\right)}^2\left(t\right)$. As a remark: for quasi-singular processes the first derivative of the cross-correlation function of zero is not equal to zero (${B^{\text{'}}}^{\left(l\right)}\left(0\right)\ne 0$), therefore the linearized algorithms equations will contain the terms with the first derivative of the cross-correlation function. In a nonlinear estimator for every ${\mathrm{\Delta }\lambda }_{\left(l\right)}$ it is necessary to know the correlation function derivatives, but using the series approximation, the value of $\frac{{\partial }^{2}B\left(\xi \right)}{\partial {\xi }^2{|}_{\xi =0}}$ can be interpreted as a constant coefficient. The linearized estimate equations (Eq. (11)) define the filtering scheme as the two-states (or two-channels) tracking system, which changes its threshold according to linearized Eq. (10).

For the class of tracking systems an adaptation mechanism of the proposed estimation algorithms represents an effective means of adaptive switching (expansion and contraction) of the effective field of view of a radar tracker or an optics (IR image) tracker, and ensures the system operation, particularly, for a wide dynamic range of target maneuvers.

The variance equations (nonlinear and the linearized one) represent the new Riccati-type differential equations first obtained for STV signal processing in STVGWN for systems with random structure (with cross correlation functions (and their derivatives) in the nonlinear term).

The adaptive filtering algorithms (nonlinear and the linearized ones) for correlation-extremum systems with random structure 1) for estimation of signal position along the axis $y$, and 2) for both components of the state vector ${\lambda }_x\left(t\right)$ and ${\lambda }_y\left(t\right)$ have been derived.

The application of the proposed correlation-extremum filtering algorithms to the systems with interrupted signal information (the filtering problem with observation process feedback adaptive control) has been investigated and the new appropriate filters have been derived for the cases of estimation problem with unobservable moments of changing structure, and for a special case of estimation with the known moments of changing structure.