Embedding non-linear filtering in autonomous underwater vehicle dynamics via the Kolmogorov backward equation

Abstract

This paper revisits the state vector of an autonomous underwater vehicle (AUV) dynamics coupled with the underwater Markovian stochasticity in the ‘non-linear filtering’ context. The underwater stochasticity is attributed to atmospheric turbulence, planetary interactions, sea surface conditions and astronomical phenomena. In this paper, we adopt the Itô process, a homogeneous Markov process, to describe the AUV state vector evolution equation. This paper accounts for the process noise as well as observation noise correction terms by considering the underwater filtering model. The non-linear filtering of the paper is achieved using the Kolmogorov backward equation and the evolution of the conditional characteristic function. The non-linear filtering equation is the cornerstone formalism of stochastic optimal control systems. Most notably, this paper introduces the non-linear filtering theory into an underwater vehicle stochastic system by constructing a lemma and a theorem for the underwater vehicle stochastic differential equation that were not available in the literature.

Keywords

Kolmogorov–Fokker–Planck equation Kolmogorov backward equation underwater non-linear Itô stochastic differential equation (SDE)underwater vehicle–ocean current random interactions

Introduction

Mumford argues the idea of the dawning of age of stochasticity. Stochastic processes are the reality. Accounting them into dynamical systems refines algorithmic procedures. Notable applications of stochastic processes can be found in mathematical finance, communications and theoretical biology, including autonomous systems. The famous papers of Mumford (2000) and Sinai (1981) are useful in this regard. They suggested that stochastic methods will change the shape of modern research. The noise equations of the underwater vehicle involve the multi-dimensional state vector as well as construction of the stochastic differential equations (SDEs) (Jazwinski, 1970; Poznyak, 2009) in lieu of ordinary differential equations. This seems to be one of the reasons why stability, estimation and control of underwater vehicles with noise processes are not available in the literature in greater detail. Non-linear filtering in combination with control signal achieves the closeness of the estimated trajectory with the actual and controlled state trajectories.

Here, we explain succinctly the non-linear filtering historic account, as the non-linear filtering is the cornerstone formalism of the paper. The Wiener filter is a signal estimation method and the Wiener filter transfer function is irrational. In 1960, Kalman sowed the seed of the filtering of dynamic systems, where he considered the linear SDE coupled with linear observation equation (Anderson and Moore, 1991). Furthermore, the notion of non-linear Kalman filtering was introduced. A remarkable paper on non-linear filtering can be traced back to a pioneering work of 1967 by HJ Kushner. That is popularly known the Kushner–Stratonovich equation, see Pugachev and Sinitsyn (1987). That is also known as the posterior density. An alternative interpretation to the Kushner–Stratonovich is the FKK filtering of Fujisaki et al. (1972). Realization of higher-order filtering on a single complex chip is still a potential research problem.

In this paper, we construct two filtering models of autonomous underwater vehicle (AUV) dynamics. The filtering model accounts for the SDE of the state vector coupled with noisy observation equation. This paper is an extension of a recently published paper, Sharma and Hirpara (2014), in the following senses: (i) this paper revisits the stochastic underwater vehicle dynamics in the continuous-discrete non-linear filtering sense. On the other hand, the Fokker–Planck setting and multi-dimensional stochastic differential rules were the major ingredients to analyse the stochasticity of the AUV dynamics in Sharma and Hirpara (2014). (ii) This paper accounts for the contributions of the process noise correction term as well as the observation noise correction term in the non-linear filtering equations of the AUV dynamics. On the other hand, the observation noise correction term is ignored in Sharma and Hirpara (2014). Secondly, this paper can be regarded as a step towards resolving an open problem stated in Aguiar and Hespanha (2003: 1993) as well as extending the work of Aguiar and Pascoal (2007) in stochastic perspectives. Filtering theory plays a central role in navigation of AUVs. Zhao et al. (2014) developed a particle filter for fault diagnosis and robust navigation of underwater robots. Allotta et al. (2016) developed a new AUV navigation system exploiting unscented Kalman filter. Rigatos et al. (2017) worked on AUV control and navigation via differential flatness theory and derivative-free non-linear Kalman filtering. Alzahrani (2018) developed an AUV navigation system using acoustic and inertial sensors. To understand underactuated AUV non-linear finite-time tracking control based on command filter and disturbance observer see Xu et al. (2019). In Demirbaş (2007, 2010), ‘non-linear discrete-time systems’ were the subject of investigations from the prediction viewpoint. On the other hand, the ‘non-linear continuous-time system’ coupled with discrete observation equation is the subject of investigations.

The major ingredients of this paper are the famous Kolmogorov backward equation as well as the evolution of conditional characteristic function of the underwater vehicle SDE. The complexity associated with non-linear filtering for the ‘continuous state-discrete measurement system’ is chiefly attributed to two factors. First, the continuous-discrete filtering algorithm, a two-stage estimation procedure, encompasses the prediction algorithm not accounting for observation terms. The filtering algorithm at the discrete-time instant accounts for observation correction terms. As a result of these, the continuous-discrete filtering involves two different sets of conditional mean and conditional variance evolution equations. On the other hand, the non-linear filtering for the ‘continuous state-continuous measurement system’ involves one set of conditional mean and conditional variance evolution equations.

Second, deriving the non-linear filtering equations under ‘nearly’ Gaussian assumptions requires replacing the higher-order conditional moment with conditional variance terms (Jazwinski, 1970: 336). The relationship involves the concept of the conditional characteristic function as well as greater permutation terms resulting in lengthy expressions. On the other hand, non-linear filtering for the continuous state-continuous measurement is a one-stage procedure. Perhaps these are the reasons that non-linear continuous-discrete filtering algorithms for potential control problems are relatively much less explored in contrast to non-linear continuous filtering algorithms.

The underwater vehicle filtering

The deterministic AUV model is available in well-known books and seminal papers, for example Aguiar and Pascoal (2007) and Fossen (1994).

M_{RB} \overset{\cdot}{v} + C_{RB} (v) v = τ_{RB}

(1)

where

\begin{matrix} M_{RB} = (\begin{matrix} m & 0 & 0 & 0 & 0 & 0 \\ 0 & m & 0 & 0 & 0 & 0 \\ 0 & 0 & m & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{x} & 0 & 0 \\ 0 & 0 & 0 & 0 & I_{y} & 0 \\ 0 & 0 & 0 & 0 & 0 & I_{z} \end{matrix}) \\ C_{RB} = (\begin{matrix} 0 & - mr & mq & 0 & 0 & 0 \\ mr & 0 & - mp & 0 & 0 & 0 \\ - mq & mp & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & I_{z} r & - I_{y} q \\ 0 & 0 & 0 & - I_{z} r & 0 & I_{x} p \\ 0 & 0 & 0 & I_{y} q & - I_{x} p & 0 \end{matrix}) \end{matrix}

\begin{matrix} C_{RB} = (\begin{matrix} 0 & 0 & 0 & 0 & m v_{z} & - m v_{y} \\ 0 & 0 & 0 & - m v_{z} & 0 & m v_{x} \\ 0 & 0 & 0 & m v_{y} & - m v_{x} & 0 \\ 0 & m v_{z} & - m v_{y} & 0 & I_{z} r & - I_{y} q \\ - m v_{z} & 0 & m v_{x} & - I_{z} r & 0 & I_{x} p \\ m v_{y} & - m v_{x} & 0 & I_{y} q & - I_{x} p & 0 \end{matrix}) \\ \overset{\cdot}{v} = {(\begin{matrix} {\overset{\cdot}{v}}_{x} & {\overset{\cdot}{v}}_{y} & {\overset{\cdot}{v}}_{\overset{\cdot}{z}} & \overset{\cdot}{p} & \overset{\cdot}{q} & \overset{\cdot}{r} \end{matrix})}^{T} \\ v = {(\begin{matrix} v_{x} & v_{y} & v_{\overset{\cdot}{z}} & p & q & r \end{matrix})}^{T} τ_{RB} = {(\begin{matrix} X & Y & Z & K & M & N \end{matrix})}^{T} . \end{matrix}

where the term $m$ denotes the mass of the body, the vector $(v_{x}, v_{y}, v_{z})^{T}$ is the linear velocity vector of the vehicle in the rotating body-fixed frame. The vector $(p, q, r)^{T}$ is the angular velocity vector of the co-ordinates of the vehicle. The vector $(X, Y, Z)^{T}$ denotes the external forces acting on the vehicle. The vector $(K, M, N)^{T}$ has an interpretation as the vector moment about the co-ordinates of the vehicle. Equation (1) holds for the underwater vehicle equations of motion, where the velocity of the underwater vehicle is relatively less and the centre of gravity of the vehicle coincides with the origin of the body-fixed frame.

After accounting for stochastic correction terms in the external forces and the moments of external forces in the white noise setting, we find the following time-varying non-linear Itô SDE in the formal setting (Sharma and Hirpara, 2014):

d ξ_{t} = a (t, ξ_{t}) dt + b (t) d B_{t},

(2)

where

\begin{matrix} ξ_{t} = (ξ_{i} (t))_{1 \leq i \leq 6} = (v_{x}, v_{y}, v_{z}, p, q, r)^{T} \\ = (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6})^{T}, \end{matrix}

(3a)

\begin{matrix} a_{t} (t, ξ_{t}) = (a_{t}^{i} (t, ξ_{t}))_{1 \leq i \leq 3} \\ = {(\frac{X}{m} + x_{2} x_{6} - x_{3} x_{5}, \frac{Y}{m} + x_{3} x_{4} - x_{1} x_{6}, \frac{Z}{m} + x_{1} x_{5} - x_{2} x_{4})}^{T}, \end{matrix}

(3b)

\begin{matrix} a_{r} (t, ξ_{t}) = (a_{r}^{i} (t, ξ_{t}))_{1 \leq i \leq 3} \\ = (\frac{K}{I_{x}} + \frac{(I_{y} - I_{z})}{I_{x}} x_{5} x_{6}, {\frac{M}{I_{y}} + \frac{(I_{z} - I_{x})}{I_{y}} x_{4} x_{6}, \frac{N}{I_{z}} + \frac{(I_{x} - I_{y})}{I_{z}} x_{4} x_{5})}^{T} . \end{matrix}

(3c)

\begin{matrix} (b_{ii} (t))_{1 \leq i \leq 6} = (γ_{1}, γ_{2}, γ_{3}, γ_{4}, γ_{5}, γ_{6})^{T}, \\ b_{ij} (t)_{i \neq j} = 0, i \neq j, d B_{t} = ({dB}_{t}^{v_{x}}, {dB}_{t}^{v_{y}}, {dB}_{t}^{v_{z}}, {dB}_{t}^{p}, {dB}_{t}^{q}, {dB}_{t}^{r})^{T} . \end{matrix}

The term $B_{t}$ is Brownian motion process and $b$ is diffusion parameter vector. Equation (3) comprises a set of three equations, (3a) to (3c). As equation (3) accounts for non-linearity and stochasticity, this suggests the ability of equation (3) to account for greater qualitative characteristics of underwater vehicle dynamics. In contrast, vehicle dynamical equations available in the literature ignore stochastic considerations. Greater detail about equations (2) and (3) can be found elsewhere (Sharma and Hirpara, 2014).

The Inertial Measurement Unit is the preliminary sensor system of underwater vehicles. It is one of the units of the Inertial Navigation System of underwater vehicles. In recent trends of measurement systems, naval dynamists can measure linear and angular velocities with the Photonics Inertial Navigation System; for more detail see Marani and Yuh (2014). In non-linear filtering sense, observations or measurements can be continuous or discrete, and that depends on the measuring instruments. If available observations are continuous, that combines continuous dynamics and continuous measurements. On the other hand, if available observations are discrete, the combination becomes a continuous-discrete filtering model of underwater vehicles. This paper considers two sets of discrete observation equations. The first set of observation equations includes linear velocity measurements of the AUV. The second set of observation equations includes angular velocity measurements. In general, the non-linear discrete observation equation is

\begin{matrix} y_{t_{k}} = h (t_{k}, ξ_{t_{k}}) + η_{t_{k},} \\ h (t_{k}, ξ_{t_{k}}) = (x_{1} (t_{k}), x_{2} (t_{k}), x_{3} (t_{k}))^{T} = (v_{x} (t_{k}), v_{y} (t_{k}), v_{z} (t_{k}))^{T} . \end{matrix}

(4)

The terms $y_{t_{k}}, h (t_{k}, ξ_{t_{k}})$ indicate the observation vector and the measurement non-linearity, respectively. The $t_{k}$ is discrete time instants at which observations are received by sensors, a more general setting that can be regarded as the measurement system. The observation noise $η_{t_{k}}$ is normally distributed, $N (0, R_{k})$ , where the term $R_{k}$ is a real-value observation noise variance matrix.

The second set of observation equations can be formalized by taking angular velocity measurements. Thus,

\begin{matrix} y_{t_{k}} = h (t_{k}, ξ_{t_{k}}) + η_{t_{k}}, \\ h (t_{k,} ξ_{t_{k}}) = (x_{4} (t_{k}), x_{5} (t_{k}), x_{6} (t_{k}))^{T} = (p (t_{k}), q (t_{k}), r (t_{k}))^{T} . \end{matrix}

(5)

Equation (2) in combination with equation (4) becomes a filtering model of the underwater vehicle. Equation (2) coupled with equation (5) becomes the second filtering model. Thus, this paper develops the non-linear filtering of the underwater vehicle SDE by considering two filtering models resulting from available discrete measurements on linear and angular velocities. The main theoretical results of the paper are summarized in the form of a Lemma and a Theorem as well as their proof.

Lemma: Consider the AUV obeys

d ξ_{t} = a (t, ξ_{t}) dt + b (t) d B_{t},

(6)

where the stochastic process

$ξ_{t} \in U,$ $U \subset R^{n},$ $a : [t_{0}, \infty) \times U \to U,$ $b : [t_{0}, \infty) \to R^{n \times r}$ and $h (t_{k}, ξ_{t_{k}}) \in R^{m} .$ The Kolmogorov backward equation becomes

\frac{\partial ϕ (t, ξ)}{\partial t} = \sum_{i} a_{i} (t, ξ,) \frac{\partial ϕ (t, ξ)}{\partial ξ_{i}} + \frac{1}{2} \sum_{i, j} (b b^{T})_{ij} (t) \frac{\partial^{2} ϕ (t, ξ)}{\partial ξ_{i} \partial ξ_{j}} .

(7)

The evolution of the conditional characteristic function evolution is

\begin{matrix} d 〈 \exp (s^{T} ξ_{t}) 〉 = (〈 \sum_{i} a_{i} (t, ξ_{t}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial ξ_{i}} 〉 \\ + \frac{1}{2} \sum_{i, j} {(b b^{T})}_{ij} (t) 〈 \frac{\partial^{2} \exp (s^{T} ξ_{t})}{\partial ξ_{i} \partial ξ_{j}} 〉) dt . \end{matrix}

(8)

After combining equation (8) with equations (2) and (3), we get the conditional characteristic function evolution of the underwater SDE. As a result of this, the conditional characteristic function evolution of the underwater vehicle SDE is

\begin{matrix} d 〈 \exp (s^{T} ξ_{t}) 〉 = (〈 (\frac{X}{m} + x_{2} x_{6} - x_{3} x_{5}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial x_{1}} + (\frac{Y}{m} + x_{3} x_{4} - x_{1} x_{6}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial x_{2}} 〉 \\ + 〈 (\frac{Z}{m} + x_{1} x_{5} - x_{2} x_{4}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial x_{3}} + (\frac{K}{I_{x}} + \frac{I_{y} - I_{z}}{I_{x}} x_{5} x_{6}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial x_{4}} 〉 \\ + 〈 (\frac{M}{I_{y}} + \frac{I_{z} - I_{y}}{I_{y}} x_{4} x_{6}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial x_{5}} + (\frac{N}{I_{z}} + \frac{I_{z} - I_{y}}{I_{z}} x_{4} x_{5}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial x_{6}} 〉 \\ + \frac{1}{2} \sum_{i} γ_{i}^{2} 〈 \frac{\partial^{2} \exp (s^{T} ξ_{t})}{\partial ξ_{i}^{2}} 〉) dt . \end{matrix}

(9)

Proof: The proof of Lemma can be constructed using the Kolmogorov backward equation and the Itô SDE. After combining with the underwater vehicle Itô SDE, we get the conditional characteristic function evolution for the specific case. The closed-form expression for the conditional characteristic function for the vector stochastic process satisfying the Itô SDE is not possible. Thus, the evolution of the conditional characteristic function is the standard formalism for the estimation of the non-linear Itô differential equation. Thus, the answer to the question is conditional characteristic function evolution. That is the inner product of the Kolmogorov backward equation and conditional probability density with time differential.

Here, we take a pause and explain the method for deriving the above expression. The above is the consequence of a multi-dimensional stochastic differential rule. Thanks to a Theorem (3.3) of Revuz and Yor (1999: 147), we arrive at the stochastic evolution of the scalar function

\begin{matrix} d \exp (s^{T} ξ_{t}) = (\sum_{i} a_{i} (t, ξ_{t}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial ξ_{i}} \\ + \frac{1}{2} \sum_{i, j} {(b b^{T})}_{ij} (t) \frac{\partial^{2} \exp (s^{T} ξ_{t})}{\partial ξ_{i} \partial ξ_{j}}) dt \\ + \sum_{i} \sum_{φ} \frac{\partial \exp (s^{T} ξ_{t})}{\partial ξ_{i}} b_{i φ} (t) d B_{φ} . \end{matrix}

After the action of the expectation operator on the stochastic evolution, we arrive at

\begin{matrix} d 〈 \exp (s^{T} ξ_{t}) 〉 = (〈 \sum_{i} a_{i} (t, ξ_{t}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial ξ_{i}} 〉 \\ + \frac{1}{2} \sum_{i, j} {(b b^{T})}_{ij} (t) 〈 \frac{\partial^{2} \exp (s^{T} ξ_{t})}{\partial ξ_{i} \partial ξ_{j}} 〉) dt . \end{matrix}

The above can be recast in the Kolmogorov backward operator setting as

\begin{matrix} d 〈 \exp (s^{T} ξ_{t}) 〉 \\ = (〈 \sum_{i} a_{i} (t, ξ_{t}) \frac{\partial \exp (s^{T} ξ_{t})}{\partial ξ_{i}} 〉 + \frac{1}{2} \sum_{i, j} {(b b^{T})}_{ij} (t) 〈 \frac{\partial^{2} \exp (s^{T} ξ_{t})}{\partial ξ_{i} \partial ξ_{j}} 〉) dt \\ = 〈 \sum_{i} a_{i} (t, ξ) \frac{\partial \exp (s^{T} ξ)}{\partial ξ_{i}} + \frac{1}{2} \sum_{i, j} (bb)_{ij}^{T} (t) \frac{\partial^{2} \exp (s^{T} ξ)}{\partial ξ_{i} \partial ξ_{j}}, p (t, ξ | t_{0}, ξ_{0}) 〉 dt, \end{matrix}

(10)

where the term $\sum_{i} a_{i} (t, ξ) \frac{\partial (.)}{\partial ξ_{i}} + \frac{1}{2} \sum_{i, j} {(b b^{T})}_{ij} (t) \frac{\partial^{2} (.)}{\partial ξ_{i} \partial ξ_{j}}$ has an interpretation as the Kolmogorov backward operator, see Feller (1971: 327). An alternative interpretation of the Kolmogorov backward equation is the evolution of conditional probability density, where time derivative and spatial derivatives are performed using the initial time, initial points and the conditional probability density is $p (x, t; y, τ) .$ That becomes (Jazwinski, 1970: 130)

- \frac{\partial p (x, t; y, τ)}{\partial τ} = f (y, τ) \frac{\partial p (x, t; y, τ)}{\partial y} + \frac{1}{2} g^{2} (y, τ) \frac{\partial^{2} p (x, t; y, τ)}{\partial y^{2}} .

Note that the notations 〈〉, 〈,〉 are different in the sense that the former denotes the expectation operator and the latter denotes the inner product. For greater detail see the Appendix of the paper. The right-hand side of equation (10) denotes the inner product. The term $p (t, ξ | t_{0}, ξ_{t_{0}})$ denotes the conditional probability density. After re-introducing the definition of the conditional expectation operator into equation (10), we get equation (8) of Lemma. From equations (8) and (2) to (3), we get equation (9). This completes the proof.

QED

Theorem: Consider the probability space $(Ω, ℑ, μ),$ where $Ω$ is a sample space, ℑ is a sigma algebra and $μ$ is a probability measure. Consider the stochastic process $ξ = {ξ_{t}, ℑ_{t} 0 \leq t < \infty},$ where ${(t, ω) | ξ_{t} (ω) \in B (R^{d})} \in [0, \infty) \times ℑ .$ Suppose the vector stochastic process $ξ = {ξ_{t}, ℑ_{t} 0 \leq t < \infty}$ denotes the underwater vehicle state vector. Equation (2) in combination with equation (4) describes the filtering model of the underwater vehicle SDE. Then, a system of conditional mean equations via the Kolmogorov backward equation is the following:

\begin{matrix} d 〈 x_{1} (τ) 〉 = 〈 (\frac{X}{m} + x_{2} x_{6} - x_{3} x_{5}) 〉 d τ, \\ d 〈 x_{2} (τ) 〉 = 〈 (\frac{Y}{m} + x_{3} x_{4} - x_{1} x_{6}) 〉 d τ, \\ d 〈 x_{3} (τ) 〉 = 〈 \frac{Z}{m} + x_{1} x_{5} - x_{2} x_{4} 〉 d τ, \\ d 〈 x_{4} (τ) 〉 = 〈 \frac{K}{I_{x}} + \frac{I_{y} - I_{z}}{I_{x}} x_{5} x_{6} 〉 d τ, \\ d 〈 x_{5} (τ) 〉 = 〈 \frac{M}{I_{y}} + \frac{I_{z} - I_{y}}{I_{y}} x_{4} x_{6} 〉 d τ, \\ d 〈 x_{6} (τ) 〉 = 〈 \frac{N}{I_{z}} + \frac{I_{z} - I_{y}}{I_{z}} x_{4} x_{5} 〉 d τ . \end{matrix}

(11)

Furthermore, the conditional mean equation of the AUV filtering at the observation instant for the measurement system of equation (4) is

\begin{matrix} 〈 x_{i}^{t_{k}} (t_{k}) 〉 = 〈 x_{i}^{t_{k - 1}} (t_{k}) 〉 + (〈 x_{i} x_{1} 〉 - 〈 x_{i} 〉 〈 x_{1} 〉 〈 x_{i} x_{2} 〉 - 〈 x_{i} 〉 〈 x_{2} 〉 \\ 〈 x_{i} x_{3} 〉 - 〈 x_{i} 〉 〈 x_{3} 〉)^{T} \\ \times (〈 (h - 〈 h 〉) (h - 〈 h 〉)^{T} 〉 + R_{k})^{- 1} (y_{t_{k}} - 〈 h (t_{k}, ξ_{t_{k}}) 〉), \end{matrix}

(12)

where the subscript $i$ satisfies the inequality $1 \leq i \leq 6$ and

\begin{matrix} 〈 (h - 〈 h 〉) (h - 〈 h 〉)^{T} 〉 \\ = (\begin{matrix} var (x_{1}) cov (x_{1}, x_{2}) cov (x_{1}, x_{3}) \\ cov (x_{1}, x_{2}) var (x_{2}) cov (x_{2}, x_{3}) \\ cov (x_{1}, x_{3}) cov (x_{2}, x_{3}) var (x_{3}) \end{matrix}) . \end{matrix}

The conditional mean equation of the AUV filtering at the observation instant for the measurement system of equation (5) is

\begin{matrix} 〈 x_{i}^{t_{k}} (t_{k}) 〉 = 〈 x_{i}^{t_{k - 1}} (t_{k}) 〉 + (〈 x_{i} x_{4} 〉 - 〈 x_{i} 〉 〈 x_{4} 〉 \\ 〈 x_{i} x_{5} 〉 - 〈 x_{i} 〉 〈 x_{5} 〉 〈 x_{i} x_{6} 〉 - 〈 x_{i} 〉 〈 x_{6} 〉)^{T} \\ \times (〈 (h - 〈 h 〉) (h - 〈 h 〉)^{T} 〉 + R_{k})^{- 1} (y_{t_{k}} - 〈 h (t_{k}, ξ_{t_{k}}) 〉), \end{matrix}

(13)

where the subscript $i$ satisfies the inequality $1 \leq i \leq 6$ and

\begin{matrix} 〈 (h - 〈 h 〉) (h - 〈 h 〉)^{T} 〉 = \\ (\begin{matrix} var (x_{4}) cov (x_{4}, x_{5}) cov (x_{4}, x_{6}) \\ cov (x_{4}, x_{5}) var (x_{5}) cov (x_{5}, x_{6}) \\ cov (x_{4}, x_{6}) cov (x_{5}, x_{6}) var (x_{6}) \end{matrix}) . \end{matrix}

Note that equation (4) is the first measurement system and equation (5) is the second measurement system. We state the filtering equations for the both measurement systems.

Proof: The conditional mean equation of the underwater SDE is a consequence of the evolution of the conditional characteristic function of the underwater SDE. Note that the conditional characteristic function evolution of the underwater SDE is a consequence of the Kolmogorov backward equation; see equation (10) that arises in the proof of Lemma of the paper. Equation (8) of Lemma can be recast using appropriate variables, i.e.

\begin{matrix} d 〈 \exp (s^{T} ξ_{τ}) 〉 = (〈 \sum_{p} a_{p} (τ, ξ_{τ}) \frac{\partial \exp (s^{T} ξ_{τ})}{\partial ξ_{p}} 〉 \\ + \frac{1}{2} \sum_{p, q} {(b b^{T})}_{pq} (τ) 〈 \frac{\partial^{2} \exp (s^{T} ξ_{τ})}{\partial ξ_{p} \partial ξ_{q}} 〉) d τ . \end{matrix}

Suppose $\exp (s^{T} ξ_{τ}) = ξ_{i},$ then

\begin{matrix} d 〈 ξ_{i} 〉 = (〈 \sum_{p} a_{p} (τ, ξ_{τ}) \frac{\partial ξ_{i}}{\partial ξ_{p}} 〉 + \frac{1}{2} \sum_{p, q} {(b b^{T})}_{pq} (τ) 〈 \frac{\partial^{2} ξ_{i}}{\partial ξ_{p} \partial ξ_{q}} 〉) d τ . \\ = (〈 \sum_{p} a_{p} (τ, ξ_{τ}) δ_{i, p} 〉 + \frac{1}{2} \sum_{p, q} {(b b^{T})}_{pq} (τ) 〈 \frac{\partial δ_{i, p}}{\partial ξ_{q}} 〉) d τ \\ = 〈 a_{i} (τ, ξ_{τ}) 〉 d τ . \end{matrix}

(14)

After introducing the state vector of equation (2) into equation (14) component wise, we get equation (11) of the Theorem. This completes the proof of the first part of the Theorem of the paper.

QED

The second part of the proof of the Theorem, conditional mean equation at the observation instant, exploits the results stated in Jazwinski (1970: 344). The proof of the second part is straightforward. A simple calculation of the variance matrix arises that is attributed to the discrete linear observation equation; see equations (4) and (5) of the paper. We omit the detail.

It is important to note that the structure of the filtering equation is judged on the basis of measurement systems. This paper adopts two measurement systems, see equations (4) and (5). Thus, we find two different conditional mean equations at the observation instant, i.e. equations (12) and (13) of the Theorem of the paper.

Remark 1: Equation (11), the conditional mean equation, can be alternatively expressed as the evolution equation, i.e.

d 〈 ξ (τ) 〉 = (A_{i} (τ, 〈 ξ_{τ} 〉, P_{τ})) d τ,

where the conditional variance matrix

P_{τ} = (P_{ij} (τ)), A (τ, 〈 ξ_{τ} 〉, P_{τ}) = (A_{i} (τ, 〈 ξ_{τ} 〉, P_{τ})) .

Since the input argument of the vector $A (.)$ embeds a conditional variance term, the AUV filtering equations account for the conditional variance evolution equation as well. For the components of $d 〈 ξ (τ) 〉 = (A_{i} (τ, 〈 ξ_{τ} 〉, P_{τ})) d τ$ and the diagonal elements of the conditional variance matrix evolution $d P (τ) = (B_{ij} (τ, 〈 ξ_{τ} 〉, P_{τ})) d τ$ , for greater detail see equation (7) and (8) of the paper Sharma and Hirpara (2014: 8808). Note that the time instant $τ$ satisfies the inequality $t_{k - 1} < τ < t_{k}$ and the instants $t_{k - 1}, t_{k}$ are observation instants.

Remark 2: Equation (12) describes the conditional mean equation at the observation instant. The conditional mean equation for the AUV filtering at the observation instant can be expressed alternatively as

\begin{matrix} 〈 ξ_{t_{k}}^{t_{k}} 〉 = G (t_{k}, 〈 ξ_{t_{k}}^{t_{k - 1}} 〉, P_{t_{k}}^{t_{k - 1}}) + H (t_{k}, 〈 ξ_{t_{k}}^{t_{k - 1}} 〉, P_{t_{k}}^{t_{k - 1}}) (y_{t_{k}} - h (t_{k}, 〈 ξ_{t_{k}}^{t_{k - 1}} 〉)), \end{matrix}

and the conditional variance equation becomes

P_{t_{k}}^{t_{k}} = C (t_{k}, 〈 ξ_{t_{k}}^{t_{k - 1}} 〉, P_{t_{k}}^{t_{k - 1}}) .

The above expressions embed two notations $〈 ξ_{t_{k}}^{t_{k}} 〉,$ $〈 ξ_{t_{k}}^{t_{k - 1}} 〉$ for the conditional mean equation at the observation instant. The former denotes ‘the conditional expectation accounts for observations accumulated up to the time instant $t_{k}' .$ The latter denotes ‘the conditional expectation accounts for observations accumulated up to the time instant $t_{k - 1} .'$ The vector $G (t_{k}, 〈 ξ_{t_{k}}^{t_{k - 1}} 〉, P_{t_{k}}^{t_{k - 1}})$ and the matrices $H (t_{k}, 〈 ξ_{t_{k}}^{t_{k - 1}} 〉, P_{t_{k}}),$ $C (t_{k}, 〈 ξ_{t_{k}}^{t_{k - 1}} 〉, P_{t_{k}}^{t_{k - 1}})$ can be calculated using the Taylor series expansion of the non-linear terms of equation (12) evaluated at the conditional mean $〈 ξ_{t_{k}}^{t_{k - 1}} 〉 .$ The approach is straightforward; a few calculations will result an alternative to ‘the AUV filtering equations (12) at the observation instant’.

The Taylor series expansion of the system non-linearity, diffusion coefficient and measurement non-linearity around the conditional mean is a mathematical technique to achieve filtering in stochastic processes. The Carleman realization is a celebrated bilinearization method for the non-linear state equation. However, the Carleman realization (Kowalski and Steeb, 1991; Rugh, 1981) to filtering equations introduces formidable complexity. The complexity is attributed to the action of the conditional expectation operator in addition to the power series expansion in the Kronecker product setting.

Note that the AUV filtering is a two-stage estimation that comprises a system of equations, equations (11) and (12) for the first measurement system of equation (4) and equations (11) and (13) for the second measurement system of equation (5). On the other hand, the AUV estimation of Sharma and Hirpara (2014) is a one-stage that comprises equation (11) of this paper. Thus, the AUV filtering of the paper also accounting for discrete observations is an extended and a refined version of the recently published results in Sharma and Hirpara (2014).

Remark 3: Here, we wish to make a sensitivity remark for non-linear filtering. As the filtering involves the process noise and observation noise correction terms, the closed-form expression of the conditional variance $P_{t_{k}}^{t_{k}}$ of the linear filters in continuous and discrete settings is unfortunately not possible. As a result of this, the notion of local error sensitivity function was introduced (Jazwinski 1970: 245–248) for linear filtering. Furthermore, for the continuous state-discrete filtering in non-linear setting, calculation of the filter error sensitivity becomes intractable. That becomes of a little interest. Thus, sensitivity of non-linear filtering is not well researched. For an alternative rigorous mathematical perspective of the non-linear filtering accuracy, Handel (2009) can be consulted.

Graphical notations: In Figures (1) to (18) of the paper, the solid line trajectory denotes the filtered trajectories of the components of the state vector. The dotted line trajectory denotes the true perturbed state trajectories of the components of the state vector.

Figure 1.

The true and estimated states: first component.

Figure 2.

The true and estimated states: second component.

Figure 3.

The true and estimated states: third component.

Figure 4.

The true and estimated states: fourth component.

Figure 5.

The true and estimated states: fifth component.

Figure 6.

The true and estimated states: sixth component.

Figure 7.

The true and estimated states: first component.

Figure 8.

The true and estimated states: second component.

Figure 9.

The true and estimated states: third component.

Figure 10.

The true and estimated states: fourth component.

Figure 11.

The true and estimated states: fifth component.

Figure 12.

The true and estimated states: sixth component.

Figure 13.

The true and estimated states: first component.

Figure 14.

The true and estimated states: second component.

Figure 15.

The true and estimated states: third component.

Figure 16.

The true and estimated states: fourth component.

Figure 17.

The true and estimated states: fifth component.

Figure 18.

The true and estimated states: sixth component.

Numerical experimentations

Here, we demonstrate simulations of underwater vehicle non-linear filtering. This accounts for the underwater vehicle–ocean current stochastic interaction as well as noisy measurements on the linear as well as angular velocities. Two different sets of initial conditions, stochastic autonomous system parameters and process noise and observation perturbations are exploited to accomplish numerical experimentations. For the numerical testing of non-linear linear filtering of this paper, we adopt the method of Germani et al. (2002) for numerical studies. In Germani et al. (2002), filtering for bilinear stochastic differential system was achieved by contrasting the filtering trajectory with the true perturbed state trajectory. For the numerical simulation of the AUV filtering, we consider two different measurement systems, equations (4) and (5) of the paper. This allows the efficacy of the AUV filtering under a variety of conditions. Thus, consider the following set of system parameters, initial states (Sharma and Hirpara, 2014; Singh et al., 2009; Smith, 2008) for the numerical simulation of filtering equations (11) and (12) of this paper in Table 1.

Table 1.

AUV system parameters.

Parameters	unit	value
The mass $m$	Kg	123.8
The inertia vector $(I_{x}, I_{y}, I_{z})^{T}$	Kg/m²	$(5.46, 5.29, 5.72)^{T}$
The linear velocity vector $(x_{1}, x_{2}, x_{3})^{T}$	m/sec	$(0.03, 0.04, 0.045)^{T}$
The angular velocity vector $(x_{4}, x_{5}, x_{6})^{T}$	rad/sec	$(0.252, - 0.189, 0.147)^{T}$

A set of the diffusion parameter vector is

\begin{matrix} (γ_{1} γ_{2} γ_{3} γ_{4} γ_{5} γ_{6})^{T} \\ = (4 \times 10^{- 2} 2 \times 10^{- 2} 4 \times 10^{- 2} 10^{- 2} 10^{- 2} 8 \times 10^{- 3})^{T} . \end{matrix}

The diffusion parameter decides the process noise intensity. The diffusion coefficient is a consequence of the process noise coefficient matrix that can be regarded as stochastic correction term in the conditional variance evolution.

As this paper is chiefly about AUV non-linear filtering, we choose the observation noise variance matrix $R_{k}$ as well. Consider

R_{k} = {(\begin{matrix} 10 & 0 & 0 \end{matrix} : \begin{matrix} 0 & 10 & 0 \end{matrix} : \begin{matrix} 0 & 0 & 10 \end{matrix})}^{T},

The other finite values of the components of the observation noise variance matrix $R_{k}$ as well as the diffusion parameter vector can be chosen conveniently. The observation noise variance matrix is a positive semi-definite symmetric matrix as well as a diagonal matrix. This denotes the covariance elements of the observation noise components will vanish, as they are independent random variables. For the simplified discussion, consider the scalar observation noise, where the observation noise variance takes finite values. For the relatively larger observation noise variance, the observation becomes valueless (Jazwinski, 1970: 179). On the other hand, for the relatively smaller noise variance, the observation becomes noiseless. Thus, we choose the finite values of the observation noise intensity. The system parameters of the stochastic differential systems are chosen using the qualitative characteristics of the stochastic differential system. Generally, the filtering behaviour is judged by considering the system parameters that lead to the bounded conditional mean trajectory. In this regard, Theorem (4.2) of Handel (2009) linking stability and accuracy of the filtering will be useful; see Zeitouni and Dembo (1988) as well. Figures (1) to (6) contrast the filtered state trajectories with true perturbed state trajectories. Figures (1) to (6) contrast the filtered state trajectories with true perturbed state trajectories. Figures (1) to (6) exploit the filtering model comprising equation (2) in combination with the first measurement equation (4).

This suggests that the filtering trajectories follow the true perturbed state trajectories. The root mean square calculation of stochastic systems is appealing for linear scalar cases, where the closed-form expression of the state is possible. On the other hand, as this paper accounts for a vector non-linear case, where the closed-form expression is not possible, we adopt the evolution of conditional variance setting. The conditional evolution equation setting is standard and popular in filtering (Jazwinski, 1970: 178) and accounts for process noise correction terms as well as observation noise correction terms.

For the given set of system parameters, the process and observation noise intensities, the AUV conditional variances are of the order $10^{- 2}$ and reveal the bounded increase and decrease. The conditional variance indicates random fluctuations in mean trajectory. Less conditional variance suggests better estimate, as well as closeness of the estimate with the actual state trajectory. The linear velocity observation case of Figures (1 –6) denotes observations are noisy, and observation noise correction terms to the filtering refine the filtering equations and the filtered state trajectories follow the true state trajectory. Thus, Figures (1 –6) and Table 2 reveal the good behaviour of the non-linear filtering equations of the paper. The difference between the filtered state trajectories and perturbed state trajectories is attributed to two terms, measurement non-linearities and measurement noise intensities. The filtering equations account for measurement non-linearities and measurement noise intensities as well. The signal to noise ratio is an indicator of the filtering algorithms (Sharma and Parthasarathy, 2008: 165). It is important to note that the signal to noise ratio is useful to signal estimation methods, e.g. Wiener filter. More detail can be found in Sharma and Parthasarathy (2008). On the other hand, the signal to noise ratio is of little interest in the state estimation method.

Table 2.

The AUV conditional variance (the observation noise variance $R_{k} = 10 m^{2} s^{- 2})$ .

Variances	The time Instant $t$ second
	0	10	20	30	40	50
$P_{11}$	$0$	$0.0212$	$0.0202$	$0.0285$	$0.0204$	$0.02$
$P_{22}$	$0$	$0.0111$	$0.0125$	$0.0120$	$0.012$	$0.011$
$P_{33}$	$0$	$0.0033$	$0.0345$	$0.0348$	$0.032$	$0.0312$
$P_{44}$	$0$	$0.001$	$0.0012$	$0.00132$	$0.0011$	$0.0065$
$P_{55}$	$0$	$0.0012$	$0.00127$	$0.00134$	$0.00121$	$0.00112$
$P_{66}$	$0$	$0.0007$	$0.00071$	$0.0022$	$0.0032$	$0.0003$

Further, we achieve the AUV filtering simulations by considering the relatively larger observation variance. The observation noise variance matrix for the second case is $R_{k} = {(\begin{matrix} 50 & 0 & 0 \end{matrix} : \begin{matrix} 0 & 50 & 0 \end{matrix} : \begin{matrix} 0 & 0 & 50 \end{matrix})}^{T} .$

The filtering trajectories and true perturbed state trajectories, Figures (7 –12), can be argued similar to the first filtering model. The linear velocity observation case of Figures (7 –12) denotes observation correction term contributions to the filtering asymptotically vanish. The predicted estimate becomes the filtered estimate and the filtered-true state difference increases (Sharma, 2009). Figures (7 –12) justify this.

We consider the third case of the AUV filtering simulation by considering the measurement system of equation (5). This considers measurements on the angular velocity vector. Note that the observation noise variance matrix $R_{k} = {(\begin{matrix} 10 & 0 & 0 \end{matrix} : \begin{matrix} 0 & 10 & 0 \end{matrix} : \begin{matrix} 0 & 0 & 10 \end{matrix})}^{T},$ is chosen the same. Other choices of the observation noise variance matrix can be exploited as well. However, the dimensional interpretation of the matrix $R_{k}$ of the second measurement system is different from the matrix $R_{k}$ of the first measurement system.

The angular velocity observation case of Figures (13 –18) denotes observables are greatly masked with noise. The contribution of observation correction terms involving the term $R_{k}^{- 1}$ to filtering is insignificant and the difference between filtered state trajectory and true state is larger. A discussion can be found in Sharma (2009). The numerical simulations of the paper confirm the standard theory and applications of the filtering equations of the paper as well.

Conclusion

In this paper, we achieve non-linear filtering of the AUV dynamics coupled with the ocean current stochasticity. This paper adopts the Itô process to account for the ocean current stochasticity. The non-linear filtering of the paper is a two-stage estimation procedure that comprises two sets of equations. The first set is a consequence of the Kolmogorov backward equation and the evolution of conditional characteristic function of the vector Itô process. The second is a consequence of Bayesian statistics.

Most notably, this paper can be regarded as a first paper that resolves an open problem recommended in Aguiar and Hespanha (2003: 1993). The open problem is ‘how to account for ocean stochasticity in the AUV dynamics as well as analyse them in non-linear filtering perspective’. The non-linear filtering equations coupled with the control signal for the underwater vehicle SDE becomes the stochastic optimal control of the underwater vehicle. This paper attempts the first part. The second part deserves investigation. The second part can be addressed by exploiting the HJB equation, a second-order non-linear partial differential equation, for the Itô SDE.

Another achievement of the paper is to introduce the notions of non-linear filtering into autonomous systems by considering a specific case involving ‘mathematical rigour’.

After investigations into the AUV filtering, the following problem merits resolution. After testing the observability, we design an observer. Second, the stochastic version of the observer is the stochastic filter that this paper intends to achieve. The closed-form observability matrix for the linear time-invariant case is convenient. The closed-form exact observability matrix for the non-linear time-invariant case is formidable. It is important to state that the AUV dynamics is non-linear and stochastic, the closed-form expression of the AUV observability matrix is an open problem.

Footnotes

Appendix

Here, we explain briefly the Kolmogorov–Fokker–Planck equation (KFPE) and the Kolmogorov Backward Equation (KBE) for Markov processes. Notably, the KFPE is also known as the KFE, the Kolmogorov Forward Equation. The operator associated with the KFE is the Kolmogorov forward operator and the operator associated with the KBE is the Kolmogorov backward operator. The cornerstone relation is

(A-1)

〈 φ, Lp 〉 = 〈 L' φ, p 〉,

where the notation $〈, 〉$ denotes the inner product, the term $φ$ is a scalar non-linear and twice continuously differentiable function. The term $p$ denotes the conditional probability density. Importantly, the operator $L'$ denotes the Kolmogorov backward operator and the operator $L$ denotes, the Kolmogorov forward operator. The structure of the Kolmogorov backward operator and the Kolmogorov forward operator are, respectively,

(A-2)

L' (\cdot) = \sum_{i} a_{i} (ξ_{t}, t) \frac{\partial (\cdot)}{\partial ξ_{i}} + \frac{1}{2} \sum_{i, j} {(b b^{T})}_{ij} (t) \frac{\partial^{2} (\cdot)}{\partial ξ_{i} \partial ξ_{j}} .

(A-3)

L (\cdot) = - \sum_{i} \frac{\partial a_{i} (ξ_{t}, t)}{\partial ξ_{i}} (.) + \frac{1}{2} \sum_{i, j} \frac{\partial^{2} {(b b^{T})}_{ij} (t)}{\partial ξ_{i} \partial ξ_{j}} (.) .

An appealing property of the Kolmogorov backward operator is that it has adjoint property. The adjoint of the backward operator is the forward operator. Here, we explain briefly the usefulness of the KBE that turns to the usefulness of the Kolmogorov backward operator. The Kolmogorov backward operator becomes the differential generator for the stability of stochastic differential systems. Furthermore, the Kolmogorov backward operator arises in the filtering evolution equation. In the stochastic control, the Kolmogorov backward operator arises in the HJB equation of the Itô process as well. This unfolds the beauty, power and universality of the Kolmogorov equations in dynamical systems and stochastic control. The Kolmogorov backward operator has found its applications in ‘estimation theory’ and ‘stability’ of Itô SDEs.

Acknowledgements

The authors express their gratitude to the anonymous reviewers and Associate Editor for fine comments on the paper that have led to improvement in the paper as well as made the paper error-less.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ravish H. Hirpara

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