Robust $H_\infty$ Filtering for Nonlinear Discrete-time Stochastic Systems

TL;DR

This paper develops a new approach for $H_$ filtering in nonlinear discrete-time stochastic systems using a stochastic bounded real lemma and Hamilton-Jacobi inequalities, providing verifiable conditions for filter design.

Contribution

It introduces a novel stochastic bounded real lemma and a Hamilton-Jacobi inequality for $H_$ filtering of nonlinear stochastic systems, advancing existing theoretical methods.

Findings

Derived a nonlinear stochastic bounded real lemma.

Established a sufficient condition for $H_$ filtering via HJI.

Validated results with practical engineering examples.

Abstract

This paper mainly discusses the $H_{\infty}$ filtering of general nonlinear discrete time-varying stochastic systems. A nonlinear discrete-time stochastic bounded real lemma (SBRL) is firstly obtained by means of the smoothness of the conditional mathematical expectation, and then, based on the given SBRL and a stochastic LaSalle-type theorem, a sufficient condition for the existence of the $H_\infty$ filtering of general nonlinear discrete time-varying stochastic systems is presented via a new introduced Hamilton-Jacobi inequality (HJI), which is easily verified. When the worst-case disturbance $\{v^*_k\}_{k\in {\mathcal N}}$ is considered, the suboptimal $H_2/H_\infty$ filtering is studied. Two examples including a practical engineering example show the effectiveness of our main results.

Tables

Table 1. Table 1:

Parameters of the Mercedes-Benz commercial vehicle
Symbol	Value	Unit
$C_R$	53071	N ms/rad
$m_s$	1700	kg
$h_{cr}$	0.25	m
$I_{xx}$	1700	kg $m^2$
$K_R$	55314	N ms/rad
$D_n$	20	N ms/rad

Equations

$$\Vert v{\Vert }_{l_{\infty }^2}=\sqrt{E\left[\sum _{k=0}^{\infty }\Vert v_k{\Vert }^2\right]}<\infty .$$

$$\{\begin{array}{l}{x}_{k+1}=f_k(x_k,w_k,v_k),f_k(0,0,0)\equiv 0,\\ y_k=g_k(x_k,v_k),g_k(0,0)\equiv 0,\\ z_k=m_k(x_k,v_k),m_k(0,0)\equiv 0,\end{array}$$

$$\{\begin{array}{l}{\hat{x}}_{k+1}={\hat{f}}_k({\hat{x}}_k)+{\hat{g}}_k(y_k),{\hat{f}}_k(0)={\hat{g}}_k(0)=0,\\ {\hat{z}}_k={\hat{m}}_k({\hat{x}}_k),{\hat{m}}_k(0)=0,{\hat{x}}_0=0,\end{array}$$

$$\begin{cases}\ \eta_{k+1}=\tilde{f}_{k}(\eta_{k},w_{k},v_{k})=\left[\begin{array}[]{cc}f_{k}(x_{k},w_{k},v_{k})\\ \hat{f}_{k}(\hat{x}_{k})+\hat{g}_{k}(y_{k})\end{array}\right],\\ y_{k}=g_{k}(x_{k},v_{k}),\\ \tilde{z}_{k}=\tilde{m}_{k}(\eta_{k},v_{k})=m_{k}(x_{k},v_{k})-\hat{m}_{k}(\hat{x}_{k}),\end{cases}$$

$${\eta }_{k+1}={\tilde{f}}_k({\eta }_k,w_k,0)$$

$$\underset{{\eta }_0\to 0}{\lim }\mathcal{P}\left(\underset{k\ge 0}{\sup }\Vert {\eta }_k^{0,{\eta }_0,0}\Vert >\epsilon \right)=0$$

$$\mathcal{P}\{\underset{k\to \infty }{\lim }{\eta }_k^{0,{\eta }_0,0}=0\}=1.$$

$$\Vert \tilde{z}{\Vert }_{l_{\infty }^2}^2\le {\gamma }^2\Vert v{\Vert }_{l_{\infty }^2}^2.$$

$$\mathcal{L}:$$

$$\Vert \mathcal{L}\Vert =\underset{{\eta }_0=0,0\ne v\in l_{\infty }^2(\Omega ,\mathcal{F},{\{{\mathcal{F}}_k\}}_{k\in \mathcal{N}},{\mathcal{R}}^{n_v})}{\sup }\frac{\Vert \tilde{z}{\Vert }_{l_{\infty }^2}}{\Vert v{\Vert }_{l_{\infty }^2}},$$

$$\Vert \mathcal{L}\Vert \le \gamma ,$$

$$V_k(x)\ge \mu (\Vert x\Vert ),\forall (k,x)\in \mathcal{N}\times {\mathcal{D}}_r.$$

$$\underset{\Vert x\Vert \to \infty }{\mathrm{liminf}}\underset{k\in \mathcal{N}}{\inf }{V}_k(x)=\infty .$$

$$v^{\ast }=\arg min\{J_{\infty }(v):=E\sum _{k=0}^{\infty }({\gamma }^2\Vert v_k{\Vert }^2-\Vert {\tilde{z}}_k{\Vert }^2)\},$$

$$E[f(\alpha ,\beta )|\mathcal{G}]=E[f(\alpha ,x){]}_{x=\beta }a.s.$$

$$E[V_{k+1}({\eta }_{k+1})|{\mathcal{F}}_k]$$

$${\Delta }_v{V}_k(\eta )$$

$$H_k(\eta ,v)-{\gamma }^2\Vert v{\Vert }^2\le 0,k\in \mathcal{N},$$

$$E[V_{k+1}({\eta }_{k+1})]-E[V_k({\eta }_k)]$$

$$=$$

$$E\{E[V_{k+1}({\tilde{f}}_k({\eta }_k,w_k,v_k)|{\mathcal{F}}_k]-V_k({\eta }_k)\}$$

$$=$$

$$E[V_{k+1}({\eta }_{k+1})]-E[V_k({\eta }_k)]$$

$$\le$$

$$E[V_{N+1}({\eta }_{N+1})]-E[V_0({\eta }_0)]$$

$$\le$$

$$\sum _{k=0}^{N}E\Vert {\tilde{m}}_k({\eta }_k,v_k){\Vert }^2=$$

$$\le$$

$$\sum _{k=0}^{\infty }E\Vert {\tilde{z}}_k{\Vert }^2\le {\gamma }^2\sum _{k=0}^{\infty }E\Vert v_k{\Vert }^2$$

$${\Delta }_{v=0}{V}_k(\eta )\le 0,\forall (\eta ,k)\in {\mathcal{D}}_h\times \mathcal{N},$$

$${\eta }_{k+1}={\tilde{f}}_k({\eta }_k,w_k,0),{\eta }_0\in {\mathcal{R}}^{\eta }$$

$$V_k(\eta )\ge \mu (\Vert \eta \Vert ),\forall (\eta ,k)\in {\mathcal{D}}_r\times \mathcal{N}.$$

Full text

Robust $H_{\infty}$ Filtering for Nonlinear Discrete-time Stochastic Systems

Tianliang Zhang 1, Feiqi Deng 1  and Weihai Zhang 2

1 School of Automation Science and Engineering,

South China University of Technology, Guangzhou 510640, P. R. China

2 College of Information and Electrical Engineering,

Shandong University of Science and Technology, Qingdao 266510, P. R. China

Email: t_lzhang@163.com(T. Zhang), aufqdeng@scut.edu.cn(F. Deng), w_hzhang@163.com(W. Zhang)

Abstract-     This paper mainly discusses the $H_{\infty}$ filtering of general nonlinear discrete time-varying stochastic systems. A nonlinear discrete-time stochastic bounded real lemma (SBRL) is firstly obtained by means of the smoothness of the conditional mathematical expectation, and then, based on the given SBRL and a stochastic LaSalle-type theorem, a sufficient condition for the existence of the $H_{\infty}$ filtering of general nonlinear discrete time-varying stochastic systems is presented via a new introduced Hamilton-Jacobi inequality (HJI), which is easily verified. When the worst-case disturbance $\{v^{*}_{k}\}_{k\in{\mathcal{N}}}$ is considered, the suboptimal $H_{2}/H_{\infty}$ filtering is studied. Two examples including a practical engineering example show the effectiveness of our main results.

Keywords: $H_{\infty}$ filtering, suboptimal $H_{2}/H_{\infty}$ filtering, stochastic LaSalle-type theorem, discrete-time stochastic systems, internal stability and external stability.

1 Introduction

$H_{\infty}$ control theory was initially formulated by G. Zames [27] in the early 1980’s for linear time-invariant systems, which has been one of the most important control approaches in the presence of external disturbances. Because in engineering practice, the system state is not always available, how to estimate the unavailable state variable or a linear combination of the state variable from the measurement output is an important issue of modern control theory. When the system noise is stationary Gaussian white noise, Kalman filtering has been shown to be one of the most celebrated estimation methods. However, in practical applications, we may not be able to accurately know the statistical properties of external disturbances. In this case, one has to turn to Robust $H_{\infty}$ filter. Robust $H_{\infty}$ filter requires one to design a filter such that the $l_{2}$-gain from the exogenous disturbance to the estimated error is less than a prescribed level $\gamma>0$. In contrast with the well-known Kalman filtering, one of the main advantages of $H_{\infty}$ filtering is that it is not necessary to know exactly the statistical properties of the external disturbance but only assumes the external disturbance to have bounded energy [31]. We refer the reader to [3, 4, 18, 21, 24, 26] for practical applications of $H_{\infty}$ filtering in signal processing and sensor networks.

Stochastic $H_{\infty}$ control of linear continuous-time Itô stochastic systems seems to start from the well-known works [11, 22]. After then, based on the stochastic bounded real lemma of [11], full- and reduced-order robust $H_{\infty}$ estimation problems for stationary continuous-time linear stochastic Itô systems were discussed in [8] and [25], respectively. All the above works are limited to the linear stationary stochastic systems. We refer the reader to the monographs [5, 20, 29] for the early development in the $H_{\infty}$ control theory of linear Itô systems. By means of completing squares and stochastic dynamic programming principle, the state-feedback $H_{\infty}$ control was extensively investigated in [30] for affine stochastic Itô systems. Based on the stochastic bounded real lemma given in [30], the reference [31] solved the the nonlinear stochastic $H_{\infty}$ filter design of nonlinear affine Itô systems by solving a second-order Hamilton-Jacobi inequality (HJI).

As said by J. P. LaSalle [13], “Today there is more and more reason for studying difference equations systematically. They are in their own right important mathematical models.”. Along the development of computer technique, it is expected that the study on discrete-time systems will become more and more important [2, 6, 7, 16, 28, 32]. $H_{\infty}$ control of linear discrete-time stochastic systems with multiplicative noise was initiated by [7], and then generalized to nonlinear discrete stochastic systems [2]. In [19], $H_{\infty}$ filtering of discrete fuzzy stochastic systems with sensor nonlinearities was studied. In [14], $H_{\infty}$ filtering for a class of nonlinear discrete-time stochastic systems with uncertainties and random Markovian delays was investigated. Although, the $H_{\infty}$ control and filtering of continuous-time Itô systems have been solved in [30] and [31], respectively. However, for a general nonlinear discrete stochastic system, its $H_{\infty}$ control and filtering problems seem more complicated than continuous-time Itô systems. The main reason lies in that discrete nonlinear stochastic systems do not have an infinitesimal generator $LV(x)$ as in Itô systems, which is a useful tool in completing squares [17]. In [2], the nonlinear discrete-time $H_{\infty}$ control was discussed based on an HJI, where the HJI depends on the supremum of a conditional mathematical expectation function, which is not easily verified. Generally speaking, the technique of completing squares used in [30] becomes invalid for general discrete-time nonlinear systems. Moreover, due to adaptiveness requirement, the method of Taylor’s series expansion is not applicable as done in deterministic nonlinear systems [15]; see [29].

As summarized above, how to give practical criteria for general discrete stochastic $H_{\infty}$ control and filtering that do not depend on the mathematical expectation of the state trajectory is a challenging work. In [32], by Doob’s super-martingale theory, a LaSalle-type stability theorem was established. A new method based on convex analysis was introduced to solve the $H_{\infty}$ control of general discrete-time nonlinear stochastic systems in [16]. In [16], the Lyapunov function is selected as a convex function, which help separate the state $x_{k}$ from the coupling of $x_{k}$ and the unknown exogenous disturbance $v_{k}$.

In this paper, our main goal is to deal with $H_{\infty}$ filtering for general nonlinear discrete stochastic systems. Firstly, by applying the smoothness of the conditional mathematical expectation, for general discrete time-varying nonlinear time-varying stochastic systems, a stochastic bounded real lemma (SBRL) on external stability is given based on a new introduced HJI, where the HJI does not depend on the mathematical expectations of the state and external disturbance. Secondly, a sufficient condition for the existence of $H_{\infty}$ filtering for general nonlinear discrete time-varying stochastic systems has been presented based on our newly developed stochastic LaSalle’s invariant principle [32] and SBRL. As corollaries, $H_{\infty}$ filtering problems of nonlinear stochastic time-invariant systems and affine nonlinear stochastic systems are discussed. Thirdly, we also discuss the suboptimal $H_{2}/H_{\infty}$ filtering problems of nonlinear stochastic systems and linear stochastic systems under worst-case disturbance $\{v^{*}_{k}\}_{k\in{\mathcal{N}}}$. In particular, for linear stochastic systems, we prove that a desired suboptimal $H_{2}/H_{\infty}$ filtering can be constructed by solving a convex optimization problem.

This paper is organized as follows: In section 2, some preliminaries are made, where a useful lemma-Lemma 2.3 on stability in probability is obtained. Section 3 is concerned about the general nonlinear $H_{\infty}$ filtering, and section 4 is about the suboptimal $H_{2}/H_{\infty}$ filtering. In section 5, we present two examples, one is a numerical example, but the other one is a practical vehicle roll example, to illustrate the validity of our main results.

For convenience, the notations adopted in this paper are as follows:

$M^{\prime}$: the transpose of the matrix $M$ or vector $M$; $M>0$ ($M<0$): the matrix $M$ is a positive definite (negative definite) real symmetric matrix; $I_{n}$: $n\times n$ identity matrix; ${\mathcal{R}}^{n}$: the $n$-dimensional real Euclidean vector space with the standard $2$-norm $\|x\|$; ${\mathcal{R}}^{n\times m}$: the space of $n\times m$ real matrices. ${\mathcal{N}}:=\{0,1,2,\cdots,\}$; $l^{2}_{\infty}(\Omega,{\mathcal{F}},{\{{\mathcal{F}}_{k}\}}_{k\in\mathcal{N}},\mathcal{R}^{n_{v}})$: the space of stochastic ${\mathcal{F}}_{k}$-adapted sequence $\{v_{k}\}_{k\in\mathcal{N}}$ with the norm

[TABLE]

A function $f(x)$ is called a positive function, if $f(x)>0$ for any $x\neq 0$, and $f(0)=0$; $\mathcal{K}$: the family of all strictly increasing continuous positive functions $\mu(\cdot)$.

2 Preliminaries

Consider the following general discrete time-varying nonlinear stochastic system

[TABLE]

where $x_{k}\in\mathcal{R}^{n}$ is the $n$-dimensional state vector, $y_{k}\in\mathcal{R}^{n_{y}}$ is the $n_{y}$-dimensional measurement output, $v_{k}\in\mathcal{R}^{n_{v}}$ stands for the exogenous disturbance signal with $\{v_{k}\}_{k\in\mathcal{N}}\in l^{2}_{\infty}(\Omega,{\mathcal{F}},{\{{\mathcal{F}}_{k}\}}_{k\in\mathcal{N}},\mathcal{R}^{n_{v}})$, $z_{k}\in\mathcal{R}^{n_{z}}$ is called the regulated output, which is the combination of the state and the exogenous disturbance to be estimated, and $\{w_{k}\}_{k\in\mathcal{N}}$ is a sequence of independent $n_{w}$-dimensional random variables defined on the complete filtered probability space $(\Omega,{\mathcal{F}},{\{{\mathcal{F}}_{k}\}}_{k\in\mathcal{N}},{\mathcal{P}})$, where ${\mathcal{F}}_{k}=\sigma(w_{0},w_{1},\cdots,w_{k-1})$, ${\mathcal{F}}_{0}=\{\phi,\Omega\}$. $f_{k}:\mathcal{R}^{n}\times\mathcal{R}^{n_{w}}\times\mathcal{R}^{n_{v}}\mapsto\mathcal{R}^{n}$ , $g_{k}:\mathcal{R}^{n}\times\mathcal{R}^{n_{v}}\mapsto\mathcal{R}^{n_{y}}$ and $m_{k}:\mathcal{R}^{n}\times\mathcal{R}^{n_{v}}\mapsto\mathcal{R}^{n_{z}}$ are continuous vector-valued functions. $\{x_{k}^{s,x,v}\}_{k\in\mathcal{N}}$ denotes the solution sequence of system (1) with the initial state $x\in{\mathcal{R}}^{n}$ starting at the initial time $k=s$ under the exogenous disturbance $\{v_{k}\}_{k\in\mathcal{N}}$. Similarly, $\{y_{k}^{s,x,v}\}_{k\in\mathcal{N}}$ and $\{z_{k}^{s,x,v}\}_{k\in\mathcal{N}}$ can also be defined.

In what follows, we construct the following filter for the estimation of $z_{k}$:

[TABLE]

where $\hat{x}_{k}\in\mathcal{R}^{n_{\hat{x}}}$ is the estimated value of $x_{k}$, $\hat{f}_{k}$, $\hat{g}_{k}$, and $\hat{m}_{k}$ are filter parameters to be determined. $\hat{z}_{k}\in\mathcal{R}^{n_{z}}$ is the estimated value of $z_{k}$ in system (1). Let \eta_{k}=\left[\begin{array}[]{cc}x_{k}\\ \hat{x}_{k}\end{array}\right], and $\tilde{z}_{k}\triangleq z_{k}-\hat{z}_{k}$, where $\tilde{z}_{k}$ denotes the estimation error of the regulated output, then we get the following augmented system:

[TABLE]

where $\eta_{k}\in\mathcal{R}^{n_{\eta}}$.

Remark 1

In practical engineering, only $\dim(\hat{x}_{k})\leq\dim(x_{k})$ is valuable. When $\dim(\hat{x}_{k})=\dim(x_{k})$, (2) is called a full-order filter; when $\dim(\hat{x}_{k})<\dim(x_{k})$, (2) is called a reduced-order filter; see [25].

Definition 1

(Internal stability) System (3) is said to be internally stable, if when $v=\{v_{k}\}_{k\in{\mathcal{N}}}\equiv 0$, the zero solution of

[TABLE]

is globally asymptotically stable in probability. In other word, for any $\varepsilon>0$, there have

[TABLE]

and

[TABLE]

When only (5) holds, system (4) is said to be stable in probability.

Definition 2

(External stability) For any given positive real number $\gamma>0$, system (3) is called externally stable, if for any nonzero $\{v_{k}\}_{k\in\mathcal{N}}\in l^{2}_{\infty}(\Omega,{\mathcal{F}},{\{{\mathcal{F}}_{k}\}}_{k\in\mathcal{N}},\mathcal{R}^{n_{v}})$ and zero initiate state $\eta_{0}=0$, we have

[TABLE]

Remark 2

Define an operator called the perturbation operator of system (3) as follows:

[TABLE]

with the norm of ${\mathcal{L}}$ defined by

[TABLE]

then the inequality (7) can be rewritten as $\|\mathcal{L}\|\leq\gamma$. $\|\mathcal{L}\|$ means the worst case effect from the stochastic disturbance $\{v_{k}\}_{k\in\mathcal{N}}$ to the controlled output $\{\tilde{z}_{k}\}_{k\in\mathcal{N}}$. Therefore, in order to determine whether the system (3) is externally stable, it is important to find a way to determine or estimate the norm $\|\mathcal{L}\|$.

The nonlinear stochastic $H_{\infty}$ filtering can be stated as follows:

Definition 3

(Nonlinear stochastic $H_{\infty}$ filtering) Find the filter parameters $\hat{f}_{k}(\hat{x})$, $\hat{g}_{k}(y)$, and $\hat{m}_{k}(\hat{x}_{k})$ such that

(i)

The augmented system (3) is internally stable, i.e., when $v_{k}=0$, ${k\in\mathcal{N}}$, system (3) is globally asymptotically stable in probability.

(ii)

The augmented system (3) is externally stable, i.e., the norm of the perturbation operator

[TABLE]

where $\gamma>0$ is the given disturbance attenuation level.

Definition 4

[10, 17]** We consider a continuous function $V_{k}(x):=V(k,x)$ defined on ${\mathcal{N}}\times\mathcal{D}_{r}$ with $\mathcal{D}_{r}:=\{x:\|x\|\leq r\}$. Let ${\mathcal{K}}$ be the family of all continuous strictly increasing functions $\mu(\cdot):{\mathcal{R}}^{+}\mapsto{\mathcal{R}}^{+}$, such that $\mu(0)=0$ and $\mu(t)>0$ for any $t>0$.

• (i)

$\{V_{k}(x)\}_{k\in{\mathcal{N}}}$* is a positive definite function sequence on ${\mathcal{N}}\times\mathcal{D}_{r}$ in the sense of Lyapunov if $V_{k}(0)\equiv 0$ for $k\in{\mathcal{N}}$, and there exists $\mu\in{\mathcal{K}}$, such that*

[TABLE]

• (ii)

$\{V_{k}(x)\}_{k\in{\mathcal{N}}}$* is said to be radially unbounded if*

[TABLE]

We call $v^{*}:=\{v_{k}=v_{k}^{*}\}_{k\in{\mathcal{N}}}$ as the worst-case disturbance sequence, if

[TABLE]

see [29].

The following property of the conditional mathematical expectation plays an important role in this paper.

Lemma 1** (Theorem 6.4 of [12])**

If ${\mathcal{R}}^{n}$-valued random variable $\alpha$ is independent of the $\sigma$-field $\mathcal{G}$, and ${\mathcal{R}}^{d}$-valued random variable $\beta$ is $\mathcal{G}$-measurable, then for any bounded or nonnegative function $f:{\mathcal{R}}^{n}\times{\mathcal{R}}^{d}\mapsto{\mathcal{R}}$,

[TABLE]

holds.

For a Lyapunov function sequence $\{V_{k}(\eta)\}_{k\in{\mathcal{N}}}$, $V_{k}:{\mathcal{R}}^{n_{\eta}}\mapsto{\mathcal{R}}^{+}$, because $w_{k}$ is independent of $\mathcal{F}_{k}$, $x_{k}$, ${\hat{x}}_{k}$ and $v_{k}$ are $\mathcal{F}_{k}$-measurable, from Lemma 1, we have

[TABLE]

We define

[TABLE]

as the difference operator of the function sequence $\{V_{k}\}_{k\in{\mathcal{N}}}$, and set $H_{k}(\eta,v):=\Delta_{v}V_{k}(\eta)+\|\tilde{m}_{k}(\eta,v)\|^{2}$.

The following lemma is referred to as a SBRL, which gives a sufficient condition for the external stability.

Lemma 2

For a given $\gamma>0$, if there exists a positive definite Lyapunov function sequence $\{V_{k}(\eta)\}_{k\in{\mathcal{N}}}$ satisfies the following HJI inequality

[TABLE]

then system (3) is externally stable.

Proof: Let $\{\eta_{k}\}_{k\in{\mathcal{N}}}$ be the solution of system (3), we have

[TABLE]

By Lemma 1, we have

[TABLE]

which together with condition (10), it follows from (2) that

[TABLE]

Taking a summation on both sides of the above inequality from $k=0$ to $k=N$, we can get that

[TABLE]

Note that $V_{k}(\eta)>0$ for $\eta\neq 0$ due to the positivity of the function sequence $\{V_{k}\}_{k\in{\mathcal{N}}}$ and $\eta_{0}=0$, then (2) leads to

[TABLE]

Let $N\rightarrow\infty$, we obtain

[TABLE]

for any $\{v_{k}\}_{k\in\mathcal{N}}\in l^{2}_{\infty}(\Omega,{\mathcal{F}},{\{{\mathcal{F}}_{k}\}}_{k\in\mathcal{N}},\mathcal{R}^{n_{v}})$. The proof is completed according to Definition 2. $\square$

Lemma 3

Suppose there exist a positive definite Lyapunov function sequence $\{V_{k}(\eta)\}_{k\in{\mathcal{N}}}$ such that

[TABLE]

then the stochastic difference equation

[TABLE]

is stable in probability.

Proof: By Definition 4-(i), there exists $\mu(\cdot)\in{\mathcal{K}}$ such that

[TABLE]

For any $\epsilon\in(0,1)$ and $0<r<h$, because $V_{k}(0)=0$, $k\in{\mathcal{N}}$, $V_{k}(\eta)$ is continuous with respect to $\eta$, we can find a positive constant $\delta>0$, $\delta<r$ such that

[TABLE]

For any fixed $\eta_{0}\in{\mathcal{D}}_{\delta}$, define

[TABLE]

Under the condition of $\Delta_{v=0}V_{k}(\eta)\leq 0$, by Lemma 1, for any $\eta_{i}\in{\mathcal{D}}_{r}$, we have

[TABLE]

Taking a summation from $i=0$ to $\tau_{r}\wedge k$, it follows that

[TABLE]

Taking a mathematical expectation operator in above, we have

[TABLE]

Because

[TABLE]

Combing (15), (16) and (17) leads to

[TABLE]

Let $k\to\infty$, then ${\mathcal{P}}(\tau_{r}<\infty)\leq\varepsilon$, or equivalently, ${\mathcal{P}}(\|\eta_{k}\|<r,\forall k\in{\mathcal{N}})>1-\varepsilon$. This theorem is proved. $\square$

The following is the so-called LaSalle-type theorem, which cites from Theorem 3.1 of [32].

Lemma 4

Suppose there exist a radially unbounded positive Lyapunov function sequence $\{V_{k}(\eta)\}_{k\in{\mathcal{N}}}$, a deterministic real-valued sequence $\{\gamma_{k}\geq 0\}_{k\in{\mathcal{N}}}$, and a nonnegative function $W:{\mathcal{R}}^{n_{\eta}}\mapsto{\mathcal{R}}^{+}$, satisfying

[TABLE]

[TABLE]

Let $\eta_{k}$ be the solution of

[TABLE]

then

[TABLE]

and

[TABLE]

3 General nonlinear $H_{\infty}$ filtering

In this section, we will discuss the $H_{\infty}$ filtering design problem for both nonlinear time-varying and time-invariant stochastic systems.

Theorem 1

For a given disturbance attenuation level $\gamma>0$, suppose that there exist a positive definite radially unbounded Lyapunov sequence $\{V_{k}\}_{k\in{\mathcal{N}}}$, and a positive radially unbounded function $W:{\mathcal{R}}^{n_{\eta}}\mapsto{\mathcal{R}}^{+}$, such that (10) and

[TABLE]

hold, then system (2) is a desired $H_{\infty}$ filter for system (1).

Proof: We first show that the augmented system (3) is internally stable. By (10) and the definition of $H_{k}(\eta,v)$, we have

[TABLE]

or equivalently,

[TABLE]

By (20), (21) gives that

[TABLE]

which, according to Lemma 3, implies that system (3) is stable in probability when $v=0$. In addition, by Lemma 4, we have $\lim_{k\to\infty}W(\eta_{k})=0,a.s..$ In view of $W(\eta)$ being a positive radially unbounded function, hence, the set of all limit points $G^{*}(\eta)$ only contains a zero point, that is,

[TABLE]

So from $\lim_{k\to\infty}W(\eta_{k})=0$, a.s., it follows that

[TABLE]

The internal stability is proved. While the external stability is obtained by Lemma 2. The proof of this theorem is completed. $\square$

In particular, for the following discrete time-invariant nonlinear stochastic system

[TABLE]

the filter equation is often taken as

[TABLE]

In this case, the augmented system can be rewritten as

[TABLE]

We choose a common positive Lyapunov function $V:{\mathcal{R}}^{n_{\eta}}\mapsto{\mathcal{R}}^{+}$, and for any $\eta\in{\mathcal{R}}^{n_{\eta}},v\in{\mathcal{R}}^{n_{v}}$, we write

[TABLE]

and reset $H_{k}(\eta,v):=\Delta_{v,k}V(\eta)+\|\tilde{m}(\eta,v)\|^{2}$. Theorem 1 immediately yields the following corollary:

Corollary 1

For a given disturbance attenuation level $\gamma>0$. Suppose there exist a positive radially unbounded Lyapunov function $V(\eta)$, and a positive radially unbounded function $W:{\mathcal{R}}^{n_{\eta}}\mapsto{\mathcal{R}}^{+}$, such that (10) and

[TABLE]

hold, then system (23) is a desired $H_{\infty}$ filter for system (22).

Remark 3

If $\{w_{k}\}_{k\in\mathcal{N}}$ is an independently identically distributed random variable sequence, then

[TABLE]

while (25) and $H_{k}(\eta,v)$ can be replaced by

[TABLE]

and

[TABLE]

respectively.

Corollary 2

For a given disturbance attenuation level $\gamma>0$, suppose $\{w_{k}\}_{k\in\mathcal{N}}$ is an independently identically distributed random variable sequence. If there exist a positive radially unbounded Lyapunov function $V(\eta)$, and a positive radially unbounded function $W:{\mathcal{R}}^{n_{\eta}}\mapsto{\mathcal{R}}^{+}$, satisfying (26) and

[TABLE]

then system (23) is a desired $H_{\infty}$ filter for system (22).

A special case of system (22) is the following affine nonlinear stochastic system

[TABLE]

where $\{w_{k}\}_{k\geq 0}$ is a sequence of one-dimensional independent white noise processes. Assume that $E[w_{i}]=0$, $E[w_{i}w_{j}]=\delta_{ij}$, where $\delta_{ij}$ is a Kronecker function defined by $\delta_{ij}=0$ for $i\neq j$ while $\delta_{ij}=1$ for $i=j$. We take the $H_{\infty}$ filter equation for system (28) as

[TABLE]

Thus the augmented system can be rewritten as

[TABLE]

where

[TABLE]

Theorem 2

Given the disturbance attenuation level $\gamma>0$. Suppose $\|\tilde{m}(\eta)\|$ is a continuous positive radially unbounded function, i.e.,

[TABLE]

If there exists a solution $(Q>0,{\hat{f}},{\hat{g}},{\hat{m}})$ satisfying

[TABLE]

where

[TABLE]

[TABLE]

Then, system (29) is the desired $H_{\infty}$ filter of system (28).

Proof: We take $V(\eta)=\eta^{\prime}Q\eta$ for all $\eta\in{\mathcal{R}}^{n_{\eta}}$. Since $Q>0$, there holds

[TABLE]

For any $\eta\in{\mathcal{R}}^{n_{\eta}},v\in{\mathcal{R}}^{n_{v}},$

[TABLE]

where $\Theta_{1}$ is defined in (42),

[TABLE]

Because

[TABLE]

[TABLE]

Hence, according to (41),

[TABLE]

where $\bar{\Theta}_{2}$ is defined in (43). So system (30) is externally stable by Lemma 2. In addition, from $H_{k}(\eta,v)-\gamma^{2}\|v\|^{2}\leq 0$, it yields that $\Delta_{v=0,k}V(\eta)\leq-\|\tilde{m}(\eta)\|^{2}$. By Lemma 4,

[TABLE]

Since $\|\tilde{m}(\eta_{k})\|$ is a positive and radially unbounded function, we must have $\lim_{k\to\infty}\eta_{k}=0$, a.s.. Hence, system (30) is internally stable. The proof is completed. $\square$

If $Q$ can be partitioned as a block diagonal matrix

[TABLE]

then, the inequality (41) can be rewritten as

[TABLE]

So the following corollary is obtained.

Corollary 3

Consider the disturbance attenuation level $\gamma>0$. If there exists the solution ($Q_{1}>0$, $Q_{2}>0$, $\hat{f}$, $\hat{g}$, $\hat{m}$) solving

[TABLE]

and

[TABLE]

then the desired $H_{\infty}$ filter for the system (28) is given by (29).

4 Suboptimal $H_{2}/H_{\infty}$ filtering

In this section, we further consider the suboptimal $H_{2}/H_{\infty}$ filtering design, that is, we design a filter that not only satisfies the robust $H_{\infty}$ performance, but also minimizes the estimation error under the worst-case disturbance.

Theorem 3

Consider system (3). For a given disturbance attenuation level $\gamma>0$, suppose there exist a positive definite Lyapunov function sequence $\{V_{k}\}_{k\in{\mathcal{N}}}$, a deterministic real-valued sequence $\{\gamma_{k}\geq 0\}_{k\in{\mathcal{N}}}$, and a nonnegative function $W:{\mathcal{R}}^{n_{\eta}}\mapsto{\mathcal{R}}^{+}$ satisfying (10), (19) and

[TABLE]

then the worst-case disturbance $\{v_{k}^{*}\}_{k\in{\mathcal{N}}}$ and the corresponding augmented system state $\{\eta_{k}^{*}\}_{k\in{\mathcal{N}}}$ satisfy

[TABLE]

Moreover,

[TABLE]

Moreover a suboptimal mixed $H_{2}/H_{\infty}$ filter can be synthesized by solving the following constrained optimization problem:

[TABLE]

Proof: Firstly, for any admissible external disturbance $v=\{v_{k}\}_{k\in{\mathcal{N}}}\in l^{2}_{\infty}(\Omega,{\mathcal{F}},{\{{\mathcal{F}}_{k}\}}_{k\in{\mathcal{N}}},\mathcal{R}^{n_{v}})$ and any initial state $\eta_{0}$, by the smoothness of the conditional mathematical expectation, we get

[TABLE]

Setting $v_{k}=v^{*}_{k}$ in (4) and considering equation (48), it follows that

[TABLE]

where $\tilde{z}^{*}_{k}$ is the estimation error corresponding to $v_{k}^{*}$. Taking the summation from $k=0$ to $k=T$ on both sides of the above, we have

[TABLE]

Obviously, for any admissible disturbance $\{v_{k}\}_{k\in{\mathcal{N}}}\in l^{2}_{\infty}(\Omega,{\mathcal{F}},{\{{\mathcal{F}}_{k}\}}_{k\in{\mathcal{N}}},\mathcal{R}^{n_{v}})$ satisfying (10), we have

[TABLE]

By (47) and Lemma 3.2 of [32], it yields that for $\forall\ v\in l^{2}_{\infty}(\Omega,{\mathcal{F}},{\{{\mathcal{F}}_{k}\}}_{k\in\mathcal{N}},\mathcal{R}^{n_{v}})$,

[TABLE]

[TABLE]

Letting $T\to\infty$ in (50) and (51), we have

[TABLE]

which shows that $\{v_{k}^{*}\}_{k\in{\mathcal{N}}}$ is the worst-case disturbance, and

[TABLE]

The theorem is proved. $\square$

Based on Theorem 3, if we consider time-invariant system (24) and assume that $w_{k}$, $k\in{\mathcal{N}}$, have the same distribution, we can get the following corollary that is easily verified.

Corollary 4

Consider system (24). For a given disturbance attenuation level $\gamma>0$, suppose there exist a positive Lyapunov function $V(\eta)$, a deterministic real-valued sequence $\{\gamma_{k}\geq 0\}_{k\in{\mathcal{N}}}$, and a nonnegative function $W:{\mathcal{R}}^{n_{\eta}}\mapsto{\mathcal{R}}^{+}$, such that, for $\forall\ k\in{\mathcal{N}},\ \forall\ (\eta,v)\in{\mathcal{R}}^{n_{\eta}}\times{\mathcal{R}}^{n_{v}}$,

[TABLE]

[TABLE]

and

[TABLE]

then the worst-case disturbance $\{v_{k}^{*}\}_{k\in{\mathcal{N}}}$ and the corresponding augmented system state $\{\eta_{k}^{*}\}_{k\in{\mathcal{N}}}$ satisfy

[TABLE]

Moreover,

[TABLE]

Simultaneously, a suboptimal mixed $H_{2}/H_{\infty}$ filter can be synthesized by solving the following constrained optimization problem:

[TABLE]

We find that for the general nonlinear stochastic system (1), to design its mixed $H_{2}/H_{\infty}$ filter, one needs to solve the constrained optimization problem:

[TABLE]

which is not an easy thing. However, for linear discrete-time stochastic systems, the above-mentioned problem can be converted into solving a convex optimization problem. In particular, the corresponding work for linear continuous time-invariant Itô systems has been done in [8].

We consider the following system

[TABLE]

Assume that $Ew_{k}=0$ and $Ew_{k}^{2}=1$ for all $k\in{\mathcal{N}}$. $\{w_{k}\}_{k\in{\mathcal{N}}}$ is the one-dimensional independent random variable sequence. We design the following filter for the estimation of $z_{k}$:

[TABLE]

Denoting \eta=\left[\begin{array}[]{ccc}x\\ x-\hat{x}\end{array}\right] and $\tilde{z}=z-\hat{z}$, we obtain

[TABLE]

where

[TABLE]

Setting $V(\eta)=\eta^{\prime}P\eta$ and the disturbance attenuation level $\gamma>0$, we have

[TABLE]

where

[TABLE]

and

[TABLE]

Thus when the following generalized algebraic Riccati inequality (GARI)

[TABLE]

admits a positive definite matrix solution $P>0$, $(\ref{2017filter-rree})$ holds. So system (56) is externally stable by Lemma 2. From (59),

[TABLE]

where,

[TABLE]

by Lemmas 3-4, system (56) is internally stable.

Summarize the above discussions, we have

Corollary 5

Consider system (56). For a given disturbance attenuation level $\gamma>0$, if GARI (59) or $\mathbb{P}<0$ has a positive definite matrix solution $P>0$, then (55) is the $H_{\infty}$ filter of (54). In this case, the worst-case disturbance $\{v_{k}^{*}\}_{k\in{\mathcal{N}}}$ satisfies

[TABLE]

Moreover, a suboptimal mixed $H_{2}/H_{\infty}$ filter can be synthesized by solving the following constraint optimization problem:

[TABLE]

Theorem 4

Consider system (54). For a given disturbance attenuation level $\gamma>0$, if there exist matrices $P_{1}>0$, $P_{2}>0$ and $P_{K}$ solving the following LMI

[TABLE]

then the filter (55) is the desired $H_{\infty}$ filter and the filter parameter is given by

[TABLE]

Moreover, a suboptimal mixed $H_{2}/H_{\infty}$ filter can be synthesized by solving the following constraint optimization problem:

[TABLE]

Proof: By Schur’s complement, $\mathbb{P}<0$ is equivalent to that

[TABLE]

If we take $P=diag(P_{1},P_{2})$, considering system (56), then this theorem is proved by Corollary 5.

5 Illustrative examples

In this section, we give two examples to illustrate the effectiveness of our obtained results.

Example 5.1: Let x_{k}=\left[\begin{array}[]{ccc}x_{k}(1)\\ x_{k}(2)\end{array}\right], y_{k}=\left[\begin{array}[]{ccc}y_{k}(1)\\ y_{k}(2)\end{array}\right]. We consider the following nonlinear discrete-time stochastic system:

[TABLE]

In the sequel, we design an $H_{\infty}$ filter for system (70). The environmental noise $\{w_{k}\}_{k\in{\mathcal{N}}}$ is assumed as a one-dimensional independent white noise process and the external disturbance $v_{k}=2\times(0.9999)^{k}$, $k\in{\mathcal{N}}$, so $\sum_{k=0}^{\infty}v_{k}^{2}<\infty$. We choose the Lyapunov function as

[TABLE]

By Corollary 4, it is easy to test that (45) and (3) hold with $Q_{1}=I$ and $Q_{2}=I$ and the appropriate $H_{\infty}$ filter for (70) is designed as

[TABLE]

Figures 1 and 2 show a sample trajectory in an experiment. Figure 1 shows trajectories of the $z_{k}$ and $\hat{z}_{k}$, and the error $\tilde{z}_{k}=z_{k}-\hat{z}_{k}$ is depicted in Figure 2. We use Matlab to simulate system (70) and system (71) for 1000 times to obtain the approximate value of $\sum_{i=0}^{k}E\tilde{z}_{i}^{2}$. In Figure 3, we can see that $\sum_{i=0}^{k}E\tilde{z}_{i}^{2}$ is always less than $\sum_{i=0}^{k}E\gamma^{2}v_{i}^{2}$, which is in accordance with our theoretical analysis.

Example 5.2: In order to verify the validity of Theorem 4, we use the vehicle model in $[1]$. A vehicle’s roll dynamic is governed by the following differential equation:

[TABLE]

where $\eta$ is the vehicle roll angle, $D_{n}$ means the noise intensity, $I_{xx}$ is the sprung mass moment of the inertia with respect to the roll axis, $m_{s}$ is the sprung mass, $h_{cr}$ is the sprung mass height about the roll axis, $C_{R}$ is the total torsional damping, $K_{R}$ is the stiffness coefficient, $a_{y}$ is the lateral acceleration at the vehicle center of gravity (COG) and $g$ is the acceleration due to gravity. $\{w_{k}\}_{k\in{\mathcal{N}}}$ represents the system internal noise driven by one-dimensional independent white noise processes with $E[w_{k}]=0$, $E[w_{k}w_{j}]=\delta_{kj}$, where $\delta_{kj}$ is a Kronecker function defined by $\delta_{kj}=0$ for $k\neq j$ while $\delta_{kj}=1$ for $k=j$. Then, by setting the length of the sampling interval $T=0.01$, the continuous-time system (5) can be discretized into the following system:

[TABLE]

where x_{k}=\left[\begin{array}[]{ccc}x_{k}(1)\\ x_{k}(2)\end{array}\right]=\left[\begin{array}[]{ccc}\eta_{k}\\ \Delta\eta_{k}\end{array}\right], $v_{k}$ is the disturbance with $v_{k}=0.01\times 0.9^{t}$, $y_{k}$ is the measurement signal, $z_{k}$ is the regulation output, and

[TABLE]

All parameters of the Mercedes-Benz commercial vehicle used in $[1]$ are presented in Table 1.

Then, in order to estimate $\eta_{k}$, we need to determine the parameter $\hat{K}$. So, by Theorem 4, we can find a set of feasible solutions to (68) as follows:

[TABLE]

and

[TABLE]

Thus we can design a proper filter as

[TABLE]

under \hat{x}_{0}=\left[\begin{array}[]{cccc}\hat{x}_{0}(1)\\ \hat{x}_{0}(2)\end{array}\right]=\left[\begin{array}[]{cccc}0\\ 0\end{array}\right]. Using Matlab to simulate systems (73)-(74) for 100 times under x_{0}=\left[\begin{array}[]{cccc}x_{0}(1)\\ x_{0}(2)\end{array}\right]=\left[\begin{array}[]{cccc}0.1\\ 1\end{array}\right], we can obtain Figures 4-5. From Figures 4 and 5, we can see that the augmented system is stable. Figure 6 displays $\tilde{z}_{k}(1)=x_{k}(1)-\hat{x}_{k}(1)$, which converges to zero quickly. So, filter (74) can track the the adjustment output of (73). Based on the data of the 100 experiments, we obtain the approximate value of $\sum^{k}_{i=0}E\|\tilde{z}_{i}\|^{2}$. In Figure 7, the red curve stands for $\sum^{k}_{i=0}E\|\tilde{z}_{i}\|^{2}$ and blue curve represents $\sum^{k}_{i=0}E\gamma^{2}\|v_{i}\|^{2}$. Figure 7 shows that $\sum^{k}_{i=0}E\|\tilde{z}_{i}\|^{2}\leq\sum^{k}_{i=0}\gamma^{2}E\|v_{i}\|^{2}$, which is in accordance with our theoretical analysis.

6 Conclusions

This paper has studied the robust $H_{\infty}$ filtering of general nonlinear discrete stochastic systems. A SBRL has been obtained based on the property of a conditional mathematical expectation (Lemma 2.2). By means of the discrete-time stochastic LaSalle’s invariance principle, it is shown that the nonlinear stochastic $H_{\infty}$ filtering can be constructed by solving an HJI. In the case of the worst-case disturbance $\{v^{*}_{k}\}_{k\in{\mathcal{N}}}$, a suboptimal $H_{2}/H_{\infty}$ filtering has also been studied. Two examples including a practical example are presented to illustrate the validity of our main results.