Stochastic boundedness filter design for Markovian jump linear systems with guaranteed H ∞ filtering performance

Abstract

This paper concerns the filtering problem for a class of continuous-time Markovian jump linear systems, where the Markovian jump is supposed to frequently occur in some short time intervals. For this class of Markovian jump system, the boundedness of estimation error deserves our investigation. By introducing the concepts of stochastic boundedness with respect to a finite-time interval, an observer ensuring the estimation error bounded in a prescribed boundary is constructed and the result is extended to the $H_{\infty}$ filtering problem with norm bounded disturbances. By formulating an optimization algorithm, we derive the optimal $H_{\infty}$ stochastic boundedness filter with an an optimized convex combination of estimation error boundary and $H_{\infty}$ performance index. We propose a design algorithm for when parameter optimization is involved. Numerical design examples are given to illustrate the effectiveness of our results.

Keywords

Finite-time stochastic boundedness Markovian jump system state estimation

Introduction

Markovian jump systems, as typical hybrid dynamical systems, are composed of a family of subsystems described by differential of difference equations and transitions between the subsystems determined by a Markov chain taking values in a finite set. This class of systems was introduced by Krasovskii and Lidskii (1961) and have attracted much attention in control theory and practice during recent decades (e.g. Boukas, 2005; Cao and Lam, 2000; Costa et al., 2005; Souza et al., 2006;Wu et al., 2006; Xiang, 2015; Xiang et al., 2014a,b; Xu et al., 2003; Zhang et al., 2009).

It is worth mentioning that such actual occurrences of Markovian jumps or switchings, caused by failures or repairs, are often supposed to appear in some short finite-time interval; most of the time the system works in a fixed mode and no Markovian jump occurs. That is to say, in many actual applications the Markovian jump doesn’t always occur in time. The Markovian jump only occurs in key short time intervals, such as failures and repair periods. There is no switching in most other intervals, such as the normal working interval. Take a practical example: the single-link robot arm in (Wu et al., 2006) is a typical nonlinear mechanical system. Under normal working conditions, we can consider the payload mass and the inertial moment to be fixed constants. This would continue for most of the running time. However, subject to some changes in the working conditions of a single-link robot arm, the payload may undergo abrupt changes over short timespans. Such changes can be modelled as a Markovian jump system with several subsystems. Since these abrupt changes usually occur in short intervals, this Markovian jump system, for a single-link robot arm, can be viewed as a typical short-time Markovian jump system.

In this paper, we call this class of jump switching a short-time Markovian jump. Two classes of filter should be considered: an asymptotic filter $F_{a}$ and a boundedness filter $F_{b}$ . The asymptotic filter $F_{a}$ for each subsystem is supposed to be activated for most of the running time in which no switching occurs. The boundedness filter $F_{b}$ ensures that the error state is stochastically bounded in prescribed limits to prevent the error from reaching unacceptably high values caused by switching during a certain interval. The most reported results focus on the convergence property of the error state, which is to say, on the asymptotic filter (Souza et al., 2006; Xu et al., 2003). Few results cover the confinement of estimation errors during the observer design process, which is also necessary and important in both theory and application. This motivates our study in this paper. Some very recent results have been reported for the boundedness filter $F_{b}$ for the discrete-time Markovian jump system, most of them consider the full-order dynamic filter (Cheng et al., 2014; He and Liu, 2013; Luan et al., 2010; Zhang et al., 2014). In this paper, we first intend to design a Luenberger-type observer for a continuous-time Markovian jump system. It is worth mentioning that the Luenberger-type observer has a simpler structure in comparison with the full-order dynamic filters in previous results. Moreover, we can also extend the Luenberger-type observer to achieve the $H_{\infty}$ disturbance attenuation performance in presence of exogenous disturbances. In this paper, we focus on a design boundedness filter $F_{b}$ . A technique will be presented to design the observer and ensure the error state is bounded in a prescribed boundary based on the concept of finite-time stochastic stability introduced in this paper. Then the result is extended to a design filter with $H_{\infty}$ disturbance attenuation for the error state. In particular, an LMI-based design algorithm is proposed to optimize some parameters related to system performance. Finally, by comparing the simulation results, the idea proposed in this note shows its effectiveness in the observer design process.

The remainder of this paper is organized as follows: in section 2 the problem formulation and some preliminaries are introduced, the main results for $H_{\infty}$ filter design is proposed in section 3. An illustrative numerical example is presented in section 4. Conclusions are given in section 5.

Notation: The notations used in this paper are fairly standard. The superscript ‘T’ stands for matrix transposition, $R^{n}$ denotes the n dimensional Euclidean space, the notation ‖·‖ refers to the Euclidean norm of vectors or the spectral norm of matrices. In addition, $diag {\dots}$ stands for a block-diagonal matrix. I is the identity matrix with appropriate dimensions. $λ_{min} (P)$ and $λ_{max} (P)$ stand for the smallest and the largest eigenvalue of matrix P. The notation $P ≻ 0$ ( $P \underline{≻} 0$ ) means P is real symmetric and positive definite (semi-positive definite). $E [\cdot]$ stands for the mathematical expectation.

Preliminaries and problem formulation

At first, the time interval sequence $S$ is introduced, which is given as follows:

S : {Γ_{1}^{s}, Γ_{1}^{l}, Γ_{2}^{s}, Γ_{2}^{l}, \dots, Γ_{k}^{s}, Γ_{k}^{l}, \dots}

where $Γ_{k}^{s} : [t_{k, 0}^{s}, t_{k, 0}^{s} + T_{k}^{s})$ denotes the short-time Markovian jump interval in which the system jumps among finite modes according to Markovian stochastic process ${θ (t)}$ , and $Γ_{k}^{l} : [t_{k, 0}^{l}, t_{k, 0}^{l} + T_{k}^{l})$ represents the relatively long interval in which the system maintains a fixed mode i. Since the system exhibits the short-time Markovian jump property, the following assumption is proposed:

Assumption 1. Consider the time interval sequence $S$ , it is assumed that for $\forall k$ , the following conditions are satisfied:

$T_{k}^{l} >> T_{k}^{s}$ ;

$t_{k, 0}^{s} + T_{k}^{s} = t_{k, 0}^{l}$ and $t_{k, 0}^{l} + T_{k}^{l} = t_{k + 1, 0}^{s}$ .

Remark 1. These two assumptions are necessary for time interval sequence $S$ to describe the short-time Markovian jump property. Condition (1) implies the Markovian jump only occurs for a short time, and for most of the time there is no jump mode. Then, Condition (2) makes sure $S$ is defined as, i.e., $⋃_{k = 1, 2, \dots} (Γ_{k}^{s} \cup Γ_{k}^{l}) = [0, \infty)$ and $Γ_{n}^{s} \cap Γ_{m}^{l} = \emptyset$ , $\forall n, m$ .

In this paper, we mainly focus on the short-time Markovian jump intervals, along with the following Markovian jump systems

\overset{\cdot}{x} (t) = A_{θ (t)} x (t) + B_{θ (t)}^{u} u (t) + B_{θ (t)}^{ω} ω (t)

(1a)

y (t) = C_{θ (t)} x (t) + D_{θ (t)}^{u} u (t) + D_{θ (t)}^{ω} ω (t)

(1b)

x (0) = x_{0}, θ (0) = θ_{0}

(1c)

where $x (t) \in R^{n}$ is the system state, $u (t) \in R^{m}$ is the control input, $ω (t) \in R^{p}$ is the disturbance signal and $y (t) \in R^{q}$ is measured output. $A_{θ (t)}$ , $B_{θ (t)}^{u}$ , $B_{θ (t)}^{ω}$ , $C_{θ (t)}$ , $D_{θ (t)}^{u}$ , $D_{θ (t)}^{ω}$ are matrix functions of random jumping process ${θ (t)}$ . $θ (t)$ is a finite-state, time homogeneous, Markovian stochastic jump process representing the system modes, taking values in a finite set $N : {1, 2, \dots N}$ . Let $Π = [π_{i, j}]$ , $(i, j) \in N \times N$ denote the transition probability matrix with

\Pr {θ (t + Δ) = j | θ (t) = i} = {\begin{matrix} π_{i, j} Δ + o (Δ) & i \neq j \\ 1 + π_{i, i} Δ + o (Δ) & i = j \end{matrix}

where $Δ > 0$ , $lim_{Δ \to 0} (o (Δ) / Δ) = 0$ , and $π_{i, j} \geq 0$ for $i \neq j$ is the transition rate from mode i at time t to mode j at time $t + δ$ and $π_{i, i} = - \sum_{j = 1, j \neq i}^{N} π_{i, j}$ for each mode $i \in N$ .

The filtering problem for dynamical systems usually concerns how to construct a filter or state observer such that the filtering or estimation error converges to zero as time goes to infinity. Naturally, most of the filter or observer design methodologies were derived based on the concepts of stability, defined in infinite-time intervals such as asymptotic stability, mean square stability and so on. As for the Markovian jump system under Assumption 1, the convergent property of the error state is absolutely dependent on each subsystem of error dynamic, so only the asymptotic observer design approach for a single system is sufficient to ensure the convergent error property. However, the convergence of the error state is not enough, since the error may reach unacceptably large values due to the Markovian jump switching behaviour in short intervals. This is shown in the motivation example, though the disturbance is not considered.

Example 1 (Motivation example): A short-time linear Markovian jump system with two subsystems without input and disturbance, as follows

\begin{matrix} A_{1} = [\begin{matrix} 0.65 & - 5.84 \\ 3.28 & - 2.32 \end{matrix}], C_{1}^{T} = [\begin{matrix} 0.5 \\ 1 \end{matrix}] \\ A_{2} = [\begin{matrix} - 0.79 & - 3.96 \\ - 3.64 & 2.79 \end{matrix}], C_{1}^{T} = [\begin{matrix} 0.5 \\ 1 \end{matrix}] \end{matrix}

The time interval sequence is $S : {Γ_{1}^{s}, Γ_{1}^{l}}$ , where $Γ_{1}^{s} : [0, 20)$ , $Π = [\begin{matrix} - 1.3 & 1.3 \\ 1.8 & - 1.8 \end{matrix}]$ and $Γ_{1}^{l} : [10, \infty)$ in which the subsystem 1 always works. If we design an asymptotic observer for each subsystem by linear system theory, the simulation is in Figure 1. In Figure 1 we see that the convergence of the error state can be established by the relatively long interval $Γ_{1}^{l}$ , but another important point we are interested in is the boundedness of the error state during the short-time Markovian jump interval $Γ_{1}^{s}$ , and how to avoid the error state reaching unacceptably large values, which is not only theoretically interesting and challenging but also significantly important in practical observer design for short-time Markovian jump systems. Motivated by Example 1, to define the boundedness property in a finite-time interval following a concept called finite-time stochastic stability for system (1) without input.

Definition 1. (Luan et al., 2010) Markov jump linear system (1) with $u (t) \equiv 0$ and is said to be finite-time stochastically bounded with respect to $(δ, ε, R, T)$ , where $0 \leq δ < ε$ , R is a positive definite matrix and $T \in R^{+}$ , if $E [x^{T} (t) Rx (t) | x_{0}, θ_{0}] \leq ε^{2}$ , $\forall t \in [0, T)$ , whenever $\forall x_{0} \in {x_{0} | x_{0}^{T} R x_{0} \leq δ^{2}}$ and $\forall θ_{0} \in N$ .

Remark 2. This definition is similar to that of finite-time stability of deterministic systems given in (Amato et al., 2006; Qin et al., 2014; Xiang and Xiao, 2011, 2013; Xiang et al., 2014b). But, there are also some differences, by the stochastic mechanism of system (1), we note here $ε$ is viewed as the boundary of average value of the state $x (t)$ in the finite-time interval $[0, T]$ .

Based on the definition of finite-time stochastic boundedness, the main problem is presented below for considering the boundedness of states in error dynamics, which plays a fundamental role in the boundedness filter design.

Problem 1. Given Markovian jump system (1), construct an observer ensuring the finite-time boundedness of error dynamics with respect to $(δ, ε, R, T)$ .

When the $H_{\infty}$ filtering performance is considered, the input disturbance is assumed to be energy bounded in $[0, T]$ , described by the following inequality

\int_{0}^{T} ω^{T} (t) ω (t) d t < d^{2}

(2)

where $d \geq 0$ . The disturbance attenuation with $H_{\infty}$ performance is considered.

Definition 2. (Luan et al., 2010) Given a real number $γ > 0$ , Markov jump linear system (1) with disturbance $ω (t)$ satisfying (2) and an input $u (t)$ which is said to be finite-time stochastically bounded with $H_{\infty}$ performance $γ$ with respect to $(δ, ε, R, T)$ , where $0 \leq δ < ε$ , R is a positive definite matrix and $T \in R^{+}$ , if the system (1) is finite-time stochastically bounded with respect to $(δ, ε, R, T)$ and the response $y (t)$ satisfies

{(E [\int_{0}^{T} y^{T} (t) y (t) d t])}^{1 / 2} < γ {(\int_{0}^{T} ω^{T} (t) ω (t) d t)}^{1 / 2}

(3)

under zero-initial condition.

Thus, the design problem for the class of boundedness filter with $H_{\infty}$ performance is summarized as

Problem 2. Given Markovian jump system (1) and a real number $γ > 0$ , construct a filter ensuring the stochastic finite-time boundedness of error dynamics with respect to $(δ, ε, R, T)$ with $H_{\infty}$ performance $γ$ .

Before ending this section, we present the well known Gronwell–Bellman Lemma.

Lemma 1. Let $m (t)$ , $r (t)$ be non-negative continuous functions on $[a, b)$ . If continuous function $y (t) : [a, b) \to R$ satisfies:

y (t) \leq m (t) + \int_{a}^{t} r (τ) y (τ) d τ, \forall t \in [a, b)

then,

y (t) \leq m (t) + \int_{a}^{t} m (τ) r (τ) e^{\int_{τ}^{t} r (s) d s} d τ, \forall t \in [a, b)

Main results

Initially, Problem 1 with $ω = 0$ is considered. Since $ω (t) \equiv 0$ is a particular case of $d = 0$ in (2), we can solve Problem 1 involving the disturbance to cover more general situations. Before giving our results, an explicit fact is recalled. For a symmetric positive definite matrix $R \in R^{n \times n}$ , it is easy to verify that R can be factorized according to $R = (R^{1 / 2})^{T} R^{1 / 2}$ , where $R^{1 / 2} \in R^{n \times n}$ is a symmetric positive definite matrix. The intial results of this paper are then derived by the following theorem, in which a stochastic boundedness observer is designed.

Theorem 1. Consider the Markovian jump system (1), if there exist matrices $P_{i}$ , $Q_{i}$ and scalars $κ > 1$ , $μ \geq 0$ , such that

R ≺ P_{i} ≺ κ R

(4)

[\begin{matrix} Ω_{i} & P_{i} B_{i}^{ω} - X_{i} D_{i}^{ω} \\ {(B_{i}^{ω})}^{T} P_{i} - {(D_{i}^{ω})}^{T} X_{i}^{T} & - Q_{i} \end{matrix}] ≺ 0

(5)

(κ δ^{2} + ρ) e^{μ T} - ε^{2} < 0

(6)

where $Ω_{i} = A_{i}^{T} P_{i} - C_{i}^{T} X_{i}^{T} + P_{i} A_{i} - X_{i} C_{i} - μ R + \sum_{j = 1}^{N} π_{i, j} P_{j}$ and $ρ = sup_{i \in N} {λ_{max} (Q_{i})} d^{2}$ . The observers for the subsystems are given as

\overset{\cdot}{\hat{x}} (t) = A_{i} \hat{x} (t) + B_{i} u (t) + L_{i} (y (t) - C_{i} \hat{x} (t) + D_{i} u (t))

(7)

where $L_{i} = P_{i}^{- 1} X_{i}$ . Ensure the finite-time error dynamics are stochastically bounded with respect to $(δ, ε, R, T)$ .

Proof. Construct the observer in the form of (7), let the error $ξ (t) = x (t) - \hat{x} (t)$ , the error dynamics is composed by

\overset{\cdot}{ξ} (t) = {\tilde{A}}_{i} ξ (t) + {\tilde{B}}_{i}^{ω} ω (t), i \in N

where ${\tilde{A}}_{i} = A_{i} - P_{i}^{- 1} X_{i} C_{i}$ , ${\tilde{B}}_{i}^{ω} = B_{i}^{ω} - P_{i}^{- 1} X_{i} D_{i}^{ω}$ .

Define the stochastic Lyapunov functional candidate for error dynamics to be $V (ξ (t), θ (t)) = ξ^{T} (t) P_{θ (t)} ξ (t)$ . Let $A$ be the weak infinitesimal operator [13] of the stochastic process ${ξ (t), θ (t)}$ . Then, for $θ (t) = i$ , $\forall i \in N$ , it can be verified that

\begin{matrix} A V (ξ (t), i) \\ = lim_{Δ \to 0} \frac{1}{Δ} [E {V (ξ (t + Δ), σ (t + Δ)) | ξ (t), i} - V (ξ (t), i)] \\ = [\begin{matrix} ξ^{T} (t) & ω^{T} (t) \end{matrix}] [\begin{matrix} Ξ_{i} & P_{i} {\tilde{B}}_{i}^{ω} \\ {({\tilde{B}}_{i}^{ω})}^{T} P & 0 \end{matrix}] [\begin{matrix} ξ (t) \\ ω (t) \end{matrix}] \end{matrix}

where $Ξ_{i} = {\tilde{A}}_{i}^{T} P_{i} + P_{i} {\tilde{A}}_{i} + \sum_{j = 1}^{N} π_{i, j} P_{j}$ .

Thus, from (4) and (5), the following inequality can be derived

\begin{matrix} A V (ξ (t), θ (t)) \\ < μ ξ^{T} (t) P_{θ (t)} ξ (t) + ω^{T} (t) Q_{θ (t)} ω (t) \\ \leq μ V (ξ (t), θ (t)) + sup_{i \in N} {λ_{max} (Q_{i})} ω^{T} (t) ω (t) \end{matrix}

Then, by Dynkin’s formula (see Kushner, 1967) and (2), we have

\begin{matrix} E [V (ξ (t), θ (t))] - V (ξ_{0}, θ_{0}) \\ = E [\int_{0}^{t} A V (ξ (τ), θ (τ)) d τ] \\ < \int_{0}^{t} E [μ V (ξ (τ), θ (τ)) + sup_{i \in N} {λ_{max} (Q_{i})} ω^{T} (t) ω (t)] d τ \\ < \int_{0}^{t} μ E [V (ξ (τ), θ (τ))] d τ + ρ \end{matrix}

where $ρ = sup_{i \in N} {λ_{max} (Q_{i})} d^{2}$ . Applying Lemma 1, it gives us

\begin{matrix} E [V (ξ (t), θ (t))] \\ < V (ξ_{0}, θ_{0}) + ρ + \int_{0}^{t} (V (ξ_{0}, θ_{0}) + ρ) μ e^{\int_{τ}^{t} μ d s} d τ \\ < (V (ξ_{0}, θ_{0}) + ρ) e^{μ t} \end{matrix}

(8)

Let $P_{θ (t)} = R^{- 1 / 2} P_{θ (t)} R^{- 1 / 2}$ , we know that

\begin{matrix} E [V (ξ (t), θ (t))] & = E [ξ^{T} (t) P_{θ (t)} ξ (t) | ξ_{0}, θ_{0}] \\ = E [ξ^{T} (t) R^{1 / 2} P_{θ (t)} R^{1 / 2} ξ (t) | ξ_{0}, θ_{0}] \\ \geq inf_{i \in N} {λ_{min} (P_{i})} E [ξ^{T} (t) R ξ (t) | ξ_{0}, θ_{0}] \end{matrix}

(9)

On the other hand, for $\forall θ_{0} \in N$ we can obtain

\begin{matrix} V (ξ_{0}, θ_{0}) = ξ_{0}^{T} P_{θ_{0}} ξ_{0} & = ξ_{0}^{T} R^{1 / 2} P_{θ_{0}} R^{1 / 2} ξ_{0} \\ \leq sup_{i \in N} {λ_{max} (P_{i})} ξ_{0}^{T} R ξ_{0} \end{matrix}

Using the fact $μ \geq 0$ , $ξ_{0}^{T} R ξ_{0} \leq δ^{2}$ and $t \in [0, T)$ we get

\begin{matrix} (V (ξ_{0}, θ_{0}) + ρ) e^{μ t} \\ \leq & (sup_{i \in N} {λ_{max} (P_{i})} ξ_{0}^{T} R ξ_{0} + ρ) e^{μ t} \\ \leq & (sup_{i \in N} {λ_{max} (P_{i})} δ^{2} + ρ) e^{μ T} \end{matrix}

(10)

Altogether, with (8)–(10), the following inequality can be derived

E [ξ^{T} (t) R ξ (t) | ξ_{0}, θ_{0}] < \frac{(sup_{i \in N} {λ_{max} (P_{i})} δ^{2} + ρ) e^{μ T}}{inf_{i \in N} {λ_{min} (P_{i})}}

(11)

Then by (4) we have $i \in N$

R ≺ P_{i} ≺ κ R \Rightarrow I ≺ R^{- 1 / 2} P_{i} R^{- 1 / 2} ≺ κ I \Rightarrow I ≺ P_{i} ≺ κ I

Since $sup_{i \in N} {λ_{max} (P_{i})} \leq κ$ , $inf_{i \in N} {λ_{min} (P_{i})} \geq 1$ and by (6),

(κ δ^{2} + ρ) e^{μ T} < ε^{2} \Rightarrow \frac{(\sup_{i \in N} {λ_{\max} (P_{i})} δ^{2} + ρ) e^{μ T}}{\inf_{i \in N} {λ_{\min} (P_{i})}} < ε^{2}

(12)

By (11) and (12), we get

E [ξ^{T} (t) R ξ (t) | ξ_{0}, θ_{0}] < ε^{2}

Thus we can conclude that error dynamics is finite-time stochastically bounded with respect to $(δ, ε, R, T)$ .

Problem 1, concerning the disturbance-free system, can be solved by corollary below as the particular case of $d = 0$ .

Corollary 1. Consider the Markovian jump system (1) with $ω (t) \equiv 0$ , if there exist matrices $P_{i}$ and scalars $κ > 1$ , $μ \geq 0$ such that

R ≺ P_{i} ≺ κ R

(13)

A_{i}^{T} P_{i} - C_{i}^{T} X_{i}^{T} + P_{i} A_{i} - X_{i} C_{i} - μ R + \sum_{j = 1}^{N} π_{i, j} P_{j} ≺ 0

(14)

κ δ^{2} e^{μ T} - ε^{2} < 0

(15)

The observers for subsystems are given as follows

\overset{\cdot}{\hat{x}} (t) = A_{i} \hat{x} (t) + B_{i} u (t) + L_{i} (y (t) - C_{i} \hat{x} (t) + D_{i} u (t))

(16)

where $L_{i} = P_{i}^{- 1} X_{i}$ . Ensure the finite-time error dynamics is stochastically bounded with respect to $(δ, ε, R, T)$ .

Proof. Since $ω (t) \equiv 0$ , we can let $B_{i}^{ω} = 0$ , $D_{i}^{ω} = 0$ and $ρ = 0$ in Theorem 1. Corollary 1 is directly established.

Remark 3. In the above results, we see that the observer design procedure is more than solving LMIs due to the nonlinearity term $e^{μ T}$ related to the parameter $μ$ . So, a one parameter search may be necessary. Nevertheless, this does not represent a hard computational problem, which will be given later. Moreover, it should be pointed out that the mean square stability of error dynamics can also established by (16) when $μ = 0$ (see Boukas(2005)). Hereby the corresponding observer can guarantee an error state, both bounded in a finite-time interval and convergent in an infinite-time interval. However, in more general cases, the problem is often solved by $μ > 0$ , which implies that only the boundedness is ensured. Then, to guarantee the convergence of the error state, another class of asymptotic observer (which can be designed according to the observer design technique for a single system) should be activated in the relatively long interval $Γ_{k}^{l}$ .

Remark 4. Theorem 1 provides a sufficient condition to ensure the existence of finite-time stochastic boundedness, as observed by the Markovian jump system (1). However, it should be clarified that this LMI condition has its conservatism, since it is a sufficient condition. The conservatism is mainly caused by the specific quadratic Lyapunov function. If advanced techniques such as sum-of-square were used to construct the Lyapunov function in the framework of Theorem 1, the conservatism could be reduced.

Then Problem 2 is considered, involving the disturbance attenuation with $H_{\infty}$ performance. The controlled response for the error state $ξ (t)$ is

\tilde{y} (t) = H_{θ (t)} ξ (t) + G_{θ (t)} ω (t)

where $H_{i}$ , $G_{i}$ , $i \in N$ are known matrices. Then the stochastic boundedness filter with guaranteed $H_{\infty}$ performance can be designed by the following theorem.

Theorem 2. Consider the Markovian jump system (1), if there exist matrices $P_{i}$ , $Q_{i}$ and scalars $κ > 1$ , $μ \geq 0$ , such that

R ≺ P_{i} ≺ κ R

(17)

[\begin{matrix} Ω_{i} & P_{i} B_{i}^{ω} - X_{i} D_{i}^{ω} & H_{i}^{T} \\ {(B_{i}^{ω})}^{T} P_{i} - {(D_{i}^{ω})}^{T} X_{i}^{T} & - γ^{2} e - μ TI & G_{i}^{T} \\ H_{i} & G_{i} & - I \end{matrix}] ≺ 0

(18)

κ δ^{2} e^{μ T} + γ^{2} d^{2} - ε^{2} < 0

(19)

where $Ω_{i} = A_{i}^{T} P_{i} - C_{i}^{T} X_{i}^{T} + P_{i} A_{i} - X_{i} C_{i} - μ R + \sum_{j = 1}^{N} π_{i, j} P_{j}$ and $ρ = sup_{i \in N} {λ_{max} (Q_{i})} d^{2}$ . The filters for the subsystems are given as

\overset{\cdot}{\hat{x}} (t) = A_{i} \hat{x} (t) + B_{i} u (t) + L_{i} (y (t) - C_{i} \hat{x} (t) + D_{i} u (t))

(20)

where $L_{i} = P_{i}^{- 1} X_{i}$ , ensures that the error dynamics finite-time is stochastically bounded with respect to $(δ, ε, R, T)$ , and the $H_{\infty}$ filtering performance is guaranteed.

Proof. Choosing the filter (20) and $ξ (t) = x (t) - \hat{x} (t)$ as the error state, the error dynamics is composed by

\overset{\cdot}{ξ} (t) = {\tilde{A}}_{i} ξ (t) + {\tilde{B}}_{i}^{ω} ω (t)

(21a)

\tilde{y} (t) = H_{i} ξ (t) + G_{i} ω (t)

(21b)

where ${\tilde{A}}_{i} = A_{i} - P_{i}^{- 1} X_{i} C_{i}$ , ${\tilde{B}}_{i}^{ω} = B_{i}^{ω} - P_{i}^{- 1} X_{i} D_{i}^{ω}$ .

Letting $Q_{i} = γ^{2} e^{- μ T} I$ and $ρ = γ^{2} d^{2} e^{- μ T}$ , by (17)–(19), error dynamics (21a) is finite-time stochastically bounded with respect to $(δ, ε, R, T)$ according to Theorem 1. Then we need to prove (3) is satisfied when $ξ_{0} = 0$ . By choosing $V (ξ (t), θ (t)) = ξ^{T} (t) P_{θ (t)} ξ (t)$ , we consider

\begin{matrix} J = & {\tilde{y}}^{T} (t) \tilde{y} (t) - γ^{2} e^{- μ T} ω^{T} (t) ω (t) \\ + A V (ξ (t), θ (t)) - μ V (ξ (t), θ (t)) \end{matrix}

From (17) and (18) it is easy to verify $J < 0$ . Then we have

\begin{matrix} E [\int_{0}^{T} {\tilde{y}}^{T} (t) \tilde{y} (t) d t] - γ^{2} e^{- μ T} \int_{0}^{T} ω^{T} (t) ω (t) d t \\ + E [\int_{0}^{T} A V (ξ (t), θ (t)) d t] - E [μ \int_{0}^{T} V (ξ (t), θ (t)) d t] < 0 \end{matrix}

By Dynkin’s formula and zero-initial condition assumption, which implies $V (ξ_{0}, θ_{0}) = 0$ , we have

\begin{matrix} E [V (ξ (T), θ (T))] \\ < \int_{0}^{T} μ E [V (ξ (t), θ (t))] d t - E [\int_{0}^{T} {\tilde{y}}^{T} (t) \tilde{y} (t) d t] \\ + γ^{2} e^{- μ T} \int_{0}^{T} ω^{T} (t) ω (t) d t \end{matrix}

Thus, by Lemma 1 it yields that

\begin{matrix} E [V (ξ (T), θ (T))] \\ < - E [\int_{0}^{T} {\tilde{y}}^{T} (t) \tilde{y} (t) d t] + γ^{2} e^{- μ T} \int_{0}^{T} ω^{T} (t) ω (t) d t \\ + \int_{0}^{T} (- E [\int_{0}^{τ} {\tilde{y}}^{T} (t) \tilde{y} (t) d t] + γ^{2} e^{- μ T} \int_{0}^{τ} ω^{T} (t) ω (t) d t) μ e^{\int_{τ}^{T} μ d s} d τ \\ = \int_{0}^{T} (- E [{\tilde{y}}^{T} (τ) \tilde{y} (τ)] e^{μ (T - τ)} + γ^{2} e^{- μ τ} ω^{T} (τ) ω (τ)) d τ \end{matrix}

Due to $μ \geq 0$ and $V (ξ (T), θ (T)) \geq 0$ , we can obtain

\begin{matrix} \int_{0}^{T} E [{\tilde{y}}^{T} (τ) \tilde{y} (τ)] e^{μ (T - τ)} d τ < γ^{2} \int_{0}^{T} e^{- μ τ} ω^{T} (τ) ω (τ) d τ \\ \Rightarrow \int_{0}^{T} E [{\tilde{y}}^{T} (τ) \tilde{y} (τ)] d τ < γ^{2} \int_{0}^{T} ω^{T} (τ) ω (τ) d τ \end{matrix}

Thus, the proof is completed.

The optimization on some parameters related to system performance is often of interest, such as the disturbance attenuation performance index $γ$ , and the value of the estimation error boundary $ε$ in the short-time Markovian jump interval. To generally cover the optimization problem mentioned above, we consider the following convex optimization problem:

\begin{matrix} min Ψ (σ) \\ s . t . & (17), (18) and (19) \end{matrix}

(22)

where $Ψ (σ) = σ γ^{2} + (1 - σ) ε^{2}$ , $0 \leq σ \leq 1$ .

In summary, the design algorithm is given as below.

Remark 5. In Step 4, the optimal solution near $\tilde{μ}$ can be found by an unconstrained nonlinear optimization approach, which can be implemented on numerical optimization software tools such as the program ‘fminsearch’ in the optimization toolbox of Matlab to ascertain a locally convergent solution.

Remark 6. By the similar techniques in (Hu and Yuan, 2009; Zhang et al.,2009), all the results in this note can be readily extended to the short-time Markovian jump system with norm-bounded parameters or polytopic uncertainties.

Algorithm 1.

Optimal finite-time robust observer design algorithm.

1: Step 1: Initialize a value of

μ = 0

, set a variation value

δ μ > 0

and termination value

\bar{μ}

;

2: Step 2: Setting

μ = μ + δ μ

, solve following optimization problem (24) with fixed

μ

3: Step 3: When the optimization problem (22) is solvable for the first time, the value of

μ

is recorded as

μ_{min}

. Then, if

μ \geq μ_{min} + \bar{μ}

, terminate procedure, otherwise record the parameters

(μ, Ψ_{min})

pair-wisely and go back to Step 2.

4: Step 4: Select the

\tilde{μ}

with smallest

Ψ_{min}

recorded in Step 3. Obtain the locally optimized observer and ascertain the local optimal value of

Ψ_{min}^{*}

near

\tilde{μ}

by an unconstrained nonlinear optimization approach.

Numerical design example

To show the advantages of our approach, we still consider the system given in Example 1 in which the error state reaches very large values. Moreover, to illustrate the design algorithm, the disturbance is considered as $ω (t) = 1 / (t + 1)$ and matrices

\begin{matrix} B_{1}^{ω} = [\begin{matrix} 0.2 \\ 0.1 \end{matrix}], H_{1} = [\begin{matrix} 0.5 & 1 \end{matrix}], G_{1} = 0.2 \\ B_{2}^{ω} = [\begin{matrix} 0.5 \\ - 0.2 \end{matrix}], H_{2} = [\begin{matrix} 1 & - 0.2 \end{matrix}], G_{2} = - 0.5 \end{matrix}

The initial state is assumed to be $x^{T} (0) x (0) < 2$ , The transition probability matrix is given as

Π = [\begin{matrix} - 1.3 & 1.3 \\ 1.8 & - 1.8 \end{matrix}]

Thus, the parameters can be ascertained as $δ = \sqrt{2}$ , $d = 1$ , $R = I$ , $T = 20$ . Using the design algorithm in (22), we obtain

L_{1} = [\begin{matrix} 1401.4 & 358.6 \end{matrix}], L_{2} = [\begin{matrix} 112.3 & - 202.5 \end{matrix}]

with $Φ_{min} = 143.9763$ where $γ = 5.8788$ and $ε = 10.4602$ . The simulations results are given in Figures 2, 3, 4.

Compared with Example 1, the boundary for the error state is explicitly reduced significantly, so that the estimated error will not reach the large and unacceptable values during the short-time Markovian jump interval $Γ_{1}^{s}$ and the required performance is obtained. In Figure 2, the benefit of the boundedness filter is obvious compared to Figure 1, where only the asymptotic filter is considered. The transient response of error $‖ e ‖$ has improved significantly, as the boundary of error has been reduced greatly and is able to meet the actual requirements. Moreover, the output also satisfies the disturbance attenuation performance of Figure 3. The switching instants are shown in Figure 4.

We should also mention that, to ensure the convergence of estimation error, another subsystem asymptotic observer should be designed and activated in $Γ_{1}^{l}$ .

Conclusions

This note has addressed the filtering problem for a class of short-time Markovian jump systems. At first, with regard to a system without disturbance, an observer guaranteeing the error state, stochastically bounded in a prescribed boundary during a finite-time interval, is designed by introducing the conception of finite-time stochastic stability. Then, the disturbance attenuation problem is considered in the presence of input noise and a $H_{\infty}$ filter is designed. Finally, an optimal filter with optimized convex combination of estimation error boundary and $H_{\infty}$ performance index is derived. A Numerical example is given to show the advantages of our approach. As Remark 4 indicates, the quadratic Lyapunov function structure could bring in some conservatism in the analysis and design results. It would be meaningful if the more general structure of the Lyapunov function, such as a polynomial structure, could be used to design an observer and filter. Moreover, in future studies, we will also aim at applying our approach to real applications, such as the single-link robot arm system, which can be modelled as a short-time Markovian jump system.

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