Robust L 2L ∞ Filtering of Time-Delay Jump Systems with Respect to the Finite-Time Interval

This paper studied the problem of stochastic finite-time boundedness and disturbance attenuation for a class of linear time-delayed systems with Markov jumping parameters. Sufficient conditions are provided to solve this problem. The L2-L∞ filters are, respectively, designed for time-delayed Markov jump linear systems with/without uncertain parameters such that the resulting filtering error dynamic system is stochastically finite-time bounded and has the finite-time interval disturbance attenuation γ for all admissible uncertainties, time delays, and unknown disturbances. By using stochastic Lyapunov-Krasovskii functional approach, it is shown that the filter designing problem is in terms of the solutions of a set of coupled linear matrix inequalities. Simulation examples are included to demonstrate the potential of the proposed results.


Introduction
Since the introduction of the framework of the class of Markov jump linear systems MJLSs by Krasovskii and Lidskii 1 , we have seen increasing interest for this class of stochastic systems.It was used to model a variety of physical systems, which may experience abrupt changes in structures and parameters due to, for instance, sudden environment changes, subsystem switching, system noises, and failures occurring in components or interconnections and executor faults.For more information regarding the use of this class of systems, we refer the reader to Sworder and Rogers 2 , Athans 3 , Arrifano and Oliveira 4 , and the references therein.It has been recognized that the time-delays and parameter uncertainties, which are inherent features of many physical processes, are very often the cause for poor performance of systems.In the past few years, considerable attention has N i<j a ij ⇔ a 12 a 13 a 23 .E{ * } stands for the mathematics statistical expectation of the stochastic process or vector and * is the Euclidean vector norm.L n 2 0 T is the space of n dimensional square integrable function vector over 0 T .P > 0 stands for a positivedefinite matrix.I is the unit matrix with appropriate dimensions.0 is the zero matrix with appropriate dimensions.In symmetric block matrices, we use " * " as an ellipsis for the terms that are introduced by symmetry.
The paper is organized as follows.In Section 2, we derive the new definitions about stochastic finite-time filtering of MJLSs.In Section 3, we give the main results of L 2 -L ∞ filtering problem of MJLSs and extend this to uncertain dynamic MJLSs in Section 4. In Section 5, we demonstrate two simulation examples to show the validity of the developed methods.

Problem Formulation
Given a probability space Ω, F, P r where Ω is the sample space, F is the algebra of events and P r is the probability measure defined on F. Let the random form process {r t , t ≥ 0} be the continuous-time discrete-state Markov stochastic process taking values in a finite set M {1, 2, . . ., N} with transition probability matrix P r {P ij t , i, j ∈ M} given by where Δt > 0 and lim Δt → 0 o Δt /Δt → 0. π ij ≥ 0 is the transition probability rates from mode i at time t to mode j i / j at time t Δt, and N j 1,j / i π ij −π ii .Consider the following time-delay dynamic MJLSs over the probability space Ω, F, P r :

2.4
Let x T t x T t e T t , the filtering error system 2.4 can be rewritten as x T t λ t λ t − ϕ t , r t ξ t , t ∈ −d 0 ,

2.6
The objective of this paper consists of designing the finite-time filter of time-delay MJLSs in 2.1 and obtaining an estimate z t of the signal z t such that the defined guaranteed L 2 -L ∞ performance criteria are minimized.For some given initial conditions 24-27 , the general idea of finite-time filtering can be formalized through the following definitions over a finite-time interval.

Assumption 1. The external disturbance w t is time-varying and satisfies
Definition 2.2.For a given time-constant T > 0, the filtering error MJLSs system 2.5 with w t 0 is stochastically finite-time stable FTS if there exist positive matrix R i ∈ R 2n×2n > 0 and scalars c 1 > 0 and c 2 > 0, such that Definition 2.3 FTB .For a given time-constant T > 0, the filtering error 2.5 is stochastically finite-time bounded FTB with respect to c 1 c 2 T R i W if condition 2.8 holds.
Remark 2.4.Notice that FTB and FTS are open-loop concepts.FTS can be recovered as a particular case of FTB with W 0 and FTS leads to the concept of FTB in the presence of external inputs.FTB implies finite-time stability, but the converse is not true.It is necessary to point out that Lyapunov stability and FTS are independent concepts.Different with the concept of Lyapunov stability 33-35 which is largely known to the control community, a stochastic MJLSs is FTS if, once we fix a finite time-interval 36, 37 , its state remain within prescribed bounds during this time-interval.Moreover, an MJLS which is FTS may not be Lyapunov stochastic stability; conversely, a Lyapunov stochastically stable MJLS could be not FTS if its states exceed the prescribed bounds during the transients.Definition 2. 5 Feng et al. 34 , Mao 35 .Let V x t , r t , t > 0 be the stochastic positive functional; define its weak infinitesimal operator as

2.9
Definition 2.6.For the filtering error MJLSs 2.5 , if there exist filter parameters A fi , B fi , and C fi , and a positive scalar γ, such that 2.5 is FTB and under the zero-valued initial condition, the system output error satisfies the following cost function inequality for T > 0 with attenuation γ > 0 and for all admissible w t with the constraint condition 2.7 , Remark 2.7.In stochastic finite-time L 2 -L ∞ filtering process, the unknown noises w t are assumed to be arbitrary deterministic signals of bounded energy and the problem of this paper is to design a filter that guarantees a prescribed bounded for the finite-time interval induced L 2 -L ∞ norm of the operator from the unknown noise inputs w t to the output error r t , that is, the designed stochastic finite-time L 2 -L ∞ filter is supposed to satisfy inequality 2.10 with attenuation γ.

Finite-Time L 2 -L ∞ Filtering for MJLSs
In this section, we will study the stochastic finite-time L 2 -L ∞ filtering problem for time-delay dynamic MJLSs 2.2 .Theorem 3.1.For a given time-constant T > 0, the filtering error MJLSs 2.5 are stochastically FTB with respect to c 1 c 2 T R i W and has a prescribed L 2 -L ∞ performance level γ > 0 if there exist a set of mode-dependent symmetric positive-definite matrices P i ∈ R 2n×2n and symmetric positive-definite matrix Q ∈ R 2n×2n , satisfying the following matrix inequalities for all i ∈ M, , and Proof.For the given symmetric positive-definite matrices P i ∈ R 2n×2n and Q ∈ R 2n×2n , we define the following stochastic Lyapunov-Krasovskii functional as Then referring to Definition 2.5 and along the trajectories of the resulting closed-loop MJLSs 2.7 , we can derive the corresponding time derivative of V x t , i as

3.5
Considering the L 2 -L ∞ filtering performance for the dynamic filtering error system 2.5 , we introduce the following cost function by Definition 2.6 with t ≥ 0,

3.6
According to relation 3.1 , it follows that J 1 t < 0, that is, Then, multiplying the above inequality by e −αt , we have I e −αt E V x t , i < e −αt w T t w t .

3.8
In the following, we assume zero initial condition, that is, x t 0, for t ∈ −d 0 , and integrate the above inequality from 0 to T ; then 3.9 Recalling to the defined Lyapunov-Krasovskii functional, it can be verified that, e −αt w T t w t dt.

3.10
By 3.2 and within the finite-time interval 0 T , we can also get w T t w t dt.

3.11
Therefore, the cost function inequality 2.10 can be guaranteed by setting γ √ e αT γ, which implies J E{ r t 2 ∞ } − γ 2 w t 2 2 < 0. On the other hand, by integrating the above inequality 3.8 between 0 to t ∈ 0 T , it yields e −αt E{V x t , i } − E{V {x 0 , r t ξ 0 }} < t 0 e −αs w T t w t ds.

3.12
Denote , σ P min i∈M σ min P i , σ P max i∈M σ max P i , and

Mathematical Problems in Engineering
From the selected stochastic Lyapunov-Krasovskii function, we can obtain 3.14 Then we can get which implies E x T t R i x t < c 2 for ∀t ∈ 0 T .This completes the proof.
Theorem 3.2.For a given time-constant T > 0, the filtering error dynamic MJLSs 2.5 are FTB with respect to c 1 c 2 T R i W with R i diag{V i V i } and has a prescribed L 2 -L ∞ performance level γ > 0 if there exist a set of mode-dependent symmetric positive-definite matrices P i ∈ R n×n , symmetric positive-definite matrix Q ∈ R n×n , a set of mode-dependent matrices X i , Y i , and C fi and positive scalars σ 1 , σ 2 satisfying the following matrix inequalities for all i ∈ M, 3.17 Moreover, the suitable filter parameters can be given as

Extension to Uncertain MJLSs
It has been recognized that the unknown disturbances and parameter uncertainties are inherent features of many physical process and often encountered in engineering systems, their presences must be considered in realistic filter design.For these, we consider the following stochastic time-delay MJLSs with uncertain parameters, point out that the unknown mode-dependent matrix Γ i t in 4.2 can also be allowed to be state-dependent, that is, Γ i t Γ i t, x t , as long as Γ i t, x t ≤ 1 is satisfied.For this case, we can get the following filtering error system by letting x T t x T t e T t :

4.3
where holds if and only if there exists a positive scalar α > 0, such that By following the similar lines and the main proofs of Theorems 3.1 and 3.2 and using the above Lemma 4.2, one can get the results stated as follows.
Theorem 4.3.For a given time-constant T > 0, the filtering error MJLSs 4.3 with uncertainties are stochastically FTB with respect to c 1 c 2 T V i W and has a prescribed L 2 -L ∞ performance level γ > 0 if there exist a set of mode-dependent symmetric positive-definite matrices P i ∈ R n×n , symmetric positive-definite matrix Q ∈ R n×n , a set of mode-dependent matricesX i , Y i , and C fi and positive scalars σ 1 , σ 1 and ε i satisfying LMIs 3.17 -3.20 , and the following matrix inequalities for all i ∈ M,  2 , and ε i .Therefore, for given V i , c 1 , T , and W, we can take γ 2 as optimal variable, that is, to obtain an optimal stochastic finite-time L 2 -L ∞ filter, the attenuation lever γ 2 can be reduced to the minimum possible value such that LMIs 3.16 -3.20 or LMIs 4.7 , 3.17 -3.20 are satisfied.The optimization problem 39 can be described as follows:

Numeral Examples
Example 5.1.Consider a class of constant time-delay MJLSs with parameters described as follows: Mode 1

5.2
Let the transition rate matrix be Π With the initial value for α 0.5, W 2, T 4, and V i I 2 , we fix γ 0.8 and look for the optimal admissible c 2 of different c 1 guaranteeing the stochastically finite-time boundedness of desired filtering error dynamic properties.Table 1 and Figure 1, respectively, give the optimal minimal admissible c 2 with different initial upper bound c 1 .

5.4
The modes, transition rate matrix, the matrices parameters and initial conditions are defined similarly as Example 5.1.By solving LMIs 4.7 , 3.17

5.6
The simulation results of jump mode the estimation of changing between modes during the simulation with the initial mode 1 , the response of system states real states and estimated states and filtering output error are shown in Figures 2-5, which show the effective of the proposed approaches.
It is clear from Figures 3-5 that the estimated states can track the real states smoothly.Furthermore, the presented L 2 -L ∞ filter guarantees a prescribed bounded for the induced finite-time L 2 -L ∞ norm of the operator from the unknown disturbance to the filtering output error with attenuation γ 0.0759, which illustrates the effectiveness of the proposed techniques.

Conclusions
In the paper, we have studied the design of stochastic finite-time L 2 -L ∞ filter for uncertain time-delayed MJLSs.It ensures the finite-time stability and finite-time boundedness for the filtering error dynamic MJLSs.By selecting the appropriate Lyapunov-Krasovskii function and applying matrix transformation and variable substitution, the main results are provided

Figure 2 :
Figure 2: The estimation of changing between modes during the simulation with the initial mode 1.

- 3 .
20 by Theorem 4.3 and Remark 4.4, we can get the optimal value γ min 0.0759, and the mode-dependent optimized L 2 -L ∞ filtering performance can be easily obtained as follows:

Figure 4 :
Figure 4: The response of the system state x 2 t .

Figure 5 :
Figure 5: The response of the output error r t .
For convenience, when r t i, we denote A r t , A d r t , B r t , C r t , D r t , and L r t as A i , A di , B i , C i , D i , and L i .Notice that the time-delays in 2.2 are constant and only dependent of the system structure, and they are not dependent on the defined stochastic process.To simplify the study, we take the initial time t 0 0 and let the initial values {λ t } t∈ −d 0 and {ξ t r t } t∈ −d 0 be fixed.At each mode, we assume that the time-delay MJLSs have the same dimension.
t ξ t , t ∈ t 0 − d t 0 , 2.2 where x t ∈ R n is the state, y t ∈ R l is the measured output, w t ∈ R p is the unknown input, z t ∈ R q is the controlled output, d > 0 is the constant time-delay, σ t is a vectorvalued initial continuous function defined on the interval t 0 − d t 0 , and ξ t is the initial mode.A r t , A d r t , B r t , C r t , D r t , and L r t are known mode-dependent constant matrices with appropriate dimensions, and r t represents a continuous-time discrete state Markov stochastic process with values in the finite set M {1, 2, . . ., N}. Remark 2.1.
For convenience, we set P i diag{P i , P i }, Q diag{Q, Q}.Then inequalities 3.1 and 3.2 are equivalent to LMIs 3.16 and 3.17 by letting X i P i A fi , Y i P i B fi .On the other hand, by setting R i diag{V i V i }, LMIs 3.18 and 3.19 imply that It can be seen that if we choose the infinite time-interval, that is, T → ∞, the main results in Theorems 3.1 and 3.2 can reduce to conclusions of regular L 2 -L ∞ filtering.And other filtering schemes, such as Kalman, H ∞ , and H 2 filtering of stochastic jump systems can be also handled, referring to 9-13, 29, 32 .When the delays in MJLSs 2.2 satisfy d 0, it reduces to a delay-free system.We can immediately get the corresponding results implied in Theorems 3.1 and 3.2 by choosing the stochastic Lyapunov-Krasovskii functional as V x t , ix T t P i x t and following the similar proofs.

7
And the suitable stochastic finite-time L 2 -L ∞ filter can be derived by 3.21 .Remark 4.4.Theorems 3.2 and 4.3 have presented the sufficient condition of designing the stochastic finite-time L 2 -L ∞ filter of time-delay MJLSs.Notice that the coupled LMIs 3.16 -3.20 or LMIs 4.7 , 3.17 -3.20 are with respect to P Remark 4.5.As we did in previous Remark 4.4, we can also fix γ and look for the optimal admissible c 1 or c 2 guaranteeing the stochastically finite-time boundedness of desired filtering error dynamic properties.

Table 1 :
The optimal minimal admissible c 2 with different initial upper bound c 1 .

Figure 1 :
The optimal minimal upper bound c 2 with different initial c 1 .
EngineeringFigure 3: The response of the system state x 1 t .