Finite-Time H ∞ Filtering for Discrete-Time Singular Markovian Jump Systems with Time Delay and Input Saturation

The paper is discussed with the problem of finite-timeH∞ filtering for discrete-time singular Markovian jump systems (SMJSs). The systems under consideration consist of time-varying delay, actuator saturation and partly unknown transition probabilities. We pay attention to the design of a H∞ filtering which ensures the filtering error systems to be singular stochastic finite-time boundedness. By employing an adequate stochastic Lyapunov functional together with a class of linear matrix inequalities (LMIs), a sufficient condition is firstly established, which guarantees the systems to achieve our goal and satisfy a prescribedH∞ attenuation level in the given finite-time interval. Considering the above conditions, a distinct presentation for the requestedH∞ filter is given. Finally, two numerical examples add to a dynamical Leontief model of economic systems are presented to illustrate the validity of the developed theoretical results.


Introduction
Over the past decades, signal estimation has received remarkable attention in the field of control, as an elementary problem in signal processing.As is known to all, the traditional Kalman filtering [1] is the most popular ways to deal with the signal estimation.However, the celebrated Kalman filtering scheme is no longer shaped while a priori information on the external noises are not accurately known.Some optimal estimation approaches are introduced to overcome this problem.One of them  ∞ filtering has a wider range of application value.Compared with the traditional kalman filtering, the main advantage of the  ∞ filtering is without assuming the statistics of the noise signal.Therefore, the  ∞ filtering problem has been investigated and widely introduced, see [2][3][4][5][6][7][8] as well as the references therein.In practice, to guarantee the error system to be asymptotically stable and satisfy a performance indicator is the main intention of  ∞ filtering design.Moreover, for the purpose of applying to the practical applications, it is also necessary to be bounded for the state of the error system.
As a special class of stochastic hybrid systems, Markovian jump systems have the advantage of better describing physical systems with sudden variations and put into use in manufacturing systems, communication systems [9][10][11], economic systems and networked control systems.In the past years, a mass of attention has been paid to the study of singular systems, which have been greatly attracted for the reason that singular systems have better than the state-space ones when to describe physical systems.However, many researchers of singular systems, often ignore the influence of Markovian jump, describing the actual systems with some limitations.With in-depth study, the researchers found that combined the singular systems with MJSs to form a new system, which not only can make up for the limitations of description, also can accurately describe more practical applications.This kind of combined singular systems and MJSs systems have been known as singular Markovian jump system (SMJSs) [12][13][14].Meanwhile, many of the subsequent researches found that the actual environment mutation and singular system parameters or the internal structure change, all can use SMJSs description, so it is mostly important to study the SMJSs.
On the other side, as a precondition, complete knowledge of the mode transition rates was asked to analyse and synthesis MJSs in lots of the studies.Therefore, it is 2 Complexity necessary and significant to further study more universal jump systems together with partly unknown transition rates, rather than spending a lot of time to measure or estimate all the transitions rates.Many attractive problems have been solved and investigated for these systems, such as stability analysis,  ∞ filtering [15][16][17] and stabilization [18,19].
It is well known that the Lyapunov stability describes the systems ranging on the dynamic behavior of the trajectories in infinite time space [20], which does not reflect the transient performance of the systems.To copie with this transient performance of control dynamics, short-time stability or finitetime stability were presented by some early researches in [21][22][23][24].Varieties of significant results were provided for discretetime or continuous-time by applying linear matrix inequality (LMIs) techniques or Lyapunov function approach, these field including singular systems [25,26], network systems [27,28], linear and nonlinear systems [29][30][31][32][33], switching systems [34][35][36], and so forth.In [37][38][39][40][41], the authors studied discrete finite-time stability, finite-time boundedness or finite-time  ∞ control.Compared with the above referred, we can find that the definition of finite-time stability have some difference in [42].When performing the controller, what is noteworthy is that the discrete-time MJSs could be more significant than continuous-time in many practical applications.However, as far as we know, the problem about discrete finite-time  ∞ filtering for stochastic systems has less been researched, so that it promotes the main purpose of our study.
To the best of our knowledge, time-delay phenomena are usually the main causes for performance deterioration and instability of systems and happened in many practical systems, such as neural networks, economics, chemical processes and mechanics [43,44].On the other hand, if we do not consider the effect of actuator saturation when designing the controller, and it can lead to poor performance and unstable of closed-loop system.Consequently, the study of control systems with actuator saturation is challenging not only theory and research value, but also in practice has important research significance and broad application prospects.Recently, Ma and Chen [45] presented singular T-S fuzzy time-varying delay systems with actuator saturation for the problem of memory dissipative control.In [46], the author talked about passive control problem for singular time-delay systems with nonlinearity and actuator saturation.In the paper, we investigates the finite-time  ∞ filter design for discrete-time SMJSs with time-varying delay, partly unknown transition probabilities and actuator saturation.Our results are a little different from those previous ones which are in [47,48] on finite-time  ∞ filtering have been presented for singular Markovian jump systems.Our primary purpose is designing a  ∞ filtering that can guarantee the filtering error discrete-time SMJSs to be stochastic finitetime stability and boundedness with a prescribed  ∞ performance index in the supplied finite-time interval.Enough conditions are supplied for the solvability of the problem, which can be worked out by making use of LMIs techniques.What is more, we consider to design filters for the systems with partly or completely known transition probabilities.Contributions from this paper are as follows (i) A class of more general systems are investigated, where we take time-varying delay, input saturation, outside disturbance, Markov switching phenomena into consideration at the same time.(ii) The singular filtering is used in this article, which makes the filtering problem more comprehensive and general.(iii) The Markovian jump system takes into account the fact that the transfer matrix is not completely known.(iv) The finite-time  ∞ filtering is investigated in this paper.
In this paper, the structure of the arrangement is as follows.In Section 2 we present the problem statement as well as preliminaries.Discussed the problem on stochastic finitetime  ∞ filtering for discrete-time SMJSs with time-varying delay, actuator saturation and partly known transition probabilities in Section 3.Moreover, the estimation of the largest domain of attraction is solved.Section 4 gives some examples to demonstrate the accuracy of the proposed approaches.Finally, we draw the conclusions and the direction of future research in Section 5.
Notations.In this paper, the following notations that will be used:   ,  × denote the sets of n-dimensional space,  ×  real matrices.For any symmetric matrix ,  min () and  max () denote the minimum and maximum eigenvalues of matrix , respectively.{.} denotes the expectation operator in the case of some probability measure.We use  stands for matrix transposition or vector and * stands for the transposed elements in the symmetric positions of a matrix.

Problem Statement and Preliminaries
Consider the following discrete-time singular Markovian jump systems (SMJSs) with time-varying delay and actuator saturation: where () ∈   is the state variable, () ∈   is the signal to be estimated, () ∈   is the control input variable and  :   →   is the standard saturation function defined as follows: where x() ∈   is the filter state variable and z() ∈   is the estimated signal,   (  ),   (  ),   (  ),   (  ) are appropriately dimensioned filter matrices to be determined.

Main Results
This section provides stochastic finite-time  ∞ boundedness analysis for the filtering error discrete-time SMJSs (13) under the given filter gains (6).LMI conditions are established to show that the filtering error discrete-time SMJSs ( 13) is SSFTB and the filtering error () and disturbance () satisfies the constraint condition (16).

Complexity
Applying Schur complement lemma.Eq.( 28) is equivalent to where From Lemma 7 and   > 0, we know that considering ( 29) and (31), it is obtained that using the similar method of the above, when  ∈ ,  ∈    , we can obtain: Then we take the unknown transition probability for an example to prove the system is regular and causal.
From (32), we can obtain that since   > 0,  = 1, 2, 3 and   > 0, then we have since  is singular and rank() =  ≤ , there exist two nonsingular matrices ,  ∈  × , such that Noting that  = 0 and rank() = 2( − ), we can obtain that  1 = 0 and rank( 2 ) = 2( − ),  2 ∈  2×2(−) , that is where * represents matrices that are not relevant in the following discussion, and from Eq.( 37), it is obtained that Now, we assume that the matrix  4 is singular for one  ∈ , then there exists a vector   ∈  − and   ̸ = 0 such that  4   = 0. Pre-multiplying and post-multiplying by   and On the other hand, for the case of  ∈ ,  ∈    , by using the similar method of above, from (20), we also can obtain that  4 is nonsingular.Therefore, by Definition 3, system ( 13) is regular and causal.
At this step we will show that system ( 13) is  with respect to (, , , (  )).
Complexity from ( 42)-( 45), we have where from Eq.( 32) and ( 33 therefore, we have noting that  ≥ 1, it follows from Eq.( 49) that On the other hand, for all  ∈ , the following is true: Remark .For all  ∈ I(), we have I() ≤  2  and from (54), we also obtain that for all  ∈ I(), we have   ()      () ≤  2 .This situation shows that began in the trajectory of I() state of () would have been in attracting field.
Remark .Though the problem of ∞ filtering for discretetime SMJS subject to time-varying delay was considered in [16].Compared with it, the finite-time and actuator saturation have been taken into the above Theorem, which is more practical and can guarantee the large values of states could not be accepted in the presence of saturations.
Base on Theorem 9, the filtering error system (13) can be proved that SSFTB, following we will propose a  ∞ finitetime sufficient condition for SMJS.
Remark .The condition (68) may be difficult to solve using the LMI toolbox of Matlab.To overcome this, we can replace the condition by the following one that may approximate this constraint: where  is a given sufficiently small positive scalar.Applying Schur complement, the constrained condition (82) is equivalent to the following LMI: Remark .Using the same method in [46], condition ℵ(    ,

Numerical Examples
Two numerical examples and a dynamical Leontief model of economic systems are shown to demonstrate the usefulness and flexibility of our theory in this section.The first example adopted to account for conservatism of the conditions shown less than [2].The second example which is employed to apply the filter we designed is feasible and effective.Finally, as the third example, we use the dynamical Leontief model of economic systems to elucidatory the practical application in our life.
Example .In order to show the advantage of the method proposed, consider the systems shown in Example 1 of [2]. Mode Mode In addition, the transition probability matrix is given by Using the method in [2], the optimal values  = 54.8648,while using the method in this paper, the optimal values  = 44.2536.We can obtain that the method in this paper is better than it in [2].The corresponding filter gains aregiven as follows: Remark .The problem of finite-time  ∞ filtering for discrete-time Markovian jump systems were investigated in [2].However, the time-delay and actuator saturation have not been considered, which have a great impact on the performance and stability of the system.Thus, we consider the singular Markovian jump systems with actuator saturation, time-varying delay and partly unknown transition probabilities.Moreover, the examples above have presented that our methods is feasible and less conservative than [2].
When we choose 3.1 ≤  ≤ 8.61 that can find feasible solution.Figure 1 shows the optimal value with different value of .Moreover, while  = 3.2 we can obtain the optimal bound  = 8.0483,  = 8.0534 and the controller gains: Under the above optimal solution, chose the jumping mode as Figure 2., let the disturbance function () = 0.5 − and assume the given initial condition  1 (0) = −0.2, 2 (0) = 0.4, x1 (0) = 0, x2 (0) = 0, and the initial mode  0 = 3, then, the error response of the resulting filtering error systems as where () is the vector of production levels,  is the inputoutput matrix and () is the amount required as direct input for the current produce,  is the capital coefficient matrix, and we know [( + 1) − ()] is the amount necessary for capacity expansion to be capable to produce (+1) in the next period.V() is the vector of final demands, according to [50], it is assumed that V() = −  ((())) −   () + (), then (101) can be rewritten as The other parameters are completely similar with Example 2.Then, by using the Matlab LMI Toolbox, the desired filter gains can be given as follows: For simulation purposes, we assume that the initial conditions (0) = [0.2−0.1]  , x(0) = [0 0]  , the exogenous disturbance signal () is the same as Example 2. Consider the jumping mode as Figure 4, and the trajectories of state are shown in Figure 5, from the picture, it is obvious that the filtering error systems is finite-time bounded.

Conclusion
The finite-time discrete SMJSs ∞ filtering problems which are considering time-varying delay, partially unknown transition probabilities and actuator saturation.First, we realized our goal, a  ∞ filter has been designed while the filtering error system is stochastic boundedness in the finite time and satisfying a prescribed  ∞ attenuation level.Following, we enumerates three examples to prove the validity of the theory that we obtained.However, in practical application, the transition probabilities are often time-varying or difficult to available.Based on the above content, lots of attentions would focus on studying Markovian jump systems with transition probabilities unknown or time-varying under actuator saturation in the near future.

Figure 1 :Figure 2 :
Figure 1: The local optimal bound of  and .