Finite-Frequency Filter Design for Networked Control Systems with Missing Measurements

This paper is concerned with the problem of robust filter design for networked control systems (NCSs) with random missing measurements. Different from existing robust filters, the proposed one is designed in finite-frequency domain. With consideration of possiblemissing data, theNCSs are firstmodeled toMarkov jump systems (MJSs). A finite-frequency stochasticH ∞ performance is subsequently given that extends the standard H ∞ performance, and then a sufficient condition guaranteeing the system to be with such a performance is derived in terms of linear matrix inequality (LMI). With the aid of this condition, a procedure of filter synthesis is proposed to deal with noises in the low-, middle-, and high-frequency domains, respectively. Finally, an example about the lateral-directional dynamicmodel of theNASAHighAlphaResearchVehicle (HARV) is carried out to illustrate the effectiveness of the proposed method.


Introduction
Due to the rapid development in communication network and computer technology, networked control systems (NCSs) have received much more research attentions.Networked control systems (NCSs) are control systems in which controller and plant are connected via a communication channel.The defining feature of an NCS is that information (reference input, plant output, control input, etc.) is exchanged using a network among control system components (sensors, controller, actuators, etc.).NCSs have many advantages such as easy diagnosis, low cost, and high mobility.And thus NCSs have been applied in many industrial systems such as automobiles, manufacturing plants, and aircrafts; see, for example, [1,2].Motivated by this wide spectrum of applications, the new problems arising from the limit resources of the communication channel are gradually taken into account when designing the NCSs.For example, the issues of network-induced delay, packet dropout, and quantization are considered in [3][4][5][6][7][8][9][10][11][12], respectively.
On the other hand, state estimation has been widely studied and has found many practical applications over the past decades [13].When a priori statistical information on the external noise signals is unknown, the celebrated Kalman filtering cannot be employed.To address this issue,  ∞ filtering is introduced, which aims to make the worst case  ∞ norm from the process noise to the estimation error minimized.More recently, there have appeared a few results on  ∞ filter design [14][15][16][17][18].However, it should be noticed that all the aforementioned methods are proposed in fullfrequency domain.Nevertheless, practical industry systems often employ large, complex, or lightweight structures, which include finite-frequency fundamental vibration modes [19].In these situations, it is more reasonable and precise to design filters in finite-frequency domain.Fortunately, the Kalman-Yakubovich-Popov (KYP) lemma is generalized in [20] to characterize frequency domain inequalities with (semi)finitefrequency ranges in terms of linear matrix inequalities (LMIs).The generalized KYP lemma is an effective tool to deal with the finite-frequency problem of linear timeinvariant systems [20][21][22][23][24][25][26].There are some results concerned with this meaningful problem, for example, [27] proposed an effective filter for fuzzy nonlinear systems.However, to the best of the author's knowledge, for system engaging random packet dropout, few results have been published for such class of NCSs.This instance motivates our present investigation.
This paper studies the robust filter design problem with finite-frequency specifications for networked control systems (NCSs) subject to random missing measurements.First, the considered systems are modeled in the framework of Markov jump systems (MJSs).Motivated by Iwasaki et al. [22], a definition of finite-frequency stochastic  ∞ norm is subsequently given to measure the robustness, which extends the standard  ∞ norm and contains the frequency information of noises.Then based on Projection Lemma, an analysis condition is presented to guarantee the MJS being with such a performance in the framework of linear matrix inequalities (LMIs).Further, a procedure of filter synthesis is designed to deal with noises in low-, middle-, and highfrequency domains, respectively.Finally, an example about the lateral-directional dynamic model of the NASA High Alpha Research Vehicle (HARV) is given to illustrate the effectiveness of the proposed method.
The rest of the paper is organized as follows.The problem statement for NCSs with random packet dropout is formulated in Section 2. Section 3 provides sufficient condition to meet the performance request and a design procedure of the robust filter.In Section 4, an example is given to illustrate the effectiveness of the proposed method.Finally, some conclusions end the paper in Section 5.
Notations.Throughout the paper, the superscripts  and −1 stand for, respectively, the transposition and the inverse of a matrix;  > 0 means that  is real symmetric and positive definite; ‖ ⋅ ‖ denotes the Euclidean norm;  2 denotes the Hilbert space of square integrable functions.In block symmetric matrices or long matrix expressions, we use * to represent a term that is induced by symmetry.The sum of a square matrix  and its transposition   is denoted by () :=  +   .

Problem Formulation
Considering the NCS depicted in Figure 1, the continuoustime plant model is where () ∈ R  is the state, () ∈ R  is the measured output, () ∈ R  is the controlled output, and () ∈ R  is the exogenous disturbance which belongs to  2 [0, ∞)., , , , and  are known real constant matrices with appropriate dimensions.
It is assumed that, as shown in Figure 1, the measurement signals will be transmitted via the networks wherein missing data may occur.Further, assume the interval between the th and the ( + 1)th successfully received measurements at the filter is   ℎ, where ℎ is the sampling period of the sensor.It is obvious that the number of the missed packets at time instant ( + 1)ℎ is   − 1, which can be modeled In this situation, the dynamics of plant (1) together with the missing measurements at time instant   can be approximated by where It can be seen that, after the above treatment, the possible missing measurements can be converted to the jumping parameter of the MJS (3) with the transition probability Λ.
In this paper, the filter is chosen as the following form: where x() ∈ R  is the filter's state, ẑ() is the estimated output, and    ,    ,    , and    are filter gains to be designed.
To ensure the achievement of filter design objective, a basic assumption, that is,   is stable, is also assumed to be valid.
Remark 1.This assumption is required to get a stable filtering error dynamics.If this assumption is not satisfied, a stabilizing output feedback controller is required.
For convenience,    ,    ,    ,    ,    , and    are notated as   ,   ,   ,   ,   , and   when   = , respectively.Denoting () = [  () x ()]  and () = () − ẑ(), the filtering error system can be described by the following system: where In order to present the objective of this paper clearly, the following definition is first given.
Definition 2 (MSS).The filter error system ( 5) is said to be mean-square stable (MSS) with holds for all   =  ∈ S.
Definition 3 (finite-frequency stochastic  ∞ norm).For all the solutions of ( 5) which satisfied the following inequalities under zero initial condition for nonzero disturbance: (ii) for the middle-frequency range where the given constant  > 0 is said to be the finite-frequency stochastic  ∞ norm of ( 5) if the following inequality holds.
Remark 4. Definition 3 is motivated by the work of [20,27], which can be regarded as an extension in finite-frequency domain of standard  ∞ norm.Noticeably, it expresses the robustness from () to () in finite-frequency, that is, the smaller it is, the more robust to () the error () becomes.
Now, the problem to be addressed in this paper can be formulated as follows: design a stable filter (4) such that the filter error system (5) is mean-square stable, and with prescribed finite-frequency stochastic  ∞ norm  for the external disturbance ().

Main Results
The filter design problem proposed in the above section will be discussed in this section.

Conditions for Robustness.
Before proceeding further, the following lemma will be recalled to help us derive our main results.
Remark 7. If all the matrices in Lemma 6 are independent on , the MJS will be reduced to a determinate linear system.In this case, Lemma 6 is equivalent to the GKYP in [20], which has been proved to be an effective tool to deal with the finitefrequency problem of linear time-invariant systems.Theorem 8. Consider system (5) for all   =  ∈ S; assume it is mean-square stable; for a given scalar  > 0, the performance of (11) is guaranteed if there exist matrices   ,   ,   , A  , B  ,   ,   , and such that the following LMIs hold: where   = ∑ ∈S     and Proof.It can be concluded from Lemma 6 that if inequality ( 13) holds for all   =  ∈ S, the performance of ( 11) can be reached.Further, ( 13) is equivalent to where with  = [ ] and   = [ ].
On the other hand, (13) implies that Combining ( 23) and ( 25), from Lemma 5, one can easily derive that (13) holds if and only if where matrix    is the slack variable with appropriate dimensions which is introduced by Lemma 5.
Rewrite   as the form of One can conclude that the following inequality provides a sufficient condition for (26): where  =   − 2 cos     + (  A  ) + C   C  .After partitioning the matrices   and   as the following form and defining the following new variables inequality ( 21) can be derived.The proof is completed.
Remark 9.In Theorem 8, by introducing a variable   , the coupling between the variable   and the filter gains will be eliminated.Such a matrix does not present any structure constraint; on the contrary, it may lead to potentially less conservative results.

Conditions for Stability.
Theorem 8 can guarantee the filtering error system to be with a specific robust performance in a certain frequency range of relevance.However, the stability has not been captured, and hence, one may wish to include a stability constraint as an additional design specification.The following theorem will give a result for stability.
Proof.Combing the knowledge of the existing stability criteria for MJSs and the lemma, following the line of the proof for Theorem 8, the conclusion can be derived easily.We do not explain it specifically here.
Remark 11.It should be pointed out that inequalities ( 28)-( 33) are all linear matrix inequalities which can be solved through LMI toolbox of MATLAB.
Based on the above analysis, a set of optimal solutions A  , B  ,   , and   can be obtained by solving the following optimization problem: min  s.t.(21) , (32) . (33) Then the filter gains can be computed by the following equalities: (34)

An Illustrative Example
In this section, an example is given to illustrate the effectiveness of the proposed method.
According to some previous researches of the aircraft dynamic model such as [28,29], it is easily concluded that the nonlinear aircraft dynamic model can be established according to Newton's Second Law of motion.Furthermore, in order to decouple the nonlinear dynamics, the model is decomposed into two models along with longitudinal-and lateral-directional motions, respectively.
The model used in this example is the lateral-directional dynamic model of the NASA High Alpha Research Vehicle (HARV) utilized in [30], that is, Assume that the sampling period is ℎ = 1s, and the sampled data are transmitted through a network, where the data packets may be lost.Further, the quantity of the lost packet (  − 1),   ∈ S = {1, 2, 3} at each sampling period is shown in Figure 2, which is subject to the following transition probability matrix: For prescribed   = 0.8, solving the optimization problem (33), we can obtain the optimal value for finite-frequency stochastic  ∞ norm, that is, the robust performance is  = 0.001 with the corresponding filter gains as follows: In order to show the advantage of the proposed method, we compare it with standard  ∞ filtering method for MJSs which can be found in many researches.In the following, the system will be simulated under zero initial condition, and the disturbance input () is  () = { sin (0.5) , 0s ≤  ≤ 40 s, 0, otherwise, which is shown in Figure 3.It is easy to see that the disturbance considered in the this paper is an instantaneous sinusoidal disturbance and its frequency is considered as zero, which belongs to the low frequency.The simulation results are shown in Figures 4 and 5, which confirm that all the expected system performance requirement are well achieved.Compared with the standard  ∞ filtering method, our method performs better even in the case of possible missing measurements.

Conclusions
In this paper, we have studied the robust filtering with finite-frequency specifications for NCSs subject to random missing measurements.Here, the NCSs are first modeled into MJLs.Then, a new robust filtering method has been proposed that makes full use of the frequency information of noises to reduce design conservatism by the introduction of finite-frequency stochastic  ∞ index for MJLSs.The design problem is formulated into solving a set of linear matrix inequalities, which can be computed by the LMI Control Toolbox.An example is included to show the effectiveness of the obtained theoretical results.

Figure 1 :
Figure 1: The structure of the NCSs.

Figure 2 :
Figure 2: The quantity of the dropped packet.