Reduced-Order l 2 − l ∞ Filter Design for a Class of Discrete-Time Nonlinear Systems with Multiple Sensor Faults

The reduced-order filtering problem for a class of discrete-time smoothnonlinear systems subject tomultiple sensor faults is studied. It is well known that a smooth complex nonlinear system can be approximated by a Takagi-Sugeno fuzzy linear system with finite number of subsystems. In this work, firstly, the discrete-time smooth nonlinear system is transferred into a Takagi-Sugeno fuzzy linear system with finite number of subsystems. Secondly, a filter with the reduced order of the original system is proposed to be designed. Different from the traditional assumption inwhich themeasurement of the output is ideal, themeasurement of the output is subject to sensor faults which are described via Bernoulli processes. By using the augmentation technique, a stochastic TakagiSugeno filtering error system is obtained. For the stochastic filtering error system, the exponential stability and the energy-to-peak performance are investigated. Sufficient conditions which can guarantee the exponential stability and the l 2 − l ∞ performance are obtained. Then, with the proposed conditions, the design procedure of the filter for the nonlinear system is proposed. Finally, a numerical example is used to show the effectiveness of the proposed design methodology.


Introduction
During the last decades, the robust filter design problem has been paid a lot of efforts due to the fact that there are a lot of practical applications.Among all the published filtering works, Kalman filtering has been shown the great effective to many systems since it was proposed in the 1960s.However, in the Kalman filter design, it requires the precise linear system model and the covariance of the measurement noise at each sampling time.In addition, the system may be not only subject to the measurement noise but also the external disturbances.Due to these facts, recently, there are many new works on the robust filtering which can attenuate the effect of the external disturbances and robust against the system uncertainties [1,2].
While in the peak-to-peak filter design, the infinity norm of the filtering error aims to be minimized when the infinity norm of the external disturbance is bounded; see [17] and the references therein.In the energy-to-peak filter design, the infinity norm of the filtering error is minimized when the energy of the disturbance is  2 bounded.If the peak value of the filtering error is constrained to be within a certain range or required to be minimized, the energy-to-peak filtering strategy is one of the best strategies in the robust filtering.The energy-to-peak performance was originally studied in [18].Since this strategy is effective in many applications, the energy-to-peak filtering has been applied to many other systems; see [19,20].
On the other hand, the research on the smooth nonlinear system which can be approximated by Takagi-Sugeno fuzzy linear system with finite number of subsystems is a hot topic  since the techniques for the linear systems can be applied to the Takagi-Sugeno fuzzy systems.Therefore, unsurprisingly, the study of the fuzzy systems based on Takagi-Sugeno [44] model has attracted a lot of attention [45,46].In [34], the filtering problem for discrete-time Takagi-Sugeno fuzzy systems with time-varying delays was studied.The parameters of the filter can be determined by solving the linear matrix inequalities.In [40,47], the decentralized fuzzy filter design problem for nonlinear interconnected systems was studied.
In traditional filtering problem, it is always assumed that the measurements of the output are perfect; that is, the output of the system is available to the filter.However, in many cases, the measurements may not be available to the filter such as the sensors are subject to faults.If these faults were not considered during the filter design procedure, it is obvious that the existing faults would affect the filter.An extreme example is that the sensors do not work at all.The filter cannot use the measurement to update the estimation.Therefore, it is necessary to consider the sensor faults during the filter design.The filtering problem with the sensor faults has attracted a lot of attention recently [48][49][50][51][52][53][54][55][56][57].In [56], the  ∞ filtering for discrete-time stochastic systems with missing measurements was studied.In [54], the robust setmembership filtering for systems with missing measurement was studied via a linear matrix inequality approach.In [53], the variance-constrained  ∞ filtering for a class of nonlinear time-varying systems with multiple missing measurements was investigated for finite-horizon case.Though there are many works on the filtering problem with sensor faults, in the literature, there are few results on the reduced-order energyto-peak filtering for discrete-time nonlinear systems with sensor faults.
In this paper, the reduced-order filtering problem for a class of discrete-time smooth nonlinear systems with multiple sensor faults is exploited.It is well known that a smooth complex nonlinear system can be approximated by a Takagi-Sugeno fuzzy linear system with finite number of subsystems.In the design, the smooth nonlinear system is firstly transferred into a Takagi-Sugeno fuzzy linear system with finite number of subsystems.Then, a reduced-order filter which is also dependent on the normalized fuzzy-weighting function is proposed to be designed.By defining the filtering error, with the augmentation skill and consideration of the sensor faults, a stochastic Takagi-Sugeno fuzzy filtering error system is obtained.Studying the exponential stability and the energy-to-peak performance of the filtering error system, the design method for the filter is proposed.Finally, a numerical example is used to show the effectiveness of the proposed design methodology.The contribution of this paper can be summarized as follows.(1) The sensor faults are considered in the energy-to-peak filtering problem, which makes the work more generalized.(2) The filter is a reduced-order filter, which makes the results more attractive in the design for high-order and complex systems.

Problem Formulation
In this work, a class of smooth discrete-time nonlinear systems which can be modeled by the following T-S fuzzy linear models is considered as follows.
where N  for  = 1, . . .,  In order to deal with the fuzzy rules, the normalized fuzzy-membership functions are defined as follows: Here,   (  ()) stands for the grade of the membership of   () in the ith fuzzy rule.It is noted that the normalized fuzzy-membership functions have the following properties: The filter problem considered in this paper is shown in Figure 1.The nonlinear system is subject to an external disturbance   .The output of the systems is   which is subject to sensor faults and the measurement of the output is described with where Ξ  = diag{ 1, , . . .,  , , . . .,  , }, and  , is a Bernoulli process with the probability of  , = 1 is   ,  = 1, . . ., , ẑ is the output of the filter; that is, ẑ is the estimation of   .
In the reduce-order filter design for T-S fuzzy systems, the filter has the following structure.
where x ∈ R  1 ( 1 < ) is the state in the filter, ẑ is the output of the filter, and   ,   ,   are filter parameters to be determined.Defining a filtering error as   :=   −ẑ  , the filtering error system is as follows: Here, the following notations are defined: Since the system parameters are dependent on parameters of each subsystem and are time-varying, we use the following notations to represents these parameters: In the filtering problem, the main objective is to minimize the filtering error.Since the external disturbance is unknown, the energy-to-peak gain  is introduced to evaluate the effect from the external input to the filtering error: for zero-initial condition, the energy-to-peak gain from the external input   to the filtering error   should be smaller than the index ; that is, where E{‖‖ ∞ } is the expectation of the ∞-norm of the filtering error   and ‖‖ 2 indicates the 2-norm of the external input   .In summary, the design objectives of the paper are as follows.
(1) The filtering error system in ( 6) is exponentially stable for zero disturbance input   .
(2) For zero-initial condition, the energy-to-peak gain from the external input to the filtering error should be lower than a prescribed level .

Main Results
In this section, by assuming that the parameters for each subfilter are given, the exponentially stability and energy-topeak performance of the filtering error system will be studied.
Based on the analysis results, the filter design method will be proposed.

Exponential Stability and Energy-to-Peak Performance
Analysis.The following theorem provides the condition for the exponential stability of the filtering error system. where Θ  = diag {0, . . ., 1 . . ., 0} .
Proof.Choose a Lyapunov function candidate for the filtering error system in (6) as follows: where (()) =   (()) > 0 is a Lyapunov matrix which is also dependent on the normalized fuzzy-weighting functions.
Since there is a stochastic variable  , in the filtering error system, the difference of the Lyapunov function is defined as In terms of Schur complement, the condition in (10) can assure that the difference of the Lyapunov function Δ(  , (), ) is negative.Thus, the filtering error system is exponentially stable.The proof is completed.
In Theorem 1, by assuming the external disturbance is zero, the exponential stability is studied.In the following, the effect of the external disturbance will be evaluated.
Theorem 2. Consider a discrete-time nonlinear system in (1) and suppose that   (()),   (()),   (()), and a positive scalar  are given.The filtering error system in (6) is exponentially stable and the energy-to-peak performance is guaranteed if there exist matrices (()) =   (()) > 0 and (( + 1)) =   (( + 1)) > 0 such that According to Theorem 1, the filtering error system is exponentially stable.Now we are going to study the energy-to-peak performance attenuation level .Considering the nonzero external input, it infers from (18) that the difference of the Lyapunov function satisfies Suppose that the initial states of the system and the filter are both zeros.Then, the Lyapunov function satisfies By using Schur complement, the inequality in ( 16) is equivalent to Recalling the dynamics of the filtering error system (6), (12), and (20), one gets Taking the supremum over time  > 0 to the previous inequality results to the following condition E{‖‖ ∞ } < ‖‖ 2 for all nonzero   ∈  2 [0, ∞).Therefore, the second objective is satisfied; that is, conditions ( 15) and ( 16) can guarantee the exponential stability and the energy-to-peak attenuation level .The proof is completed.
In Theorem 2, the energy-to-peak performance has been investigated.However, the obtained conditions are only sufficient.In order to derive less conservative results, the following theorem is introduced.

Filter Design Approach.
Based on the results in Theorem 3, the filter design method will be developed in this subsection Theorem 4. Consider a discrete-time nonlinear system in (1) and give a positive scalar .Then, the filtering error system in (6) is exponentially stable with a reduced-order filter in the form of (5) where Moreover, the parameters for each subfilter can be determined by the following equations: Proof.The proof of the theorem can be done by partitioning these matrices in Theorem 3 as This completes the proof.
It is noted that the filter error can be smaller if the performance level  is smaller when the system is subject to the same disturbance.So it is meaningful to optimize the value for .The minimum value for  can be derived by the following corollary.
Corollary 5.The minimum energy-to-peak performance index  for the filtering error system in (6) can be found by solving the following convex optimization problem: (32)

Comparison with
Robust  ∞ Filter.In order to compare with the  ∞ filtering, we are going to study the  ∞ performance of the filtering error system and propose the robust  ∞ filter.The  ∞ performance index is defined as follows: where E{‖‖ 2 } is the expectation of the Proof.The proof can be done as similar lines in [58,59] and the proof in the previous subsections.
Similarly, we have the following filter design method.
Theorem 7. Consider a discrete-time nonlinear system in (1) and give a positive scalar .Then, the filtering error system in (6) is exponentially stable with a reduced-order filter in the form of (5) Moreover, the parameters for each subfilter can be determined by the following equations: Corollary 8.The minimum  ∞ performance index  for the filtering error system in (6) can be found by solving the following convex optimization problem: ∀ ≤ , , ,  = 1, . . .,  1 . (38)

Illustrative Example
In this section, a numeric example is given to show the design procedure of the proposed design method.Consider a discrete-time smooth nonlinear system which is approximated by the following discrete-time T-S fuzzy system with two subsystems.
Considering the sensor fault, the probability of the available measurement is assumed to be 0.5.Since the order of the previous discrete-time nonlinear system is two, the order of the reduced-order filter is supposed to be one.By using Corollary 5, the obtained minimal energy-to-peak  is 0.
In order to verify the performance of the designed filter and compare it with the  ∞ filtering, it is assumed that the initial state of the system is [0.5, 0.5]  .Figure 2 shows the filtering errors with different filters when the sensor is subject to the faults described in Figure 3.It can be seen that the reduced-order filter can track the signal well though the sensor is subject to faults.Moreover, the filtering error of the designed energy-to-peak filter is smaller than the one of  ∞ filter.The 2-norm of the filtering error with the energy-topeak filter is 0.7246.But the 2-norm of the filtering error with the  ∞ filter is 0.8420.The percentage of improvement is 13.94%.

Conclusions
In this paper, the reduced-order energy-to-peak filter design problem for a class of discrete-time smooth nonlinear systems which can be modeled as T-S fuzzy linear systems was studied.The filter with the reduced order of the plant is dependent on the normalized fuzzy-weighting functions.The measurements of the output are subject to sensor faults which are described via a Bernoulli process.The exponential stability and the energy-to-peak performance for the filtering error system were studied.Based on the results, the filter design method was proposed.The parameters of the fuzzy filters can be obtained by solving a set of linear matrix inequalities.In the simulation section, a numeric example was used to show the procedure of the proposed design method.From the simulation results, the performance of the design filter is good.

Figure 1 :
Figure 1: Filtering problem for nonlinear systems with sensor faults.

Figure 2 :
Figure 2: Signal to be estimated and the estimation.
1,  = 1, . . .,  are the fuzzy sets,   = [ 1 ,  2 , . . .,   ] is the premise variable vector,  denotes the number of the premise variable,  1 represents the number of fuzzy rule,   ∈ R  is the state vector,   is the external disturbance,   ∈ R  is the systems output which is measured by corresponding sensors,   stands for the signal to be filtered, and   ,   ,   ,   , and   are given constant matrices.
2-norm of the filtering error   and ‖‖ 2 indicates the 2-norm of the external input   .