Filter Design for Continuous-Time Linear Systems Subject to Sensor Saturation

This paper presents new sufficient conditions to copewith the filtering problem for continuous-time linear systems subject to sensor saturation. A generalized sector condition has been used to handle the saturation in themeasured output.TheH∞ performancewas considered and a quadratic Lyapunov function was employed in order to derive the design conditions.The conditions are presented in terms of matrix inequalities that become linear when a scalar parameter is fixed. The efficiency of the proposed conditions and their capability to deal with different levels of saturation are illustrated by numerical examples.


Introduction
The study of nonlinear phenomena occurring in control systems has attracted a lot of attention in the last years, mainly because of the effects these types of phenomena may cause [1].Among others, one may cite quantization, time-delay, polynomial systems, hysteresis, and saturation [2][3][4][5][6].The presence of those effects can degrade the performance and even bring instability to the control systems.Therefore, one must be cautious when analyzing systems that are subjected to nonlinear effects.The saturation phenomenon is related to the physical limitations presented in actuators, sensors, and other systems components.When the presence of saturation is not taken into account in the design process, it can lead to very conservative performances [4,7,8].
In the control literature, one of the most studied topics is the filter design problem [9].In the filtering problem, we are interested in obtaining a good approximation of a desired signal in the presence of noise.The filter to be designed makes use of the output measurements of the system to provide an estimation of the desired signal.The desired signal can be a combination of the states or the states themselves.The problems have been studied under different conditions such as linear systems [10], uncertain systems [11,12], timedelay systems [13,14], polynomial systems [2,15], and 2D systems [16].Although the filter design depends on the output measurements of the system, which are obtained by using sensors that can present saturation, this scenario has not been fully explored in the literature.
In [17], the H ∞ filtering problem has been studied for continuous-time systems subject to sensor nonlinearities, including sensor saturation.The main difference between the proposed technique and the method presented in [17] resides in the sector condition that has been employed.This paper addresses the problem of robust linear filter design for continuous-time linear systems subject to sensor saturation.The H ∞ performance will be used as performance criteria and a generalized sector condition will be employed to handle the saturation.A quadratic Lyapunov function is considered and the filter design is accomplished by partitioning the Lyapunov matrix and its inverse.New sufficient conditions in the form of matrix inequalities are presented for both nominal case and uncertain case.Numerical experiments from the literature illustrate the potential of the proposed conditions and indicate that the saturation in the measured output does affect the H ∞ performance of the robust linear filter.
This paper is organized as follows.Section 2 introduces the problem under consideration and the main goal of this paper.Section 3 presents some preliminary results and the main results are given in Section 4. Section 5 is devoted to present the numerical experiments and Section 6 concludes the paper.

Mathematical Problems in Engineering
Notation 1.For two symmetric matrices of the same dimensions  and ,  >  means that − is positive definite.R + is the set of positive real numbers.For matrices and vectors, (  ) indicates the transpose.Identity matrices are denoted by  and null matrices are denoted by 0. The symbol ⋆ indicates a symmetric block in matrices.

Problem Statement
Consider the following continuous-time system: where  ∈ R  is the state vector,  ∈ R   is the noise input,  ∈ R   is the signal to be estimated,  ∈ R   is the measured output, and  ∈ R + is the time domain.The constant matrices that describe the system have the following dimensions:  ∈ R × ,  ∈ R ×  ,   ∈ R   × ,   ∈ R   ×  , and   ∈ R   × .The matrix  is supposed to be Hurwitz stable.Due to physical limitations, the sensor output  is supposed to be bounded in amplitude; that is, Furthermore, the signal  is supposed to be energy bounded; that is,  ∈ L 2 .Without loss of generality, we assume that the signal  is L 2 -normalized; that is, it satisfies The problem to be addressed in this paper is finding a full order stable robust continuous-time linear filter given by where   ∈ R   ,   = , is the estimated state and   ∈ R   is the estimated output, and the filtering matrices are Defining the decentralized dead-zone nonlinearity  ∈ R   as system (1) reads Connecting filter (4) with system (8) and defining the augmented state vector x()  = [()    ()  ] and the output error () = () −   (), one has that can be written in a compact form with where The H ∞ performance will be used as the performance criteria, assuring that the augmented system ( 10) is asymptotically stable and the energy gain from the disturbance input () to the error () = () −   () is minimized.
Definition 2 (see [18]).If there exists a Lyapunov function (x()) > 0, then a bound  2 to the H ∞ performance of the augmented system (10), from the noise input () to the error output (), can be obtained by

Preliminaries
The preliminary results presented here follow the lines in [19,20].
] ≥ 0,  = 1, . . .,   (14) are satisfied, then the H ∞ performance is limited by  for every initial condition belonging to Proof.Consider the quadratic Lyapunov function (x()) = x()   −1 x(), with  =   > 0. The H ∞ performance bound from () to () for system (10) can be obtained by By using equations from system (10), one can write Given that x() ∈ ( 0 ) with ( 0 ) = {x() ∈ R  ; − 0 ≤  −1 x() ≤  0 }, one can use Lemma 1 from [6], to verify that where  1 is a positive diagonal matrix.It is possible to rewrite (17) as where Applying a Schur complement on inequality (14), one can write Pre-and postmultiplying the last inequality by  −1 , one has or equivalently In this way, one concludes that the ellipsoid E( −1 ) = {x ∈ R 2 ; x  −1 x ≤ 1} is contained in ( 0 ).Moreover, if (18) is positive definite, one can write To ensure that (15) holds, it suffices to verify that the right side of inequality ( 22) is negative definite.This fact can be guaranteed by the following inequality: where By applying Schur complement in (23), one has condition (13):

Main Results
The following theorem presents a sufficient condition for the filter design problem based on Proposition 3.

Theorem 4.
If there exist positive definite symmetric matrices  ∈ R × and  ∈ R × , a positive definite diagonal matrix then are the matrices of the robust filter that guarantee an H ∞ performance bounded by  2 .The matrices  and  are obtained from the relation  +    = .
where , Ỹ, , and X are positive definite symmetric matrices.
Remark 5. Note that Theorem 4 presents a bilinear matrix inequality condition, because the variable  1 appears multiplying some other variables of the problem.However, since matrix  1 is diagonal, this problem can be overcome by considering and performing a line search in  > 0. In this way, condition (26) becomes a linear matrix inequality for each fixed value of .Moreover, since matrix  1 is precisely known, matrix   will be recovered directly from condition (26) and no change of variables is necessary in this case.
The proposed approach can also be extended to deal with time-varying uncertainties and state-dependent polytopic uncertainties as introduced in [22].In this case, the system is described as ẋ () =  ()  () +  ()  () ,  () =   ()  () +   ()  () ,  () =  +   ()  () . (34) The matrices in (34) belong to a polytopic domain parameterized in terms of the states () and in terms of a time-varying parameter , being generically represented by where () represents any matrix of the system in (34),   ,  = 1, . . ., , are the vertices,  is the number of vertices of the polytope, and Λ  is the unit simplex, given by Connecting the filter as in (4) with system (34), one can use the same procedure as presented before to obtain sufficient conditions to design a robust filter for system (34).The next theorem presents the conditions.Theorem 6.If there exist positive definite symmetric matrices  ∈ R × and  ∈ R × , a positive definite diagonal matrix for  = 1, . . ., , then are the matrices of the robust filter that guarantee an H ∞ performance bounded by  2 .The matrices  and  are obtained from the relation  +    = .
Proof.Multiplying condition (37) by   for  = 1, . . .,  and summing up, one has a parameter dependent condition.Then, the proof follows the same steps as the ones in proof of Theorem 4. The comments presented in Remark 5 are also valid for this case.

Numerical Experiments
The main goal of the experiments is to illustrate the potential of the proposed conditions and show the effect of the saturation level  0 in the H ∞ performance of the robust filter designed by Theorems 4 and 6.The routines were implemented in Matlab, version 7.1.0.246 (R14) SP 3 using the packages Yalmip [23] and SeDuMi [24].
Example 7. Consider the continuous-time system borrowed from [25] that describes the longitudinal dynamics of the F-8 aircraft, whose system matrices are  In this example, the matrix  1 has been considered as  1 = diag(, ),  > 0, and three different situations for the saturation level  0 have been considered: (40) Figure 1 presents the behavior of  with the variation of  for a strictly proper filter (i.e., considering   = 0) obtained with Theorem 4, considering three different levels of sensor saturation in the measured output .For this system, the method in [17] provides a proper filter with  = 0.4693 for   0 = [1 1].It can be seen that Theorem 4 can provide smaller bounds to the H ∞ performance than [17] even for more restrictive situations in the output ().Moreover, for each fixed value of , the curve for  1 yields the small values for .This is expected since the case  1 is the less restrictive situation.
In order to perform a time-domain simulation, let us consider the following input noise: Figure 2 shows the error signal obtained with a time simulation of the augmented system (10)  directly the performance of the augmented system.Moreover, the filter obtained for  1 provides a more attenuated error signal.
Figure 3 shows the behavior of the saturated output () considering case  1 with  = 0.7 in Theorem 4. Note that the component  2 () is the most affected by the saturation level.
Example 8. Consider the following example adapted from [22] with matrices:  z = 0. (42) Figure 4 depicts the behavior of the bounds , for different values of  and considering different saturation levels  0 .One can see that the saturation level has an important role in the H ∞ performance.Moreover, the parameter  can be used as an extra degree of freedom to search for better solutions.As can be seen,  = 2 provided the lower bounds to the H ∞ performance in this case.

Conclusion
This paper has proposed new sufficient conditions for the design of full order robust linear filters for systems that are subject to sensor saturation.The designed filters guarantee an H ∞ performance for the augmented system.The proposed conditions were based on the use of a quadratic Lyapunov matrix and a generalized sector condition.Two different types of systems have been taken into consideration: precisely known systems and systems with time-varying uncertainties.It has been shown by numerical experiments that the saturation in the output () does affect the performance of the H ∞ robust filter.Moreover, the proposed technique proved to be efficient in taking into account the sensor saturation in the filter design problem.As future research, the author is investigating how to consider LPV filters to treat the systems with time-varying uncertainties.

Figure 1 :
Figure 1: Behavior of  with the variation of  for a strictly proper filter obtained with Theorem 4 for different levels of sensor saturation.

Figure 2 :Figure 3 :
Figure2shows the error signal obtained with a time simulation of the augmented system(10) considering two different levels of sensor saturation.The dashed red line depicts the error for a filter obtained with Theorem 4 with  1 and  = 0.7, while the solid blue line shows the error-time response for a filter designed by Theorem 4 with  3 and  = 0.8.It can be noted that the sensor saturation in the output  can affect

𝐴 1 Figure 4 :
Figure 4: Behavior of  with the saturation level  0 for different values of .
and   ∈ R   ×  , and a scalar  such that inequality (27) and the following inequalities are satisfied,