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

Authors: Márcio J. Lacerda
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper presents new sufficient conditions to cope with the filtering problem for continuous-time linear systems subject to sensor saturation. A generalized sector condition has been used to handle the saturation in the measured output. TheH∞ performance was 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.


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## Body

## 1. 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 [2–6]. 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], time-delay 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. Section2 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.Notation 1.
For two symmetric matrices of the same dimensionsX and Y, X>Y means that X-Y is positive definite. R+ is the set of positive real numbers. For matrices and vectors, ′ indicates the transpose. Identity matrices are denoted by I and null matrices are denoted by 0. The symbol ⋆ indicates a symmetric block in matrices.


## 2. Problem Statement

Consider the following continuous-time system:(1)x˙t=Axt+Bwt,zt=Czxt+Dzwt,yt=saty0Cyxt,where x∈Rn is the state vector, w∈Rnw is the noise input, z∈Rnz is the signal to be estimated, y∈Rny is the measured output, and t∈R+ is the time domain. The constant matrices that describe the system have the following dimensions: A∈Rn×n, B∈Rn×nw, Cz∈Rnz×n, Dz∈Rnz×nw, and Cy∈Rny×n. The matrix A is supposed to be Hurwitz stable. Due to physical limitations, the sensor output y is supposed to be bounded in amplitude; that is,(2)-y0i≤yi≤y0i,y0i>0,i=1,…,ny.Furthermore, the signal w is supposed to be energy bounded; that is, w∈L2. Without loss of generality, we assume that the signal w is L2-normalized; that is, it satisfies(3)w22=∫0∞wτ′wτdτ≤1.The problem to be addressed in this paper is finding a full order stable robust continuous-time linear filter given by(4)x˙ft=Afxft+Bfyt,zft=Cfxft+Dfyt,where xf∈Rnf, nf=n, is the estimated state and zf∈Rnz is the estimated output, and the filtering matrices are Af∈Rnf×nf, Bf∈Rnf×ny, Cf∈Rnz×nf, and Df∈Rnz×ny. As a consequence of the bounds (2), the output y that will be used by the filter is a saturated one(5)yt=saty0Cyxtand each component of satCyxt is defined, ∀i=1,…,ny, by(6)satCyxt≜signCyixtmin⁡Cyixt,y0i.Defining the decentralized dead-zone nonlinearity ϕ∈Rny as(7)ϕx=saty0Cyxt-Cyxt,system (1) reads(8)x˙t=Axt+Bwt,zt=Czxt+Dzwt,yt=ϕ+Cyxt.Connecting filter (4) with system (8) and defining the augmented state vector x~t′=xt′xft′ and the output error et=zt-zft, one has (9)x˙tx˙ft=A0BfCyAfxtxft+B0wt+0Bfϕ,et=Cz-DfCy-Cfxtxft+Dzwt+-Dfϕ that can be written in a compact form with(10)x~˙t=A~x~t+B~wt+B~sϕ,et=C~x~t+D~wt+D~sϕ,where(11)A~=A0BfCyAf∈R2n×2n,B~=B0∈R2n×nw,B~s=0Bf∈R2n×ny,C~=Cz-DfCy-Cf∈Rnz×2n,D~=Dz∈Rnz×nw,D~s=-Df∈Rnz×ny.TheH∞ 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 wt to the error et=zt-zft is minimized.Definition 2 (see [18]).
If there exists a Lyapunov functionVx~t>0, then a bound γ2 to the H∞ performance of the augmented system (10), from the noise input wt to the error output et, can be obtained by(12)V˙x~t+1γ2et′et-wt′wt<0.


## 3. Preliminaries

The preliminary results presented here follow the lines in [19, 20].Proposition 3.
If there exist a matrixQ=Q′>0∈R2n×2n, a diagonal matrix S1>0∈Rny×ny, and F∈Rny×2n such that the inequalities(13)QA~′+A~QB~B~sS1-QC~s′-F′QC~′⋆-Inw0D~′⋆⋆-2S1S1D~s′⋆⋆⋆-γ2Inz<0,(14)QF′j⋆y0j2≥0,j=1,…,nyare satisfied, then the H∞ performance is limited by γ for every initial condition belonging to EQ-1=x~t∈R2n;x~t′Q-1x~t≤1.Proof.
Consider the quadratic Lyapunov functionVx~t=x~t′Q-1x~t, with Q=Q′>0. The H∞ performance bound from wt to et for system (10) can be obtained by(15)V˙x~t+1γ2et′et-wt′wt<0.By using equations from system (10), one can write(16)x~twtϕ′A~′Q-1+Q-1A~Q-1B~Q-1B~sB~′Q-1-Inw0B~s′Q-100+C~′D~′D~s′1γ2InzC~D~D~sx~twtϕ<0.Given that x~t∈Sy0 with Sy0=x~t∈Rn;-y0≤FQ-1x~t≤y0, one can use Lemma 1 from [6], to verify that(17)-2ϕ′S1-1ϕ+Cyxt-FQ-1x~t≥0,where S1 is a positive diagonal matrix. It is possible to rewrite (17) as(18)-2ϕ′S1-1ϕ+C¯sx~t-FQ-1x~t≥0,where C¯s=Cy0ny×n. Applying a Schur complement on inequality (14), one can write(19)Q-F′1y02F≥0. Pre- and postmultiplying the last inequality by Q-1, one has(20)Q-1-Q-1F′1y02FQ-1≥0or equivalently(21)x~t′Q-1x~t≥x~t′Q-1F′1y02FQ-1x~t. In this way, one concludes that the ellipsoid EQ-1=x~∈R2n;x~′Q-1x~≤1 is contained in Sy0. Moreover, if (18) is positive definite, one can write(22)V˙x~t+1γ2et′et-wt′wt≤V˙x~t+1γ2et′et-wt′wt-2ϕ′S1-1ϕ+C¯sx~t-FQ-1x~t.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:(23)Q-1x~twtS1-1ϕ′R1B~R2B~′-Inw0R2′0-2S1+QC~′D~′S1D~s′1γ2InzC~QD~D~sS1Q-1x~twtS1-1ϕ<0,where(24)R1=QA~′+A~Q,R2=B~sS1-QC~s′-F′. By applying Schur complement in (23), one has condition (13): (25)R1B~R2QC~′⋆-Inw0D~′⋆⋆-2S1S1D~s′⋆⋆⋆-γ2Inz<0.


## 4. Main Results

The following theorem presents a sufficient condition for the filter design problem based on Proposition3.Theorem 4.
If there exist positive definite symmetric matricesZ∈Rn×n and X∈Rn×n, a positive definite diagonal matrix S1∈Rny×ny, matrices G∈Rn×n, H∈Rnz×n, L∈Rn×ny, W1∈Rn×ny, W2∈Rn×ny, and Df∈Rnz×ny, and a scalar γ such that the following inequalities are satisfied,(26)A′Z+ZAA′X+ZA+Cy′L′+G′ZB-Cy′-W1Cz′-Cy′Df′-H′⋆A′X+XA+Cy′L′+LCyXBLS1-Cy′-W2Cz′-Cy′Df′⋆⋆-Inw0Dz′⋆⋆⋆-2S1-S1Df′⋆⋆⋆⋆-γ2Inz<0,(27)ZZW1j⋆XW2j⋆⋆y0j2>0,then(28)Af=U′-1GVZ-1,Bf=U′-1L,Cf=HVZ-1,Dfare the matrices of the robust filter that guarantee an H∞ performance bounded by γ2. The matrices U and V are obtained from the relation XY+U′V=I.Proof.
Define(29)Q=YV′VY~,Q-1=XU′UX~,Q-1Q=I, where Y, Y~, X, and X~ are positive definite symmetric matrices.(30)XY+U′V=I,XV′+U′Y~=0,UV′+X~Y~=I,UY+X~V=0. Define the matrices(31)S=YIV0,S-1=0V-1I-YV-1,R=Y00I,R-1=Y-100I. By multiplying (13) on the left by T1=diagR-1S′Q-1,I,I,I and on the right by T1′, one has condition (26) where(32)Z=Y-1,L=U′Bf,G=U′AfVZ,H=CfVZ,W1=ZF1′,W2=XF1′+U′F2′. By multiplying (14) on the left by T2=diagR-1S′Q-1,I and on the right by T2′, one has condition (27). For details about the filter reconstruction, the reader is referred to [21].Remark 5.
Note that Theorem4 presents a bilinear matrix inequality condition, because the variable S1 appears multiplying some other variables of the problem. However, since matrix S1 is diagonal, this problem can be overcome by considering(33)S1=diagλ,λ,…,λ︸ptimes,λ>0 and performing a line search in λ>0. In this way, condition (26) becomes a linear matrix inequality for each fixed value of λ. Moreover, since matrix S1 is precisely known, matrix Df 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(34)x˙t=Aσxt+Bσwt,zt=Czσxt+Dzσwt,yt=ϕ+Cyσxt.The matrices in (34) belong to a polytopic domain parameterized in terms of the states xt and in terms of a time-varying parameter α, being generically represented by(35)Zσ=∑i=1Nσixi,αiZi,σ∈ΛN,where Zσ represents any matrix of the system in (34), Zi,  i=1,…,N, are the vertices, N is the number of vertices of the polytope, and ΛN is the unit simplex, given by(36)ΛN=σ∈RN:∑i=1Nσixi,αi=1,σixi,αi≥0,i=1,…,N. 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 matricesZ∈Rn×n and X∈Rn×n, a positive definite diagonal matrix S1∈Rny×ny, matrices G∈Rn×n, H∈Rnz×n, L∈Rn×ny, W1∈Rn×ny, W2∈Rn×ny, and Df∈Rnz×ny, and a scalar γ such that inequality (27) and the following inequalities are satisfied,(37)Ai′Z+ZAiAi′X+ZAi+Cyi′L′+G′ZBi-Cyi′-W1Czi′-Cyi′Df′-H′⋆Ai′X+XAi+Cyi′L′+LCyiXBi+LDyiLS1-Cyi′-W2Czi′-Cyi′Df′⋆⋆-Inw0Dzi′⋆⋆⋆-2S1-S1Df′⋆⋆⋆⋆-γ2Inz<0for i=1,…,N, then (38)Af=U′-1GVZ-1,Bf=U′-1L,Cf=HVZ-1,Df are the matrices of the robust filter that guarantee an H∞ performance bounded by γ2. The matrices U and V are obtained from the relation XY+U′V=I.Proof.
Multiplying condition (37) by σi for i=1,…,N 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 Remark5 are also valid for this case.


## 5. Numerical Experiments

The main goal of the experiments is to illustrate the potential of the proposed conditions and show the effect of the saturation levely0 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(39)A=-2.47010.3430-1.2680-4.3661-2.56180.0539-17.2462-36.5734-2.01720.0698-2.5839-5.10061000,B=10010000,Cy=000100-11,Cz=0001,Dz=0. In this example, the matrix S1 has been considered as S1=diagλ,λ, λ>0, and three different situations for the saturation level y0 have been considered:(40)C1⟶y0′=10-110-1,C2⟶y0′=10-210-2,C3⟶y0′=10-310-3.Figure 1 presents the behavior of γ with the variation of λ for a strictly proper filter (i.e., considering Df=0) obtained with Theorem 4, considering three different levels of sensor saturation in the measured output y. For this system, the method in [17] provides a proper filter with γ=0.4693 for y0′=11. 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 yt. Moreover, for each fixed value of λ, the curve for C1 yields the small values for γ. This is expected since the case C1 is the less restrictive situation.Figure 1
Behavior ofγ with the variation of λ for a strictly proper filter obtained with Theorem 4 for different levels of sensor saturation.In order to perform a time-domain simulation, let us consider the following input noise:(41)wt=sin⁡0.5texp⁡-0.1t11. Figure 2 shows 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 C1 and λ=0.7, while the solid blue line shows the error-time response for a filter designed by Theorem 4 with C3 and λ=0.8. It can be noted that the sensor saturation in the output y can affect directly the performance of the augmented system. Moreover, the filter obtained for C1 provides a more attenuated error signal.Figure 2
Error-time response for the augmented system (10) obtained for different levels of sensor saturation with initial condition x~0=0.Figure3 shows the behavior of the saturated output yt considering case C1 with λ=0.7 in Theorem 4. Note that the component y2t is the most affected by the saturation level.Figure 3
Behavior of the outputy with a saturation level y0′=10-110-1.Example 8.
Consider the following example adapted from [22] with matrices:(42)A1=-9.150-1-10,A2=-0.150-1-10,A3=-0.150-1-16,A4=-0.150-1-4,Bi=01′,Cyi=10,Czi=10,Dzi=0. Figure 4 depicts the behavior of the bounds γ, for different values of λ and considering different saturation levels y0. One can see that the saturation level has an important role in the H∞ performance. Moreover, the parameter λ can be used as an extra degree of freedom to search for better solutions. As can be seen, λ=2 provided the lower bounds to the H∞ performance in this case.Figure 4
Behavior ofγ with the saturation level y0 for different values of λ.


## 6. 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 anH∞ 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 yt 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.



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