# Robust Observer Design for Discrete Descriptor Systems with Packet Losses

Authors: Yan Liu; Weifeng Zhong; Qiang Fan; Bo You; Jiazhong Xu
Journal: Complexity
 (2021)

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## Abstract

This paper considers observer design for discrete-time descriptor systems with packet losses. By taking packet loss into consideration, the error dynamic of the proposed observer becomes a stochastic switched system. Consequently, the proposed observer is synthesized in a stochastic switched system framework. Sufficient conditions for the stochastic stability with a prescribed robust performance of the error dynamic system are derived and converted into linear matrix inequalities. Not only can the proposed observer deal with packet losses, but it also attenuates the effect of process disturbance and measurement noise. A numerical simulation of a truck-trailer is given to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Descriptor systems, which are also referred to as singular systems, differential-algebraic equation systems, or generalized state-space systems, appear in many practical systems such as electrical circuits [1, 2], power systems [3], mechanical systems [4], and robots [5, 6]. Moreover, the descriptor system approach has recently been used for synchronization [7] and fault diagnosis [8, 9]. Therefore, in the past decades, the studies on descriptor systems have attracted considerable attention. Many results on descriptor systems have been reported in the literature, such as the analysis and design [10, 11], robust estimation and control [12–14], and fault diagnosis and fault-tolerant control [15–17].It is known that state feedback is very important in control design [18–20]. However, it may be too expensive or even impossible to directly measure all of the states in many applications. In these situations, state estimation from output measurements is necessary to implement the control algorithm. Moreover, the estimated output provided by observers can be used to generate residuals that contain information about fault. Therefore, observers have also been widely used in fault detection and diagnosis [21, 22]. Therefore, the observer design problem is of important practical significance and has been extensively investigated in the literature; see [23–26], just to name a few. In the past decades, many significant results have been reported on observer design for descriptor systems. Observer design for discrete linear descriptor systems was studied in [27]. For the continuous-time linear descriptor systems, full-order and reduced-order observer design methods were proposed in [28]. By using the linear matrix inequality (LMI) technique, Lu and Ho in [29] proposed full-order and reduced-order observer design methods for continuous Lipschitz descriptor systems. Observer design for descriptor systems with unknown disturbance has also been considered in the literature. The unknown input observer design method for continuous linear descriptor systems was considered in [30, 31]. For continuous-time Lipschitz nonlinear descriptor systems with unknown input, Koenig in [32] proposed an unknown input observer design method via convex optimization. In [33], Darouach et al. presented an H∞ observer design method for a class of Lipschitz nonlinear descriptor systems. It is noted that most of the existing works focus on continuous-time descriptor systems while the results on observer design for discrete descriptor systems are limited. In [34], Wang et al. proposed an LMI-based approach to design observers for discrete-time linear and Lipschitz nonlinear descriptor systems and the methodology in [34] has been used to design fault diagnosis observers for linear time-invariant descriptor systems [35] and linear parameter-varying descriptor systems [36].It should be noted that all the aforementioned methods assume that the communication link between the measurements and observer is perfect. However, it is not the case in practice, especially in networked control systems that are often subject to packet dropouts and other nonideal phenomena. For example, the GPS signal may be intermittently available due to system malfunctions or obstacles between the receiver and satellite signal [37]. In the past few years, networked control systems have attracted much attention, but to the best of the authors’ knowledge, no work has been done on observer design for descriptor systems with packet losses. The main contribution of this paper lies in the following aspects. First, a novel observer is proposed to deal with packet losses in the descriptor systems. Second, the convergence of the proposed observer is guaranteed by using a stochastic switched system framework. Moreover, the H∞ technique is used in the design of the proposed observer such that the effect of process disturbance and measurement noise is attenuated.The notation used throughout the paper is standard. The superscriptT stands for matrix transposition. ℝn and ℝm×n denote an n-dimensional and m×n-dimensional Euclidean space, respectively. ℂ represents the one-dimensional complex space. In denotes the n×n identity matrix; 0 represents the zero matrix with appropriate dimensions. For a square matrix M, λminM denotes the minimum eigenvalue of matrix M. For a real symmetric matrix P, P>0 and P<0 mean that P is positive definite and negative definite, respectively. ℒ20,∞ is the space of square-integrable vector functions over 0,∞, and x2=∑k=0∞xTkxk stands for the ℒ2 norm. In symmetric block matrices, we use an asterisk (∗) to represent a term that is induced by symmetry. In addition, PA denotes the probability if the event A occurs, and ℰx and ℰx|y represent expectation of x and expectation of x conditional on y, respectively.


## 2. Problem Formulation

Consider the following discrete descriptor system:(1)Exk+1=Axk+Buk+Ddk,yk=σkCxk+vk,where xk∈ℝn is the state vector, uk∈ℝp is the input vector, dk∈ℝl and vk∈ℝm, respectively, represent the process disturbance and measurement noise, and yk∈ℝm is the measured output vector. Without loss of generality, it is assumed that dk and vk are square summable sequences; i.e., dk,vk∈ℒ20,∞. The matrix E∈ℝn×n may be singular; i.e., rankE=r≤n. A∈ℝn×n, B∈ℝn×p, C∈ℝm×n, and D∈ℝn×l are known constant matrices. In this paper, it is assumed that system (1) is observable, i.e.,(2)rankEC=n,(3)rankzE−AC=n,for anyz∈ℂ,z≥1.In this paper, the considered descriptor system is in a network environment where the measured output is affected by packet losses, which is described byσk∈0,1. It is assumed that σk is a stochastic variable with the probability distribution as follows:(4)Pσk=1=σ¯,Pσk=0=1−σ¯,where σ¯∈0,1 is a priori probability. There is no packet loss at the instant k if σk=1. Otherwise, the measurement is missed. Without loss of generality, this paper assumes the data packet has a timestamp so that σk is available in real time.The aim of this paper is to design an observer to estimate the statexk even in the presence of packet losses. Moreover, the state estimate should be robust against the disturbance and measurement noise. The proposed observer takes the form(5)Ex^k+1=Ax^k+Buk+Ly^k−Cx^k,y^k=σkyk+1−σkCx^k,where x^k∈ℝn is the state estimate vector, y^k∈ℝm is introduced to deal with the packet losses, and L∈ℝn×m is the gain matrix to be synthesized.Remark 1.
The observer in (5) is a generalization of the Luenberger observer to the descriptor system with packet losses.
If we define the estimation error as(6)ek=xk−x^k,and subtract 5 from 1, the error dynamic can be obtained as follows:(7)Eek+1=A−σkLCek+Ddk−σkLvkσkA−LCek+1−σkAek+Ddk−σkLvk.
Note that the introduction of the stochastic variableσk makes the error dynamic system (7) stochastic instead of deterministic. Therefore, before proceeding further, it is necessary to introduce the notion of stochastic stability. Similar to stochastic stability in [38], we propose the following definition of stochastic stability for discrete-time stochastic descriptor systems.Definition 1.
A discrete-time stochastic descriptor systemExk+1=gxk is said to be stochastically stable if there exists a finite Q≥0 independent of x0 such that the following holds:(8)ℰ∑k=0∞xTkxk|x0≤xT0Qx0,for any initial condition x0.
Based on the definition of stochastic stability, the robust observer design problem in this paper is stated as follows.
Consider the descriptor system in (1). Design an observer in the form of (5) such that(1)
The error dynamic system in (7) is stochastically stable.(2)
The state estimation errorek is stochastically robust to the process disturbance and measurement noise, i.e.,(9)ℰe22≤γd2ℰd22+γv2ℰv22+V0,where γd>0 and γv>0 are given scalars and V0 is a quadratic function of e0.


## 3. Main Results

In this section, an observer design method for descriptor systems (1) is formulated as an LMI feasibility problem. First, the following theorem is proposed to guarantee the stochastic stability and stochastic robustness of the error dynamic system in (7).Theorem 1.
Consider the descriptor system (1) and observer (5). The error dynamic system in (7) is stochastically stable and satisfies the robust performance condition (9) if there exist matrices P∈ℝn×n and L∈ℝn×m satisfying(10)ETPE≥0,(11)Φ+In∗∗σ¯DTPA−LC+1−σ¯DTPADTPD−γd2Il∗−σ¯LTPA−LC−σ¯LTPDσ¯LTPL−γv2Im<0,where(12)Φ=σ¯A−LCTPA−LC+1−σ¯ATPA−ETPE.Proof.
We first prove the stochastic stability of the error dynamic system in (7). Consider the following Lyapunov candidate:(13)Vk=eTkETPEek,with P satisfying (10), and define(14)ℰΔV=ℰeTk+1ETPEek+1|ek−eTkETPEek.
If we consider the nominal part of the error dynamic in (7) (i.e., dk=0 and vk=0), we obtain(15)ℰΔV=ℰeTkσkA−LC+1−σkATPσkA−LC+1−σkAek|ek−eTkETPEek=σ¯eTkA−LCTPA−LCek+1−σ¯eTkATPAek−eTkETPEek=eTkΦek.
IfΦ<0, we obtain(16)ℰΔV≤−λmin−ΦeTkek.
It follows that(17)ℰeTk+1ETPEek+1​−ℰeTkETPEek≤−λmin−ΦℰeTkek.
Summing up both sides of (17) from k=0,2,…,∞ leads to(18)ℰeT∞ETPEe∞−ℰeT0ETPEe0≤−λmin−Φℰ∑k=0∞eTkek,which implies(19)ℰ∑k=0∞eTkek≤1λmin−ΦℰeT0ETPEe0−ℰeT∞ETPEe∞​≤1λmin−ΦℰeT0ETPEe0.
Then, we obtain(20)ℰ∑k=0∞eTkek|e0≤xT0Qx0,where Q=1/λmin−ΦETPE. It is easy to see that the (1, 1) block of (11) gives(21)Φ+In<0,which implies that Φ<0. Thus, according to Definition 1, it is known that the error dynamic system in (7) is stochastically stable in mean square.
Next, we prove that the performance defined in (9) is guaranteed. To this end, a criterion function is introduced as(22)J=ℰ∑k=0∞ΔVk+eTkek−γd2dTkdk−γv2vTkvk.
By using the fact thatΔVk=Vk+1−Vk, we obtain(23)J=ℰV∞−V0+∑k=0∞eTkek−γd2dTkdk−γv2vTkvk.
Note thatV∞≥0. Consequently, if J<0 holds, we have(24)ℰe22−γd2d22−γv2v22−V0<0.
That is, criterion (9) is satisfied. On the other hand, it is clear that J<0 holds if(25)ℰΔVk+eTkek−γd2dTkdk−γv2vTkvk<0.
Therefore, criterion (9) is satisfied if (25) holds.
Using (7), it gives(26)ℰΔVk+eTkek−γd2dTkdk−γv2vTkvk=ℰσkA−LCek+1−σkAek+Ddk−σkLvkTPσkA−LCek+1−σkAek+Ddk−σkLvk−eTkETPEek+eTkek−γd2dTkdk−γv2vTkvk=ℰσ2keTkA−LCTPA−LCek+2σk1−σkA−LCTPAek+2σkeTkA−LCTPDdk−σkLvk+1−σk2eTkATPAek+21−σkeTkATPDdk−σkLvk+dTkDTPDdk−2σkdTkDTPLvk+σ2kvTkLTPLvk−eTkETPEek+eTkek−γd2dTkdk−γv2vTkvk.
From the probability distribution ofσk in (4), it is obtained that(27)ℰσkσk=σ¯,ℰσk1−σk=0,ℰ1−σk2=1−σ¯.
Substituting (27) into (26) yields(28)ℰΔVk+eTkek−γd2dTkdk−γv2vTkvk=σ¯eTkA−LCTPA−LCek+1−σ¯eTkATPAek−eTkETPEek+eTkek+2σ¯eTkA−LCTPDdk−Lvk+21−σ¯eTkATPDdk+dTkDTPDdk−γd2dTkdk−2σ¯dTkDTPLvk+σ¯vTkLTPLvk−γv2vTkvk≜ekdkvkTΩ11Ω12Ω13∗Ω22Ω23∗∗Ω33ekdkvk,where(29)Ω11=σ¯A−LCTPA−LC+1−σ¯ATPA−ETPE+In+q,Ω12=σ¯A−LCTPD+1−σ¯ATPD,Ω13=−σ¯A−LCTPL,Ω22=DTPD−γd2Il,Ω23=−σ¯DTPL,Ω33=σ¯LTPL−γv2Im.
Now, (25) is satisfied if (11) holds, and then criterion (9) is fulfilled. This completes the proof.Remark 2.
Without loss of generality, it is assumed thatE,A is regular, which guarantees the existence and the uniqueness of a solution to (1). According to Lemma 2.1 in [12], there exist two nonsingular matrices M1 and N1 such that M1EN1=diagI,J, where J is a nilpotent matrix. Consequently, we can find a matrix P satisfying (10). However, the conditions in Theorem 1 are not strict LMIs. In order to facilitate the design of the observer (5), further procedures are needed to transform the conditions in (10) and (11) into LMIs.
Before proposing the main result, we recall the following useful lemma.Lemma 1 (see [39]).
For matricesX, Y, and J>0 with appropriate dimensions, the following inequality holds:(30)XY+YTXT≤XJXT+YTJ−1Y.
Based on Theorem1 and Lemma 1, the main result is proposed in the following theorem.Theorem 2.
The error dynamic system in (7) is stochastically stable with the robust performance (9) if there exist matrices P∈ℝn×n, G∈ℝn×n, M∈ℝn+l+m×n, and L∈ℝn×m and a scalar ϵ>0 such that(31)ELTPEL>0,(32)Ψ+MA¯+A¯TMT∗∗∗−MT+GA¯−WC¯σ¯P−G−GT∗∗MT0εIn−G−GT∗−WC¯00−1εIm<0,where(33)Ψ=1−σ¯ATPA−ETPE+In∗∗1−σ¯DTPA1−σ¯DTPD−γd2Il∗00−γv2Im,(34)A¯=AD0,(35)C¯=C0I,and EL∈ℝn×r is a matrix satisfying(36)ELER=E,with ER∈ℝr×nand rankEL=rankER=r. Moreover, if the LMIs in (31) and (32) are solvable, then the matrix L can be determined by L=G−1W. Proof. First, (31) implies that(37)ERTELTPELER≥0.
That is, the condition in (10) is satisfied.
By the definitions ofΨ, A¯, and C¯, inequality (11) can be written as(38)Ψ+σ¯A¯−LC¯TPA¯−LC¯<0.
Moreover, (38) holds if there exist a matrix M and a nonsingular matrix G such that(39)Ψ+MA¯−LC¯+A¯−LC¯TMT∗−MT+GA¯−LC¯σ¯P−G−GT<0.
This can be easily shown by pre- and postmultiplying 38 withIA¯−LC¯T and its transpose, respectively.
Then, (39) is written as(40)Ψ+MA¯+A¯TMT∗−MT+GA¯−LC¯σ¯P−G−GT+MG−10−GLC¯T0T+−GLC¯T0MG−10T<0.
Using Lemma1, we obtain(41)Ψ+MA¯+A¯TMT∗−MT+GA¯−LC¯σ¯P−G−GT+MG−10−GLC¯T0T+−GLC¯T0MG−10T≤Ψ+MA¯+A¯TMT∗−MT+GA¯−LC¯σ¯P−G−GT+MG−10ϵ−1MG−10T+−GLC¯T0ϵ−GLC¯T0T.
SinceG is a nonsingular matrix, by using (41) and Schur complement lemma, it is shown that (40) holds if(42)Ψ+MA¯+A¯TMT∗∗∗−MT+GA¯−LC¯σ¯P−G−GT∗∗MT0−εGTG∗−GLC¯00−1εIm<0.
Note that the inequalityIn−GTIn−G≥0 implies that(43)−GTG≤In−G−GT.
Therefore, inequality (42) holds if(44)Ψ+MA¯+A¯TMT∗∗∗−MT+GA¯−LC¯σ¯P−G−GT∗∗MT0εIn−G−GT∗−GLC¯00−1εIm<0.
LetW=GL, then (44) becomes (32). This completes the proof.Remark 3.
It is noted that the condition in (39) is a nonconvex problem, which is inconvenient to solve. By using Lemma 1, this condition is converted into the matrix inequality in (32). By choosing a scalar ϵ in advance, (32) becomes a linear matrix inequality, which can be solved efficiently.


## 4. Simulation Results

In this section, a truck-trailer model from [40] is used to show the state estimation performance of the proposed method. The dynamic equation of the truck-trailer is given by(45)x˙1t=vltanu0t−θL0sinx1t,x˙2t=vL0sinx1t,x˙3t=vcosx1tsinx2t+x2t−Δt2,where Δt is the sampling period. The discrete-time version of (45) is(46)x1k+1=x1k+vΔtltanu0k−θktL0sinx1k,x2k+1=x2k+vΔtL0sinx1k,x3k+1=x3k+vΔtcosx1ksinx2k+1+x2k2,where x1k is the angle difference between truck and trailer, x2k is the angle of trailer, x3k is the vertical position of the rear end of the trailer, and u0k is the steering angle. l=2.8m is the length of truck, L0=5.5m is the length of trailer, Δt=0.2s is the sampling time, and v=−1m/s represents the reversing speed.Under the assumption thatx1k is small, the truck-trailer model (46) can be simplified as(47)x1k+1=1−vΔtL0x1k+vΔtltanu0k,x2k+1=x2k+vΔtL0x1k,x3k+1=x3k+vΔt⋅sinx2k+1+x2k2.By introducing the variables(48)x4k=vΔt2x2k,uk=tanu0k,the simplified track-trailer model (47) becomes(49)Exk+1=Axk+Buk+Ddk,where(50)E=10000100001−10000,A=1−vΔtL000vΔtL10000110−vΔt201,B=vΔtL000,D=00vΔt0,dk=sinx2k+1+x2k2−x2k+1+x2k2.Remark 4.
It can be seen thatrankE=3. This implies that the truck-trailer is a descriptor system, rather than a regular one. Therefore, the observer design method for a descriptor system is necessary in this situation. In the simulation, the measurement equation takes the following form:(51)yk=σkCxk+vk,with(52)C=010000100001,Pσk=1=0.4,Pσk=0=0.6.
It is assumed that the measurements are corrupted by zero-mean Gaussian noises, with the standard deviations of noises in the measurements ofx2 and x3 being 0.0087rad (0.5∘) and 0.05m, respectively.
By choosingϵ=0.1, γd=3, and γv=5 and solving the LMIs in (31) and (32), we obtain(53)L=−0.42810.17380.01530.2867−0.1343−0.0068−0.22840.6497−0.05330.02450.00080.0035.
Note that the state estimationx^k should be used in the controller since the actual state xk is not available. The control input uk in the simulation is given by uk=Kx^k, with K=3−60.7. In the simulation, the initial conditions are x0=−0.1745,1.2217,3,−0.1222T and x^0=0,1.138,3.5,−0.105T. For the packet loss scenario shown in Figure 1, the state estimation result is depicted in Figure 2. It can be seen from Figure 2 that the designed observer is able to provide accurate state estimates despite the presence of packet losses. Although the initial estimation error and long-time packet losses at the beginning of the simulation render some estimation error, the state estimates quickly converge to the actual states.
The observer in [34] can also be used to estimate the states of system (1) if it is slightly modified as follows:(54)zk+1=TAx^k+TBuk+Ly^k−Cx^k,x^k=zk+Ny^k,y^k=σkyk+1−σkCx^k,where T and N should satisfy TE+NC=I. Using the design method presented in [34], we can obtain the following parameters:(55)T=10000100000.6667000−0.33331,N=00000000.33330.666700.33330.6667,L=−1.4658001.050000.66670.66670.1−0.33330.6667.
The state estimation results are shown in Figure3. From Figures 2 and 3, it can be seen that the method in [34] can obtain a little better estimation results of x1k and x2k than the proposed method, but the estimation results of x3k and x4k by the proposed method are much better than that provided by the method in [34]. We can conclude that the proposed method can obtain more accurate estimation results than the method in [34].Figure 1
The sequence ofσk.Figure 2
The state estimation responses by the proposed method.
(a)(b)(c)(d)Figure 3
The state estimation responses by the method in [34].
(a)(b)(c)(d)


## 5. Conclusion

This paper proposed an observer design method for descriptor systems with packet losses. To deal with packet losses, the observer changes its structure depending on whether the measurements are successfully received. As a result, the error dynamic becomes a stochastic switched system. With the definition of stochastic stability, sufficient conditions to ensure the stochastic stability and stochastic robust performance of the error dynamic were derived and transformed into linear matrix inequalities. The simulation results of a truck-trailer system are given to demonstrate that the proposed method has sufficient performance. Future work will focus on extending the proposed method to nonlinear systems, such as Lipschitz nonlinear systems and Takagi–Sugeno systems.



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