A Note on Observers for Discrete-Time Lipschitz ... - Semantic Scholar
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 59, NO. 2, FEBRUARY 2012
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A Note on Observers for Discrete-Time Lipschitz Nonlinear Systems Wei Zhang, Housheng Su, Fanglai Zhu, and Dong Yue, Senior Member, IEEE
Abstract—This brief considers observer design for a class of discrete-time nonlinear systems with Lipschitz nonlinearities. We first remark some statements and results in a recent brief by Zemouche and Boutayeb. In particular, we show that their results are more conservative than an existing one, rather than less conservative as claimed. Moreover, most of the existing results are only applicable to some particular classes of Lipschitz systems with a Lipschitz constant less than one. In order to obtain less conservative results, the concept of a one-sided Lipschitz condition, which is an extension of its well-known Lipschitz counterpart, is introduced. Sufficient conditions ensuring the existence of state observers for one-sided Lipschitz nonlinear systems are then presented. A numerical example is included to illustrate the advantages and effectiveness of the proposed design. Index Terms—Discrete-time nonlinear systems, linear matrix inequality (LMI), Lyapunov stability, observer design, one-sided Lipschitz condition.
I. I NTRODUCTION
R
ECENT literature has witnessed an increasing interest in the study of state estimation for continuous-time or discrete-time nonlinear systems [1]–[21]. As we know, however, it is often difficult or even impossible to design a state observer for a general nonlinear system. Therefore, up to now, most of the works on observer synthesis were carried out for some special classes of nonlinear systems. For the continuoustime cases, the works on the study of observers include [1]– [11], where the nonlinearities were restricted to the cases satisfying the monotone or Lipschitz conditions. Arcak and Kokotovic [1] and Fan and Arcak [2] investigated the observer design for systems with monotone nonlinearities. Thau [3] first
Manuscript received June 11, 2011; revised September 12, 2011; accepted October 25, 2011. Date of publication November 16, 2011; date of current version February 23, 2012. This work was supported in part by the National Natural Science Foundation of China under Grants 61104140, 61074009, 61074025, 61074003, and 60834002; by the Innovation Program of Shanghai Municipal Education Commission under Grant 12YZ156; by the Excellent Young Teachers Program of Shanghai Higher Education under Grant gjd11009; by the Fundamental Research Funds for the Central Universities (Huazhong University of Science and Technology under Grants 2011JC055, 2010QN040, and 20112292); and by the Research Fund for the Doctoral Program of Higher Education (RFDP) under Grant 20100142120023. This paper was recommended by Associate Editor G. Grassi. W. Zhang is with the Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai 201620, China (e-mail: [email protected]). H. Su and D. Yue are with the Key Laboratory for Image Processing and Intelligent Control of the Ministry of Education of China, Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]). F. Zhu is with the College of Electronics and Information Engineering, Tongji University, Shanghai 200092, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSII.2011.2174671
studied the observer design of Lipschitz nonlinear systems and proposed a sufficient condition ensuring the asymptotic stability of the observer error dynamics. Inspired by [3], many other researchers addressed the observer synthesis of Lipschitz nonlinear systems by using various approaches [4]–[11]. As the discrete-time counterpart, observers for nonlinear systems have been recently investigated in [12]–[16]. A linear matrix inequality (LMI)-based observer synthesis for discrete-time Lipschitz nonlinear systems was proposed in [13]–[15]. In [16], the discrete-time nonlinear observer was designed by using a circle criterion approach. Recently, in [12], by introducing a new Lyapunov function, the existence conditions of observers have been derived for discrete-time Lipschitz systems. The idea is inspiring, but it seems that there exist some errors in the conclusions in [12]. In particular, we find that their results are more conservative than those in [13], rather than less conservative as claimed. Moreover, all the results in [12] and [13] are only applicable to some particular classes of Lipschitz systems with a Lipschitz constant less than one. Indeed, most of the existing results are restricted to some classes of nonlinear systems with small Lipschitz constant. When the Lipschitz constant is large or when the equivalent Lipschitz constant has to be chosen large due to the non-Lipschitz nature of the nonlinearity, most existing observer design results fail to provide a solution [11]. In order to obtain less conservative results, the so-called one-sided Lipschitz condition is introduced to design nonlinear observers for the continuous-time case [17]–[21]. The nonlinear observers for one-sided Lipschitz systems are first introduced by Hu [17] and are further studied in [18]. In [19], reduced-order observers for such systems are given via using a similar method of Zhu and Han [5]. The existence condition of the observer is further discussed by Zhao et al. [20]. Very recently, Abbaszadeh and Marquez [21] have proposed a solution via solving a class of matrix inequalities. However, it should be noted that only continuous-time systems were considered in these references. To the best of our knowledge, until now, no results have been given on the study of observers of discrete-time nonlinear systems with one-sided Lipschitz condition; this motivates our research. In this note, we deal with the problem of observer design for discrete-time one-sided Lipschitz nonlinear systems. The major contributions of this brief are divided into three parts. First, it modifies some statements and results in [12]. Second, it deals with the one-sided Lipschitz condition, which covers a broad family on nonlinear systems and includes the well-known Lipschitz condition as a special case. Third, an LMI approach is proposed to design the state observer. The obtained conditions are less conservative and can be applied to design observers for discrete-time nonlinear systems with a Lipschitz constant γ ≥ 1.
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 59, NO. 2, FEBRUARY 2012
Moreover, the observer design approach based on the onesided Lipschitz condition may have potential applications in chaos synchronization [22], [23] and in the study of coordinated control of multiagent systems [24]. Notations: Rn represents the n-dimensional real Euclidean space. ·, · is the inner product in Rn , i.e., given x, y ∈ Rn , then x, y = xT y, where xT is the transpose of the column vector x ∈ Rn . · denotes the Euclidean norm in Rn . For a symmetric matrix S, S > 0 means that the matrix is positive definite. In symmetric block matrices, an asterisk “∗ ” represents a term induced by symmetry. I is an identity matrix with appropriate dimension. II. P ROBLEM S TATEMENT As studied in [12], we consider the following discrete-time nonlinear systems described by: x(k + 1) = Ax(k) + f (x(k), y(k)) (1) y(k) = Cx(k) where x(k) ∈ Rn is the state vector, and y(k) ∈ Rp is the measured output. A and C are the constant matrices of appropriate dimensions. The continuous function f : Rn × Rp → Rn is a nonlinear map. It is said to be Lipschitz with respect to x(k) if for all x1 , x2 ∈ Rn f (x1 , y(k)) − f (x2 , y(k)) ≤ γx1 − x2
(2)
where γ > 0 is independent of y(k) and is called the Lipschitz constant. Similar to the definition of the continuous case [21], nonlinear function f is said to be one-sided Lipschitz if there exists ρ ∈ R such that, for arbitrary x1 , x2 ∈ Rn f (x1 , y(k)) − f (x2 , y(k)), x1 − x2 ≤ ρx1 − x2 2
(3)
where ρ ∈ R is called the one-sided Lipschitz constant. Note that, while Lipschitz constant γ must be positive, one-sided Lipschitz constant ρ can be positive, zero, or even negative. It is easy to see that any Lipschitz function is also one-sided Lipschitz, but the converse is not true [21]. Moreover, in the observer synthesis of one-sided Lipschitz systems, the concept of quadratic inner-boundedness of f [21] is useful. Recall that f is quadratic inner-boundedness if, for arbitrary x1 , x2 ∈ Rn , there exist σ, ϕ ∈ R such that T
2
Δf Δf ≤ σx1 − x2 + ϕx1 − x2 , Δf
(4)
where Δf = f (x1 , y(k)) − f (x2 , y(k)). It is clear that any Lipschitz function is also quadratically inner bounded with ϕ = 0 and σ > 0. Thus, the Lipschitz continuity implies quadratic inner-boundedness. However, the converse is not true [21]. Note that ϕ ∈ R is not necessarily positive. In fact, if ϕ is restricted to be positive, then it can be shown that f must be Lipschitz. As usual, we consider the following observer for system (1): x ˆ(k + 1) = Aˆ x(k) + f (ˆ x(k), y(k)) + L (y(k) − yˆ(k)) (5) yˆ(k) = C x ˆ(k) where x ˆ(k) denotes the estimate of x(k). Our design goal is to find a gain matrix L such that the estimation error, i.e., e(k) = x(k) − x ˆ(k)
(6)
asymptotically converges toward zero. Note that the dynamics of the estimation error is governed by e(k + 1) = (A − LC)e(k) + Δfk
(7)
where Δfk = f (x(k), y(k)) − f (ˆ x(k), y(k)). In [12], the authors used a new Lyapunov function, i.e., Vk = V (e(k)) = eT (k)P e(k) + ΔfkT QΔfk
(8)
where P > 0 and Q > 0 to deduce sufficient conditions achieving the convergence of the estimation errors. This new function is enlightening. However, it seems that there exist some errors in the conclusions in [12]. In the next section, we will give the modified results and compare them with the corresponding results in [13]. III. R EMARKS ON S OME E XISTING R ESULTS This section gives some remarks on the conclusions and statements in [12]. More specially, it will show that theorems 1–4 in [12] are not valid, but they can be modified by following the main idea of [12]. Using the same Lyapunov functions used in the proof of theorems 1–3 in [12], it is easy to obtain the following Propositions 1–3. Here, we only give a proof for Proposition 2. Proposition 1: Assume that system (1) satisfies condition (2) with constant γ > 0 and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist scalars ε > 0, α > 0, and β ∈ R and matrices P > 0, Q = QT , and R of appropriate dimensions such that the following inequalities are feasible: − P ⎡+ βQ + (1 + ε)αγ 2 I < 0 β ε ˜ − 1+ε P − 1+ε Q R Ω=⎣ ∗ P − αI ∗ ∗
⎤ ˜ R 0 ⎦ 0 and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist scalars α > 0 and β > 0 and matrices P > I, Q > 0, and R of appropriate dimensions such that the following inequality is feasible: ⎡ ˜ ˜ ⎤ λR 0 λR −P + βγ 2 I ∗ λP − Q − βI 0 0 ⎥ ⎢ ⎦ < 0 (11) ⎣ ∗ ∗ Q − αI 0 ∗ ∗ ∗ −λP ˜ = AT P − C T R. When (11) admits where λ = 1 + αγ 2 and R a solution, gain matrix L is given by L = P −1 RT .
ZHANG et al.: NOTE ON OBSERVERS FOR DISCRETE-TIME LIPSCHITZ NONLINEAR SYSTEMS
Proof: Let Vk be given in (8). Then, the difference of Vk is given by T ΔVk = eT (k + 1)P e(k + 1) + Δfk+1 QΔfk+1 T −e (k)P e(k) − ΔfkT QΔfk . (12)
Moreover, from Lipschitz condition (2), we have βγ 2 eT (k)e(k) − βΔfkT Δfk ≥ 0.
(13)
Similarly, by (2) and P ≥ I, we obtain T αγ 2 eT (k + 1)P e(k + 1) − αΔfk+1 Δfk+1 ≥ 0.
(14)
Adding the left-sided terms in (13) and (14) to ΔVk yields ΔVk ≤ ξkT Θξk where ξkT = [e(k) Δfk Δfk+1 ]T , and ⎡ λA¯T P A¯ − P¯ λA¯T P Θ=⎣ ∗ λP − Q − βI ∗ ∗
(15) ⎤ 0 ⎦. 0 Q − αI
with A¯ = A − LC, and P¯ = P − βγ 2 I. Applying the Schur complement, Θ < 0 is equivalent to ⎤ ⎡ ¯ 0 λA¯T P −P λA¯T P λP − Q − βI 0 0 ⎥ ⎢ ∗ Γ=⎣ ⎦ < 0. ∗ ∗ Q − αI 0 ∗ ∗ ∗ −λP By denoting R = LT P , condition (11) implies that Γ < 0. Thus, we have ΔVk < 0 for all e(k) = 0 if (11) is satisfied. Therefore, error dynamics (7) is asymptotically stable. Remark 2: It is easy to see that the sufficient condition in [12, Th. 2] is not equivalent to Θ < 0. Consequently, [12, Th. 2] should be modified as Proposition 2. On the other hand, if (11) is satisfied, it follows that −P + βγ 2 I < 0
λP − Q − βI < 0
Q − αI < 0.
Hence, we have (λ − 1)P + β(γ 2 − 1)I − αI < 0. Since P > I and λ = 1 + αγ 2 , one can obtain (α + β)(γ 2 − 1)I < 0, which implies that γ < 1. Proposition 3: Assume that system (1) satisfies condition (2) with constant γ > 0 and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist scalars α > 0 and β > 0 and matrices P > αI, Q > 0, and R of appropriate dimensions such that the following LMI is feasible: ⎡ ˜ ˜ ⎤ ηR 0 ηR −P + βγ 2 I ∗ ηP − Q − βI 0 0 ⎥ ⎢ ⎦ < 0 (16) ⎣ ∗ ∗ Q − αI 0 ∗ ∗ ∗ −ηP ˜ = AT P − C T R. When LMI (16) is where η = 1 + γ 2 , and R feasible, gain matrix L is given by L = P −1 RT . Remark 3: Note that, if (16) is satisfied, we have γ < 1. Moreover, in [12], the authors also presented a sufficient condition that ensures the existence of a reduced-order observer for system (1). Notice that the condition is based on Theorem 3. Therefore, the corresponding result, i.e., [12, Th. 4] , can be
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easily modified via using Proposition 3. Here, we omit it due to page limitation. Propositions 1–3 are obtained by following the same lines in [12]. One of the main features of [12] is the use of new particular Lyapunov functions to deduce the stability condition of error dynamics. However, it seems that such treatment cannot reduce conservatism. On the contrary, we will show that Propositions 1–3 are more conservative than the existing one. For the purpose of comparison, we list the main result in [13] (i.e., [13, Th. 3.1] ) as follows. Proposition 4 (see [13]): Assume that system (1) satisfies condition (2) with constant γ > 0 and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist scalar τ > 0 and matrices P > 0 and R of appropriate dimensions such that the following LMI is feasible: ⎤ ⎡ ˜ ˜ R R −P + τ γ 2 I ∗ P − τI 0 ⎦ 0, α > 0, and β ∈ R and matrices P > 0, Q = QT , and R such that inequalities (9) and (10) are satisfied, then matrices P , R, and τ = α satisfy condition (17). Proof: Assume that ε > 0, α > 0, and β ∈ R, and P > 0, Q = QT , and R satisfy (9) and (10). Let Υ and Ω be defined as in (17) and (10), respectively. Let τ = α. Then, from (9) ⎡ β ⎤ 1 2 0 0 1+ε Q − 1+ε P + αγ I Υ−Ω=⎣ 0 0 0 ⎦ ≤ 0. 0 0 0 Thus, Υ ≤ Ω < 0, which completes the proof. Proposition 6: If there exist scalars α > 0 and β > 0 and matrices P > I, Q > 0, and R such that inequality (11) is satisfied, then matrices P , R, and τ = (α + β/λ) satisfy condition (17), where λ = 1 + αγ 2 . Proof: Assume that there exist some scalars α > 0 and β > 0 and matrices P > I, Q > 0, and R such that (11) is satisfied. Then, from (11), we have Q − αI < 0 and ⎡ ⎤ ˜ ˜ −P + βγ 2 I λR λR Ξ=⎣ ∗ λP − Q − βI 0 ⎦ < 0. ∗ ∗ −λP Let Υ be defined by (17). Let τ = (α + β/λ). Then ⎡ ⎤ (1 − λ)P + (λτ − β)γ 2 I 0 0 λΥ − Ξ = ⎣ ∗ Q − αI 0 ⎦ . ∗ ∗ 0 Since (1−λ)P + (λτ − β)γ 2 I = −αγ 2 (P −I) ≤ 0, we have λΥ ≤ Ξ < 0, which completes the proof. Similar to Proposition 6, we have the following conclusion. Proposition 7: If there exist scalars α > 0 and β > 0 and matrices P > αI, Q > 0, and R such that LMI condition (16) is satisfied, then matrices P , R, and τ = (α + β/η) satisfy condition (17), where η = 1 + γ 2 .
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Remark 4: Propositions 5–7 indicate that LMI condition (17) is less conservative than those given by following the idea used in [12] although it is simpler than them. In other words, Proposition 4 is better than Propositions 1–3. However, it is easy to see that condition (17) is only applicable to systems with a Lipschitz constant γ < 1, as pointed in [13]. Therefore, it is important to reduce the conservatism of most existing results by avoiding the constraint on the Lipschitz constant.
where ε2 is a positive scalar. Then, adding the left-sided terms in (20) and (21) to the difference of Vk yields T e(k) e(k) ΔVk ≤ Π (22) Δfk Δfk
IV. O BSERVER D ESIGN FOR D ISCRETE O NE -S IDED L IPSCHITZ S YSTEMS
Using the Schur complement and notation R = LT P , (18) is equivalent to Π < 0. Therefore, if LMI (18) holds a feasible solution, then ΔVk < 0 for all e(k) = 0, which implies that error dynamics (7) is asymptotically stable. Remark 5: Note that, if ε1 = 0, ϕ = 0, and σ > 0, conditions (4) and (18) reduce to (2) and (17), respectively. In this sense, Proposition 8 is an extension of Proposition 4. Remark 6: For Lipschitz systems with constant γ > 0, one can also find some constants ρ, σ, and ϕ such that conditions (3) and (4) are satisfied. When γ > 1, Propositions 1–4 cannot work, but Proposition 8 may be applied to deal with this situation if ρ < 0. Moreover, if the nonlinear system satisfies only conditions (3) and (4) but does not satisfy Lipschitz condition (2), Proposition 8 is also applicable. In this scene, the condition given in Proposition 8 makes the applicable class of systems larger.
The above discussions indicate that, under the setting of Lipschitz nonlinearities, it is difficult to design observers in the case of Lipschitz constant γ ≥ 1. Then, a natural idea is to generalize the classical Lipschitz condition to a more broad family of nonlinearities, e.g., one-sided Lipschitz condition (3). Such treatment may reduce the conservatism of the existing results, as we will see below. Therefore, this section discusses the observer design for discrete-time one-sided Lipschitz nonlinear systems. Proposition 8: Assume that system (1) satisfies conditions (3) and (4) with constants ρ, σ, and ϕ and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist matrices P > 0 and R and scalars ε1 > 0 and ε2 > 0 such that the following LMI is feasible: ⎤ ⎡ ˜ + ϕε2 −ε1 I R ˜ −P + ε1 ρI + ε2 σI R 2 ⎣ 0 ⎦ 1, the proposed conditions of Propositions 1–3, as well as Proposition 4, are not applicable to this case. On the other hand, this system satisfies onesided Lipschitz condition (3), for which the one-sided Lipschitz constant is much smaller than the Lipschitz constant. In fact, using the mean value theorem, one can obtain
ΔfkT Δfk
From (3), we get ρeT (k)e(k) − eT (k)Δfk ≥ 0. Therefore, for any positive scalar ε1
Π=
eT (k)Δfk ≤ − e(k)2
A˜ = (A − LC)T P (A − LC) − P.
ε1
where
(21)
2
≤ 9 e(k) .
(23) (24)
Hence, conditions (3) and (4) are satisfied with ρ = −1, σ = 9, and ϕ = 0. Thus, we can use Proposition 8 to design a state observer for the system. Solving LMI (18) yields 0.0783 −0.0628 P = R = [ −0.0768 0.1527 ] . −0.0628 0.2467 Therefore, gain matrix L is given by −0.6074 −1 T L=P R = . 0.4642
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ACKNOWLEDGMENT The authors would like to thank Associate Editor Prof. G. Grassi and the anonymous reviewers for their insightful comments and helpful suggestions and Prof. Z. Han for various fruitful discussions. R EFERENCES
Fig. 1. Simulation result of state x1 (k) and its estimate x ˆ1 (k) with initial values x1 (0) = 2 and x ˆ1 (0) = −2.
Fig. 2. Simulation result of state x2 (k) and its estimate x ˆ2 (k) with initial values x2 (0) = −1 and x ˆ2 (0) = 3.
Figs. 1 and 2 show the simulation results for states x1 (k) and x2 (k), respectively. From the simulation, we know that the effect of state trajectory tracking is satisfactory. VI. C ONCLUSION We have addressed the observer design problem for discretetime Lipschitz nonlinear systems. In particular, we modified some statements and conclusions in [12] and pointed out that the proposed conditions cannot provide less conservative results than some existing ones. Moreover, a major limitation of the existing results is that they work only for the case of nonlinear systems with Lipschitz constant γ < 1. In order to obtain less conservative results, we introduced the concept of a one-sided Lipschitz condition to an observer synthesis. An LMI condition is given to achieve the convergence of the error dynamics of discrete-time one-sided Lipschitz nonlinear systems. A numerical example is included to illustrate the effectiveness of the proposed design. The application of the proposed design to chaos synchronization and input recovery is a subject for further research.
[1] M. Arcak and P. Kokotovic, “Nonlinear observers: A circle criterion design and robustness analysis,” Automatica, vol. 37, no. 12, pp. 1923– 1930, Dec. 2001. [2] X. Fan and M. Arcak, “Observer design for systems with multivariable monotone nonlinearities,” Syst. Control Lett., vol. 50, no. 4, pp. 319–330, Nov. 2003. [3] F. Thau, “Observing the state of nonlinear dynamic systems,” Int. J. Control, vol. 17, no. 3, pp. 471–479, 1973. [4] R. Rajamani, “Observers for Lipschitz nonlinear systems,” IEEE Trans. Autom. Control, vol. 43, no. 3, pp. 397–401, Mar. 1998. [5] F. Zhu and Z. Han, “A note on observers for Lipschitz nonlinear systems,” IEEE Trans. Autom. Control, vol. 47, no. 10, pp. 1751–1754, Oct. 2002. [6] M. Chen and C. Chen, “Robust nonlinear observer for Lipschitz nonlinear systems subject to disturbances,” IEEE Trans. Autom. Control, vol. 52, no. 12, pp. 2365–2369, Dec. 2007. [7] G. P. Lu and D. W. C. Ho, “Full-order and reduced-order observers for Lipschitz descriptor systems: The unified LMI approach,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 7, pp. 563–567, Jul. 2006. [8] A. Zemouche, M. Boutayeb, and G. I. Bara, “Observers for a class of Lipschitz systems with extension to H∞ performance analysis,” Syst. Control Lett., vol. 57, no. 1, pp. 18–27, Jan. 2008. [9] A. Zemouche and M. Boutayeb, “A unified H∞ adaptive observer synthesis method for a class of systems with both Lipschitz and monotone nonlinearities,” Syst. Control Lett., vol. 58, no. 4, pp. 282–288, Apr. 2009. [10] A. M. Pertew, H. J. Marquez, and Q. Zhao, “H∞ observer for Lipschitz nonlinear systems,” IEEE Trans. Autom. Control, vol. 51, no. 7, pp. 1211– 1216, Jul. 2006. [11] G. Phanomchoeng and R. Rajamani, “Observer design for Lipschitz nonlinear systems using Riccati equations,” in Proc. Amer. Control Conf., Baltimore, MD, Jul. 2010, pp. 6060–6065. [12] A. Zemouche and M. Boutayeb, “Observer design for Lipschitz nonlinear systems: The discrete-time case,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 8, pp. 777–781, Aug. 2006. [13] G. I. Bara, A. Zemouche, and M. Boutayeb, “Observer synthesis for Lipschitz discrete-time systems,” in Proc. IEEE Int. Symp. Circuits Syst., Kobe, Japan, May 2005, pp. 3195–3198. [14] S. Ibrir, W. F. Xie, and C. Y. Su, “Observer-based control of discretetime Lipschitzian nonlinear systems: Application to one-link flexible joint robot,” Int. J. Control, vol. 78, no. 6, pp. 385–395, Apr. 2005. [15] M. Abbaszadeh and H. J. Marquez, “Robust H∞ observer design for sampled-data Lipschitz nonlinear systems with exact and Euler approximate models,” Automatica, vol. 44, no. 3, pp. 799–806, Mar. 2008. [16] S. Ibrir, “Circle-criterion approach to discrete-time nonlinear observer design,” Automatica, vol. 43, no. 8, pp. 1432–1441, Aug. 2007. [17] G. Hu, “Observers for one-sided Lipschitz non-linear systems,” IMA J. Math. Control Inf., vol. 23, no. 4, pp. 395–401, Dec. 2006. [18] G. Hu, “A note on observer for one-sided Lipschitz nonlinear systems,” IMA J. Math. Control Inf., vol. 25, no. 3, pp. 297–303, Sep. 2008. [19] M. Xu, G. Hu, and Y. Zhao, “Reduced-order observer for one-sided Lipschitz nonlinear systems,” IMA J. Math. Control Inf., vol. 26, no. 3, pp. 299–317, Sep. 2009. [20] Y. Zhao, J. Tao, and N. Z. Shi, “A note on observer design for one-sided Lipschitz nonlinear systems,” Syst. Control Lett., vol. 59, no. 1, pp. 66–71, Jan. 2010. [21] M. Abbaszadeh and H. J. Marquez, “Nonlinear observer design for onesided Lipschitz systems,” in Proc. Amer. Control Conf., Baltimore, MD, Jul. 2010, pp. 5284–5289. [22] G. Grassi and D. A. Miller, “Theory and experimental realization of observer-based discrete-time hyperchaos synchronization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 3, pp. 373–378, Mar. 2002. [23] M. Boutayeb, M. Darouach, and H. Rafaralahy, “Generalized state-space observers for chaotic synchronization and secure communication,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 3, pp. 345–349, Mar. 2002. [24] H. Su, G. Chen, X. Wang, and Z. Lin, “Adaptive second-order consensus of networked mobile agents with nonlinear dynamics,” Automatica, vol. 47, no. 2, pp. 368–375, Feb. 2011.
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A Note on Observers for Discrete-Time Lipschitz Nonlinear Systems Wei Zhang, Housheng Su, Fanglai Zhu, and Dong Yue, Senior Member, IEEE
Abstract—This brief considers observer design for a class of discrete-time nonlinear systems with Lipschitz nonlinearities. We first remark some statements and results in a recent brief by Zemouche and Boutayeb. In particular, we show that their results are more conservative than an existing one, rather than less conservative as claimed. Moreover, most of the existing results are only applicable to some particular classes of Lipschitz systems with a Lipschitz constant less than one. In order to obtain less conservative results, the concept of a one-sided Lipschitz condition, which is an extension of its well-known Lipschitz counterpart, is introduced. Sufficient conditions ensuring the existence of state observers for one-sided Lipschitz nonlinear systems are then presented. A numerical example is included to illustrate the advantages and effectiveness of the proposed design. Index Terms—Discrete-time nonlinear systems, linear matrix inequality (LMI), Lyapunov stability, observer design, one-sided Lipschitz condition.
I. I NTRODUCTION
R
ECENT literature has witnessed an increasing interest in the study of state estimation for continuous-time or discrete-time nonlinear systems [1]–[21]. As we know, however, it is often difficult or even impossible to design a state observer for a general nonlinear system. Therefore, up to now, most of the works on observer synthesis were carried out for some special classes of nonlinear systems. For the continuoustime cases, the works on the study of observers include [1]– [11], where the nonlinearities were restricted to the cases satisfying the monotone or Lipschitz conditions. Arcak and Kokotovic [1] and Fan and Arcak [2] investigated the observer design for systems with monotone nonlinearities. Thau [3] first
Manuscript received June 11, 2011; revised September 12, 2011; accepted October 25, 2011. Date of publication November 16, 2011; date of current version February 23, 2012. This work was supported in part by the National Natural Science Foundation of China under Grants 61104140, 61074009, 61074025, 61074003, and 60834002; by the Innovation Program of Shanghai Municipal Education Commission under Grant 12YZ156; by the Excellent Young Teachers Program of Shanghai Higher Education under Grant gjd11009; by the Fundamental Research Funds for the Central Universities (Huazhong University of Science and Technology under Grants 2011JC055, 2010QN040, and 20112292); and by the Research Fund for the Doctoral Program of Higher Education (RFDP) under Grant 20100142120023. This paper was recommended by Associate Editor G. Grassi. W. Zhang is with the Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai 201620, China (e-mail: [email protected]). H. Su and D. Yue are with the Key Laboratory for Image Processing and Intelligent Control of the Ministry of Education of China, Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]). F. Zhu is with the College of Electronics and Information Engineering, Tongji University, Shanghai 200092, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSII.2011.2174671
studied the observer design of Lipschitz nonlinear systems and proposed a sufficient condition ensuring the asymptotic stability of the observer error dynamics. Inspired by [3], many other researchers addressed the observer synthesis of Lipschitz nonlinear systems by using various approaches [4]–[11]. As the discrete-time counterpart, observers for nonlinear systems have been recently investigated in [12]–[16]. A linear matrix inequality (LMI)-based observer synthesis for discrete-time Lipschitz nonlinear systems was proposed in [13]–[15]. In [16], the discrete-time nonlinear observer was designed by using a circle criterion approach. Recently, in [12], by introducing a new Lyapunov function, the existence conditions of observers have been derived for discrete-time Lipschitz systems. The idea is inspiring, but it seems that there exist some errors in the conclusions in [12]. In particular, we find that their results are more conservative than those in [13], rather than less conservative as claimed. Moreover, all the results in [12] and [13] are only applicable to some particular classes of Lipschitz systems with a Lipschitz constant less than one. Indeed, most of the existing results are restricted to some classes of nonlinear systems with small Lipschitz constant. When the Lipschitz constant is large or when the equivalent Lipschitz constant has to be chosen large due to the non-Lipschitz nature of the nonlinearity, most existing observer design results fail to provide a solution [11]. In order to obtain less conservative results, the so-called one-sided Lipschitz condition is introduced to design nonlinear observers for the continuous-time case [17]–[21]. The nonlinear observers for one-sided Lipschitz systems are first introduced by Hu [17] and are further studied in [18]. In [19], reduced-order observers for such systems are given via using a similar method of Zhu and Han [5]. The existence condition of the observer is further discussed by Zhao et al. [20]. Very recently, Abbaszadeh and Marquez [21] have proposed a solution via solving a class of matrix inequalities. However, it should be noted that only continuous-time systems were considered in these references. To the best of our knowledge, until now, no results have been given on the study of observers of discrete-time nonlinear systems with one-sided Lipschitz condition; this motivates our research. In this note, we deal with the problem of observer design for discrete-time one-sided Lipschitz nonlinear systems. The major contributions of this brief are divided into three parts. First, it modifies some statements and results in [12]. Second, it deals with the one-sided Lipschitz condition, which covers a broad family on nonlinear systems and includes the well-known Lipschitz condition as a special case. Third, an LMI approach is proposed to design the state observer. The obtained conditions are less conservative and can be applied to design observers for discrete-time nonlinear systems with a Lipschitz constant γ ≥ 1.
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Moreover, the observer design approach based on the onesided Lipschitz condition may have potential applications in chaos synchronization [22], [23] and in the study of coordinated control of multiagent systems [24]. Notations: Rn represents the n-dimensional real Euclidean space. ·, · is the inner product in Rn , i.e., given x, y ∈ Rn , then x, y = xT y, where xT is the transpose of the column vector x ∈ Rn . · denotes the Euclidean norm in Rn . For a symmetric matrix S, S > 0 means that the matrix is positive definite. In symmetric block matrices, an asterisk “∗ ” represents a term induced by symmetry. I is an identity matrix with appropriate dimension. II. P ROBLEM S TATEMENT As studied in [12], we consider the following discrete-time nonlinear systems described by: x(k + 1) = Ax(k) + f (x(k), y(k)) (1) y(k) = Cx(k) where x(k) ∈ Rn is the state vector, and y(k) ∈ Rp is the measured output. A and C are the constant matrices of appropriate dimensions. The continuous function f : Rn × Rp → Rn is a nonlinear map. It is said to be Lipschitz with respect to x(k) if for all x1 , x2 ∈ Rn f (x1 , y(k)) − f (x2 , y(k)) ≤ γx1 − x2
(2)
where γ > 0 is independent of y(k) and is called the Lipschitz constant. Similar to the definition of the continuous case [21], nonlinear function f is said to be one-sided Lipschitz if there exists ρ ∈ R such that, for arbitrary x1 , x2 ∈ Rn f (x1 , y(k)) − f (x2 , y(k)), x1 − x2 ≤ ρx1 − x2 2
(3)
where ρ ∈ R is called the one-sided Lipschitz constant. Note that, while Lipschitz constant γ must be positive, one-sided Lipschitz constant ρ can be positive, zero, or even negative. It is easy to see that any Lipschitz function is also one-sided Lipschitz, but the converse is not true [21]. Moreover, in the observer synthesis of one-sided Lipschitz systems, the concept of quadratic inner-boundedness of f [21] is useful. Recall that f is quadratic inner-boundedness if, for arbitrary x1 , x2 ∈ Rn , there exist σ, ϕ ∈ R such that T
2
Δf Δf ≤ σx1 − x2 + ϕx1 − x2 , Δf
(4)
where Δf = f (x1 , y(k)) − f (x2 , y(k)). It is clear that any Lipschitz function is also quadratically inner bounded with ϕ = 0 and σ > 0. Thus, the Lipschitz continuity implies quadratic inner-boundedness. However, the converse is not true [21]. Note that ϕ ∈ R is not necessarily positive. In fact, if ϕ is restricted to be positive, then it can be shown that f must be Lipschitz. As usual, we consider the following observer for system (1): x ˆ(k + 1) = Aˆ x(k) + f (ˆ x(k), y(k)) + L (y(k) − yˆ(k)) (5) yˆ(k) = C x ˆ(k) where x ˆ(k) denotes the estimate of x(k). Our design goal is to find a gain matrix L such that the estimation error, i.e., e(k) = x(k) − x ˆ(k)
(6)
asymptotically converges toward zero. Note that the dynamics of the estimation error is governed by e(k + 1) = (A − LC)e(k) + Δfk
(7)
where Δfk = f (x(k), y(k)) − f (ˆ x(k), y(k)). In [12], the authors used a new Lyapunov function, i.e., Vk = V (e(k)) = eT (k)P e(k) + ΔfkT QΔfk
(8)
where P > 0 and Q > 0 to deduce sufficient conditions achieving the convergence of the estimation errors. This new function is enlightening. However, it seems that there exist some errors in the conclusions in [12]. In the next section, we will give the modified results and compare them with the corresponding results in [13]. III. R EMARKS ON S OME E XISTING R ESULTS This section gives some remarks on the conclusions and statements in [12]. More specially, it will show that theorems 1–4 in [12] are not valid, but they can be modified by following the main idea of [12]. Using the same Lyapunov functions used in the proof of theorems 1–3 in [12], it is easy to obtain the following Propositions 1–3. Here, we only give a proof for Proposition 2. Proposition 1: Assume that system (1) satisfies condition (2) with constant γ > 0 and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist scalars ε > 0, α > 0, and β ∈ R and matrices P > 0, Q = QT , and R of appropriate dimensions such that the following inequalities are feasible: − P ⎡+ βQ + (1 + ε)αγ 2 I < 0 β ε ˜ − 1+ε P − 1+ε Q R Ω=⎣ ∗ P − αI ∗ ∗
⎤ ˜ R 0 ⎦ 0 and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist scalars α > 0 and β > 0 and matrices P > I, Q > 0, and R of appropriate dimensions such that the following inequality is feasible: ⎡ ˜ ˜ ⎤ λR 0 λR −P + βγ 2 I ∗ λP − Q − βI 0 0 ⎥ ⎢ ⎦ < 0 (11) ⎣ ∗ ∗ Q − αI 0 ∗ ∗ ∗ −λP ˜ = AT P − C T R. When (11) admits where λ = 1 + αγ 2 and R a solution, gain matrix L is given by L = P −1 RT .
ZHANG et al.: NOTE ON OBSERVERS FOR DISCRETE-TIME LIPSCHITZ NONLINEAR SYSTEMS
Proof: Let Vk be given in (8). Then, the difference of Vk is given by T ΔVk = eT (k + 1)P e(k + 1) + Δfk+1 QΔfk+1 T −e (k)P e(k) − ΔfkT QΔfk . (12)
Moreover, from Lipschitz condition (2), we have βγ 2 eT (k)e(k) − βΔfkT Δfk ≥ 0.
(13)
Similarly, by (2) and P ≥ I, we obtain T αγ 2 eT (k + 1)P e(k + 1) − αΔfk+1 Δfk+1 ≥ 0.
(14)
Adding the left-sided terms in (13) and (14) to ΔVk yields ΔVk ≤ ξkT Θξk where ξkT = [e(k) Δfk Δfk+1 ]T , and ⎡ λA¯T P A¯ − P¯ λA¯T P Θ=⎣ ∗ λP − Q − βI ∗ ∗
(15) ⎤ 0 ⎦. 0 Q − αI
with A¯ = A − LC, and P¯ = P − βγ 2 I. Applying the Schur complement, Θ < 0 is equivalent to ⎤ ⎡ ¯ 0 λA¯T P −P λA¯T P λP − Q − βI 0 0 ⎥ ⎢ ∗ Γ=⎣ ⎦ < 0. ∗ ∗ Q − αI 0 ∗ ∗ ∗ −λP By denoting R = LT P , condition (11) implies that Γ < 0. Thus, we have ΔVk < 0 for all e(k) = 0 if (11) is satisfied. Therefore, error dynamics (7) is asymptotically stable. Remark 2: It is easy to see that the sufficient condition in [12, Th. 2] is not equivalent to Θ < 0. Consequently, [12, Th. 2] should be modified as Proposition 2. On the other hand, if (11) is satisfied, it follows that −P + βγ 2 I < 0
λP − Q − βI < 0
Q − αI < 0.
Hence, we have (λ − 1)P + β(γ 2 − 1)I − αI < 0. Since P > I and λ = 1 + αγ 2 , one can obtain (α + β)(γ 2 − 1)I < 0, which implies that γ < 1. Proposition 3: Assume that system (1) satisfies condition (2) with constant γ > 0 and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist scalars α > 0 and β > 0 and matrices P > αI, Q > 0, and R of appropriate dimensions such that the following LMI is feasible: ⎡ ˜ ˜ ⎤ ηR 0 ηR −P + βγ 2 I ∗ ηP − Q − βI 0 0 ⎥ ⎢ ⎦ < 0 (16) ⎣ ∗ ∗ Q − αI 0 ∗ ∗ ∗ −ηP ˜ = AT P − C T R. When LMI (16) is where η = 1 + γ 2 , and R feasible, gain matrix L is given by L = P −1 RT . Remark 3: Note that, if (16) is satisfied, we have γ < 1. Moreover, in [12], the authors also presented a sufficient condition that ensures the existence of a reduced-order observer for system (1). Notice that the condition is based on Theorem 3. Therefore, the corresponding result, i.e., [12, Th. 4] , can be
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easily modified via using Proposition 3. Here, we omit it due to page limitation. Propositions 1–3 are obtained by following the same lines in [12]. One of the main features of [12] is the use of new particular Lyapunov functions to deduce the stability condition of error dynamics. However, it seems that such treatment cannot reduce conservatism. On the contrary, we will show that Propositions 1–3 are more conservative than the existing one. For the purpose of comparison, we list the main result in [13] (i.e., [13, Th. 3.1] ) as follows. Proposition 4 (see [13]): Assume that system (1) satisfies condition (2) with constant γ > 0 and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist scalar τ > 0 and matrices P > 0 and R of appropriate dimensions such that the following LMI is feasible: ⎤ ⎡ ˜ ˜ R R −P + τ γ 2 I ∗ P − τI 0 ⎦ 0, α > 0, and β ∈ R and matrices P > 0, Q = QT , and R such that inequalities (9) and (10) are satisfied, then matrices P , R, and τ = α satisfy condition (17). Proof: Assume that ε > 0, α > 0, and β ∈ R, and P > 0, Q = QT , and R satisfy (9) and (10). Let Υ and Ω be defined as in (17) and (10), respectively. Let τ = α. Then, from (9) ⎡ β ⎤ 1 2 0 0 1+ε Q − 1+ε P + αγ I Υ−Ω=⎣ 0 0 0 ⎦ ≤ 0. 0 0 0 Thus, Υ ≤ Ω < 0, which completes the proof. Proposition 6: If there exist scalars α > 0 and β > 0 and matrices P > I, Q > 0, and R such that inequality (11) is satisfied, then matrices P , R, and τ = (α + β/λ) satisfy condition (17), where λ = 1 + αγ 2 . Proof: Assume that there exist some scalars α > 0 and β > 0 and matrices P > I, Q > 0, and R such that (11) is satisfied. Then, from (11), we have Q − αI < 0 and ⎡ ⎤ ˜ ˜ −P + βγ 2 I λR λR Ξ=⎣ ∗ λP − Q − βI 0 ⎦ < 0. ∗ ∗ −λP Let Υ be defined by (17). Let τ = (α + β/λ). Then ⎡ ⎤ (1 − λ)P + (λτ − β)γ 2 I 0 0 λΥ − Ξ = ⎣ ∗ Q − αI 0 ⎦ . ∗ ∗ 0 Since (1−λ)P + (λτ − β)γ 2 I = −αγ 2 (P −I) ≤ 0, we have λΥ ≤ Ξ < 0, which completes the proof. Similar to Proposition 6, we have the following conclusion. Proposition 7: If there exist scalars α > 0 and β > 0 and matrices P > αI, Q > 0, and R such that LMI condition (16) is satisfied, then matrices P , R, and τ = (α + β/η) satisfy condition (17), where η = 1 + γ 2 .
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Remark 4: Propositions 5–7 indicate that LMI condition (17) is less conservative than those given by following the idea used in [12] although it is simpler than them. In other words, Proposition 4 is better than Propositions 1–3. However, it is easy to see that condition (17) is only applicable to systems with a Lipschitz constant γ < 1, as pointed in [13]. Therefore, it is important to reduce the conservatism of most existing results by avoiding the constraint on the Lipschitz constant.
where ε2 is a positive scalar. Then, adding the left-sided terms in (20) and (21) to the difference of Vk yields T e(k) e(k) ΔVk ≤ Π (22) Δfk Δfk
IV. O BSERVER D ESIGN FOR D ISCRETE O NE -S IDED L IPSCHITZ S YSTEMS
Using the Schur complement and notation R = LT P , (18) is equivalent to Π < 0. Therefore, if LMI (18) holds a feasible solution, then ΔVk < 0 for all e(k) = 0, which implies that error dynamics (7) is asymptotically stable. Remark 5: Note that, if ε1 = 0, ϕ = 0, and σ > 0, conditions (4) and (18) reduce to (2) and (17), respectively. In this sense, Proposition 8 is an extension of Proposition 4. Remark 6: For Lipschitz systems with constant γ > 0, one can also find some constants ρ, σ, and ϕ such that conditions (3) and (4) are satisfied. When γ > 1, Propositions 1–4 cannot work, but Proposition 8 may be applied to deal with this situation if ρ < 0. Moreover, if the nonlinear system satisfies only conditions (3) and (4) but does not satisfy Lipschitz condition (2), Proposition 8 is also applicable. In this scene, the condition given in Proposition 8 makes the applicable class of systems larger.
The above discussions indicate that, under the setting of Lipschitz nonlinearities, it is difficult to design observers in the case of Lipschitz constant γ ≥ 1. Then, a natural idea is to generalize the classical Lipschitz condition to a more broad family of nonlinearities, e.g., one-sided Lipschitz condition (3). Such treatment may reduce the conservatism of the existing results, as we will see below. Therefore, this section discusses the observer design for discrete-time one-sided Lipschitz nonlinear systems. Proposition 8: Assume that system (1) satisfies conditions (3) and (4) with constants ρ, σ, and ϕ and that the observer holds the form of (5). Then, error dynamics (7) is asymptotically stable if there exist matrices P > 0 and R and scalars ε1 > 0 and ε2 > 0 such that the following LMI is feasible: ⎤ ⎡ ˜ + ϕε2 −ε1 I R ˜ −P + ε1 ρI + ε2 σI R 2 ⎣ 0 ⎦ 1, the proposed conditions of Propositions 1–3, as well as Proposition 4, are not applicable to this case. On the other hand, this system satisfies onesided Lipschitz condition (3), for which the one-sided Lipschitz constant is much smaller than the Lipschitz constant. In fact, using the mean value theorem, one can obtain
ΔfkT Δfk
From (3), we get ρeT (k)e(k) − eT (k)Δfk ≥ 0. Therefore, for any positive scalar ε1
Π=
eT (k)Δfk ≤ − e(k)2
A˜ = (A − LC)T P (A − LC) − P.
ε1
where
(21)
2
≤ 9 e(k) .
(23) (24)
Hence, conditions (3) and (4) are satisfied with ρ = −1, σ = 9, and ϕ = 0. Thus, we can use Proposition 8 to design a state observer for the system. Solving LMI (18) yields 0.0783 −0.0628 P = R = [ −0.0768 0.1527 ] . −0.0628 0.2467 Therefore, gain matrix L is given by −0.6074 −1 T L=P R = . 0.4642
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ACKNOWLEDGMENT The authors would like to thank Associate Editor Prof. G. Grassi and the anonymous reviewers for their insightful comments and helpful suggestions and Prof. Z. Han for various fruitful discussions. R EFERENCES
Fig. 1. Simulation result of state x1 (k) and its estimate x ˆ1 (k) with initial values x1 (0) = 2 and x ˆ1 (0) = −2.
Fig. 2. Simulation result of state x2 (k) and its estimate x ˆ2 (k) with initial values x2 (0) = −1 and x ˆ2 (0) = 3.
Figs. 1 and 2 show the simulation results for states x1 (k) and x2 (k), respectively. From the simulation, we know that the effect of state trajectory tracking is satisfactory. VI. C ONCLUSION We have addressed the observer design problem for discretetime Lipschitz nonlinear systems. In particular, we modified some statements and conclusions in [12] and pointed out that the proposed conditions cannot provide less conservative results than some existing ones. Moreover, a major limitation of the existing results is that they work only for the case of nonlinear systems with Lipschitz constant γ < 1. In order to obtain less conservative results, we introduced the concept of a one-sided Lipschitz condition to an observer synthesis. An LMI condition is given to achieve the convergence of the error dynamics of discrete-time one-sided Lipschitz nonlinear systems. A numerical example is included to illustrate the effectiveness of the proposed design. The application of the proposed design to chaos synchronization and input recovery is a subject for further research.
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