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State estimation for discretetime systems with generalized Lipschitz nonlinear dynamics
Advances in Difference Equations volume 2015, Article number: 307 (2015)
Abstract
This paper considers the state estimation problem for a class of discretetime systems with generalized Lipschitz nonlinear dynamics. Under the assumption that the system nonlinearities satisfy a quadratically innerboundedness condition, we design both the fullorder observer and the reducedorder observer for the discretetime nonlinear system. Sufficient conditions ensuring the existence of fullorder observers as well as reducedorder observers for such systems are established and formulated in terms of linear matrix inequality (LMI). Compared with some existing results, we remove the onesided Lipschitz restrict and extend the classical Lipschitz observer design to a larger class of discretetime nonlinear systems. A numerical example is included to illustrate the effectiveness of the proposed design.
Introduction
During the past two decades, the state estimation or observer design problem for nonlinear dynamic systems has received extensively research attention; see [1–14] and the references therein. This is partly due to the fact that knowledge of the state of a dynamic system plays a key role in many control problems. It is well known that state estimation can be used for control design, diagnosis or synchronization and unknown input recovery. However, designing a state observer for a general nonlinear system is not easy or even impossible. Many current research efforts are focused on some specialized classes of nonlinear systems. For instance, Arcak et al. [1, 2] developed a circlecriterion approach to design observer for sector nonlinear systems. For Lipschitz nonlinear systems, the existence conditions of the fullorder as well as the reduceorder observers were established in Rajamani [3] and Zhu and Han [4], respectively. Robust observers for Lipschitz nonlinear systems subject to disturbances were proposed in [5, 6]. Nonlinear observer for neutral uncertain timedelay systems was addressed in [7]. Very recently, the classical Lipschitz nonlinear observer design has been extended to the onesided Lipschitz case; see e.g. [8–14].
It should be noted that most of the abovementioned works are concerned on continuoustime nonlinear systems. Generally, the state estimation problem for discretetime nonlinear systems has received little attention. Moreover, in the existing literature there have been several useful observer design approaches for some specialized classes of discretetime nonlinear systems [15–23]. For example, Ibrir [15] proposed the circlecriterion approach to discretetime nonlinear observer design. In [16] and [17], the authors considered the observer design for discretetime Lipschitz nonlinear systems. Motivated by the Arcaktype observer design [1, 2], Zemouche and Boutayeb [18] provided a unified observer design method for discretetime Lipschitz systems and extended it to \(H_{\infty}\) synchronization and unknown input recovery. An LMI approach was proposed by Wang et al. [19] to design state observer for discretetime Lipschitz descriptor systems. In [20] the authors considered an observer design for discretetime epidemic models. A new reducedorder observer normal form for nonlinear discretetime systems was provided in [21].
Very recently, several authors have considered the observer design for onesided Lipschitz nonlinear systems in the discretetime case. Both fullorder and reducedorder observer designs were studied in Benallouch et al. [22]. In fact, they have developed an LMIbased design approach to deal with the state estimation problem of onesided Lipschitz discretetime systems. Zhang et al. [23] investigated the same problem and proposed a simple observer synthesis condition to ensure the asymptotic convergence. It should be emphasized that the systems considered by Benallouch et al. [22] and Zhang et al. [23] are actually a subset of onesided Lipschitz nonlinear systems (see Figure 1 below). More precisely, the systems are assumed to simultaneously satisfy the onesided Lipschitz condition and the quadratically innerbounded condition. This assumption may lead to more conservative results and bring additional restrictions on the system model. How to reduce the conservatism in the existing results of observer design of nonlinear systems is still an open problem. This motivates our present research.
In this paper, we focus on state observer design for a general class of nonlinear discretetime systems that satisfies the quadratically innerbounded condition only. The main contributions of this paper are two folds. First, we remove the onesided Lipschitz restriction and only need the assumption of quadratically innerbounded condition. Note that the quadratically innerbounded condition includes the classical Lipschitz condition as a special case; see e.g. Figure 1 below. Therefore, we extend the state observer design to a larger class of discretetime nonlinear systems. Second, some simple stability conditions are obtained for both fullorder and reducedorder observer designs. In our approach, the observer designs are formulated as an LMI feasible problem, which is easily solved by standard convex optimization algorithms. An example on the singlelink flexible joint robot is given to illustrate the effectiveness of the proposed design.
Notations: \(\mathbb {R}^{n}\) denotes the ndimensional real Euclidean space. \(\langle\cdot, \cdot\rangle\) represents the inner product in \(\mathbb {R}^{n}\), i.e., for given \(x,y \in \mathbb {R}^{n}\), then \(\langle x,y\rangle= x^{T}y\), where \(x^{T}\) is the transpose of the column vector \(x \in \mathbb {R}^{n}\). \(\Vert { \cdot} \Vert \) denotes the Euclidean norm on \(\mathbb {R}^{n}\). For a symmetric matrix P, \(P > 0\) (\(P < 0\)) means that the matrix is positive definite (negative definite). In symmetric block matrices, we use an asterisk ∗ to represent a term induced by symmetry. I represents an identity matrix with appropriate dimension.
Problem statement and preliminaries
In this paper, we consider the class of discretetime nonlinear systems described by
where \(x(k) \in \mathbb {R}^{n}\) is the state vector and \(y(k) \in \mathbb {R}^{p}\) is the linear measured output. A, B and C are constant matrices of appropriate dimensions. \(f: \mathbb {R}^{n} \times \mathbb {R}^{p} \to \mathbb {R}^{n}\) is a real nonlinear vector field, which is assumed to satisfy the following quadratically innerbonded condition [22].
Assumption 1
f is quadratically innerbounded with respect to \(x(k)\), i.e., for all \(x,\hat{x} \in \mathbb {R}^{n}\), there exist \(\beta,\gamma \in \mathbb {R}\) such that
It is clear that any Lipschitz function is also quadratically innerbounded corresponding to \(\beta> 0\) and \(\gamma= 0\). Consequently, Lipschitz continuity implies quadratic innerboundedness, but the converse is not true [8, 9]. It should be emphasized that β and γ in (2) can be any real number and are not necessarily positive. Therefore, the system considered in the paper includes the wellknown Lipschitz nonlinear system as a special case (see Figure 1).
For the purpose of comparison, we introduce the following two assumptions, which are commonly used in the recent literature for observer design of nonlinear systems. For further details, we refer the interested reader to [8, 9, 22].
Assumption 2
(see e.g. [9])
f is Lipschitz with respect to \(x(k)\), i.e., for all \(x,\hat{x} \in \mathbb {R}^{n}\), there exists a scalar \(\lambda> 0\) such that
Assumption 3
f is onesided Lipschitz with respect to \(x(k)\), i.e., for all \(x,\hat{x} \in \mathbb {R}^{n}\), there exists a scalar \(\rho\in \mathbb {R}\) such that
Notice that Assumption 2 is the wellknown Lipschitz condition, while Assumption 3 is the socalled onesided Lipschitz condition. It is worth mentioning that the onesided Lipschitz condition has been frequently employed in the study of synchronization of complex networks [25, 26]. Moreover, as shown in [8] and [9], the onesided Lipschitz condition implies the Lipschitz condition but the converse is not true. Figure 1 shows the relation between the Lipschitz, onesided Lipschitz, and quadratically innerbounded function sets [24].
We end this section by introducing a useful lemma.
Lemma 1
(The Schur complement lemma; see e.g. [27])
For a real symmetric matrix Σ, the following assertions are equivalent:

(1)
\(\Sigma:=\bigl [ {\scriptsize\begin{matrix}{} \Sigma_{11} & \Sigma_{12} \cr \Sigma_{12}^{T} & \Sigma_{22} \end{matrix}} \bigr ]<0\).

(2)
\(\Sigma_{11} < 0\), and \(\Sigma_{22}  \Sigma_{12}^{T}\Sigma_{11}^{  1}\Sigma_{12} < 0\).

(3)
\(\Sigma_{22} < 0\), and \(\Sigma_{11}  \Sigma_{12} \Sigma_{22} ^{1}\Sigma_{12}^{T} < 0\).
Fullorder observer design
In this section, we consider the fullorder observer design for system (1) under Assumption 1. As usual, we consider a Luenbergerlike observer for system (1) in the form of
where \(\hat{x}(k)\) denotes the estimate of the state \(x(k)\). Our design goal is to find a gain matrix L such that the estimation error \(e(k): = x(k)  \hat{x}(k)\) converges asymptotically toward zero. From (1) and (5), the dynamics of the estimation error is governed by
where \(\Delta f_{k} : = f(x(k),y(k))  f(\hat{x}(k),y(k))\).
Now, we have the following conclusion.
Theorem 1
Suppose that system (1) satisfies Assumption 1 and the observer has the form of (5). Then the error dynamics is asymptotically stable if there exist matrices \(P > 0\) and R with appropriate dimensions and a scalar \(\omega> 0\) such that the following LMI is feasible:
The resulting observer gain matrix L is given by \(L = P^{  1}R^{T}\).
Proof
For the estimation error dynamics (6), let us consider the Lyapunov function candidate \(V(k) = e^{T}(k)Pe(k)\). Then the difference of \(V(k)\) along the trajectories of (6) is given by
By Assumption 1,
It then follows from (9) that
where \(\omega> 0\) is a scalar. Adding the lefthand side of (10) to \(\Delta V_{k} \) yields
where \(\xi_{k}^{T} = [ {e(k)} \ {\Delta f_{k} } ]^{T}\) and
Applying Lemma 1, \(\Omega< 0\) is equivalent to
By denoting \(R = L^{T}P\), the condition (7) implies \(\Gamma< 0\). Therefore, we have \(\Delta V_{k} < 0\) for all \(e(k) \ne0\) if (7) is satisfied. This completes the proof. □
Since the quadratically innerbounded condition include the Lipschitz condition as a special case, we immediately have Corollary 1.
Corollary 1
Suppose that system (1) satisfies Assumption 2 and the observer has the form of (5). Then the error dynamics is asymptotically stable if there exist matrices \(P > 0\) and R with appropriate dimensions and a scalar \(\omega> 0\) such that the following LMI is feasible:
The resulting observer gain matrix L is given by \(L = P^{  1}R^{T}\).
Reducedorder observer design
In this section, we address the reducedorder observer design problem for system (1) under Assumption 1. Note that our design is inspired by the approach developed in [17] and [22], but we remove the onesided Lipschitz restriction and provide a simple observer synthesis condition. Let \(\xi(k)\) denote the reduced state vector to be estimated. Without loss of generality, assume
where \(H \in \mathbb {R}^{(n  p) \times n}\) is a matrix so that \(\bigl [ {\scriptsize\begin{matrix}{} H \cr C \end{matrix}} \bigr ]\) is nonsingular with
We then have
From (1), (14), and (16), we obtain the following nonlinear reduced form:
where \(A_{Z} : = HAN\), \(B_{Z} : = HAM\), and \(g(z(k),y(k)): = f(Nz(k) + My(k),y(k))\).
Inspired by [22], we design a reducedorder observer corresponding to (17) as follows:
Denoting the estimator error by \(\varepsilon(k): = z(k)  \hat {z}(k)\) and letting \(C_{Z} : = CAN\), we have
where \(\Delta g_{k} : = g(z(k),y(k))  g(\hat{z}(k),y(k))\).
From (17)(19), we know that the dynamics of the estimation error is governed by
Now, we have the following theorem.
Theorem 2
Under Assumption 1, the proposed reducedorder observer (18) is an asymptotic observer for system (1) if there exist matrices \(P > 0\) and K of appropriate dimensions and a scalar \(\omega> 0\) such that the following matrix inequality is feasible:
Proof
Notice that \(e(k) = N\varepsilon(k)\). For the error dynamics (20), we also consider the Lyapunov function candidate \(V(k) = e^{T}(k)Pe(k)\), i.e., \(V(k) = \varepsilon ^{T}(k)N^{T}PN\varepsilon(k)\). Then the difference of \(V(k)\) along the trajectories of (20) is given by
By Assumption 1, the nonlinear function \(f(x(k),y(k))\) is quadratically innerbounded, then also the function \(g(z(k),y(k))\) is quadratically innerbounded with constants \(\beta_{g} \) and \(\gamma_{g} \). In fact, from Assumption 1, we can deduce
where \(\omega> 0\) is a scalar. Note that \(e(k) = N\varepsilon(k)\) and \(\Delta f_{k} = \Delta g_{k}\). It follows from (23) that
Adding the lefthand side of (24) to \(\Delta V_{k} \) yields
where \(\chi_{k}^{T} = [ {\varepsilon(k)} \ {\Delta g_{k} } ]^{T}\), and
Note that \(\Delta V_{k} < 0\) if \(\Pi< 0\). Using Lemma 1, \(\Pi< 0\) is equivalent to
Therefore, if the matrix inequality (21) has a feasible solution, we have \(\Delta V_{k} < 0\) for all \(\varepsilon(k) \ne0\). By the standard Lyapunov theorem, we know that the estimation error system is asymptotically stable, which means (18) is an asymptotic reducedorder observer for system (1). This completes the proof. □
Remark 1
Compared with the fullorder or the reducedorder observer design in [22], the paper removes the onesided Lipschitz restriction, which significantly reduces the conservatism and complexity of the designs. In fact, in Theorems 1 and 2, we only assume that f satisfies the quadratically innerbounded condition (2) and do not employ the onesided Lipschitz condition (3).
Remark 2
It should be noted that (21) is not an LMI. To make it more tractable, we can formulate it into an LMI by letting \(P = \alpha I\) for a prior given scalar \(\alpha> 0\). In this case, (21) becomes
Similarly, we have Corollary 2, since the quadratically innerbounded condition includes the Lipschitz condition as a special case.
Corollary 2
Under Assumption 2, the proposed reducedorder observer (18) is an asymptotic observer for system (1) if there exist matrices \(P > 0\) and K of appropriate dimensions and a scalar \(\omega> 0\) such that the following matrix inequality is feasible:
Illustrative example
In this section gives a numerical example to illustrate the applications of the proposed observer design. For convenience, we take the wellknown singlelink flexible joint robotic system as an example [3, 4]. The continuoustime model of the system is described by
where
and
Let \(T_{e} \) be the sample time. Then by using the Euler discretized approach on system (28), we can derive the following discretetime system model:
where
It is easy to verify that \(f(x(k),y(k))\) is quadratically innerbounded with \(\beta= (0.333T_{e} )^{2}\) and \(\gamma= 0\). Let the sample time \(T_{e} = 0.1[s]\). To design the fullorder observer, we need to solve the LMI (7). By using the Matlab LMI tool, we get
Therefore, the fullorder observer gain matrix L is given by
On the other hand, the reducedorder observer can be designed by using Theorem 2. With
we have
Let \(\alpha= 0.001\). By solving the LMI (21), we obtain the reducedorder observer gain matrix
Conclusion
We have addressed the state estimation problem for a general class of nonlinear discretetime systems that satisfies the quadratically innerbounded condition. The system under consideration need not satisfy the onesided Lipschitz restriction, which is a common assumption in some recent literature on observer design for nonlinear discretetime systems. We considered both the fullorder and the reducedorder observer designs and formulated the observer synthesis condition as an LMI formulation. Finally, we used an example on the singlelink flexible joint robotic system to illustrate the effectiveness of the proposed design.
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant 51505273, the State Key Laboratory of Robotics and System (HIT) under Grant SKLRS2014MS10, the Jiangsu Provincial Key Laboratory of Advanced Robotics Fund Projects under Grant JAR201401, and the Foundation of Shanghai University of Engineering Science under Grant nhky201506.
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Wang, D., Song, F. & Zhang, W. State estimation for discretetime systems with generalized Lipschitz nonlinear dynamics. Adv Differ Equ 2015, 307 (2015). https://doi.org/10.1186/s136620150645x
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Keywords
 observer design
 quadratically innerboundedness
 Lipschitz condition
 linear matrix inequality (LMI)
 discretetime nonlinear systems