#### Abstract

The issue of estimation for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input is addressed. First, through integrating the filtering performance index with the Lipschitz conditions of the nonlinearity, the design of robust estimator is formulated as a positive minimum problem of indefinite quadratic form. Then, by introducing the Krein space model and applying innovation analysis approach, the minimum of the indefinite quadratic form is obtained in terms of innovation sequence. Finally, through guaranteeing the positivity of the minimum, a sufficient condition for the existence of the estimator is proposed and the estimator is derived in terms of Riccati-like difference equations. The proposed algorithm is proved to be effective by a numerical example.

#### 1. Introduction

In control field, nonlinear estimation is considered to be an important task which is also of great challenge, and it has been a very active area of research for decades [1–7]. Many kinds of methods on estimator design have been proposed for different types of nonlinear dynamical systems. Generally speaking, there are three approaches widely adopted for nonlinear estimation. In the first one, by using an extended (nonexact) linearization of the nonlinear systems, the estimator is designed by employing classical linear observer techniques [1]. The second approach, based on a nonlinear state coordinate transformation which renders the dynamics driven by nonlinear output injection and the output linear on the new coordinates, uses the quasilinear approaches to design the nonlinear estimator [2–4]. In the last one, methods are developed to design nonlinear estimators for systems which consist of an observable linear part and a locally or globally Lipschitz nonlinear part [5–7]. In this paper, the problem of estimator design is investigated for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input.

In practice, most nonlinearities can be regarded as Lipschitz, at least locally when they are studied in a given neighborhood [6]. For example, trigonometric nonlinearities occurring in many robotic problems, non-linear softening spring models frequently used in mechanical systems, nonlinearities which are square or cubic in nature, and so forth. Thus, in recent years, increasing attention has been paid to estimator design for Lipschitz nonlinear systems [8–19]. For the purpose of designing this class of nonlinear estimator, a number of approaches have been developed, such as sliding mode observers [8, 9], optimization techniques [10–13], adaptive observers [14, 15], high-gain observers [16], loop transfer recovery observers [17], proportional integral observers [18], and integral quadratic constraints approach [19]. All of the above results are obtained in the assumption that the Lipschitz nonlinear systems are delay free. However, time delay is an inherent characteristic of many physical systems, and it can result in instability and poor performances if it is ignored. The estimator design for time-delay Lipschitz nonlinear systems has become a substantial need. Unfortunately, compared with estimator design for delay-free Lipschitz nonlinear systems, less research has been carried out on the time-delay case. In [20], the linear matrix inequality-(LMI-) based full-order and reduced-order robust observers are proposed for a class of Lipschitz nonlinear discrete-time systems with time delay. In [21], by using Lyapunov stability theory and LMI techniques, a delay-dependent approach to the and filtering is proposed for a class of uncertain Lipschitz nonlinear time-delay systems. In [22], by guaranteeing the asymptotic stability of the error dynamics, the robust observer is presented for a class of uncertain discrete-time Lipschitz nonlinear state delayed systems; In [23], based on the sliding mode techniques, a discontinuous observer is designed for a class of Lipschitz nonlinear systems with uncertainty. In [24], an LMI-based convex optimization approach to observer design is developed for both constant-delay and time-varying delay Lipschitz nonlinear systems.

In this paper, the estimation problem is studied for a class of Lipschitz nonlinear discrete time-delay systems with disturbance input. Inspired by the recent study on fault detection for linear discrete time-delay systems in [25], a recursive Kalman-like algorithm in an indefinite metric space, named the Krein space [26], will be developed to the design of estimator for time-delay Lipschitz nonlinear systems. Unlike [20], the delay-free nonlinearities and the delayed nonlinearities in the presented systems are decoupling. For the case presented in [20], the observer design problem, utilizing the technical line of this paper, can be solved by transforming it into a delay-free system through state augmentation. Indeed, the state augmentation results in a higher system dimension and, thus, a much more expensive computational cost. Therefore, this paper based on the presented time-delay Lipschitz nonlinear systems, focuses on the robust estimator design without state augmentation by employing innovation analysis approach in the Krein space. The major contribution of this paper can be summarized as follows: (i) it extends the Krein space linear estimation methodology [26] to the state estimation of the time-delay Lipschitz nonlinear systems and (ii) it develops a recursive Kalman-like robust estimator for time-delay Lipschitz nonlinear systems without state augmentation.

The remainder of this paper is arranged as follows. In Section 2, the interest system, the Lipschitz conditions, and the estimation problem are introduced. In Section 3, a partially equivalent Krein space problem is constructed, the estimator is obtained by computed Riccati-like difference equations, and sufficient existence condition is derived in terms of matrix inequalities. An example is given to show the effect of the proposed algorithm in Section 4. Finally, some concluding remarks are made in Section 5.

In the sequel, the following notation will be used: elements in the Krein space will be denoted by **boldface** letters, and elements in the Euclidean space of complex numbers will be denoted by normal letters; denotes the real -dimensional Euclidean space; denotes the Euclidean norm; means ; the superscripts “−1” and “” stand for the inverse and transpose of a matrix, resp.; is the identity matrix with appropriate dimensions; For a real matrix, (, resp.) means that is symmetric and positive (negative, resp.) definite; denotes the inner product in the Krein space; denotes a block-diagonal matrix; denotes the linear space spanned by sequence .

#### 2. System Model and Problem Formulation

Consider a class of nonlinear systems described by the following equations: where , and the positive integer denotes the known state delay; is the state, is the measurable information, and are the disturbance input belonging to , is the measurement output, and is the signal to be estimated; the initial condition is unknown; the matrices , , , and , are real and known constant matrices.

In addition, and are assumed to satisfy the following Lipschitz conditions: for all , and . where and are known Lipschitz constants, and , are real matrix with appropriate dimension.

The estimation problem under investigation is stated as follows. Given the desired noise attenuation level and the observation , find an estimate of the signal , if it exists, such that the following inequality is satisfied: where is a given positive definite matrix function which reflects the relative uncertainty of the initial state to the input and measurement noises.

*Remark 2.1. *For the sake of simplicity, the initial state estimate is assumed to be zero in inequality (2.3).

*Remark 2.2. *Although the system given in [20] is different from the one given in this paper, the problem mentioned in [20] can also be solved by using the presented approach. The resolvent first converts the system given in [20] into a delay-free one by using the classical system augmentation approach, and then designs estimator by employing the similar but easier technical line with our paper.

#### 3. Main Results

In this section, the Krein space-based approach is proposed to design the estimator for Lipschitz nonlinear systems. To begin with, the estimation problem (2.3) and the Lipschitz conditions (2.2) are combined in an indefinite quadratic form, and the nonlinearities are assumed to be obtained by at the time step . Then, an equivalent Krein space problem is constructed by introducing an imaginary Krein space stochastic system. Finally, based on projection formula and innovation analysis approach in the Krein space, the recursive estimator is derived.

##### 3.1. Construct a Partially Equivalent Krein Space Problem

It is proved in this subsection that the estimation problem can be reduced to a positive minimum problem of indefinite quadratic form, and the minimum can be obtained by using the Krein space-based approach.

Since the denominator of the left side of (2.3) is positive, the inequality (2.3) is equivalent to where .

Moreover, we denote where and denote the optimal estimation of and based on the observation , respectively. And, let From the Lipschitz conditions (2.2), we derive that Note that the left side of (3.1) and (3.4), , can be recast into the form where

Since , it is natural to see that if then the estimation problem (2.3) is satisfied, that is, . Hence, the estimation problem (2.3) can be converted into finding the estimate sequence such that has a minimum with respect to and the minimum of is positive. As mentioned in [25, 26], the formulated estimation problem can be solved by employing the Krein space approach.

Introduce the following Krein space stochastic system where ; the initial state and , , , and are mutually uncorrelated white noises with zero means and known covariance matrices , , , , , and ; , and are regarded as the imaginary measurement at time for the linear combination , , and , respectively.

Let

*Definition 3.1. *The estimator denotes the optimal estimation of given the observation ; the estimator denotes the optimal estimation of given the observation ; the estimator denotes the optimal estimation of given the observation .

Furthermore, introduce the following stochastic vectors and the corresponding covariance matrices And, denote

For calculating the minimum of , we present the following Lemma 3.2.

Lemma 3.2. * is the innovation sequence which spans the same linear space as that of .*

*Proof. *From Definition 3.1 and (3.9), , and are the linear combination of the observation sequence , , and , respectively. Conversely, , and can be given by the linear combination of and , respectively. Hence,
It is also shown by (3.9) that , and satisfy
Consequently,
This completes the proof.

Now, an existence condition and a solution to the minimum of are derived as follows.

Theorem 3.3. *Consider system (2.1), given a scalar and the positive definite matrix , then has the minimum if only if
**
In this case the minimum value of is given by
**
where
** is obtained from the Krein space projection of onto , is obtained from the Krein space projection of onto , and is obtained from the Krein space projection of onto .*

*Proof. *Based on the definition (3.2) and (3.3), the state equation in system (2.1) can be rewritten as
In this case, it is assumed that and are known at time . Then, we define
By introducing an augmented state
we obtain an augmented state-space model
where
Additionally, we can rewrite as
where
Define the following state transition matrix
and let
Using (3.20) and (3.24), we have
where
The matrix is derived by replacing in with .

Thus, can be reexpressed as
where

Considering the Krein space stochastic system defined by (3.7) and state transition matrix (3.24), we have
where matrices , , and are the same as given in (3.26), vectors and are, respectively, defined by replacing Euclidean space element and in and given by (3.25) with the Krein space element and , vectors and are also defined by replacing Euclidean space element and in and given by (3.23) with the Krein space element and , and vector is given by replacing Euclidean space element in given by (3.19) with the Krein space element when .

Using the stochastic characteristic of , and , we have
where .

In the light of Theorem 2.4.2 and Lemma 2.4.3 in [26], has a minimum over if and only if and have the same inertia. Moreover, the minimum of is given by

On the other hand, applying the Krein space projection formula, we have
where
where is derived by replacing in with, is derived by replacing in with Furthermore, it follows from (3.33) that
where
Since matrix is nonsingular, it follows from (3.35) that and are congruent, which also means that and have the same inertia. Note that both and are block-diagonal matrices, and
is given by (3.23). It follows that and have the same inertia if and only if , and .

Therefore, subject to system (2.1) with Lipschitz conditions (2.2) has the minimum if and only if , and . Moreover, the minimum value of can be rewritten as
The proof is completed.

*Remark 3.4. *Due to the built innovation sequence in Lemma 3.2, the form of the minimum on indefinite quadratic form is different from the one given in [26–28]. It is shown from (3.15) that the estimation errors , and are mutually uncorrelated, which will make the design of estimator much easier than the one given in [26–28].

##### 3.2. Solution of the Estimation Problem

In this subsection, the Kalman-like recursive estimator is presented by using orthogonal projection in the Krein space.

Denote Observe from (3.8), we have

*Definition 3.5. *Given , the estimator for denotes the optimal estimate of given the observation , and the estimator for denotes the optimal estimate of given the observation . For simplicity, we use to denote , and use to denote throughout the paper.

Based on the above definition, we introduce the following stochastic sequence and the corresponding covariance matrices Similar to the proof of Lemma 2.2.1 in [27], we can obtain that , is the innovation sequence which is a mutually uncorrelated white noise sequence and spans the same linear space as or equivalently .

Applying projection formula in the Krein space, is computed recursively as Note that where Substituting (3.44) into (3.43), we have Moreover, taking into account (3.7) and (3.46), we obtain Consequently, where . Thus, can be computed recursively as

Similarly, employing the projection formula in the Krein space, the optimal estimator can be computed by where Then, from (3.7) and (3.50), we can yield Thus, we obtain that(1)if , we have (2)if , we have It follows from (3.53), (3.54), and (3.55) that can be computed by