Estimation for a Class of Lipschitz Nonlinear Discrete-Time Systems with Time Delay

and Applied Analysis 3 will be denoted by normal letters; R denotes the real n-dimensional Euclidean space; ‖ · ‖ denotes the Euclidean norm; θ k ∈ l2 0,N means ∑N k 0 θ T k θ k < ∞; the superscripts “−1” and “T” stand for the inverse and transpose of a matrix, resp.; I is the identity matrix with appropriate dimensions; For a real matrix, P > 0 P < 0, resp. means that P is symmetric and positive negative, resp. definite; 〈∗, ∗〉 denotes the inner product in the Krein space; diag{· · · } denotes a block-diagonal matrix; L{· · ·} denotes the linear space spanned by sequence {· · ·}. 2. System Model and Problem Formulation Consider a class of nonlinear systems described by the following equations: x k 1 Ax k Adx kd f k, Fx k , u k h k,Hx kd , u k Bw k ,


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 H ∞ estimator design is investigated for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input.will be denoted by normal letters; R n denotes the real n-dimensional Euclidean space; • denotes the Euclidean norm; θ k ∈ l 2 0, N means N k 0 θ T k θ k < ∞; the superscripts "−1" and "T " stand for the inverse and transpose of a matrix, resp.;I is the identity matrix with appropriate dimensions; For a real matrix, P > 0 P < 0, resp.means that P is symmetric and positive negative, resp.definite; * , * denotes the inner product in the Krein space; diag{• • • } denotes a block-diagonal matrix; L{• • • } denotes the linear space spanned by sequence {• • • }.

System Model and Problem Formulation
Consider a class of nonlinear systems described by the following equations: where k d k − d, and the positive integer d denotes the known state delay; x k ∈ R n is the state, u k ∈ R p is the measurable information, w k ∈ R q and v k ∈ R m are the disturbance input belonging to l 2 0, N , y k ∈ R m is the measurement output, and z k ∈ R r is the signal to be estimated; the initial condition x 0 s s −d, −d 1, . . ., 0 is unknown; the matrices A ∈ R n×n , A d ∈ R n×n , B ∈ R n×q , C ∈ R m×n and L ∈ R r×n , are real and known constant matrices.
In addition, f k, Fx k , u k and h k, Hx k d , u k are assumed to satisfy the following Lipschitz conditions: where α > 0 and β > 0 are known Lipschitz constants, and F, H are real matrix with appropriate dimension.The H ∞ estimation problem under investigation is stated as follows.Given the desired noise attenuation level γ > 0 and the observation {y j } k j 0 , find an estimate z k | k of the signal z k , if it exists, such that the following inequality is satisfied: Remark 2.1.For the sake of simplicity, the initial state estimate x 0 k k −d, −d 1, . . ., 0 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.

Main Results
In this section, the Krein space-based approach is proposed to design the H ∞ estimator for Lipschitz nonlinear systems.To begin with, the H ∞ 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 {y i } k i 0 at the time step k.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.

Construct a Partially Equivalent Krein Space Problem
It is proved in this subsection that the H ∞ 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 zf k | k and zh k d | k denote the optimal estimation of z f k and z h k d based on the observation {y j } k j 0 , respectively.And, let

3.3
From the Lipschitz conditions 2.2 , we derive that Note that the left side of 3.1 and 3.4 , J N , can be recast into the form 3.5 where

3.6
Since J N ≤ J * N , it is natural to see that if J N > 0 then the H ∞ estimation problem 2.3 is satisfied, that is, J * N > 0. Hence, the H ∞ estimation problem 2.3 can be converted into finding the estimate sequence } such that J N has a minimum with respect to {x 0 , w} and the minimum of J N is positive.As mentioned in 25, 26 , the formulated H ∞ estimation problem can be solved by employing the Krein space approach.
Introduce the following Krein space stochastic system  Let

3.8
Definition 3.1.The estimator y i | i − 1 denotes the optimal estimation of y i given the observation L{{y z j } i−1 j 0 }; the estimator z m i | i denotes the optimal estimation of zm i | i given the observation L{{y z j } i−1 j 0 ; y i }; the estimator Furthermore, introduce the following stochastic vectors and the corresponding covariance matrices 3.9 And, denote

3.10
For calculating the minimum of J N , we present the following Lemma 3.2.

Lemma 3.2. {{ y z i } k i 0 } is the innovation sequence which spans the same linear space as that of
Proof.From Definition 3.1 and 3.9 , y i | i − 1 , z m i | i and z h i d | i are the linear combination of the observation sequence {{y z j } i−1 j 0 ; y i }, {{y z j } i−1 j 0 ; y i , zm i | i }, and {{y z j } i j 0 }, respectively.Conversely, y i , zm i | i and zh i d | i can be given by the linear combination of {{ y z j } i−1 j 0 ; It is also shown by 3.9 that

3.13
This completes the proof.
Now, an existence condition and a solution to the minimum of J N are derived as follows.

3.14
In this case the minimum value of J N is given by

3.15
where Proof.Based on the definition 3.2 and 3.3 , the state equation in system 2.1 can be rewritten as 3.17 In this case, it is assumed that

3.18
By introducing an augmented state we obtain an augmented state-space model

3.21
Additionally, we can rewrite J N as where

3.23
Define the following state transition matrix

3.25
Using 3.20 and 3.24 , we have where

3.27
The matrix Ψ wN is derived by replacing B u,a in Ψ uN with B a .Thus, J N can be reexpressed as where

3.29
Considering the Krein space stochastic system defined by 3.7 and state transition matrix 3.24 , we have where matrices Ψ 0N , Ψ uN , and Ψ wN are the same as given in 3.26 , vectors y zN and u N are, respectively, defined by replacing Euclidean space element y z and u in y zN and u N given by 3.25 with the Krein space element y z and u, vectors w N and v z,aN are also defined by replacing Euclidean space element w and v z,a in w N and v z,aN given by 3.23 with the Krein space element w and v z,a , and vector x a 0 is given by replacing Euclidean space element x in x a k given by 3.19 with the Krein space element x when k 0.
Using the stochastic characteristic of x a 0 , w N and v z,a , we have

3.32
On the other hand, applying the Krein space projection formula, we have where

3.34
where ϕ m,ij is derived by replacing C z,a in ϕ ij with

3.36
Since matrix Θ N is nonsingular, it follows from 3.35 that R y zN and R y zN are congruent, which also means that R y zN and R y zN have the same inertia.Note that both R y zN and Q v z,aN are blockdiagonal matrices, and Therefore, J N subject to system 2.1 with Lipschitz conditions 2.2 has the minimum if and only Moreover, the minimum value of J N can be rewritten as

3.38
The proof is completed.
Remark 3.4.Due to the built innovation sequence {{ y z i } k i 0 } in Lemma 3.2, the form of the minimum on indefinite quadratic form J N is different from the one given in 26-28 .It is shown from 3.15 that the estimation errors y k | k −1 , z m k | k and z h k d | k are mutually uncorrelated, which will make the design of H ∞ estimator much easier than the one given in 26-28 .

Solution of the H ∞ Estimation Problem
In this subsection, the Kalman-like recursive H ∞ estimator is presented by using orthogonal projection in the Krein space.Denote y 0 i y i ,

3.39
Observe from 3.8 , we have Based on the above definition, we introduce the following stochastic sequence and the corresponding covariance matrices
Applying projection formula in the Krein space, x i, 2 i 0, 1, . . ., k d is computed recursively as 3.42

3.43
Note that

3.44
Abstract and Applied Analysis 15 where

3.46
Moreover, taking into account 3.7 and 3.46 , we obtain

3.48
where Q w i I. Thus, P 2 i, i i 0, 1, . . ., k d can be computed recursively as

3.49
Similarly, employing the projection formula in the Krein space, the optimal estimator x i, 1 i k d 1, . . ., k can be computed by

3.51
Then, from 3.7 and 3.50 , we can yield

3.52
Thus, we obtain that 1 if i − j ≥ k d , we have

3.56
Next, according to the above analysis, z m k | k as the Krein space projections of zm k | k onto L{{y z j } k−1 j 0 ; y 0 k } can be computed by the following formula where

3.58
And, z h k d | k as the Krein space projections of zh k d | k onto L{{y z j } k−1 j 0 ; y 1 k } can be computed by the following formula

3.59
Based on Theorem 3.3 and the above discussion, we propose the following results.

3.61
R y 0 k, 0 , P 1 i, j , and R y 1 j, 1 are calculated by 3.58 , 3.56 , and 3.51 , respectively.Moreover, one possible level-γ H ∞ estimator is given by where E 0 I , and z m k | k is computed by 3.57 .
Proof.In view of Definitions 3.1 and 3.5, it follows from 3.9 and 3.41 that R y k | k − 1 R y 0 k, 0 .In addition, according to 3.7 , 3.9 , and 3.57 , the covariance matrix R z m k | k can be given by the second equality in 3.61 .Similarly, based on 3.7 , 3.9 , and 3.59 , the covariance matrix R z h k d | k can be obtained by the third equality in 3.61 .Thus, from Theorem 3.3, it follows that J N has a minimum if 3.60 holds.
On the other hand, note that the minimum value of J N is given by 3.15 in Theorem 3.3 and any choice of estimator satisfying min J N > 0 is an acceptable one.Therefore, Taking into account

A Numerical Example
Consider the system 2.   parameter L −0.3243 0.0945 T of 5 in 20 .In this numerical example, we compare our algorithm with the one given in 20 in case of γ 1.6164.Figure 1 shows the true value of signal z k , the estimate using our algorithm, and the estimate using the algorithm given in 20 .Figure 2 shows the estimation error of our approach and the estimation error of the approach in 20 .It is shown in Figures 1 and 2 that the proposed algorithm is better than the one given in 20 .Figure 3 shows the ratios between the energy of the estimation error and input noises for the proposed H ∞ estimation algorithm.It is shown that the maximum energy ratio from the input noises to the estimation error is less than γ 2 by using our approach.Figure 4 shows the value of indefinite quadratic form J N for the given estimation algorithm.It is shown that the value of indefinite quadratic form J N is positive by employing the proposed algorithm in Theorem 3.6.

Conclusions
A recursive H ∞ filtering estimate algorithm for discrete-time Lipschitz nonlinear systems with time-delay and disturbance input is proposed.By combining the H ∞ -norm estimation condition with the Lipschitz conditions on nonlinearity, the H ∞ estimation problem is converted to the positive minimum problem of indefinite quadratic form.Motivated by the observation that the minimum problem of indefinite quadratic form coincides with Kalman filtering in the Krein space, a novel Krein space-based H ∞ filtering estimate algorithm is developed.Employing projection formula and innovation analysis technology in the Krein space, the H ∞ estimator and its sufficient existence condition are presented based on Riccatilike difference equations.A numerical example is provided in order to demonstrate the performances of the proposed approach.Future research work will extend the proposed method to investigate more general nonlinear system models with nonlinearity in observation equations.Another interesting research topic is the H ∞ multistep prediction and fixed-lag smoothing problem for time-delay Lipschitz nonlinear systems.

2 . 3 where
Π k k −d, −d 1, . . ., 0 is a given positive definite matrix function which reflects the relative uncertainty of the initial state x 0 k k −d, −d 1, . . ., 0 to the input and measurement noises.
; the initial state x 0 s s −d, −d 1, . . ., 0 and w k , v k , v z f k , v z k and v z h k are mutually uncorrelated white noises with zero means and known covariance matrices and zh k d | k are regarded as the imaginary measurement at time k for the linear combination Fx k , Lx k , and Hx k d , respectively.

3 . 40 Definition 3 . 5 .
Given k ≥ d, the estimator ξ i | j, 2 for 0 ≤ j < k d denotes the optimal estimate of ξ i given the observation L{{y 2 s } j s 0 }, and the estimator ξ i | j, 1 for k d ≤ j ≤ k denotes the optimal estimate of ξ i given the observationL{{y 2 s } k d −1 s 0 ; {y 1 τ } j τ k d }.For simplicity, we use ξ i, 2 to denote ξ i | i − 1, 2 , and use ξ i, 1 to denote ξ i | i − 1, 1 throughout the paper.

1 with time delay d 3 Then we have α β 1 .
Set x k −0.2k 0.1k T k −3, −2, −1, 0 , and Π k I k −3, −2, −1, 0 .Both the system noise w k and the measurement noise v k are supposed to be band-limited white noise with power 0.01.By applying Theorem 3.1 in 20 , we obtain the minimum disturbance attenuation level γ min 1.6164 and the observer algorithm Estimate using algorithm of[20]

Figure 1 :
Figure1: Signal z k solid , its estimate using our algorithm star , and its estimate using algorithm in 20 dashed .

Figure 2 :
Figure 2: Estimation error of our algorithm solid and estimation error of algorithm in 20 dashed .

Figure 3 :Figure 4 :
Figure 3:The energy ratio between estimation error and all input noises for the proposed H ∞ estimation algorithm.
31where y zN y zN − Ψ uN u N .In the light of Theorem 2.4.2 and Lemma 2.4.3 in 26 , J N has a minimum over {x a 0 , w N } if and only if R y zN y zN , y zN and Q v z,aN v z,aN , v z,aN have the same inertia.Moreover, the minimum of J N is given by a k is given by 3.23 .It follows that R y zN and Q v z,aN have the same inertia if and only if R y 3.60 , one possible estimator can be obtained by setting zm k | k z m k | k and zh k d | k z h k d | k .This completes the proof.Remark 3.7.It is shown from 3.57 and 3.59 that z m k | k and z k d | k are, respectively, the filtering estimate and fixed-lag smoothing of zm k | k and z k d | k in the Krein space.Additionally, it follows from Theorem 3.6 that zm k | k and zh k d | k achieving the H ∞ estimation problem 2.3 can be, respectively, computed by the right side of 3.57 and 3.59 .Thus, it can be concluded that the proposed results in this paper are related with both the H 2 filtering and H 2 fixed-lag smoothing in the Krein space.Recently, the robust H ∞ observers for Lipschitz nonlinear delay-free systems with Lipschitz nonlinear additive uncertainties and time-varying parametric uncertainties have been studied in 10, 11 , where the optimization of the admissible Lipschitz constant and the disturbance attenuation level are discussed simultaneously by using the multiobjective optimization technique.In addition, the sliding mode observers with H ∞ performance have been designed for Lipschitz nonlinear delay-free systems with faults matched uncertainties and disturbances in 8 .Although the Krein space-based robust H ∞ filter has been proposed for discrete-time uncertain linear systems in 28 , it cannot be applied to solving the H ∞ estimation problem given in 10 since the considered system contains Lipschitz nonlinearity and Lipschitz nonlinear additive uncertainty.However, it is meaningful and promising in the future, by combining the algorithm given in 28 with our proposed method in this paper, to construct a Krein space-based robust H ∞ filter for discrete-time Lipschitz nonlinear systems with nonlinear additive uncertainties and time-varying parametric uncertainties.