Predictor-Based Global Sampled-Data Output Feedback Stabilization for Nonlinear Uncertain Systems Subject to Delayed Output

In this article, we address the problem of global sampled-data output feedback stabilization for a class of nonlinear uncertain systems with delayed output using the continuous-discrete method. Thanks to the prediction technique and feedback domination method, a novel coupled design method of predictor-based continuous-discrete observer and linear controller is proposed when only delayed sampled-data output is accessible. The proposed predictor-based observer can effectively estimate the unknown state by compensating the influences of sampling and output delay. The main advantage of the proposed control method is that the full state information and accurate model nonlinearities do not need to be known any more. The global exponential stability of the overall hybrid control system can be ensured when there hold some sufficient conditions with respect to the maximum allowable sampling period and output delay.


I. INTRODUCTION
Recent decades have witnessed the remarkable development of sampled-data control due to the wide applications of digital and communication technologies in control systems [1], [2]. Roughly speaking, available results on the sampled-data control for continuous-time nonlinear systems in the literature can be categorized into three main methods. The first is the discretization method [3], [4], where the continuous-time nonlinear system is approximately discretized at first, and under a well-designed discrete-time controller, the corresponding stability analysis is then conducted on the approximated discrete-time system. The idea of the second one, called the emulation method [5]- [7], is that by meticulously selecting the sampling period, the sample-and-hold implementation of a predesigned continuous-time control method is given to guarantee the certain stability property of the continuous-time nonlinear control system. The last is named the continuous-discrete method (also called the hybrid method), where the sampled-data controller via holder is directly designed for continuous-time nonlinear system, and the stability analysis needs to be rigorously deduced for the hybrid control system [8]- [12]. By using the continuous-discrete method, this article is concerned with the global sampled-data stabilization problem of a class of continuous-time nonlinear uncertain systems. The case that only delayed sampled-data output is accessible is quite common in many practical applications, such as vision-based control system [13] and networked control system [14]. Neglecting the influence of output delay significantly degrades the control performance, and even leads to the instability of sampled-data control system [15]. However, there are a few existing results on sampled-data stabilization problem when only the delayed sampled-data output is known. For examples, Karafyllis and Krstic [11] considered the problem of sampleddata stabilization for nonlinear delay systems, but did not take nonlinear uncertainties into account. The problem of global sampled-data stabilization was considered for triangular nonlinear systems via output feedback in [12]. By using sampled-data output and continuous-time model, continuous-discrete observer is often designed to estimate the unknown state for continuous-time nonlinear systems. Different from the sample-and-hold implementation of output in [9], [10], and [12], a new continuous-discrete observer design method has been proposed via predicting the output over the intersample time interval, the aim of which is to compensate the influences of the sampling and/or output delay, see [16]- [20] when only the sampling is considered and see [21]- [25] when both the sampling and output delay are considered. Note that the aforementioned results were specifically developed to handle the continuous-discrete observer design problems and cannot be directly utilized to address the coupling design problem of state observer and controller when the separation principle is not applicable any more.
Motivated by the prediction technique from [21]- [23], this article considers the global sampled-data output feedback stabilization problem for a class of nonlinear uncertain systems subject to delayed sampled-data measurement output. By using the feedback domination and prediction techniques, a novel high-gain continuous-discrete observer is first proposed along with an output predictor. Thereafter, a new sampled-data controller with holder is developed so as to globally stabilize the considered nonlinear uncertain system. For the overall hybrid control system, a Lyapunov function is finally constructed, and the global exponential stability (GES) is guaranteed by using a small-gain argument and feedback domination method. Different from existing results [21]- [24], where the proposed continuous-discrete observer is separately designed from the controller construction for nonlinear systems, this article, however, couples the predictor-based observer construction together with the controller design, instead of designing the observer and controller separately. Moreover, the precise knowledge of system model that was employed in [21]- [24] is no longer needed in this article. In [25], the predictor-based state feedback was considered to handle the global stabilization problem of nonlinear systems, based on the assumptions that the nonlinear system considered is locally Lipschitz and the full information of state needs to be known. In contrast with [25], the main advantages of this work are that the proposed control method does not need to know the information of full state and the precise knowledge of system model, and the global/local Lipschitz condition on the nonlinearities is no longer required.
As a conclusion, this article has the following three main contributions.
1) The proposed method in this article does not need to know the precise knowledge of the model nonlinearities in contrast with [21]- [25]. To handle the non-Lipschitz nonlinear uncertainties emerged in the considered nonlinear systems, the article employs a scaling gain in both the observer and controller designs by using the feedback domination technique.
2) The predictor-based continuous-discrete observer and controller proposed in the article are designed in a coupling manner, instead of using the separate design method.
3) The stability analysis tools shown in [21]- [25] cannot be directly utilized to solve the global stabilization problem considered in the work, since the aforementioned results did not consider non-Lipschitz nonlinear uncertainties or assumed to know the full state information. To analyze the GES of the resultant overall closed-loop control system, the article develops a new Lyapunov stability analysis with the help of feedback domination and small-gain argument.
The rest of this article is organized as follows. Sections II and III show the problem formulation and control method design, respectively. Section IV gives the rigorous stability analysis and some sufficient conditions to guarantee the GES of the resultant hybrid control system. In Sections V and VI, we present the practical example simulation and conclusions, respectively.

A. Notations
The sets of nonnegative integers and real numbers are represented by N and R, respectively. For a given matrix A, the transpose of A is represented by A T . I denotes the identity matrix with appropriate dimensions. || · || stands for the 2-norm. For a symmetric matrix Q, its maximum and minimum eigenvalues are defined as λ M (Q) and λ m (Q). Let S ∈ R denote the nonempty set, we call f ∈ C 0 (S; R n ) if f : S → R n is a continuous function.

B. Problem Description
In this article, we consider the following nonlinear uncertain systems: . , x n (t)] T ∈ R n is the system state and u(t) ∈ R is the system control input. y(t) andȳ(t) are the delay-free measurement output and delayed measurement output, respectively.
is the unknown nonlinear uncertainty.
In this article, we suppose that the nonlinear uncertain system (1) is forward complete (see [22]), which ensures that the corresponding solution of (1) exists in the time interval t ∈ [0, ∞) for every initial condition and every measurable locally essentially bounded input signal. In addition, we assume that the nonlinear uncertainty is restricted by the following condition.
Assumption 1: Given a scaling gain L > 1, there exist two constants c ≥ 0 and α ∈ [0, 1) such that for i = 1, . . . , n, the growth condition holds. Remark 1: The nonlinear functions satisfying Assumption 1 cover some nonsmooth ones as special cases, such as 2 , and the global/local Lipschitz condition is no longer required in this article. In addition, the nonlinear restricting condition is quite common in many existing works on nonlinear control systems (see [8], [9], and [12]). Since L > 1, inequality (2) leads to Hence, it can be concluded that the condition given in Assumption 1 is more general than the nonlinear restricting condition (2). For system (1), the positive constant T denotes the sampling period, and the sampled-data controller u(t) with zero-order holder updates at a constant period T . Due to the existence of the output delay, only the delayed sampled-dataȳ(kT ) = x 1 (kT − τ ), k ∈ N, is accessible in the controller design. The aim of this article is to develop a new global sampled-data control law via output feedback, such that the nonlinear uncertain system (1) can achieve GES.

III. PREDICTOR-BASED SAMPLED-DATA OUTPUT FEEDBACK CONTROL LAW DESIGN
To begin with, we define a scaling gain L > 1 for the purpose of dominating the nonlinear uncertainties in system (1).
where the triple (A, B, C) has the cascade form with compatible dimensions, and For system (4), the initial condition z(s) = z 0 (s) ∈ C 0 ([−τ, 0]; R n ) can be obtained by the change of coordinates. The novel output predictor-based continuous-discrete observer is designed for system (4) aṡ T is the observer gain, y p denotes the predicted value of the delay-free output y,ȳ p designates the predicted value of the continuous-time delayed outputȳ over the intersample time interval [kT, (k + 1)T ), andẑ 0 (s) ∈ C 0 ([−τ, 0]; R n ) denotes the initial condition for the proposed observer.
To make it clear, we give some more detailed explanations for the proposed observer (5) by the following. 1) The dynamics (5c) is an intersample output predictor, which is designed for obtaining the predicted valueȳ p of the delayed output y over the intersample time interval [kT, (k + 1)T ). The dynamics (5c) is reinitialized whenever the delayed sampled-data output y(kT ) is available.
2) The dynamics (5b) is a new output predictor, which uses the predicted valueȳ p of the continuous-time delayed outputȳ and provides the predicted value y p of the delay-free output y.
3) The dynamics (5a) is a high-gain observer for estimating the variable z by using the predicted value y p of y. For system (4), in t ∈ [kT, (k + 1)T ), we design the sampled-data controller with zero-order holder as with the feedback control gain K = [k 1 , . . . , k n ]. Remark 2: In the proposed observer (5), the intersample output predictor (5c) and predictor (5b) are utilized to compensate the influences of the sampling and output delay, respectively. In the absence of the sampling and output delay, it can be obtained from (5) that y =ȳ =ȳ p = y p , which indicates that the proposed observer (5) is changed into the traditional continuous-time one.
Remark 3: It should be underlined that the state estimateẑ in the proposed observer (5) is continuous, even though the predicted valuē y p is discontinuous since at every sampling time instant t = kT , the predictor (5c) needs to be reinitialized.
Remark 4: When the output delay τ = 0, the proposed observer (5) is changed intȯ It can be seen that the intersample output dynamics is considered in a manner of predicting the output over the intersample time intervals for the proposed observer. Such a design method is totally different from the existing works [9], [10], where the intersample output dynamics is omitted since the sampled output y(kT ) is simply utilized by holding it constant over the intersample time intervals.
Remark 5: It can be observed that the proposed control method does not require to know the precise knowledge of nonlinearities and information of full state, which is totally different from the existing results developed based on known accurate model in [21]- [25]. Moreover, different from the existing results [21]- [24], where the proposed continuous-discrete observer is separately designed from the controller construction for nonlinear systems, this article, however, couples the predictor-based observer construction together with the controller design, instead of designing the observer and controller separately.

IV. MAIN RESULTS
Letting e = [e 1 , . . . , e n ] T = z −ẑ, e y = y − y p , andē y =ȳ −ȳ p , one combines (4) and (5) together and gets the estimation error system asė Substituting the proposed controller (6) into system (4), we get the following: Define ω = [z T , e T ] T and With the help of the abovementioned definitions and combining (7) and (8), one obtains the overall hybrid control system aṡ where (t) : [0, ∞) → R 2n is described as follows: Obviously, the control gains K and H can be selected to make A − BK and A + HC Hurwitz, which indicates that the matrix Π 1 is also Hurwitz according to its definition. Let the matrix P = P T ∈ R 2n×2n denotes the positive definite solution of Π T 1 P + P Π 1 = −I. In the hybrid control system (9), the error term e y is induced by the sampling and output delay, and the error term is caused by the sampling. The estimates of these two terms play an important role for the main results in this article. To this end, we introduce two important lemmas about the estimates of e y and as follows.
By using the result presented in Lemma 1, the estimate of (t) can be obtained by the following conclusion.
Proof: For the detailed proof, see Appendix B.
In Lemmas 1 and 2, the parameter c 0 can be viewed as an auxiliary parameter. The aim of defining the parameter c 0 is to deduce the inequalities (11) and (12), which play a key role in the proof for the main results of this article.
Theorem 1: Under Assumption 1, consider system (1) with the proposed control methods (5) and (6). For every initial conditions , the overall closed-loop control system (9) can achieve GES if the positive constant c 0 used in Lemmas 1 and 2 is selected such that c 0 < c 4 , and where and the parameters c 1 , c 2 , and c 3 have been given in the statements of Lemmas 1 and 2.
Remark 6: In Theorem 1, it should be pointed out that the sufficient conditions L > 2||P ||c 2 , T < 1 c 0 ln( c 0 c 3 + 1), and c 0 < c 4 can be achieved by choosing appropriate sampling period T and high gain L. Moreover, by the definition of λ 2 in (13), we have that λ 2 can be regarded as a continuous function, which has two arguments T and τ , i.e., λ 2 (T, τ ) : Obviously, λ 2 (0, 0) = 0. Thus, λ 2 < 1 can be guaranteed when T and τ are not so large.
Remark 7: It should be stated that the stability conditions presented in Theorem 1 are conservative to some extent, since they are only sufficient rather than necessary to guarantee GES of the overall closed-loop control system (9). The development of sufficient and necessary stability conditions for these kind of robust control problems is quite challenging and might be actually infeasible (see [5], [26], and [10]). In order to mitigate the conservativeness, in simulation and/or experimental studies, it is suggested to tune the control parameters in a manner of trial and error to get the desirable control performance.
By Theorem 1, the conclusions on the prediction errorsē y , e y , and are obtained as follows. Corollary 1: If the overall closed-loop control system (9) can achieve GES, then the prediction errorsē y , e y , and globally exponentially converge to the origin as time goes to infinity.
Proof: By the proof of Lemma 1, one has Note that e c 0 t ||ē y (t)|| ≤ sup 0≤s≤t (e c 0 s ||ē y (s)||). From (24) which means that the prediction errorē y globally exponentially converges to the origin as t → ∞. Similarly, with the proof given previously, it is easily deduced that both e y and globally exponentially converge to the origin as time goes to infinity, and thus, the rest of proof procedure is omitted here.

V. SIMULATION RESULTS
In this section, a practical example of a single-link robot arm system is employed in order to verify the effectiveness of the proposed control method. Derived from [26], [27], and by a coordinate transformation, the robot arm system is modeled as the following fourth-order differential equation: where K m , N , J 1 , J 2 , d, m, and g are known parameters. The viscous friction coefficients F i (t), i = 1, 2, are unknown but bounded.
The proposed control method for system (27) is designed as follows: high-gain observer: output predictor: In this simulation, the parameters are set as follows: the high gain L = 1. . After simple calculation, one can obtain that the scaling gain L is less than 2||P ||c 2 , thus c 4 = L−2||P ||c 2 2λ M (P ) < 0, which means that it is impossible to select a positive constant c 0 satisfying c 0 < c 4 and find appropriate sampling period T and output delay τ satisfying the stability condition (13) given in Theorem 1. However, as stated in Remark 7, the stability conditions presented in Theorem 1 are only sufficient rather than necessary. In this simulation, we select the control parameters in a manner of trial and error to get the desirable control performance. It shows that both the state and estimation errors asymptotically converge to the origin when t approaches infinity. The prediction errors e y andē y are plotted in Fig. 2. From Fig. 2, we can observe that the prediction errorē y =ȳ −ȳ p is equal to zero at each sampling time instant, sinceȳ p is reset to be equal toȳ whenever the delayed sampled-data outputȳ is available.

VI. CONCLUSION
This article has investigated the problem of output feedback based global sampled-data stabilization for a class of nonlinear uncertain systems with sampled and delayed output. By using the techniques of feedback domination and prediction, a novel coupled design of predictor-based observer and controller has been proposed to ensure GES of the considered nonlinear uncertain systems. The prediction technique used in this article can effectively predict the current output so as to compensate the undesirable influences of the sampling and output delay. The main benefit of this work is that the proposed control method does not need to know the full state information and the accurate system model information. The practical example results have shown the effectiveness of the proposed control method. Regarding the future work, we will consider the global output feedback stabilization for nonlinear systems with both time-varying input and output delays.

A. Proof of Lemma 1
For a given positive constant c 0 , the proof can be split into two main steps. The estimate of sup 0≤s≤t (e c 0 s ||ē y (s)||) is deduced, and the estimate of sup 0≤s≤t (e c 0 s ||e y (s)||) is synthesized. The detailed proof is depicted as follows.
Step 1: With (9) in mind, we obtain the following: where δ = √ n + 1 max(L, cL α ) has been given in the statement of Lemma 1.