## Mathematical Theories and Applications for Nonlinear Control Systems

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# Disturbance Attenuation via Output Feedback for Nonlinear Time-Delay Systems with Input Matching Uncertainty

**Academic Editor:**Weihai Zhang

#### Abstract

This paper studies the problem of output feedback disturbance attenuation for a class of uncertain nonlinear systems with input matching uncertainty and unknown multiple time-varying delays, whose nonlinearities are bounded by unmeasured states multiplying unknown polynomial-of-output growth rate. By skillfully combining extended state observer, dynamic gain technique, and Lyapunov-Krasovskii theorem, a delay-independent output feedback controller can be developed with only one dynamic gain to guarantee the boundedness of closed-loop system states and the achievement of global disturbance attenuation in the -gain sense.

#### 1. Introduction

The problem of disturbance attenuation via output feedback for nonlinear systems is a relatively meaningful problem in control theory and applications. Compared with the stabilization control and tracking control, fewer results on output feedback disturbance attenuation design have been obtained until now, such as [1–5] and the references therein. It is worth mentioning that, for nonlinear systems with known polynomial-of-output growth rate, the problem of output feedback disturbance attenuation was studied in [5].

In this paper, we consider output feedback disturbance attenuation problem of uncertain nonlinear systems as follows: where , , and are the system state, control input, and output, respectively. is an unknown constant, representing the input matching uncertainty, and let for notational convenience. is a vector of continuous time-varying parameters belonging to an unknown bounded set. , represent the bounded time-varying delays satisfying for an unknown positive constant , and the initial condition is with and being a continuous function vector defined on . is disturbance satisfying . For , functions and function vectors are continuous in the first argument and locally Lipschitz with respect to the rest of variables.

*Assumption 1. *For , there is an unknown positive constant and a known positive integer such that

During the past decade years, the problem of global output feedback control for uncertain nonlinear or nonlinear time-delay systems with unknown growth rate has been extensively studied with the aid of the dynamic gain technique, and a series of interesting results have been obtained; see [6–11] and the references therein. Specifically, for nonlinear time-delay systems with unknown polynomial-of-output growth rate, [10] achieved the global output feedback stabilization based on only one dynamic gain.

However, these results do not consider the input matching uncertainty. In many practical control systems, since input matching uncertainty often causes instability or serious deterioration in the performance of systems, output feedback control of nonlinear systems with input matching uncertainty is an attractive topic in recent years; see [12–17] and the references therein. Reference [12] achieved global output feedback regulation of nonlinear systems with zero dynamics and input matching uncertainty, whose nonlinearities are bounded by unmeasured states multiplying known function of output growth rate. References [13, 14] investigated the problem of global adaptive output feedback stabilization of nonlinear systems with input matching uncertainty, whose uncertain nonlinearities only depend on system output. For a class of uncertain time-varying nonlinear systems with input matching uncertainty, whose nonlinearities are strictly restricted, [15] achieved global output feedback stabilization based on two dynamic gains. Lately, a compact design scheme for nonlinear systems with unknown polynomial-of-output growth rate and input matching uncertainty was proposed in [16] based on only one dynamic gain. Reference [17] studied the output tracking problem for a class of stochastic nonlinear systems with unstable modes.

As far as we know, the problem of output feedback disturbance attenuation of uncertain nonlinear systems with input matching uncertainty and unknown multiple time-varying delays, whose nonlinearities are bounded by unmeasured states multiplying unknown constant and polynomial-of-output growth rates, has not yet been considered until now. In this paper, we make an attempt to handle this interesting problem by skillfully combining extended state observer, dynamic gain technique, and Lyapunov-Krasovskii theorem.

Since there simultaneously exist three types of uncertainties in system (1) for the problem of disturbance attenuation, input matching uncertainty, two types of growth rates (unknown constant and polynomial-of-output growth rates), and unknown multiple time-varying delays, some essential technical difficulties to control design will be inevitably produced. (i) The observer in [5, 10] is inapplicable to systems of this paper due to the existence of input matching uncertainty, so a rather difficult work is how to construct a feasible observer. (ii) The analysis method in [16] is unsuitable due to the existence of unknown multiple time-varying delays; hence, another difficulty is how to give a new analysis method. This paper will focus on solving these two difficulties.

This paper is organized as follows. Section 2 gives preliminaries. In Section 3, the design and analysis of output feedback controller are presented, following a simulation example in Section 4. Section 5 concludes this paper.

#### 2. Mathematical Preliminaries

In this paper, the argument of function will be omitted whenever no confusion can arise from the context. , , and denote the set of real numbers, the set of all nonnegative real numbers, and the real -dimensional space, respectively. For any real vector or matrix , denotes its transpose; denotes that matrix is a positive definite matrix; denotes the minimal eigenvalue of the symmetric matrix . For any vector , and denote its 1-norm and 2-norm, respectively. Clearly, , where is the dimension of . denotes diagonal matrix whose element is and others are zero. denotes the -dimensional identity matrix. and denote the appropriate dimension space of square integrable functions and the appropriate dimension space of uniformly bounded functions on , respectively, where .

Lemma 2 (see [12]). *For any positive real number , there exist real number , symmetric positive definite matrices and , and column vectors and satisfying the following set of inequalities: where , and *

Lemma 3 (Young’s inequality). *Let real numbers and satisfy , then for any and any given positive number , .*

Lemma 4 (see [18]). *For and is a constant, then .*

Lemma 5 (Barbalat’s lemma, see [19]). *For a continuously differentiable function , if and for some , then .*

#### 3. Design and Analysis of Output Feedback Controller

##### 3.1. Control Objective of This Paper

The objective of this paper is to construct an output feedback controller for system (1) under Assumption 1 such that, by suitably choosing the design parameters,

(i) when or , all the states of the closed-loop system are bounded and the original system states and their corresponding observer states all converge to zero, and the estimation of the input matching uncertainty converges to its actual value.

(ii) when , for any pregiven small real number , the system output has the following property: where is a nonnegative bounded function.

*Remark 6. *Compared with the problem of disturbance attenuation of free-delay systems in [5], where is a known constant, and compared with the problem of stabilization control of time-delay systems in [10], where is an unknown time-varying delay; it is worth mentioning that none of the systems in [5, 10] take into account the input match uncertainty. Furthermore, compared with the problem of stabilization control of free-delay systems in [16] with input matching uncertainty, where is an unknown constant. This paper considers the problem of disturbance attenuation for the case in which there simultaneously exist input matching uncertainty, unknown polynomial-of-output growth rate, and unknown multiple time-varying delays; all these factors lead to some essential technical difficulties to control design of the more general systems in this paper.

##### 3.2. The Design of Observer and Controller for System (1)

Motivated by [14–16], we design the following extended state observer to rebuild the unmeasured states and estimate the input matching uncertainty and construct a coupled controller: with being a dynamic gain updated by where and are the observer states. is a design parameter and will be first selected such that , is the same as in Assumption 1. Then, according to Lemma 2, a set can be determined to satisfy the inequalities in Lemma 2, and the vectors and are selected as the gains of the extended state observer and controller, respectively. and are positive design parameters to be determined. The dynamic gain has the following properties for all :

*Remark 7. *Since the existence of the input matching uncertainty in system (1), the introduction of in the observer (9) is indispensable to compensate the input matching uncertainty . In Theorem 8, we will prove that ultimately converge to the actual value of the input matching uncertainty .

Introduce coordinate transformation By (1), (9), and (12), we obtain where , , , . Let and with . By (11), (13), Lemma 2, and the fact that and , we have By Lemma 3, it is easy to get By (2), (12), and the fact that is a monotone nondecreasing function for , Using (17) and Lemma 3, it follows that where and are unknown positive constants related to and .

Choose the Lyapunov-Krasovskii functional and select the design parameters and to satisfy Then, using (14)-(16), (18)-(20), and , we arrive at

##### 3.3. Stability and Convergence Analysis

We state the main result in this paper.

Theorem 8. *Consider system (1) satisfying Assumption 1, and under the output feedback controller (9) and (10), the closed-loop system consisting of (1), (9), and (10) achieves global disturbance attenuation in the -gain sense. Moreover, if or , then .*

*Proof. *It is observed that the right-hand side of the closed-loop system consisting of (1), (9), and (10) is continuous and locally Lipschitz in ; hence, the closed-loop system has a unique solution on the maximal interval with . Next, we divide the proof into two steps.*Step I* (the boundedness of , , and on )*(i) Boundedness of ** on *. We prove the boundedness of on by a contradiction argument. Suppose that is unbounded on ; note the monotone nondecreasing property of ; there holds . Hence, there is a finite time such that . Then, from (21), it follows that which, together with , implies that and are bounded on and By (12), Lemma 4, and the fact that , it is obvious that Then, by and the boundedness of and on , there is a finite time , such that on , which, together with (10) and (12), implies that This contradicts with (23). Thus, is bounded on and suppose with being a constant.*(ii) Boundedness of ** on *. By (11)-(12), (15)-(16), and (20), we obtain Integrating both sides of (26) with being a monotone nondecreasing function and , leads to, on , from which it follows that, , which implies that and are bounded on .*(iii) Boundedness of ** on *. To prove the boundedness of on , we introduce a new change of coordinate where the constant is with being a positive constant to be defined. By (13) and (29), we arrive at where , , and . Choose the Lyapunov function . By (11), (30), Lemma 2, and the fact that , Using (11), (12), (29), and Lemma 3, we obtain Similar to the proof of (18) with , one gets where , are unknown positive constants related to and .

Choose the Lyapunov-Krasovskii functional By (20), (26), (31)-(34), the definition of , and the fact that , we derive where is a positive constant. Similar to the proof of (27), integrating both sides of (35) yields, on , from which it follows that, , which together with implies that and are bounded on . From (29) with the fact that is a constant, we obtain and are bounded on .*Step II* (we prove that the closed-loop system consisting of (1), (9), and (10) achieves global disturbance attenuation in the -gain sense). From (12), it is easy to get When , by the boundedness of , , on and the fact that is a constant, it can be derived from (38) that , , and are bounded on . By (9), it follows that is also bounded on .

Then, we prove that . The conclusion follows again by a contradiction argument. Suppose , then would be the finite-escape time of the closed-loop system, which means that at least one component of , , , and would tend to infinity at . However, , , , and are bounded on the maximal interval and hence also bounded at due to the continuity of the solution; this is a contradiction.

means that , , , , are bounded on , which indicates that , , , and are bounded on , and , . Meanwhile, from (38) with , we obtain .

Using (21), for any pregiven small real number , Integrating both sides of (39) leads to, , choose . Obviously, is a nonnegative bounded function. Then the global disturbance attenuation of the closed-loop system is achieved in the -gain sense.

Moreover, if or , by the boundedness of all signals on , from (13), it is obvious that and are also bounded on . Then, by Lemma 5, we conclude that . Therefore, from (9) and (38) with the boundedness of , we deduce .

#### 4. Simulation Example

Consider a simple system where is an unknown constant representing the input matching uncertainty, and let . , are unknown constants, and and represent unknown time-varying delays with , being unknown constants. The system disturbance is ; obviously, . Since , Assumption 1 holds with , .

From Lemma 2, we choose , , , . Then, by (20), select and . According to Section 3, we get the output feedback controller

In the simulation, we choose , , and the initial values for , . Figure 1 demonstrates the effectiveness of the control scheme.

#### 5. Conclusions

By skillfully combining extended state observer, dynamic gain technique, and Lyapunov-Krasovskii theorem, the problem of output feedback disturbance attenuation for nonlinear systems with input matching uncertainty and unknown multiple time-varying delays is solved in this paper based on only one dynamic gain.

Some interesting problems still remained; e.g., (i) for system (1) with the unknown output function [20–24], can we design an output feedback controller? (ii) Another work is to consider more general input matching uncertainty such as an uncertain harmonic signal.

#### Data Availability

The data supporting the conclusions of this manuscript are some open access articles that have been properly cited, and the readers can easily obtain these articles to verify the conclusions, replicate the analysis, and conduct secondary analyses. Therefore, we do not create a publicly available data repository.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the Taishan Scholar Project of Shandong Province of China (no. ts201712040), National Natural Science Foundation of China (no. 61673242), and Shandong Provincial Natural Science Foundation of China (no. ZR2016FM10).

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#### Copyright

Copyright © 2018 Chao Guo and Kemei Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.