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

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 developedwith only one dynamic gain to guarantee the boundedness of closed-loop system states and the achievement of global disturbance attenuation in the L2-gain sense.


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][2][3][4][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].
Assumption 1.For  = 1, ⋅ ⋅ ⋅ , , there is an unknown positive constant  and a known positive integer  ≥ 1 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][7][8][9][10][11] and the references therein.Specifically, for nonlinear time-delay systems with unknown polynomialof-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][13][14][15][16][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-ofoutput 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.

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;  > 0 denotes that matrix  is a positive definite matrix;  min () denotes the minimal eigenvalue of the symmetric matrix .For any vector , ‖‖ 1 and ‖‖ denote its 1-norm and 2-norm, respectively.Clearly, ‖‖ ≤ ‖‖ 1 ≤ √‖‖, where  is the dimension of .diag{ 1 , ⋅ ⋅ ⋅ ,   } denotes  ×  diagonal matrix whose element (, ) is   and others are zero.  denotes the -dimensional identity matrix. 2 [0, ) and  ∞ [0, ) denote the appropriate dimension space of square integrable functions and the appropriate dimension space of uniformly bounded functions on [0, ), respectively, where 0 <  ≤ +∞.

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 () = 0 or () ∈  2 [0, +∞) ∩  ∞ [0, +∞), 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.
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.

The Design of Observer and Controller for System (1).
Motivated by [14][15][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 x = [x 1 , ⋅ ⋅ ⋅ , x ] ⊤ ∈   and x+1 ∈  are the observer states. > 0 is a design parameter and will be first selected such that 0 <  < 1/2,  is the same as in Assumption 1.
Then, according to Lemma 2, a set ( 1 , , , , ) 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. 1 and  2 are positive design parameters to be determined.The dynamic gain () has the following properties for all  ≥ 0: Remark 7. Since the existence of the input matching uncertainty ] in system (1), the introduction of x+1 in the observer ( 9) is indispensable to compensate the input matching uncertainty ].In Theorem 8, we will prove that x+1 ultimately converge to the actual value of the input matching uncertainty ].

Stability and Convergence Analysis.
We state the main result in this paper.
Proof.It is observed that the right-hand side of the closedloop system consisting of ( 1), ( 9), and ( 10) is continuous and locally Lipschitz in (, x, x+1 , ); hence, the closed-loop system has a unique solution on the maximal interval [0,   ) with 0 <   ≤ +∞.Next, we divide the proof into two steps.
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 (), x(), x+1 (), and () would tend to infinity at  =   .However, (), x(), x+1 (), and () are bounded on the maximal interval [0,   ) and hence also bounded at  =   due to the continuity of the solution; this is a contradiction.Using (21), for any pregiven small real number  > 0, Integrating both sides of (39) leads to, ∀ ≥ 0, Obviously, (⋅) is a nonnegative bounded function.Then the global disturbance attenuation of the closedloop system is achieved in the  2 -gain sense.

Simulation Example
Consider a simple system where ] is an unknown constant representing the input matching uncertainty, and let

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][21][22][23][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.
Figure 1   demonstrates the effectiveness of the control scheme.