Disturbance Attenuation via Output Feedback for Uncertain Nonlinear Systems with Output and Input Depending Growth Rate

The problem of output feedback disturbance attenuation is investigated for a class of uncertain nonlinear systems.The uncertainties of the considered systems are bounded by unmeasured stateswith growth rate function of output and inputmultiplying an unknown constant. Based on a dynamic gain observer, an adaptive output feedback controller is proposed such that the states of the closedloop system are globally bounded, and the disturbance attenuation is achieved in the L 2 -gain sense. An example is provided to demonstrate the effectiveness of the proposed design scheme.


Introduction
The problem of output feedback stabilization is one of the important problems in the field of nonlinear control and has attracted much attention [1][2][3].In particular, the output feedback control design has received great attention for nonlinear systems with linearly bounded unmeasurable states in [4][5][6][7][8][9][10] recently.For the class of systems that are bounded by a low-triangular-type condition, when the growth rate is a polynomial function of output multiplying an unknown constant, the design of output feedback controller was proposed in [7].Furthermore, when the growth function depends polynomially on input and output, the problem of global output feedback regulation was investigated in [9].For feedforward nonlinear time-delay systems satisfying linear growth condition, the problem of global stabilization by state feedback and output feedback was studied in [10].
On the other hand, disturbance attenuation of nonlinear systems is a very meaningful problem for both control theory and applications.And the problem of almost disturbance decoupling for nonlinear systems has received considerable attention during the past decades.Several researchers have presented various approaches for the problems of disturbance attenuation of nonlinear systems with different forms and assumptions in [11][12][13][14][15].For a class of nonlinear systems depending on unmeasured states with an unknown constant or polynomial-of-output growth rate, the problems of adaptive disturbance attenuation via output feedback were considered in [13,15].However, up to now, for a class of feedforward uncertain nonlinear systems with linearly bounded unmeasurable states, the problem of global disturbance attenuation by output feedback has seldom been studied.
Motivated by [7,9,10,13,15], in this paper, we consider the problem of output feedback disturbance attenuation for a class of uncertain nonlinear systems.The main contribution of this paper lies in the following.(i) For a class of feedforward uncertain nonlinear systems with linearly bounded unmeasurable states, an adaptive output feedback controller is proposed such that the states of the closed-loop system are globally bounded, and the disturbance attenuation is achieved in the  2 -gain sense.(ii) The assumptions in [8,10] are relaxed; see Remark 2.
Notation 1.In this paper, R, R + , and R  denote the set of real numbers, the set of nonnegative real numbers, and the set of real -dimensional column vectors, respectively. denotes the identify matrix with appropriate dimension.For a vector

Problem Statement and Preliminaries
Consider a class of nonlinear systems that can be written in the following form: where Remark 2. An adaptive output feedback controller to globally stabilise system (1) satisfying (2) with ℎ() = 1 and  = () = 0 was proposed in [8].The problem of output feedback stabilization for system (1) satisfying (2), where  > 0 is a known constant and  = () = 0, was solved in [10].However, to the best of our knowledge, the problem of output feedback disturbance attenuation has not been investigated for system (1) satisfying (2).
In this paper, our objective is to design, under Assumption 1, an adaptive output feedback controller for system (1), such that (i) when () = 0, the state of system (1) converges to zero, and the other signals of the closed-loop system are bounded on [0, +∞); (ii) for every disturbance () ∈  2 [0, +∞) and any pregiven small real number  > 0, the output  has the following property, where (⋅) is a nonnegative bounded function.
To prove our main result, we need the following lemma.

Main Result
Theorem 4. Considering system (1) satisfying Assumption 1, we design the output feedback controller as follows: where ,   , and   ,  = 1, 2, . . .,  are the appropriately chosen parameters such that Lemma 3 holds.Then, the closed-loop system ( 1) and ( 6)-( 9) achieve global disturbance in the  2gain sense.Furthermore, if Proof.The proof process can be separated into the following three steps.
Step 1.The change of coordinates and Lyapunov functions.

Mathematical Problems in Engineering 3
Note that the dynamic gains  and  have the following properties from ( 8) and ( 9): Define the state transformation Then in the rescaled coordinates, the dynamics of  and  can be written as the following compact form: where , , , and  are given by Lemma 3, ]  , and we have  = −  .
Remark 7. From the proof procedure of Theorem 4, we see that the dynamic gains  and  are introduced to deal with the unknown growth rate  and the function ℎ(), respectively, and both are required.Remark 8.It is worth pointing out that, for any known constant  > 0 and any known continuous function () satisfying () ≥ ℎ(), if we define Ṁ = max {(1 + ||  ) 2 ()/ − /2, 0} /() in ( 8), Theorem 4 also holds.Moreover, when  is a sufficiently large constant, we can get the better state properties of the closed-loop system; that is, the values of , x in the transient phase are getting smaller, and the convergence to zero of  and x is getting faster when () ∈  2 [0, +∞) ∩  ∞ [0, +∞).

Simulation Example
Consider the following uncertain nonlinear system:  Remark 9.In [17], Jo et al. have shown that the nonlinear LLC resonant circuit system, through appropriate transformation, can be changed into system (42) with  1 () =  2 () =  3 () = 0. Considering that disturbance and the uncertainty, which are unavoidable, are frequently encountered in real engineering systems, we decide to adopt system (42) to verify our theoretical analysis.
Then, according to Remark 8 and Theorem 4, we design the observer dynamics and the output feedback controller for (42) as follows:

Conclusion
In this paper, we have studied the problem of output feedback disturbance attenuation for a class of uncertain nonlinear systems.By using a linear observer with two dynamic gains and introducing the transformation of coordinates, we propose an adaptive output feedback controller such that the states of the closed-loop system are globally bounded, and the  disturbance attenuation is achieved in the sense of  2 -gain.Furthermore, the system is globally asymptotically regulated when the system disturbance () ∈  2 [0, +∞)∩ ∞ [0, +∞).