Semiglobal Stabilization via Output-Feedback for a Class of Nontriangular Nonlinear Systems with an Unknown Coefficient

This paper is devoted to the semiglobal stabilization via output-feedback for a class of uncertain nonlinear systems. Remark that the systems in question contain an unknown control coefficient which inherently depends on the system output and allow largerthan-two order growing unmeasurable states which is the obstruction of global stabilization via output-feedback. By introducing a recursive reduced-order observer and combining with saturated state estimate, a desired output-feedback controller is explicitly constructed for the systems. Under the appropriate choice of design parameters, the controller can make the closed-loop system semiglobally attractive and locally exponentially stable at the origin. A simulation example is provided to illustrate the effectiveness of the proposed approach.

In this paper, a semiglobal stabilization scheme via output-feedback is proposed for uncertain nontriangular nonlinear system (1) with serious nonlinearities and the unknown control coefficient depending on the system output.Specifically, a state-feedback controller is first constructed for a nominal system (where   (⋅) ≡ 0,  = 1, . . ., ), which ensures that the closed-loop system (corresponding to the nominal system) is globally exponentially stable.Then, a recursive reduced-order observer is introduced to recover the unmeasurable states of system (1).Based on these and combining with saturated state estimate [3,4], a semiglobal output-feedback stabilizer is explicitly constructed for system (1).By appropriately choosing design parameters, the controller can guarantee that the closed-loop system (corresponding to system (1)) is semiglobally attractive and locally exponentially stable at the origin.
The remainder of this paper is organized as follows.Section 2 presents semiglobal output-feedback control design for system (1).Section 3 provides the main results and the rigorous performance analysis of the closed-loop system.Section 4 gives a numerical example to illustrate effectiveness of the proposed method.Section 5 addresses some concluding remarks.This paper ends with an appendix that collects two proofs of important propositions.

Semiglobal Output-Feedback Control
The section is to design a semiglobal stabilizer via outputfeedback for system (1) under Assumptions 1 and 2.
To achieve this, we introduce the coordinate transformation which changes system (1) into the following: where  = [ 1 , . . .,   ] T and  ≥ 1 is a design parameter to be determined later.
Moreover, it is worth stressing that the semiglobal stabilization of system (1) is implied by that of system (7).In the sequel, we turn to the controller design of system (7).We first establish the following proposition, which gives a state-feedback controller for system (7) without considering the nonlinearities φ (⋅)'s.

Proposition 3. Under Assumption 1, consider the following nominal system:
There exist positive constants   ,  = 1, . . ., , such that system ( 9) is globally exponentially stabilized by the state-feedback controller: Then, we rewrite system (9) as where Noting that  1 −  1  1 is a Hurwitz matrix, there exists a positive definite matrix such that Then, we define the Lyapunov function whose derivative along system ( 11) is as follows: Using Young's inequality, we have where Substituting the above estimation into (15) yields Choose   ≥  −1 +  + 1 and construct the state feedback controller Then, by Assumption 1, we derive which, together with (14), implies that the closed-loop system consisting of ( 9) and ( 10) is globally exponentially stable.This completes the proof.

Main Results
This section is devoted to the performance analysis of the closed-loop system consisting of ( 7), (20), and (21) and summarizes the main results of this paper.
In view of (25) and (26), we recursively determine the observer gains ℓ  's as follows: such that where ℓ is a positive constant to be determined later and   () = α  2  + ∑ −1 =2 α (ℓ +1 , . . ., ℓ  ) 2  .Now, we are ready to address the main results of this paper, which are summarized in the following theorem.

Concluding Remarks
In this paper, the semiglobal stabilization via output-feedback has been investigated for a class of uncertain nontriangular nonlinear systems.Essentially different from the existing related works, the control coefficient of the system is unknown and inherently depends on the system output, and, hence, the scope of the nonlinear systems is considerably broadened.By introducing a recursive reduced-order observer and combining with saturated state estimate, a semiglobal output-feedback controller has been constructed for the uncertain system.Under the appropriate choice of design parameters, the controller can guarantee that the closed-loop system achieves semiglobal attractivity and locally exponential stability.Along this direction, another interesting research problem is how to design a semiglobal finite-time stabilizer via output-feedback for system (1).

Appendices
The appendix provides the rigorous proofs of Propositions 4 and 5, which are collected here for the sake of compactness.

A. The Proof of Proposition 4
By (7)   In what follows, we estimate the right-hand side of (A.1).

B. The Proof of Proposition 5
By system (24), we have