Abstract
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 larger-than-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.
1. Introduction
In this paper, we consider the semiglobal stabilization via output-feedback for a class of nonlinear systems described by where is the system state with the initial value ; and are the input and output of the system, respectively; , , and are unknown but continuous functions, called nonlinearities and control coefficient of the system, respectively. In what follows, suppose that only the output is measurable.
Rigorously speaking, the control objective of this paper is, for any given constant (may be arbitrarily large), to seek a dynamic output-feedback controller for system (1) as follows: where and are continuous functions satisfying and , such that the closed-loop system is (i)semiglobally attractive; that is, by starting from the compact set , all the trajectories of the closed-loop system converge to the origin, where and ;(ii)locally exponentially stable; that is, the closed-loop system is locally exponentially stable at the origin .
As in [1, 2], we introduce the dilation with being any positive constant. Based on this, the following assumptions are made on the nonlinearities and the control coefficient of system (1).
Assumption 1. There exist known positive continuous functions and such that, for any ,
Assumption 2. There exist a known constant and known nonnegative continuous function , , such that, for any ,
For nonlinear systems, semiglobal stabilization via output-feedback has attracted a great deal of attention (see, e.g., [3–13] and references therein), since it has a control objective meeting the needs of practical application and, compared to the global case, it requires rather weak restrictions on the system nonlinearities. We would like to stress that our problem is nontrivial and rather difficult to solve, which is mainly due to the generality of Assumptions 1 and 2. On the one hand, system (1) allows additional unknowns and nonlinearities in the control coefficient. Assumption 1 shows that the control coefficient of system (1) is unknown and inherently depends on the system output, essentially different from [7–10, 14–16] where the control coefficients are known/unknown constants or known functions of the system output. On the other hand, system (1) can be of nontriangular structure and has inherent nonlinearities. Assumption 2 indicates that, for any , inherently depends on all the states, rather than merely on like [8, 10, 11]. Also, due to the presence of ’s, system (1) allows more serious nonlinearities than those in [10, 11, 17]. Particularly, in [10], the system nonlinearities ’s are required to satisfy where , , and , , are known nonnegative smooth functions and . This is a special case of Assumption 2 with , . Therefore, one can see that the nonlinearities ’s of system (1) permit larger-than-two order growing unmeasurable states, which is the obstruction of global stabilization via output-feedback.
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 , ), 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.
2. Semiglobal Output-Feedback Control
The section is to design a semiglobal stabilizer via output-feedback for system (1) under Assumptions 1 and 2.
To achieve this, we introduce the coordinate transformationwhich changes system (1) into the following: where and is a design parameter to be determined later.
For simplicity, we denote and , . By Assumption 2, it can be verified that, for ,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 ’s.
Proposition 3. Under Assumption 1, consider the following nominal system:There exist positive constants , , such that system (9) is globally exponentially stabilized by the state-feedback controller:
Proof. Choose constants , , such that polynomial is Hurwitz. Then, we rewrite system (9) aswhere , ,Noting that is a Hurwitz matrix, there exists a positive definite matrix such thatThen, we define the Lyapunov functionwhose derivative along system (11) is as follows:Using Young’s inequality, we have where . Substituting the above estimation into (15) yields Choose and construct the state feedback controller Then, by Assumption 1, we derivewhich, together with (14), implies that the closed-loop system consisting of (9) and (10) is globally exponentially stable.
This completes the proof.
Based on controller (10) of nominal system (9), we construct a desired output-feedback controller for system (7). Motivated by [8, 18, 19], we introduce the following recursive reduced-order observer to recover unmeasurable states , , of system (7): where and , , are the gains to be determined later. For the later use, we denote and .
Furthermore, using saturated state estimate, we construct the following output-feedback controller for system (7):where , , are the same as those in Proposition 3; the saturation function is defined as with , , and standing for the -norm of vector .
3. 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.
Define , , and denote . Then, by (7) and (20), we deduce
Define the Lyapunov function and choose the same Lyapunov function used in Proposition 3. On the set , there hold Propositions 4 and 5 (whose detailed proofs are given in Appendices A and B, resp.), which will play a key role in the later performance analysis.
Proposition 4. For system (7) on the set , there exist constants and , such thatwhere is a constant independent of ’s and , , are certain polynomial functions of their arguments.
Proposition 5. For system (24) on the set and constant given in Proposition 4, there holdswhere is a constant independent of ’s and , , are certain polynomial functions of their arguments.
In view of (25) and (26), we recursively determine the observer gains ’s as follows:such thatwhere is a positive constant to be determined later and .
Now, we are ready to address the main results of this paper, which are summarized in the following theorem.
Theorem 6. Consider system (1) under Assumptions 1 and 2. For any given constant (may be arbitrarily large), there exist appropriate and , , depending on , such that the closed-loop system, consisting of (1), (20), and (21), is locally exponentially stable, and, by starting from the given compact set , all the trajectories of the closed-loop system converge to the origin.
Proof. Motivated by [5, 6, 13], for the closed-loop system consisting of (7), (20), and (21), we choose Lyapunov function where . From the definitions of and , we see that is positive definite with respect to . Moreover, by (25) and (28), we obtain that, on the set , From , , it follows that which implies that, on the set ,From the choice of ’s, we see that ’s are polynomial functions of . Then, we can find to be sufficiently large such thatwhich, together with (32), implies that, on the set ,Moreover, define the compact set . Then, it is easy to verify the following relation:We now estimate the last term in the right-hand side of (34) on the compact set . By (8) and Young’s inequality and noting the boundedness of ’s and on , there exist positive constants and depending on ’s, such that, on the set , Substituting this into (34), we obtain that, on the set , By , we can choose satisfyingThen, we obtain that, on the set ,By the definitions of ’s and , we derive on , which implies that, by starting from , remains in the compact set , and By and the continuity of on , there exists a neighborhood of the origin, such that for all . Then, by (39), we obtain that, on the set ,By the definition of and , we deduce that, on the set , with some positive constants and . Substituting this into (41) and noting the definitions of , we have that, on the set , with positive constant . Therefore, the closed-loop system consisting of (7), (20), and (21) is locally exponentially stable.
By (6), (35) and noting , there holds Furthermore, by the invertibility of (6), we arrive at the fact that the closed-loop system, consisting of (1), (20), and (21), is locally exponentially stable, and, by starting from , the trajectories converge to the origin.
This completes the proof.
4. A Simulation Example
In this section, we consider the following nonlinear system to illustrate the correctness and effectiveness of the theoretical results:with . It can be verified that system (45) satisfies Assumption 2 with , , and .
According to the design procedure of the state feedback controller, we choose and and then, by solving the matrix inequality (13), yield Furthermore, by Proposition 3, we choose .
Let . Then, by the above , (14), and (23), we choose . Moreover, from (33) and (38), we choose and and, in terms of (27), recursively determine and . Thus, by the design procedure in Section 2, we construct the following observer: and design the following output-feedback controller:
Let , and let the initial values of the states be and . Using MATLAB, Figures 1–3 are obtained to exhibit the trajectories of the closed-loop system states and the controller, from which the effectiveness of the design method is indeed demonstrated.
5. 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
In what follows, we estimate the right-hand side of (A.1). By , we obtainwhere .
By the definitions of ’s and ’s, we havewhich impliesUsing (10) and (21) yields where . By this and (A.4) and noting that, for any , on , we can find a positive constant such that, on , where , . Then, by Assumption 1, we derive that, on ,where , , and with .
Moreover, by (8), there exists a positive constant independent of ’s, such that, on ,
By substituting (A.2), (A.7), and (A.8) into (A.1), we conclude that Proposition 4 holds.
B. The Proof of Proposition 5
By system (24), we haveBy (A.4) and Young’s inequality, the second and third terms in the right-hand side of (B.1) satisfywhere and , .
By (A.3) and (A.4), we have that which, together with Young’s inequality and Assumption 1, implieswhere , , , and , .
Substituting (B.2) and (B.4) into (B.1), we conclude that (26) holds with , and , .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundations of China (61325016, 61273084, and 61233014), the National Science Foundation of Distinguished Young Scholar of Shandong Province of China (JQ200919), and the Independent Innovation Foundation of Shandong University of China (2012JC014).