Research Article | Open Access
An Analysis and Design for Nonlinear Quadratic Systems Subject to Nested Saturation
This paper considers the stability problem for nonlinear quadratic systems with nested saturation input. The interesting treatment method proposed to nested saturation here is put into use a well-established linear differential control tool. And the new conclusions include the existing conclusion on this issue and have less conservatism than before. Simulation example illustrates the effectiveness of the established methodologies.
In the past decades, there has been significant interest in the study of the quadratic systems due to such systems’ widely present from engineering systems to economic phenomena [1–5]. To some extent, such systems can accurately describe the interaction dynamics of various species, for example, in enzyme kinetics  or population models . In many practical control applications, saturation nonlinearities almost emerge everywhere in the real control process. In particular, the input saturation is an impactive source resulting in instability of the control systems. Numerous researches on the various control problems containing saturation nonlinearity have been conducted; for example, see [8, 9] and the references therein. In general, we have two primary approaches to deal with saturation nonlinearity; one is to deal with the saturation nonlinearity as a local sector bound nonlinearity with different multipliers [10, 11]; the other [12, 13] is to conduct the saturation nonlinearity as a polytopic representation, which reduces less conservatism than the first one.
Over the years, several papers have focused on the study of the quadratic systems subject to saturation input. The region of attraction (RA) is an interesting issue in the stability analysis of nonlinear quadratic systems. A Lyapunov-based procedure is presented in  to compute an ellipsoidal estimate of the RA of a second-order nonlinear systems containing either linear and quadratic or linear and cubic terms. More recently, the problem of estimating the RA of quadratic systems subject to saturation input has been solved as a transformative linear matrix inequalities (LMIs) feasibility problem [15, 16], although in practice it is sometimes fairly difficult to construct an appropriate Lyapunov function. References [17, 18] consider linear systems subject to nested actuator saturation and extend the corresponding conditions.
Based on the Lyapunov function and a particular presentation for the quadratic terms, the purpose of this paper is in a sense to be made precise of the sufficient conditions for local stabilization for the quadratic systems subject to nested saturation input in terms of LMIs. Our research is provoked by the work of . It is clear that the sufficient conditions presented in  are special cases of our newly built treatment of nest saturation nonlinearity. Moreover, this paper is organized as follows. The system discussed is presented in detail in Section 2. Section 3 states the main results of the local stabilization for the quadratic systems in terms of LMIs. An example is given in Section 4 to illustrate the proposed methodologies, and Section 5 concludes the whole paper.
Notation. Throughout this paper, standard notation will be adopted. For a matrix , means that is a symmetric and (semi-) positive definite matrix, and . For a matrix , we define , where denotes the -norm of the vector . The symbol denotes , where is the transpose of ; denotes the th eigenvalue of , and denotes the eigenvalue set of matrix . If , and mean the maximal and minimal eigenvalues of , respectively. Moreover, denotes the th row of matrix , and denotes the th diagonal element of the diagonal matrix . means the Kronecker product of the pair of . In addition, we need some special notation to present the main results. For a given integer , we define , . It is easy to see that there are elements in the set . Denote the number of elements in by . Assume that ; then, for arbitrary , we can denote with .
Definition 1 (see ). Let with . Define as a matrix such that the th rows of are nonzero while the other rows are zeros, and define as a diagonal matrix such that the th diagonal elements are 1 and the others are zeros. Furthermore, define . For two integers and , one has
Definition 2 (see ). Let be a given vector, and denote the th element of by . The matrix is defined as if , for all .
2. Plant Analysis
Consider the quadratic system subject to nested saturated-input: where , is the control input, is a given matrix, and denotes the nested saturation nonlinearity of . Also , , and , . Define matrices as follows: where denotes the th row of matrix . The quadratic system (2) can be read as In the next section, we consider three types of and draw the corresponding conclusions for the quadratic systems (4). The following lemmas are essential for the development of our paper.
Lemma 3 (see ). Consider matrix , , and a vector such that . Every point on the boundary of an ellipsoid, , can be parameterized by where .
Lemma 4 (see ). Let be two arbitrary scalars; then
Lemma 6 (see ). For two vectors , if , then , where denotes the convex hull of a set.
3. Main Results
In this section, through the new treatment of saturation nonlinearity given in Lemma 5, we divide three cases to estimate the RA for the quadratic system (4) subject to nested input saturation with conditional control design.
At first, we consider the quadratic systems subject to conventional actuator saturation. For a saturation level , the standard saturation function is defined as follows: and it meets the following sufficient condition.
Proposition 7. If there exist a positive scalar and matrices and , for all , satisfying the following inequality: then the region is an estimate of the RA for the nonlinear quadratic system (4) with conditional actuator saturation (7).
Proof. Consider the quadratic Lyapunov function , where . Let , for all . Then by Lemma 5, the time derivative of along the trajectories of system (4) is given as
with . To guarantee that at each point of , it suffices to verify the following inequality:
One can write
Then one get
Thus if the inequality
holds, then inequality (10) is satisfied.
If the relation (8) is right, obtained from (14) by the Schur complement, it follows that (10) holds for every and satisfying (12). From Lemma 3, if (10) is satisfied, then, for every , we have . As is a convex set and from the fact that terms appear linearly in matrix , any point in , for all , also satisfies , and thus is an estimate of the RA for the quadratic system (4) with conventional actuator saturation (7). The proof is completed.
In the following, we consider the quadratic systems subject to diagonal nested actuator saturation: where . is the nested saturation input function as follows: in which , , are diagonal matrices and , , are feedback gain matrices.
To express concisely, we denote Then we propose a new result for the local stability of the quadratic system (15).
Theorem 8. If there exist a positive scalar and matrices , and , for all , defined in Definition 2, such that then the quadratic system (15) is locally asymptotically stable and is an estimate of the RA.
Proof. Consider the Lyapunov quadratic function with . Its time derivative along the trajectories of the system (15) is given by
where . We define with . From Lemma 4, we have
Repeating the above procedure and substituting (21) into (20) yield
Let be an arbitrary vector. Then it follows from (22) that
Assume that , for all . For , we obtain the following inequality from (23) and the property of : Therefore, if (18) is satisfied, we deduce from (20) and (24) that , for all , for all , where is a sufficiently small scalar; namely, , for all , is a contractively invariant set. The proof is completed.
Finally, we study the estimate of the RA of the quadratic system with general nested saturation as follows: where and in which , , and , , the matrices , , are not necessarily diagonal, and , , are not necessarily the same.
Theorem 9. If there exist a positive scalar and matrices and , , for all such that the following inequality holds: where then the quadratic system (25) is locally asymptotically stable for every initial condition belonging to the region .
Proof. Consider the Lyapunov quadratic function with . Its time derivative along the quadratic system (25) is given as
where, for , we define with . Let , . Then it follows from Lemma 4 that
Then, from the above inequalities, we have that
Repeating the above process, we finally obtain
Applying Lemma 3 and (27) for , each point on the boundary of , we obtain where and and satisfying (12). Thus if the following inequality holds then, from the Schur complement, inequality (27) is satisfyied for every and satisfying (12). The ellipsoid is therefore the estimate of the RA for the quadratic system (25).
Remark 10. In the case of , Theorem 9 is a set invariance condition on stability analysis for nonlinear quadratic systems subject to traditional saturation input. One can verify that condition (27) is equivalent to the set invariance condition of Proposition 1 presented in ; thus we can say that set invariance condition in Theorem 9 here contains the one given in .
4. Illustrative Example
In this section, we will present an example to show the effectiveness of the above approaches in Theorem 9. Consider a nonlinear quadratic system in  with : where the coefficient matrices are given as follows:
To measure the impact of the quadratic term of control law on the RA, we compare the value of obtained using the conditions from Theorem 9, resulting in a control law having gains and . The RA of such an equilibrium point obtained by applying Theorem 9 is given by corresponding to , , and . Figure 1 shows the estimate of the RA of this example with an initial condition .
In this paper, we have studied the stability of the quadratic systems subject to nested saturation through the polytopic representation of saturation nonlinearity. The interesting treatment of saturation nonlinearity here, which includes the linear differential conclusion, presents more universal results in the form of LMIs through the quadratic Lyapunov functions. Our results effectively reduce the conservatism of the previous study and contain the existing results. The future study will extend the treatment of nested saturation nonlinearity to other control problems and more general classes of the control systems.
This work is supported by the Graduate Innovation and Creativity Foundation of Jiangsu Province under Grant no. CXZZ12-0202.
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Copyright © 2013 Minsong Zhang and Fu Chen. 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.