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 awell-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.


Introduction
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][2][3][4][5].To some extent, such systems can accurately describe the interaction dynamics of various species, for example, in enzyme kinetics [6] or population models [7].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 [14] 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 [19].It is clear that the sufficient conditions presented in [4] 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.
Definition 1 (see [18]).Let S = { 1 ,  2 , . . .,  (S) } ∈ V  with 1 ≤  1 <  2 < ⋅ ⋅ ⋅ <  (S) ≤ .Define  S ∈  × as a matrix such that the   th rows of  S are nonzero while the other rows are zeros, and define  S as a diagonal matrix such that the   th diagonal elements are 1 and the others are zeros.Furthermore, define  − S =   −  S .For two integers  ≥ 1 and  ≥ 1, one has Definition 2 (see [18]).Let  ∈ W be a given vector, and denote the th element of  by   .The matrix H  ∈  × is defined as H   = 0 if   = 0, for all  ∈ Φ  .

Plant Analysis
Consider the quadratic system subject to nested saturatedinput: where () ∈   ,  =  is the control input,  ∈  × is a given matrix, and sat() denotes the nested saturation nonlinearity of .Also  ∈  × ,  ∈  × , and   ∈  × ,  = 1, . . ., .Define matrices A 0 ∈  × 2 as follows: where  () ∈  1× denotes the th row of matrix   ∈  × .The quadratic system (2) can be read as In the next section, we consider three types of sat(()) and draw the corresponding conclusions for the quadratic systems (4).The following lemmas are essential for the development of our paper.

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 (>  1 ), 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  > 0 ∈  × and  S ∈  × , for all S ∈ V  , satisfying the following inequality: then the region E() ⊆ L( S ) is an estimate of the RA for the nonlinear quadratic system (4) with conditional actuator saturation (7).
Then by Lemma 5, the time derivative of () along the trajectories of system ( 4) is given as with ‖V‖ 2 = 1.To guarantee that V() < 0 at each point of E(), it suffices to verify the following inequality: One can write where  > 0 and Then one get Thus if the inequality He ( ( +  − S  +  S )) holds, then inequality (10) is satisfied.
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  > 0 ∈  × , and H  ∈  × , for all  ∈ V, defined in Definition 2, such that then the quadratic system (15) is locally asymptotically stable and E() ⊆ L(H  ) is an estimate of the RA.
Proof.Consider the Lyapunov quadratic function () =    with  =   > 0. Its time derivative along the trajectories of the system ( 15) is given by where  ∈ Φ  \ {1}.We define From Lemma 4, we have Repeating the above procedure and substituting ( 21) into (20) yield Let  ∈ W be an arbitrary vector.Then it follows from ( 22) that Assume that  ∈ L(H  ), for all  ∈ W. For E() ⊆ L(H  ), we obtain the following inequality from (23) and the property of L(H  ): Therefore, if ( 18) is satisfied, we deduce from (20) and (24) that V() ≤ −‖‖ 2 , for all  ∈ L(H  ), for all  ∈ W, where  > 0 is a sufficiently small scalar; namely, E() ⊆ L(H  ), for all  ∈ W, is a contractively invariant set.The proof is completed.

Theorem 9.
If there exist a positive scalar  and matrices  =   > 0 ∈  × and ) such that the following inequality holds: where then the quadratic system (25) is locally asymptotically stable for every initial condition belonging to the region E() ⊆ L( S  ).
Proof.Consider the Lyapunov quadratic function () =    with  =   > 0. Its time derivative along the quadratic system (25) is given as where Then, from the above inequalities, we have that where Repeating the above process, we finally obtain where Applying Lemma 3 and (27) for  ∈ E, each point on the boundary of E, we obtain where  > 0 and  and V satisfying (12).Thus if the following inequality holds He ( ( + Ξ + A  X)) ≤ He ( ( + Ξ) then, from the Schur complement, inequality (27) is satisfyied for every  and V satisfying (12).The ellipsoid E() ⊆ L( S  ) is therefore the estimate of the RA for the quadratic system (25).
Remark 10.In the case of  = 1, 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 [4]; 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 [4] with  = 1: (39) To measure the impact of the quadratic term of control law on the RA, we compare the value of trace() obtained using the conditions from Theorem 9, resulting in a control law having gains  and  0 .The RA of such an equilibrium point obtained by applying Theorem 9 is given by corresponding to  = 0.0020,  = 0.1, and  = 0.01. Figure 1 shows the estimate of the RA of this example with an initial condition  0 = [0.2,0.1]  .

Conclusion
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.

2 Figure 1 :
Figure 1: Phase portrait of the system (38) with gains (40) and the estimate of the RA.