An Analysis and Design for Nonlinear Quadratic Systems Subject to Nested Saturation

Abstract

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.

1. 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 [1–5]. 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.

Notation. Throughout this paper, standard notation will be adopted. For a matrix 𝑃, 𝑃>(≥)0 means that 𝑃 is a symmetric and (semi-) positive definite matrix, and ℰ(𝑃)={𝑥∈𝑅^𝑛∶𝑥^𝑇𝑃𝑥≤1}. For a matrix 𝐻∈𝑅^𝑚×𝑛, we define ℒ(𝐻)={𝑥∈𝑅^𝑛∶‖𝐻𝑥‖_∞≤1}, where ‖𝑣‖_∞ denotes the ∞-norm of the vector 𝑣. The symbol He(𝐴) denotes 𝐴^𝑇+𝐴, where 𝐴^𝑇 is the transpose of 𝐴; 𝜆_𝑖(𝐴) denotes the 𝑖th eigenvalue of 𝐴, and 𝜆(𝐴) denotes the eigenvalue set of matrix 𝐴. If 𝑃≥0, 𝜆_max(𝑃) and 𝜆_min(𝑃) mean the maximal and minimal eigenvalues of 𝑃, respectively. Moreover, 𝐹^𝑖 denotes the 𝑖th row of matrix 𝐹, and 𝐵^(𝑖) denotes the 𝑖th diagonal element of the diagonal matrix 𝐵. 𝐴⊗𝐵 means the Kronecker product of the pair of (𝐴,𝐵). In addition, we need some special notation to present the main results. For a given integer 𝑚≥1, we define Φ_𝑚={1,2,…,𝑚}, 𝒱_𝑚={𝒱_𝑠∶𝒱_𝑠⊂Φ}. It is easy to see that there are 2^𝑚 elements in the set 𝒱_𝑚. Denote the number of elements in 𝒮∈𝒱_𝑚 by 𝜎(𝒮). Assume that 𝜎(∅)=0; then, for arbitrary 𝒮∈𝒱_𝑚, we can denote 𝒮={𝑖₁,𝑖₂,…,𝑖_𝜎(𝒮)} with 1≤𝑖₁<𝑖₂<⋯<𝑖_𝜎(𝒮)≤𝑚 [18].

Definition 1 (see [18]). Let 𝒮={𝑖₁,𝑖₂,…,𝑖_𝜎(𝒮)}∈𝒱_𝑚 with 1≤𝑖₁<𝑖₂<⋯<𝑖_𝜎(𝒮)≤𝑚. Define 𝐻_𝒮∈𝑅^𝑚×𝑛 as a matrix such that the 𝑖_𝑗th rows of 𝐻_𝒮 are nonzero while the other rows are zeros, and define 𝐷_𝒮 as a diagonal matrix such that the 𝑖_𝑗th diagonal elements are 1 and the others are zeros. Furthermore, define 𝐷⁻_𝒮=𝐼_𝑚−𝐷_𝒮. For two integers 𝑚≥1 and 𝑞≥1, one has 𝒲=𝜔=𝜔₁𝜔₂⋯𝜔_𝑚^𝑇∶𝜔_𝑖∈0,1,…,𝑞. (1)

Definition 2 (see [18]). Let 𝜔∈𝒲 be a given vector, and denote the 𝑖th element of 𝜔 by 𝜔_𝑖. The matrix ℋ_𝜔∈𝑅^𝑚×𝑛 is defined as ℋ^𝑖_𝜔=0 if 𝜔_𝑖=0, for all 𝑖∈Φ_𝑚.

2. Plant Analysis

Consider the quadratic system subject to nested saturated-input: 𝑥̇(𝑡)(𝑡)+𝑥(𝑡)=𝐴𝑥^𝑇𝐴_𝑞1𝑥(𝑡)𝑥(𝑡)^𝑇𝐴_𝑞2𝑥(𝑡)⋮𝑥(𝑡)^𝑇𝐴_𝑞𝑛𝑥(𝑡)+𝐵sat(𝑢(𝑡)), (2) where 𝑥(𝑡)∈𝑅^𝑛, 𝑢=𝐹𝑥 is the control input, 𝐹∈𝑅^𝑚×𝑛 is a given matrix, and sat(𝑢) denotes the nested saturation nonlinearity of 𝑢. Also 𝐴∈𝑅^𝑛×𝑛, 𝐵∈𝑅^𝑛×𝑚, and 𝐴_𝑞𝑖∈𝑅^𝑛×𝑛, 𝑖=1,…,𝑛. Define matrices 𝒜_𝑞0∈𝑅^𝑛×𝑛2 as follows: 𝒜_𝑞0=𝐴_𝑞1(1)𝐴_𝑞1(2)⋯𝐴_𝑞1(𝑛)𝐴⋮⋮⋱⋮_𝑞𝑛(1)𝐴_𝑞𝑛(2)⋯𝐴_{𝑞𝑛(𝑛)}, (3) where 𝐴_{𝑞𝑖(𝑗)}∈𝑅^1×𝑛 denotes the 𝑗th row of matrix 𝐴_𝑞𝑖∈𝑅^𝑛×𝑛. The quadratic system (2) can be read as 𝑥̇(𝑡)=𝐴+𝒜_𝑞0𝐼_𝑛⊗𝑥(𝑡)𝑥(𝑡)+𝐵sat(𝑢(𝑡)). (4) In the next section, we consider three types of sat(𝑢(𝑡)) and draw the corresponding conclusions for the quadratic systems (4). The following lemmas are essential for the development of our paper.

Lemma 3 (see [15]). Consider matrix 𝑃∈𝑅^𝑛×𝑛, 𝑃=𝑃^𝑇>0, and a vector 𝑣 such that ‖𝑣‖=1. Every point on the boundary of an ellipsoid, 𝜕ℰ(𝑃)={𝑥∈𝑅^𝑛∶𝑥^𝑇𝑃𝑥=1}, can be parameterized by 𝑥=𝑃^−1/2𝑈𝑣 where 𝑈^𝑇𝑈=𝐼.

Lemma 4 (see [19]). Let 𝑎,𝑏∈𝑅 be two arbitrary scalars; then 𝑎sat(𝑏)≤max{𝑎𝑏,−|𝑎|}. (5)

Lemma 5 (see [18]). Let 𝐹∈𝑅^𝑛×𝑚 be a given matrix and 𝑢∈𝑅^𝑚 an arbitrary vector. If 𝑥∈ℒ(𝐻)_𝒮, for all 𝒮∈𝒱_𝑚, where 𝒮, 𝒱_𝑚, and 𝐻_𝒮∈𝑅^𝑚×𝑛 are defined in Definition 1, then 2𝑥^𝑇𝐹sat(𝑢)≤max_{𝒮∈𝒱𝑚}2𝑥^𝑇𝐹𝐷⁻_𝒮𝑢+𝑥^𝑇He𝐹𝐻_𝒮𝑥. (6)

Lemma 6 (see [18]). For two vectors 𝑢,𝑣∈𝑅^𝑚, if ‖𝑣‖_∞≤1, then sat(𝑢)∈co{𝐷⁻_𝒮𝑢_𝑖+𝐷_𝒮𝑣_𝑖,𝒮∈𝒱_𝑚}, where co{⋅} 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 𝜇(>𝜀₁), the standard saturation function is defined as follows: [sat𝑢(𝑡)]_𝑙≜[𝑢(𝑡)]_𝑙if[𝑢(𝑡)]_𝑙[𝑢(𝑡<𝜇,𝜇if)]_𝑙[𝑢(𝑡≥𝜇,−𝜇if)]_𝑙≤−𝜇, (7) and it meets the following sufficient condition.

Proposition 7. If there exist a positive scalar 𝜀 and matrices 𝑃>0∈𝑅^𝑛×𝑛 and 𝐻_𝒮∈𝑅^𝑚×𝑛, for all 𝒮∈𝒱_𝑚, satisfying the following inequality: 𝑃He𝐴+𝐵𝐷⁻_𝒮𝐹+𝐵𝐻_𝒮+𝜀2𝐼𝑃𝒜_𝑞0𝜀0−2𝐼_𝑛⊗𝑃<0, (8) then the region ℰ(𝑃)⊆ℒ(𝐻_𝒮) is an estimate of the RA for the nonlinear quadratic system (4) with conditional actuator saturation (7).

Proof. Consider the quadratic Lyapunov function 𝑉(𝑥)=𝑥^𝑇𝑃𝑥, where 𝑃=𝑃^𝑇>0. Let 𝑥∈ℒ(𝐻_𝒮), for all 𝒮∈𝒱_𝑚. Then by Lemma 5, the time derivative of 𝑉(𝑥) along the trajectories of system (4) is given as 𝑉̇(𝑥)=𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛(𝑡)+𝐼⊗𝑥_𝑛⊗𝑥(𝑡)^𝑇𝒜^𝑇_𝑞0𝑃𝑥+2𝑥^𝑇𝑃𝐵sat(𝐹𝑥)≤𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛(𝑡+𝐼⊗𝑥)_𝑛⊗𝑥(𝑡)^𝑇𝒜^𝑇_𝑞0𝑃𝑥+max_{𝒮∈𝒱𝑚}2𝑥^𝑇𝑃𝐵𝐷⁻_𝒮𝐹𝑥+𝑥^𝑇He𝑃𝐵𝐻_𝒮𝑥=𝑥^𝑇He𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃^−1/2𝐼_𝑛⊗𝑈𝐼_𝑛⊗𝑣𝑥+max_{𝒮∈𝒱𝑚}2𝑥^𝑇𝑃𝐵𝐷⁻_𝒮𝐹𝑥+𝑥^𝑇He𝑃𝐵𝐻_𝒮𝑥 (9) with ‖𝑣‖²=1. To guarantee that 𝑉̇(𝑥)<0 at each point of 𝜕ℰ(𝑃), it suffices to verify the following inequality: He𝑃𝐴+𝐵𝐷⁻_𝒮𝐹+𝐵𝐻_𝒮+𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃^−1/2𝐼_𝑛⊗𝑈𝐼_𝑛⊗𝑣<0. (10) One can write He𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃^−1/2𝐼_𝑛⊗𝑈𝐼_𝑛<1⊗𝑣𝜀𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃⁻¹𝒜^𝑇_𝑞0𝑃+𝜀𝐼_𝑛⊗𝑣^𝑇𝐼_𝑛⊗𝑈^𝑇𝐼_𝑛⊗𝑈𝐼_𝑛⊗𝑣, (11) where 𝜀>0 and 𝐼_𝑛⊗𝑈^𝑇𝐼_𝑛⊗𝑈=𝐼_𝑛2,𝐼_𝑛⊗𝑣^𝑇𝐼_𝑛⊗𝑣=𝐼_𝑛. (12) Then one get He𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃^−1/2𝐼_𝑛⊗𝑈𝐼_𝑛<1⊗𝑣𝜀𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃⁻¹𝒜^𝑇_𝑞0𝑃+𝜀𝐼. (13) Thus if the inequality He𝑃𝐴+𝐵𝐷⁻_𝒮𝐹+𝐵𝐻_𝒮+1𝜀𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃⁻¹𝒜^𝑇_𝑞0𝑃+𝜀𝐼<0 (14) 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 𝑈 and 𝑣 satisfying (12). From Lemma 3, if (10) is satisfied, then, for every 𝑥∈𝜕ℰ(𝑃), we have 𝑉̇(𝑥)<0. As ℰ(𝑃) is a convex set and from the fact that terms 𝑥=𝑃^−1/2𝑈𝑣 appear linearly in matrix He(𝑃(𝐴+𝐵𝐷⁻_𝒮𝐹+𝐵𝐻_𝒮)+𝑃𝒜_𝑞0(𝐼_𝑛⊗𝑥(𝑡))), any point in ℰ(𝑃)={𝑥∈𝑅^𝑛∶𝑥^𝑇𝑃𝑥≤1}⊆ℒ(𝐻_𝒮), for all 𝒮∈𝒱_𝑚, also satisfies 𝑉̇(𝑥)<0, and thus ℰ(𝑃) 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: 𝑥̇(𝑡)=𝐴+𝒜_𝑞𝐼_𝑛⊗𝑥(𝑡)𝑥(𝑡)+𝐵sat(𝑢(𝑡)), (15) where 𝐵=[𝑏₁,𝑏₂,…,𝑏_𝑚]∈𝑅^𝑛×𝑚. 𝑢(𝑡) is the nested saturation input function as follows: 𝑢(𝑡)=𝐹_𝑞−1𝑥(𝑡)+𝐵_𝑞−1×𝐹sat_𝑞−2𝑥(𝑡)+⋯+𝐵₂sat𝐹₁𝑥(𝑡)+𝐵₁sat𝐹₀𝑥(𝑡)⋯ (16) in which 𝐵_𝑖∈𝑅^𝑚×𝑚, 𝑖∈{1,2,…,𝑞−1}, are diagonal matrices and 𝐹_𝑗∈𝑅^𝑚×𝑛, 𝑗∈{0,1,…,𝑞−1}, are feedback gain matrices.

To express concisely, we denote ℬ_𝑞ℬ=𝐼,_𝑞−1=𝐵_𝑞−1,ℬ_𝑞−2=𝐵_𝑞−1𝐵_𝑞−2,⋮ℬ₁=𝐵_𝑞−1𝐵_𝑞−2,…,𝐵₂𝐵₁,ℬ₀ℱ=0,_𝑞ℱ=0,_𝑞−1=𝐹_𝑞−1,ℱ_𝑞−2=𝐹_𝑞−1+^𝑞−2_{𝑖=𝑞−2}ℬ_𝑖+1𝐹_𝑖,⋮&ℱ₀=𝐹_𝑞−1+^𝑞−2_𝑖=0ℬ_𝑖+1𝐹_𝑖. (17) Then we propose a new result for the local stability of the quadratic system (15).

Theorem 8. If there exist a positive scalar 𝜀 and matrices 𝑃>0∈𝑅^𝑛×𝑛, and ℋ_𝜔∈𝑅^𝑚×𝑛, for all 𝜔∈𝒱, defined in Definition 2, such that 𝑃He𝐴+^𝑚_𝑖=1𝑏_𝑖ℱ^𝑖_𝜔𝑖+ℬ_𝜔^(𝑖)_𝑖ℋ^𝑖_𝜔+𝜀2𝐼𝑃𝒜_𝑞0𝜀0−2𝐼_𝑛⊗𝑃<0, (18) then the quadratic system (15) is locally asymptotically stable and ℰ(𝑃)⊆ℒ(ℋ_𝜔) is an estimate of the RA.

Proof. Consider the Lyapunov quadratic function 𝑉(𝑥)=𝑥^𝑇𝑃𝑥 with 𝑃=𝑃^𝑇>0. Its time derivative along the trajectories of the system (15) is given by 𝑉̇(𝑥)=𝑥^𝑇He𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛(𝑡𝑥+⊗𝑥)_𝑚𝑖=12𝑥^𝑇𝑃𝑏_𝑖sat𝜙^𝑖_𝑗−1, (19) where 𝑗∈Φ_𝑞⧵{1}. We define 𝜙^𝑖_𝑗−1=𝐹^𝑖_𝑗−1𝑥+𝐵^(𝑖)_𝑗−1𝜌^𝑖_𝑗−1 with 𝜌^𝑖_𝑗−1=sat(𝐹^𝑖_𝑗−2𝑥+⋯+𝐵₂sat(𝐹₁𝑥(𝑡)+𝐵₁sat(𝐹₀𝑥(𝑡)))⋯). From Lemma 4, we have 2𝑥^𝑇𝑃𝑏_𝑖sat𝜙^𝑖_𝑗−1≤max2𝑥^𝑇𝑃𝑏_𝑖𝜙^𝑖_𝑞−1,−2𝑥^𝑇𝑃𝑏_𝑖,2𝑥^𝑇𝑃𝑏_𝑖𝐵^(𝑖)_𝑞−1𝜌^𝑖_𝑞−1=2𝑥^𝑇𝑃𝑏_𝑖𝐵^(𝑖)_𝑞−1sat𝜙^𝑖_𝑞−2≤max2𝑥^𝑇𝑃𝑏_𝑖𝐵^(𝑖)_𝑞−1𝜙^𝑖_𝑞−2,−2𝑥^𝑇𝑃𝑏_𝑖𝐵^(𝑖)_𝑞−1. (20)(21) Repeating the above procedure and substituting (21) into (20) yield 2𝑥^𝑇𝑃𝑏_𝑖sat𝜙^𝑖_𝑞−1≤max2𝑥^𝑇𝑃𝑏_𝑖ℱ^𝑖₀𝑥,2𝑥^𝑇𝑃𝑏_𝑖ℱ^𝑖₁𝑥−2𝑥^𝑇𝑃𝑏_𝑖ℬ₁^(𝑖),…,2𝑥^𝑇𝑃𝑏_𝑖ℱ^𝑖_𝑞−1𝑥−2𝑥^𝑇𝑃𝑏_𝑖ℬ^(𝑖)_𝑞−1,−2𝑥^𝑇𝑃𝑏_𝑖=max2𝑥^𝑇𝑃𝑏_𝑖ℱ^𝑖₀𝑥−2𝑥^𝑇𝑃𝑏_𝑖ℬ₀^(𝑖),2𝑥^𝑇𝑃𝑏_𝑖ℱ^𝑖₁𝑥−2𝑥^𝑇𝑃𝑏_𝑖ℬ₁^(𝑖),…,2𝑥^𝑇𝑃𝑏_𝑖ℱ^𝑖_𝑞−1𝑥−2𝑥^𝑇𝑃𝑏_𝑖ℬ^(𝑖)_𝑞−1,2𝑥^𝑇𝑃𝑏_𝑖ℱ^𝑖_𝑞𝑥−2𝑥^𝑇𝑃𝑏_𝑖ℬ_𝑞^(𝑖). (22) Let 𝜔∈𝒲 be an arbitrary vector. Then it follows from (22) that ^𝑚_𝑖=12𝑥^𝑇𝑃𝑏_𝑖sat𝜙^𝑖_𝑞−1≤max_{𝒮∈𝒱𝑚}^𝑚_𝑖=12𝑥^𝑇𝑃𝑏_𝑖ℱ^𝑖_𝜔𝑖𝑥−2𝑥^𝑇𝑃𝑏_𝑖ℬ_𝜔^(𝑖)_𝑖. (23)
Assume that 𝑥∈ℒ(ℋ_𝜔), for all 𝜔∈𝒲. For ℰ(𝑃)⊆ℒ(ℋ_𝜔), we obtain the following inequality from (23) and the property of ℒ(ℋ_𝜔): ^𝑚_𝑖=12𝑥^𝑇𝑃𝑏_𝑖sat𝜙^𝑖_𝑞−1≤max_{𝒮∈𝒱𝑚}^𝑚_𝑖=1𝑥^𝑇He𝑃𝑏_𝑖ℱ^𝑖_𝜔𝑖+ℬ_𝜔^(𝑖)_𝑖ℋ^𝑖_𝜔𝑥. (24) Therefore, if (18) is satisfied, we deduce from (20) and (24) that 𝑉̇(𝑥)≤−𝜀‖𝑥‖², for all 𝑥∈ℒ(ℋ_𝜔), for all 𝜔∈𝒲, where 𝜔>0 is a sufficiently small scalar; namely, ℰ(𝑃)⊆ℒ(ℋ_𝜔), for all 𝜔∈𝒲, 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: 𝑥̇(𝑡)=𝐴+𝒜_𝑞0𝐼_𝑛⊗𝑥(𝑡)𝑥(𝑡)+𝐵sat(𝑢(𝑡)), (25) where 𝐵_𝑞∈𝑅^{𝑛×𝑚𝑞} and 𝑢(𝑡)=𝐹_𝑞−1𝑥(𝑡)+𝐵_𝑞−1sat×𝐹_𝑞−2𝑥(𝑡)+⋯+𝐵₂sat𝐹₁𝑥(𝑡)+𝐵₁sat𝐹₀𝑥(𝑡)⋯ (26) in which 𝐹₀∈𝑅^𝑚1×𝑛, 𝐹_𝑖∈𝑅^{𝑚𝑖+1×𝑛}, and 𝐵_𝑖∈𝑅^{𝑚𝑖+1×𝑚𝑖}, 𝑖∈Φ_𝑞−1, the matrices 𝐵_𝑖, 𝑖∈Φ_𝑞−𝑖, are not necessarily diagonal, and 𝑚_𝑖, 𝑖∈Φ_𝑞, are not necessarily the same.

Theorem 9. If there exist a positive scalar 𝜀 and matrices 𝑃=𝑃^𝑇>0∈𝑅^𝑛×𝑛 and 𝐻_𝒮𝑖∈𝑅^{𝑚𝑖×𝑛}, 𝑖∈Φ_𝑞, for all (𝒮₁,𝒮₂,…,𝒮_𝑞)∈(𝒱_𝑚1×𝒱_𝑚2×⋯×𝒱_𝑚𝑞) such that the following inequality holds: 𝑃()+𝜀He𝐴+Ξ2𝐼𝑃𝒜_𝑞0𝜀0−2𝐼_𝑛⊗𝑃<0, (27) where Ξ=^𝑞_𝑗=1^𝑗_𝑖=𝑞𝐵_𝑖𝐷⁻_𝒮𝑖𝐹_𝑗−1+^𝑞−1_𝑗=1^𝑗+1_𝑖=𝑞𝐵_𝑖𝐷⁻_𝒮𝑖𝐵_𝑗𝐻_𝒮𝑗+𝐵_𝑞𝐻_𝒮𝑞, (28) then the quadratic system (25) is locally asymptotically stable for every initial condition belonging to the region ℰ(𝑃)⊆ℒ(𝐻_𝒮𝑖).

Proof. Consider the Lyapunov quadratic function 𝑉(𝑥)=𝑥^𝑇𝑃𝑥 with 𝑃=𝑃^𝑇>0. Its time derivative along the quadratic system (25) is given as 𝑉̇(𝑥)=𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛(𝑡+𝐼⊗𝑥)_𝑛⊗𝑥(𝑡)^𝑇𝒜^𝑇_𝑞𝑃𝑥+2𝑥^𝑇𝑃𝐵_𝑞sat𝜙_𝑗, (29) where, for 𝑗∈Φ_𝑞−1, we define 𝜙_𝑗=𝐹_𝑗𝑥+𝐵_𝑗𝜌_𝑗 with 𝜌_𝑗=sat(𝐹_𝑗−1𝑥+⋯+𝐵₂sat(𝐹₁𝑥+𝐵₁sat(𝐹₀𝑥))…). Let 𝑥∈ℒ(ℋ_𝒮𝑖), 𝒮_𝑖∈𝒱_𝑚𝑖. Then it follows from Lemma 4 that 𝑉̇(𝑥)≤𝑥^𝑇He𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛⊗𝑥(𝑡)𝑥+max_{𝒮𝑞∈𝒱𝑚𝑞}2𝑥^𝑇𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝜙_𝑞−1+𝑥^𝑇He𝑃𝐵_𝑞𝐻_𝒮𝑞𝑥=max_{𝒮𝑞∈𝒱𝑚𝑞}𝑥^𝑇He𝑃𝐵_𝑞𝐻_𝒮𝑞+𝐷⁻_𝒮𝑞𝐹_𝑞−1+𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛⊗𝑥(𝑡)𝑥+2𝑥^𝑇𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝜌_𝑞−1, (30) where 2𝑥^𝑇𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝜌_𝑞−1=2𝑥^𝑇𝑃𝐵_𝑞−1𝐷⁻_𝒮𝑞𝐵_𝑞−1sat𝜙_𝑞−2≤max_{𝒮𝑞−1∈𝒱𝑚𝑞−1}2𝑥^𝑇𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝐷⁻_𝒮𝑞−1𝜙_𝑞−2+𝑥^𝑇He𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝐻_𝒮𝑞−1𝑥=max_{𝒮𝑞−1∈𝒱𝑞−1}2𝑥^𝑇𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝐷⁻_𝒮𝑞−1𝐵_𝑞−2𝜌_𝑞−2+𝑥^𝑇He𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1×𝐻_𝒮𝑞−1+𝐷⁻_𝒮𝑞−1𝐹_𝑞−2𝑥. (31) Then, from the above inequalities, we have that 𝑉̇(𝑥)≤max_{𝒮𝑞∈𝒱𝑚𝑞}𝑥^𝑇He𝑃𝐵_𝑞𝐻_𝒮𝑞+𝐷⁻_𝒮𝑞𝐹_𝑞−1+𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛⊗𝑥(𝑡)𝑥+max_{𝒮𝑞−1∈𝒱𝑚𝑞−1}2𝑥^𝑇𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝐷⁻_𝒮𝑞−1𝐵_𝑞−2𝜌_𝑞−2+𝑥^𝑇He𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1×𝐻_𝒮𝑞+𝐷⁻_𝒮𝑞−1𝐹_𝑞−2𝑥=max_{𝒮𝑞∈𝒱𝑚𝑞}max_{𝒮𝑞−1∈𝒱𝑚𝑞−1}𝑥^𝑇Ξ_𝑞−1𝑥+2𝑥^𝑇𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1×𝐷⁻_𝒮𝑞−1𝐵_𝑞−2𝜌_𝑞−2𝑥, (32) where Ξ_𝑞−1=He𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝐻_𝒮𝑞−1+𝐷⁻_𝒮𝑞−1𝐹_𝑞−2+𝑃𝐵_𝑞𝐻_𝒮𝑞+𝐷⁻_𝒮𝑞𝐹_𝑞−1+𝑃𝐴+𝑃𝒜_𝑞0𝐼_𝑛⊗𝑥(𝑡). (33) Repeating the above process, we finally obtain 𝑉̇(𝑥)≤max_{𝒮𝑞∈𝒱𝑚𝑞}max_{𝒮𝑞−1∈𝒱𝑚𝑞−1}⋯max_{𝒮1∈𝒱𝑚1}×𝑥^𝑇He𝑃𝐴+𝒜_𝑞0𝐼_𝑛⊗𝑥(𝑡)+Ξ𝑥, (34) where Ξ=He𝑃𝐵𝐻_𝒮𝑞+𝐷⁻_𝒮𝑞𝐹_𝑞−1+He𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝐻_𝒮𝑞−1+𝐷⁻_𝒮𝑞−1𝐹_𝑞−2+⋯+He𝑃𝐵_𝑞𝐷⁻_𝒮𝑞𝐵_𝑞−1𝐷⁻_𝒮𝑞−1𝐵_𝑞−2⋯𝐷⁻_𝒮2𝐵₁𝐻_𝒮1+𝐷⁻_𝒮1𝐹₀. (35)
Applying Lemma 3 and (27) for 𝑥∈𝜕ℰ, each point on the boundary of ℰ, we obtain He𝑃𝒜_𝑞0𝐼_𝑛⊗𝑥(𝑡)=He𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃^−1/2𝐼_𝑛⊗𝑈𝐼_𝑛≤1⊗𝑣𝜀𝑃𝒜_𝑞0𝐼_𝑛⊗𝑃⁻¹𝒜^𝑇_𝑞0𝑃+𝜀𝐼, (36) where 𝜀>0 and 𝑈 and 𝑣 satisfying (12). Thus if the following inequality holds He𝑃𝐴+Ξ+𝒜_𝑞𝒳𝑃()+1≤He𝐴+Ξ𝜀𝑃𝒜_𝑞𝐼_𝑛⊗𝑃⁻¹𝒜^𝑇_𝑞𝑃+𝜀𝐼<0, (37) then, from the Schur complement, inequality (27) is satisfyied for every 𝑈 and 𝑣 satisfying (12). The ellipsoid ℰ(𝑃)⊆ℒ(𝐻_𝒮𝑖) is therefore the estimate of the RA for the quadratic system (25).

Remark 10. In the case of 𝑞=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].

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 𝑞=1: 𝑥̇(𝑡)=𝐴+𝒜_𝑞𝐼_𝑛(𝑡𝑥(𝑡)𝐹⊗𝑥)+𝐵sat₁𝑥(𝑡)+𝐵₁sat𝐹₀𝑥(𝑡), (38) where the coefficient matrices are given as follows: ,𝐵𝐴=0100,𝐵=−2111₁=0.2000.1,𝒜_𝑞=.0000.22110 (39)

To measure the impact of the quadratic term of control law on the RA, we compare the value of trace(𝑃) obtained using the conditions from Theorem 9, resulting in a control law having gains 𝐾 and 𝐾₀. The RA of such an equilibrium point obtained by applying Theorem 9 is given by ,𝐾𝑃=0.5889−0.1651−0.16510.1259,𝐾=−0.1083−1.3299₀=0.0760−0.0142, (40) corresponding to 𝜆=0.0020, 𝜀=0.1, and 𝜂=0.01. Figure 1 shows the estimate of the RA of this example with an initial condition 𝑥₀=[0.2,0.1]^𝑇.

Figure 1 

Phase portrait of the system (38) with gains (40) and the estimate of the RA.

5. 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.

Acknowledgment

This work is supported by the Graduate Innovation and Creativity Foundation of Jiangsu Province under Grant no. CXZZ12-0202.