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

Authors: Minsong Zhang; Fu Chen
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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∈Rn:xTPx≤1}. For a matrix H∈Rm×n, we define ℒ(H)={x∈Rn:∥Hx∥∞≤1}, where ∥v∥∞ denotes the ∞-norm of the vector v. The symbol He(A) denotes AT+A, where AT is the transpose of A; λi(A) denotes the ith eigenvalue of A, and λ(A) denotes the eigenvalue set of matrix A. If P≥0, λmax(P) and λmin(P) mean the maximal and minimal eigenvalues of P, respectively. Moreover, Fi 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 Φm={1,2,…,m}, 𝒱m={𝒱s:𝒱s⊂Φ}. It is easy to see that there are 2m elements in the set 𝒱m. Denote the number of elements in 𝒮∈𝒱m by σ(𝒮). Assume that σ(∅)=0; then, for arbitrary 𝒮∈𝒱m, we can denote 𝒮={i1,i2,…,iσ(𝒮)} with 1≤i1<i2<⋯<iσ(𝒮)≤m [18].Definition 1 (see [18]).
Let𝒮={i1,i2,…,iσ(𝒮)}∈𝒱m with 1≤i1<i2<⋯<iσ(𝒮)≤m. Define H𝒮∈Rm×n as a matrix such that the ijth rows of H𝒮 are nonzero while the other rows are zeros, and define D𝒮 as a diagonal matrix such that the ijth diagonal elements are 1 and the others are zeros. Furthermore, define D𝒮-=Im-D𝒮. For two integers m≥1 and q≥1, one has
(1)𝒲={ω=[ω1ω2⋯ωm]T}:ωi∈{0,1,…,q}.Definition 2 (see [18]).
Letω∈𝒲 be a given vector, and denote the ith element of ω by ωi. The matrix ℋω∈Rm×n is defined as ℋωi=0 if ωi=0, for all i∈Φm.


## 2. Plant Analysis

Consider the quadratic system subject to nested saturated-input:(2)x˙(t)=Ax(t)+(x(t)TAq1x(t)x(t)TAq2x(t)⋮x(t)TAqnx(t))+Bsat(u(t)),
where x(t)∈Rn, u=Fx is the control input, F∈Rm×n is a given matrix, and sat(u) denotes the nested saturation nonlinearity of u. Also A∈Rn×n, B∈Rn×m, and Aqi∈Rn×n, i=1,…,n. Define matrices 𝒜q0∈Rn×n2 as follows:
(3)𝒜q0=(Aq1(1)Aq1(2)⋯Aq1(n)⋮⋮⋱⋮Aqn(1)Aqn(2)⋯Aqn(n)),
where Aqi(j)∈R1×n denotes the jth row of matrix Aqi∈Rn×n. The quadratic system (2) can be read as
(4)x˙(t)=(A+𝒜q0(In⊗x(t)))x(t)+Bsat(u(t)).
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 matrixP∈Rn×n, P=PT>0, and a vector v such that ∥v∥=1. Every point on the boundary of an ellipsoid, ∂ℰ(P)={x∈Rn:xTPx=1}, can be parameterized by x=P-1/2Uv where UTU=I.Lemma 4 (see [19]).
Leta,b∈R be two arbitrary scalars; then
(5)asat(b)≤max{ab,-|a|}.Lemma 5 (see [18]).
LetF∈Rn×m be a given matrix and u∈Rm an arbitrary vector. If x∈ℒ(H)𝒮, for all 𝒮∈𝒱m, where 𝒮, 𝒱m, and H𝒮∈Rm×n are defined in Definition 1, then
(6)2xTFsat(u)≤max𝒮∈𝒱m{2xTFD𝒮-u+xT(He(FH𝒮))x}.Lemma 6 (see [18]).
For two vectorsu,v∈Rm, if ∥v∥∞≤1, then sat(u)∈co{D𝒮-ui+D𝒮vi,𝒮∈𝒱m}, where co{·} denotes the convex hull of a set.


## 3. Main Results

In this section, through the new treatment of saturation nonlinearity given in Lemma5, 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 levelμ(>ε1), the standard saturation function is defined as follows:
(7)[satu(t)]l≜{[u(t)]lif|[u(t)]l|<μ,μif[u(t)]l≥μ,-μif[u(t)]l≤-μ,
and it meets the following sufficient condition.Proposition 7.
If there exist a positive scalarε and matrices P>0∈Rn×n and H𝒮∈Rm×n, for all 𝒮∈𝒱m, satisfying the following inequality:
(8)He([P(A+BD𝒮-F+BH𝒮)+ε2IP𝒜q00-ε2(In⊗P)])<0,
then the region ℰ(P)⊆ℒ(H𝒮) is an estimate of the RA for the nonlinear quadratic system (4) with conditional actuator saturation (7).Proof.
Consider the quadratic Lyapunov functionV(x)=xTPx, where P=PT>0. Let x∈ℒ(H𝒮), for all 𝒮∈𝒱m. Then by Lemma 5, the time derivative of V(x) along the trajectories of system (4) is given as
(9)V˙(x)=xT(+(In⊗x(t))T𝒜q0TPATP+PA+P𝒜q0(In⊗x(t))+(In⊗x(t))T𝒜q0TP)x+2xTPBsat(Fx)≤xT(+(In⊗x(t))T𝒜q0TPATP+PA+P𝒜q0(In⊗x(t))+(In⊗x(t))T𝒜q0TP)x+max𝒮∈𝒱m{2xTPBD𝒮-Fx+xT(He(PBH𝒮))x}=xTHe(PA+P𝒜q0(In⊗P)-1/2(In⊗U)(In⊗v))x+max𝒮∈𝒱m{2xTPBD𝒮-Fx+xT(He(PBH𝒮))x}
with ∥v∥2=1. To guarantee that V˙(x)<0 at each point of ∂ℰ(P), it suffices to verify the following inequality:
(10)He{+P𝒜q0(In⊗P)-1/2(In⊗U)(In⊗v)P(A+BD𝒮-F+BH𝒮)+P𝒜q0(In⊗P)-1/2(In⊗U)(In⊗v)}<0.
One can write
(11)He{P𝒜q0(In⊗P)-1/2(In⊗U)(In⊗v)}<1εP𝒜q0(In⊗P)-1𝒜q0TP+ε(In⊗v)T(In⊗U)T(In⊗U)(In⊗v),
where ε>0 and
(12)(In⊗U)T(In⊗U)=In2,(In⊗v)T(In⊗v)=In.
Then one get
(13)He(P𝒜q0(In⊗P)-1/2(In⊗U)(In⊗v))<1εP𝒜q0(In⊗P)-1𝒜q0TP+εI.
Thus if the inequality
(14)He(P(A+BD𝒮-F+BH𝒮))+1εP𝒜q0(In⊗P)-1𝒜q0TP+εI<0
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 V˙(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+BD𝒮-F+BH𝒮)+P𝒜q0(In⊗x(t))), any point in ℰ(P)={x∈Rn:xTPx≤1}⊆ℒ(H𝒮), for all 𝒮∈𝒱m, also satisfies V˙(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:(15)x˙(t)=(A+𝒜q(In⊗x(t)))x(t)+Bsat(u(t)),
where B=[b1,b2,…,bm]∈Rn×m. u(t) is the nested saturation input function as follows:
(16)u(t)=Fq-1x(t)+Bq-1sat×(Fq-2x(t)+⋯+B2sat(F1x(t)+B1sat(F0x(t)))⋯Fq-2x)
in which Bi∈Rm×m, i∈{1,2,…,q-1}, are diagonal matrices and Fj∈Rm×n, j∈{0,1,…,q-1}, are feedback gain matrices.To express concisely, we denote(17){ℬq=I,ℬq-1=Bq-1,ℬq-2=Bq-1Bq-2,⋮ℬ1=Bq-1Bq-2,…,B2B1,ℬ0=0,{ℱq=0,ℱq-1=Fq-1,ℱq-2=Fq-1+∑i=q-2q-2‍ℬi+1Fi,⋮&ℱ0=Fq-1+∑i=0q-2‍ℬi+1Fi.
Then we propose a new result for the local stability of the quadratic system (15).Theorem 8.
If there exist a positive scalarε and matrices P>0∈Rn×n, and ℋω∈Rm×n, for all ω∈𝒱, defined in Definition 2, such that
(18)He([P(A+∑i=1m‍bi(ℱωii+ℬωi(i)ℋωi))+ε2IP𝒜q00-ε2(In⊗P)])<0,
then the quadratic system (15) is locally asymptotically stable and ℰ(P)⊆ℒ(ℋω) is an estimate of the RA.Proof.
Consider the Lyapunov quadratic functionV(x)=xTPx with P=PT>0. Its time derivative along the trajectories of the system (15) is given by
(19)V˙(x)=xTHe(PA+P𝒜q0(In⊗x(t)))x+∑i=1m‍2xTPbisat(ϕj-1i),
where j∈Φq∖{1}. We define ϕj-1i=Fj-1ix+Bj-1(i)ρj-1i with ρj-1i=sat(Fj-2ix+⋯+B2sat(F1x(t)+B1sat(F0x(t)))⋯). From Lemma 4, we have
(20)2xTPbisat(ϕj-1i)≤max{2xTPbiϕq-1i,-2|xTPbi|},2xTPbiBq-1(i)ρq-1i=2xTPbiBq-1(i)sat(ϕq-2i)(21)≤max{2xTPbiBq-1(i)ϕq-2i,-2|xTPbiBq-1(i)|}.
Repeating the above procedure and substituting (21) into (20) yield
(22)2xTPbisat(ϕq-1i)≤max{2xTPbiℱ0ix,2xTPbiℱ1ix-2|xTPbiℬ1(i)|,…,2xTPbiℱq-1ix-2|xTPbiℬq-1(i)|,-2|xTPbi|}=max{2xTPbiℱ0ix-2|xTPbiℬ0(i)|,2xTPbiℱ1ix-2|xTPbiℬ1(i)|,…,2xTPbiℱq-1ix-2|xTPbiℬq-1(i)|,2xTPbiℱqix-2|xTPbiℬq(i)|}.
Let ω∈𝒲 be an arbitrary vector. Then it follows from (22) that
(23)∑i=1m‍2xTPbisat(ϕq-1i)≤max𝒮∈𝒱m{∑i=1m‍2xTPbiℱωiix-2|xTPbiℬωi(i)|}.
Assume thatx∈ℒ(ℋω), for all ω∈𝒲. For ℰ(P)⊆ℒ(ℋω), we obtain the following inequality from (23) and the property of ℒ(ℋω):
(24)∑i=1m‍2xTPbisat(ϕq-1i)≤max𝒮∈𝒱m{∑i=1m‍xT(He(Pbi(ℱωii+ℬωi(i)ℋωi)))x}.
Therefore, if (18) is satisfied, we deduce from (20) and (24) that V˙(x)≤-ε∥x∥2, for all x∈ℒ(ℋω), for all ω∈𝒲, where ω>0 is a sufficiently small scalar; namely, ℰ(P)⊆ℒ(ℋω), 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:(25)x˙(t)=(A+𝒜q0(In⊗x(t)))x(t)+Bsat(u(t)),
where Bq∈Rn×mq and
(26)u(t)=Fq-1x(t)+Bq-1sat×(Fq-2x(t)+⋯+B2sat(F1x(t)+B1sat(F0x(t)))⋯)
in which F0∈Rm1×n, Fi∈Rmi+1×n, and Bi∈Rmi+1×mi, i∈Φq-1, the matrices Bi, i∈Φq-i, are not necessarily diagonal, and mi, i∈Φq, are not necessarily the same.Theorem 9.
If there exist a positive scalarε and matrices P=PT>0∈Rn×n and H𝒮i∈Rmi×n, i∈Φq, for all (𝒮1,𝒮2,…,𝒮q)∈(𝒱m1×𝒱m2×⋯×𝒱mq) such that the following inequality holds:
(27)He(P(A+Ξ)+ε2IP𝒜q00-ε2(In⊗P))<0,
where
(28)Ξ=∑j=1q‍(∏i=qj‍BiD𝒮i-)Fj-1+∑j=1q-1‍(∏i=qj+1‍BiD𝒮i-)BjH𝒮j+BqH𝒮q,
then the quadratic system (25) is locally asymptotically stable for every initial condition belonging to the region ℰ(P)⊆ℒ(H𝒮i).Proof.
Consider the Lyapunov quadratic functionV(x)=xTPx with P=PT>0. Its time derivative along the quadratic system (25) is given as
(29)V˙(x)=xT(ATP+PA+P𝒜q0(In⊗x(t))+(In⊗x(t))T𝒜qTP)x+2xTPBqsat(ϕj),
where, for j∈Φq-1, we define ϕj=Fjx+Bjρj with ρj=sat(Fj-1x+⋯+B2sat(F1x+B1sat(F0x))…). Let x∈ℒ(ℋ𝒮i), 𝒮i∈𝒱mi. Then it follows from Lemma 4 that
(30)V˙(x)≤xTHe(PA+P𝒜q0(In⊗x(t)))x+max𝒮q∈𝒱mq{2xTPBqD𝒮q-ϕq-1+xTHe(PBqH𝒮q)x}=max𝒮q∈𝒱mq{xTHe(PBq(H𝒮q+D𝒮q-Fq-1)+PA+P𝒜q0(In⊗x(t))(H𝒮q+D𝒮q-Fq-1))x+2xTPBqD𝒮q-Bq-1ρq-1},
where
(31)2xTPBqD𝒮q-Bq-1ρq-1=2xTPBq-1D𝒮q-Bq-1sat(ϕq-2)≤max𝒮q-1∈𝒱mq-1{2xTPBqD𝒮q-Bq-1D𝒮q-1-ϕq-2+xT(He(PBqD𝒮q-Bq-1H𝒮q-1))x}=max𝒮q-1∈𝒱q-1{2xTPBqD𝒮q-Bq-1D𝒮q-1-Bq-2ρq-2+xT(He(PBqD𝒮q-Bq-1×(H𝒮q-1+D𝒮q-1-Fq-2)))x}.
Then, from the above inequalities, we have that
(32)V˙(x)≤max𝒮q∈𝒱mq{xTHe(PBq(H𝒮q+D𝒮q-Fq-1)+PA+P𝒜q0(In⊗x(t))(H𝒮q+D𝒮q-Fq-1))x+max𝒮q-1∈𝒱mq-1{2xTPBqD𝒮q-Bq-1D𝒮q-1-Bq-2ρq-2+xTHe(PBqD𝒮q-Bq-1×(H𝒮q+D𝒮q-1-Fq-2))x}}=max𝒮q∈𝒱mqmax𝒮q-1∈𝒱mq-1{×D𝒮q-1-Bq-2ρq-2xxTΞq-1x+2xTPBqD𝒮q-Bq-1×D𝒮q-1-Bq-2ρq-2x},
where
(33)Ξq-1=He{PBqD𝒮q-Bq-1(H𝒮q-1+D𝒮q-1-Fq-2)+PBq(H𝒮q+D𝒮q-Fq-1)+PA+P𝒜q0(In⊗x(t))PBqD𝒮q-Bq-1(H𝒮q-1+D𝒮q-1-Fq-2)}.
Repeating the above process, we finally obtain
(34)V˙(x)≤max𝒮q∈𝒱mqmax𝒮q-1∈𝒱mq-1⋯max𝒮1∈𝒱m1×{xTHe(P(A+𝒜q0(In⊗x(t)))+Ξ)x},
where
(35)Ξ=He(PB(H𝒮q+D𝒮q-Fq-1))+He(PBqD𝒮q-Bq-1(H𝒮q-1+D𝒮q-1-Fq-2))+⋯+He(PBqD𝒮q-Bq-1D𝒮q-1-Bq-2⋯)D𝒮2-B1(H𝒮1+D𝒮1-F0).
Applying Lemma3 and (27) for x∈∂ℰ, each point on the boundary of ℰ, we obtain
(36)He(P𝒜q0(In⊗x(t)))=He(P𝒜q0(In⊗P)-1/2(In⊗U)(In⊗v))≤1εP𝒜q0(In⊗P)-1𝒜q0TP+εI,
where ε>0 and U and v satisfying (12). Thus if the following inequality holds
(37)He(P(A+Ξ+𝒜q𝒳))≤He(+1εP𝒜q(In⊗P)-1𝒜qTPP(A+Ξ)+1εP𝒜q(In⊗P)-1𝒜qTP)+εI<0,
then, from the Schur complement, inequality (27) is satisfyied for every U and v satisfying (12). The ellipsoid ℰ(P)⊆ℒ(H𝒮i) is therefore the estimate of the RA for the quadratic system (25).Remark 10.
In the case ofq=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 Theorem9. Consider a nonlinear quadratic system in [4] with q=1:
(38)x˙(t)=(A+𝒜q(In⊗x(t)))x(t)+Bsat(F1x(t)+B1sat(F0x(t))),
where the coefficient matrices are given as follows:
(39)A=(0100),B=(-2111),B1=(0.2000.1),𝒜q=(0000.22110).To measure the impact of the quadratic term of control law on the RA, we compare the value oftrace(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
(40)P=[0.5889-0.1651-0.16510.1259],K=[-0.1083-1.3299],K0=[0.0760-0.0142],
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.



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