Global Output Control for a Class of Inherently Higher-Order Nonlinear Time-Delay Systems Based on Homogeneous Domination Approach

Abstract

This paper addresses the problem of global output feedback stabilization for a class of inherently higher-order uncertain nonlinear systems subject to time-delay. By using the homogeneous domination approach, we construct a homogeneous output feedback controller with an adjustable scaling gain. With the aid of a homogeneous Lyapunov-Krasovskii functional, the scaling gain is adjusted to dominate the time-delay nonlinearities bounded by homogeneous growth conditions and render the closed-loop system globally asymptotically stable. In addition, we also show that the proposed approach is applicable for time-delay systems under nontriangular growth conditions.

1. Introduction

This paper addresses the global stabilization problem for a class of uncertain systems with delay which is described by 𝑥̇₁(𝑡)=𝑥₂(𝑡)^𝑝1+𝜙₁((𝑡)()(𝑡,⋮𝑥̇𝑡,𝑥,𝑥𝑡−𝜏,𝑢))_𝑛−1(𝑡)=𝑥_𝑛(𝑡)^𝑝𝑛−1+𝜙_𝑛−1((𝑡)()(𝑡,𝑥̇𝑡,𝑥,𝑥𝑡−𝜏,𝑢))_𝑛(𝑡)=𝑢(𝑡)+𝜙_𝑛((𝑡)()(𝑡,𝑦(𝑡)𝑡,𝑥,𝑥𝑡−𝜏,𝑢))=𝑥₁(𝑡),𝑥(𝑡)=Ψ(𝑡),−𝜏≤𝑡≤0, (1) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈ℝ^𝑛 are the system states, 𝑢(𝑡)∈ℝ is the control input, 𝑦(𝑡)∈ℝ is the system output with 𝑝_𝑖∈ℝ^odd_≥1, 𝜏>0 is a given time-delay constant, and Ψ(𝑡) is the initial function of the system state vector. The terms 𝜙_𝑖(⋅), 𝑖=1,…,𝑛, represent nonlinear perturbations that are not guaranteed to be precisely known.

It has been known that the problem of global output feedback stabilization for uncertain nonlinear systems is very challenging compared to the state feedback case. In the past decade, global stabilization by output feedback domination method has been proved to be achievable for a series of nonlinear systems. For the system of a five-spot pattern reservoir, a nonlinear reduced-order model is identified and an asymptotically stabilizing controller is proposed based on the circle criterion in [1]. With the help of linear feedback domination design [2], some interesting results have been established under a linear growth condition [2] and under a higher-order growth condition [3]. Recently, the homogeneous domination approach proposed in [3] has been used as a universal tool to solve the problem of global output feedback stabilization for inherently nonlinear systems. As a consequence, fruitful research results have been achieved in [3–9].

However, the aforementioned results have not considered the time-delay effect which is actually very common in state, input, and output due to the time consumed in sensing, information transmitting, and controller computing. In the case when the nonlinearities contain time-delay, some interesting results have been obtained. For instance, in [10], the global asymptotic stability analysis problem is investigated for a class of stochastic bidirectional associative memory (BAM) networks with mixed time-delays and parameter uncertainties. The paper [11] investigated the state estimation problem for a class of discrete time-delay nonlinear complex networks with randomly occurring phenomena from the sensor measurements. In [12], an adaptive approach was employed to design a state feedback controller to globally stabilize a class of upper-triangular systems with time-delay. The work [13] relaxed the growth condition imposed in [12] by employing a dynamic gain. In [14], a state feedback stabilizer was constructed for a class of time-delay higher-order nonlinear systems. In [15], the problem of state-feedback stabilization for a class of lower-triangular stochastic time-delay nonlinear systems was investigated.

In the case when only output is available, the problem of output feedback stabilization is more challenging and fewer results have been achieved for nonlinear systems with time-delay. For a linear system with time-delay in the input, the problem of output feedback stabilization of was solved in [16, 17]. For nonlinear systems (1) subject to time-delay and uncertainties, the problem of output feedback stabilization has not been extensively investigated.

In this paper, we aim to tackle the problem by using the output feedback domination approach. First, based on homogeneous domination approach [3, 9], output feedback controllers are constructed to globally stabilize higher-order nonlinear time-delay systems whose nonlinearities are bounded by homogeneous growth conditions. Then, we construct a Lyapunov-Krasovskii functional and use it to choose an appropriate scaling gain in the output feedback controller to guarantee the closed-loop system globally asymptotic stability. In addition, homogeneous output stabilization controllers are extended to the nontriangular time-delay systems. The simulation results show the effectiveness of the proposed method.

2. Homogeneous Output Feedback Controllers of Inherently Nonlinear Systems

In this section, we show that under a lower-triangular homogeneous growth condition, the nonlinear time-delay system (1) can be globally stabilized by a homogeneous output feedback controller. First, we consider the nonlinear continuous functions 𝜙_𝑖∶ℝ×ℝ^𝑛×ℝ^𝑛×ℝ→ℝ, 𝑖=1,2,…,𝑛, which satisfy the following higher-order growth condition.

Assumption 1. Consider the following: 𝜙_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑢(𝑡))≤𝑐₁𝑥₁^{(𝑟𝑖+𝑚)/𝑟1}+⋯+𝑥_𝑖^{(𝑟𝑖+𝑚)/𝑟𝑖}+𝑐₂𝑥₁(𝑡−𝜏)^{(𝑟𝑖+𝑚)/𝑟1}+𝑥+⋯_𝑖(𝑡−𝜏)^{(𝑟𝑖+𝑚)/𝑟𝑖}, (2) where 𝑖=1,…,𝑛, 𝑟₁=1, 𝑚≥0, 𝑟_𝑖𝑝_𝑖−1=𝑚+𝑟_𝑖−1, for 𝑖=2,…,𝑛+1, and 𝑝_𝑛=1. Denoting 𝜅₁=0 and 𝜅_𝑖=(𝜅_𝑖−1+1)/𝑝_𝑖−1, for 𝑖=2,…,𝑛+1. The objective of this note is to prove that the growth condition, namely, Assumption 1, guarantees the existence of a linear dynamic homogeneous output feedback controller based on observer: 𝑢(𝑡)=−𝐿^𝜅𝑛+1𝛽_𝑛1𝐿^𝜅𝑛𝑥^{𝜇/𝑟𝑛}_𝑛(𝑡)+𝛽_𝑛−1×1𝐿^𝜅𝑛−1𝑥^{𝜇/𝑟𝑛−1}_𝑛−1(𝑡)+⋯+𝛽₂1𝐿^𝜅2𝑥^𝜇/𝑟2₂(𝑡)+𝛽₁𝑥^𝜇/𝑟1₁(𝑡)⋯^{(𝑟𝑛+𝑚)/𝜇},𝑥₁=𝑥₁,𝜂̇₂=−𝑙₁𝑥^𝑝1₂,𝑥₂=𝐿^𝜅2𝜂₂+𝑙₁𝑥₁^𝑟2/𝑟1,𝜂̇_𝑖=−𝑙_𝑖−1𝑥^𝑝𝑖−1_𝑖,𝑥_𝑖=𝐿^{𝜅𝑖−(𝜅𝑖−1𝑟𝑖/𝑟𝑖−1)}𝜂_𝑖+𝑙_𝑖−1𝑥_𝑖−1^{𝑟𝑖/𝑟𝑖−1},𝑖=3,…,𝑛, (3)(4) where 𝜇≥max_{𝑖=2,…,𝑛+1}{𝑟_𝑖𝑝_𝑖−1} and 𝑙_𝑖, 𝑖=1,…,𝑛−1 are the gains selected by Theorem 3.1 in [5] renders system (7) globally asymptotically stable, with 𝑙_𝑛−1=14+1+𝛼+𝑙𝛼,_𝑖−1=𝑚_𝑖⁻¹14+𝛼_𝑖+1+𝛼+𝛼+𝑔_𝑖+1𝑙_𝑖+ℎ_𝑖+1𝑙_𝑖,𝑙𝑖=𝑛−1,…,3,₁=𝑚₂⁻¹14+𝛼₂+1+𝛼+𝛼+𝑔₃𝑙₂+ℎ₃𝑙₂, (5) where 𝛼≥0, 𝛼, and 𝛼_𝑖 are constants, 𝑚_𝑖 for 𝑖=2,…,𝑛−1 are known powers of 2, and 𝑔_𝑖+1(𝑙_𝑖), ℎ_𝑖+1(𝑙_𝑖) are continuous functions.

Remark 2. When 𝑚>0, (2) is a higher-order growth condition which is actually homogeneous (see Definition A.1 in the appendix) with the following dilation (For simplicity, in this paper we assume 𝑚=𝑞/𝑑 with an even integer 𝑞 and an odd integer 𝑑. Therefore, 𝑟_𝑖 is a ratio of two odd integers.) 𝑟₁=1,𝑟_𝑖𝑝_𝑖−1=𝑚+𝑟_𝑖−1,𝑖=2,…,𝑛+1. (6)
First, we construct a output feedback stabilizer for the following linear system 𝑧̇_𝑖=𝑧^𝑝𝑖_𝑖+1𝑧̇,𝑖=1,…,𝑛−1,_𝑛=𝑣,𝑦=𝑧₁, (7) where 𝑣∈ℝ, 𝑦∈ℝ are the control input and system output, respectively. Using the approach in [5], we can design for (7) a homogeneous output feedback stabilizer, which can be described in the following lemma.

Lemma 3. There exists a dynamic homogeneous output feedback controller based on observer: 𝑢(𝑡)=−𝛽_𝑛𝑧^{𝜇/𝑟𝑛}_𝑛(𝑡)+𝛽_𝑛−1×𝑧^{𝜇/𝑟𝑛−1}_𝑛−1(𝑡)+⋯+𝛽₂𝑧^𝜇/𝑟2₂(𝑡)+𝛽₁𝑧^𝜇/𝑟1₁(𝑡)⋯^{(𝑟𝑛+𝑚)/𝜇},𝑧₁=𝑥₁,𝜂̇₂=−𝑙₁𝑧^𝑝1₂,𝑧₂=𝜂₂+𝑙₁𝑧₁^𝑟2/𝑟1,𝜂̇_𝑖=−𝑙_𝑖−1𝑧^𝑝𝑖−1_𝑖,𝑧_𝑖=𝜂_𝑖+𝑙_𝑖−1𝑧_𝑖−1^{𝑟𝑖/𝑟𝑖−1},𝑖=3,…,𝑛, (8)(9) where 𝜇≥max_{𝑖=2,…,𝑛+1}{𝑟_𝑖𝑝_𝑖−1}, and 𝑙_𝑖, 𝑖=1,…,𝑛−1 are the same with those in (4).

It can be verified that the closed-loop system (7)–(9) is homogeneous according to Definition A.1 in the appendix. As a matter of fact, by defining the compact notation 𝑍∶=(𝑧₁,…,𝑧_𝑛,𝜂₂,…,𝜂_𝑛)^𝑇, the closed-loop system (7)–(9) can be rewritten as the following compact form: 𝑍̇=𝐹(𝑍)=𝑧^𝑝1₂,…,𝑧^𝑝𝑛−1_𝑛,𝑣,𝑓₁,…,𝑓_𝑛−1^𝑇, (10) where 𝑓_𝑖=𝜂̇_𝑖+1, 𝑖=1,…,𝑛−1. Moreover, by introducing the dilation weight Δ=(𝑟₁,𝑟₂,…,𝑟_𝑛,𝑟₁,𝑟₂,…,𝑟_𝑛−1),for𝑧₁,…,𝑧_𝑛,for𝜂₂,…,𝜂_𝑛, (11) system (10) is homogeneous of degree 𝑚 according to Definition A.1 in the appendix. Since system (10) is globally stable by Lemma 3, therefore, by Lemmas A.2 and A.3 in the appendix, there exists a homogeneous Lyapunov function 𝑈 with the degree 2𝑟_𝑛+1−𝑚 such that 𝑈̇(𝑍)|₍₁₀₎=𝜕𝑈(𝑍)𝜕𝑍⋅𝐹≤−𝑐₁‖𝑍‖_Δ^2𝜇,𝑐₁>0, (12) where ‖𝑍‖_Δ=∑^2𝑛−1_𝑖=1|𝑍_𝑖|^2/𝑟𝑖. Moreover, by Lemma A.4 in the appendix, for 𝑖=1,…,𝑛, the following holds: 𝜕𝑈𝜕𝑍_𝑖≤𝑐₂‖𝑍‖^{2𝜇−𝑚−𝑟𝑖}_Δ,𝑐₂>0. (13)

Theorem 4. Consider system (1) under Assumption 1. For any given delay constant 𝜏, there exists a homogeneous output feedback controller (3) based on observer (4) such that system (1) can be globally stabilized.

Proof. The output feedback controller is constructed by introducing a scaling gain into the output feedback controller obtained in Lemma 3. First, we define a change of coordinates as 𝐿^𝜅𝑖𝑧_𝑖=𝑥_𝑖,𝐿^𝜅𝑖𝑧_𝑖=𝑥_𝑖,𝑖=1,…,𝑛,𝐿^𝜅𝑛+1𝜂𝑣=𝑢,₂=𝜂₂,𝐿^𝜅𝑖−1𝜂_𝑖=𝜂_𝑖,𝑖=3,…,𝑛, (14) where the constant gain 𝐿≥1 will be determined later. Under (14), system (1) can be rewritten as 𝑧̇₁=𝐿𝑧^𝑝1₂+𝜙₁,⋮𝑧̇_𝑛−1=𝐿𝑧^𝑝𝑛−1_𝑛+𝜙_𝑛−1𝐿^𝜅𝑛−1,𝑧̇_𝑛𝜙=𝐿𝑣+^𝑛𝐿^𝜅𝑛. (15) The observer (9) can be rewritten as follows: 𝜂̇₂=−𝐿𝑙₁𝑧^𝑝1₂,𝑧₂=𝜂₂+𝑙₁𝑧₁^𝑟2/𝑟1,𝜂̇_𝑘=−𝐿𝑙_𝑘−1𝑧^𝑝𝑘−1_𝑘,𝑧_𝑘=𝜂_𝑘+𝑙_𝑘−1𝑧_𝑘−1^{𝑟𝑘/𝑟𝑘−1},𝑣(𝑡)𝑘=3,…,𝑛,=−𝛽_𝑛𝑧^{𝜇/𝑟𝑛}_𝑛(𝑡)+𝛽_𝑛−1𝑧^{𝜇/𝑟𝑛−1}_𝑛−1(𝑡)+⋯+𝛽₂𝑧^𝜇/𝑟2₂(𝑡)+𝛽₁𝑧^𝜇/𝑟1₁(𝑡)⋯^{(𝑟𝑛+𝑚)/𝜇}, (16) where the same 𝛽_𝑖, 𝑖=1,…,𝑛 as in (9). Now, the closed-loop system (15) and (16) can be rewritten as 𝑍̇=𝐿𝐹(𝑍)+𝜙₁(⋅),1𝐿^𝜅2𝜙₂1(⋅),…,𝐿^𝜅𝑛𝜙_𝑛(⋅),0,…,0^𝑇, (17) where the vector field 𝐹(𝑍) has the exactly same construction of (10). Therefore, we adopt the same Lyapunov function 𝑈 used in (12), whose derivative along (17) is 𝑈̇=𝜕𝑈(𝑍)+𝜕𝑍𝐿𝐹𝜕𝑈×𝜙𝜕𝑍₁(⋅),1𝐿^𝜅2𝜙₂(⋅)1,…,𝐿^𝜅𝑛𝜙_𝑛(⋅),0,…,0^𝑇≤−𝐿𝑐₁‖𝑍‖_Δ^2𝜇+^𝑛_𝑖=1𝜕𝑈𝜕𝑍_𝑖𝜙_𝑖𝐿^𝜅𝑖. (18) According to Assumption 1, there exists a constant 𝑙_𝑖>0 such that 1𝐿^𝜅𝑖𝜙_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑢(𝑡))≤𝑐₁𝑧₁^{(𝑟𝑖+𝑚)/𝑟1}+𝐿^𝜅2𝑧₂^{(𝑟𝑖+𝑚)/𝑟2}+⋯+𝐿^𝜅𝑖𝑧_𝑖^{(𝑟𝑖+𝑚)/𝑟𝑖}1𝐿^𝜅𝑖+𝑐₂𝑧₁(𝑡−𝜏)^{(𝑟𝑖+𝑚)/𝑟1}+𝐿^𝜅2𝑧₂(𝑡−𝜏)^{(𝑟𝑖+𝑚)/𝑟2}+⋯+𝐿^𝜅𝑖𝑧_𝑖(𝑡−𝜏)^{(𝑟𝑖+𝑚)/𝑟𝑖}1𝐿^𝜅𝑖≤𝑙_𝑖𝐿^1−𝜈𝑖𝑧₁^{(𝑟𝑖+𝑚)/𝑟1}+⋯+𝑧_𝑖^{(𝑟𝑖+𝑚)/𝑟𝑖}+𝑧₁(𝑡−𝜏)^{(𝑟𝑖+𝑚)/𝑟1}+⋯+𝑧_𝑖(𝑡−𝜏)^{(𝑟𝑖+𝑚)/𝑟𝑖}≤𝑙_𝑖̃𝐿^1−𝜈𝑖‖𝑍‖^𝑟𝑖_Δ^+𝑚+‖𝑍(𝑡−𝜏)‖^𝑟𝑖_Δ^+𝑚, (19) for a constant 𝜈_𝑖>0, since it can be seen that by definition 𝑟_𝑗=𝑚𝜅_𝑗+1/(𝑝₁⋯𝑝_𝑗−1), so 𝜅_𝑗𝑟_𝑖+1𝑝_𝑖𝑟_𝑗−𝜅_𝑖=𝜅_𝑗𝑚𝜅_𝑖+1/𝑝₁⋯𝑝_𝑖−1+𝑚𝑚𝜅_𝑗+1/𝑝₁⋯𝑝_𝑖−1−𝜅_𝑖=𝑚𝜅_𝑗+𝜅_𝑗/𝑝₁⋯𝑝_𝑖−1−𝜅_𝑖/𝑝₁⋯𝑝_𝑖−1𝑚𝜅_𝑗+1/𝑝₁⋯𝑝_𝑖−1≤𝑚𝜅_𝑗𝑚𝜅_𝑗+1/𝑝₁⋯𝑝_𝑖−1<1,𝑗=2,…,𝑖,𝑖=1,…,𝑛. (20) Substituting (13) and (19) into (18) yields 𝑈̇≤−𝐿𝑐₁−𝑐₂^𝑛_𝑖=1𝑙_𝑖̃𝐿^−𝜈𝑖‖𝑍‖_Δ^2𝜇+𝑐₄^𝑛_𝑖=1𝑙_𝑖̃𝐿^−𝜈𝑖‖𝑍‖^{2𝜇−𝑟𝑖}_Δ^−𝑚⋅‖𝑍(𝑡−𝜏)‖^𝑟𝑖_Δ^+𝑚. (21) By Lemma A.5 in the appendix, there exists a constant 𝑐₃>0 such that 𝑐₂‖𝑍‖^{2𝜇−𝑟𝑖}_Δ^−𝑚⋅‖𝑍(𝑡−𝜏)‖^𝑟𝑖_Δ^+𝑚≤‖𝑍‖_Δ^2𝜇+𝑐₃‖𝑍(𝑡−𝜏)‖_Δ^2𝜇, (22) which yields 𝑈̇≤−𝐿𝑐₁−𝑐₂+1^𝑛_𝑖=1𝑙_𝑖̃𝐿^−𝜈𝑖‖𝑍‖_Δ^2𝜇+𝑐₃^𝑛_𝑖=1𝑙_𝑖̃𝐿^−𝜈𝑖‖𝑍(𝑡−𝜏)‖_Δ^2𝜇. (23) Construct a candidate Lyapunov functional as follows: 𝑊=𝑈+^𝑡_𝑡−𝜏‖𝑍(𝜃)‖_Δ^2𝜇𝑆𝑑𝜃, (24) where 𝑆 is a positive constant. Let 𝑆=𝑐₃∑^𝑛_𝑖=1𝑙_𝑖̃𝐿^1−𝜈𝑖, so it follows from (23) and (24) that 𝑊̇≤−𝐿𝑐₁−1+𝑐₂+𝑐₃^𝑛_𝑖=1𝑙_𝑖̃𝐿^−𝜈𝑖‖𝑍‖_Δ^2𝜇. (25) Hence, by choosing a large enough 𝐿 as 𝐿>max{(((1+𝑐₂+𝑐₃)/𝑐₁)∑^𝑛_𝑖=1𝑙_𝑖̃)^1/𝜈,1}, where 𝜈=min_{𝑖=1,…,𝑛}{𝜈_𝑖}, the right-hand side of (25) is negative definite, that is, there exists a constant 𝜌>0, such that 𝑊̇≤−𝜌𝐿‖𝑍‖_Δ^2𝜇. (26) As a conclusion, we know that the system described by (1) under Assumption 1 can be globally stabilized by the output controller (8).

3. Extension to Nontriangular Systems

Assumption 5. Consider the following 1𝐿^𝜅𝑖𝜙_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑢(𝑡))≤𝐿^1−𝜈𝑖𝑥₁𝐿^{𝜅1(𝑟𝑖+𝑚)/𝑟1}𝑥+⋯+^𝑖𝐿^{𝜅𝑖(𝑟𝑖+𝑚)/𝑟𝑖}+𝑥₁(𝑡−𝜏)𝐿^{𝜅1(𝑟𝑖+𝑚)/𝑟1}𝑥+⋯+^𝑖(𝑡−𝜏)𝐿^{𝜅𝑖(𝑟𝑖+𝑚)/𝑟𝑖}. (27) By (19), it is apparent that Assumption 5 includes Assumption 1 as a special case. The next theorem is a more general result achieved under Assumption 5.

Theorem 6. Under Assumption 5, there exist constants 𝑙_𝑖 and 𝛽_𝑖, 𝑖=1,…,𝑛 and 𝐿>1 such that the homogeneous output feedback controller (3) globally stabilizes system (1).

Proof of Theorem 6. The proof is very similar to that of Theorem 4. We can use the exactly same observer (4) and control law (3). Although the nonlinear function is not in the triangular form, Assumption 5 will lead directly to (19) by using the change of coordinates (14). Then, the global stabilization can be concluded with an appropriate choice of gain 𝐿, which is similar to that in (25). The detailed proof is omitted here for brevity.

4. Examples and Simulations

Consider the following inherently nonlinear time-delay system.

Example 1. Consider 𝑥̇₁(𝑡)=𝑥₂^5/3(𝑡)+0.25𝑥²₁(𝑡)sin𝑥₂(𝑡),𝑥̇₂(𝑡)=𝑢(𝑡)+0.125𝑥₂^5/3(𝑡−0.2),𝑦=𝑥₁(𝑡). (28) Therefore, Assumption 1 holds for 𝑝₁=5/3, 𝑚=2/3, 𝑟₁=1, 𝑟₂=1, and 𝜇=5/3. By (3), the output controller can be constructed as 𝜂̇₂=−𝐿^3/5𝜂₂+𝑙₁𝑦,𝑢=−𝐿^8/5𝛽₂𝜂₂+𝑙₁𝑦^5/3+𝛽₁𝑦^5/3, (29) where we choose 𝑙₁=0.55, 𝛽₁=1.06, 𝛽₂=1.93, and 𝐿=4. The computer simulation results of the closed-loop systems are given in Figures 1 and 2 with the following initial functions ^𝑥1_𝑥^(𝑡)_2(𝑡)=^{−5sin(𝑡−0.2)}_{−3sin(𝑡−0.2)}, 𝜂₂(𝑡)=0, for −0.2≤𝑡≤0.

Figure 1 

State trajectories of𝑥(𝑡)in (28).

Figure 2 

Time history of𝑢(𝑡)in (29).

5. Conclusion

In this paper, we have studied the problem of global output feedback stabilization for a class of higher-order time-delay nonlinear systems under a homogeneous condition. First, homogeneous output feedback controllers have been constructed with adjustable scaling gains. Then, with the help of a homogeneous Lyapunov-Krasovskii functional, we’ve redesigned the homogeneous domination approach to tune the scaling gain for the overall stability of the closed-loop systems. The output feedback controllers proposed in this paper are memoryless and, therefore, can be easily implemented in practice.

Appendix

This appendix collects the definition of homogeneous function and several useful lemmas.

Definition A.1 (see [18]). For a set of coordinates (𝑥₁,…,𝑥_𝑛)^𝑇∈ℝ^𝑛 and an 𝑛-tuple of positive real numbers 𝑟=(𝑟₁,…,𝑟_𝑛).(i)The dilation Δ^𝑟_𝜀(𝑥) is defined by Δ^𝑟_𝜀(𝑥)∶=(𝜀^𝑟1𝑥₁,…,𝜀^𝑟𝑛𝑥_𝑛), ∀𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈ℝ^𝑛, ∀𝜀>0, with 𝑟_𝑖 being called as the weights of the coordinates. For simplicity of notation, we define dilation weight Δ=(𝑟₁,…,𝑟_𝑛).(ii)A function 𝑉∶ℝ^𝑛→ℝ is said to be Δ^𝑟-homogeneous of degree 𝑚  (𝑚∈ℝ) if 𝑉(Δ^𝑟_𝜀(𝑥))=𝜀^𝑚𝑉(𝑥₁,…,𝑥_𝑛), ∀𝑥∈ℝ^𝑛, ∀𝜀>0.(iii)A vector field 𝑓=(𝑓₁,…,𝑓_𝑛)^𝑇∶ℝ^𝑛→ℝ^𝑛 is said to be Δ^𝑟-homogeneous of degree 𝑘 if the component 𝑓_𝑖 is Δ^𝑟-homogeneous of degree 𝑘+𝑟_𝑖 for each 𝑖, that is, 𝑓_𝑖(𝜀^𝑟1𝑥₁,…,𝜀^𝑟𝑛𝑥_𝑛)=𝜀^{𝑘+𝑟𝑖}𝑓_𝑖(𝑥), ∀𝑥∈ℝ^𝑛, ∀𝜀>0, 𝑖=1,…,𝑛.

Lemma A.2 (see [18]). If the trivial solution 𝑥=0 of the Δ_𝜀-homogeneous system 𝑥̇=𝑓(𝑥),𝑓(0)=0,𝑥∈ℝ^𝑛 (A.1) is globally asymptotically stable, there exists a Δ_𝜀-homogeneous Lyapunov function 𝑉, which is positive definite and proper, such that 𝑉̇|_(A.1)=𝜕𝑉𝑓(𝑥)𝜕𝑥<0,∀𝑥≠0. (A.2)

Lemma A.3 (see [18]). Denote Δ as dilation weight, and suppose 𝑉₁(𝑥) and 𝑉₂(𝑥) are homogeneous functions with degree 𝑚₁ and 𝑚₂ respectively. Then 𝑉₁(𝑥)𝑉₂(𝑥) is still a homogeneous function with degree of 𝑚₁+𝑚₂ with same dilation weight Δ.

Lemma A.4 (see [18]). Suppose 𝑉∶ℝ^𝑛→ℝ is a homogeneous function of degree 𝑚 with respect to dilation weight Δ, then the following holds.(1)𝜕𝑉/𝜕𝑥_𝑖 is homogeneous of degree 𝑚−𝑟_𝑖 with 𝑟_𝑖 being the homogeneous weight of 𝑥_𝑖.(2)There is a constant 𝑐 such that 𝑉(𝑥)≤𝑐‖𝑥‖^𝑚_Δ. Moreover, if 𝑉(𝑥) is positive definite, 𝑉(𝑥)≥𝑐‖̃𝑥‖^𝑚_Δ, for a constant 𝑐̃>0.

Lemma A.5 (see [5]). Let 𝑐, 𝑑 be positive real numbers. The following holds for 𝑥∈ℝ, 𝑦∈ℝ, and any positive real-valued function 𝛾(𝑥,𝑦)|𝑥|^𝑐𝑦^𝑑≤𝑐𝛾𝑥|𝑐+𝑑𝑥,𝑦|^𝑐+𝑑+𝑑𝛾𝑐+𝑑^−𝑐/𝑑𝑥,𝑦𝑦^𝑐+𝑑. (A.3)

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (61374038, 61273119, 61174076), Natural Science Foundation of Jiangsu Province of China (BK2011253), and Research Fund for the Doctoral Program of Higher Education of China (20110092110021).