ƯỚC LƯỢNG MIỀN HẤP DẪN CHO HỆ Ô-TÔ-NÔM BẰNG HÀM LYAPUNOV LIÊN TỤC, AFFINE TỪNG MẢNH

Từ khóa: hệ ô-tô nôm; miền hấp dẫn; lý thuyết Lyapunov; hàm Lyapunov; hàm Lyapunov liên tục, affine từng mảnh.[r]

ESTIMATING THE REGION OF ATTRACTION FOR AN AUTONOMOUS SYSTEM WITH CPA LYAPUNOV FUNCTIONS

Tran Thi Hue, Dinh Van Tiep *

University of Technology - TNU

ABSTRACT

Lyapunov Converse Theorems only tell us the sufficient conditions for the existence of a Lyapunov function in a nonconstructive way They are helpless to construct such a function Recently, constructing Continuous Piecewise Affine (CPA) Lyapunov functions has been developed Basing on this technique, we can construct one Then, this construction can help us to find a very exact estimate of the region of attraction This is the main result of the paper We use the method to estimate the region of attraction for the case of the asymptotical stability We study this technique for the case autonomous systems

Keywords: autonomous system; the region of attraction; Lyapunov theory; Lyapunov functions;

CPA Lyapunov functions.

INTRODUCTION*

We study the autonomous system

𝒙̇ = 𝑓(𝒙) (1)

where 𝑓: 𝑹𝒏→ 𝑹𝒏 belongs to 𝐶2(𝑹𝒏, 𝑹𝒏),

and 𝑓(𝟎) = 𝟎, i.e 𝒙∗= 𝟎 is an equilibrium

point

Lyapunov Theorem asserts that if we can find

a neighborhood 𝐷 ⊂ 𝑹𝒏 of the origin 𝟎 ∈ 𝑹𝒏,

and a continuously differentiable, positive

definite function 𝑉: 𝑹𝒏→ 𝑹, 𝑉(𝟎) = 0, called

a Lyapunov function, which possesses a

negative definite derivative along the

trajectory of (1), then 𝒙∗= 𝟎 is

asymptotically stable Moreover, if 𝑉 satisfies

that

𝑎‖𝒙‖𝛼 ≤ 𝑉(𝒙) ≤ 𝑏‖𝒙‖𝛼,

𝑉̇(𝒙) = ∇𝑉(𝒙) ⋅ 𝑓(𝒙) ≤ −𝑐‖𝒙‖𝛼, (2)

for some a, b, c, 𝛼 > 0, ∀𝒙 ∈ 𝑹𝒏, then 𝒙∗= 𝟎

is exponentially stable The existence of such

a Lyapunov function 𝑉 was also already

affirmed if 𝒙∗= 𝟎 is exponentially stable

However, the construction of 𝑉 is recently

known (cf [1]) The method of constructing

𝑉 was based on the Linear Programming with

the constraints guaranteeing the condition (2),

*

Tel: 0968 599033, Email: tiepdinhvan@gmail.com

relaxing the condition of continuous differentiability to only locally Lipschitz continuity The constraints here are represented linearly, and function 𝑉 now is required only continuously piecewise affine (CPA) To generalize the concept of derivative, we introduce the notation of Dini’s derivative (the upper Dini’s derivative) along the trajectory of the system (1) That is,

𝐷+𝑉(𝒙) ≔ lim

𝑡→0sup𝑉(𝒙 + 𝑡𝑓(𝒙)) − 𝑉(𝒙)

So, if we have a CPA function 𝑉 satisfying the following condition, instead of (2), for

𝛼 = 1, 𝑎‖𝒙‖ ≤ 𝑉(𝒙) ≤ 𝑏‖𝒙‖,

𝐷+𝑉(𝒙) ≤ −𝑐‖𝒙‖, (2)′ then 𝒙∗= 𝟎 is exponentially stable (cf

Theorem 4.10 [4]) Conversely, if 𝒙∗= 𝟎 is

exponentially stable for (1), (cf [1]) we can construct a CPA Lyapunov function 𝑉 defining on a neighborhood 𝐷 of the origin The main contribution of this article is to provide a method to estimate the region of attraction,

ℛ{𝒙 ∈ 𝑹𝒏| lim𝑡→∞𝑠𝑢𝑝 𝜙(𝑡, 𝒙) = 𝟎}, where

𝜙(𝑡, 𝒙) is the solution of (1) satisfying that

𝜙(0, 𝒙) = 𝒙 The idea for this estimation is

based on Proposition 1 (cf [2] for detail)

Proposition 1 If 𝑉: 𝐷 → 𝑹 is a positive

definite function, and 𝐷+𝑉(𝜙(𝑡, 𝒙)) < 0 for

all 𝜙(𝑡, 𝒙) ∈ 𝐷, then every compact,

connected component of 𝑉−1([0, 𝑐]), ∀ 𝑐 > 0,

is a subset of ℛ

LINEAR PROGRAMMING PROBLEM

This section, we are going to discuss the

problem of constructing a CPA Lyapunov

function based on a linear programming To

do so, first, we consider a neighborhood

𝐷 ⊂ 𝑹𝒏 of the origin, which can be

partitioned into n-simplices Δ𝜈 This simplical

partition is called a triangulation Λ That is,

𝐷 = ⋃Δ𝜈∈ΛΔ𝜈

For each Δ𝜈∈ Λ, assume that 𝒙𝟎, 𝒙𝟏, … , 𝒙𝒏

are all its vertices We require the

triangulation satisfying that 𝟎 = 𝒙𝟎 if 𝟎 ∈ Δ𝜈,

and that there exists Δ𝜈 ∈ Λ containing 𝟎

Now, we assign each vertex 𝒙𝒊 ∈ Δ𝜈 a value

𝑉𝒙𝒊∈ 𝑹 This setting naturally defines

uniquely a linear function 𝑉𝜈 on Δ𝜈, given by

𝑉𝜈(𝒙𝒊) = 𝑉𝒙𝒊 So, for every 𝒙 ∈ Δ𝜈, 𝒙 =

∑𝑛𝑖=0𝜆𝑖𝒙𝒊, (0 ≤ 𝜆𝑖 ≤ 1, ∑𝑛𝑖=0𝜆𝑖 = 1), we

have 𝑉ν(𝒙) = ∑𝑛𝑖=0𝜆𝑖𝑉𝒙𝒊= ∇𝑉𝜈⋅ 𝒙 + 𝑎𝜈 (for

some constant vectors ∇𝑉𝜈 , 𝑎𝜈 of 𝑹𝒏) Let 𝐷Λ

be the set of all vertices of all n-simplices Δ𝜈

in Λ Then, we can use the set {𝑉𝒙|𝒙 ∈ 𝐷Λ} ⊂

𝑹 to parameterize a CPA function 𝑉 defined

on 𝐷 by 𝑉|Δ𝜈 = 𝑉𝜈 Now, we set up the

constraints for these parameters under which

the obtained function 𝑉 fulfilled (2)′

Linear programming problem (LPP): The

variables of the LPP are 𝑉𝒙 (∀𝒙 ∈ 𝐷Λ) and

𝑊𝜈,𝑖 (∀𝑖 = 1,2, … , 𝑛, ∀Δ𝜈∈ Λ) The

constraints are

Constraint 1 Set 𝑉𝟎= 0, and ∀Δ𝜈∈ Λ, with

all its simplices 𝒙𝟎, 𝒙𝟏, … , 𝒙𝒏, we need

𝑉𝒙𝒊≥ ‖𝒙𝒊‖2

Constraint 2 ∀𝑖 = 1,2, … , 𝑛, and ∀∆𝜈∈ Λ,

we need |∇𝑉𝜈,𝑖| ≤ 𝑊𝜈,𝑖, where ∇𝑉𝜈,𝑖 is the i-th

component of the constant vector ∇𝑉𝜈

Constraint 3 ∀Δ𝜈 ∈ Λ, with all its simplices

𝒙𝟎, 𝒙𝟏, … , 𝒙𝒏, and ∀𝑖 = 0,1, … , 𝑛, we require

that

∇𝑉𝜈⋅ 𝑓(𝒙𝒊) + 𝑘𝜈,𝑖∑ 𝑊𝜈,𝑗

𝑛 𝑗=1

≤ −‖𝒙𝒊‖2, where the constant 𝑘𝜈,𝑖 is defined by

𝑘𝜈,𝑖 ≔ 𝑛𝐾𝜈

2 ‖𝒙𝒊− 𝒙𝟎‖2( max

2

− ‖𝒙𝒊− 𝒙𝟎‖2), with 𝐾𝜈 is an upper bound of all second order partial derivatives of 𝑓, on ∆𝜈

In Constraint 2, ∇𝑉𝜈,𝑖 are a linear combination of (𝑉𝒙𝒊− 𝑉𝒙𝟎), ∀𝑖 = 1,2, … , 𝑛 (cf [3]) Therefore, it is indeed a linear constraint We are now going to prove that a feasible solution of LPP succeed in parameterizing a CPA Lyapunov function 𝑉 First, we need some other results

Lemma 1 Let Δ𝜈 = 𝑐𝑜𝑛𝑣{𝒙𝟎, 𝒙𝟏, … , 𝒙𝒏} is

an n-simplex Then, for every 𝒙 ∈ Δ𝜈, assuming that 𝒙 = ∑𝑛 𝜆𝑖𝒙𝒊

𝑖=0 , (0 ≤ 𝜆𝑖 ≤

1, ∑𝑛 𝜆𝑖

𝑖=0 = 1), we have

‖𝑓(𝒙) − ∑ 𝜆𝑖𝑓(𝒙𝒊)

𝑛 𝑖=0

‖

∞

≤ ∑ 𝜆𝑖𝑘𝜈,𝑖

𝑛

𝑖=0

(The proof is available in [1], Lemma 4.16)

Lemma 2 Assume that ∇𝑉𝜈⋅ 𝑓(𝒙) ≤

−‖𝒙‖, ∀𝒙 ∈ Δ𝑂𝜈 (Δ𝜈𝑂 is the interior of Δ𝜈) Then, 𝐷+𝑉(𝒙) ≤ −‖𝒙‖, ∀𝒙 ∈ Δ𝑂𝜈

(For the proof, cf [1], Theorem 4.17)

Now, we are ready to prove the above

statement

Theorem 1 Suppose that LPP has a feasible

solution 𝑉𝒙 (∀𝒙 ∈ 𝐷Λ) and 𝑊𝜈,𝑖 (∀𝑖 = 1,2, … , 𝑛, ∀Δ𝜈 ∈ Λ) Then, the CPA function

𝑉 parameterized by {𝑉𝒙|𝒙 ∈ 𝐷Λ}, (that is, 𝑉|Δ𝜈 = 𝑉𝜈, ∀Δ𝜈∈ Λ) is a CPA Lyapunov function

Proof We are going to show that the CPA

function 𝑉 parameterized a feasible solution

{𝑉𝒙|𝒙 ∈ 𝐷Λ} fulfills (2)′ Indeed, ∀Δ𝜈 ∈ Λ,

∀𝒙 ∈ Δ𝜈, 𝒙 = ∑𝑛𝑖=0𝜆𝑖𝒙𝒊, (0 ≤ 𝜆∑ 𝜆𝑖 ≤ 1,

𝑖

𝑛

By Hölder′s inequality, Lemma 1 and

Constraint 2,3, we have

∇𝑉𝜈⋅ 𝑓(𝒙) = ∇𝑉𝜈⋅ ∑ 𝜆𝑖𝑓(𝒙𝒊)

𝑛 𝑖=0

+ ∇𝑉𝜈⋅ (𝑓(𝒙) − ∑ 𝜆𝑖𝑓(𝒙𝒊)

𝑛 𝑖=0

)

≤ ∇𝑉𝜈⋅ ∑𝑛 𝜆𝑖𝑓(𝒙𝒊)

+‖∇𝑉𝜈‖1⋅ ‖𝑓(𝒙) − ∑𝑛 𝜆𝑖𝑓(𝒙𝒊)

≤

∑𝑛 𝜆𝑖∇𝑉𝜈⋅ 𝑓(𝒙𝒊)

(∑𝑛𝑖=0𝜆𝑖𝑘𝜈,𝑖)

≤ ∑ 𝜆𝑖

𝑛

𝑖=0

(∇𝑉𝜈⋅ 𝑓(𝒙𝒊) + 𝑘𝜈,𝑖(∑|∇𝑉𝜈,𝑖|

𝑛 𝑖=1

))

≤ − ∑ 𝜆𝑖‖𝒙𝒊‖2

𝑛

𝑖=0

≤ − ‖∑ 𝜆𝑖𝒙𝒊

𝑛 𝑖=0

‖

2

= −‖𝒙‖2 Hence, ∇𝑉𝜈⋅ 𝑓(𝒙) ≤ −‖𝒙‖2, ∀𝒙 ∈ Δ𝜈 So, by

Lemma 2,

𝐷+𝑉(𝒙) ≤ −‖𝒙‖, ∀𝒙 ∈ Δ𝑂𝜈

Moreover, by Constraint 1,

𝑉(𝒙) = ∑ 𝜆𝑖𝑉𝒙𝒊

𝑛 𝑖=0

≥ ∑ 𝜆𝑖

𝑛 𝑖=0

‖𝒙𝒊‖2

≥ ‖∑ 𝜆𝑖

𝑛

𝑖=0

𝒙𝒊‖

2

= ‖𝒙‖2, ∀𝒙 ∈ Δ𝜈

The condition that 𝑉(𝒙) ≤ 𝑏‖𝒙‖2 is quiet

obvious by taking 𝑏 = maxΔ𝜇∈Λ‖∇𝑉𝜇‖2 in the

case 𝒙 ∈ Δ𝜈, where 𝟎 ∈ Δ𝜈, and 𝑏 =

(maxΔ𝜇∈Λ‖∇𝑉𝜇‖2+ ‖𝑎𝜈 ‖2

𝟎 ∉ Δ𝜈 So, 𝑉 satisfies (2)′ □

SIMPLICIAL PARTITIONS OF 𝑹𝒏

This section is sacrificed to introduce the

method of triangulate a neighborhood

𝐷 ⊂ 𝑹𝒏 of the origin 𝟎 We focus on a

strategy of partition which provides a feasible solution for LPP, in the case 𝒙∗= 𝟎 is an

exponentially stable equilibrium point Such a strategy exists (cf [1], [3])

First, consider 𝐷 = [−𝑏, 𝑏]𝑛 for some 𝑏 > 0 Assume that 0 = 𝑦0< 𝑦1< < 𝑦𝑁 = 𝑏 Let {𝒆𝟏, 𝒆𝟐, … , 𝒆𝒏} be the standard basis of 𝑹𝒏

Piecewise Scaling Functions: Consider a

function 𝑷𝑺: [−𝑁, 𝑁]𝑛⟶ [−𝑏, 𝑏]𝑛, given by 𝑷𝑺(𝒙) = ∑ 𝑠𝑖𝑔𝑛(𝑥𝑖) 𝑃(|𝑥𝑖|)𝒆𝑖

𝑛 𝑖=1

, (3) where 𝒙 = ∑𝒏 𝑥𝑖𝒆𝒊

𝒊=𝟏 and 𝑃: [0, 𝑁] → [0, 𝑏] is

a CPA function, defined by 𝑃(𝑖) = 𝑦𝑖, ∀𝑖 = 0,1, … , 𝑁, and 𝑃|[𝑖,𝑖+1] is affine 𝑷𝑺 is called a piecewise scaling function

The Reflection Function: Let 𝒥 be a subset

of the set {1,2, … , 𝑛}, and 𝒳𝒥: {1,2, … , 𝑛} → {0,1} be the characteristic function of 𝒥 We call the following function 𝓡𝒥 a reflection function, 𝓡𝒥: 𝑹𝒏⟶ 𝑹𝒏, given by

𝓡𝒥(𝒙) = ∑𝑛 (−1)𝒳𝒥(𝑖)𝑥𝑖𝒆𝒊

The Basic Triangulation: For each 𝜎 in the permutation group 𝑆𝑦𝑚𝑛 of {1,2, … , 𝑛}, and for each 𝒛 ∈ 𝒁≥0𝒏 , define an n-simplex

∆𝒥𝒛,𝜎≔ 𝑐𝑜𝑛𝑣{𝓡𝒥(𝒛), 𝓡𝒥(𝒛 + ∑𝑗𝑖=1𝒆𝜎(𝑖)), ∀𝑗 =

1,2, … , 𝑛}

We get naturally a partition of 𝑹𝒏, called the

basic triangulation of 𝑹𝒏 That is,

𝑹𝒏= ⋃ {∆𝒥𝒛,𝜎|𝒛 ∈ 𝒁≥𝟎𝒏 , 𝜎 ∈ 𝑆𝑦𝑚𝑛, 𝒥

⊂ {1,2, … , 𝑛}}

Proposition 2 The region 𝐷 = [−𝑏, 𝑏]𝑛 can

be triangulated as follows:

𝐷 = ⋃ {𝑷𝑺(∆𝒥𝒛,𝜎)|𝒛 ∈ [0, 𝑁]𝒏∩ 𝒁≥𝟎𝒏 , 𝜎

∈ 𝑆𝑦𝑚𝑛, 𝒥 ⊂ {1,2, … , 𝑛}}

Proof Because each component 𝑃𝑆𝑖(𝑥𝑖) ≔ 𝑠𝑖𝑔𝑛(𝑥𝑖) 𝑃(|𝑥𝑖|) of the scaling function 𝑷𝑺

are CPA on [−𝑁, 𝑁], so 𝑷𝑺 is CPA on

[−𝑁, 𝑁]𝑛 Therefore, the image of an

n-simplex though 𝑷𝑺 is also an n-n-simplex in 𝐷

Moreover, 𝑷𝑺 is a surjection, and

[−𝑁, 𝑁]𝑛 = ⋃ {∆𝒥𝒛,𝜎|𝒛 ∈ [0, 𝑁]𝒏∩ 𝒁≥𝟎𝒏 , 𝜎

∈ 𝑆𝑦𝑚𝑛, 𝒥 ⊂ {1,2, … , 𝑛}} ,

𝐷 = 𝑷𝑺([−𝑁, 𝑁]𝑛) = ⋃{𝑷𝑺(∆𝒥𝒛,𝜎)|

𝒛 ∈ [0, 𝑁]𝒏∩ 𝒁≥𝟎𝒏 , 𝜎 ∈ 𝑆𝑦𝑚𝑛, 𝒥

⊂ {1,2, … , 𝑛}} □

Fan-like Triangulation: For each 𝐾 ∈ 𝑵,

consider the basic triangulation 𝒯 of

[−2𝐾, 2𝐾]𝑛 For every n-simplex ∆𝒥𝒛,𝜎∈ 𝒯

which intersects the boundary of [−2𝐾, 2𝐾]𝑛,

the intersection is exactly an (n-1)-simplex

𝑐𝑜𝑛𝑣{𝒙𝟏, 𝒙𝟐, … , 𝒙𝒏} = ∆𝒥𝒛,𝜎∩ [−2𝐾, 2𝐾]𝑛,

where 𝒙𝟎, 𝒙𝟏, 𝒙𝟐, … , 𝒙𝒏 are all vertices of

∆𝒥𝒛,𝜎 Construct a new n-simplex by

𝑐𝑜𝑛𝑣{𝟎, 𝒙𝟏, 𝒙𝟐, … , 𝒙𝒏} The set of all such

n-simplices forms a new triangulation 𝒮 of

[−2𝐾, 2𝐾]𝑛, called a fan-like triangulation

If we define a CPA function 𝑃: [0,2𝐾] ⟶

[0, 𝑐], (𝑐 > 0), given by 𝑥 ↦ 𝑐2−𝐾𝑥, we get

correspondingly a scaling function

𝑷𝑺: [−2𝐾, 2𝐾]𝑛⟶ [−𝑐, 𝑐]𝑛 defined as (3)

Hence, a fan-like triangulation 𝒮𝐾,𝑐 of

[−𝑐, 𝑐]𝑛 can be revealed by [−𝑐, 𝑐]𝑛=

⋃{𝑷𝑺(∆)|∀∆∈ 𝒮}

Proposition 3 Assume that 𝒙∗= 𝟎 is an

exponential equilibrium point of the system

(1) Consider the sequence of fan-like

triangulations {𝒮

𝐾≥0

There exists

𝐾 ∈ 𝑵 such that LPP has a feasible solution

on the region 𝐷𝐾≔ ⋃ {∆|∆∈ 𝒮

Proof Refer to [3]

ATTRACTION

Assume that 𝒙∗= 𝟎 is an exponential

equilibrium point of the system (1) We state

the algorithm

Step 1 Find a region 𝐷𝐾 in Proposition 3 Then, set 𝑦0= 0, 𝑦1= (3/4)𝐾

Step 2 For each 2 ≤ 𝑁 ∈ 𝑵, take arbitrarily a positive number 𝑦𝑁 Define a CPA function 𝑃: [0, 𝑁] ⟶ [0, 𝑦𝑁], given by 𝑃 (𝑖 (83)𝐾) =

𝑦𝑖 and 𝑃|

a scaling function 𝑷𝑺 as (3), and establish the triangulation 𝒯𝑁 of [– 𝑦𝑁, 𝑦𝑁]\[−𝑦1, 𝑦1] as in Proposition 2 Finally, triangulate [−𝑦𝑁, 𝑦𝑁]

as [– 𝑦𝑁, 𝑦𝑁] = ⋃ {∆|∆∈ 𝒯𝑁∪ 𝒮

𝐾,(34)𝐾} We denote this triangulation 𝔗𝑁

Step 3 Check that whether the triangulation

𝔗𝑁 of [−𝑦𝑁, 𝑦𝑁] guarantees the existence of a feasible solution for LPP or not If it does, then [−𝑦𝑁, 𝑦𝑁] is a subset of the region of attraction ℛ, increase 𝑁 to 𝑁 + 1 and repeat Step 2, 3 If it does not, stop the algorithm The region returned at Step 3 is an estimate of the region of attraction ℛ Note that, in order for the choose of the sequence {𝑦𝑁}𝑁≥2 to be automatically performed, we take 𝑦𝑁 ≔

𝑁, ∀𝑁 ≥ 2

Theorem 2 The algorithm always succeeds

in finding an estimate of the region of attraction

Proof By Proposition 3, the region 𝐷𝐾 in Step 1 always exists Moreover, by Theorem

1 and Proposition 1, the region returned in Step 3 is a subset of ℛ, so it gives a lower estimate for ℛ □

COMPARISON OF THE METHOD AND THE INDIRECT METHOD

This section aims to evaluate the advantage of the above method of estimating ℛ by comparing it with one secured by the indirect method First we introduce the estimation of

ℛ secured by the indirect method In this method, we find a Lyapunov function of the form 𝑉(𝒙) = 𝒙𝑇𝑃𝒙, where 𝑃 is positive

definite matrix solving the Lyapunov

equation 𝑃𝐴 + 𝐴𝑃𝑇 = −𝐼𝑛, with 𝐴 is

Jacobian of 𝑓 at 𝟎 Let 𝜙(𝑡, 𝒙) be a solution

of (1), then 𝑉̇(𝜙) = −‖𝜙‖22+ 𝜙𝑇𝑃(𝑓(𝜙) −

𝐴𝜙) + (𝑓(𝜙) − 𝐴𝜙)𝑇𝑃𝜙 ≤ −‖𝜙‖2(‖𝜙‖2−

2‖𝑃‖2‖𝑓(𝜙) − 𝐴𝜙‖2) So, 𝑉̇(𝜙) < 0, ∀𝜙

such that

𝜙(𝑡, 𝒙) ∈ {𝝎 ∈ 𝑹𝒏|‖𝑓(𝝎) − 𝐴𝝎‖2< ‖𝝎‖2

By Taylor’s Theorem, with

𝛼𝑖𝑗𝑘 ≥ sup

𝜕𝑥𝑘𝜕𝑥𝑗(𝝎)| , ∀𝑖 = 1, 𝑛̅̅̅̅̅,

we have

‖𝑓(𝝎) − 𝐴𝝎‖2 ≤ ‖12 ∑ 𝑥𝑗𝑥𝑘𝛼𝑖𝑗𝑘𝒆𝒊

𝑛 𝑖,𝑗,𝑘=1

‖

2

≤‖𝝎‖∞2

𝑛 𝑗,𝑘=1

)

𝑛 𝑖=1

2

< ‖𝝎‖2 2‖𝑃‖2 The last inequality is valid only if ‖𝝎‖∞<

(‖𝑃‖2√∑ (∑𝑛 𝛼𝑖𝑗𝑘

𝑛

𝑖=1

2

)

−1

=: 𝛼 Hence,

the set Ωc≔ {𝝎 ∈ 𝑹𝒏|𝝎𝑻𝑃𝝎 < 𝑐} ⊂

(−𝛼, 𝛼)𝑛 for some 𝑐 > 0 is the lower

estimate of ℛ Here, 𝑐 should be taken such

that 𝜕Ωc intersects 𝜕(−𝛼, 𝛼)𝑛, so that Ωc is

the best estimate of ℛ secured by the indirect

method

EXAMPLES

Example 1[6] Consider the system:

𝑥̇1= −𝑥2, 𝑥̇2 = 𝑥1− 𝑥2+ 𝑥12𝑥2− 0.1𝑥14𝑥2

Choose 𝑁 = 8, 𝑦0= 0, 𝑦1 = 0.078, 𝑦2 =

0.28, 𝑦3 = 0.51, 𝑦4 = 0.696, 𝑦5 = 0.842,

𝑦6 = 0.96, 𝑦7= 1.024, 𝑦8= 1.056 The

indirect method returns a Lyapunov function

𝑉(𝒙) = 𝒙𝑇𝑃𝒙, where

𝑃 = [ 1.5−0.5 −0.51 ]

The calculation reveals the values 𝛼211=

2.112, 𝛼212= 𝛼221= 1.641 are all non-zero

𝛼𝑖𝑗𝑘

So, 𝛼 = 0.1025, Ω0.00867 is the best lower estimate of ℛ (cf Figure 1, the small ellipse

is ∂Ω0.00867, while the bigger loop represents the boundary of the estimate of ℛ secured by the CPA Lyapunov function.)

Example 2[6] Consider the system: 𝑥̇1=

𝑥2, 𝑥̇2= −𝑥1− 𝑥2+13𝑥12 Take 𝑁 = 8,

𝑦0= 0, 𝑦1 = 0.156, 𝑦2 = 0.513, 𝑦3 = 0.88, 𝑦4= 1.204, 𝑦5= 1.427, 𝑦6= 1.58, 𝑦7 = 1.662,

𝑦8= 1.686

By the calculation, the indirect method reveals the Lyapunov function 𝑉(𝒙) =32𝑥1 −

𝑥1𝑥2+ 𝑥22, while 𝛼 = 0.52 So, we can find out that Ω0.225 is the best lower estimate of ℛ secured by the indirect method (cf Figure 2, the small ellipse is 𝜕Ω0.225, the bigger loop is the boundary of the estimate of ℛ secured by the CPA Lyapunov function.)

SUMMARY

By constructing a CPA Lyapunov function,

we can find a good estimation for the region

of attraction The algorithm we state in this paper always succeeds in finding such an estimate if the operating equilibrium is exponentially stable The method of

constructing a CPA Lyapunov function

undoubtedly has the upper hand of giving a

better estimate for the region of attraction

However, the drawbacks of the method

includes the complexity of the calculation, so

it takes so much time Besides, a computer

program required to support the calculation of

the algorithm for this method seems quite

sophisticated In the future, we hope to dial

with these problems, or even reduce the

disadvantage of the method

ACKNOWLEDGEMENT

I am very grateful to College of Technology

(Thai Nguyen University), who supports my

paper to public this work in Journal of

Science and Technology, TNU

REFERENCES

1 S Marinosson (2002), Stability Analysis of Nonlinear Systems with Linear Programming: A Lyapunov Functions Based Approach

Gerhard-Mercator-University, Duisburg

2 P Giesl, and S.F Hafstein (2012), “Existence of piecewise linear Lyapunov functions in arbitrary

dimensions”, Discrete and Continuous Dynamical Systems - Series A, 32-10, pp 3539-3565

3 P Giesl, and S.F Hafstein (2014), “Revised CPA method to compute Lyapunov functions for

nonlinear systems”, Journal of Mathematic Analysis and Applications, 410, pp 292-306

4 H Khalil (1992), Nonlinear Systems, New

York: Macmillan

5 S.F Hafstein (2007), “An algorithm for constructing Lyapunov functions”, Monograph,

Electronic Journal of Differential Equations

6 S.F Hafstein (2004), “A constructive converse Lyapunov theorem on exponential stability”, Discrete and Continuous Dynamical Systems - Series A, 10(3), pp 657–678

TÓM TẮT

ƯỚC LƯỢNG MIỀN HẤP DẪN CHO HỆ Ô-TÔ-NÔM BẰNG HÀM LYAPUNOV LIÊN TỤC, AFFINE TỪNG MẢNH

Trần Thị Huê, Đinh Văn Tiệp *

Trường Đại học Kỹ thuật Công nghiệp – ĐHTN

Các Định lý đảo Lyapunov chỉ cho ta biết các điều kiện đủ để suy ra sự tồn tại của các hàm Lyaponov, nhưng không cho biết cách xây dựng các hàm này Gần đây, việc xây dựng các Lyapunov liên tục, affine từng mảnh, đã được phát triển Dựa vào các kết quả này, ta có thể xây dựng được một hàm như vậy Việc xây dựng này sẽ được áp dụng để ước lượng miền hấp dẫn của

hệ Đây là kết quả chính của bài báo Ta sẽ sử dụng phương pháp này để ước lượng miền hấp dẫn cho trường hợp ổn định tiệm cận Hiện tại, phương pháp này vẫn chỉ là một phương pháp mò mẫm Ta nghiên cứu phương pháp này cho hệ ô-tô nôm

Từ khóa: hệ ô-tô nôm; miền hấp dẫn; lý thuyết Lyapunov; hàm Lyapunov; hàm Lyapunov liên tục,

affine từng mảnh

Ngày nhận bài: 15/3/2018; Ngày phản biện: 03/5/2018; Ngày duyệt đăng: 31/5/2018

*

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