86

Chapter IІI. Study of stability of solutions of dynamic systems with cylindrical phase space

The values

3, 4

0, 5

0

satisfied. Consequently,

are selected so, that conditions of Theorem 3 are

									T
	I			lim						[ (				(t))			(t)			2		(t)	2	( (t))]dt C ,					| C			| .
		5
					T 1						3							5				4						1		1
									0
Then improper integral
I	6	lim					T [			1		2	(	2		1	)]g( (t), (t)) (t)								[M	y 2 (t) ... M				n	y 2		(t)]
				T																				1		1 1						n
							0
4 2 ( (t)) dt I5														lim 1				1	2		(T ) y* (T )Ny(T )
																		2
															T
					1	2		(0)		y* (0)Ny(0) | C |,											| C | ,			| C | 0.
			1															1				1			1
				2
From here taking into account that																					I	5 C1 | C1		|,	| C1 | I		5 ,	0 ,					we get an

assessment (3.92). The theorem is proved.

Global asymptotic stability. The next Theorem gives the solution of Problem 2.

Theorem 9. Let the following conditions be satisfied:

1. there exists a vector

*

R

n

such that

B 0,

AB 0, ... ,

n 2

B 0,

A

n 1

B 0;

A

2. the value

0;

1

n;

3. the rank of the matrix

R ||

*

,

*

*

*n 1

*

|| is equal to

A

, ... , A

4.

( )d ; 0; ,

R

.

1

5. Values M

i

0, i 1, n; matrix

N 0.

Then a stationary set

of the system (3.60), (3.61) is globally asymptotically

stable.

Proof. As conditions of Theorem 8 are satisfied, then we have evaluation (3.92).

As conditions of the theorem are satisfied (3.65), then improper integral

I7 0.

Then it

follows from (3.92) that functions

y(t), t I

is not limited, then 0

can be written as

(t), y(t

I7 .

), t I

are limited. If

(t), t I

either

It is impossible. Now evaluation (3.92)

0 I	7	lim	T	[		1		2	(	2		1	)]g( (t), (t)) (t)				[M	1	y 2	(t) ...
		T														1			1
			0																	(3.93)
								12 2
		2													2
... M n yn (t)]																( (t)) dt ,
					4 2 ( 2 2						1 1 )

201

Lectures on the stability of the solution of an equation with differential inclusions

From here we have (v. (3.93))

where

y	(t),
i

							T
			0 lim				[ 1 2 (											2		1 )]g( (t), (t)) (t
					T		0
									T
					0 lim								[M				y	2	(t) .... M							y	2	(t)]
																1									n		n
						T					1						1
									0
											2				2					T
																							2
					0						1								lim					( (t))dt
						4	2	(		2				2				) T
																	1		1	0
	1		(	2		) 0, M						i		0,			i 1, n,					2		0,
		2			1															2					1		1

i 1, n, t I are uniformly continuous.

) ,

,

,

functions

	(3.94)
	(3.95)
	(3.96)
(t),	(t),

From (3.94)–(3.96), in a similar way as in the proof of Theorem 4, we get

lim y(t) 0,	lim (t) 0,	lim (t)	* ,	where ( * ) 0.	This means that the statio-
t	t	t

nary set		of the system (3.60), (3.61) is globally asymptotically stable. The theorem

is proved.

In the theory of synchronous machines there are processes described by equations of the following kind

d	,	d	*	df ( )	( ),	(3.97)
dt		dt	g( , ) x C	d
		dx	Ax Bf ( ) a,			(3.98)
		dt

where the function g( , ),

f ( ), ( ) satisfies the conditions (3.62), (3.63).

The system of differential equations (37, (3.98) by nonsingular transformation

(Lienard transforms)

x(t) z(t) A

1

Bf ( ) A

1

a,

Is reduced to a system of differential equations (3.60), (3.61).

Indeed,

x*C

df ( )

z*C

df ( )

f ( )

* A* 1C

df ( )

a* A*C

df ( )

,

B

d

d

d

d

dx

dz

A 1

df ( )

A[z A 1

B

Bf ( ) A 1a] Bf ( ) a.

dt

dt

d

202

Chapter IІI. Study of stability of solutions of dynamic systems with cylindrical phase space

From here it follows

d	,	d	g( , ) z	Cf ( ) ( ),
			*
dt		dt

where

B A 1B,	f ( )	df ( )	, ( ) ( )	df ( )
						C* A 1[Bf ( ) a].
		d			d

Various applied tasks for synchronous machines can be found in the work [2].

Comments

Global asymptotic stability of dynamic systems with countable equilibrium state was studied for two cases: 1) when the value of the integral of a periodic function on period is equal to zero; 2) when the value of the integral is not equal to zero. A method of selection of the area of global asymptotic stability in space of constructive parameters of the system was developed. The method effectiveness is shown in two examples: the problem of phase synchronization, the motion of the mathematical pendulum. The proposed method of study allows us to highlight a wider stability area in space of parameters of the system than the other known methods.

A distinctive feature of the proposed method distinguishing it from the known methods (frequency and periodic Lyapunov functions) is that the condition of global asymptotic stability follows from estimates of improper integrals along the solution of the system.

In the work the following basic results were obtained: equations of motion of the system via nonsingular transformation were reduced to a special form; identities along the solution of the system regarding nonlinearity and angular coordinates were obtained; evaluation of the solutions of the phase system was made; the asymptotic properties of functions were studied related to the limitations of an improper integral; a new representation of a quadratic form with respect to phase variables as a sum of two terms (the first term is a quadratic form reduced to a diagonal form, and the second term is a full differential function by time) were obtained; based on an assessment of improper integrals along the solution of the system, theorems of global asymptotic stability of a stationary set of the phase system were proved.

Literature

1.Barbashin E.A., Tabueva V.A. Dinamicheskie sistemy s cilindricheskim fazovym prostranstvom.

–M.: Nauka, 1969. – 380 s.

2.Bakaev Ju.N., Guzh A.A. Optimal'nyj priem signalov chastotnoj moduljacii v uslovijah jeffekta Dopplera // Radiotehnika i jelektronika, 1965. – Т. 10. – № 1. – S. 15-27.

3.Shahgil'djan V.V., Lihovkin A.A. Sistemy fazovoj avtopodstrojki chastoty. – M.: Svjaz', 1972.

–364 s.

4.Janko-Trinickij A.A. Novyj metod analiza rabot sinhronnyh dvigatelej pri rezko-peremennyh nagruzkah. – M.-L.: Gosjenergoizdat, 1958. – 240 s.

5.Blehman I.I. Sinhronizacija dinamicheskih sistem. – M.: Nauka, 1971. – 320 s.

6.Triomi F. Integrazione di unequazione differenziale presentatasi in electrotechnica // Annali della Roma schuola Normale Superiore de Pisa Scienza Physiche e Matematiche. – V. 2. – № 2, 1933.

203

Lectures on the stability of the solution of an equation with differential inclusions

7.Amerio L. Determihazione delle condizioni de stabilita per gli integrali di un’equazione interessante l’electrotecnica // Anali di Matematica puza ed applicata. – T. 30, 1949.

8.Seifert G. On the existence of certain solutions of nonlinear differential equations // Zeitschrift fur angewandte Mathematik und Physuk. – V. 3. – № 3, 1952.

9.Leonov G.A. Ob ustojchivosti fazovyh system // Sibirskij matematicheskij zhurnal, 1976. – № 1.

–S. 105-120.

10.Leonov G.A. Ob odnom klasse dinamicheskih sistem s cilindricheskim fazovym prostranstvom // Sibirskij matematicheskij zhurnal, 1976. – № 1. – S. 10-20.

11.Leonov G.A. Teorema svedenija dlja nestacionarnyh nelinejnostej // Vestnik LGU, 1977. – № 7.

–S. 51-62.

12.Andronov A.A., Vitt A.A. Hajkin S.Je. Teorija kolebanij. – M.: Fizmatgiz, 1959. – 570 s.

13.Ajsagaliev S.A., Ajpanov Sh.A. K teorii global'noj asimptoticheskoj ustojchivosti fazovyh sistem.

–Differencial'nye uravnenija. – Minsk – Moskva, 1999. – T. 35. – № 8. – S. 37-49.

14.Ajsagaliev S.A., Imankul T.Sh. Teorija fazovyh sistem. – Almaty: Qazaq universitetі, 2005.

–272 s.

15.Ajsagaliev S.A. Teorija ustojchivosti dinamicheskih sistem. – Almaty: Qazaq universitetі, 2012.

–216 s.

16.Aisagaliev S.A., Kalimoldayev M.N. Certain problems of synchronization theory // Journal Inverse ill-Posed Problems, 21 (2013). – Р. 159-175.

17.Abenov B.K., Ajsagaliev S.A., Kalimoldaev M.N. K absoljutnoj ustojchivosti reguliruemyh sistem v prostom kriticheskom sluchae // Matematicheskij zhurnal, 2014. – Tom 14. – № 1(3.49). – S. 5-33.

18.Abenov B.K., Ajsagaliev S.A., Ajazbaeva A.M. K absoljutnoj ustojchivosti reguliruemyh sistem v kriticheskom sluchae. Vestnik KazNU, ser. mat., meh., inf. 2014. – № 4(83). – S. 12-30.

19.Gelig A.H., Leonov G.A., Jakubovich V.A. Ustojchivost' nelinejnyh sistem s needinstvennym sostojaniem ravnovesija. – M.: Nauka, 1978. – S. 400.

20.Beljustina L.N., Bykov V.V, Kiveleva K.G., Shalfeev V.D. O velichine polosy zahvata sistemy FAP s proporcional'no integrirujushhim fil'trom // Izvestija vuzov, Radiofizika, 1970. – T. 13.

–№ 14. – S. 95-103.

21.Ajsagaliev S.A., Kalimoldaev M.N. Teorija dinamicheskih sistem s cilindricheskim fazovym prostranstvom. – Almaty: Gylym, 2000. – S. 97.

22.Ajsagaliev S.A., Abenov B.K. K global'noj asimptoticheskoj ustojchivosti dinamicheskih sistem s

cilindricheskim fazovym prostranstvom // Vestnik KazNU. Serija mat., meh., inf. – 2010.

– № 2(65). – 7 s.

204

Chapter IV. Global asymptotic stability of multidimensional dynamic systems with cylindrical …

Chapter IV

GLOBAL ASYMPTOTIC STABILITY OF MULTIDIMENSIONAL DYNAMIC SYSTEMS WITH CYLINDRICAL PHASE SPACE

A general theory of global asymptotic stability of multidimensional dynamic systems with cylindrical phase space with a countable equilibrium state is developed. Limits are imposed on solutions of multidimensional phase systems and their derivateives. Conditions under which solution and its derivative have asymptotic properties are found. Conditions of global asymptotic stability of multidimensional phase systems with equal to zero in period values of integrals from components of periodic nonlinearities, and also conditions of global asymptotic stability of phase systems with not equal to zero in period values of integrals from components of nonlinear periodic functions are obtained.

Lecture 32.

Problem statement. Nonsingular transformation

Problem statement. Consider a dynamic system with a cylindrical phase space described by the equation of the following kind:

x Ax B ( ), Cx R (x), (0) = 0, (0) = 0, t I (0, ), (4.1)

where A, B, C, R are constant matrices of orders n×n, n×m, m×n, m×m, respectively,

		____	____
matrix A is a Hurwitz matrix, i.e.	Re j (A) 0,	j 1, n, j	(A) j 1, n	are eigenvalues of

the matrix

A,

x0 , 0 , ( ) ( 1( 1),..., m ( m )),

(

,...,

m

).

1

The function

( )

= { ( ) = ( (

), ,

(

)) C1 (Rm , Rm )/

d i ( i )

0

m

m

2i

1 1

1i

d i

(4.2)

i ( i ) = i ( i i ), i , i R1, i = 1, m},

where i

is a period of the function i ( i ),

2i, i = 1, m are given numbers, | 1 |< ,

| 2 |< ,

1 = ( 11, , 1m ), 2 = ( 21, , 2m ).

The system equilibrium state (4.1), (4.2) is determined from the solution of

algebraic equations. Ax* B ( * ) = 0,

Cx* R ( * ) = 0.

205

Lectures on the stability of the solution of an equation with differential inclusions

As	x* = A	1	B ( * ),

matrix of order m m the

1	B) ( * ) = 0,		1
(R CA		then when	R CA	B is a nonsingular

system (4.1), (4.2) has a stationary set.

= {(x

,

) R

n m

/x = 0, (

) = 0}.

*

*

*

*

As (

*

) = (

*

k ) = 0,

k = 0, 1, 2, ,

then the equilibrium state of the

system (4.1), (4.2) is a countable set,

* = ( 1*, , m* ),

= ( 1, , m ).

Definition 1. Stationary set

of the system (4.1), (4.2)

is globally asympto-

tically

stable, if

for any

functions

( ) 0 and

any

initial

state

(x ,

) R

n m

,

0

0

| x0 |< , | 0

|<

solution

of the

system

x(t) = x(t;

0, x0

,

0 , ),

(t) = (t;

0, x0 , 0

, ),

t I

has the property x(t) x

= 0,

(t)

*

when

t ,

where (

*

) = 0.

*

Definition 2. Conditions of global asymptotic stability of the system (4.1), (4.2)

are

called

relations,

binding

constructive

parameters

of

the

system

( A, B,C, R, 1, 2 ), under which set

is globally asymptotically stable.

It is necessary to examine separately two cases:

i

i

1.

= 0,

, R

,

i = 1, m;

( )d

1

i

i

i

i

i

i

i

i

2.

0,

, R

,

i = 1, m.

( )d

1

i

i

i

i

i

i

The following tasks are proposed:

Problem 1. Find estimates of improper integrals, along with the solution of the

system (4.1), (4.2), for cases 1, 2.

Problem 2. Find new effective condition of global asymptotic stability of a

stationary set of the system (4.1), (4.2)

for the case, when

i i

i ( i )d i = 0, i , i R1, i = 1, m;

i

based on an assessment of improper integrals for case 1.

Problem 3. Find new effective condition of global asymptotic stability of a stationary set of the system (4.1), (4.2) for the case, when

i i

i ( i )d i 0, i , i R1, i = 1, m;

i

based on an assessment of improper integrals for case 2.

206

Chapter IV. Global asymptotic stability of multidimensional dynamic systems with cylindrical …

The first work devoted to qualitatively-numerical methods of study of phase systems was the article of F. Tricomi [1]. Application of the method of point mappings to phase systems was reviewed in the works of A.A. Andronov and his followers [2, 3]. The next stage of development of qualitative numerical methods was the application of Lyapunov’s periodic functions to the study of phase systems. Fundamentals of the theory of Lyapunov’s periodic functions are given in the works [4, 5]. Methods of constructing various Lyapunov’s periodic functions, providing stability in most of phase systems, can be found in [2]. Approximate nonlocal methods of study of phase systems are presented in [6].

The original approach to phase systems is asymptotic stability of frequency conditions, based on Bakaev – Guzh procedure. Such an approach was firstly proposed in the work of G.A. Leonov [7]. In subsequent works of Leonov and his students [8–10] the limitations of the solution of phase systems, asymptotic properties of solutions of integro-differential equations with periodic nonlinearities, as well as stability and fluctuations of phase systems were studied. Bibliographic review of scientific literature on phase systems can be found in monographs [11], and also in the author’s works outlined in [12–16].

Pendulum systems in mechanics, navigation systems in radio engineering, synchronous machines in the energy sector, vibrating systems in technology are dynamic systems with cylindrical phase space. The mathematical model of phase systems is a class of ordinary differential equations, having a countable set of equilibrium states with periodic nonlinearities from a given set. Consequently, equations of motion of phase systems belong to the class of equations with differential inclusions. As the equilibrium state is a countable set, then for the stability of the system, it is necessary that every solution asymptotically approaches an equilibrium state from a countable set.

In the article we propose a completely new approach to the solution of global asymptotic stability of phase systems based on a priori estimation of improper integrals along the solution of the system.

Nonsingular transformation. The purpose of nonsingular transformation is to reduce equations of motion (4.1) to a special form, using the nonlinearity property

from (4.2) and	to obtain estimates of improper integrals.
Consider	differential equation (4.1) with respect to phase variables	x = x(t),

= (t), t I. Represent the matrix

B

as B = B

, , B

,

1

m

columns n 1. Let the matrix = , ,

m

, where

Rn ,

1

i

orders

n m.

Lemma 1. Let the vectors

Rn ,

i = 1, m such that:

i

*B = 1, *B

j

= 0, j = 1, m, i j,

i

i

i

where

i = 1, m.

B	,	i = 1, m
i

Matrices

vectors

B,

of

(4.3)

where Bi Bj ,	i j, (*)	is a transpose sign. Then along the solution of the first
equation from (4.1) an identity holds
	* x(t) = * Ax(t)								(4.4)
				(	(t)),	t [0, ), i = 1,m.
	i	i	i	i

207

Lectures on the stability of the solution of an equation with differential inclusions

If in addition rank B* = m and Gram determinant

			<		,	>	<	2	,	2	>		<		,		m		>
				1	1									1
(	, ,	m	) =																0,
1
			<	m	,	>	<	m	,	2	>		<	m		,		m	>
					1

(4.5)

then vectors i ,	i = 1, m				satisfying		conditions (4.3), (4.4) exist and are linearly
independent, where		<	,	j	>	is a scalar product of vectors i , j ,i, j = 1, m.
		i
*	= ( i1		, , in ), i = 1, m, then by multiplying the identity from the left
Proof. As i
						t I on	*
x(t) = Ax(t) B ( (t)),							i we have

*	B = (
where i

that ratio (4.3)

	* x(t) = * Ax(t) * B ( (t)),
				t I, i = 1, m,
	i	i	i
*	*	Bm ). From here taking into account (4.3), we get (4.4). Notice
i	B1, , i

can be written as linear algebraic equations.

B

i 2

B

in

B

= 0,

i1

11

12

1n

B

i 2

B

in

B

= 1

i1

i1

i 2

in

B

i 2

B

B

= 0,

i = 1, m.

i1

m1

m2

in

mn

If rank B

*

= m, then this system of equations has a solution, i ,

i = 1, m. . From the

condition (4.5) it follows that vectors i ,

i = 1, m

are linearly independent. The lemma

is proved.

Lemma 2. Let the

vectors

0

Rn

such

that: *B = 0, i = 1, m, where

B B

,

0

i

i

j

i j. Then along the solution of the first equation from (4.1) an identity holds

*

x(t) =

*

Ax(t),

t I.

(4.6)

0

0

If in addition rank

B

*

= m and Gram determinant

< 01, 02 >

< 01, 0n m >

( 01, , 0n m ) =

0,

(4.7)

< 0n m , 01 >

< 0n m , 0n m >

208

Chapter IV. Global asymptotic stability of multidimensional dynamic systems with cylindrical …

then vectors

	01	, ,	0n m

exist and they are linearly independent, where

	0i	,

i = 1, n m

are obtained from		0 by choosing (n m) arbitrary components of the vector
			0 = ( 10 , , n0 ) R	n	.		*	= 0,
Proof. Let the		vector				Then ratio	0 Bi		i = 1, m

written as

	.
0

can be

	B		n0	B	= 0, ,	B		n0	B	= 0.
10	11			in	10	m1			mn

(4.8)

If rank

m 1,0 , , n0

B	*	= m, then the system (4.8) has a solution			=
				0

are any numbers. Define the vectors			R	n	,
		01

	0	(	m 1,0

,	0n m

,
R	n

,	n,0	),	where

by choosing

arbitrary numbers

		, ,	n,0	.
	m 1,0

In particular

	01	=	(1,0, ,0),
		0

	02	=	(0,1,0, ,0),
		0

0n m = (0, ,0,1).

Multiplying identity

x(t) = Ax(t) B ( (t)),

t I

from the left by

*

0

we

get (4.6). By

the condition

of

the

lemma,

the

following inequality holds

(4.7).

Consequently,

vectors

n

i = 1, n m

are linearly independent. As

*

B = 0,

0i R

,

0i

i

i = 1, n m, then identity (4.6) is equivalent to

*

*

Ax(t),

t

I,

i = 1, n m.

(4.9)

0i x(t) = 0i

The lemma is proved.

Lemma 3. Let the conditions of the lemmas 1, 2 be satisfied, and let, moreover, the rank of the matrix

P =

1 , , m , 01, , 0n m

(4.10)

of order

n n be equal to

n.

Then equation (4.1) is equivalent to the following system

of equations

y

= c

y

c

y

n

(

1

), , y

m

= c

y

c

mn

y

n

m

(

m

),

1

11

1

1n

1

m1

1

= cn1 y1

cnn yn ,

(4.11)

ym 1 = cm 1,1 y1

cm 1,n yn , , yn

1

= d

y

d

y

n

R ( ), ,

m

= d

m1

y

d

mn

y

n

R ( ).

11 1

1n

1

1

m

*

y

=

*

x,

*

*

*

where yi

= i x, i = 1, m,

m

i

0i

i = 1, n m,

R = (R1

, , Rm ).

Proof. As rank P = n, then from (4.10) it follows that vectors i Rn , i =

1, m

,

Rn ,

0i

i = 1, n m form a basis in Rn . Then vectors

209

Lectures on the stability of the solution of an equation with differential inclusions

Ax = c

x c

x c

x c

x,

*

*

*

*

*

1

n 1

1m m

1m 1

01

m

0n m

x,

Ax = c

x c

x c

x c

*

*

*

*

*

m

m1

1

m,m

m

m,m 1 01

mn

0n m

x,

*

Ax = c

x c

x c

x c

*

*

*

*

01

m 1,1

1

m 1,m

m

m 1,m 1 01

m 1,n

0n m

x

Ax = c

x c

x c

x c

*

*

*

*

*

0n m

n1

1

n,m

m

n,m 1

01

n,n

0n m

(4.12)

c

,

i,

j = 1, n are decomposition coefficients

*

*

,

i = 1, n m

where

ij

A i , i = 1, m,

A 0i

by

bases

i

R

n

,

i = 1, m,

0i

R

n

, i = 1, n m.

From (4.12) taking into account (4.4),

(4.6), (4.9) we get a system of equations (4.11) with respect to variables

y

, , y

.

1

n

*

n

by bases i ,

i = 1, m,

0i ,

Similarly, by decomposing a vector

ci

R

i = 1, n m

we get

= C x R ( ) = d

x d

x d

x d

x R ( ),

1

*

*

*

*

1

1

1

11

1

1m

m

1m 1

01

1n

0n m

x R

( ),

= C

x R

m

( ) = d

*

x d

*

x d

*

x d

*

m

m

m

m1

1

mm

m

m,m 1

01

mn

0n m

where

C

*

*

*

R

*

*

*

The lemma is proved.

= (C1

, ,Cm ),

= (R1

, , Rm ).

The system of equations (4.11) in the vector form has the form

y = Ay B ( ),

= Cy R ( ), ( ) 0 ,

(4.13)

where

A,

B,

C

are constant matrices of orders n n,

n m,

m n

, respectively,

I

m

A = cij , i, j = 1, n;

C =

dij

,

i = 1, m,

j = 1, n,

B =

,

O

n m,m

order

m m,

O

is a matrix of order

(n m) m with

n m,m

matrix

*

,

then

*

x,

x = K

1

*

)

1

y,

matrix

K = P

y = Kx = P

y = (P

I	m	is a unit matrix of

zero elements. If the

A,	B,
		C are equal

1

*

*

1

*

C = CK

1

*

1

.

A = KAK

= (P )A(P )

,

B = KB = P B,

= C(P )

Thus, differential equa-

tions (4.1) with nonlinearities (4.2) with nonsingular transformation x = K 1 y = (P* ) 1 y are reduced to (4.13).

Lecture 33.

Solution properties. Improper integrals

Solution properties. To obtain estimates of improper integrals along the solution of the system (4.1), we need to show limitation of the solution of the system (4.1), (4.2) and get identities along the solution of the equation (4.13).

210