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Chapter II. Absolute stability and Aizerman’s problem of multidimensional regulated systems


								* (t)P y											* (t)P y(t) y* (t)P y(t)]dt = 0,
				I	3	= [ y		* (t)P y							(t) y				* (t)P y(t) y* (t)P y(t)]dt = 0,
					3								1							2							3
							0
																										m			m
																												11	12
																												11	12
where	P = N		H ,			P = H		*		N	*	N				A12 ,		P = N A12,							N =		m		m	, N		=
where	P = N		H ,			P = H				N	1	N			2	A12 ,		P = N A12,							N =		m		m	, N	2	=
	1	2		1		2	1				1				2				3		1				1			21	22		2
																											m		m
																											m		m
																												31	32
then improper integral
																	I	5	= I	I			3	=
																		5	1				3
						1										1								1				( )		1
			*			1						*				1			*					1						1
	=	[ y		(t)R y(t)				y					(t) y(t) y							(t)W y(t)]dt =									( ) d
		0																										(0)
where	R	= P ,							=				2	P ,			W =				3	P .
where	1		1			1	1						2			2			1		3			3

n		n
	11	12
	11	12
	n	n	,
	21	22
	n	n
	n	n
	31	32

< ,

Transformation of matrix

W =		P .
1	3	3

If we select elements of the matrix

equal (v. (2.155), (2.162)):

m	= k,
11

m	= k	(	),
21	1	1	1

m	= k,
31

m	= 0,
12

m	= k		(
22	1	1	1	1
				k
		W =		k
		W =		k
			1
				k
				k

	),	m
1		32
		k
k k
		1
	k

= 0,

kk k

> 0,	k
	1
*
0, y W
	1

> 0,

y = k

then the matrix

( y	y		y	)	2	k y	2	.
		2					2
1			3			1

Matrices	R	,		are equal:	R =		P ,	=	2	P .
Matrices	1		1	are equal:	1	1	1	1	2	2

												=
From the condition of symmetry of the matrix														*	we get:
From the condition of symmetry of the matrix													1	1	we get:
n		m		= n			n	,n	n	n			=		m	=				1	k,
21		11			1	11	12	21	11		12			1	11			1		1
n	m		=		( ) n , n n =						(		) m			=	1	( ),
31		12		1	1	1	11	31	11	1		1		1	12		1		1		1

n31 n32 m22 = n21 m31,n31 n21 n32 = m31 m22 = k k1 1 ( 1 1 1 ) .

(2.173)

Notice that surface	y*W y = 0 does not	contain whole trajectories, besides
	1
y(t) 0,t I.
Г. Iserman’s problem. From (2.173) when n11 = 0, n12 = 0, n31 = n22 , we
have n21 = k 1 1, n32	= k 1 ( 1 1),	n31 = 1 ( 1 1) . Matrix

161

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

	1 ( 1 1 )		0	0
					= R1* 0
R1	=	0	n21	0	= R1* 0
		0	0	n
				32

when				(								) 0					,			n		0			,	n				0					. Then set of values																			(				,		,	1	)		for which
when			1			1					1						,				21				,		32								. Then set of values																					1			1		1			for which
Iserman’s problem has a solution belongs to the set
							= {(									,				,		) \| < 0,								1		< 0,					1	(							) 0,n								0,n						0.}
															1				1		1			1						1							1			1				1						21							32
	Here				<						0		,		1		< 0				follows from the Hurwitz of the matrix																																				A = A						( )			for any
	Here						1						,		1						follows from the Hurwitz of the matrix																																					1			1					for any
arbitrarily small value																					> 0,			inequalities															(							)		0,					n					0,		n					0		ensure
arbitrarily small value																								inequalities													1				1				1								21							32							ensure
the fulfillment of the condition																										R = R								*		0				. Values									n		,		n			= n					,			n			found
																										R = R										0													n		,		n			= n					,			n	32
																											1						1																21					31						22					32
																										*	,										k > 0,							k			> 0
from the symmetric matrix																								=			,		where								k > 0,							k			> 0			are some numbers.
from the symmetric matrix																						1				1			where																1					are some numbers.
	In		particular											triple								(		= 0,5,													= 1,										1	= 1)									.		Indeed								1	< 0,
	In		particular											triple								1														1											1										.		Indeed								1
	< 0,						(								) = 1,5								when								1		< 0				;			n			= k 0,5												1	0				when							1	< 0		;
1						1				1				1									when								1						;				21														1					when							1			;
n	= k									(0,5 1) = k													0,5					1		> 0						when								1	< 0				. In				other						words					there are
32				1				1														1						1								when								1					. In				other						words					there are
numbers						1		< 0,						k > 0,							k > 0					for which															(								) 0,					n				0,				n				0.			As
numbers						1																1				for which														1			1					1							21									32					As
follows from ratios (2.172) the limit value																																									0	is determined from the inequalities
follows from ratios (2.172) the limit value																																									0	is determined from the inequalities
a (		0	) = 1 1,5											0	>			0,														a			(		0	) = 2							0	> 0,														a				=			0	> 0,
1		0												0																			2				0								0																	3					0
a ( )a					(			0	) a (									0	) > 0.					From here it follows that																										0		= .				And thus, Iserman’s
1		0		2				0						3				0						From here it follows that																										0						And thus, Iserman’s
problem has a solution in the sector																												[ , )								.
problem has a solution in the sector																																				.

Comments

A new method of study of absolute stability of equilibrium state of regulated systems with limited nonlinearities in a critical case based on nonsingular transformation was developed, properties of solutions and evaluations of improper integrals along the solution of the system were studied.

The proposed method allows us to significantly expand the area of absolute stability in space of constructive parameters of the system in comparison with the known results and in some cases we can get necessary and sufficient conditions of absolute stability.

The nonlinear class of regulated systems with limited resources in a critical case for which Iserman’s problem has a solution was determined. For this class of regulated systems necessary and sufficient conditions for absolute stability are obtained.

162

Chapter II. Absolute stability and Aizerman’s problem of multidimensional regulated systems

Literature

1.Lur'e A.I. Nekotorye nelinejnye zadachi teorii avtomaticheskogo regulirovanija. – M.: Gostehizdat, 1951. – 216 s.

2.Ajzerman M.A., Gantmaher F.R. Absoljutnaja ustojchivost' reguliruemyh sistem. – Izdatel'stvo AN SSSR, 1963. – 139 s.

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

4.Ajsagaliev S.A. Ob opredelenii oblasti absoljutnoj ustojchivosti vynuzhdennyh dvizhenij v nelinejnyh sistemah // Izv. AN SSSR. Tehnicheskaja kibernetika. – 1969. – № 5. – S. 38-48.

5.Ajsagaliev S.A. Ob opredelenii oblasti absoljutnoj ustojchivosti sistemy upravlenija s neskol'kimi nelinejnymi jelementami // AN SSSR. Avtomatika i telemehanika. – 1970. – № 12. – S. 83-94.

6.Ajsagaliev S.A. K teorii absoljutnoj ustojchivosti reguliruemyh sistem // Differencial'nye uravnenija. – 1994. – T. 30. – № 5. – S. 748-757.

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

//Differencial'nye uravnenija. Minsk-Moskva. MGU. – 1999. – T. 8. – № 30. – S. 3–11.

8.Ajsagaliev S.A. Problemy kachestvennoj teorii differencial'nyh uravnenij. – Almaty: Qazaq universiteti, 2015. – 420 s.

9.Aisagaliev S.A., Kalimoldayev M.N. Certain problems of Synchronization theory // Journal of Inverse and Ill-Posed Problems, 21 (2013). – Pp. 159-175.

10.Ajzerman M.A. Ob odnoj probleme, kasajushhejsja ustojchivosti v "bol'shom" dinamicheskih sistem // UMN, 1949. – T. 4. – № 4. – S. 186-188.

11.Ajsagaliev S.A. Problema Ajzarmana v teorii absoljutnoj ustojchivosti reguliruemyh sistem // Matematicheskij sbornik, In-ta matematiki im. V.A. Steklova RAN, 2018. – T. 209. – № 6.

– S. 3-24.

12.Pliss V.A. O probleme Ajzermana dlja sluchaja sistemy treh differencial'nyh uravnenij // Dokl. AN SSSR. – 1958. – T. 3. – № 121.

13.Zubov V.I. Lekcii po teorii upravlenija. – M.: Nauka, 1975. – 495 s.

14.Demidovich B.P. Lekcii po matematicheskoj teorii ustojchivosti. – M.: Nauka, 1967. – 472 s.

163

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

Chapter III

STUDY OF STABILITY OF SOLUTIONS

OF DYNAMIC SYSTEMS WITH CYLINDRICAL PHASE SPACE

Mathematical models of pendulum systems (connected pendulums, double pendulums and other) in mechanics [1], navigation systems in radio engineering [2, 3], synchronized machines in energetics and electronics [4], vibration systems in engineering [5] are dynamic systems with cylindrical phase space (or simply phase systems).

The source of study of phase systems is the study of properties of solutions of equations of the mathematical pendulum

		(3.1)
( ) 0,

where

( ) sin

. Thanks to the works of F. Tricomi [6], L. Amerio [7],

G. Seifert [8] estimates of the critical value

	кр

were obtained: 1) in the case

1	кр		кр	( )
	кр		кр

for the system (3.1) there exist both stable and not stable states of

equilibrium; 2) in case

	кр	( )
	кр

any solution of the system (3.1) when

tends to

some state of equilibrium from countable set; 3) in the case

the system (3.1) there are numbers	and	such that	d (t
			dt



)

1 for any solutions of

when	t	(circular

motion).

The next stage in the development of methods of study of phase systems was the application of Lyapunov’s periodic functions [2]. The original research approach to phase systems is frequency conditions, based on the Bakaev-Guzh procedure followed by the application of the frequency theorem of V.A. Yakubovich. Such an approach was first proposed in the work of G.A. Leonov [9].

In the works [10, 11] the method of reduction of phase systems of high order to the phase system of the second-order (3.1) was proposed.

Quality and qualitative numerical methods, point mapping method [12], and also methods, based on the application of Lyapunov’s periodic functions (including frequency methods), form a group of exact methods of study of phase systems. Main disadvantages of the exact methods are as follows:

a)exact methods, besides the method of Lyapunov’s periodic functions, are applicable to second-order systems;

b)the most common among the methods, using periodic functions of Lyapunov, are frequency methods. However, the frequency conditions of the global asymptotic stability of phase systems, are hard to check.

164

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

Therefore the development of new effective conditions of global asymptotic stability of phase systems is an actual issue. One of such new approaches to the theory of global asymptotic stability of phase systems is outlined in this chapter.

In work [13] the basic idea of the proposed method to study the stability of phase systems is proposed. Solutions to more general problems (limit cycles, controllability, optimal control) of phase systems are presented in the monograph [14]. Studies of the dynamics of complex systems are contained in [15, 16].

Lecture 26.

Problem statement. Nonsingular transformation. Properties of the solution

We obtain a new effective condition of global asymptotic stability of a countable set of the equilibrium state of a dynamic system with cylindrical phase space, by evaluating improper integrals along with the solution of the system. The proposed method of study of global asymptotic stability allows us to get the stability area in space of parameters of the system wider than the known methods. The effectiveness of the proposed method is shown in the example.

Consider a dynamic system with cylindrical phase space described by the equation of the following kind

x Ax B ( ), Cx R ( ),	x(0) x	, (0)	0	,
	0		0

t I

[0, ]

(3.2)

where A, B, C, tively. Function

R are constant matrices of orders n n,								n 1,	1 n,
( )		{ ( ) C	(R	, R	) /	d ( )				,
		1	1	1
	0				1		d		2
	( ) ( ), , R						d
	( ) ( ), , R						},
						1

, respec-

(3.3)


where		is the period		of the function	( ),

\| 1	\| , \| 2	\| . Matrix	A	is a Hurwitz matrix i.e.

	,			2
1				2
Re			(A)
		j

are given numbers

0,	j 1,n,		(A)
		j

are eigenvalues of matrix

А.

Equilibrium state of the system (3.2), (3.3) is determined from the solution of algebraic equations

Ax* B ( * ) 0, Cx* R ( * ) 0.

As x* system (3.2),

1	B ( * ),	1	B) ( * ) 0 , then when
A		(R CA

(3.3) has a stationary set

{( x*, * ) Rn 1 / x* 0, ( * ) 0}.

1	B 0
R CA

the

As ( *) ( * k ) 0, k 00, 1±, 1,2,...± 2, … then the equilibrium state of the system (3.2), (3.3) is a countable set.

165

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

Definition 1. Stationary set

stable, if for any functions

( )



of the system (3.1), (3.3) is globally asymptotically

0	and any initial state	(x ,	0	) Rn 1	, \| x	0	\| , \|	0	\|
0		0	0			0		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 (3.2), (3.3)
are called relations, binding constructive parameters of the system								(A,B,C,R, , )
								1	2

under which the stationary set is globally asymptotically stable. We need to study separately two cases:


1.



2.

( )d 0,	,
( )d 0,	,

(case with «zero» load);

(non-zero load case);

The following tasks are proposed:

Problem 1. Find a new effective condition of global asymptotic stability of a

stationary set		of the system (3.2), (3.3) for the case, when

Problem stationary set

( )d 0, , R1.

Find a new effective condition of global asymptotic stability of a of the system (3.2), (3.3) for the case, when


( )d 0, , R .
	1

The special case of the system (3.2), (3.3) is the equation of motion of a simple
pendulum (3.1) with a function ( ) sin ,	0 1.

Problem 3. Find the value

	кр		кр	( ).
	кр		кр

This work is a continuation of research from [17, 18]. The article [17] contains results of a study of absolute stability of regulated systems with a single position of equilibrium in a simple critical case, and in [18] absolute stability of regulated systems in a critical case.

Nonsingular transformation. To obtain the main results it is necessary to transform the initial system (3.2) by nonsingular transformation to a special form.

Let the characteristic polynomial matrices A have the form

( ) det( In A) n an 1 n 1 ... a1 a0 .

As follows from Theorem of Hamilton-Cayley the matrix ( A) 0 . Then

166

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

An an 1 An 1 an 2 An 2 ... a1 A a0 In ,

where	I	n	is a unit matrix of order n n.
where		n	is a unit matrix of order n n.
Lemma 1. Let the constant vector				*	R		n	such that
Lemma 1. Let the constant vector					R			such that
			n 2			B		n 1	B	0,
			B 0, AB 0,..., A			B		0, A	B	0,
where	(*) is a transpose sign, is a row vector.

Then the first equation from (3.2) can be represented as

(3.4)

where

,..., y

n 1

a y

... a

n 1

B ( ),

n 1

Ax,..., y

n 1

x(t),

y (t),

i 1,n.

(3.5)

The proof of a similar lemma is given in Chapter I.

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

*n 1

,..., A

(3.6)

of order n n be equal to n . Then:

There is a row vector

( 0

, 1,..., n 1 ) such as

...

n 1

R ( );

(3.7)

0 1

y x

0, y

Ax 0,..., y

n 1

, then

x 0;

If 1

The equation (3.2) is equivalent to the system of equations

y2 , y2

y3 ,..., yn 1

yn ,

a0 y1

a1 y2 ... an 1 yn

An 1B ( ),

(3.8)

0 y1 1 y2

... n 1 yn R ( ),

( ) 0 .

The proof of a similar lemma is given in Chapter I.

Solution properties. Of interest is the study of the general property of solution of the system (3.2), (3.3), and also of the system (3.8).

Theorem 1. Let the matrix A be a Hurwitz matrix, i.e. Re j ( A) 0,
	j 1,n ,

the function ( ) 0 , and let, moreover, the equalities (3.4) be satisfied and rank= . Then the following estimates are true

167

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

where	l	, l	const
where	1	2

Moreover, the nuous.

\|\| x(t) \|\| l , \|\| x(t) \|\| l							,	t I [0, ),
		1			2
\|\| y(t) \|\| m		, \|\| y	(t) \|\| m					, i 1,n, t I
	i1	i				i2
	\| (t) \| c			,	t,		t I ,
			0
, m	const ,				m		const ,
i1						i2

functions	x(t),	yi (t),		i 1,n,				(t), t I

(3.9)

,	(3.10)
	(3.10)
	(3.11)
c	const .
0

are uniformly conti-

Proof. Notice that periodic limited, i.e. | ( ) | , 0 equation (3.2) has the form


continuously differentiable function				( )	0	is

, ,	R	1	. The solution of the differential

		0	t
	At	0			A(t )
x(t) e		x		e		B ( ( ))d ,	t I.

From here taking


\|\| e	At	\|\| c( )e		(a )t	,
\|\| e		\|\| c( )e			,
a max Re				A 0,		\|
	1 j n		j
	1 j n

into t( )

0
account that		matrix
I , 0 is		an
\| , t I,	we get

A is a Hurwitz matrix and arbitrarily small number,

|| x(t) || c( ) || x

|| e

( a )t

c( )e

( a )t

|| B ||

( a )t

l ,

where l

const, 0 l

. So, || x(t) || l , t,

t I ,

. As

x Ax

B ( )

, then

|| B || l2.

Consequen-

|| x(t) || || A || || x(t) || || B ||| ( ) | || A || l1

tly,

|| x(t) || l

, t,

t I, ,

. From

|| x||(xt()t||) || l ,l ,|| x||(xt()t||) || l

,l , t,t, t I I

there follows evaluation (3.9) and uniform continuity of the function

x(t),

t I.

As the systems (3.2) and (3.8) are equivalent,

y1 x,

Ax,...

..., yn

An 1x ,

then

| y1 | || || || x || || || l1 m11,...,|| yn || || || || A || || x || mn1 . From

(3.5)

it fol-

| yi (t) | mi2 ,

lows that

i 1,n,

t I . Thus, proven estimates (3.10).

Finally from (3.8) it follows that | (t) | | 0 | | y1(t) | ... | n 1 | | yn | | R |

c0 ,

t I . Consequently, we have an evaluation (3.11). From the limitation of deriva-

(t), t I , uniform continuity of functions x(t),

tives

x(t),

yi (t), i 1,n,

y(t), (t), t I

follows. The theorem is proved.

168

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

Theorem 2. Let the conditions of the lemmas 1, 2 be satisfied. Then along the solution of the system (3.8) the following identities hold

( (t))

[ y

(t) a

y (t) a y

(t) ... a

(t)],

t I ,

n 1

1 2

(t)

n 1

(t) y (t)

(t) ...

(t),

t I ,

1 1

(t)

d ( (t))

[ y

(t) a y

(t)

a y

(t) ...

n 2

1 3

n 2

(t) a

n 1

(t)], t I ,

(3.12)

(3.13)

(3.14)

(t)

(t) ...

n 1

(t) ...

n 1

(t),

t I,

(3.15)

(t),

(t) yn 2 (t), t I, 0

n 1

j j 1

, j 1,n.

where yn (t) yn 1

yn 1

, A

aj 1

Proof.

follows

from the

second

equation

(3.8)

derivative

yn (t) yn 1(t) a0 y1(t) ... an 1 yn (t) ( (t)),

t I

, where

B .

n 1

From here

we get

identity

(3.12). Substituting

the

values

( (t)) from identities

(3.12) on the right side of the third equation in (3.8), we get the identity (3.13). Diffe-

rentiating by	t	identities (3.12), (3.13) we get (3.14), (3.15) respectively. The theorem
rentiating by
is proved.

Lemma 3. Let the conditions of the lemmas 1, 2 be satisfied, the matrix


Hurwitz matrix, the function	( ) 0		. Then for any constant matrix		Q
(n 2) (n 2) quadratic form	*	(t)Q(t) (t),		(t) ( y1 (t), y2 (t),... ..., yn (t), yn 1 (t), yn 2

can be represented as

А is a of order

(t)), t I

(t)Q (t) q y

(t) q y

(t)

... q

n 1

[

(t)F (t)],

I [0, ),

where F is a constant matrix of order (n 2) (n 2) .

Proof. It is easy to verify that

(

n 2

n 1

( y

) y

( y

n 2

n 1

y (t) y

(t) y (t) y

(t)

( y2 ),

y (t) y

(t)

2 dt

q	y	2
		2
n 2		n 2

) y2 ,...,

n 1

( y1 y2 ) y22 ,... .

(3.16)

169

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

In particular when

n 2

we have

y		y		y			y		d	(		1	y			2		);
																2
	4		3			3		3	dt			2				3
	4		3			3		3								3

y		y		y			y		d	( y				y				) y				y				d		( y		y		) y			2	;
																																			2
	4		2			3		2					2							2				3					2						3
	4		2			3		2	dt				2			3				2				3		dt			2		3				3
									dt																	dt
y		y		y			y		d	( y y							) y			y						d	( y y				) y			y			d	( y y		) y		y		d	( y y		1	y	2	);
y	4	y		y	3		y			( y y					3		) y			y			3				( y y				) y		2	y		3		( y y		) y	2	y	2		( y y	3		y	2	);
	4		1		3		1		dt			1			3			1					3		dt			1		3			2			3	dt	1	3		2		2	dt	1	3	2		2
									dt																dt												dt							dt			2
y		y		y			y		d	(		1		y		2		);
y	3	y	2	y		2	y	2	dt	(		2		y		2		);
	3		2			2		2								2

y	y			y			y		d	( y y							) y		2		;
																			2
					2										2				2
	3	1			2		1		dt			1			2				2
									dt
y		y		y			y		d	(	1		y		2		).
															2
	2
	2		1		1		1		dt		2			1
									dt		2

As quadratic form		*	(t)Q (t)	contains terms with constant coefficients at the pro-
		*

duct of components of the vector (t) , then we get a representation of the form (3.16).

The lemma is proved.
Lemma 4. Let the conditions of the lemmas 1, 2 be satisfied, the matrix	А be a

Hurwitz matrix, the function ( ) 0	. Then for any constant matrix	Q	of order
		1
(n 1) (n 1) the improper integral

(t)Q z(t)dt

[q y

(t)

... q

(t)]dt l

n 1

1 1

n 1

(t)F z(t)]dt z

;

| l | ,

(t)F z(t) |

where

z(t) ( y1(t),..., yn 1(t)),

t I ,

F1 is a constant matrix of order

max(| 1 |, | 2 |,),

| 1 | ,

| 2 | .

Proof. As follows from formula (3.8), the function

(

(3.17)

(3.18)

n 1)

yn 1(t) yn (t

where \| yi (t) \| mi1,		(t)
	\| yi
the function yn 1 (t),	t I

rentiable by , moreover

the function ( (t)), t the fact that the function

) a0 y1(t) ... an 1 yn (t) ( (t)), t I ,

\| mi2 , i 1,n, t I, \| ( (t)) \| , t I							. Consequently,
is limited. The function				( ) 0 is continuously diffe-
\|	d	\| ,	max(\|	\|, \|		\|,), \| \| ,	\|		\| . Then
					2			2
	d		1			1

I is limited and uniformly continuous by t, t I , due to (t), t I is uniformly continuous.

170

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