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2016 ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА Сер. 10 Вып. 4
ПРОЦЕССЫ УПРАВЛЕНИЯ
УДК 517.977.1 A. D. Khamitova
CHARACTERISTIC POLYNOMIALS FOR A CYCLE OF NON-LINEAR DISCRETE SYSTEMS WITH TIME DELAYS
Odessa National Polytechnic University, 1, Shevchenko Ave, Odessa, 65044, Ukraine
We study a method associated with constructing of delayed feedback for local stabilization of periodic orbits of nonlinear discrete systems. An alternative approach to the construction of characteristic polynomial for the delay system linearized in the neighborhood of T-cycle is suggested. It is proven that our new alternative approach is equivalent to the standard one, however, it allows us to produce directly new forms of polynomials. These forms are convenient in applications to the problems of chaos control and allow us to apply methods of geometric complex function theory. This article is an extension of the results, which received D. Dmitrishin, P. Haglstein, A. Khamitova and A. Stokolos to the vector case. Refs 6. Fig 1. Keywords: Non-linear systems, asymptotic stability of cycles, DFC methods.
А. Д. Хамитова
ХАРАКТЕРИСТИЧЕСКИЕ ПОЛИНОМЫ ЦИКЛОВ НЕЛИНЕЙНЫХ ДИСКРЕТНЫХ СИСТЕМ С ЗАПАЗДЫВАНИЕМ
Одесский Национальный политехнический университет, Украина, 65044, Одесса, пр. Шевченко, 1
Изучается один из методов локальной стабилизации периодических орбит нелинейных дискретных систем, связанный с построением запаздывающей обратной связи. Предлагается альтернативный подход к построению характеристического полинома для линеаризованной в окрестности T-цикла системы с запаздыванием. Доказано, что альтернативный подход эквивалентен стандартным, однако напрямую дает возможность получать новые формы характеристических полиномов. Эти формы оказываются удобными в решении проблемы управления хаосом и позволяют применить методы геометрической теории функции комплексного переменного. В статье приведены и обcуждены результаты, полученные Д. Дмитришиным, П. Хагелстейном, А. Хамитовой и А. Стоколосом на векторный случай. Библиогр. 6 назв. Ил. 1.
Ключевые слова: нелинейные системы, асимптотическая стабильность циклов, ДФК методы.
1. Introduction. The goal of the current paper is to introduce a modified approach to the problem of local asymptotic stability of cycles in special non-linear systems and to
Khamitova Anna Dmitrievna — postgraduate student; [email protected]
Хамитова Анна Дмитриевна — аспирант; [email protected] © Санкт-Петербургский государственный университет, 2016
make a comparison. The standard approach to this problem is well outlined in [1, 2] which is a further development of the method describe by O. Morgiil [3].
We will distinguish between the systems with no time delays
Xk+1 = f (xk), f e = 0,1,..., (1)
and with time delays of the form
xk+1 = aif (xk) + a2f (xk-T) + ... + aN f (xk-(N-1)T), f e Rm, k = 0, 1, ...,T e Z + . (2)
Note that the systems (2) appear in the problems of localization of unknown periodic orbits and control of chaos.
In scalar case the standard approach to investigate stability with no delays is to construct a new map that has points of the cycle as equilibriums and then linearize about the equilibrims. Our approach is to construct an auxiliary system that has the cycle elements as coordinates of the equilibrium vector. So, instead of studying cycles of a scalar system we study the equilibrium of a vector system.
In vector case the standard approach produces a characteristic equation as a result of applying the chain rule to the new system, while our approach produces the same equation as a consequence of the non-degenerate property for the auxiliary system linearized around equilibrium.
At this level the advantages of our approach are not visible, but later on they will be more transparent.
For a system with time delays the standard approach leads to a system with dimension that increases with delays. The characteristic polynomials of such systems appear in a standard form that makes impossible analysis on Schur stability, thus useless. Our approach leads to a very well structured system where the characteristic polynomial appears factorized with each factor in a special form that enables stability analysis. Moreover, this new form (see (22) on p. 113) is very compact and is convenient in applications.
2. Main results. A. Standard approach. Let us consider the scalar case of the system (1). Based on a cycle {q1, ...,vt} let us construct a new dynamical system
xk+T = f(T\xk), (3)
which iterates T times the initial map. Therefore
f(T )(x)= f (f(T-1)(x)),...,f (1)(x) = f (x).
Now, let F(x) = f(T)(x) (see the figure below) and denote x0 = y0, xT = y1, ..., xkT = yk, ... . Then the system (3) has changed to
yk+1 = F (yk). (4)
In fact for any j = 0,..., T — 1 we can write
xj = yo, xj+T = y1,..., xj+kT = yk,...
and obtain the same dynamical system (4).
Note that the system (4) has T equilibriums n1, ...,nT. Then the cycle {q1, ...,r)T} of the system (1) is asymptotically locally stable if and only if all equilibriums n1,...,nT of the system (4) are asymptotically locally stable.
/ / / Vi^Vi^Vi ...... т
Definition of F map
The multiplier of every equilibrium nj of the system (4) is F' (nj), j = 1, ...,T. By the chain rule
f'(m) = f'(f(f-f(m))) • f'(f-f(m))• • • f'(m) = it■■■ mi
where ¡j = ff(nj). It is clear that F'(nj) = ¡iT ■ ■ ■ Mi, j = 1, - -,T. The quantity ¡iT ■■■ mi is called a multiplier for the cycle {ni, ...,nT}- The condition of the local asymptotic stability is \lT ■ ■ ■ Mi | < 1.
B. Alternative approach. We suggest considering a system
{Xk+i = f (xk),
Xk+2 = f(xk+i), (5)
Xk+T = f (Xk+T-i).
Let
zi =
( xi \
X2
\XT J
Z2 =
( XT+1 \
XT+2
X2T
, zk =
( X(k-1)T+1 X(k-2)T+2
V
XkT
zk+1 =
/
( XkT+1 \ XkT+2
\ X(k+1)T )
A very simple but a very important observation is that (5) can be written in the following form:
zk+1 =
1 f(XkT) \ f {XkT +1)
\ f (XkT+T-1) )
(6)
Let us note that (6) is a dynamical system zk+i = F(zk) where
F(zk) =
f(1) (XkT) ^ f(2) (XkT )
f(T) (XkT) )
(7)
Our novelty is that the asymptotically local stability of the cycle {ni, ...,nT} of the system (1) corresponds the asymptotically local stability of the equilibrium
\ Пт
of the system (5).
In the standard approach the cycle corresponds to T equilibriums of the scalar dynamical systems. In our approach this cycle corresponds to one equilibrium of the vector system. So far these two approaches seems absolutely equivalent. The difference will be visible when we start to consider the systems with time delays, especially when the number of the used delays is much larger compared to the size of the cycle.
To compute multipliers we have to linearize the system (5) in the neighborhood of the equilibrium (8)
\ /
< №
')
and extract the linear part. One can linearize either system (6) or system (7). It turns out that it is more convenient to linearize system (6) because in that case we can avoid dealing with a superposition of the maps. The linearized system has the form
(т)
= Zk
ni
\ Пт
f = 1 °k+i " 1 = fПт ¿T), f '(ni)sk+.i,
1 ¿т -^ ¿k+i " f (Пт-i )S(.T+-1i)
sj (sj = 0) we obtain
Xsi Xs2 = VtSt , = viXsi, f Xsi = J XS 2 - Vt • Vi ■ ■ ■ ■ Visi, ■ ■ V2S2,
XSt = Vt-iXSt-i 1 XSt = Vt - -i ■ ■ ■ vt St
Thus, we come up with the same representation for the multipliers A = mT • • • m ■ C. Vector case. It is amazing that there is no major changes in the vector case. Namely, in standard approach the Jacobi matrices Mj = f '(nj) came up and the multipliers are the roots of the characteristic polynomial
\XI - MT ■ ■ ■ Mi\ =0,
Xsi = MT ■ ■ ■ MiSi, Xs2 = Mi ■ ■ ■ M2S2,
while in our case
Ast = Mt• • • MtSt,
where Sj are non-zero vectors. Since the eigenvalues of the product of square matrices AB and BA are the same (c. f. [4, Ex. 9, p. 55]) we come out with the characteristic equation
\XI - MT ■ ■ ■ M-i_\ =0.
U) \ k
k
In this case the eigenvalues of the matrix MT ■ ■ ■ Mi are called multipliers of the cycle. If all multipliers in absolute values are less then one then the cycle is asymptotically stable. D. Systems with time delay. Let us consider a system with time delay in a general
form
Xk+i = f (Xk, Xk-i,..., Xk—T), f e Rm, T eZ +. (9)
And let us study a local stability of the cycle {ni,..., nT} where nj e IRm. In other words for all k > t + 1 the following equations are valid:
n(k+i) mod T = f (Vk mod T ,n(k-i) mod T ,...,n(k-T) mod T )-
There abusing notation we assume that T mod T = T.
Standard. An auxiliary system with respect to the vector of size m(T + 1) is
' z^ \
J2)
Zk =
Then
Zk+1
zk
I z(1) \ Zfc+1
zfc+1
(т+1)
\ z(++ ) /
(t +1)
^ Xk—T
Xk-т +1
/ f
\ f (
Xk
z(2)
zk J3)
>+1) Jt ) 'k
z(1) , zk , , zk
) /
l. e.
zk+i = F (zk).
Note, that F e IRm(T+i) in the above formula and is different from the one on the page 2.
Further, let zk+T = F(...F(zk)) be the T-times iterated map F. This map can be written as a system
yk+i =HVk). (10)
Let us periodically repeat the elements of the cycle: {ni, n2,... ,rT, nt, n2, ■■■,nT ,...}. The first t +1 elements of this sequence form a vector
/
У1 =
П1
П2 \ .
In the same way we define the vectors
У2
/ П2 П3
V .
yT
( nT
П1 \ .
It is clear that the vectors yl,..., yT are equilibriums of the system (10).
Then the cycle {ni, ..,Vt} of the system (9) is asymptotically locally stable if and only if all equilibriums y\,..., yT of the system (10) are asymptotically locally stable.
For the equilibrium of the system y* of the system (10) the Jacobi matrix is defined by the formula with T factors
&(y*) = F '(yT*y)...F' (y(l)), where the matrix F'(y*) has dimensions m(r + 1) x m(r + 1) and is equal to
(11)
f '(v*) =
( O O I O 0. 1. .. O .. O
O V Qj O Qj O. Qj . . qT+1
(12)
/
There the matrices O and I are zero and unit matrices correspondingly and are of the dimensions m x m. Further,
4r dzir)
r = T + 1, j = 1,...,T,
i. e. the value of the derivative evaluated at the point y*.
For all other equilibriums y* the Jacobi matrices F'(y*) can be computed in the same manner and
*'(yj )= F' (y$+3) mod T)...F' (yj>l
which can be obtained from (11) by a cyclic permutation of the factors, and therefore the eigenvalues of &(y*) coincide for all j = 1, ...,T (c. f. [4]).
If all eigenvalues of the matrix ^'(y*) which are roots of the polynomial
P!(X) = \XI - Ф'(у*
(13)
are less then one in absolute values then the cycle of the system (9) locally assymptotical stable.
Note that in the scalar case m = 1 the matrices F'(yj) are in the Frobenius form. Therefore, the matrix (12) is a generalized form of companion matrix. If system is of special case below, then the product is manageable and the characteristic equation was found in scalar case by means of induction in [1].
Our approach. The system (9) generates the system
Xk+1 = f (xk, ..., Xk—т ), Xk+2 = f (Xk+1, ...,Xk-T +1 ),
Xn+T = f (xk+T — 1, ..., Xk—т+T-1 ).
(14)
Note that (14) is a dynamical system, which might be seeing by replacing in the right hand side each of xk+i, ...,xk+T-i by the functions of xk-T, ...,xk. If
( ¿1) \ / \
zk / x(fc-1)T+1 \
x(k-1)T+2
Zk =
,(2)
(T) \zk )
XkT
Zk e
dT m
then
zk+i
z(2)
Zk+1
Z(T ) Zk+1
f (z
(T) (T-1) k , zk
_ f (zkïi,zkT U,
_ f ( (T-1) (T-2) )
_ J (Zk+1 ,Zk+1
The system (15) can be written in the form
Zk+1 _ !>(zk,zk-b ■■■), Ф e
~>mT
The system (16) has an equilibrium
/ ni
\ Пт
nj e IRm, j _1,..,T.
(16)
(17)
To study the behavior of the solutions to the system (16) in the neighborhood of the equilibrium (17) let us make a substitution
Skj) _ zj - nj,
к _ 0,1,..., j _!,..,T.
So, instead of linearizing of the system (16) one can linearize the equivalent system (15). As a result we get
Sk+i = AoSk+i + AiSk + ..., (18)
where
( 3( 1) \
5k _
€ > /
and Aj are square matrices of the dimension mT x mT. The matrix A0 on the main diagonal and above have zero entries. That means that matrix E — A0 is non-degenerate, therefore the system (18) is a dynamical system. Its characteristic equation is
\XI — A0A — Ai — ...\ =0.
If all roots of this equation in absolute values are less then one, then the cycle of the system (14) is locally asymptotically stable.
The polynomial Am(T+i)-i\AI — A0A — Ai — ...\ has the same degree, same leading coefficient and the same roots as polynomial (13) therefore coinside with (13).
E. Example. Let us consider a system (9) written in a coordinate form for (Xk,yk)T e R2, k = 0,1,..., where T means transposition
Xk+i = f(Xk, yk, Xk—i, yk—i, Xk — 2, yk — 2),
yk+i = g(Xk ,yk ,Xk—i,yk—i,Xk—2,yk—2). Let us assume that the system (19) has a cycle ni,n2 of the length 2
П1
n11 n12
П2
П21
n22
(19)
(20)
l. e.
V21 = f (V11,V12 ,V21 ,122,111,112 ), 122 = g(n11 ,112,121,122,111,112),
111 = f (l21,l22 ,111 ,112,121,122 ), 112 = g(V21 ,122,111,112,121 ,122), and f)1 = 12.
Thus, for the system (19) we have m = 2,t = 2 and T = 2. We came up with the following problem: write the characteristic equation for checking the local asymptotic stability of the cycle (20).
Standard approach. Define the vector of the size m(T + 1) = 6
Zk =
Z(1)
Zk Z(2)
Zk
.(3)
where
(1)
Xk—2 Vk — 2
(2)
Xk— Vk—
„(3)
Xk Vk
Then the system (19) is equivalent to the system
Z(1) Zk+1
Zk+1 z(3) Zk+1
This system define the map
Z(2)
Zk
(3)
Z
к
f (Xk,Vk, Xk—1, Vk—1, Xk—2, Vk—2) g(Xk ,Vk, Xk—1,Vk—1,Xk—2,Vk—2)
' С1' Сз
ь С4
Сз С5
С4 Св
С5 f (С5 ,Св,С3,С4,С1,С2)
Св д(С5,Св,С3,С4 ,С1,С2)
which Jacobi matrix evaluated at the points
I I11 \ 112
V1 =
121 122 П11 \ П12 )
and y*2 =
correspondingly are
F '(V*
( 0 0 0 0
Фа1 Фа2 Фа3 Фа4 Фа5 Фав \ Ya1 Ya2 Ya3 Ya5 1ав Ya2 )
l 121 \
122 111 112 121 \ П22 )
0 0 0 1
1, 2.
k
k
Above
ф1,3,
VI
дд_
71,;
Vi
df_
ф2,з,
V2
dg_
Y2,j, j = 1, •••, 6.
V2
Then
Ф'(У1 ) = F\y*1 )F '(y2 )•
The characteristic polynomial is
Pi(A) := \AI — F'(yi)F\y%)\,
where I is a unit matrix of the dimensions 6 x 6.
Alternative approach. The system (14) has a form
= f (Xk ,yk,Xk-1,yk-1,Xk-2,yk-2),
= g(Xk,yk ,Xk-1 ,yk-1, Xk-2,yk-2),
= f (Xk+1,yk+1,Xk ,yk,Xk-1,yk-1),
= g(Xk+1, yk+1, Xk , yk, Xk-1 ,yk-1 )•
Define a vector
zk
z(1)
zk .(2)
where
then
v(1)
X2(k-1)+1 y2(k-1)+1
(2)
X2k y2k
zk+1 =
/ X2k+i \ / f (X2k ,y2k,X2k—i ,y2k—i,X2k—2 ,y2k — 2) \
y2k+i = g(X2k ,y2k,X2k — i,y2k—i ,X2k — 2,y2k—2)
X2k+2 f (X2k+i ,y2k+i,X2k ,y2k, X2k — i,y2k—i)
V y2k+2 J \ g(X2k+i,y2k+i,X2k,y2k,X2k—i,y2k — i) J
After substituting in the third and the forth equations the values for X2k+i and y2k+i by their values from the first two equations we get a system
zk+i = !>(zk ,zk—i).
Making a substitution = nj — z^K j = 1, 2, we linearize the system. Then (18) takes the form
°k+1 °k+1
ф13 Ф14
Y13 Y14
Ф25 Ф26
Y25 Y26
^ + С +
Ф15 Ф16
Y15 Y16 Ф23 Ф24 Y23 Y24
42)+ 42)+
ф11 ф12 \ j(2)
7п Y12 J k-1
ф21 ф22 A ^(1)
Y21 Y22 / k
From there we get the characteristic polynomial
/ 0 0 0 0 \
P2(X) :=
A2I-
0 0 0 0
ф25 ф26 0 0
\ Y25 Y26
0
A2 -
0
k
k
( Ф13 Фи Ф15 Ф16 \
Y13 Y14 Y15 Y16
Ф21 Ф22 Ф23 Ф24
V Y21 Y22 Y23 Y24 /
A-
( 0 0 Ф11 Ф12 \
0 0 Y11 Y12
0 0 0 0
У 0 0 0 0 J
One can verify that pi(X) and X-2p2(X) are same.
3. Special case. Our approach in contrast to the standard one is allowed for the systems of the special type (2) which is
Xk+1 = aif (xk) + 0,2 f (xk-T) + ... + aN f (xk_(N _i)T),
immediately write the characteristic polynomial. Then
z
z
(1) k+1 (.2)
k+1 (+)
k+1
= a1f {zf )) + a,2f (zk1_)1) +
(T)
a1f {zi1+)1) a1f {zkh)
a2f (zk1]) a2f {z™)
Zk+1 = a1f {zkT+11) ) + a2f {zkT 1) ) + ••• + aN f {zk1-N1++2)-Let us denote Jacobi matrices f'{nj) of the dimension m x m by Mj, j = 1,...,T. Let
• + aN f {zk-N+1)' + aN f {zk-N+2)' + aN f {zk-N+2)'
ST-1)
(T-1h
ST-1)
Sj)
(j)
nj + S(j). By linearizing we get
= Mt (01 S(T) + avSfl + ... + ON S(T)
g(1) °k+1
°k+1 S3)
k+1 + ••• + aN gk-N +1 ^
M1{a1gi+1 + a2gk+2 + ••• + aN ¿k-N+J'
<k+1 = M2{a15k+1 + a2S)+2 + ••• + aN б^+Л,
(2)
5k?
g(T) Jk+1
MT -1{a15k+1 ^ + a2SikF+21) + ••• + aN gkT—N+1) •
k+2
k-N+1)
Letting 6kj) = Ak Sj we obtain ( Sj = 0)
Ak+1 SX = MT {a1Ak + ... + aN Ak-N+1)ST, Ak+1 S2 = M1{a1Ak+1 + ... + aN Ak-N+2 )S1
Ak+1 ST = MT-1{a1Ak+1 + ... + aN Ak-N+2 )ST-1. Denote p{A) = a1
AN
+ ... + aN. Then
AN S1 AN-1S2
Mt p{A) st, M1 p{A)S1,
XN-1 sT = MT-l p(X)sT-1.
Since the eigenvalues of the matrix MT ■ ■ ■ Mi are independent on the cyclic reordering of the product matricies [4], we have
\A1+(n-1)t I - Mt Mt-1 • • • M1{p{A))T \ = 0.
(21)
Let the eigenvalues of the matrix MTMT-l ■ ■ ■ M\ be m, ...,^m. Then changing to the product to the Jordan canonical form one get from (21)
]J{A1+(N-1)T - »j {p{A))T) = 0. (22)
j=1
z
k
In special case and m = 1 the equivalence of the characteristic polynomials obtain by the standard and by our approach was established in [1]. 4. Conclusion. Assume that the system
xk+i = f (xk), f : A ^ A, A c IRm, (23)
has an unstable T-cycle (n1,...,nr). The cycle multipliers are zeros of the
characteristic polynomial
det ^J-n f '(Vj )j =0.
Assume that the multipliers are known only approximately, i. e. located in a region M c €. Now, let us close the system by a control
N-1
Un
= - E ^ (f (Xn-jT+T ) - f (xn-jT )), \£j \ < 1,j = 1,...,N - 1. (24)
j=i
The closed-loop system xn+1 = f (xn ) + un can be written as
NN
Xn+1 = ak f (xn-kT+t ), = 1, (25)
k=i k=i
where ak and ek are in bijection
N
£i = E ak, j = 1, ...,N - 1.
k=j+i
Note that the T-cycles of the systems (23) and (25) coincide.
The characteristic equation of the linearized around the cycle system is
П
j=1
AT(N-1)+1 - ak X
NT
N-k
ak A
k=1
= 0, Vj e M, j = 1, ..., m.
It is required to choose the gain £j in the control (24) such that
i) T-cycle of the system (25) to be locally stable;
ii) the depth of the prehistory T(N — 1) in the control (24) to be minimal. Thanks to formula (22) the above problem can be treated by the use of geometric
function theory of complex variables. This theory allows to find necessary metric properties of the exceptional sets of the polynomial mappings of the unit disc
N
F : D ^ С, F(z) = z | ^ ajzj-1
j=1
From there one can find the minimal values of N and the optimal coefficients. The scalar case of T = 1,2 was treated in [5, 6].
The author would like to express her deep gratitude to Dmitriy Dmitrishin and Paul Hagelstein for the guidance through the problem, useful discussions and for the help in preparation of manuscript.
References
1. Dmitrishin D., Hagelstein P., Khamitova A., Stokolos A. On the stability of cycles by delayed feedback control. Linear and Multilinear Algebra, 2016, vol. 64, issue 8, pp. 1538—1549.
2. Dmitrishin D., Khamitova A., Stokolos A., Tohaneanu M. Extremal trigonometric polynomials and the problem of optimal stabilization of chaos. Harmonic analysis, partial differential equations, complex analysis, banach spaces, and operator theory. Association for Women in Mathematics Series, 2016, vol. 2 (unpublished).
3. Morgül O. On the stability of delayed feedback controllers. Physics Letters A, 2003, vol. 314, pp. 278-285.
4. Horn R. A., Johnson C. Matrix analysis. Cambridge, Cambridge University Press, 1985, 561 p.
5. Dmitrishin D., Khamitova A. Methods of harmonic analysis in nonlinear dynamics. Comptes rendus mathematique, 2013, vol. 351, issues 9-10, pp. 367-370.
6. Dmitrishin D., Khamitova A., Stokolos A. Fejer polynomials and Chaos. Springer Proceedings in Mathematics and Statistics, 2014, vol. 108, pp. 49-75.
For citation: Khamitova A. D. Characteristic polynomials for a cycle of non-linear discrete systems with time delays. Vestnik of Saint Petersburg University. Series 10. Applied mathematics. Computer science. Control processes, 2016, issue 4, pp. 104-115. DOI: 10.21638/11701/spbu10.2016.410
Статья рекомендована к печати доц. А. П. Жабко. Статья поступила в редакцию 15 октября 2015 г. Статья принята к печати 29 сентября 2016 г.
