Стабилизация неустойчивых состояний равновесия динамических систем. Часть 1

Шумафов Магомет Мишаустович

МАТЕМАТИКА MATHEMATICS

УДК 517.977 ББК 22.19 Ш 96

Шумафов М.М.

Доктор физико-математических наук, профессор, зав. кафедрой математического анализа и методики преподавания математики факультета математики и компьютерных наук Адыгейского государственного университета, Майкоп, тел (8772) 593905, e-mail: [email protected]

Стабилизация неустойчивых состояний равновесия динамических систем. Часть 1

(Рецензирована)

Аннотация. Рассматривается проблема стабилизации неустойчивых состояний равновесия управляемых динамических систем обратной связью. Работа состоит из трех частей. В первой части дается краткий обзор по стабилизации неустойчивых состояний равновесия. Используются различные типы обратной связи по выходу (по состоянию): стационарная и нестационарная; классическая и в форме Пирагоса (Pyragas) с запаздыванием. Сформулированы задачи стабилизации неустойчивых состояний равновесия динамических систем обратной связью перечисленных выше типов. Эти задачи возникают в различных приложениях теории управления, их постановки принадлежат известным ученым. Во второй и третей частях представлены основные результаты, полученные в работах из приведенного в статье списка литературы. Даны эффективные необходимые и/или достаточные условия стабилизации неустойчивых состояний равновесия двумерных и трехмерных динамических систем в терминах параметров систем. Эти условия показывают, что введение в рассматриваемую систему нестационарной обратной связи или обратной связи с запаздыванием в целом расширяет возможности обычной стационарной стабилизации. Результаты могут быть использованы при исследовании вопросов устойчивости нелинейных управляемых систем в окрестности неустойчивого состояния равновесия, а также при стабилизации неустойчивых состояний равновесия, встроенных в хаотические аттракторы нелинейных динамических систем.

Ключевые слова: асимптотическая устойчивость, стабилизация, назначение полюсов, неустойчивое состояние равновесия, управляемая система, обратная связь по выходу, обратная связь с запаздыванием.

Shumafov M.M.

Doctor of Physics and Mathematics, Professor, Head of Department of Mathematical Analysis and Methodology of Teaching Mathematics of Mathematics and Computer Science Faculty, Adyghe State University, Maikop, ph. (8772) 593905, e-mail: [email protected]

Stabilization of unstable steady states of dynamical systems. Part 1

Abstract. In the paper a problem of stabilization of unstable steady states (unstable equilibria) of controllable dynamical systems by feedback control is considered. The work consists of three parts. In the first part a short survey on the feedback control stabilization of unstable steady states of dynamical systems is presented. Different types of output (state) feedback control are used: stationary and nonstationary ones; classical and Pyragas' time-delayed ones. The problems of stabilization of unstable steady states of dynamical systems by the mentioned above types of feedback control are formulated. These problems have originated in a variety of control theory applications, and are stated by the famous scholars. In the second and third parts main results are presented along with a brief literature review. Effective necessary and/or sufficient conditions for stabilization of unstable steady states of two-and three-dimensional dynamical systems in terms of the system parameters are given. These conditions show that an introduction in the system considered nonstationary feedback control or time-delayed feedback one, in general, extends the possibilities of the ordinary stationary stabilization. The results can be used for stability analysis of nonlinear control systems in the neighborhood of an unstable equilibrium point, and also for stabilization of unstable steady states embedded in chaotic attractors of nonlinear dynamical systems.

Keywords: asymptotic stability, stabilization, pole assignment, unstable steady state, controllable system, output feedback, delayed feedback control.

Работа представляет собой расширенный текст пленарного доклада на Первой Международной научной конференции «Осенние математические чтения в Адыгее», посвященной памяти профессора К.С. Мамия, 8-10 октября 2015 г. Адыгейский государственный университет, Майкоп, Республика Адыгея.

1. Introduction

One of the most fundamental topics of control theory is a stabilization problem of dynamical systems. Within last 140 years the methods of stabilization have been constructed, developed and improved: from the creation of Watt's regulator to the analysis and synthesis of the rocket stabilization systems and the controlling chaos.

At present the various methods of stabilization have become classical ones in control theory, and have entered into many books and surveys ([1,2]: see also bibliography in [3]). But in the last thirty years a rapid growth of publications devoted to the methods of stabilization of control systems occurred. The increasing interest to stabilization problems is motivated both the needs of the practice of control, and the formulation open problem by many famous scholars ([4-9]).

One of the problems stimulated a number of publications was the Brockett problem [9] on stabilizability of an unstable linear stationary system by means of a nonstationary output feedback control. In mathematical terms the Brockett's stabilization problem is stated as follows.

Consider a linear time-invariant controllable dynamical system described by the differential equation

where x € Rn is a state vector, u € Rm is an input (control) vector, y £ ~B! is an output

vector, and A,B,C are real constant (n x n) — , (n x m) — , (I x n)- matrices , respectively. The Brookett's problem consists in finding a time-varying (nonstationary) output feedback

is asymptotically stable.

If in the feedback (1.2) the matrix K(t) is constant K(t) = K(= const), then the Brockett problem turns into the classical stationary feedback stabilization one.

As is well-known the solution of the classical stabilization problem by stationary full state feedback u — Kx follows from Zubov's and Wonham's theorem on pole assignment [4,5].

Note that the proof of Zubov's-Wonham's theorem in multi-input (m > 1) case is rather tedious. Therefore after publication of works [4, 5] there were offered alternative proofs to simplify them (see, bibliography in [3]). A simple and new direct proof of Zubov-Wonham theorem is proposed in works [10, 11].

First, the solution of the Brockett problem in a number of cases, important for practice, was given in G.A. Leonov's [12,13] and L. Moreau & D. Aeyels' [14] works. In particular, in these works necessary and sufficient conditions for nonstationary low- and high-frequency stabilizations of two- and three-dimensional dynamical systems are obtained. It was shown that nonstationary feedback control strategy can achieve results that cannot be obtained by stationary feedback, and this approach essentially extends the domains of stabilization derived by stationary feedback.

x — Ax + Bu, у — Cx

(1.1)

u = K(t)y

with real (m x Z)-matrix K(t) such that the closed-loop system (1.1), (1.2)

(1.2)

x = (A + BK(t)C)x

The problem of controlling chaos and its stabilization in deterministic chaotic dynamical systems was other one which caused the enormous number of publications. Starting with the pioneering works of Ott, Grebogi, Yorke [15] and K.Pyragas [16] this problem is intensively studied by many researchers for last more than twenty years ( see. for instance, surveys [17-20]). In these works stabilization of a chaotic system is achieved by stabilization of unstable periodic orbits (UPOs) embedded in a strange attractor of the system.

Various methods of control have been developed in order to stabilize UPOs embedded in a chaotic attractors of dynamical systems. One of these methods is the method of Pyragas [16], called delayed feedback control (DFC), in which the control input is constructed by the difference between the current output (or state) of a given system and the delayed output (or state): u(t) — K[y(t) — y(t — T)]. Here r > 0 is a delay time, and K is a feedback gain(weight).

The Pyragas' DFC is a simple and powerful efficient method for stabilization UPOs of dynamical systems. The DFC and its various modifications have been successfully implemented in a large variety of systems in physics, chemistry, biology, medicine and engineering (see reviews [18,19]). It turned out that the DFC scheme, which was originally invented for stabilization of UPOs, is also suitable to stabilize unstable steady states (USSs) of dynamical systems ([21-33]).

One of the central issues in applied sciences is stabilization of USS or equilibria of dynamical systems. Steady states play an important role in studying of a large variety of optical, electronic, chemical, biological, and other nonlinear systems (see, for instance, [18,19]). According to Pyragas [18,22] the problem of stabilizing steady states by DFC (or other its extensions) techniques "is, maybe, more important for various applications that the problem of stabilizing UPOs".

Although the effects of DFC schemes on the stabilization of UPOs are studied in a large number of works (see bibliography in [18,19]), much less is known in the case of USSs.

It should be noted that the theory of DFC is rather difficult since the equations describing the closed-loop system, including DFC, are delayed differential equations. Even linear stability analysis of such systems is quite complicated because of existing of infinite number of Floquet exponents (in the case of the problem of UPOs stabilization) or infinite number of roots of transcendental characteristic equation associated with delayed linear differential equations being the linearized ones of the nonlinear closed-loop system equations (in the case of USSs stabilization). It makes difficult to a considerable extent obtaining effective stabilization analytic criteria. Nevertheless some analytical approaches have been developed [21-38].

In particular, it was shown that DFC is subject to a substantial limitation, which is now referred to as the odd number limitation (ONL). It turned out that DFC can stabilize only a certain class of USSs and UPOs.

Namely, assume that the system (1.1) is the linearized system around the USS x — 0 (corresponding to uncontrolled system: u = 0) of a nonlinear system

x = f(x,u), x e Rn, u e Km, (1.3)

where function f is continuously differentiable, /(0, 0) = 0 , so that

A = (d/dx)f(0, 0), B = (d/du)f(0, 0).

The ONL for stabilizing USSs of continuous-time systems [21,22]: If the Jacobian matrix A of system (1.3) evaluated at the target USS x = 0(m = 0) has an odd number of real eigenvalues that are greater than zero, then the linearized system

x = Ax + BK[x(t) - x(t - r)]

of the closed-loop system

x = f(x, K[x(t) — x(t — r)])

around point x = 0 cannot be stabilized by the DFC u(t) = K[x(t) — x(t — r)] with any choices of the constant feedback gain (m x n)-matrix K and positive number r > 0.

At first ONL was established for stabilizing USSs of discrete-time systems in the Ushio's work [37]. In this case the Jacobian matrix A of nonlinear discrete-time system must have an odd number of real eigenvalues that are greater than unity. For stabilizing UPOs of continuous-time system the ONL was established in Nakajima's work [35] for non-autonomous systems, and in the Hooton's and Amann's work [38] for autonomous ones.

There are also some interesting analytical results [23-30]. derived by using numerical simulations, but a detailed theoretical (analytical) investigation on the whole is still missing.

In the work [30] necessary and/or sufficient conditions of the stabilization of USSs of the linearized system of a two-dimensional nonlinear dynamical system are obtained by DFC and by using an eigenvalue optimization approach in combination with a continuation argument and numerical simulation. Such approach allowed to "guess" the analytical expressions for the boundary of the domains of stabilization, that then one are verified by means of numerical computations.

Other approach of solving the USSs stabilization problem based on the method of D-decomposition [39] of the space of system parameters is proposed in [31-33]. The advantage of this approach consist in that, the used method is purely analytical and yet much less mathematical tools are exploited, and therefore the corresponding stabilization algorithms turn out to be more simple. Necessary and/or sufficient analytical conditions of the stabilization of the USSs of two-and three-dimensional dynamical systems are obtained it terms of the system parameters. These conditions show that the introduction of a delay in the feedback of the linearized systems of nonlinear ones on the whole enlarges the opportunities of stationary stabilization by feedback without delay.

The results obtained can be used in the linear stability analysis of nonlinear control systems in the neighborhood of an equilibrium point, and for stabilization of unstable equilibria of nonlinear dynamical systems with chaotic behavior.

2. Problem Statements

Consider a continuous-time nonlinear control dynamical system described by an ordinary differential equation

x{t) = f{x(t),u{t)), (2.1)

where x(t) denotes a state vector at the time I in the n-dimensional state space, x(t) £ R":

u(t) denotes an input (control) vector at the time t, u(t) € Rm: / : Rn x Rm —y Rn, / : (x,u) —y f(x,u), is continuously differentiate function.

Assume that from the state vector x(t) £ Rn one can calculate an output vector y(t) £ M.1 via a continuously differentiate function g : Kn —y M1, g : x —y y — g(x), which measures the state x(t) to create an output signal y(t) in the Z-dimensional signal space:

y(t)=g(x(t)). (2.2)

In particular, the output signal y could be. for instance, a single component of the state vector x. In the case g(x) = x the output y(t) is the state vector x(t).

Suppose that the system (2.1) with u(t) = 0 has an unstable steady state x0 (equilibrium), which may be assumed without loss of generality that it is the origin, i.e.. x0 — 0. and /(0, 0) = 0. Let g(0) = 0.

General Stabilization Problem

Given system (2.1). (2.2). find an output feedback control

u(t) = h(t,y{t)), (2.3)

where h : R x W —y Rm, h : (t,y) —>• u — h{t, y) is a continuously differentiable function, h(t, 0) = 0 for allt, such that the origin (equilibrium) of the closed-loop system (2.1). (2.3). i.e.,

x(t) = f(x(t),h{t,g{x{t))) (2.4)

is asymptotically stable.

Here in (2.3) h is a varying function, which should be appropriately chosen. In the case y(t) = x(t) it is required to find a state feedback control u(t) = h(t, x(t)), such that the origin of the system

i(t) = f(x(t),h(t,x(t))) (2.5)

is asymptotically stable.

In order to investigate the stability of the steady state x — 0 of equation (2.4) or (2.5) it is performed a linear stability analysis. The latter is carried out by linearization of the equation (2.4)/(2.5) around the origin x — 0. This linerazation can be maked by considering the linearized equations (2.1). (2.2) about x — 0 with combination of linearized equation (2.3). We have

x(t)=Ax(t) + Bu(t), (2.6)

y{t) = Cx{t), (2.7)

where A - Dxf(0, 0), B = Duf(0, 0), C = Dxg(0).

Here Dxf(0,0). Duf(0,0) and Dxg{0) denote the Jacobian (n x n) — , (n x m) — , and (I x n)— matrices with respect to vector variables x and u: A £ RnXn, B £ Q ^

In the following we assume without loss of generality, that rankB — m, rankC — I. In what follows we will have need of the notions of controllability and observability of system (2.6),(2.7). Recall that:

(a) the linear system (2.6) or the pair (A,B) is controllable if and only if (Kalman [40])

rank(B AB ... An~1B) = n:

(b) the linear system (2.6), (2.7) or the pair (A,C) is observable if and only if (Kalman [40])

rank(C* A*C*...{A*)n~lC*) = n. - 17-

(Here the sign * denotes transposition.)

In what follows we consider four types of feedback control (2.3):

(1) time-invariant output feedback

u(t) = Ky(t), (2.8)

where K € Kmx* is a control gain constant matrix;

(2) time-varying output feedback

u(t) = K{t)y(t), (2.9)

where the control gain matrix K{t) is an real variable time-dependent matrix;

(3) time-delayed output feedback control of the classical form

u(t) = Ky{t-T), (2.10)

where matrix K £ Rm>^ and r > 0 is a positive parameter;

(4) time-delayed output feedback control of the Pyragas' form

u(t) = -K[y{t)-y(t-r)], (2.11)

where K € Mmxi and r > 0 is a positive parameter.

We mean that above-mentioned output feedbacks (2.8)-(2.11) are static ones. i.e.. the control (input) u(t) at time t is a linear function of the output y(t) also at time t. (Here we do not consider stabilizability by dynamic output feedback.)

The linearized closed-loop system (2.4) with control u(t) defined by (2.8). (2.9). (2.10). and (2.11) is described by differential equations

x(t) = (A + BKC)x{t), (2.12)

±{t) = (A + BK{t)C)x(t), (2.13)

±(t) = Ax(t)+BKCx(t-r), (2.14)

x(t) = Ax(t) + BKC[x(t - r) - z(i)], (2.15)

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respectively.

In the case y(t) — xit) (C — I - indentity matrix) the output of system (2.6). (2.7) is the full state, and instead of output feedbacks (2.8)-(2.11) we have state ones.

In conformity to equations (2.12)-(2.15) the stabilization problem consists in finding appropriate matrices K and K(t) in (2.12) and (2.13). and a matrix K and a number r > 0 in (2.14) and (2.15). respectively, so that the steady state x=0 of systems (2.12)-(2.15) would be asymptotically stable.

Later on for convenience of references we formulate separately the stabilization problem for system (2.6), (2.7) with feedbacks (2.8)-(2.11).

Problem 1. Stationary Output Feedback Stabilization

Given a system (2.6),(2.7), find an real (mx I)-matrix K such that the origin of the system (2.12) would be asymptotically stable.

In other words Problem 1 can be more exactly reformulated as follows:

Given a triple real matrices (A,B,C). Determine necessary and sufficient conditions under which there exists a real matrix K such that the matrix A + BKC is stable, i.e., all eigenvalues Xj(A + BKC), j — 1,..., n, of the matrix A + BKC lie in the open left-half plane: Xj(A + BKC) < 0.

The following problem is a generalized one of Problem 1.

Problem 2. Pole Assignment

Given a triple real matrices (A,B,C) and an arbitrary set {^j}nj=l of complex numbers Hj closed under complex conjugation. Find an real (m x I) -matrix K such that the spectrum a (A + BKC) of the matrix A + BKC coincides with the set :

a(A + BKC) =

Note that in the full state feedback case y(t) — x(t) (C — I) the Problem 2 was set and solved by V.I. Zubov [4] and W.M. Wonham [5].

Problem 3. Nonstationary Output Feedback Stabilization (Brockett [9])

Given a system (2.6), (2.7), find a time-varying (nonstationary) output feedback (2.9) such that the origin of the system (2.13) would be asymptotically stable.

In other words Problem 3 is more precisely formulated as follows.

Given a triple real matrices (A, B, C) of dimensions nxn, nxm and Ixn, respectively. Under what conditions (necessary and sufficient) does there exist a time-dependent matrix K(t) of dimension mx I, such that the origin of the system (2.13) is asymptotically stable?

Thus. Brocket Problem consists in that how much the introduction of time-dependent matrices K(t) in feedback enlarges the possibilities of stationary stabilization by (2.8).

Note that, since for linear systems asymptotical stability of every solution is equivalent to asymptotical stability of the origin, in the following we may say of the asymptotical stability of the systems (2.12) and (2.13).

Problem 4. Delayed Feedback Control (DFC)

Given a system (2.6), (2.1), find necessary and sufficient conditions under which there exist an real (m x I)— matrix K and a number r > 0 such that the origin of the system (2.14)/(2.15) would be asymptotically stable.

The approaches for solution of problems formulated above will be presented in the second part and third parts (Part II). (Part III) of this paper.

This work was supported by the Ministry of Education and Science of the Russian Federation under project no. 451.

Примечания:

1. Static Output Feedback. - A Survey / V.L. Syrmos, C.T. Abd-Allah, P. Dorato, K. Grigoriadis // Automatica. 1997. Vol. 33, No. 2. P. 125-137.

2. Polyak B.T., Shcherbakov P.S. Hard problems of linear control theory. Some approaches to solving // Avtomatika i Telemekhanika. 2005. No. 5. P. 7-46.

3. Leonov G.A., Shumafov M.M. Stabilization of Linear Systems // Cambridge Scientific Publishers. Cambridge, 2012. 408 pp.

4. Зубов В.И. Теория оптимального управления. Л.: Судостроение, 1966. 352 с.

5. Wonham W.M. On Pole Assignment in Multi-Input Controllable Linear Systems // IEEE Trans. Aut. Contr. 1967. Vol. AC-12, No. 6. P. 660-665.

6. Wonham W.M. Linear Multivariable Control: a Geometric Approach. New York; Heidelberg; Berlin: Springer-Verlag, 1979. 326 pp.

7. Bernstein D.S. Some Open Problems in Matrix Theory Arising in Linear Systems and Control // Linear Algebra and its Applications. 1992. Vol. 162-164. P. 409-432.

8. Rosenthal J., Willems J.C. Open problems in the area of pole placement // Open Problems in Mathematical Systems and Control Theory. Springer, 1999. 288 pp.

9. Brockett R. A stabilization problem // Open Problems in Mathematical Systems and Control Theory. Springer, 1999. 288 pp.

10. Леонов Г.А., Шумафов М.М. Элементарное доказательство теоремы о стабилизируемости линейных управляемых систем // Вестник С.-Петербургского университета. Сер. Математика, механика и астрономия. 2003. Вып. 3 (№ 17). С. 56-68.

11. Леонов Г. А., Шумафов М.М. Алгоритм пошаговой стабилизации линейного объекта управления // Известия вузов. Сев.-Кавказский регион. Естественные науки. 2005. № 2. С. 14-19.

12. Леонов Г.А. Проблема Брокетта в теории устойчивости линейных дифференциальных уравнений // Алгебра и анализ. 2001. Т. 13, вып. 4. С. 134-155.

13. Леонов Г. А. Стабилизационная проблема Бро-кетта // Автоматика и телемеханика. 2001. № 5. С. 190-193.

14. Moreau L., Aeyels D. Periodic output feedback stabilization of single-input single-output continuous-time systems with odd relative degree // Systems & Control Letters. 2004. Vol. 51, No. 5. P. 395-406.

15. Ott E., Grebogi C., Yorke J.A. Controlling Chaos // Phys. Rev. Lett. A. 1990. P. 1196-1199.

16. Pyragas K. Continuous control of chaos by selfcon-trolling feedback // Phys. Lett. A. 1992. Vol. 170. P. 421-428.

17. Tian Yu., Zhu J., Chen Gu. A survey on delayed feedback control of chaos // Journ. of Control Theory and Application. 2005. No. 4. P. 311-319.

18. Pyragas K. Delayed feedback control of chaos // Phil. Trans. Royal Soc. A. Vol. 369. 2006. P. 20392334.

19. Pyragas K. A Twenty-Year Review of Time-Delay Feedback Control and Recent Developments // International Symposium on Nonlinear Theory and its Applications. Spain, October 22-26, 2012. P. 683-686.

20. Leonov G.A., Shumafov M.M., Kuznetsov N.V. A short survey of delayed feedback stabilization // 1st IFAC Conference on Modeling, Identification and Control of Nonlinear Systems. S.Petersburg, June 24-26, 2015. P. 716-719.

References:

1. Static Output Feedback. - A Survey / V.L. Syrmos, C.T. Abd-Allah, P. Dorato, K. Grigoriadis // Automatica. 1997. Vol. 33, No. 2. P. 125-137.

2. Polyak B.T., Shcherbakov P.S. Hard problems of linear control theory. Some approaches to solving // Avtomatika i Telemekhanika. 2005. No. 5. P. 7-46.

3. Leonov G.A., Shumafov M.M. Stabilization of Linear Systems // Cambridge Scientific Publishers. Cambridge, 2012. 408 pp.

4. Zubov V.I. Theory of optimal control. L.: Sudostroenie, 1966. 352 pp.

5. Wonham W.M. On Pole Assignment in Multi-Input Controllable Linear Systems // IEEE Trans. Aut. Contr. 1967. Vol. AC-12, No. 6. P. 660-665.

6. Wonham W.M. Linear Multivariable Control: a Geometric Approach. New York; Heidelberg; Berlin: Springer-Verlag, 1979. 326 pp.

7. Bernstein D.S. Some Open Problems in Matrix Theory Arising in Linear Systems and Control // Linear Algebra and its Applications. 1992. Vol. 162-164. P. 409-432.

8. Rosenthal J., Willems J.C. Open problems in the area of pole placement // Open Problems in Mathematical Systems and Control Theory. Springer, 1999. 288 pp.

9. Brockett R. A stabilization problem // Open Problems in Mathematical Systems and Control Theory. Springer, 1999. 288 pp.

10. Leonov G.A., Shumafov M.M. Elementary proof of the theorem on stabilizability of linear controllable systems // The Bulletin of the Saint-Petersburg University. Ser. Mathematics, Mechanics and Astronomy. 2003. Iss. 3 (No. 17). P. 56-68.

11. Leonov G.A., Shumafov M.M. Algorithm of step-by-step stabilization of linear object of control // News of Higher Schools. North Caucasian Region. Natural Sciences. 2005. No. 2. P. 14-19.

12. Leonov G.A. Brockett's problem in stability theory of linear differential equations // Algebra and Analysis. 2001. Vol. 13, Iss. 4. P. 134-155.

13. Leonov G.A. Brockett's problem of stabilization // Automatics and Telemechanics. 2001. No. 5. P. 190-193.

14. Moreau L., Aeyels D. Periodic output feedback stabilization of single-input single-output continuous-time systems with odd relative degree // Systems & Control Letters. 2004. Vol. 51, No. 5. P. 395-406.

15. Ott E., Grebogi C., Yorke J.A. Controlling Chaos // Phys. Rev. Lett. A. 1990. P. 1196-1199.

16. Pyragas K. Continuous control of chaos by selfcon-trolling feedback // Phys. Lett. A. 1992. Vol. 170. P. 421-428.

17. Tian Yu., Zhu J., Chen Gu. A survey on delayed feedback control of chaos // Journ. of Control Theory and Application. 2005. No. 4. P. 311-319.

18. Pyragas K. Delayed feedback control of chaos // Phil. Trans. Royal Soc. A. Vol. 369. 2006. P. 20392334.

19. Pyragas K. A Twenty-Year Review of Time-Delay Feedback Control and Recent Developments // International Symposium on Nonlinear Theory and its Applications. Spain, October 22-26, 2012. P. 683-686.

20. Leonov G.A., Shumafov M.M., Kuznetsov N.V. A short survey of delayed feedback stabilization // 1st IFAC Conference on Modeling, Identification and Control of Nonlinear Systems. S.Petersburg, June 24-26, 2015. P. 716-719.

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