144

ISSN 1064–5624, Doklady Mathematics, 2009, Vol. 79, No. 1, pp. 144–146. © Pleiades Publishing, Ltd., 2009.Original Russian Text © G.A. Leonov, 2009, published in Doklady Akademii Nauk, 2009, Vol. 424, No. 4, pp. 462–464.

When the method of harmonic linearization [1–4] isapplied to the system

(1)

where

P

is a constant

n

×

n

matrix,

q

and

r

are constant

n

-vectors,

ϕ

(

σ

)

is a piecewise continuous function, andthe asterisk denotes transposition, it is usually assumedthat the matrix

P

has a pair of purely imaginary eigen-values

±

i

ω

0

(

ω

0

> 0)

and the other eigenvalues have neg-ative real parts. Under these assumptions, system (1)can be written in the form

(2)

Here,

A

is a constant

(

n

– 2)

×

(

n

– 2)

matrix all ofwhose eigenvalues have negative real parts,

b

and

c

are(

n

– 2)-vectors, and

b

1

and

b

2

are numbers.The development of numerical methods, computers,

and applied bifurcation theory suggests revisiting andrevising early ideas on the application of the smallparameter method and the harmonic linearization pro-cedure to control systems. The combined application ofharmonic linearization, the classical small parametermethod, and numerical methods makes it possible tocalculate periodic oscillations by a multistep procedureinvolving the application of harmonic linearization atthe first step. Under this approach, it is very natural todefine a special Poincaré map for system (2). Thispaper studies such a map.

In the basic, noncritical, case, we assume that

ϕ

(

σ

) =

εψ

(

σ

)

, where

ε

is a small positive parameter. In whatfollows, without loss of generality, we also assume that,for given

A

, there exists a number

α

> 0 such that

Consider the following set in the phase space of sys-tem (2):

dxdt------ Px qϕ r*x( )+= ,

x1 ω0x2– b1ϕ x1 c*x3+( ),+=

x2 ω0x1 b2ϕ x1 c*x3+( ),+=

x3 Ax3 bϕ x1 c*x3+( ).+=

x3* A A*+( )x3 α x32, x3 Rn 2– .∈∀–≤

Here,

D

,

a

1

,

and

a

2

are positive numbers.

Lemma 1.

For any point x

1

(0),

x

2

(0),

x

3

(0)

from

Ω

,there exists a number

for which

x

1

(

T

) > 0

and

x

2

(

T

)

= 0.

Moreover, the rela-tions

x

1

(

t

) > 0

and

x

2

(

t

)

= 0

do not hold for

t

∈

(0,

T

)

.

Lemma 2.

There exists a number D not dependingon

ε such that, for any point x1(0), x2(0), x3(0) from Ω,

In what follows, we consider such numbers D.We set

Lemma 3. If

(3)

(4)

then the point x1(0) = a1, x2(0) = 0, x3(0) from Ω satisfiesthe condition x1(T) > a1, and the point x1(0) = a2, x2(0) = 0,x3(0) from Ω satisfies the condition x1(T) < a2.

Lemmas 1–3 imply the following theorem.Theorem 1. If (3) and (4) hold, then, for any suffi-

ciently small ε > 0, the Poincaré map

of the set Ω is a self-map, that is, FΩ ⊂ Ω.This implies the following result.Theorem 2. If

Ω x3 Dε, x2≤ 0, x1 a1 a2,[ ]∈= .=

T T x1 0( ) x3 0( ),( ) 2πω0------ O ε( ),+= =

x3 T( ) Dε.≤

K a( ) ψ ω0t( )acos( ) ω0t( )cos t.d

0

2π/ω0

∫=

b1K a1( ) 0,>

b2K a2( ) 0,<

Fx1 0( )

0

x3 0( )⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ x1 T( )

0

x3 T( )⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

K a( ) 0, b1dK a( )

da--------------- 0,<=

On the Harmonic Linearization MethodCorresponding Member of the Russian Academy of Sciences G. A. Leonov

Received July 8, 2008

DOI: 10.1134/S1064562409010426

St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034 Russia

CONTROLTHEORY

DOKLADY MATHEMATICS Vol. 79 No. 1 2009

ON THE HARMONIC LINEARIZATION METHOD 145

then, for a sufficiently small ε > 0, system (1) has a T-periodic solution such that

This periodic solution is stable in the sense that it hasan ε-neighborhood such that all solutions with initialdata from this ε-neighborhood remain in it withincreasing time t.

The standard, basic method of harmonic lineariza-tion described in Theorem 2 turns out to be too roughfor detecting periodic oscillations in nonlinear systemssatisfying the generalized Routh–Hurwitz conditions.However, a generalization of Theorem 1 in the spirit ofthe classical analysis of critical cases in motion stabilitytheory [5] gives effective bounds for periodic oscilla-tions in systems satisfying the generalized Routh–Hur-witz conditions.

Consider the class of functions ϕ(σ) of the form

(5)

where µ and M are positive numbers and ε is a smallpositive parameter.

Consider the following set in the phase space of sys-tem (2):

where D, a1, and a2 are positive numbers.

The Poincaré map FΩ of the set Ω can be defined ina similar way. Here, the number T is chosen so that

and these relations do not hold for t ∈ (0, T).Theorem 3. If

(6)

(7)

then there exists a number D not depending on ε forwhich

(8)

This result implies the following theorem.Theorem 4. If b1 < 0 and

then system (1) with nonlinearity (5) has a periodicsolution with initial data

r*x t( ) a ω0t( )cos O ε( ), T+ 2πω0------ O ε( ).+= =

ϕ σ( ) µσ, σ ε– ε,( ),∈∀=

ϕ σ( ) Mε3, σ ε,>∀=

ϕ σ( ) Mε3– , σ ε,–<∀=

Ω = x3 Dε2, x1 c*x3 0= , x2 a1 a2–,–[ ]∈+≤ ,

x1 T( ) c*x3 T( )+ 0, x2 T( ) 0<=

µ3ω0a2--------------- b2 c*b b1+( )µ b1ω0+( ) Mb1a2 0,>+

µ3ω0a1--------------- b2 c*b b1+( )µ b1ω0+( ) Mb1a1 0,<+

FΩ Ω.⊂

µb2r*q b1ω0 0,>+

This solution is stable in the sense of inclusion (8).

The periodic solutions described in Theorems 2 and4 can be treated as “support” periodic oscillations, andsystem (1) with nonlinearities ϕ(σ) satisfying the aboveassumptions can be regarded as a “generator” of basicsystems in algorithms for seeking periodic solutions tothe other system

(9)

In this case, we can construct a finite sequence of func-tions ϕj (σ) such that the graphs of each pair ϕj, ϕj + 1 areclose to each other. For the system

(10)

with ϕ1(σ) = ϕ(σ) and small ε, we take the periodicsolution g1(t) described in Theorem 2 or 4. Either allpoints of this periodic solution belong to the domain ofattraction of a stable periodic solution g2(t) to system (10)with j = 2, or, in the passage from system (10) with j = 1 tosystem (10) with j = 2, the stability loss bifurcationoccurs and the periodic solution disappears. In theformer case, we can numerically determine g2(t) by cal-culating the trajectory of system (10) with j = 2 startingat the initial point x(0) = g1(0).

After a start at the point g1(0) and a transition pro-cess, the computational procedure reaches the periodicsolution g2(t) and computes it, provided that the compu-tational interval [0, T] is sufficiently large.

After g2(t) is computed, we pass to the next system (10)with j = 3 and perform a similar procedure for computingthe periodic solution g3(t), considering the trajectory fromthe initial point x(0) = g2(T), which approaches the peri-odic trajectory g3(t) with increasing t.

Continuing this procedure and successively calcu-lating gj(t) by using the trajectories of system (10) withinitial data x(0) = gj − 1(T), we either obtain a periodicsolution ϕm(σ) = f(σ) of system (10) with j = m orobserve the bifurcation of stability loss and disappear-ance of a periodic solution at some step.

Consider an example of such a procedure for com-puting periodic oscillations and the stability loss bifur-cation.

Example 1. Let ψ(σ) = k1 + k3σ3. Then, K(a) = k1a +

. This implies

x1 0( ) O ε2( )= , x3 0( ) O ε2( ),=

x2 0( )µ µb2r*q b1ω0+( )

3ω0M b1–( )--------------------------------------------- O ε( ).+–=

dxdt------ Px qf r*x( )+= .

dxdt------ Px qϕ j r*x( )+=

3k3a3

4-------------

146

DOKLADY MATHEMATICS Vol. 79 No. 1 2009

LEONOV

and the stability condition takes the form b1k1 > 0.Suppose that ω0 = 1, b1 = –1, b2 = 1, c = 1, b = –1, A =

–1, k1 = –3, and k3 = 4. The classical harmonic lineariza-tion method [1–4] yields the existence at any ε > 0 of astable periodic solution to system (1) satisfying thecondition σ(t) = r*x(t) ≈ cos t.

According to Theorem 2, system (1) has a stableperiodic solution of the form

for small ε > 0. Considering discrete increments of ε(ε1 = 0.01, ε2 = 0.02, …) and applying the numericalprocedure ϕj(σ) = εjψ(σ) described above, we see that,for ε ∈ (0, ρ), there exist stable periodic solutions, andat ρ ≈ 0.33 the stability loss bifurcation occurs and theperiodic solution disappears. For ε ∈ (ρ, 1), the domaincontaining the periodic trajectory is the domain ofattraction of a stable equilibrium.

Example 2. Consider system (1) with nonlinearity (5),where ε = εj, ε1 is a small parameter, ε2 = 0.1, …, ε8 =0.7, µ = 2, M = 1, r*q = –1, ω0 = 1, b1 = –1, b2 = 1, c = 1,b = –1, and A = –1. In the linear case of ϕ(σ) = kσ, thesolution is stable for k ∈ (0, +∞). At the first step, wehave x1(0) = O(ε2), x3(0) = O(ε2), and x2(0) =

. Continuing the numerical procedure

described above, we obtain periodic solutions for j =2, …, 7; for j = 8, the disappearance of the periodicsolution and the attraction to the stable equilibrium areobserved.

Example 3. For n = 3, the absolute stability criteriagiven in [6–8] and Theorem 4 completely solve Aizer-man’s problem [9]. In this case, the conditions in The-orem 4 are necessary, and the condition

which follows from Popov’s criterion, is sufficient forabsolute stability. This question was considered indetail in monograph [10].

REFERENCES1. E. P. Popov and I. P. Pal’tov, Approximate Methods for

Studying Nonlinear Automatic Systems (Fizmatgiz,Moscow, 1960) [in Russian].

2. E. P. Popov, The Theory of Nonlinear Automatic Controland Adjustment Systems (Nauka, Moscow, 1979) [inRussian].

3. A. A. Pervozvanskii, A Course in the Theory of Auto-matic Control (Nauka, Moscow, 1986) [in Russian].

4. H. K. Khalil, Nonlinear Systems (Prentice Hall, Engle-wood Cliffs, N.J., 2002).

5. I. G. Malkin, Stability of Motion (Nauka, Moscow, 1966)[in Russian].

6. M. A. Aizerman and F. B. Gantmakher, The AbsoluteStability of Adjustable Systems (Akad. Nauk SSSR,Moscow, 1963) [in Russian].

7. G. A. Leonov, I. M. Burkin, and A. I. Shepelyavy, Fre-quency Methods in Oscillation Theory (Kluwer, Dor-drecht, 1996).

8. G. A. Leonov, D. V. Ponomarenko, and V. B. Smirnova,Frequency-Domain Methods for Nonlinear Analysis.Theory and Applications (World Sci., Singapore, 1996).

9. B. A. Aizerman, Usp. Mat. Nauk 4 (4), 186–188 (1949).10. V. A. Pliss, Some Problems of the Theory of Global Sta-

bility of Motion (Leningr. Gos. Univ., Leningrad, 1958)[in Russian].

a4k1

3k3--------– ,=

x1 t( ) tcos O ε( ), x2 t( )+ t O ε( ),+sin= =

x3 t( ) O ε( )=

23--- O ε( )+–

µb2r*q b2ω0 0,<+