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On the stability bifurcation of a nonlinear time delay system*
Automatica, Vol. 34, No. 3, pp. 375—378, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00
PII: S0005–1098(97)00208–2
Brief Paper
On the Stability Bifurcation of a Nonlinear Time Delay System* MUSTAPHA S. FOFANA- and DAVID J. LAMB Jr.Key Words—Hopf bifurcation; time delay; averaging method; stability.
Abstract—A computational burden has always been associated with the stability analysis of a time delay system over its entire parameter ranges. Indeed, this has led many previous investigators to develop control schemes under the assumption of small or negligible time delay. In this paper we have attempted to explore an alternative approach, which has been previously called as Critical or Single Points, Hale’s Method, Projection Series and Centre Manifold. Here we have deliberately omitted the relevant theorems, lemmas or corollaries of this approach and rather focus mainly on a calculation spirit to make the underlying ideas accessible to investigators seeking more insight into the wealth of time delay dynamics. The nonlinear delay equation around which we present this approach is a variational form of equations studied earlier by Bentsman et al. (1991), Lehman et al. (1992) and Lehman and Bentsman (1992, 1994). Furthermore, using the integral averaging method explicit bifurcation equations of the form g(a, k)"0 are derived and from the signs of their characteristic exponents, the stability of the trivial and nontrivial solutions is obtained. ( 1998 Elsevier Science Ltd. All rights reserved.
to a family of ordinary differential equations (ODEs) for fixed and positive time delay r. More details about the formulation of this equation and including many other delay equations in engineering and science, the books of Burton (1985), Chukwu (1992), Gopalsamy (1992), Kolmanovskii and Nosov (1986), Kuang (1993) and Salamon (1984) are excellent references. In equation (1.1), the bifurcation parameter k is written as the sum k "k : #ek8 of a small change ekJ from criticality k , in which e is # # a small scaling coefficient with value 04e;1 and the real coefficients c "ec : 6 , c "ec : 6 characterize the nonlinear 03 03 13 13 nature of the restoring forces. The functional elements x (h),col [x (t!r), x (t!r)] defined in C "C([!r, : 0], R2), t 1 2 are the restoring and damping forces at time t!r, whose trajectories coincide with those of the state variables x(t),col [x (t), x (t)]LR2 at h"t"0 through the definitions 1 2 x (h) "x(t#h), : xR (h) "x : R (t#h), for hL[!r, 0]. C is the t t Banach space of all continuously differentiable functions on [!r, 0] having values in R2 with the usual uniform supremum norm E x (/(h))E"suph L[!r, 0] Dx (/(h))D, where D ) D is any vector t t norm in R2. A function x "x (/(h), k)LC is a unique solution t t of equation (1.1) for all values of time t3[!r, R), if one can specify an initial continuously differentiable function, say /(h) on [!r, 0] with value / at h"0. The statements and proofs of 0 results associated with the alternative approach and in general, with solutions of delay differential equations for any fixed and positive time delay can be found in the following references: Banks and Manitius (1975), Chafee and LaSalle (1971), Chow and Mallet-Paret (1977), Claeyssen (1976), De Oliveira (1980), Hale and Perelo´ (1964), Hale (1977), Hale and Lunel (1993), Hausrath (1973), Henry (1974), Nussbaum (1973) and Stech (1977). The linear part of equation (1.1)
1. Introduction One of the predominant themes of stability analysis of dynamical systems of engineering and science interest, is the problem of controlling time delay effects caused by the presence of large displacements and forces. Several techniques have been suggested in the expository papers of Bentsman et al. (1991), Lehman and Bentsman (1992), Youcef-Toumi and Bobbert (1991) and Youcef-Toumi and Reddy (1992) as possible schemes for the controlling of time delay effects. Their techniques although useful in some control systems, may have a disadvantage because of the restriction on the magnitude of the time delay. Incidentally, there is a great deal of evidence (Driver, 1977; El’sgol’ts and Norkin, 1973; Gopalsamy, 1992; Kurzweil, 1971) in support that the investigations of delay systems under the assumption of a small or negligible time delay are not likely to provide satisfactory understanding about the possible dynamics of such systems. In a different direction, we adopt an alternative approach which rests on the solid mathematical foundation of the classical Hopf Bifurcation theorem (Golubitsky and Schaeffer, 1985), in order to transform a nonlinear delay differential equation of the form
C D
0 1 xR (t) 1 " n2 n xR (t) ! ! 2 8 5
C
!
C D
0 x (t) 1 ! n2 x (t) k 2 5
D
0 , c x3 (t)#c x3 (t!r) 03 1 13 1
0 n 4
C
C D
0 1 xR (t) 1 " n2 n xR (t) ! ! 2 8 5
#
D
x (t!r) 1 x (t!r) 2
C D x (t) 1 x (t) 2
0 0 n2 n ! k ! 5 4
C
D
x (t!r) 1 , x (t!r) 2
(1.2a)
generates a strongly continuous semigroup of bounded linear operator J(t, k), t50, k'0 with infinitesimal generator
A(h, k)/(h)"
(1.1)
* Received 15 May 1995; revised 30 July 1996; received in final form 31 October 1997. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Alberto Isidori under the direction of Editor Tamer Bas,ar. Corresponding author Prof. M. S. Fofana. Tel. 001 508 831 5966; Fax 001 508 831 5680; E-mail [email protected]. - Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609-2280, U.S.A. 375
d/(h) , !r4h(0, dh 0 1 0 0 /(0)# /Q (!r) n2 n n ! ! 0 ! (1.2b) 8 5 4 #
0 0 k/(!r), h"0, n2 ! 0 5
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Brief Papers
in C([!r, 0], R2) for /(h)L[!r, 0]. It is known that the semi group J(t, k), which maps the infinite-dimensional space C into itself J(t, k): RxCPC, t50, k'0 by the relation x (/(h), k)" t J(t, k)/(h), h"0, has exponential solutions of the form J(t, k)"ejt, where the eigenvalues j are those of the point spectrum p(A(h, k)) associated with the generator A(h,k). Interestingly, these eigenvalues coincide also with values that satisfy the transcendental characteristic equation n n2 n *(j, k)"j2# j# # M5j#4nkNe~jr"0, 5 8 20
(1.3)
corresponding to the linear delay equation (1.2a). This equation has infinite number of eigenvalues lying on the left side of the complex plane and may also have a finite number of eigenvalues that occur in complex conjugate pairs or lie on the right side of the complex plane. Here we will assume that *(j, k) is continuously differentiable in k and there is a critical value k of k, for # which the characteristic equation (1.3) has the imaginary conjugate pairs §(j, k)"t (k)$iu(k) satisfying t (k)'0, u(k)O0, Re Md* (j, k)/dkNO0. The Banach space C is decomposed into a direct sum as C"P(j, k) = Q(j, k) by all the eigenvalues of equation (1.3) with the two-dimensional generalized eigenspace P,P(j, k) and the infinite-dimensional complementary subspace Q,Q(j, k) both invariant under J(t, k) and A(h, k). Such a decomposition serves as a coordinate system for the projections of all the eigenvalues of equation (1.3) onto the disjoint subspaces P, Q and of course, for the unique solution representation x (/(h), k)"xP(/(h), k)#xQ(/(h), k) of the nonlinear delay t t t equation (1.1) through /(h). The superscripts designate the projected solution of x (/(h), k)LC in P and Q, respectively, for all t values of t3[!r,R). To this end, we make the substitution j"t (k)#iu(k) into the equation (1.3). Then equating the real and imaginary parts of the resulting equation to zero leads to n n2 Re M*((t#iu), k)N"t2# t!u2# 5 8 n n # M5t#4nkNe~t rcos ur# ue~t r sin ur"0, (1.4a) 20 4 n n Im M*((t#iu), k)N"2ut# u# ue~t r cos ur 5 4 n ! M5t#4nkN e~t r sin ur"0. 20
(1.4b)
It is now a straightforward algebraic exercise to readily see from these equations that there is a Hopf bifurcation at k"k "1 , # 2 for u(k )"n/2r and r"1. Moreover, the eigenvalues of equa# tion (1.3) lying on the left side in the complex plane cross the imaginary axis at $in/2r to the right side with the nonvertical velocity:
G
sign Re
G
d*(j, k) dk
sign Re
H
" k/k#
H
8n M(4n#30)!i(5n!8)N (4n#30)2#(5n!8)2
'0.
(1.5)
j/*n@2r,k#/1@2 Thus, the linearly independent row of periodic functions '(h) "[/ : (h), / (h)], in which / (h)"(cos (n/2r)h# 1 2 1 sin (n/2r)h), / (h)"(!sin (n/2r)h#cos (n/2r)h), hL[!r, 0], 2 corresponding to the simple eigenvalues $in/2r form a basis for PLC. Moreover, there is the 2]2 constant real matrix B(k)"(a ), j, k"1, 2, whose elements: a "a "0, a " jk 11 22 11 a "0, a "!n/2r, a "n/2r at criticality are determined 22 12 21 by the relation A(h, k) '(h)"'(h)B(k) such that '(h)"'(0) eBh. This latter expression combined with the relation J(t, k)'(h)"'(0)eB(t`h), t50 yields the solution operator J(t,k)'(h)b"'(0)eB(t`h)b of equation (1.2a) for some constant vector b. What all this boils down to is that, in C either of the solution x (/(h),k)"'(h) beB(t`h), or equivalently, x(/ , t, k)" t 0 '(h)beB(t`h) of equation (1.2a) behaves as an ODE. The projection of the remaining eigenvalues of the characteristic equation (1.3) onto the complementary subspace Q proceeds by considering the semigroup Jª (q, k),q50, k'0 and its infinitesimal
generator dt(s) ! , 04s(r, ds n2 0 8 0 0 t(0)# n n !1 0 ! 4 5
Aª (s, k)t(s)"
tQ (r)
(1.6a)
n2 5 k/(r), s"0, 0 0 0
#
in Cª "C : K ([0, r], Rª 2), which are both generated by the formal adjoint equation 0 yR (q) 1 " yR (q) !1 2
C D
n2 8 n 5
n2 k 5 n 5
0 y (q) 1 # y (q) 0 2
C D
C
D
y (q#r) 1 , y (q#r) 2 (1.6b)
of (1.2), where t(s) is an initial continuously differentiable function in CK . Despite a unique solution y "y (t(s), k) of equation q q (1.6b) is obtained by specifying the function t(s) on [0, r], however, the properties of Jª (q, k), Aª (s, k) in CK ([0, r], RK 2) are identical with those of the operators J(t, k), A(h, k) in C([!r, 0], R2). Also, the characteristic equation *K (j, k)"0 associated with equation (1.6b) has exactly the same eigenvalues j as those of *(j, k)"0. Then, in exactly the same way as previously, the solution y (t(s), q) corresponding to $in/2r are the column of s periodic functions ((s) "[t : (s), t (s)], where t (s)" 1 2 1 (cos(n/2r)s!sin (n/2r)s), t (s)"(sin (n/2r) s#cos (n/2r) s), 2 sL[0, r], that forms a dual basis for the two-dimensional generalized eigenspace PK "P : K (j, k)LCK . The bases '(h) of PLC and ((s) of PK LCK supply initial continuously differentiable functions /(h), t(s) for the solutions x (/(h), k)LC, y (t(s), k)LCK , t q respectively and furthermore, they are also in some sense adjoint with respect to the associated bilinear relation n (t (s),/ (h))"t (0)/ (0)# t (0)/ (!r) j i j i j 4 i n # 4
P G P 0
H
dt i (f#r) / (f) df j df
~r n2 0 ! Mt (f#r)N/ (f) df, i, j"1, 2, (1.7) i j 10 ~r on C([!r, 0], R2)xCK ([0, r], RK 2). In particular, by substituting the elements
A A A A
BA BA BA BA
B B B B
n n (t (s), / (h))" cos s!sin s 1 1 2r 2r
cos
n n (t (s), / (h))" cos s!sin s 1 2 2r 2r
n n cos h!sin h , 2r 2r
n n (t (s), / (h))" sin s#cos s 2 1 2r 2r n n (t (s), / (h))" sin s#cos s 2 2 2r 2r
n n h#sin h , 2r 2r
cos
n n h#sin h , 2r 2r
cos
n n h!sin h) , 2r 2r
(1.8a)
of the inner product (((s),'(h)) into equation (1.7) and integrating, yields
((, ') " nsg
A
B A B A
9n n2 1! # 20 10
A
n2 1! 8
for k "1, r"1, # 2
B
n2 1! 8
B
n n2 1# ! 20 10
(1.8b)
Brief Papers which is a nonsingular matrix. That is the determinant of ((, ') is non zero, which contradicts the topological nsg properties of the semigroups J(t, k)LC([!r, 0], R2) and Jª (q,k)LCK ([0, r], RK 2). Without any loss of generality, one can normalize the dual basis ((s) of the subspace PK LCK to a new column of periodic functions (1 (s)LCK by computing (1(s) "col : [t1 (s) t1 (s)]"((, ')~1 ((s), namely 1 2 nsg n n2 n n t1 (s)"R 1# ! cos s!sin s 1 111 20 10 2r 2r
A
GA
BA
n2 ! 1! 8 t1 (s)"!R 2 111
A
GA
BA
cos
BH
n n s#sin s 2r 2r
BA BA
n2 1! 8
9n n2 ! 1! # 20 10
cos
B
,
(1.9a)
B BH
1600 R "! , 111 41n4!80n3!364n2#640n
,
1 ((1, ') "R *$ 111
111 0
(1.9b)
(1.9c)
0 1
.
(1.10)
R 111
Up until now we have merely established the functional prerequisites for the reduction of the original nonlinear delay equation (1.1) to a two-dimensional family of ODEs in the Banach space C([!r, 0], R2) for the given fixed time delay r. The next crucial step is to carefully establish a link between these linear results and equation (1.1). This is achieved by exponentially estimating the linear solution of equation (1.2b) in C. Indeed, for any real number, say t " : t (k) with positive number c , where 0 0(c (t, then there exists a constant K(c )5l such that the 0 0 estimates of x (/(h), k) in P, QLC by the semigroup J(t, k), t t50, k"k '0, satisfy the inequalities DJ(t, k)/PD4K(c ) exp # 0 [Mt#c Nt]D/PD, t40, DJ(t, k)/QD4K(c ) exp [!Mt#c Nt]D/QD, 0 0 0 t50. These inequalities imply that the projected solution xP(/(h), k) of PLC, for t3(!R, R) is bounded as tP!R t and unbounded as tPR, while the solution xQ (/(h), k) of t QLC remains bounded for all values of t50. In other words, without any loss of generality, the solution x (/(h), k) in Q can be t neglected and while that in P is equivalent to a family of ODEs. These remarks are consequence of an extensive general results for delay differential equations and more details are in the classical books of Hale (1977), Hale and Lunel (1993). Consequently, at the expense of the change of variables xP(/(h), k)"'(h)z(t), where z(t)"((1(s), xP(/(h), k)) and z(t)3R2 t t is an element in the Euclidean space of two-dimensional column vectors, will give rise to the following: n n cos (!r) !sin (!r) x (t!r) z (t) 2r 2r 1 1 " , (1.11a) n n x (t!r) z (t) 2 2 sin (!r) cos (!r) 2r 2r
C
D
n n cos (0) !sin (0) x (t) 2r 2r 1 " n n x (t) 2 sin (0) cos (0) 2r 2r
C D
C D
C D
z (t) 1 , z (t) 2
(1.12b) for all values of t3[!r, R), where the quantities n(n#2) R , t1 (0)" 1 111 4
9n(n!2) t1 (0)" R , (1.12c) 2 111 40
of (1(0) are obtained by putting s"0 in equations (1.9a)—(1.9c). These are precisely the equivalent ODEs corresponding to equation (1.1), which are further written in terms of amplitude a(t) and phase b(t) in the required standard form:
G
03
G
03
] c6
so that the new inner product matrix ((1 (s), '(h))"I ] is the 2 2 identity. Again the substitution of the elements (t1 (s), / (h)), i j for i, j"1, 2, of the matrix ((1(s),'(h)) into the bilinear relation (1.7) will thus lead to the identity matrix
R
n n2 zR (t)" z (t)#et1 (0) Mc6 z3 (t)#c6 z3(t)# kJ z (t)N, 2 2 03 1 13 2 2r 1 5 2
aR (t)"ea Mt1 (0) sin #!t1 (0) cos #N 1 2
n n s!sin s 2r 2r
n n cos s#sin s 2r 2r
377
(1.11b)
for / "'(h)b3C([!r, : 0], R2), b "(( : 1 (s),/(h)). Hence z(t) satisfies the ODEs n n2 zR (t)"! z (t)#et1 (0) Mc6 z3 (t)#c6 z3 (t)# kJ z (t)N, 1 1 03 1 13 2 2r 2 5 2 (1.12a)
bQ (t)"eMt1 (0) cos ##t1 (0) sin #N 1 2 ] c6
H
n2 a2 sin3 h!c6 a2cos3 #! k8 cos # , 13 5
(1.13a)
H
n2 a2 sin3 #!c6 a2 cos3 #! k8 cos # , (1.13b) 13 5
by means of the convenient polar coordinates z (t)"a sin #, 1 z (t)"!a cos #, #"(n/2r)t#b(t). Carrying out the averaging 2 method (Bogoliubov and Mitropolsky, 1960) on these equations subsequently leads to the uncoupled averaged equations: ea aR (t)" M30 (t1 (0)c6 #t1 (0)c6 ) a2#8n2t1 (0)k8N, (1.14a) !7' 1 03 2 13 2 80 e bQ (t)"! M30 (t1 (0)c6 !t1 (0)c6 ) a2#8n2t1 (0)k8N. 1 13 2 03 1 !7' 80 (1.14b) The bifurcation equation g(a, k)"0 of equation (1.14a) and its characteristic exponents, which identify essentially the conditions of stability and instability can now be determined as follows. 2. Stability of the trivial and nontrivial solutions The stability of the trivial and nontrivial solutions of the nonlinear delay equations (1.1a) and (1.1b) or equivalently the averaged amplitude equation (1.14a) is examined by letting a(t)"a #o(t), where the variable o(t) is the perturbation from 0 the steady state value a which is determined by putting 0 aR (t)"bQ (t)"0 in equations (1.14a) and (1.14b). The linear !7' !7' variational equations governing o(t), which yields apart from the trivial solution a "0 0 en2t1 (0) 2 kJ o(t), oR (t)" (1.15a) 10 the nontrivial solutions a O0 0 e o5 (t)" M90(t1 (0) c6 #t1 (0)c6 ) a2#8n2t1 (0) k8N o(t), (1.15b) 1 03 2 13 0 2 80 are solved both by seeking exponential solutions for o(t) of the form o(t)"o ejI (a0,k)t where jI (a , k) is the characteristic expo0 0 nent and o is an arbitrary constant. Thus from these equations 0 (1.15a) and (1.15b), one can deduce that the characteristic exponent j (a , kJ ), or the bifurcation equation g(a, k) 0"0 for the T 0 a/a trivial solution (a "0) is of the form 0 en2t1 (0) 1 2 kJ "0, k" #ekJ , (1.16a) g(a , k)" 0 10 2 while that of the nontrivial solution (a O0) is 0 e g(a ,k)" M90(t1 (0)c6 #t1 (0)c6 )a2#8n2t1 (0)k8N"0. 0 1 03 2 13 0 2 80 (1.16b) The types of stability bifurcations corresponding to the equations g(a , k)"0 of (1.16) for the defined parameter values 0
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Brief Papers
Fig. 1. (a) Stability of the supercritical bifurcation type. (b) Stability of the subcritical bifurcation type. c6 ,c6 "!0.05 (supercritical) and c6 ,c6 "0.05 (subcriti03 13 03 13 cal) are depicted qualitatively in Fig. 1. As seen in these figures we can ascertain that the trivial solutions loose stability through Hopf bifurcation at k"k "1/2. # 3. Conclusions This investigation has explored an alternative theoretic approach, which leads to the transformation of a nonlinear delay equation to a two-dimensional ODEs in the Banach space C([!r, 0], R2), for a fixed time delay r'0 through Hopf bifurcation theorem. Explicit bifurcation equations of the form g(a, k)"0 for the trivial and nontrivial solutions of the nonlinear delay equation have been derived using a first approximation of the integral averaging method. From the signs of the characteristic exponents of g(a, k), the stability and instability regions as k varies near k critical have been identified. # References Banks, H. T. and A. Manitius (1975). Projection series for retarded functional differential equations with applications to optimal control problems. J. Diff. Eqns., 18, 296—332. Bentsman, J., K. Hong and J. Fakhfakh (1991). Vibrational control of nonlinear time lag systems: vibrational stabilization and transient behaviour. Automatica, 27, 491—500. Bogoliubov, N. N. and Y. A. Mitropolsky (1960). Asymptotic Methods in ¹heory of Nonlinear Oscillations. Gordon and Breach, New York.
Burton, T. A. (1985). Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, Orlando. Chafee, N. and J. P. LaSalle (1971). A bifurcation problem for a functional differential equation of finitely retarded type. J. Math. Anal. Appl., 35, 312—348. Chow, S. N. and J. Mallet-Paret (1977). Integral averaging and bifurcation. J. Diff. Eqns., 26, 112—159. Claeyssen, J. R. (1976). Effects of delays on functional equations. J. Diff. Eqns., 20, 404—440. Chukwu, E. N. (1992). Stability and ¹ime Optimal Control of Hereditary Systems. Academic Press. De Oliveira, J. C. F. (1980). Hopf bifurcation for functional differential equations. NonlinearAnalysis, 4, 217—229. Driver, R. D. (1977). Ordinary and Delay Differential Equations. Springer, New York. El’sgol’ts, L. E. and S. B. Norkin (1973). Introduction to the ¹heory and Application of Differential Equations ¼ith Deviating Arguments. Academic Press, New York. Golubitsky, M. and D. G. Schaeffer (1985). Singularities and Groups in Bifurcation ¹heory. Springer, New York. Gopalsamy, K. (1992). Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, Dordrecht. Hale, J. K. and C. Perello´ (1964). The neighbourhood of a singular point of functional differential equations. Control Diff. Eqns., 3, 351—375. Hale, J. K. (1977). ¹heory of Functional Differential Equations. Springer, New York. Hale, J. K. and S. M. V. Lunel (1993). Introduction to Functional Differential Equations. Springer, New York. Hausrath, A. (1973). Stability in the critical case of purely imaginary roots for neutral functional differential-difference equations. J. Diff. Eqns., 13, 329—357. Henry, D. (1974). Linear autonomous neutral functional differential equations. J. Diff. Eqns., 15, 106—128. Kolmanovskii, V. B. and V. R. Nosov (1986). Stability of Functional Differential Equations. Academic Press, New York. Kuang, Y. (1993). Delay Differential Equations ¼ith Applications in Population Dynamics. Mathematics in Science and Engineering, Academic Press, New York. Kurzweil, J. (1971). Small delays don’t matter. ¸ecture Notes in Mathematics, Springer, Berlin, Vol. 206, pp. 47—49. Lehman, B. and J. Bentsman (1992). Vibrational control of linear time lag systems with arbitrarily large but bounded delays. IEEE ¹rans. Automat. Control, 37, 1576—1582. Lehman, B., J. Bentsman, S. Verduyn Lunel and E. Verriest (1992). Vibrational control of linear time lag systems with arbitrarily large but bounded delays: averaging theory, stabilizability and transient behaviour. Proc. IEEE ¹rans. Automat Control 31st CDC, 1, 1287—1294. Lehman, B. and J. Bentsman (1994). Vibrational stabilization and calculation formulas for nonlinear time delay systems: linear multiplicative vibrations. Automatica, 30, 1207—1211. Nussbaum, R. D. (1973). Periodic solutions of some nonlinear autonomous functional differential equations II. J. Diff. Eqns., 14, 360—394. Salamon, D. (1984). Control and Observations of Neutral Systems. Pitman Advanced Publishing Program, Boston. Stech, H. W. (1979). The Hopf bifurcation: a stability result and application. J. Math. Anal. Appl., 71, 525—546. Youcef-Toumi, K. and J. Bobbert (1991). Stability of uncertain linear systems with time delay. J. Dyn. Syst. Meas. Control, 113, 558—567. Youcef-Toumi, K. and S. Reddy (1992). Analysis of linear time invariant systems with time delay. J. Dyn. Syst. Meas. Control, 114, 544—555.
PII: S0005–1098(97)00208–2
Brief Paper
On the Stability Bifurcation of a Nonlinear Time Delay System* MUSTAPHA S. FOFANA- and DAVID J. LAMB Jr.Key Words—Hopf bifurcation; time delay; averaging method; stability.
Abstract—A computational burden has always been associated with the stability analysis of a time delay system over its entire parameter ranges. Indeed, this has led many previous investigators to develop control schemes under the assumption of small or negligible time delay. In this paper we have attempted to explore an alternative approach, which has been previously called as Critical or Single Points, Hale’s Method, Projection Series and Centre Manifold. Here we have deliberately omitted the relevant theorems, lemmas or corollaries of this approach and rather focus mainly on a calculation spirit to make the underlying ideas accessible to investigators seeking more insight into the wealth of time delay dynamics. The nonlinear delay equation around which we present this approach is a variational form of equations studied earlier by Bentsman et al. (1991), Lehman et al. (1992) and Lehman and Bentsman (1992, 1994). Furthermore, using the integral averaging method explicit bifurcation equations of the form g(a, k)"0 are derived and from the signs of their characteristic exponents, the stability of the trivial and nontrivial solutions is obtained. ( 1998 Elsevier Science Ltd. All rights reserved.
to a family of ordinary differential equations (ODEs) for fixed and positive time delay r. More details about the formulation of this equation and including many other delay equations in engineering and science, the books of Burton (1985), Chukwu (1992), Gopalsamy (1992), Kolmanovskii and Nosov (1986), Kuang (1993) and Salamon (1984) are excellent references. In equation (1.1), the bifurcation parameter k is written as the sum k "k : #ek8 of a small change ekJ from criticality k , in which e is # # a small scaling coefficient with value 04e;1 and the real coefficients c "ec : 6 , c "ec : 6 characterize the nonlinear 03 03 13 13 nature of the restoring forces. The functional elements x (h),col [x (t!r), x (t!r)] defined in C "C([!r, : 0], R2), t 1 2 are the restoring and damping forces at time t!r, whose trajectories coincide with those of the state variables x(t),col [x (t), x (t)]LR2 at h"t"0 through the definitions 1 2 x (h) "x(t#h), : xR (h) "x : R (t#h), for hL[!r, 0]. C is the t t Banach space of all continuously differentiable functions on [!r, 0] having values in R2 with the usual uniform supremum norm E x (/(h))E"suph L[!r, 0] Dx (/(h))D, where D ) D is any vector t t norm in R2. A function x "x (/(h), k)LC is a unique solution t t of equation (1.1) for all values of time t3[!r, R), if one can specify an initial continuously differentiable function, say /(h) on [!r, 0] with value / at h"0. The statements and proofs of 0 results associated with the alternative approach and in general, with solutions of delay differential equations for any fixed and positive time delay can be found in the following references: Banks and Manitius (1975), Chafee and LaSalle (1971), Chow and Mallet-Paret (1977), Claeyssen (1976), De Oliveira (1980), Hale and Perelo´ (1964), Hale (1977), Hale and Lunel (1993), Hausrath (1973), Henry (1974), Nussbaum (1973) and Stech (1977). The linear part of equation (1.1)
1. Introduction One of the predominant themes of stability analysis of dynamical systems of engineering and science interest, is the problem of controlling time delay effects caused by the presence of large displacements and forces. Several techniques have been suggested in the expository papers of Bentsman et al. (1991), Lehman and Bentsman (1992), Youcef-Toumi and Bobbert (1991) and Youcef-Toumi and Reddy (1992) as possible schemes for the controlling of time delay effects. Their techniques although useful in some control systems, may have a disadvantage because of the restriction on the magnitude of the time delay. Incidentally, there is a great deal of evidence (Driver, 1977; El’sgol’ts and Norkin, 1973; Gopalsamy, 1992; Kurzweil, 1971) in support that the investigations of delay systems under the assumption of a small or negligible time delay are not likely to provide satisfactory understanding about the possible dynamics of such systems. In a different direction, we adopt an alternative approach which rests on the solid mathematical foundation of the classical Hopf Bifurcation theorem (Golubitsky and Schaeffer, 1985), in order to transform a nonlinear delay differential equation of the form
C D
0 1 xR (t) 1 " n2 n xR (t) ! ! 2 8 5
C
!
C D
0 x (t) 1 ! n2 x (t) k 2 5
D
0 , c x3 (t)#c x3 (t!r) 03 1 13 1
0 n 4
C
C D
0 1 xR (t) 1 " n2 n xR (t) ! ! 2 8 5
#
D
x (t!r) 1 x (t!r) 2
C D x (t) 1 x (t) 2
0 0 n2 n ! k ! 5 4
C
D
x (t!r) 1 , x (t!r) 2
(1.2a)
generates a strongly continuous semigroup of bounded linear operator J(t, k), t50, k'0 with infinitesimal generator
A(h, k)/(h)"
(1.1)
* Received 15 May 1995; revised 30 July 1996; received in final form 31 October 1997. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Alberto Isidori under the direction of Editor Tamer Bas,ar. Corresponding author Prof. M. S. Fofana. Tel. 001 508 831 5966; Fax 001 508 831 5680; E-mail [email protected]. - Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609-2280, U.S.A. 375
d/(h) , !r4h(0, dh 0 1 0 0 /(0)# /Q (!r) n2 n n ! ! 0 ! (1.2b) 8 5 4 #
0 0 k/(!r), h"0, n2 ! 0 5
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Brief Papers
in C([!r, 0], R2) for /(h)L[!r, 0]. It is known that the semi group J(t, k), which maps the infinite-dimensional space C into itself J(t, k): RxCPC, t50, k'0 by the relation x (/(h), k)" t J(t, k)/(h), h"0, has exponential solutions of the form J(t, k)"ejt, where the eigenvalues j are those of the point spectrum p(A(h, k)) associated with the generator A(h,k). Interestingly, these eigenvalues coincide also with values that satisfy the transcendental characteristic equation n n2 n *(j, k)"j2# j# # M5j#4nkNe~jr"0, 5 8 20
(1.3)
corresponding to the linear delay equation (1.2a). This equation has infinite number of eigenvalues lying on the left side of the complex plane and may also have a finite number of eigenvalues that occur in complex conjugate pairs or lie on the right side of the complex plane. Here we will assume that *(j, k) is continuously differentiable in k and there is a critical value k of k, for # which the characteristic equation (1.3) has the imaginary conjugate pairs §(j, k)"t (k)$iu(k) satisfying t (k)'0, u(k)O0, Re Md* (j, k)/dkNO0. The Banach space C is decomposed into a direct sum as C"P(j, k) = Q(j, k) by all the eigenvalues of equation (1.3) with the two-dimensional generalized eigenspace P,P(j, k) and the infinite-dimensional complementary subspace Q,Q(j, k) both invariant under J(t, k) and A(h, k). Such a decomposition serves as a coordinate system for the projections of all the eigenvalues of equation (1.3) onto the disjoint subspaces P, Q and of course, for the unique solution representation x (/(h), k)"xP(/(h), k)#xQ(/(h), k) of the nonlinear delay t t t equation (1.1) through /(h). The superscripts designate the projected solution of x (/(h), k)LC in P and Q, respectively, for all t values of t3[!r,R). To this end, we make the substitution j"t (k)#iu(k) into the equation (1.3). Then equating the real and imaginary parts of the resulting equation to zero leads to n n2 Re M*((t#iu), k)N"t2# t!u2# 5 8 n n # M5t#4nkNe~t rcos ur# ue~t r sin ur"0, (1.4a) 20 4 n n Im M*((t#iu), k)N"2ut# u# ue~t r cos ur 5 4 n ! M5t#4nkN e~t r sin ur"0. 20
(1.4b)
It is now a straightforward algebraic exercise to readily see from these equations that there is a Hopf bifurcation at k"k "1 , # 2 for u(k )"n/2r and r"1. Moreover, the eigenvalues of equa# tion (1.3) lying on the left side in the complex plane cross the imaginary axis at $in/2r to the right side with the nonvertical velocity:
G
sign Re
G
d*(j, k) dk
sign Re
H
" k/k#
H
8n M(4n#30)!i(5n!8)N (4n#30)2#(5n!8)2
'0.
(1.5)
j/*n@2r,k#/1@2 Thus, the linearly independent row of periodic functions '(h) "[/ : (h), / (h)], in which / (h)"(cos (n/2r)h# 1 2 1 sin (n/2r)h), / (h)"(!sin (n/2r)h#cos (n/2r)h), hL[!r, 0], 2 corresponding to the simple eigenvalues $in/2r form a basis for PLC. Moreover, there is the 2]2 constant real matrix B(k)"(a ), j, k"1, 2, whose elements: a "a "0, a " jk 11 22 11 a "0, a "!n/2r, a "n/2r at criticality are determined 22 12 21 by the relation A(h, k) '(h)"'(h)B(k) such that '(h)"'(0) eBh. This latter expression combined with the relation J(t, k)'(h)"'(0)eB(t`h), t50 yields the solution operator J(t,k)'(h)b"'(0)eB(t`h)b of equation (1.2a) for some constant vector b. What all this boils down to is that, in C either of the solution x (/(h),k)"'(h) beB(t`h), or equivalently, x(/ , t, k)" t 0 '(h)beB(t`h) of equation (1.2a) behaves as an ODE. The projection of the remaining eigenvalues of the characteristic equation (1.3) onto the complementary subspace Q proceeds by considering the semigroup Jª (q, k),q50, k'0 and its infinitesimal
generator dt(s) ! , 04s(r, ds n2 0 8 0 0 t(0)# n n !1 0 ! 4 5
Aª (s, k)t(s)"
tQ (r)
(1.6a)
n2 5 k/(r), s"0, 0 0 0
#
in Cª "C : K ([0, r], Rª 2), which are both generated by the formal adjoint equation 0 yR (q) 1 " yR (q) !1 2
C D
n2 8 n 5
n2 k 5 n 5
0 y (q) 1 # y (q) 0 2
C D
C
D
y (q#r) 1 , y (q#r) 2 (1.6b)
of (1.2), where t(s) is an initial continuously differentiable function in CK . Despite a unique solution y "y (t(s), k) of equation q q (1.6b) is obtained by specifying the function t(s) on [0, r], however, the properties of Jª (q, k), Aª (s, k) in CK ([0, r], RK 2) are identical with those of the operators J(t, k), A(h, k) in C([!r, 0], R2). Also, the characteristic equation *K (j, k)"0 associated with equation (1.6b) has exactly the same eigenvalues j as those of *(j, k)"0. Then, in exactly the same way as previously, the solution y (t(s), q) corresponding to $in/2r are the column of s periodic functions ((s) "[t : (s), t (s)], where t (s)" 1 2 1 (cos(n/2r)s!sin (n/2r)s), t (s)"(sin (n/2r) s#cos (n/2r) s), 2 sL[0, r], that forms a dual basis for the two-dimensional generalized eigenspace PK "P : K (j, k)LCK . The bases '(h) of PLC and ((s) of PK LCK supply initial continuously differentiable functions /(h), t(s) for the solutions x (/(h), k)LC, y (t(s), k)LCK , t q respectively and furthermore, they are also in some sense adjoint with respect to the associated bilinear relation n (t (s),/ (h))"t (0)/ (0)# t (0)/ (!r) j i j i j 4 i n # 4
P G P 0
H
dt i (f#r) / (f) df j df
~r n2 0 ! Mt (f#r)N/ (f) df, i, j"1, 2, (1.7) i j 10 ~r on C([!r, 0], R2)xCK ([0, r], RK 2). In particular, by substituting the elements
A A A A
BA BA BA BA
B B B B
n n (t (s), / (h))" cos s!sin s 1 1 2r 2r
cos
n n (t (s), / (h))" cos s!sin s 1 2 2r 2r
n n cos h!sin h , 2r 2r
n n (t (s), / (h))" sin s#cos s 2 1 2r 2r n n (t (s), / (h))" sin s#cos s 2 2 2r 2r
n n h#sin h , 2r 2r
cos
n n h#sin h , 2r 2r
cos
n n h!sin h) , 2r 2r
(1.8a)
of the inner product (((s),'(h)) into equation (1.7) and integrating, yields
((, ') " nsg
A
B A B A
9n n2 1! # 20 10
A
n2 1! 8
for k "1, r"1, # 2
B
n2 1! 8
B
n n2 1# ! 20 10
(1.8b)
Brief Papers which is a nonsingular matrix. That is the determinant of ((, ') is non zero, which contradicts the topological nsg properties of the semigroups J(t, k)LC([!r, 0], R2) and Jª (q,k)LCK ([0, r], RK 2). Without any loss of generality, one can normalize the dual basis ((s) of the subspace PK LCK to a new column of periodic functions (1 (s)LCK by computing (1(s) "col : [t1 (s) t1 (s)]"((, ')~1 ((s), namely 1 2 nsg n n2 n n t1 (s)"R 1# ! cos s!sin s 1 111 20 10 2r 2r
A
GA
BA
n2 ! 1! 8 t1 (s)"!R 2 111
A
GA
BA
cos
BH
n n s#sin s 2r 2r
BA BA
n2 1! 8
9n n2 ! 1! # 20 10
cos
B
,
(1.9a)
B BH
1600 R "! , 111 41n4!80n3!364n2#640n
,
1 ((1, ') "R *$ 111
111 0
(1.9b)
(1.9c)
0 1
.
(1.10)
R 111
Up until now we have merely established the functional prerequisites for the reduction of the original nonlinear delay equation (1.1) to a two-dimensional family of ODEs in the Banach space C([!r, 0], R2) for the given fixed time delay r. The next crucial step is to carefully establish a link between these linear results and equation (1.1). This is achieved by exponentially estimating the linear solution of equation (1.2b) in C. Indeed, for any real number, say t " : t (k) with positive number c , where 0 0(c (t, then there exists a constant K(c )5l such that the 0 0 estimates of x (/(h), k) in P, QLC by the semigroup J(t, k), t t50, k"k '0, satisfy the inequalities DJ(t, k)/PD4K(c ) exp # 0 [Mt#c Nt]D/PD, t40, DJ(t, k)/QD4K(c ) exp [!Mt#c Nt]D/QD, 0 0 0 t50. These inequalities imply that the projected solution xP(/(h), k) of PLC, for t3(!R, R) is bounded as tP!R t and unbounded as tPR, while the solution xQ (/(h), k) of t QLC remains bounded for all values of t50. In other words, without any loss of generality, the solution x (/(h), k) in Q can be t neglected and while that in P is equivalent to a family of ODEs. These remarks are consequence of an extensive general results for delay differential equations and more details are in the classical books of Hale (1977), Hale and Lunel (1993). Consequently, at the expense of the change of variables xP(/(h), k)"'(h)z(t), where z(t)"((1(s), xP(/(h), k)) and z(t)3R2 t t is an element in the Euclidean space of two-dimensional column vectors, will give rise to the following: n n cos (!r) !sin (!r) x (t!r) z (t) 2r 2r 1 1 " , (1.11a) n n x (t!r) z (t) 2 2 sin (!r) cos (!r) 2r 2r
C
D
n n cos (0) !sin (0) x (t) 2r 2r 1 " n n x (t) 2 sin (0) cos (0) 2r 2r
C D
C D
C D
z (t) 1 , z (t) 2
(1.12b) for all values of t3[!r, R), where the quantities n(n#2) R , t1 (0)" 1 111 4
9n(n!2) t1 (0)" R , (1.12c) 2 111 40
of (1(0) are obtained by putting s"0 in equations (1.9a)—(1.9c). These are precisely the equivalent ODEs corresponding to equation (1.1), which are further written in terms of amplitude a(t) and phase b(t) in the required standard form:
G
03
G
03
] c6
so that the new inner product matrix ((1 (s), '(h))"I ] is the 2 2 identity. Again the substitution of the elements (t1 (s), / (h)), i j for i, j"1, 2, of the matrix ((1(s),'(h)) into the bilinear relation (1.7) will thus lead to the identity matrix
R
n n2 zR (t)" z (t)#et1 (0) Mc6 z3 (t)#c6 z3(t)# kJ z (t)N, 2 2 03 1 13 2 2r 1 5 2
aR (t)"ea Mt1 (0) sin #!t1 (0) cos #N 1 2
n n s!sin s 2r 2r
n n cos s#sin s 2r 2r
377
(1.11b)
for / "'(h)b3C([!r, : 0], R2), b "(( : 1 (s),/(h)). Hence z(t) satisfies the ODEs n n2 zR (t)"! z (t)#et1 (0) Mc6 z3 (t)#c6 z3 (t)# kJ z (t)N, 1 1 03 1 13 2 2r 2 5 2 (1.12a)
bQ (t)"eMt1 (0) cos ##t1 (0) sin #N 1 2 ] c6
H
n2 a2 sin3 h!c6 a2cos3 #! k8 cos # , 13 5
(1.13a)
H
n2 a2 sin3 #!c6 a2 cos3 #! k8 cos # , (1.13b) 13 5
by means of the convenient polar coordinates z (t)"a sin #, 1 z (t)"!a cos #, #"(n/2r)t#b(t). Carrying out the averaging 2 method (Bogoliubov and Mitropolsky, 1960) on these equations subsequently leads to the uncoupled averaged equations: ea aR (t)" M30 (t1 (0)c6 #t1 (0)c6 ) a2#8n2t1 (0)k8N, (1.14a) !7' 1 03 2 13 2 80 e bQ (t)"! M30 (t1 (0)c6 !t1 (0)c6 ) a2#8n2t1 (0)k8N. 1 13 2 03 1 !7' 80 (1.14b) The bifurcation equation g(a, k)"0 of equation (1.14a) and its characteristic exponents, which identify essentially the conditions of stability and instability can now be determined as follows. 2. Stability of the trivial and nontrivial solutions The stability of the trivial and nontrivial solutions of the nonlinear delay equations (1.1a) and (1.1b) or equivalently the averaged amplitude equation (1.14a) is examined by letting a(t)"a #o(t), where the variable o(t) is the perturbation from 0 the steady state value a which is determined by putting 0 aR (t)"bQ (t)"0 in equations (1.14a) and (1.14b). The linear !7' !7' variational equations governing o(t), which yields apart from the trivial solution a "0 0 en2t1 (0) 2 kJ o(t), oR (t)" (1.15a) 10 the nontrivial solutions a O0 0 e o5 (t)" M90(t1 (0) c6 #t1 (0)c6 ) a2#8n2t1 (0) k8N o(t), (1.15b) 1 03 2 13 0 2 80 are solved both by seeking exponential solutions for o(t) of the form o(t)"o ejI (a0,k)t where jI (a , k) is the characteristic expo0 0 nent and o is an arbitrary constant. Thus from these equations 0 (1.15a) and (1.15b), one can deduce that the characteristic exponent j (a , kJ ), or the bifurcation equation g(a, k) 0"0 for the T 0 a/a trivial solution (a "0) is of the form 0 en2t1 (0) 1 2 kJ "0, k" #ekJ , (1.16a) g(a , k)" 0 10 2 while that of the nontrivial solution (a O0) is 0 e g(a ,k)" M90(t1 (0)c6 #t1 (0)c6 )a2#8n2t1 (0)k8N"0. 0 1 03 2 13 0 2 80 (1.16b) The types of stability bifurcations corresponding to the equations g(a , k)"0 of (1.16) for the defined parameter values 0
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Brief Papers
Fig. 1. (a) Stability of the supercritical bifurcation type. (b) Stability of the subcritical bifurcation type. c6 ,c6 "!0.05 (supercritical) and c6 ,c6 "0.05 (subcriti03 13 03 13 cal) are depicted qualitatively in Fig. 1. As seen in these figures we can ascertain that the trivial solutions loose stability through Hopf bifurcation at k"k "1/2. # 3. Conclusions This investigation has explored an alternative theoretic approach, which leads to the transformation of a nonlinear delay equation to a two-dimensional ODEs in the Banach space C([!r, 0], R2), for a fixed time delay r'0 through Hopf bifurcation theorem. Explicit bifurcation equations of the form g(a, k)"0 for the trivial and nontrivial solutions of the nonlinear delay equation have been derived using a first approximation of the integral averaging method. From the signs of the characteristic exponents of g(a, k), the stability and instability regions as k varies near k critical have been identified. # References Banks, H. T. and A. Manitius (1975). Projection series for retarded functional differential equations with applications to optimal control problems. J. Diff. Eqns., 18, 296—332. Bentsman, J., K. Hong and J. Fakhfakh (1991). Vibrational control of nonlinear time lag systems: vibrational stabilization and transient behaviour. Automatica, 27, 491—500. Bogoliubov, N. N. and Y. A. Mitropolsky (1960). Asymptotic Methods in ¹heory of Nonlinear Oscillations. Gordon and Breach, New York.
Burton, T. A. (1985). Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, Orlando. Chafee, N. and J. P. LaSalle (1971). A bifurcation problem for a functional differential equation of finitely retarded type. J. Math. Anal. Appl., 35, 312—348. Chow, S. N. and J. Mallet-Paret (1977). Integral averaging and bifurcation. J. Diff. Eqns., 26, 112—159. Claeyssen, J. R. (1976). Effects of delays on functional equations. J. Diff. Eqns., 20, 404—440. Chukwu, E. N. (1992). Stability and ¹ime Optimal Control of Hereditary Systems. Academic Press. De Oliveira, J. C. F. (1980). Hopf bifurcation for functional differential equations. NonlinearAnalysis, 4, 217—229. Driver, R. D. (1977). Ordinary and Delay Differential Equations. Springer, New York. El’sgol’ts, L. E. and S. B. Norkin (1973). Introduction to the ¹heory and Application of Differential Equations ¼ith Deviating Arguments. Academic Press, New York. Golubitsky, M. and D. G. Schaeffer (1985). Singularities and Groups in Bifurcation ¹heory. Springer, New York. Gopalsamy, K. (1992). Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, Dordrecht. Hale, J. K. and C. Perello´ (1964). The neighbourhood of a singular point of functional differential equations. Control Diff. Eqns., 3, 351—375. Hale, J. K. (1977). ¹heory of Functional Differential Equations. Springer, New York. Hale, J. K. and S. M. V. Lunel (1993). Introduction to Functional Differential Equations. Springer, New York. Hausrath, A. (1973). Stability in the critical case of purely imaginary roots for neutral functional differential-difference equations. J. Diff. Eqns., 13, 329—357. Henry, D. (1974). Linear autonomous neutral functional differential equations. J. Diff. Eqns., 15, 106—128. Kolmanovskii, V. B. and V. R. Nosov (1986). Stability of Functional Differential Equations. Academic Press, New York. Kuang, Y. (1993). Delay Differential Equations ¼ith Applications in Population Dynamics. Mathematics in Science and Engineering, Academic Press, New York. Kurzweil, J. (1971). Small delays don’t matter. ¸ecture Notes in Mathematics, Springer, Berlin, Vol. 206, pp. 47—49. Lehman, B. and J. Bentsman (1992). Vibrational control of linear time lag systems with arbitrarily large but bounded delays. IEEE ¹rans. Automat. Control, 37, 1576—1582. Lehman, B., J. Bentsman, S. Verduyn Lunel and E. Verriest (1992). Vibrational control of linear time lag systems with arbitrarily large but bounded delays: averaging theory, stabilizability and transient behaviour. Proc. IEEE ¹rans. Automat Control 31st CDC, 1, 1287—1294. Lehman, B. and J. Bentsman (1994). Vibrational stabilization and calculation formulas for nonlinear time delay systems: linear multiplicative vibrations. Automatica, 30, 1207—1211. Nussbaum, R. D. (1973). Periodic solutions of some nonlinear autonomous functional differential equations II. J. Diff. Eqns., 14, 360—394. Salamon, D. (1984). Control and Observations of Neutral Systems. Pitman Advanced Publishing Program, Boston. Stech, H. W. (1979). The Hopf bifurcation: a stability result and application. J. Math. Anal. Appl., 71, 525—546. Youcef-Toumi, K. and J. Bobbert (1991). Stability of uncertain linear systems with time delay. J. Dyn. Syst. Meas. Control, 113, 558—567. Youcef-Toumi, K. and S. Reddy (1992). Analysis of linear time invariant systems with time delay. J. Dyn. Syst. Meas. Control, 114, 544—555.