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[6] M. di Bernardo, F. Garolfo, L. Glielmo, and F. Vasca, “Switching, bifurcation, and chaos in dc/dc converters,” IEEE Trans. Circuits Syst. I, vol. 45, pp. 133–141, Feb. 1998. [7] F. B. Cunha and D. J. Pagano, “On dynamic phenomena in a dc–dc boost converter subject to variable structure control,” in Proc. Int. Federation Automatic Control 15th World Congr. IFAC’02, Barcelona, Spain, 2002. [8] H. Dankowicz and A. Nordmark, “On the origin and bifurcations of stick-slip oscillations,” Phys. D, vol. 136, pp. 280–302, 2000. [9] F. Dercole, A. Gragnani, and S. Rinaldi, “Sliding bifurcation in relay control systems: An appllication to natural resources management,” in Proc. Int. Federation Automatic Control 15th World Congr. IFAC’02, Barcelona, Spain, 2002. [10] F. B. Cunha and D. J. Pagano, “Bifurcation analysis of the lotka-volterra model subject to variable structure control,” in Proc. Int. Federation Automatic Control 15th World Congr. IFAC’02, Barcelona, Spain, 2002. [11] M. di Bernardo, P. Kwalczyk, and A. Nordmark, “Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings,” Phys. D, pp. –, 2002, to be published. [12] Y. A. Kuznetsov, S. Rinaldi, and A. Gragnani, “One-parameter bifurcations in planar filippov systems,” Dep. of Mathematics, Universitet Utrecht, , Utrecht, Denmark, Preprint no. 1227, 2002, to be published. [13] M. Costa, E. Kaskurewicz, A. Bhaya, and L. Hsu, “Achieving global convergence to an equilibrium population in predator-prey systems by the use of a discontinuous harvesting policy,” Ecol. Model., vol. 128, pp. 89–99, 2000. [14] W. C. Y. Chan and C. K. Tse, “Study of bifurcations in current-programed dc/dc boost converters: From quasi-periodicity to period-doubling,” IEEE Trans. Circuits Syst. I, vol. 44, pp. 1129–1142, Dec. 1997. [15] D. C. Hamill, J. H. B. Deane, and D. J. Jefferies, “Modeling of chaotic dc–dc converters by iterated nonlinear mappings,” IEEE Trans. Power Electron., vol. 7, pp. 25–36, Jan. 1992.
A View on Limit Cycle Bifurcations in Relay Feedback Systems Fig. 4. State–space trajectories for different regions of the parameter space.
It is also important to remark that, despite the fact that the bifurcations analyzed in this work were detected for 2-D system, all the concepts and ideas are extendable to systems of higher dimensions with little changes. Of course, the difficulty of the analysis of higher dimension systems can be largely increased. The dc–dc boost converter used in this brief as an exemple was described by an autonomous piecewise linear state equation. Unlike other types of bifurcations on switching systems, for example, those reported on the references [6], [14], and [15], the ones discussed in this brief do not result of the use of pulsewidth-modulated switching. Besides that, the example presented in this brief, features the three types of local bifurcations involving the switching surface analyzed in this work.
R. Genesio and G. Bagni
Abstract—The paper is concerned with the study of oscillations in linear dynamic systems with relay feedback. The specific interest is about the bifurcations of these periodic solutions, with regard to phenomena which also occur in smooth systems and to others due to the relay discontinuity. The followed approach moves from the describing function method, leading to results which are approximate in nature but which express in a simple and correct way the essential mechanisms of the studied phenomena, as shown in the proposed examples. Index Terms—Bifurcations, describing function, limit cycles, relay control systems, sliding modes.
I. INTRODUCTION The importance of relay systems in the study and applications of control engineering is well known. This is due to their use in analysis, for modeling several physical phenomena, and to the common use of
REFERENCES [1] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides. Dordrecht, The Netherlands: Kluwer, 1988. [2] A. A. Andronov, E. A. Vitt, and S. E. Khaikin, Theory of Oscillators. Oxford, U.K.: Pergamon, 1966. [3] V. I. Utkin, Sliding Modes and their Application in Variable Structure Systems. Moscow, U.S.S.R.: MIR, 1978. [4] M. di Bernardo, K. Johanson, and F. Vasca, “Sliding orbits and their bifurcations in relay feedback systems,” in Proc. 38th Circuits and Devices Conf., 1999, pp. 708–713. [5] C. K. Tse, “Flip bifurcation and chaos in a three-state boost switching regulator,” IEEE Trans. Circuits Syst. I, vol. 42, pp. 16–23, Jan. 1994.
Manuscript received August 26, 2002; revised April 14, 2003. This work was supported by the Ministero dell Istruzione, dell Universitá e della Ricera under Project FIRB 2002 RBNE01CW3M and Project COFIN 2002 “Tecniche e Applicazioni della Dinamica Nonlineare e del Caos nell’Ingegneria dell’Informazione”. This paper was recommended by Guest Editor M. di Bernardo. The authors are with the Dipartimento di Sistemi e Informatica, Università di Firenze, 50139 Firenze, Italy (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCSI.2003.815224
1057-7122/03$17.00 © 2003 IEEE
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 50, NO. 8, AUGUST 2003
their peculiar behavior, the switching action, for suitably controlling dynamic processes. The significant interest for relays in control arises from their early utilization for simple mechanical and electromechanical devices (see, e.g., [1]), but a number of recent applications have caused renewed attention on them. We cite, among others, the automatic tuning of process controllers, the design of variable-structure systems, the use in hybrid systems, and in delta–sigma converters for signal processing (see [2], [3] and references therein). The study of relay-systems dynamics has also a long history and many methods, rigorous and approximate (see, e.g., [4], [1]), have been developed for investigating different phenomena, also due to the special features of the involved nonlinearities. In particular, a topic of actual interest concerns the oscillations of feedback relay systems for which many results have been obtained (see, e.g., [5], [6]) but also a number of questions remain unsolved. A part of these questions, which have received recent attention, is about the existence of periodic solutions and their interaction with the system sliding modes [7], giving rise to a number of complicated phenomena typical of nonsmooth dynamics [2], [3], [8], [9]. Moreover, complex and chaotic behavior has been revealed in relay systems (see, e.g., [10]–[12] and, recently, [13]). In such a framework, the interest of this brief is focused on the limit cycles which can be exhibited in linear systems with relay feedback and in related bifurcation phenomena. So, a part of the usual bifurcations [14] which can occur for periodic solutions of smooth systems will be considered, as well as situations, for example, sliding bifurcation [2], which is characteristic of nonsmooth systems. Without discussing new behaviors, this brief tries to give a unifying approach to describe limit cycles and their bifurcations in terms of simple properties of the system linear part, due to the fact that the relay nonlinearity is fixed, without any free parameter. Of course, this kind of results is possible since an approximate method is used, namely the describing function method (see, e.g., [15]–[17]), also employed in previous studies of such systems with other frequency techniques [4], [1]. This is a classical approach coming from the area of control engineering and it consists in a first harmonic analysis of the system, while preserving the nonlinear effects on the amplitude oscillations, and it leads to results which are not rigorous in nature. On the other hand, it is possible to justify the method and to give conditions on the reliability of its results (see, for example [15]–[17]) which are often obtained in closed form, so indicating the essential features of the system for the onset of many possible dynamic behaviors. The empirical evidence over many years has proven the power of describing function method in predicting and locating limit cycles. This paper shows that such a method can be also useful for deriving more subtle phenomena, as a number of related bifurcations which arise in relay feedback systems, often leading to simple and intuitive results. Even if a number of complicated bifurcations and phenomena, as described for example in [2], [3], [13], remains out of the possibilities of the method, the obtained results can be useful, for example, in the study of design problems for relay systems. The paper is organized as follows. Section II gives the background on describing function method by applying it for a general prediction of limit cycles in relay feedback systems. Section III considers such results for perturbing them in order to derive some limit cycle bifurcations (known as cyclic fold and symmetry breaking bifurcations), while Section IV extends the method for analyzing the important phenomenon of period doubling of periodic solutions. Section V introduces, in our model, the occurrence of sliding modes and derives the condition for the so-called sliding bifurcation [2] of limit cycles in the frequency terms of Section II. Section VI is devoted to some examples of applica-
Fig. 1.
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Relay feedback system.
tion of the proposed technique, and the brief comments of Section VII are at the end. II. GENERAL CONDITIONS FOR OSCILLATIONS Consider the time-invariant system depicted in Fig. 1, where a linear dynamic block of transfer function N (s) bm sm + bm01 sm01 + 1 1 1 + b0 L(s) = (1) = D(s) sn + an01 sn01 + 1 1 1 + a0 is feedback connected to a relay defined by
n(y(t)) = sgn(y(t)) 2
f1g; 0 f01g;
[ 1; 1];
y(t) > 0 y(t) = 0 y(t) < 0:
(2)
The output of the linear system is denoted by y and a negative feedback is considered, while it is assumed m n. In order to study the existence of periodic solutions in the system of Fig. 1 by means of the well-known describing function method [15]–[17], assume
y(t) = A + B cos !t;
B; ! > 0
(3)
and compute the bias and harmonic gains of the relay as 1 2
N0 (A; B ) = N1 (A; B ) =
2A 1
0
B 0
n(A + B cos !t)d!t =
A
'
n(A + B cos !t)e0j!t d!t =
(4) 4
B
cos '
(5)
where
' = arcsin
A B
2 02 ; 2
:
(6)
Then, the first-order harmonic balance, applied to the considered feedback loop to have the solutions of the form (3), easily results in the general conditions
A[1 + L(0)N0 (A; B )] =0 1 + L(j! )N1 (A; B ) =0
(7) (8)
which become, by means of (4) and (5)
A = 0 L(0)' 4 B = 0 L(j!) cos ': 2
(9) (10)
These equations can be restated as Im[L(j! )] =0
(11)
L(0) sin 2' = (12) ' L(j!) to be solved in the unknown variables ! and ' (and then A and B ). We remark that the solutions ! of (11), denoted by !0 where necessary, can be in any number depending on the form of transfer function
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III. SOME BIFURCATIONS OF LIMIT CYCLES
Fig. 2.
Limit cycle detection by the describing function method.
(1), while the structure of (12) implies that the solutions ' must be one, ' , or three, ' , and two others which are symmetric with respect to 0.
=0
=0
The analysis of conditions (11) and (12), and the preceding equations allows one to have a general view of the possible limit cycles which can be exhibited by a feedback relay system as that of Fig. 1. Certainly, this holds under the approximation guaranteed by the describing function method and outlined below.
The different situations are determined only by the values L(0) and L(j!0 ) which are the harmonic gains of the linear subsystem at the frequencies zero and !0 , respectively. Concerning the existence of limit cycles, it can be observed that a necessary (and sufficient) condition is that L j!0 , at least for one !0 , is real and negative [see (5) and (8)].
( )
Therefore, the possible situations are the following. •
•
L(0) 0, L(j!0 ) < 0. The system has the only output equilibrium at y = 0 and there is one symmetric limit cycle (' = 0, and A = 0), for any negative L(j!0 ). L(0) < 0, L(j!0 ) < 0. The system has three output equilibria at y = 0 and y = 6L(0), and there is one symmetric limit cycle for any negative L(j!0 ) > L(0)=2, and three limit cycles for any negative L(j!0 ) < L(0)=2.
It is well known that the describing function method can have a meaningful graphical interpretation in terms of (8). The solutions defining amplitude and frequency of limit cycles correspond to the intersections of Nyquist plot L j! with the curve 0 =N1 . This can be viewed as a function of the only variable , by substituting, in general, the bias A A B as given by (7). An indicative situation related to the considered relay system is shown in Fig. 2, where the curve 0 =N1 coincides with the negative real axis and goes monotonically from 0 to 01 for increasing amplitude B (this happens both in the symmetric and asymmetric cases [12]).
( )
= ( )
B
1
1
For what concerns the validity of the method, apart from its empirical evidence over many years, it can be said that the related accuracy is essentially based on a suitable attenuation (filtering effect) of higher frequency components !; !; along the loop of Fig. 1. Some measures of error (the distortion [4], [18], for example), can be evaluated for any obtained result and also rigorous arguments can be used to guarantee that, in certain conditions, a real limit cycle exists in a computable neighborhood of the predicted one [15]–[17], [19]. On the other hand, the angle of intersection between the loci of Fig. 2 can also be important in the accuracy of the results [20], [21]. Moreover, on the basis of the Loeb criterion and its extensions [4], [18], the above graphical representation can give indications on the stability of derived limit cycles, depending on the stability features of L s and on the way of crossing of the two plots. Indeed, such indications often result more uncertain than the predictions concerning the existence of limit cycles.
2 3 ...
()
The study of (local) bifurcations of a limit cycle can be performed in the neighborhood of the considered solution. In the sense that the trajectories which are close to the limit cycle in the state space exhibit particular behaviors when such a limit cycle is in the proximity of a bifurcation (see, e.g., [22]–[24]). We refer, of course, to modifications of system trajectories occurring under quasi-static variations of a suitable parameter. These special trajectories are typical of the bifurcation in study, and can be analyzed considering small perturbations of the nominal solution. So, a general way of analyzing the occurrence of various bifurcations could be that of substituting the system nonlinearity of Fig. 1 by its linearization, which would become now a time-varying periodic gain, and studying the corresponding perturbation dynamics [25]. Unfortunately, in the relay case, there is the discontinuous nonlinearity of (2), so that different approaches must be followed to investigate such dynamics. In the framework of the describing function approach as presented in Section II, we first follow the idea of giving small perturbations denoted by A, B , and ! to the parameters—bias, amplitude, and frequency—which define the studied limit cycle of the form (3), so having the modified cycle as
1 1
1
y(t) = (A + 1A) + (B + 1B )cos(! + 1!)t:
(13)
Now, taking into account the harmonic balance equations (7) and (8), the related general conditions which ensure that the variation can be sustained result in
L01 (0) + N0 + A @N0 @A @N1 @A
1A +
@N1 @B
1A + 1B +
A @N0 @B
1B+
A @N0 @!
@ L01 (j!) + N 1 @!
1! =0 (14)
1! = 0: (15)
Here, the inverse transfer function L01 is used for simplicity and any term must be computed at the values A, B , and ! which are solutions of (7) and (8), and define a predicted limit cycle of the system. The problem of the discontinuous nonlinearity is clearly avoided, due to the fact that its models N0 and N1 as given by (4) and (5) result to be smooth. Moreover, the relay function is not dynamic and N0 and N1 are independent by ! , so that it results in @N0 =@! @N1 =@! . Therefore, (14) and (15), where L01 j! is now the only complex variable, simply reduce to
( )
Im d!d
L01 (j!)
=
= 0:
=0
(16)
According to the followed approach, this relation gives the general condition, in the sense that any variation A, B , and ! is usually different from 0, for the existence of another limit cycle in the neighborhood of that in study, with small modifications of its bias, amplitude, and frequency. This situation corresponds to the well-known cyclic fold bifurcation of a limit cycle [22]–[24] and in the considered case of relay systems, according to (11) and (16), it corresponds to the indicative scenario depicted in Fig. 3 where the transfer function plot L results in being tangent to the real axis. This condition corresponds in itself to a critical prediction of the describing function [21], but it must be viewed as a
1 1
1
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The response of the nonlinearity of Fig. 1 to signal (19) cannot be directly derived only by the gains N0 and N1 introduced in (4) and (5). Since the nonsmoothness of relay function n y , as given by (2), does not allow its linearization, we extend (see also [26]) the describing function approach of Section II by defining, in addition to the usual gains N0 and N1 (now related to the signal (19)), also the complex gain N1=2 as
()
(
N1=2 A; B;
Fig. 3.
Cyclic fold bifurcation of a limit cycle.
transition, moving a suitable bifurcation parameter, between the cases of zero and two intersections of the loci, respectively. Indeed, it can be easily seen that the systems of (14) and (15) admit, and for A , in our case, another particular solution for ! corresponding to the condition
1 =0
=0
01 + N0 = 0: L(0)
(17)
( ) = L(0) 2 :
(18)
Clearly, this represents the so-called symmetry breaking condition for a limit cycle [22]–[24] and this occurrence could be simply evaluated by considering (12), which is basic to derive all the system limit cycles, and looking for the asymmetric solutions onset from a symmetric one. The results of Section II allow one to derive in a closed form, although approximate, the limit cycles of the relay system. This means that the coefficients (A, B , and ! ) of the harmonic solution y t of (3) are expressed in terms of the parameters affecting the system. In a similar way, in this section, we have derived the qualitative conditions, depending on structural elements of the system, which characterize the cyclic fold and the symmetry-breaking bifurcations of the above limit cycles. These conditions, first deduced in general terms and then applied to the considered relay feedback system, result in the relations (16) and (18), respectively, which are in addition to the main limit cycle equations (11) and (12). By assuming that the limit cycle of interest is stable, the above conditions put in evidence the situations where, through a suitable parameter variation: 1) another unstable limit cycle tends to the considered one and they collide with the disappearance of any periodic solutions (cyclic fold bifurcation), and 2) two asymmetric stable limit cycles stem out from the original one which becomes unstable (symmetry-breaking bifurcation).
In the previous section, we have considered the possible existence of typical perturbed trajectories close to a limit cycle (assumed in the form of (3)) in order to reveal some of its bifurcations. Indeed, the admissible variations taken into account have the same structure of the original limit cycle, as shown in (13). This fact reduces the possibility of investigating different perturbation forms which can denote, on the other hand, the occurrence of other bifurcations. In particular, one important case of such a kind refers to the so-called flip bifurcation, when a period doubling component arises over the nominal limit cycle [22]–[24]. Just to consider this phenomenon we assume the perturbed output signal as
( ) = A + B cos !t + cos !2 t +
y t
where is a small number.
(19)
0
( ( )) 0 j !=
n y t e
( (
2)t+ )
d
!
2 t:
(20)
Due to smallness of the amplitude perturbation , the form of N0 and remains unchanged, while it can be computed that
(
N1=2 A; B;
2 ) = B cos 1 0 sin ' e0j ' 2
:
(21)
Then, the harmonic balance concerning the system of Fig. 1, for the existence of a period doubling limit cycle of expression (19) yields the complex equation
1 + L j!2
(
N1=2 A; B;
)=0
(22)
in addition to the same relations (7) and (8) of Section II. Again, (7) and (8) have to be solved in A, B , and ! , in terms of system parameters, and then substituted in (22) to derive the possible values of such parameters corresponding to the bifurcation in study, and also to give the delay appearing in (19). For the relay system the condition (22), by eliminating the inessential delay, becomes
) = sin 1 0 2cos ' LL(j! j!
()
IV. PERIOD DOUBLING BIFURCATION
0
N1
Taking into account (4) and (10), for A going to 0 in (6), such a relation becomes L j!
1 ) = lim !
2
2
'
(23)
2
and it is added to the main limit cycle equations (11) and (12) for denoting the flip bifurcation. We remark that these last equations give ! !0 and ' as a function of L and L j!0 , while (23) requires the knowledge of the linear gain L j!0= to derive, according to the method, the flip bifurcation of system limit cycles. In this sense, the condition (23) appear more complicated than those of Section III for cyclic fold and symmetry-breaking bifurcations, since they have been obtained just by the same elements introduced in Section II for finding limit cycles. After the flip bifurcation, the original limit cycle, if stable, generally becomes unstable, while the stability is transferred to the period doubling solution, here assumed in the form of (19). In the corresponding range of parameters, the stability prediction of the simple limit cycle as obtained in Section II has to be considered essentially not correct. Indeed, observe that the occurrence of the flip bifurcation can be of interest also as an important indicator of possible presence of chaotic dynamics in the considered system. In fact, the first period doubling often results in the beginning of a Feigenbaum cascade which is one of the more frequent routes to chaos of dynamical systems [22]–[24], [27].
(0) (
( ) 2)
=
V. SLIDING BIFURCATION Sections III and IV have been devoted to discuss some bifurcations of limit cycles which are usually present in nonlinear dynamical systems. In our case, some general considerations, approximate in nature, have been applied to the specific case of the nonsmooth system given by relay feedback of Fig. 1 Indeed, nonsmooth dynamical systems can also exhibit a number of special phenomena and bifurcations depending on their discontinuous features [2], [3], [13].
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In particular, the system motions can partly remain on sliding surfaces, which are intuitively associated with an infinite number of switches between two different system configurations. These (first-order) sliding surfaces are clearly defined for our relay feedback system by y = 0 and by the relative reaching condition y y_ < 0 [2], [3]. In the case of their existence, it can happen that a limit cycle of the system reaches a sliding surface and that part of the periodic orbit develops on it. The onset of this situation, when the limit cycle begins its interaction with one of the above surfaces, determines a so-called sliding bifurcation [2], which is obviously specific of nonsmooth systems. Indeed, the periodic solution generated via a sliding bifurcation of a simple orbit (two crossing per period) can undergo a class of successive intricate bifurcations, called multisliding, when more sliding sections appear in the system orbit [2]. These solutions imply complicated behaviors which are impossible to be detected by a first harmonic technique as that employed in this paper. Then, the objective of this section is the study of the sliding bifurcation, the simplest one of those concerning limit cycles in nonsmooth systems [2], [3], by the same frequency techniques introduced in the paper. With reference to (1), assume that m = n 0 1, i.e., that the relative degree of the linear subsystem is equal to 1. It is now convenient to rewrite the related transfer function as
L(s) =
bn01 + b s + 1 1 1 + s b s + an01 + a s + 1 1 1 + s a
(24)
Fig. 4. Cyclic fold bifurcation diagrams: prediction (dashed) and numerical computation (continuous).
VI. EXAMPLES A. Cyclic Fold Bifurcation Consider that the linear subsystem, connected to the relay as in Fig. 1, has the transfer function [2]
L(s) =
1 1 1 0 bn02
t 0
t 0
n[y( )]d 1 1 1 =: z (t): (25)
By imposing y = 0 with y y_ < 0, and taking into account (2), which defines the relay, the sliding surface, in terms of the variable z, defined by (25), corresponds to the condition (26)
This formula puts in evidence that, for the existence of the above sliding mode, the coefficient bn01 , known as the first Markov parameter of L(s), must be positive. On the other hand, by considering again (25) and (2), (26) corresponds to
j y_ jy=0 j < 2bn01
(27)
which is a useful relation for revealing, by our technique, the sliding bifurcation of a limit cycle. In fact, since such a cycle has been approximated in the form of (3), it results in j y_ jy=0 j = !B cos ' and then, at the equality limit of condition (27), through (10), we obtain 4!
j L(j!)j cos2 ' = 2 bn01:
< 0:
(s + 1)3
2 + 6 + 3 6
!0 =
( + 9)( + 1)3 2
while
y( ) d + 1 1 1
jz(t)j < bn01:
0 )2 ;
L(0) = 2 L(j!0 ) =
3
02 : 0 !02
(30)
(31)
For suitable values of , the Nyquist plot of L has two intersections with the negative real axis, so indicating, in accordance with the results of Section II, the existence of two symmetric limit cycles (the larger stable and the smaller unstable) in the system dynamics. In particular, (16) easily gives = 09, with the related frequency !0 = 3:873. This situation is similar to that drawn in Fig. 3 and a cyclic fold bifurcation is then predicted for the limit cycle solving (11) and (12) (or (9) and (10)) with A = 0 and B = 1:91. Fig. 4 compares the diagrams obtained by the proposed prediction method (dashed line) and by numerical computations (continuous line) done by using the software package AUTO, showing a good agreement of the results. The amplitude of the periodic solutions is reported as a function of the parameter , and the unstable branches are depicted as dotted. B. Symmetry Breaking Bifurcation In the scheme of Fig. 1, consider now the system of transfer function
L(s) =
(28)
We remark that the last equation takes into account the onset of sliding periodic motions in symmetric as well as in asymmetric cycles, and that the corresponding relation (27) puts in evidence that this situation occurs when the slope at y = 0 of the time signal y (t) is suitably small. Finally, it can be observed that after the sliding bifurcation, denoted by (28), the limit cycle increases its distortion, since it remains for an interval at y = 0, so that the prediction of Section II necessarily becomes less accurate.
(29)
It results (see (11)) in
so that the relay feedback system of Fig. 1 can be considered to agree the integrodifferential equation
y_ + an01 y(t) + bn01 n[y(t)] = 0an02
(s
1 ; (s + )(s + 1)2
0
(33)
so that
(a)
!0 and consequently
L(0) = L j
!0 2
=
01
=
1
L(j!0 ) =
0 0:05 0:05(2
0
01 0 0:05 + 1)
8 (0:12 + 5:995 + 0:1) + j 3(1
0 0:05)3=2 :
For suitable values of , according to Section II, and in particular to (11) and (12), a pair of asymmetric limit cycles (with bias 6A and equal amplitude and frequency) are predicted to exist. For such cycles, the flip condition (23) has a solution for = 0:173, while it results in !0 = 0:996 and ' = 1:364 (A = 65:008, B = 5:117). Fig. 6 reports the related bifurcation diagram obtained by numerical computations and shows the good accuracy of the above prediction. It is to be observed that the stability prediction done by the Loeb criterion is now incorrect, since an unstable limit cycle is indicated instead of a stable one, differently from the other examples where such a result agrees with the real system behavior. This confirms that the stability prediction based on the describing function method is more inaccurate than the prediction concerning the existence and location of the same limit cycle. Finally, we remark that this period doubling of the considered limit cycles is also followed by a Feigenbaum cascade, leading to chaotic dynamics of the system approximately for > 0:19. D. Sliding Bifurcation The linear subsystem is again defined by [2]
L(s) =
(s
0 )2 ;
(s + 1)3
>0
(34)
(b) Fig. 7. Time signals y (t) by prediction (dashed) and by simulation (continuous), (a) before ( = 2:5) and (b) after ( = 0:5) the sliding bifurcation.
and we have the same !0 , L(0), and L(j!0 ) given by (30) and (31). The application of condition (28) for the sliding bifurcation leads to = 1:854, and to the corresponding frequency !0 = 0:916, with A = 0 and B = 2:184, concerning a stable limit cycle. The time diagrams of Figs. 7 and 8 show the behavior of y (t) according to the describing function method (dashed line) and to numerical simulations (continuous line). In particular, Fig. 7 reports two periodic solutions far from the sliding bifurcation: 1) before, at = 2:5, and 2) after, at = 0:5, while Fig. 8 shows the periodic solution close to the bifurcation value, i.e., at = 1:85. Indeed, accurate numerical computations [2] indicate that the actual bifurcation occurs at 2:1 and then, the approximation in de-
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 50, NO. 8, AUGUST 2003
Fig. 8. Time signals y (t) by prediction (dashed) and by simulation (continuous) close ( = 1:85) to the sliding bifurcation.
termining the bifurcation value results, which can be expected by the method, while the qualitative phenomenon is correctly revealed. Moreover, as just outlined, after the bifurcation, the cycle clearly increases its distortion, so that for = 0:5 [Fig. 7(b)], the predicted limit cycle becomes inaccurate in shape and frequency. VII. CONCLUSION A general view of limit cycles in linear-feedback relay systems has been given by the classical describing function method, and the obtained results have been used for studying some bifurcations of such oscillations. By suitable perturbations, closed-form conditions are first derived for cyclic fold, symmetry-breaking, and flip bifurcations. Then, in the same frequency context, the simplest case of the sliding bifurcation of a limit cycle has been investigated and a related existence condition is given. The application examples indicate that the proposed technique provides a good qualitative prediction of the studied phenomena, in the common approximation of describing function methods. REFERENCES [1] Y. Z. Tsypkin, Relay Control Systems. Cambridge, UK: Cambridge Univ. Press, 1984. [2] M. di Bernardo, K. H. Johansson, and F. Vasca, “Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations,” Int. J. Bifurcation Chaos, vol. 11, pp. 1121–1140, 2001. [3] K. H. Johansson, A. Barabanov, and K. J. Åström, “Limit cycles with chattering in relay feedback systems,” IEEE Trans. Automat. Contr., vol. 47, pp. 1414–1423, Sept. 2002. [4] D. P. Atherton, Nonlinear Control Engineering: Describing Function Analysis and Design. London, U.K.: Van Nostrand Reinhold , 1975. [5] K. J. Åström, “Oscillations in systems with relay feedback,” in Adaptive Control, Filtering, and Signal Processing, K. J. Åström, G. C. Goodwin, and P. R. Kumar, Eds. New York: Springer-Verlag, 1995, vol. 74, IMA Volumes in Mathematics and its Applications, pp. 1–25. [6] A. Megretski, “Global stability of oscillations induced by a relay feedback,” in Proc. IFAC 13th World Congress, vol. E, San Francisco, CA, 1996, pp. 49–54. [7] V. I. Utkin, Sliding Modes in Control Optimization. Berlin, Germany: Springer-Verlag, 1992. [8] M. di Bernardo, F. Garofalo, L. Glielmo, and F. Vasca, “Switchings, bifurcations and chaos in dc/dc converters,” IEEE Trans. Circuits Syst. I, vol. 45, pp. 133–141, Feb. 1998. [9] K. H. Johansson, A. Rantzer, and K. J. Åström, “Fast switches in relay feedback systems,” Automatica, vol. 35, pp. 539–552, 1999. [10] P. A. Cook, “Simple feedback systems with chaotic behavior,” Syst. Contr. Lett., vol. 6, pp. 223–227, 1985. [11] D. Amrani and D. P. Atherton, “Designing autonomous relay systems with chaotic motion,” Proc. 28th IEEE Conf. Decision and Control, vol. 1, pp. 512–517, Dec. 1989.
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