9 6th International Multi-Conference on Systems, Signals and Devices COEXISTENCE OF REGULAR AND CHAOTIC BEHAVIOR IN THE TIME-DELAYED FEEDBACK CONTROLLED TWO-CELL DC/DC CONVERTER K. Kaoubaa, M. Feki, A. El Aroudi and B.G.M. Robert 3 Research Unit ICOS, École Nationale d Ingénieurs de Sfax (ENIS), BP 73, 338 Sfax, Tunisia. e-mail: karama484@gmail.com, moez.feki@enig.rnu.tn Research group GAEI, Technical Engineering School of the Rovira i Virgili University, Spain. e-mail: abdelali.elaroudi@urv.cat 3 Laboratoire CReSTIC, Université de Reims-Champagne-Ardenne, France. e-mail: bruno.robert@univ-reims.fr ABSTRACT This work deals with the behavior of the controlled twocell DC/DC converter. It has been reported in literature that the two-cell DC/DC converter may exhibit chaotic behavior under a badly tuned proportional controller. To stabilize the periodic orbit sann the chaotic attractor we propose to apply the discrete time-delayed feedback controller. We first present conditions of stability using the Lyapunov theory. Next, we investigate the coexistence of a chaotic attractor and a regular behavior in a particular region of the parameter space. Finally, we attempt to bring some theoretical explanation to the coexistence of attractors approved by numerical simulations in which we depict the sensitivity of the attractors to the initial condition. Index Terms DC/DC converter; Chaotic behavior; Chaos control; Time-delayed state feedback; Coexisting attractors.. INTRODUCTION Research on power converters is noticing a boosting interest among researchers during the last decade motivated by energy optimization reasons. Simple DC/DC converters, in all their simple types: buck, buck-boost and boost, have been thoroughly studied in recent years and different types of behaviors have been perceived. Indeed, what has been considered as strange unknown behavior few decades ago, is now proved to be chaotic behavior with all its universal features [,, 3, 4]. Knowing that chaotic behavior yields to high power losses, it has been worthy to control chaos into regular behavior [, 6, 7]. More recently, there has been an interest in analyzing the nonlinear behavior of the multi-cell power electronic converters. As a matter of fact, multi-cell power converters This wors supported by the The Spanish Agency for International Co-operation and Tunisian Ministry of Higher Education with the Grant number A/698/8. have been widely used in real applications since they circumvent shortcomings of ordinary switching devices due to their ability to support high-voltages. Research on the multi-cell dc/dc converters has shown that these devices may exhibit sub-harmonic modes and also chaotic behaviors [8, 9, ]. This work represents a tiny step towards the exploration of the richness of the two-cell DC/DC converter and helps in understanding its behavior. To accomplish our aim, we suggest to stabilize the desired regular behavior sann the chaotic attractor using the discrete timedelayed control method. Using the Lyapunov theory, we establish the stability conditions in the control parameter space. Then, we investigate the coexistence of a chaotic attractor and a regular behavior in a vary particular region of the parameter space which is supposed to conduct only regular behavior. Finally, we attempt to bring some theoretical explanation to coexistence of attractors. The paper is organized as follows. In the second section we present the discrete model of the two-cell DC/DC converter. In section three we explore the effect of the simple proportional controller and in section four we propose the discrete time-delayed feedback (DTDF) controller and establish stability conditions. The fifth section is devoted to numerical simulations backing the theoretical analysis. In section six, we present some further numerical simulations which revealed unpredicted behavior of the two-cell converter and finally the seventh section will contain the concluding remarks.. DISCRETE TIME MODEL OF A TWO-CELL CONVERTER The converter that we will deal with in this paper is depicted in Figure-. It is based on a buck chopper modified in order to allow a higher input voltage by using two serial switches (transistors or diodes). The role of the capacitor is to balance the switch voltages so that they underlay 978--444-4346-8/9/$. 9 IEEE
S S + V C U C U E + D D Figure. A basic two-cell dc/dc buck converter. equal potential differences. U and U are the outputs of a PWM modulator; they are square signals with period T and duty cycles d and d that we will define in this work with respect to the OFF state of the switches. The aim of the wors to find the sequences of the duty cycles that achieve a constant voltage V c = E and a constant output current i = I r. In this paper, we will consider a dimensionless state space model, where the state vector x = [v, i] t is scaled with the maximum vector [E, E R ]t and time will be scaled with the switching period T. Therefore, the duty cycles d and d should vary in the unit interval [, ]. Knowing that the switches and the diodes operate in a complementary way, we can define four different topologies in terms of the states of the switches []. Topologies are defined in Table. L i L R By integrating on each interval and stacking up solutions, we can define the state x ( (n+)t ) as a function of x(nt) and the duty cycles d and d in addition to the circuit parameters. It has been shown in [] that the simplified discrete dynamic model is the same for all operating modes and it is given by: [ ] [ ][ ] [ ] x (n + ) δc x = + x (n + ) ( ) x ( d ) () where = (d d ), = RT L and δ C = T RC. For practical considerations, we need to reduce the ripple current through the load and the ripple voltage across the capacitor, therefore, circuit parameters should satisfy, δ C. 3. THE EFFECT OF A PROPORTIONAL CONTROLLER In this section we present some numerical simulations with a simple control strategy. Herein, the duty cycles are calculated on the basis of a proportional controller: d = sat(k v e + e ), d = sat( k v e + e ), (a) (b) where e = x V r, e = x I r and sat(x) is shown in Fig. 3. Due to the piecewise aspect of the duty sat(x) Table. Two-cell DC/DC converter different topologies. Topologies state of S state of S Topology (T ) OFF ON Topology (T ) ON ON Topology 3 (T 3 ) ON OFF Topology 4 (T 4 ) OFF OFF x Obviously, the dynamic of each topology can be described by a linear continuous system of the form ẋ = A k x+b k. Since the signals U and U are shifted by half a period, then according to the duty cycles length we can have six different operating modes. Let s consider for instance a scaled reference current I r >., then in normal operating mode we should have < d, d <. at steady state and the operating cycle is split into four intervals as shown in Figure-. U U Figure 3. the saturation function. cycles, we can easily verify that the system can mathematically have several fixed points, among which only one belongs to its domain of definition. This fixed point arises from non-saturation of both duty cycles and it is given by (x, x ) = ( V r, +kiir + ). We understand that the reference voltage V r can be achieved while the reference current I r can only be approached if the gain is very high. Unfortunately, increasing may yield to a chaotic behavior if it is raised beyond a critical value [8]. The Bifurcation diagram depicted in Fig. 4 confirms our claim. Figure. mode. d + d Different topologies at the normal operating t 4. TIME-DELAYED FEEDBACK CONTROLLER To stabilize the normal period T operating behavior, sank in the chaotic attractor, we suggest to apply a DTDF controller. Since we can choose a gain k v to achieve V r, and
be included in future work. To tackle our analysis, we define the error vector as e = x V r, e = x I r and e 3 = x 3 I r and thus we obtain the error system: e (n + ) = ( k v δ C x ) e (7) ( e (n + ) = k v e V r ) e Figure 4. Bifurcation diagram under the action of a proportional controller. only the current behavior bifurcates until it reaches chaos by increasing the gain, we thought it might be obvious to apply the DTDF controller part in terms of current only. Hence, the duty cycle expressions become: d = k v e + e + η i (x x (n ))(3a) d = k v e + e + η i (x x (n )) (3b) The application of the DTDF controller increases by one the order of the discrete dynamical system. We here note that this is one great advantage of using a discrete model. When applying TDF controller to a continuoustime dynamical system, the system becomes infinite dimensional. Assuming the new vector notation x = [x, x, x 3 ] t where x 3 = x (n ) Then in case of non-saturation, the closed loop system is described by the following recurrent map: ( ) x (n + ) = k v δ C x V r x + x (4) ( x (n + ) = k v x V r ) x + ( ( + η i + ) ) x + η i x 3 + ( + I r + k v V r ) () x 3 (n + ) = x (6) The fixed point x of the normal operating mode is obtained by equating x (n + ) = x. Since the third equation ( gives x 3 = ) x, then we obtain: (x, x, x 3 ) = Vr, +kiir +, +kiir +. 4.. Stability analysis We should note that the stability analysis we will present herein only concerns the non saturating case. A thorough analysis including the saturation of the duty cycles will + ( ( + η i + ) ) e + η i e 3 + ( I r ) (8) e 3 (n + ) = e (9) we remark that the error system is a nonlinear system in a cascade triangular form: z (n + ) = f (z ) () z (n + ) = f (z, z ) () with z = e and z = [e, e 3 ] t. It has been proved [3] that under mild conditions, the stability of system ()- () can be deduced from the simultaneous stability of the following subsystems: z (n + ) = f (z ) () z (n + ) = f (z, z ) (3) where z is the fixed point of (). Now, the first subsystem is described by: e (n + ) = ( k v δ C x ) e (4) Hence, we may consider the Lyapunov function candidate V (e ) = e. In discrete systems we need to show that the variation of the Lyapunov function is negative i.e. V (e ) = V (e (n + )) V (e ) <. V (e ) = ( k v δ C x ) e e () = k v δ C x ( k v δ C x ) e (6) Knowing that x is always positive, then we can guarantee that V is negative if the voltage gain k v is chosen to satisfy condition (7). < k v < δ C x = k v,cri (7) We can also optimize the choice of k v to obtain fast convergence of the error to zero. This is done by minimizing V (e ), therefore we need to minimize h(k v ) = k v δ C x ( k v δ C x ) with respect to k v and we get k v,opt = δ C x (8) Until now, we can guarantee the stability of subsystem (4) at the origin by the choice of k v, that is we obtain e = equivalent to x = V r. Now, by substituting e = in the second subsystem i.e. z (n + ) = f (, z ) we get: e (n + ) = ( ( + η i + ) ) e + η i e 3 + ( I r ) (9) e 3 (n + ) = e ()
This a linear system with characteristic equation: ( ) Z ( + η i + ) Z η i = () According to Jury criteria of stability, the control parameters and η i should satisfy the following conditions: η i < () > (3) which is satisfied when restricting the design to positive values of and finally η i < + (4) Figure depicts the stability zone defined in the parameter space (, η i ) according to Jury test. The current x The voltage x.7..4.3.. 3 3 4 4 Number of periodes..4.3.. Current gain: ki=38., Current gain: η i = 9.8, Voltage gain: kv= 3 3 4 4 Number of periodes Figure 6. Evolution of the states under a DTDF controller. η i η i = + To investigate the optimality relation presented in () we present two different curves of the current with = 3 while η i is fixed first at η i = 8 and then at η i = η i,opt. The initial conditions where fixed at +% of the fixed point values. The results are shown in Fig. 7. Clearly, the current settles to the steady state faster when usingη i,opt. 6 Current gain: ki=3 Current gain: η i = 8, Voltage gain: kv= Figure. Stability zone in the parameter space. To summaries the analysis carried out here, we can say that if we choose k v < δ C and any pair of (, η i ) in the shaded area of Fig., then the system evolves towards the fixed point (x, x, x 3) = ( ) V r, +kiir +, +kiir +. To obtain a fast evolution, we might prefer to choose k v = δ C and choose the pair of (, η i ) that gives eigenvalues with the least norm, that should be the pair that leads to equal real eigenvalues which in fact verifies: η i,opt = + ( + ) ( + ) (). NUMERICAL SIMULATIONS In this section we present numerical simulations that confirm the theoretical results obtained in the foregoing section. In all the presented simulations, we have chosen = δ C =., V r =. and I r =. Following the theoretical analysis presented earlier, we have shown that k v can be chosen to achieve x = V r, thus we have fixed k v = δ C =. Next, we will present simulations with different pairs of (, η i ). First, in Fig. 6 we present the evolution of the current x and the voltage x with pushed next to its maximum = 38. and η i = 9.8 thus the obtained steady state error is about %. The current x The current x 4.8 6 4.8 3 3 4 4 Number of periods Current gain: ki=3 Current gain: η i = 6.9, Voltage gain: kv= 3 3 4 4 Number of periods Figure 7. Evolution of the current under a DTDF controller with and without optimized parameters. Figure 8 depicts a D-bifurcation diagram where the behavior of the converter is elucidated for each pair of parameter (, η i ). The colors indicate different periodic behaviors and black stands for chaotic motion. For easiness of reading, we have used the electronic color code with which researcher on electronic devices are very familiar. Hence, brown stands for the T -periodic behavior and it confirms the triangular zone obtained theoretically. If we look at the horizontal line η i =, we observe that the behavior is exactly that obtained on Fig. 4 with red standing for T -periodic behavior and so on. In Fig. 8 we also plot in dashed line the curve of op-
timal pairs. Although, one may intuitively choose a pair situated in the middle of the stability zone thinking to have faster response when situated far from the stability limits. We notice that the optimality curve is in fact and surprisingly very close to the upper limit of the stability zone. We should point out here that this specific region changes slightly its shape according to the initial conditions and the value of k v. kv= kv= 3 4 3 η i 6 4 7 η i 8 6 9 7 chaos 8 3 3 4 9 3 3 4 chaos Figure. D-Bifurcation diagram of the converter behavior under the action of a DTDF controller. Figure 8. D-Bifurcation diagram of the converter behavior under the action of a DTDF controller. 6. UNPREDICTED BEHAVIOR To make an in depth investigation we plotted a bifurcation diagram with η i = 9.8 using simulations that start at +% of the fixed point values. The diagram of Fig. shows a chaotic attractor with three separate regions around = 3. When exploring the optimality condition obtained in (), we have chosen to start the simulation in the vicinity of the fixed point with the aim to observe the effect of the parameter choice and eliminate any other effects such as transients due to far initial conditions. Amazingly, we have obtained a rather unpredictable behavior for some very particular pairs. We present in Fig. 9 the evolution of the current with (, η i ) = (3, 9). We notice that the current evolution looks like 3T -periodic with values wondering about three different values. 6 Current gain: ki=3 Current gain: η i = 9, Voltage gain: kv= The current x 4.8 Number of periods Figure 9. Evolution of the current under a DTDF controller with particular parameters. Unlike the D-bifurcation diagram presented in Fig. 8 where simulations start at the origin of the state space, the D-bifurcation diagram presented in Fig. is obtained by carrying simulations that start at +% of the fixed point values. There, we have a slight difference in the region of = 3. Indeed, chaotic behavior is obtained from pairs that according to section 4, should give stable fixed point! Figure. coexistence of chaotic attractor and periodic behavior under the action of a DTDF controller. The sensitivity of the current evolution to initial condition implies here the coexistence of periodic motion and chaotic behavior. In Fig., we present the basins of attraction of the fixed point (in brown) and of the chaotic attractor (in black) using some fixed parameters. We first note that the basins of attraction are intertwined. The separatrices are mainly vertical; making the basins independent of x 3, except in some region where they have a different shape. Actually, in the regions where the basins of attraction are formed from vertical strips, we note that the duty cycles are both saturated either at zero d = d = for lower values of initial conditions (call that region R ) or saturated at one d = d = for higher values of initial
conditions (call that region R ). The boundaries of saturation of d and d are plotted in yellow and red respectively. x 3.9.8.7..4.3. =3 η i = 9.8 K v = V()=. L R R L will reach it while spiraling around it since we can check that the eigenvalues associated to the fixed point are complex conjugate. Now, in absence of saturation of the duty cycles, the fixed point is globally attractive. However, in presence of the saturation, points which are still far from the fixed point; that is next to the boundaries of the region R fp, those points might be attracted to another virtual fixed point following different trajectories. This is the key point of the problem. If a trajectory gets close to the fixed point such that its future evolution does not leave the nonsaturation region then it converges to the fixed point. On the other hand, if a trajectory approaches the fixed point but still relatively far so that its future steps are outside the non-saturation region then it will go farther by the effect of the virtual fixed points but cannot leave the region R fp. That s how a new chaotic attractor has been created.....3.4..7.8.9 x Figure. Riddled basins of attraction of the coexisting attractor and the fixed point. Now, when the system starts evolving in R, it is attracted towards a virtual fixed point in R and vice-versa. The evolution of the current follows the line L : x 3 = 9 x 9 in R (see the yellow circles and white asterisks). In R the evolution of the current follows the line L : x 3 = 9 x (see the cyan triangles and magenta diamonds). Eventually, the evolution should reach a region delimited by the duty cycle saturation lines and the lines L and L (call that region R fp ). Once the trajectories are in R fp, the system will either converge to the fixed point or to the three-regions chaotic attractor. 7. CONCLUSION In this work, we have presented an analysis of an unpredictable local behavior exhibited by the two-cell DC/DC converter which has been controlled by a DTDF controller in order to stabilize its T -periodic motion that was sank in a chaotic attractor. Surprisingly, the two-cell converter has shown a coexistence of periodic motion and a chaotic behavior with controller parameters that are theoretically supposed to yield only to stable periodic behavior. Our analysis to explain the phenomenon was based on the time evolution of the trajectories. In future work, we intend to characterize the type of bifurcation the system undergoes in that particular region. It is also worthy to find relations between control parameters and the boundaries of the region of coexisting attractors. A thorough stability analysis including the saturation of the duty cycle is also due..7 =3 η i = 9.8 K v = V()=. 8. REFERENCES 8 6 4 R fp R [] I. Nagy, Nonlinear dynamics in power electronics, in Proc. Electrical drives and power Electronics Conf.,, pp.. x 3.8 R [] S. Banerjee and G. Verghese, Nonlinear phenomena in power electronics: Attractors, bifurcations, chaos and nonlinear control. New York: IEEE Press,..6.4...4.6.8 4 6 8.7 x Figure 3. Zoom about the fixed point and the coexisting chaotic attractor. In Fig. 3, we present a zoom of the region R fp, there we observe that trajectories attracted by the fixed point [3] M. Di Bernardo and C. Tse, Chaos in power electronics: An overview, in Chaos in circuits and systems, G. Chen and T. Ueta, Eds. New York: World Scientific, June, vol., ch. 6, pp. 37 34. [4] E. Fossas and G. Olivar, Study of chaos in the buck converter, IEEE Trans. on Circuits and Systems I, vol. 43, pp. 3, 996. [] G. Poddar, K. Chakrabarty, and S. Banerjee, Control of chaos in the boost converter, Electronics Letters, vol. 3, no., pp. 84 84, 99.
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