CONTROL OF BIFURCATION OF DC/DC BUCK CONVERTERS CONTROLLED BY DOUBLE - EDGED PWM WAVEFORM

Transcription

1 ENOC-8, Saint Petersburg, Russia, June, 3 July, 4 8 CONRO OF BIFURCAION OF DC/DC BUCK CONVERERS CONROED BY DOUBE - EDGED PWM WAVEFORM A Elbkosh, D Giaouris, B Zahawi & V Pickert School of Electrical, Electronic and Comuter Engineering, Newcastle University, UK DamianGiaouris@nclacuk S Banerjee Centre for heoretical Studies and Deartment of Electrical Engineering, Indian Institute of echnology, Kharagur, India soumitro@eeiitkgernetin Abstract he nonlinear behaviour of the buck converter controlled by double-edged PWM waveform is studied in this aer he stability of the system is analyzed using the state transition matrix over one switching cycle (the monodromy matrix) including the state transition matrices during each switching (saltation matrices) hree suervising control methods are alied to extend the normal eriod one oeration of the system Good agreement is demonstrated between theoretical analysis, numerical simulations and the results are exerimentally verified Key words DC/DC converters, PWM, monodromy matrix, control of bifurcations Introduction Pulse width modulation (PWM) is the most frequently used control method for switching converters Conventionally a saw-tooth signal is comared with a control signal to roduce a series of ulses that control the states of the ower electronic switches he nonlinear effects of these switchings have been extensively studied in the ast and various methods have been roosed for their analysis and control [Verghese, Elbuluk and Kassakian, 986], [Chakrabarty, Poddar and Banerjee, 996], [Di Bernando, Garofalo, Glielmo and Vasca, 998], and [se, 3] Most of these methods assume that the saw-tooth signal is discontinuous at t =, where is the eriod of the signal his, however, is not a valid aroximation at high frequencies where the signal looks more like an asymmetric triangular wave Such a system demands a fresh look at its bifurcations and chaotic behaviour he control of a chaotic resonse has been the focus of extensive research such as the Ott-Grebogi-Yorke (OGY) method [Ott, Grebogi and Yorke, 99] and linear ime Delayed Feedback Control (DFC) method [Bleich, and Socolar, 996] and [Pyragas, ] which are designed to stabilize one of the unstable eriodic orbits that exist in a chaotic system However, these controllers may require high rocessing ower and are not always alicable to real systems with small signal to noise ratio An alternative method has been roosed by the authors which is simler, easier to imlement, and greatly extends the stable oerating region of the converter [Giaouris, Banerjee, Zahawi and Pickert, 8] and [Giaouris, Elbkosh, Pickert, Zahawi and Banerjee, 6] his method is based on changing the state transition matrices during the switchings in such a way that the eigenvalues of the system remain within the unit cycle In this aer, the analysis is extended by alying this method to DC/DC buck converters controlled by double-edged PWM waveform he aer is organized as follows: Section two briefly outlines the oeration of the DC/DC buck converter and its dynamic equations Section three shows the bifurcation behaviour of the system In Section four, we analyse the stability of the system based on the comlete cycle solution matrix of the system (the monodromy matrix) Finally, we aly three control methods based on the roosed analysis to maintain the stability of the system over a much wider range of inut arameters Analytical, numerical and exerimental results validate the roosed controllers System Descritions and Oeration Fig (a) shows a simlified block diagram that describes the voltage mode controlled buck converter a) b) Figure (a) Block diagram of the buck converter with voltage controller, (b) the generation of switching signal

2 A s x + B E, S = Off x = A s x + B E, S = On () where x = [vout i ] = [x x ], E is the inut voltage and As, B are the system matrices given by: / C A s = RC, B = and B = / Bifurcation Behaviour In general, the circuit gives an average dc outut voltage close to the desirable value with a eriodic rile equal to the eriod of the driving clock (the triangular signal) as shown in Fig However, by increasing the inut voltage a smooth eriod doubling bifurcation renders the system unstable and almost instantly a nonsmooth bifurcation forces the system into a chaotic region (Figs 3 & 4) Inductor Current (A) Outut Voltage(V) ime (S) Figure Exerimental result, Period- oeration at inut voltage of volt, Vref = 3 V, K = 84, = H, C = 47µF, R= Ω, =4 µs, V = 38 V, VU = 8 V Figure 4 he eriod- resonse of the system at inut voltage of 34 V his non-smooth bifurcation occurs when the eriodic solution of the system touches the non-smooth hyersurface (ie, the uer or lower art of the triangular signal) [Banerjee, Ott, yorke and Yuan, 997], [Yuan, Banerjee, Ott and yorke, 998], [Banerjee and Verghese, ] and [se and Di Bernardo, ] Stability Analysis of the System he stability analysis of a saw-tooth controlled DC/DC converter is usually based on the derivation of the Poincaré ma of the system [Verghese, Elbuluk and Kassakian, 986], [Banerjee and Chakrabarty, 998], [Di Bernando and Vasca, ] and [Fang and Abed, ] Other methods include trajectory sensitivity [Hiskens and Pai, ] and auxiliary vectors [Dranga, Buti, Nagy and Funato, 5] A different aroach, based on the system saltation matrices [Aizerman and Gantmakher, 958], [Filiov, 988], [Baushev, Zhusubaliev, Kolokolov, and erekhin, 99] and [Zhusubaliyev, Soukhoterin, and Mosekilde, ] as reviously alied to non-smooth mechanical systems [eine, Camen and Vrande, ] and [eine and Nijmeijer, 4], has been roosed by the authors and alied to DC/DC converters with fruitful results [Giaouris, Banerjee, Zahawi and Pickert, 8], [Giaouris, Elbkosh, Pickert, Zahawi and Banerjee, 6] and [Giaouris, Elbkosh, Banerjee, Zahawi and Pickert, 6] One of the advantages of this method is that it treats each switching searately and hence the overall analysis is simler than the conventional Poincaré ma aroach Furthermore, the analysis it offers an insight into the oeration of the system and its loss of stability, and hence makes it ossible to derive new control laws to extend the stable oerating region [Giaouris, Banerjee, Zahawi and Pickert, 8] he alication of the scheme to converters that use a triangular carrier wave imoses new challenges as the trajectory crosses two switching manifolds at t = d and t = d (Fig 5) 4 Border collision 6 5 Period doubling f f- X ()=X() h 54 h 5 f Figure 3 Bifurcation diagram with inut voltage as bifurcation arameter Switch OFF X (d) ime (S) inductor current (A) Samled Values of i 7 Outut Voltage (V) V < t < 5 t + VU () Vtri = V t + V + VU 5 t < where VU, V are the uer and lower limit of the triangular signal and V = VU V he state equations that reresent the states during a switching cycle are: 5 Inductor Current (A) he switch S is controlled by a comarator which comares the control signal Vcon with a suitable eriodic triangular waveform Vtri and a switching occurs when these two signals become equal (Fig b) he triangular signal can be written as: Switch ON 9 9 f- X (d) outut voltage (V) Figure 5 Period- orbit of the system

3 he monodromy matrix of the system consists of the fundamental solution matrices during the smooth areas and the saltation matrices during the switchings [eine and Nijmeijer, 4]: A s ( d ) A s ( d d) A sd, X,) = e S e S e M ( where S and S are the saltation matrices at the switching time d and d, resectively [eine and Nijmeijer, 4], and can be calculated from: ( f+ ), ) f ), )) n S = I + (4) h n f ), ) + ), ) t where h is the switching manifold, n is the normal to h, I is the identity matrix, t is the switching time, f and f + Σ (3) are the two smooth vector fields before and after the switching From the circuit toology, it can be deduced that the two switching manifolds are: h h ( x, t) = K( x () t Vref ) ( VU + V d) ( x, t) = K( x ( t) Vref ) ( V + VU ) Vd he erendicular vectors onto these surfaces are: h = x h n = x h = x h n = x [ k ] [ k ] At the time instant t = d the switching condition h x, d = is satisfied and hence a switching takes lace ( ) from f ( x ( d )) = A x + B E to f (5) s + ( x ( d )) = Asx + BE he saltation matrix at this oint is calculated from (4): E S = + I ( x ( ) ( ) ) ( ) d x d R VU V (6) C 5 K h x, d = At t= d, there is another switching as ( ) he two vector fields before and after the switching are f + ( x ( d )) = A s x + B E & f ( x ( d )) = A s x + B E he state transition matrix at this oint is E S = + I ( x ( ) ( ) ) ( ) d x d R VU V (7) + C 5 K he stability of the system can be determined by calculating the Floquet multiliers or the eigenvalues of the fundamental solution matrix M Common roblems that have to be addressed here are the location of the limit cycle and the times at which the switchings take lace his can be achieved numerically by deriving a nonlinear function g ( d i ) whose root will define the switching instant [Fossas, and Olivar, 996] and [Giaouris, Banerjee, Zahawi and Pickert, 8] Once the switching instances have been identified, utilizing the fact that the system is linear time invariant (I) before and after the switching, it is ossible to locate the limit cycle Once these values are found, the monodromy matrix can be exressed as a function of the inut voltage using (3), (6) and (7), and its eigenvalues can be calculated Fig 6 shows the eigenvalues of the system for different values of the inut voltage clearly indicating the loss of stability through a smooth eriod doubling bifurcation around 33 V as redicted by the results resented in Figs -4 At this oint we would like to note that these results could have also been obtained by using the Poincare ma but then the analysis would have been more comlicated as the overall formula for the Jacobian of the Poincare ma would have to include all the switchings [Fang and Abed, ] Imaginary art of the eigenvalues E= Real artof the eigenvaues Figure 6 oci of the eigenvalues of M 5 Control of Bifurcation In addition to its simlicity, the stability analysis resented in the revious section offers an insight into the behaviour of the system that can be used to evolve control strategies to avoid instabilities he control requirement here is not only to stabilise the eriod- orbit but also to kee the same steady state value, which imlies keeing the location of the orbit almost unchanged First of all we note that the stability of the system is governed by the eigenvalues of the state transition matrix over a clock cycle (the monodromy matrix), and this matrix is in turn influenced by the state transition matrices across the switching events (the saltation matrices) Hence it is ossible to stabilize the orbit without changing its location by maniulating the saltation matrices hus we aim at a control action that will change M but not di, x( di ), x( ) S can be influenced by changing the smooth vector fields before and after the switching, and the switching manifold In ractice we cannot change the vector fields as this would imly the hysical re-design of the converter But it is ossible to alter the switching manifold(s) o avoid a big change that would change the location of the limit cycle we aim at small changes in h that will cause significant changes in h/ x and h/ t 5 Control Based on the Change of the Uer imit of the Ram Signal he system can be stabilized and the eigenvalues of the monodromy matrix ushed back inside of the unit circle by

4 modifying the eak value of the ram signal (V U ) to (a V U ) with the value of a chosen to determine the desired location he effect of this change is to alter the time derivative of h to influence the saltation matrices Using the theory resented in the last section it is ossible to create an analytical exression of the Floquet multiliers as a function of a Hence, by numerically solving the nonlinear transcendental equation eig ( M (,)) 84 =, it is ossible to locate the eigenvalues at a redefined location (in this case at a circle of radius 84) which indicates stable eriod- behaviour Corresonding values of a for different values of inut voltage can then be calculated (Fig 7) Exerimental waveforms of the outut voltage and the inductor current at 34 V are resented in Fig 8 showing that the system is oerating in eriod- oeration Note that according to Fig 4 the system is unstable at an inut voltage of 34V without the roosed control Otimum Values of a Figure 7 Values of a calculated to lace the eigenvalues at a circle of radius 84-change of the ram signal Outut Voltage (V) ime (S) Figure 8 Exerimental results with the controller at inut voltage of 34 V 5 Control Based on an Additional Sinusoidal Signal Another control method is based on adding a sinusoidal signal to the switching manifold h, this can be made by changing the value of V ref to V ref (+ a sin(ωt)), where ω=π/ and the values of a are chosen to determine the desired location he effect of this change on the saltation matrix can be seen by deriving the time derivative of h X ( d ) : ( 5 5 Inductor Current (A) the eigenvalues at any chosen location he relation between the change of the inut voltages and the required values of a for a radius of 84 are shown in Fig 9 he resonse of the system while the inut voltage changes suddenly from 3 to 35V (at sec) is shown in Fig It is clear that the system, after a very small transient, will settle down quickly to the stable eriod limit cycle his method of stabilising an unstable orbit by another signal at the same frequency is referred to as resonant control and it has been emirically alied to a PWM controlled buck converter [Zhou, se, Qiu, and au, 3] But, by using the saltation matrix aroach it is ossible to carefully design the controller and avoid using trial and error aroaches which may lead to instability Otimum Values of a x Figure 9 Values of a calculated to lace the eigenvalues at a circle of radius 84-additional sinusoidal signal Outut Vltage (volt) ime (sec) Figure Resonse of the controller when the inut voltage is changed 53 Control Based on Additional Control Signal Proortional to the Outut Voltage Another control method is based on introducing a change to the normal to the switching manifold In this method, we slightly change the magnitude of the normal vector n so that [ ] n = k ) ( + a3 his will have an influence on the saltation matrices S and S thus changing the osition of the eigenvalues of the system he relation between the change of the inut voltages and the required values of a 3 are shown in Fig he resonse of the system when the inut voltage changes from 3 to 35V is shown in Fig he system settles to the stable eriod limit cycle after a brief transient h ( X ( d )) ( VU V ) ω ) (8) 5 K = Vref a cos( ω t t It is obvious from (8) that altering the value of a will have an effect on the time derivative and hence S and the eigenvalues of the monodromy matrix Again we can locate

5 Otimum Values of a Figure Values of a 3 calculated to lace the eigenvalues at a circle of radius 84-roortional control signal Outut Voltage (volt) ime (sec) Figure Resonse of the controller when the inut voltage is changed It is clear that all three controllers extend the range of eriod one oeration However the third controller which is based on additional control signal roortional to the outut voltage has a higher steady-state error 6 Conclusion In this aer, the stability of the buck converter controlled by double-edged PWM waveform was analysed using the comlete cycle solution matrix Based on the exression of the saltation matrices, we have roosed and demonstrated three controllers to control the first eriod doubling to extend the arameter range for stable eriod- oeration he suervising control laws are based on suitably changing the switching manifolds so that the orbit will be stabilised without altering its location, ie by keeing the same average value All controllers were analytically, numerically, and exerimentally validated References Aizerman, M A and Gantmakher, F R (958) On the stability of eriodic motions, Journal of Alied Mathematics and Mechanics (translated from Russian), Banerjee, S, Ott, E, Yorke, J A, Yuan, G-H, (997) Anomalous bifurcations in dc-dc converters: Borderline collisions in iecewise smooth mas, IEEE Power Electronics Secialists' Conference, St ouis, Missouri, USA, Banerjee, S and Chakrabarty, K (998) Nonlinear modeling and bifurcations in the boost converter'', IEEE ransactions on Power Electronics, vol3, no, 5-6 Banerjee, S and Verghese, G C, () Nonlinear henomena in ower electronics: Attractors, bifurcations, chaos, and nonlinear control New York, USA: IEEE Press Baushev, V S, Zhusubaliev, Zh, Kolokolov, Yu V, and erekhin, I V, (99) ocal stability of eriodic solutions in samled data control systems, Automation and Remote Control, vol 53, no, Bleich, M E and Socolar, J E S (996) Stability of eriodic orbits controlled by time-delayed feedback, Phys ett A, vol, Chakrabarty, K, Poddar, G and Banerjee, S (996) Bifurcation Behaviour of the Buck Converter, IEEE ransactions on Power Electronics, vol, no3, Di Bernardo, M, Garofalo, F, Glielmo, and Vasca, F, (998) Switchings, bifurcations and chaos in DC/DC converters, IEEE ransactions on Circuits and Systems-I, vol 45, no, Di Bernardo, M and Vasca, F () On discrete-time mas for the analysis of bifurcations and chaos in DC/DC converters, IEEE ransactions on Circuits and Systems-I, vol 47, no, 3 43 Dranga, O, Buti, B, Nagy, I and Funato, H, (5) Stability analysis of nonlinear ower electronic systems utilizing eriodicity and introducing auxiliary state vector, IEEE ransactions on Circuits and Systems-I, vol 5, no Fang, C-C and Abed, E H, () Samled-data modelling and analysis of the ower stage of PWM DC/DC converters, International Journal of Electronics, vol 88, no 3, Filiov, A F, (988) Differential equations with discontinuous righthand sides Dortrecht: Kluwer Academic Publishers Fossas, E and Olivar, G (996) Study of chaos in the buck converter, IEEE ransactions on Circuits and Systems-I, vol 43, no, 3 5 Giaouris, D, Elbkosh, A, Pickert, V, Zahawi, B and Banerjee, S, (6) Control of eriod doubling bifurcations in dc-dc converters, in International Conference Control 6, (Glasgow - 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