Multiple Resonance Networks

Transcription

1 4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 49, NO, FEBRUARY [4] Y-Y Cao, Y-X Sun, and J Lam, Delay-deendent robust H control for uncertain systems with time-varying delays, IEE Proc D: Contr Theory Al, vol 45, no 3, , 998 [5] B-S Chen, S-S Wang, and H-C Lu, Stabilization of time-delay systems containing saturating actuators, Int J Contr, vol 47, , 988 [6] J-H Chou, I-R Horng, and B-S Chen, Dynamical feedback comensator for uncertain time-delay systems containing saturating actuator, Int J Contr, vol 49, , 989 [7] K Gu, Discretized Lyaunov functional for uncertain systems with multile time-delay, Int J Contr, vol 7, no 6, , 999 [8] K Gu and S I Niculescu, Additional dynamics in transformed timedelay systems, IEEE Trans Automat Control, vol 45, , Mar [9] J Hale, Theory of Functional Differential Equations New York: Sringer, 977 [] Q L Han and B Ni, Delay-deendent robust stabilization for uncertain constrained systems with ointwise and distributed time-varying delays, in Proc IEEE 38th Conf DC, 999, 5 [] T Hu and Z Lin, Control Systems with Actuator Saturation: Analysis and Design Boston, MA: Birkhäuser, [] T Hu, Z Lin, and B M Chen, An analysis and design method for linear systems subject to actuator saturation and disturbance, in Proc ACC,, [3] B Lehman, J Bentsman, S V Lunel, and E I Verriest, Vibrational control of nonlinear time lag systems with bounded delay: Averaging theory, stabilizability and transient behavior, IEEE Trans Automat Control, vol 39, 898 9, May 994 [4] Z Lin, Low Gain Feedback London, UK: Sringer, 998 [5] D Liu and A Michel, Dynamical Systems With Saturation Nonlinearities: Analysis and Design London, UK: Sringer-Verlog, 994 [6] S I Niculescu, J M Dion, and L Dugard, Robust stabilization for uncertain time-delay systems containing saturating actuators, IEEE Trans Automat Control, vol 4, , May 996 [7] S-I Niculescu, E I Verriest, L Dugard, and J-D Dion, Stability and robust stability of time-delay systems: A guided tour, in Stability and Control of Time-Delay Systems, L Dugard and E I Verriest, Eds London, UK: Sringer-Verlag, 997, vol 8, 7 [8] S Oucheriah, Global stabilization of a class of linear continuous timedelay systems with saturating controls, IEEE Trans Circuits Syst I, vol 43, 5, 996 [9] J-K Park, C-H Choi, and H Choo, Dynamic anti-windu method for a class of time-delay control systems with inut saturation, Int J Robust Nonlin Contr, vol, no 6, , [] G Stean, Retarded Dynamical Systems: Stability and Characteristic Functions Harlow, UK, 989, Pitman Research Notes in Mathematics, Longman Scientific and Technical [] T-J Su and C-G Huang, Robust stability of delay deendence for linear uncertain systems, IEEE Trans Automat Control, vol 37, , Oct 99 [] S Tarbouriech and J M Gomes da Silva Jr, Synthesis of controllers for continuous-time delay systems with saturating controls via LMI s, IEEE Trans Automat Contr, vol 45, 5, Jan Multile Resonance Networks Antonio Carlos M de Queiroz Abstract This brief shows how multile resonance networks of any order and with many ossible structures can be systematically designed using standard lossless imedance synthesis techniques These networks are comosed of linear lumed or distributed caacitors, inductors, and transformers, with a switch searating one of the caacitors from the remaining circuit They have the roerty of transferring comletely the energy initially stored in the caacitor insulated by the switch, to another, much smaller, caacitor in the circuit, through a linear transient when the switch is closed These circuits find alications in the roduction of very high voltages for ulsed ower systems Index Terms Linear network synthesis, ower converters, resonance I INTRODUCTION Multile resonance networks [] is a name that generalizes the double resonance [], [3], trile resonance [4] [6], and the higher order networks discussed in this brief These circuits are usually comosed of a transformer and some extra caacitors and inductors and work by transferring the energy initially stored in a caacitor at one side of the transformer to another, much smaller, caacitor at the other side of the transformer, through a linear transient comosed (in the ideal lossless case) of a sum of several cosinusoidal waveforms (Fig ) The double resonance case is long known [], [7] as the Tesla coil [3] In this case, only two caacitors and one transformer are used, resulting in a fourth-order system with a transient formed by two oscillatory modes (Fig ) With the system roerly designed, after some cycles all the initial energy in C is transferred to C, and the obtained voltage is given, by energy conservation, by vout max vin() (with ) This same equation fixes the maximum outut voltage for all the systems of this tye More recently, trile resonance systems were develoed [4] [6] for instrumentation used in high-energy hysics An additional caacitor and an inductor were added to the outut side (Fig 3), with the aim of reducing the voltage stress over the transformer and of taking into consideration the outut caacitance of the transformer With only the extra inductor added, the system is still a double resonance system, long known as the Tesla magnifier With the extra caacitor the system is of sixth order and the transient has three oscillatory modes, but oeration with comlete energy transfer is equally ossible In all the cases found in the literature, the design of these systems is based on the analysis of a fixed structure The following sections show that the design can be made by synthesis, can be alied to a wide range of structures, and can be extended to systems of any order C C () II SYNTHESIS APPROACH The transformer can be left out of the roblem, because it can be inserted after the synthesis of a ladder structure comosed of series 57 7/$7 IEEE Manuscrit received November 9, ; revised July 6,, and Setember 9, This aer was recommended by Associate Editor P K Rajan The author is with the Electrical Engineering Program COPPE and the Electronic and Comuter Engineering Deartment, Federal University of Rio de Janeiro, Rio de Janeiro , Brazil (e mail: acmq@coeufrjbr) Publisher Item Identifier S 57-7()85-6

2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 49, NO, FEBRUARY 4 Fig Multile resonance network An initial energy in C is totally transferred to C through the transformer and two ossible LC networks, during the transient after the closure of the switch Fig 3 Tyical trile resonance network and voltages in C, C, and C after the closure of the switch The voltage gain was also designed as C nf, C 6:3 F, C F, L H, L 78:7 H, L :3 mh, k :83, and v () kv Fig Tyical double resonance network, and voltages in C and C after the closure of the switch The voltage gain was designed as C nf, C F, L 9:8 H, L :98 mh, and k :95 v () kv inductors and shunt caacitors, with a shunt inductor somewhere, as shown in Fig 4(a) An ideal transformer with turns ratio : n is inserted at the left side of where the shunt inductor aears and is then converted into a real transformer by using the equivalence shown in Fig 4(b), where L a L x n k ab L b L x + L y L x L x + L y Z (s) Z(s) n : () The turns ratio can be chosen as convenient for the desired voltage gain It multilies the gain in () directly because the inut caacitor is multilied by n k ab is the couling coefficient of the resulting real transformer that can be quite small if the energy transfer occurs in many cycles If the shunt inductor is at the low-voltage end (as is the case for the fourth- and sixth-order cases in Figs and 3, with the transformer eliminated, the roblem is reduced to the synthesis of the outut imedance of the circuit by a succession of comlete ole removals at infinity, or an imedance synthesis in Cauer s first form The following discussion shows how to find the required imedance With the switch closed, the imedance seen across any of the caacitors has a denominator of order, even, and a numerator of order, with a zero at s The voltage resonse of one of these imedances to a current imulse alied in arallel with the corresonding caacitor is roortional to the resonse to a charged caacitor there It aears as a sum of ure cosinusoidal oscillations with ositive Fig 4 (a) General structure without transformer (b) Equivalence that allows the insertion of a transformer where a shunt inductor aears multilying factors Sinusoidal comonents don t aear and the multilying factors must be ositive, due to the roortionality between the Lalace transform of the voltage waveform and the imedance at that oint in Foster s first form With the oscillation frequencies considered as distinct integer multiles of a basic frequency! by factors k j ;j ; ;, all the caacitor voltages have the forms (Lalace transform) and A Z in;i (s) / ij s V i (s) s + k j! v i (t) (3) A ij cos (k j! t) (4) (time domain) The currents in all the inductors are then roortional to the derivatives of the caacitor voltages (4), and so are all sums of sinusoids at the same frequencies i i (t) B ij sin (k j! t): (5) It is convenient to work from the outut of the network and comute the outut imedance Considering C initially charged to v, the energy there is transferred to C using the return art of the (eretual) transient waveform As all the k j are different ositive integers, all the currents reduce to at t!

3 4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 49, NO, FEBRUARY If the outut is excited by a unit imulse source, the roortionality in (3) becomes an identity From (3) and the structure of the network [Fig 4(a)], when s!: A j : (6) C At t!, the backward energy transfer is comlete, and v From (4) we have v A j cos(k j ) A j () k : (7)! At the same instant, v, v,, v At any time after t, considering the currents i i in the inductors in Fig 4(a) going to the right, we have di i (t) v i(t) v i (t) +L i ;i ; ; ; 3: (8) dt Exanding these exressions as functions of v (t), it can be shown that these voltages are all zero at t! if all the even derivatives of v (t) u to order 4 are null at this instant Combining this condition with (6) and (7), and eliminating owers of! and of that multily the derivatives of v (t), the following system of equations results: () k () k () k k () k k () k k () k k 4 () k k 4 () k k 4 () k A A A C : (9) It is observed that ositive solutions for all the A j are only obtained if the owers of have alternate signs for increasing k j This adds a condition on the k j, mentioned in [], [4] [6], and that extends for higher orders: Given a ositive integer as k j, the next value k j+ is obtained by adding an odd ositive integer to k j Valid sequences are then,, 3, ;, 3, 4, ;,, 5, ;, 4, 5, ; etc This rule is stated here without a formal roof, but no excetions could be found Even differences between all the successive k j or identical k j result in a singular system Sequences with mixed odd and even differences result also in solvable systems, but roduce negative residues The rule allows a further simlification of (9) [] with the elimination of the owers of, and is assumed in the deduction of the formulas resented in the following sections A articularly interesting alternative method for the calculation of the residues of the outut imedance of the network, that does not require the solving of a system of equations, is discussed in [8] With the A j comuted, an LC network can be obtained by the exansion of the outut imedance (3) in ladder form Alternative forms for the exansion of the imedance are also ossible, for examle, extracting the shunt inductor at other oints of the exansion, or extracting more than one shunt inductor With this, a transformer can be inserted at other oints, or more than one transformer can be inserted III EXAMPLES A Fourth-Order Case This is the classic double resonance circuit, but without a transformer The system of equations in (9) with C normalized to reduces to two equations that give A A The outut imedance Fig 5 Structures for transformerless multile resonance networks of: (a) fourth, (b) sixth, and (c) eighth orders of the network is then, normalizing! to and naming k k and k l: Z out s + k + s 3 s + s + l k + l s s 4 +(k + l ) s + k l : () This imedance, exanded in Cauer s first form, results in the structure in Fig 5(a), with the values C ; L k + l C k + l l k ; L l k (k + l ) k l () This circuit roduces the voltage gain [from ()] v Cmax v C () C k + l C l k : () If a transformer is inserted through the use of the relations shown in (), the relations for the elements in Fig reduce to C L a C L b k + l k l k l k k + l : (3) The turns ratio n, actually just a number because the coils can have different geometries, affects only the voltage gain (), multilying it directly The second equality in the first equation just sets the energy transfer time to seconds B Sixth-Order Case For the trile resonance case, the system (9) has three equations Symbolic exressions for the element values can be obtained as A 3 l m (k m ) ; A 3 ; A 33 k l (k m ) C 3 ; L 3 l ; C l 4 (l m )(k l ) l m k l L l (k (l + m ) l (l m )) C k l + m l l m (k l )(l m ) k l l m L (4) k l m (k (l + m ) l (l m )) for the structure in Fig 5(b), with k k, k l, and k 3 m The voltage gain is given by v C3 max v C() C k l + m l l m C 3 (k l )(l m ) : (5)

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 49, NO, FEBRUARY 43 TABLE I NORMALIZED ELEMENT VALUES FOR QUADRUPLE RESONANCE NETWORKS [FIG 5(C)], AS FUNCTIONS OF THE FREQUENCY MULTIPLIERS k, k, k, AND k IN ALL CASES THE TOTAL ENERGY TRANSFER OCCURS IN SECONDS Fig 6 Exerimental voltage waveforms obtained for an eighth-order network [Fig 5(c)] in mode,, 3, 4 C nf, L 46 H, C 9nF, L 99 H, C 338nF, L 3 H, C 76 nf, and L 3 H Measured voltage gain: 3 Energy transfer time: 5 s The dark traces are v and v, and the light traces are v and v For the structure with a transformer (see Fig 3), convenient design equations, adated and normalized to!, are L l b m k l L 3 k m ; C C 3 l 4 (l m )(k l ) L b L a C (L b + L 3 ) C 3 ;k : (6) L b + L 3 These equations show several curious deendencies among the comonents of the structure, similar to what haens in the fourth-order case [see (3)] C Eighth-Order Case The extension to higher orders results in higher voltage gain for the same basic frequencies of oeration, and maybe smaller voltage differences across the series inductors (with the excetion of the last, that always has to sustain the full outut voltage) No attemts of design or of alications for networks with orders greater than 6 could be found in the literature Symbolical exressions continue to be relatively easy to derive for the element values, but become rather imractical for the quadrule resonance case and above Table I lists numerical normalized (C ;! ) element values for the structure in Fig 5(c) for some of the ossibly more ractical combinations of frequency multiliers An exerimental circuit was constructed, oerating in mode,, 3, 4 The values in Table I were denormalized for resonances at,, 3, and 4 khz, with C 4 nf The element values were adjusted for maximum error of %, and the resistances of the inductors, wound on ferrite ot-cores, were ket around Ohm In this examle, it was not necessary to comensate for arasitic caacitances in the inductors and all the caacitances were lumed A mechanical switch was used to start the energy transfer Fig 6 shows the resulting voltage waveforms obtained The measured voltage gain was just 84% of the ideal, mainly due to the losses, but the waveforms are very similar to the ideal ones IV CONCLUSION A systematic rocedure for the design of LC voltage multiliers that are a generalization of the double resonance and trile resonance networks was resented The rocedure first obtains a secial LC imedance that is then exanded in ladder form A transformer is not necessary, but can be easily included Only lossless circuits were considered, but the alications of these circuits generally require low losses, and they are designed to behave as lossless circuits The resence of small losses does not significantly affect the waveforms in ractical circuits of this tye, adding essentially only a decay with time in the waveforms REFERENCES [] A C M de Queiroz, Synthesis of multile resonance networks, in Proc IEEE ISCAS, vol V, Geneva, Switzerland, May, [] D Finkelstein, P Goldberg, and J Shuchatowitz, High voltage imulse system, Rev Sci Instrum, vol 37, no, 59 6, 966 [3] K D Skeldon, A I Grant, and S A Scott, A high otential Tesla coil imulse generator for lecture demonstrations and science exhibitions, Amer J Phys, vol 65, no 8, , Aug 997 [4] F M Bieniosek, Trile resonance high voltage ulse transformer circuit, in Proc 6th IEEE Pulsed Power Conf, 987, 7 7

5 44 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 49, NO, FEBRUARY [5], Trile resonance ulse transformer circuit, Rev Sci Instrum, vol 6, no 6, 77 79, June 99 [6], Trile resonance ulse transformer circuit, U S Patent , May 3, 989 [7] N Tesla, High frequency oscillators for electro-theraeutic and other uroses, Proc IEEE, vol 87, no 7, 8 9, July 999 [8] A C M de Queiroz, A simle design technique for multile resonance networks, in Proc ICECS, vol I, Malta, Set, 69 7 Advanced Feedback Control of the Chaotic Duffing Equation Zhong-Ping Jiang Abstract This brief deals with the celebrated chaotic Duffing equation with external control force It is shown that Lyaunov direct method in conjunction with recent develoments in nonlinear control yields a romising way of engineering chaotic dynamics Among the three tyes of feedback controllers introduced in the aer, we articularly emhasize the value of linear feedback strategy in controlling chaos For the forced Duffing equation, it is shown that linear feedback control laws are inherently robust to (even large) sensor errors Index Terms Adative nonlinear control, Duffing equation, exonential convergence, global stability, linear feedback I INTRODUCTION Chaos control has been an active research field in recent years Various control methodologies have been develoed by many researchers from a oint of view of dynamic system theory and traditional feedback control Among these creative control algorithms are the celebrated OGY method of small time-deendent ertubations of an available system arameter [6] and Lyaunov control methods [3], [5], [7], [4] (see the books [], [], [], [5] for a rather comlete list of references in this quickly exanding area) Possibilities of alying chaotic system theory to secure communication have also been considered and justified by exerimental work (see, eg, [], [4], [8], [7], [] and references therein) The urose of this brief is to make novel contributions to Lyaunov control of chaotic continuous-time dynamic systems Because of the tremendous comlexity of chaotic dynamics, we will restrict ourselves to Duffing s equation which has been investigated as a benchmark chaotic system in several articles [3], [6], [7], [3], [4] It is hoed that our methodologies develoed for this ecular chaotic system will be alicable to other tyes of chaotic dynamic systems such as Chua s circuits and Lorenz chaotic attractor [], [] In this aer, we consider a general form of Duffing s equation with external control inut u x + _x + x + x 3 u + q cos(!t): () The control u is added in order to order or guide the chaotic dynamics to meet our secific requirements We are interested in driving the state Manuscrit received December 5, ; revised Setember 6, This work has been suorted in art by the US National Science Foundation under Grant INT , Grant ANI-857, and Grant ECS-9376 This aer was recommended by Associate Editor M Di Bernardo The author is with the Deartment of Electrical and Comuter Engineering, Polytechnic University, Brooklyn, NY USA ( zjiang@contrololyedu) Publisher Item Identifier S 57-7()8-7 x to an aroriately defined reference signal x d This issue is widely known as the tracking roblem in the control community The first contribution of this aer is that new solutions to the tracking roblem are obtained with the hel of advanced nonlinear control theory Previous work of others [3], [6], [7], [3], [4] have resented interesting results on the tracking of a more restrictive form of the forced Duffing equation () For examle, the seemingly first solution develoed by Chen and Dong [3] is alicable to Duffing s equation only when > and In addition, their result solves the local tracking roblem, ie, only those trajectories of () starting from a small neighborhood of the desired reference orbit can be asymtotically controlled to the desired trajectory The assumtion that > was relaxed by Nijmeijer and Berghuis [4] following classical Lyaunov direct method Global tracking results were obtained in [3], [4] In this aer, we do not require that > and When all system arameters are known, we resent two different feedback controllers to solve the global tracking roblem with arbitrary rate of exonential convergence Such a roerty of stability is introduced in Section II The first such controller is a linear time-varying state-feedback control law The second one is derived from the first one in conjunction with a nonlinear observer without assuming that _x is measurable When the system arameters,, and q are unknown, we develo a nonlinear adative controller to achieve the global tracking task It is simly shown that the oular method of adative backsteing [] roves useful for chaos control, in articular for the forced Duffing equation in the general form () The second contribution of this aer is to argue the imortance of linear feedback strategy in the context of controlling chaos Linear feedback control laws are simler to imlement in ractice than nonlinear controllers and therefore more accetable by racticing engineers More imortantly, linear controllers are often less sensitive to sensor errors and are inherently robust against measurement errors It is well-known in the nonlinear control community that a globally stabilizing nonlinear controller may not be robust in front of (even small) sensor errors see [9] and references therein for detailed discussions Intuitively, the effect of measurement errors can be amlified through the nonlinearity of the control law in question, therefore leading to instability A redesign of nonlinear control is needed to guarantee the robustness roerty In case when the full-state information is available, we construct a linear feedback controller to guarantee the roerty of global exonential convergence with arbitrary rate for the tracking error In articular, we show that this linear feedback control law enjoys the inherent robustness to (even large) measurement errors (see Proosition below) The rest of this aer is organized as follows In Section II, we recall some definitions from the literature of stability theory and introduce a new notion of stability The statement of the control roblem is also given in Section II Section III resents two tracking algorithms for the controlled Duffing equation () Linear state-feedback and nonlinear observer-based outut-feedback controllers are obtained It should be mentioned that our observer structure is quite different from the mechanism of observing chaos roosed in [] In Section IV, we show how to remove the assumtion of requiring the recise knowledge of system arameters and roose nonlinear adative controllers Simulation results are offered in Section V to suort our theoretic findings Some concluding remarks are contained in Section VI II DEFINITIONS AND PROBLEM STATEMENT First of all, recall some definitions that we will frequently use throughout this aer A new concet of stability is introduced The goal of this aer is to show that we can achieve this tye of stability 57 7/$7 IEEE