Adaptive Control of the Boost Inverter with Load RL

Transcription

1 Proceeings of the 7th Worl Congress The International Feeration of Automatic Control Seoul, Korea, July 6-, 8 Aaptive Control of the Boost Inverter with Loa RL Carolina Albea Francisco Gorillo Carlos Canuas-e-Wit Universia e Sevilla, Dpto. Ingeniería e Sistemas y Automática, Camino e los Descubrimientos, s/n, 9 Sevilla, Spain ( calbea@cartuja.us.es, gorillo@esi.us.es CNRS, Gipsa-lab Grenoble. ENSIEG-BP 6, 8 Saint Martin Hères Ceex, France, ( carlos.canuas-e-wit@resulting inpg.fr Abstract: This paper proposes an aaptive control for the nonlinear boost inverter with nonresistive loa in orer to cope with unknown resistive loa as well as unknown inuctive loa. The aaptive control is accomplishe by using a state observer to one sie of the inverter with a quaratic Lyapunov function an by measuring the states variables. The stability of the complete system is analyze by means of singular perturbation analysis. The aaptation of the both parameters is teste by using simulations.. INTRODUCTION The control of switche-moe power converters (SMPC with AC output is usually accomplishe tracking a reference (sinusoial signal Sanchis et al. [], Vazquez et al. []. The use of this external signal makes from the mathematicalpointofview the close-loopcontrolsystem to be non-autonomous an thus, making its analysis involve. A ifferent approach was use in Pagano et al. []: a control law was esigne for stabilizing a limit cycle corresponing to the esire behavior. No external signals were neee. In Albea et al. [6] this iea was applie to the boost inverter Caceres an Barbi [999]. A Phase-Lock Loop (PLL is ae to the control law in orer to achieve synchronizationbetween both parts of the circuit as well as synchronize the voltage output with a pre-specifiesignal, e.g. synchronizationwiththe electrical gri. Only the case of known resistive loa was consiere. The unknown loa problem in SMPC is usually ealt with aaptation mechanisms combine with other techniques such as iscontinuous feeback regulators, backstepping with passivity-base controllers, sliing moe, El Fail et al. [], Tan et al. []. In Pagano et al. [] is compute an aaptive control for the case of the boost converter. The controller use in such paper follows the iea given above. The fact that the boost inverter moel is -imensional an nonlinear makes the esign of the aaptation law more involve. A state observer for some of the converter variables is esigne even when the state variables are measure. In orer to analyze the stability of the full system singular perturbation analysis is use Khalil []. For simplicity, the PLL is not consiere in this analysis. The case of non-resistive loa was stuie in Albea an Gorillo [7], but only for known loa. In this paper, This work was supporte by the MEC-FEDER grant DPI6-78. the previous results are extene to case of unknown an non-resistive loa using the aaptation mechanism from Albea et al. [7], aapting the parameters of the loa at the same time. Singular perturbation analysis was use to prove the stability of the full system. The resultant aaptive control is teste by means of simulations. The rest of the paper is organize as follows: in Sect. the moel of the ouble boost converter (boost inverter is presente. Section states the problem, which is solve in Sect. by means of the esign of the aaptation mechanism. Section is evote to the stability analysis an Sect. 6 presents some simulation results. The paper closes with a section of conclusions.. BOOST INVERTER MODEL The boost inverter Caceres an Barbi [999] is specially interesting because it can generates an ac output voltage larger than the its c input Caceres an Barbi [999]. It is compose of two c-c converters an a loa connecte across of them. Each converter prouces a c-biase sine wave output, v an v, so that each source generates an unipolar voltage. Voltages v an v shoul present a phase shift equal to 8, which maximizes the voltage excursion across the loa. The circuit implementation is shown in Fig.. By this mean it is possible to generate an oscillatory signal without bias. In Albea et al. [6] the loa was resistive, an in Albea an Gorillo [7] was stuie this same circuit with a RL loa. Here it is assume that: all the components are ieal an the currents of the converter are continuous, the inuctances L = L = L, an the capacitances C = C = C, are known an symmetric, the loa R an are unknown, an it has to be estimate /8/$. 8 IFAC 6.8/876--KR-.9

2 7th IFAC Worl Congress (IFAC'8 Seoul, Korea, July 6-, 8 v Q V in loa C L L Q Fig.. Boost inverter moel V o The boost c-ac converter can be simplifie as shown in Fig.. This simplification lets see clearer the biirectional V in L i L Q C Q V o R v i LR Fig.. Simplifie Boost inverter moel current of each boost c-c converter. The circuit in Fig. is riven by the transistor ON/OFF inputs Q i. This yiels two moes of operation illustrate in Appenix A. Formally this yiels a switche moel which is more involve. For control purposes, it is common to use an averagemoel escribein terms of the meancurrents an voltages values. This moel is more suite for control because it is escribe by a continuous time smooth an nonlinear ODE. Following Albea et al. [6], this averaging process yiels the normalize moel escribe below.. Normalize average moel In orer to simplify the stuy, we can yiel a normalize moel by using the change of variables: x = L V in C i L x = v V in x = ˆ V in C i L Q C Q v v an efining the new time variable with t = LC t. Assuming an unknown RL loa with a average current x, a normalize moel in terms of the average current x an the average voltage x, for one sie of the inverter (see, Albea an Gorillo [7], is: ẋ = u x ( ẋ = u x bx ( ẋ = θ x θ cx θ cx a ˆθ x ( where θ = R LC, θ =,ˆθ =, a = ˆL R V in C, b = Ḽ an c = LˆL an the respective normalize R moel for the other sie is obtaine similarly by symmetry ẋ = u x ( ẋ = u x bx ( where u i [, ], i =, escribes the control inputs. Note that they are also normalize an reflect the mean uty-cycle activation percent of each circuit. They are here treate as continuous variables. Parameters a, b an c epens on the estimation of an is assume to be known. θ an θ epens on the unknown loa which is unknown.. PROBLEM FORMULATION The control problem is to esign a control law for u, an u, for the system ( ( an ( ( in orer to make the output y to oscillate as a sinusoial signal with a given amplitue i.e. y = x x y r = Acos(ωt ϕ with a pre-specifie value for A an ω. The phase shift ϕ is not specifie. Uner the assumption that a, b, c, θ an θ are constants an knowns, in Albea an Gorillo [7] a nonlinear control law base on Hamiltonian approach was propose. The esign is base on the following change of coorinates: η = x x (6 η = x bx x η (7 η = x x (8 η = x bx x η (9 The controller recalle further below, has as an objective to rener the following functions ten to zero Γ ω (η η (η η µ = ( Γ ω (η η (η η µ = ( orbitally stable. The parameters η, η an η, η efine the respective ellipse centers an ω, µ are relate to their size. Base on this efinition, the nonlinear control law as propose in Albea an Gorillo [7] has the following form: [ ] [ ] u k (x, θ u = k(x, θ, θ = =, θ u R u k (x, θ, θ with k (x, θ, θ, k (x, θ, θ given in Appenix B. The esign is complete with an aitional outer loop (PLL that has the function of achieving a phase shift equal to 8 between the two voltages v, an v reaching in that way the esire objective. The goal here is to extent this work to the case of unknown loa.. ADAPTATION LAW LOAD DESIGN In this section we propose an aaptive law (or a loa observer to cope with loa variations an/or uncertainties 7

3 7th IFAC Worl Congress (IFAC'8 Seoul, Korea, July 6-, 8 on the loa parameters θ, an θ. This observer is esigne base only on one-sie of the circuit, which contains enough information to make this parameter observable. Therefore the use of the full two-sie circuit is not necessary at this stage. The one-sie (left circuit (-(, can be rewritten compactly as: ẋ l = U l x l bb l x cθ C l y (aˆθ θ C l x E l ( y = x x ( with x l = [x, x, x ] T, an [ ] [ u U l = u, B l = ], E l = [ where θ an θ are isturbe parameters. ], C l = [ In what follows, we assume that both voltages an currents are measurable (either analogically or numerically, an thus accessible for control use.. Aaptation law The propose aaptation law is compose by: state observers, plus aaptation laws for θ an θ. It has the following structure: ˆx = U lˆx bb l x cˆθ C l y (aˆθ ˆθ C l x ( E l K(x l ˆx ( ˆθ = β (x l, ˆx (6 ˆθ = β (x l, ˆx (7 where K R is a constant esign matrix, an θ (x l, ˆx an θ (x l, ˆx are the aaptation laws to be esigne. Note, that even if x is accessible, the aaptation law esigne here requires the aitional (or extene state observer. This will become clear uring the analysis of the error equation system, as stuie below.. Error equation Assume that θ an θ are constant parameters ( θ = an θ = (or that change slowly θ an θ an efine the following error variables: x = x l ˆx, θ = θ ˆθ, θ = ˆθ θ = θ ˆθ, ] θ = ˆθ Error equation are now erive from ( ( together with ( (7 x = K x c θ C l y θ C l x (8 θ = β (x l, ˆx (9 θ = β (x l, ˆx ( ( Let K be of the form, K = αi, α > an P = I be the trivial solution of PK T KP = Q, with Q = αi. Now introucing it follows that V = x T Q x θ ( x T PC l x = x T Q x θ ( x T PC l x V = x T P x θ γ θ γ ( θ θ θ ( x T PcC l y γ γ θ θ θ ( x T PcC l y γ γ The aaptation laws are now esigne by canceling the terms in square brackets, i.e. ˆθ = γ ( x T PC l x ( ˆθ = γ x T PcC l y ( The stability properties of the observer an the aaptive law error equations were iscusse in Albea et al. [7] reaching to: Lemma. Consier the open-loop system ( (, an assume that its solutions are boune. The extene observer ( (7 has the following properties: i The estimate states ˆx, ˆθ, ˆθ are boune. ii lim t ˆx(t = x(t. iii lim t ˆθ (t = θ y(t, x (t t. iv lim t ˆθ (t = θ y(t, x (t t.. STABILITY OF THE FULL CLOSED-LOOP EQUATIONS The stability of the complete system (the system state variables plus the observer extene is analyze in this section. The open-loop two-sies inverter (-( an (-(, can be compactly rewritten as: ẋ = Ux bbx cθ Cy (aˆθ θ Cx E ( y = x x (6 with x = [x, x, x, x, x ] T, an u u U = u, B =, C =, E = u. Tune System The tune system is efine as the ieal close-loop system uner the action of the tune feeback law u = k(x, θ, θ, compute with the exact value of x, θ an θ. The tune systems given in Albea an Gorillo [7] writes ẋ = U(u x bbx cθ Cy (aˆθ θ Cx E = U(k(x, θ, θ x bbx cθ Cy (aˆθ θ Cx E = f(x 8

4 7th IFAC Worl Congress (IFAC'8 Seoul, Korea, July 6-, 8 an it achieves an asymptotically orbitallystable perioic solutions, i.e. x (t = x (t T In Albea et al. [6] it has been shown that the functions Γ an Γ efine in ( ( ten to zero. They correspon to perioic sinusoial solutions with perio T = π/ω. Consequently, y = x x is also sinusoial.. Close-loop system How it is seen in Albea et al. [7], note that control law epens on the estimation as û = k(x, ˆθ, ˆθ. Note that this control law epens on the state x an not on its estimation ˆx, because the state x is irectly measure. The role of ˆx is then just to make possible the esign of the aaptation law for θ an θ. The close-loop equation resulting from the use of û = k(x, ˆθ, ˆθ writes, as ẋ = U(ûx bbx cθ Cy (aˆθ θ Cx E ± U(u x = f(x [U(û U(u ] x = f(x U(ũx where ũ = u û. Note that U(ũ = U( x, θ, θ, captures the mismatch between the estimate an the true value of the loa (see Albea et al. [7]. This term has the following property: Property. Let M = {(x, θ, θ : x x < ǫ x, θ < ǫ θ, θ < ǫ θ }, be a compact omain incluing the asymptotic perioic solutions of the tune system an the exact loa. Then, the function U(ũ = U(x, θ, θ has (x, θ, θ M, the following properties: i it is continuous, analytic, an free of singularities ii it has the following limits: lim U(ũ = lim ũ θ θ x x U(x, θ, θ = Puttingtogetherthe close-loopequationresultingfrom ( with the observer error system give the complete set, with y = y(x ẋ = f(x U(x, θ, θ x (7 x = α x c(ˆθ θ C l y θ C l x (8 θ = γ ( x T PC l x (9 θ = γ x T Pc(ˆθ C l y ( we have substitute K = αi. The stability consieration iscusse here will be base on the time-scale separation. The main iea is that with the suite choice of gains (as iscusse latter the observer equation (8-( can be seen as the fast variables an the equation (7 as the slow subsystem. Note again, that this time-scale separation shoul be enforce by a particular choice of the observer an aaption gains: α, γ an γ.. Singular perturbe form To put the system above in the stanar singular perturbation form, we follow the next steps: introuce θ = θ α, θ = θ α select γ = γ = γ = α efine ε = α With these consierations, we achieve, ẋ = f(x U(x, b, cx ε x = x c( θ θ C l y θ C l x ε θ = x T PC l x ε θ = x T Pc( θ C l y where ε > being the small parameters. Note that this particular selection of gains imposes relationships for the aaptation γ. The target system for the slow variables, efine after the change of coorinates (6 (9 Albea et al. [6], is η = ωη η = ωη kη Γ(η, η. Diviing this equations by ω they achieve a similar form to fast variables equations. As we want that variable x is much slower than x, θ an θ, we have to impose ε ω, ε k. This means that the aaptation gain γ as well as the tuning parameterk shoulbe relateto the esirefrequencyas: γ ω, γ k Letting z = [ x, θ, θ ] T gives the general form ẋ = f(x U(ũx ( εż = g(x, z ( with, x(t = x, x R, z(t = z, z R, an g(z, x = ε[ x, θ, θ ] T Accoring to the singular perturbation analysis, we nee to follow the next steps: ( Fin a stationary solution of the fast subsystem ( by fining roots of the equation g(x, z =, i.e. z = φ(x ( Substitute this solution in the slow subsystem (, an fin a the resulting slow system ẋ = f(x U(ũ(x, φ(xx ( Check the bounary layer properties of the fast subsystem along one particular solution of ẋ = f(x U(ũ(x, φ(xx.. Slow sub-system Proceeing to the steps an above requires to fin the roots of g(x, z =, which are calculate from x = c( θ θ C l y θ C l x = (c( θ θ C l y θ C l x T PC l x = (c( θ θ C l y θ C l x T Pc( θ C l y If the initial conitions are such that y, x, then z = φ(x = [ x, θ, θ ] T = become an isolate root. Then for this particular solution, an noticing that 9

5 7th IFAC Worl Congress (IFAC'8 Seoul, Korea, July 6-, 8 θ = θˆθ α = an θ = θˆθ α =, e.i. ˆθ = θ an ˆθ = θ the slow moel writes as: ẋ = f(x U(x, x = f(x, ( which is nothing else than the tune system whose solutions x(t = x (t are sinusoial.. Bounary layer fast subsystem For the evaluation of the bounary layer system in the finite time interval t [t, t ] Albea et al. [7], we consier a particular solution of y an x expresse in the stretche time coorinates τ = (t t /ε: y = y (τ, ω, ε, t an x = x (τ, ω, ε, t, the fast subsystem ( evaluate along such particular solution is, τ x = x τ x = x τ x = x c( θ θ C l y θ C l x θ = x x τ θ = c( θ x y τ we can compactly rewrite them as: τ z = J(y, x z = J(τ, ω, ε z ( with x c( θ y αl θ y J = θ L θ x c( θ y αl x y θ L θ An the autonomous linear system τ z = J(τ, ω,, z = J(y, x z ( Consier the y, x D x, with D x {x : y = x x > δ >, ˇx > δ > }, being δ an δ are constants. the above system has the following properties. Property. The eigenvalues of J(y, x, for [t, ˇx, z] [t, t ] D x R, are all strictly negative. By the Routh stability criterion is prove that the eigenvalues are strictly negative. Tikhonov s theorem, see Khalil [], can now be avocate to summarize the previous result. Theorem. There exists a positive constants ε such that for all y D x, an < ε < ε, the singular perturbation problem of (-( has a unique solution x(t, ε, z(t, ε on [t, t ], an x(t, ε x (t = O(ε (6 z(t, ε ẑ (t/ε = O(ε (7 hol uniformly for t [t, t ], where ẑ (τ is the solution of the bounary layer moel (. Moreover, given any t b > t, there is ε ε such that z(t, ε = O(ε hols uniformly for t [t θ, t ] whenever, ε < ε. Extension of this result to infinite time interval, requires prove that the bounary layer system is exponential stable in a neighborhoo of the tune slow solution x (t for all t t. This may not be a trivial emonstration, an it will be leftforfurtherinvestigation. Instea, we showebelow using simulation the effectiveness of this approach. An intuitive yet not completely rigorous explanation for the goo resultant behavior can be given with the help of Fig.. Notice that the Hurwitz character of Jacobian (. is only lost when y = an x = at the same instants. Since the fast motion, z, evolves with almost constant y an x (see Fig., y an x will not reach the value zero uring this motion provie that the initial conition is such that y an x are far enough from zero. Once the slow manifol is reache, the slow variable will evolve in the omain z =. This omain correspons to the case when the aaptation mechanism has reache its objective an parameter θ an θ are correctly estimate. In this omain y an x may reach the value zero but, intuitively, we can think that the system, once the aaptation law has reache the correct value, will present a behavior that is similar to the case of known loa. For this last case stability is prove in Albea et al. [6]. z 6 6 x. x.. Fig.. Evolution of (x, x, z. The last part of the trajectory is in the plane z = x 6. SIMULATIONS The following simulations are mae consiering V in = V, R = Ω, L = L =.mh, C = C = µf. The esire output of the circuit is V out = sint V. In orer to obtain this voltage, the parameters are θ =., θ = 66.66, ε = 8, ω =., A =, k =. an η = η =. The ellipse parameters result accoring to Albea an Gorillo [7] are η = η =.87, µ =.7. The loa parameters are perturbe in two times, corresponing to the transitory an stationary states. In the instant t = s, the estimate value of parameter θ an θ will be ˆθ =. an ˆθ = 7. (R = Ω an = mh, i.e., a.% an % error, respectively. Later, in the instant t = 6.s the parameters take the real value of R = Ω an = mh..

6 7th IFAC Worl Congress (IFAC'8 Seoul, Korea, July 6-, 8 Fig. shows the output voltage evolution at the instant of the perturbation. Note that the output oes not suffer any perturbation. The aaptation of the parameters θ an θ are represente in the Fig. an Fig. 6, respectively. V oltage(v Fig.. Output voltage with aaptation of a perturbation of a.% for θ an % for θ at the instant of the perturbation. θ a θ b Fig.. Time-evolution of the fast variable θ θ θ a θ b Fig. 6. Time-evolution of the fast variable θ θ 7. CONCLUSIONS An aaptive control for an unknown RL loa is presente for a nonlinear boost inverter. The metho is base on using a state observer an a quaratic Lyapunov function to one sie of the inverter an by knowing that the state variables are measure. The stability of the complete system is prove putting the system in the stanar singular perturbation form, hence we obtaine a relationshipbetween the aaptationgains, γ, the observermatrixparameter, α, an the perturbe variable parameters, ε. Another important relationship between the perturbe variable parameter, ε, an the system frequency, ω, was achieve in the analysis of the bounary layer fast subsystem. Finally, the stability is establishe by means of Tikhonov s theorem. Open problem is the extension of this result to infinite time interval. REFERENCES C. Albea an F. Gorillo. Control of the boost DC-AC converter with RL loa by energy shaping. Decision an Control, 7 6th IEEE Conference on, pages 7, 7. C. Albea, C. Canuas-e Wit, an F. Gorillo. Aaptive Control of the Boost DC-AC Converter. pages 6 66, 7. Carolina Albea, Francisco Gorillo, an Javier Aracil. Control of the boost DC-AC converter by energy shaping. IEEE Inustrial Electronics, IECON 6-n Annual Conference on, pages 7 79, 6. RO Caceres an I. Barbi. A boost DC-AC converter: analysis, esign, an experimentation. Power Electronics, IEEE Transactions on, (:, 999. H. El Fail, F. Giri, M. Haloua, H. Ouai, I. LAP, an F. Caen. Nonlinear anaaptive controlof buck power converters. Decision an Control,. Proceeings. n IEEE Conference on,,. HassanK. Khalil. Nonlinear Systems. Prentice Hall, thir eition eition,. D. J. Pagano, J. Aracil, an F. Gorillo. Autonomous oscillation generation in the boost converter. In Proceeings of the 6 th IFAC Worl Congress,. P. Sanchis, A. Ursaea, E. Gubia, an L. Marroyo. Boost DC AC Inverter: A New ControlStrategy. Power Electronics, IEEE Transactions on, (:,. S.C. Tan, YM Lai, ancktse. Aaptive controlschemes for stabilizing switching frequency of sliing moe controlle power converters. Power Electronics an Applications, European Conference on, page 8,. N. Vazquez, D. Cortes, C. Hernanez, J. Alvarez, J. Arau, anj. Alvarez. Anew nonlinearcontrolstrategyforthe boost inverter. PowerElectronicsSpecialist Conference,. PESC. IEEE th Annual,,. Appenix A. OPERATION MODES The equations of two operation moes of the boost inverter system are L i L t C v C t i LR t = uv V in = ui L i R = v v i R R Appenix B. CONTROL LAW k (x, θ, θ = ( b x bcθ x (x x bθ θ x x ba θ x x (x bx x c θ x x kγ(η η ω (η η (x bx x k (x, θ, θ = ( b x bcθ x (x x bθ θ x x ba θ x x (x bx x c θ x x kγ(η η ω (η η (x bx x