Adaptive Control of the Boost DC-AC Converter

Transcription

1 Aaptive Control of the Boost DC-AC Converter Carolina Albea-Sanchez, Carlos Canuas De Wit, Francisco Gorillo Alvarez To cite this version: Carolina Albea-Sanchez, Carlos Canuas De Wit, Francisco Gorillo Alvarez. Aaptive Control of the Boost DC-AC Converter. IEEE n International Symposium on Intelligent Control, ISIC 7, Oct 7, Singapour, Singapore. <hal-9487> HAL I: hal Submitte on 7 Dec 7 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.

2 Aaptive Control of the Boost DC-AC Converter Carolina Albea, Carlos Canuas-e-Wit*, Francisco Gorillo Abstract In this paper an aaptive control is esigne for the nonlinear boost inverter in orer to cope with unknown resistive loa. This aaptive control is accomplishe by using a state observer to one sie of the inverter an by measuring the state variables. In orer to analyze the stability of the full system singular perturbation analysis is use. The resultant aaptive control is teste by means of simulations. I. Introuction The control of boost DC-AC converters is usually accomplishe tracking a reference (sinusoial) signal. The use of this eternal signal makes the close-loop control system to be non-autonomous an thus, making its analysis involve. In [], [] a ifferent approach was use: a control law was esigne for the boost converter in orer to stabilize a limit cycle corresponing to the esire behavior. No eternal signals were neee. Nevertheless, the use of a boost converter prevents the achievement of zero-crossing signals an, thus, AC current was not achieve. This problem was solve in [3] with the use of a ouble boost converter as was propose in [4]. A phase-lock loop was necessary for the correct operation of the circuit as well as for synchronization with the electrical gri. Only the case of known resistive loa was consiere. In this paper the previous results are etene to the case of unknown loa using an aaptation mechanism. Aaptation mechanism for similar controllers were use in [] for the case of the buck converter which can be moelle by linear equations. The fact that the boost converter moel is 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. For simplicity, the phase-lock system is not consiere in this analysis. The resultant aaptive control is teste by means of simulations. The rest of the paper is organize as follows: in Sect. II the moel of the ouble boost converter (boost inverter) is presente. Section III states the problem, which is solve in Sect. IV by means of the esign of the aaptation mechanism. Section V is evote to the stability Universia e Sevilla, Dpto. Ingeniería e Sistemas y Automática, Camino e los Descubrimientos, s/n, 49 Sevilla, Spain, calbea@cartuja.us.es, gorillo@esi.us.es *Laboratoire Automatique e Grenoble. ENSIEG-BP 46, 384 Saint Martin Hères Cee, France, carlos.canuas-ewit@lag.ensieg.inpg.fr analysis an Sect. VI presents some simulation results. The paper closes with a section of conclusions. II. Boost inverter moel The boost inverter is specially interesting because it generates an AC output voltage larger than the its DC input [4]. It is compose of two DC-DC converters an a loa connecte as shown in Fig.. Each converter prouces a DC-biase sine wave output, V a an V b, so that each source generates an unipolar voltage. The circuit implementation is shown in Fig.. Voltages V a an V b shoul present a phase shift equal to 8, which maimizes the voltage ecursion across the loa [4]. It is here assume that: all the components are ieal an the currents of the converter are continuous, the inuctances L = L, an the capacitances C = C, are known an symmetric, the loa R is unknown, an it is neee to be estimate. converter A loa V a V b converter B Fig.. Basic representation of the boost inverter. Q V o R L L V C C V Q Q 4 Q 3 Fig.. Ieal Boost DC-AC Converter. The circuit in Fig. is riven by the transistor ON/OFF inputs Q i. This yiels two moes of operations illustrate in Appeni A. Formally this yiels a switche moel which is more involve. For control purposes, it is common to use an average moel escribe in terms of the mean currents an voltages values. This moel is more suite for control because it is escribe

3 by a continuous time smooth an nonlinear ODE. Following [3], this averaging process yiels the normalize moel escribe below. A. Normalize average moel Assuming resistive loa, a normalize moel in terms of the average current an the average voltage, for one sie of the inverter (see, [3]) is: ẋ = u () ẋ = u a a 4 () an the respective normalize moel for the other sie is obtaine similarly by symmetry ẋ 3 = u 4 (3) ẋ 4 = u 3 a 4 a (4) where here 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. Parameter a = L R C epens on the loa charge an is assume to be known. III. Problem formulation The control problem is to esign a control law for u, an u, for the system () () an (3) (4) in orer to make the output y to oscillate as a sinusoial signal with a given amplitue i.e. y = 4 y r = Acos(ωt ϕ) with a pre-specifie value for A, an ω. The phase shift ϕ is no specifie. Uner the assumption that a is constant an known, in [3] a nonlinear control law base on Hamiltonian approach was propose. The esign is base on the following change of coorinates: η = (5) η = a a 4 η (6) η 3 = 3 4 (7) η 4 = 3 a 4 a 4 η 4 (8) 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 3 η 3 ) (η 4 η 4) µ = orbitally stable. Parameters η, η an η 3, η 4 efine the respective ellipse centers an ω, µ are relate to their size. Base on this efinition, the nonlinear control law as propose in [3] has the following form: [ ] [ ] u k (, a) u = k(, a) = = u R u k (, a) with k (, a), k (, a) given in Appeni 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 etent this work to the case of unknown loa. IV. Aaptation law loa esign In this section we propose an aaptive law (or a loa observer) to cope with loa variations an/or uncertainties on the loa parameter a. 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: l = U l l ab l y E l (9) y = 4 () with l = [, ] T, an [ ] [ u U l =, B u l = ] [, E l = In what follows, we assume that both voltages an currents are measurable (either analogically or numerically), an thus accessible for control use. A. Aaptation law The propose aaptation law is compose by: an state observer for one sie of the inverter boost, plus an aaptation law for a. It has the following structure: ˆ = U lˆ âb l y E l K( l ˆ) () â = β( l, ˆ) () where K R is a constant esign matri, an β( l, ˆ) is the aaptation law to be esigne. Note, that even if l is accessible, the aaptation law esigne here requires the aitional (or etene) state observer. This will become clear uring the analysis of the error equation system, as stuie below. B. Error equation Assume that a is a constant parameter (ȧ = ) (or that change slowly ȧ ) an efine the following error variables: = l ˆ, ã = a â, ã = â. Error equation are now erive from (9) () together with () () Let K be of the form, = K ãb l y (3) ã = β( l, ˆ) (4) K = αi, α > an P = I be the trivial solution of PK T KP = Q, with Q = αi. ]

4 Now introucing V = T P ã γ it follows that ( V = T Q ã T PB l y ã ) γ ( = T Q ã T PB l y â ) γ (5) The aaptation law is now esigne by canceling the terms in square brackets, i.e. C. Stability properties â = γ(bl T P )y (6) The observer an the aaptive law error equations are now fully efine. These equations are: = K ãb l y (7) ã = γ(b T l P )y (8) The stability properties of these equations follows from the Lyapunov function V efine above. Note that with the choice (6) it follows that V = T Q From stanar Lyapunov arguments, it follows that the error variable an ã are boune. In aition by LaSalle invariant principle, we easily conclue that, which implies from (8) that ã. From (7), an by the property, an, we have that ãb l y Note that if y behaves as a sinusoial as epecte from the control problem formulation, the unique asymptotic solution for ã, is ã =, as y, t. Note, that in the instants that y =, (9) oes not epens on parameter a. The following lemma summarizes the results: Lemma : Consier the open-loop system (9) (), an assume that its solutions are boune. The etene observer () () has the following properties: i) The estimate states ˆ, â are boune. ii) lim t ˆ(t) = (t). iii) lim t â(t) = a, if an only if y(t), t. V. Stability of the full close-loop equations In the previous section we have presente the stability properties of the etene observer. These properties are inepenent to the evolution of the system state variables. The stability of the complete system is analyze in this section. The open-loop two-sies inverter ()-() an (3)-(4), can be compactly rewritten as: ẋ = U(u) aby E (9) y = 4 () with = [,, 3, 4 ] T, an u u U = u u A. Tune System,B =,E = The tune system is efine as the ieal close-loop system uner the action of the tune feeback law u = k(, a), compute with the eact value of a. The tune systems given in [3] writes ẋ = U(u ) aby E () = U(k(, a)) aby E () = f() (3) an it achieves an asymptotically orbitally stable perioic solutions, i.e. (t) = (t T) In [3] it has been shown that the funtions Γ an Γ efine in (9) (9) ten to zero. They correspon to perioic sinusoial solutions with perio T = π/ω. Consequently, y = 4 is also sinusoial. B. Close-loop system In practice, the control law which is effectively applie epens on the estimation of parameter a. We enote this control law as û = k(, â). Note that this control law epens on the state an not on its estimation ˆ, because the state is irectly measure. The role of ˆ is then just to make possible the esign of the aaptation law for a. The close-loop equation resulting from the use of û = k(, â) writes, as ẋ = U(û) aby E ± U(u ) (4) = f() [U(û) U(u )] (5) = f() U(ũ) (6) where ũ = u û. Note that U(ũ) = U(, ã), can be partitione as follows: ( ) Iϕ (, a U(ũ) = U(, ã) = m )ã Iϕ (, a m )ã where ϕ i (, a m ) = a k i (, ã) ã=am, i =,. This epression results from the application of the mean-value theorem, with a m [a min, a ma ], being a value of a in the allowe physical interval. The screw-symmetric matri S = S T is efine as ( ) S = iag{i, I}, I = The term U(ũ) = U(, ã) captures the mismatch between the estimate an the true value of the loa. In view of the iscussion above, this term has the following property: Property : Let M = {(, ã) : < ǫ, ã < ǫ a }, be a compact omain incluing the asymptotic perioic

5 solutions of the tune system an the eact loa. Then, the function U(ũ) = U(, ã) has (, ã) M, the following properties: i) it is continuous, analytic, an free of singularities ii) it has the following limits: lim U(ũ) = lim U(, ã) =. ũ ã Putting together (6) with the observer error system give the complete set of close-loop equation, with y = y() ẋ = f() U(, ã) (7) = α ãby (8) ã = γ(b T P )y (9) where 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)- (9) can be seen as the fast variables an the equation (7) as the slow subsystem. Note again, that this timescale separation shoul be enforce by a particular choice of the observer an aaption gains: K an γ. C. Singular perturbe form To put the system above in the stanar singular perturbation form, we follow the net steps: introuce ā = ã α, select γ = α efine ε = α With these consierations, we achieve, ẋ = ε = ε ā = f() U(, ā) āby, (B T P )y. where ε > being the small parameter. Note that this particular selection of gains imposes relative gains for the aaptation γ, an efines precisely how the observer gain are relate to γ. The target system for the slow variables, efine after the change of coorinates (5) (8) [3], is η = ωη η = ωη k Γ. Diviing this equations by ω they achieve a similar form to fast variables equations. As we want that variable is much slower than z we have to impose ε ω, ε k. This means that the aaptation gain γ as well as the tuning parameter k shoul be relate to the esire frequency as: γ ω, γ k Letting z = [, ā] T gives the general form ẋ = f() U(ũ) (3) εż = g(, z) (3) with, (t ) =, R 3, z(t ) = z, z R 5, an [ ] āby g(z, ) = (B T P )y Accoring to the singular perturbation analysis, we nee to follow the net steps: ) Fin a stationary solution of the fast subsystem (3) by fining roots of the equation g(, z) =, i.e. z = φ() ) Substitute this solution in the slow subsystem (3), an fin a the resulting slow system ẋ = f() U(ũ(, φ())) 3) Check the bounary layer properties of the fast subsystem along one particular solution of ẋ = f() U(ũ(, φ())). D. Slow sub-system Proceeing to the steps an above requires to fin the solution for the algebraic equation [ ] āby = g(, z) = (B T P )y whose roots are calculate from = āby = āb T PBy note that B T PB =, an that the above equation has multiple solutions, i.e = āy () = which means that if y, there one solution for =, an infinite solutions for ā. However, if y, for instance the particular tune solution y = Acos(ωt), then [ ] z = φ() = = ā become an isolate root. Then for this particular solution, an noticing that ā = aâ α =, e.i. â = a the slow moel writes as: ẋ = f() U(, ) = f(), (3) which is nothing else than the tune system whose solutions (t) = (t) are sinusoial.

6 E. Bounary layer fast subsystem Now, the net step is to evaluate the stability of the bounary layer system in the finite time interval t [t, t ]. This is obtaine by evaluating the fast subsystem (3) along one particular solution of the quasi-steaystate (t), t [t, t ] (tune system solutions), an by re-scaling time t to τ = (t t )/ε. As a particular solution we consier y = 4 = Acos(ωt ϕ), which epresse in the stretche time coorinates τ = (t t )/ε is: y = y (τ, ω, ε, t ) = Acos(ω(ετ t ) ϕ) the fast subsystem (3) evaluate along such solution is, τ ˆ = ˆ τ ˆ = ˆ ˆāy τ ˆā = ˆ y which can be rewritten as: τ ẑ = J(y )ẑ = J(τ, ω, ε)ẑ (33) with J = y y Uner this conitions, system (33) is reuce to the autonomous linear system τ ẑ = J(τ, ω, )ẑ = J(y )ẑ (34) Consier the y D, with D { : y = 4 > α > }, the above system has the following properties. Property : The eigenvalues of J(y ), for [t,, z] [t, t ] D R 3, are all strictly negative, i.e. λ = (35) { } 4y λ = Re < (36) { } 4y λ 3 = Re < (37) where c > is a constant. Therefore J(y ) is Hurwitz in the consiere omain. As a consequence, there eists a matri P(y ) = P(y ) T > an a Q(y ) > such that the stanar Lyapunov equation hols: P(y )J(y ) J(y ) T P(y ) = Q(y ) From stanar Lyapunov arguments, it follows that for all t [t, t ] { ( )} t ẑ(t, ε) c ep λ min (Q(y t )) ε Tikhonov s theorem, see [5], can now be avocate to summarize the previous result. Theorem : There eists a positive constant ε such that for all y D, an < ε < ε, the singular perturbation problem of (3)-(3) has a unique solution (t, ε), z(t, ε) on [t, t ], an (t, ε) (t) = O(ε) (38) z(t, ε) ẑ (t/ε) = O(ε) (39) hol uniformly for t [t, t ], where ẑ (τ) is the solution of the bounary layer moel (34). Moreover, given any t b > t, there is ε ε such that z(t, ε) = O(ε) hols uniformly for t [t b, t ] whenever, ε < ε. Etension of this result to infinite time interval, requires prove that the bounary layer system is eponential stable in a neighborhoo of the tune slow solution (t) for all t t. This may not be a trivial emonstration, an it will be left for further investigation. Instea, we emonstrate using simulation the effectiveness of this approach. VI. Simulations The following simulations are mae consiering = V, R = Ω, L = L = µh, C = C = µf. The esire output of the circuit is V out = 4 sin5t V. In orer to obtain this voltage, the parameters are a =., ω =.34, A =, k =. an η = η 4 =. The ellipse parameters result accoring to [3] are η = η 3 = 5.4, µ =.395. The estimate value of parameter a will be â =.77 (R = 3Ω), i.e., a 3% error. Fig. 3 shows the output voltage evolution. Note that the circuit achieves the esire behavior. The time scale is the real time scale, without variable change. The aaptation of the parameter a is represente in the Fig. 4, moreover, it tests the equation (39). The convergence of slow state variable (t, ε) (t) to O(ε) with ifferent ε is shown in the Fig. 5, (equation (38)). Notice the scale value for the vertical ais. V oltage(v ) Fig. 3. Output voltage with aaptation of a perturbation of a 3%

7 ā a) ā b) Appeni A. operation moes The two operation moes of the boost inverter are shown in Fig. 6. L i L V c C V o R Q Q V ON OFF Fig. 4. Time-evolution of the fast variable ā (soli) an ā (ashe) with ε =. in a) an ε =. in b) L i L V c V o R C Fig. 6. Operation status. Q Q V OFF ON a) b) Fig. 5. Convergence of with ε =. in a) an ε =. in b) VII. Conclusions An aaptive control for unknown loa is presente for a nonlinear boost inverter. The metho is base on using a state observer 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 relationship between the aaptation gain, γ, the observer matri parameter, α, an the perturbe variable parameter, 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 etension of this result to infinite time interval. Acknowlegments This research has been partially supporte by the MCyT-FEDER grant DPI The equations of the system are L i L t = u v C (4) v C C = u i L v C t R v C (4) R where u is a continuous variable that control the transistor states. In orer to simplify the stuy, the system (4) (4) is normalize as () () by using the change of variables: = L C i L, = vc an efining the new time variable with t = LC t an a = R L C. B. Control law propose in [3] k (, a) = a 3a 4 a 4 a ẋ 4 a a 4 kγ (η η ) ω (η η ) a a 4 k (, a) = a 4 3a 4 a a 4 ẋ 4 a 3 4 a 3 kγ (η 4 η 4 ) ω (η 3 η 3 ) 4 a 3 4 a 3 References [] F. Gorillo, D. Pagano, an J. Aracil, Autonomous oscillation generation in electronic converters, in Proceeings of the 4 International Workshop on Electronics an System Analysis, IWESA 4, 4, pp [] D. J. Pagano, J. Aracil, an F. Gorillo, Autonomous oscillation generation in the boost converter, in Proceeings of the 6 th IFAC Worl Congress, 5. [3] C. Albea, F. Gorillo, an J. Aracil, Control of the boost cac converter by energy shaping, in Proceeings of the 3 th Annual Conference of the IEEE Inustrial Electronics Society, IECON6, 6. [4] R. O. Cáceres an I. Barbi, A boost DC-AC converter: analysis, esign, an eperimentation, in IEEE Transactions on Power Electronics, vol. 4, 999, pp [5] H. K. Khalil, Nonlinear Systems, thir eition e. Prentice Hall,.