Sliding Dynamics Bifurcations in the Control of Boost Converters

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1 Sliding Dynamics Bifurcations in the Control of Boost Converters Enrique Ponce Daniel J. Pagano Dep. Applied Math., Univ. de Sevilla, Spain ( Dep. of Automation and Systems, Univ. Federal de Santa Catarina, Florianópolis, Brazil ( Abstract: Dynamicalsystemstechniquesareexploited toassessthe qualityofswitchingcontrol strategies leading to sliding model controllers (SMC). A boost converter controlled by SMC through a washout filter is considered. The crucial role of possible bifurcations in the ideal sliding dynamics is stressed.the possible reduction ofthe 3D-sliding dynamics to that ofcertain topologically equivalent planar standard vector field is emphasized so that the existence of Hopf bifurcations in the ideal sliding dynamics can be rigorously shown. Both a deeper insight in the global behavior of the controlled system and valuable design criteria are obtained. Keywords: Sliding mode control, qualitative analysis, converters. 1. INTRODUCTION In the analysis of nonlinear control systems, as dynamical systems, one naturally looks first for constant solutions associated to equilibrium points, since one of these equilibriaisthedesiredoperatingpoint.infact,otherinvariant sets for the dynamics, namely limit cycles, homoclinic or heteroclinic connections, are also relevant from the point of view of robustness issues and global dynamics. Once designed a controller, the occasional separation of parametersfromtheir nominal values can lead to dramatic changes in the dynamical behavior through different bifurcations, see for instance Kuznetsov [004. Thus, the parameter ranges where the local (and global, hopefully) dynamics is satisfactory should be found out. Extremes of these parameter ranges are typically bifurcation values: thedeterminationofsuchcriticalparametervaluesandthe characterization of the involved bifurcations are crucial to getaperspectiveofthedynamicalbehaviortobeexpected. Roughly speaking, the parameter nominal values farther from their critical values, the system is more robust. The geometric theory of dynamical systems and in particular bifurcation theory for smooth systems have achieved a rather satisfactory maturity degree in the last decades. This is not the case for discontinuous or non-smooth differential systems, even the research effort is nowadays increasing, see Kuznetsov et al. [003, di Bernardo et al. [008, di Bernardo et al. [008. In the case of sliding mode control, few practitioners are aware of the information that can be gained through the bifurcation analysis of the slidingdynamics,perhapsdue tothe difficulties arising in the analysis of the sliding vector field. In this paper, we take advantage of a methodology previously developed in Pagano and Ponce [010, to facilitate This work was supported in part by the spanish Ministry of Science and Innovation under grant MTM and by CNPq - National CouncilforScientific andtechnological development/brazil under grant PQ / the quoted analysis and qualify the robustness of a sliding modecontrollerforaboost converter,usingamorerealistic model than those previouslyanalyzedin the literature (see Sira-Ramírez [006). As usual in bifurcation analysis, we pay attention mainly to stability and robustness issues. Oncedefined the controlstrategythroughawashoutfilter, we arrive at a 3D discontinuous dynamical system; the relevant dynamics is to be confined to the sliding surface, which is in fact a plane. Next, it is shown how to reduce the dimensional sliding dynamics to another topologically equivalent dynamics determined by a standard planar vector field, getting so practical design criteria to choose control parameters. The existence of a subcritical Hopf bifurcation in the sliding dynamics is rigorously shown, in a similar way to what happens for a different, less efficient control system, see Ponce and Pagano [009a,b. The paper is organized as follows. In Section we give a short review of the relevant concepts in the analysis of discontinuous control systems through dynamical systems theory and bifurcation theory, fixing also the notation to be followed in the rest of the paper. Next, in Section 3 we recall the model of the boost converter and the proposed sliding mode controller. In Section 4, we show how the methodology proposed permits to arrive at useful design criteria. Some conclusions are offered in last section.. SMC CONCEPTS AND NOTATION In this section, we introduce the notation followedthrough the paper along with some elementary concepts about discontinuous control systems, see Pagano and Ponce [010 for more details. We start by considering affine control systems of the form ẋ = f(x)+g(x)u (1) where x R n and the functions f(x) and g(x) 0, are smooth and the control signal u is supposed to be a scalar discontinuous function. Assume a smooth non-constant scalar function h : R n R that defines the discontinuity Copyright by the International Federation of Automatic Control (IFAC) 1393

2 manifold Σ = {x R n : h(x) = 0}, supposed to be regular, that is, h(x) 0, x R n, and splitting the state space into two open regions S = {x R n : h(x) < 0} and S + = {x R n : h(x) > 0}. Accordingly,the switching control law is taken { u u = u(x) = (x), if h(x) < 0, i.e. x S, u + (x), if h(x) > 0, i.e. x S + (), where u (±) are scalar smooth functions of x (typically constant ones) to be later specified. System (1) endowed withthecontrollaw()definesthenon-smoothvectorfield f (±) (x) = f(x)+g(x)u (±) (x) for x S (±). As usual, we look for a stable operating point ˆx, belonging to the discontinuity manifold Σ. We will need the derivative of h along the different orbits when extended continuously to the boundary of the open regions S (±), that is, the orbital derivative of h or Lie derivative given by L f h(x) = h(x),f (x), for all x S = S Σ and the corresponding one for all x S + = S + Σ. Then it is natural to define the crossing part of Σ as the Σ- open set Σ c = {x Σ : L f h(x) L f +h(x) > 0}, and its complement in Σ, that is the Σ-closed set Σ s = {x Σ : L f h(x) L f +h(x) 0}, (3) which is normally called the sliding part Σ s. Of course, we are mainly interested in its attractive part, namely the Σ-open set Σ as = {x Σ : L f h(x) > 0, and L f +h(x) < 0}, (4) where the two vector fields from both sides out of Σ push orbits towards Σ. Clearly, points in Σ as satisfy L g h(x)u (x) < L f h(x) < L g h(x)u + (x). (5) A first goal is to define the controller for the desired operating point ˆx to belong to Σ as. The two conditions in (4) are not sufficient for stability purposes however, since there appears in Σ s a sliding dynamics induced by the interaction of the two vector fields. For robustness purposes, as stressed in the introduction, we must know as deeper as possible this sliding dynamics around the point ˆx Σ s and its possible bifurcations. According to (Filippov [1988) the sliding dynamics induced by the discontinuous vector field (1)-(), see also Sedghi [003, is given by the vector field f s (x) = λf (x)+(1 λ)f + (x), (6) where for each x Σ s the value of λ should be selected such that L fs h(x) = h(x),f s (x) = 0. Imposing such condition, the sliding vector field becomes f s (x) = L f +h(x)f (x) L f h(x)f + (x). (7) L f +h(x) L f h(x) Note that f s (x) = f(x)+g(x)u eq where u eq = L fh(x) L g h(x) = h(x),f(x) (8) h(x),g(x) is the so called equivalent control, see Utkin [199. We note that the transversality condition L g h(x) 0 is a necessary condition for the existence of u eq. Discontinuous system (1) inherits the equilibria of each vector field f ± (x), which can be real or virtual equilibria. In particular, we call (i) admissible or real equilibrium points to both the solutions of f (x) = 0 that belong to S and the solutions of f + (x) = 0 that belong to S + ; (ii) virtual equilibrium points are both the solutions of f (x) = 0 that belong to S +, and the solutions of f + (x) = 0 that belong to S. Virtual equilibria are not true equilibrium points, but they can play a role in the dynamics for the corresponding region. Regarding now the dynamical system corresponding to the vector field f s (x) induced on the sliding set Σ s, and following Kuznetsov et al. [003 we call pseudoequilibrium points to the solutions of f s (x) = 0, with x Σ s. Pseudo-equilibrium points are, in some sense, almost true equilibria for system (1). If L g h(x) = 0 for some x Σ s, then we have L f h(x) = L f +h(x), and from (3) this common value vanishes. Thus the point belongs to the boundary of Σ as, where the two tangency sub-manifolds of Σ intersect, and is called a singular sliding point. See Colombo et al. [009, Jeffrey and Colombo [009 for details about possible intricate dynamics around these points. Another interesting possibility from the point of view of bifurcation analysis is the appearance of boundary equilibrium points, that is, solutions of f (x) = 0 or f + (x) = 0 that simultaneously satisfy h(x) = 0. Such boundary equilibrium bifurcations give rise to changes in the sliding dynamics, and can be responsible for the appearance or disappearance of pseudo-equilibrium points, see for instance di Bernardo et al. [008. When boundary equilibrium bifurcations are not possible, pseudo-equilibrium points can also enter or escape from the attractive sliding set Σ as through singular sliding points, see Section 3. The ideal sliding dynamics is governed by the vector field (7) and typically involves fractional expressions with nonconstant denominators. We note that from (5) L f +h(x) L f h(x) = L g h(x)(u + u )(x) < 0, (9) so that, as usuallly u + (x) > u (x), we can assume that L g h(x) < 0 in Σ as, the attractive part of Σ s. Then, the following result easily follows. Proposition 1. Under the assumption L g h(x) < 0 for all x Σ as, the de-singularized sliding vector field f ds (x) = L f h(x)g(x) L g h(x)f(x), (10) and the sliding vector field (7) are topologically equivalent in Σ as, that is they have identical orbits and the systems are distinguished only by the time parametrization along theorbits.therefore,pseudo-equilibriaof (7)arealsoequilibria for (10) and the discontinuity manifold Σ remains invariant under the flow generated by f ds. Furthermore, if x is a singular sliding point of the vector field (7), that is, L f h(x) = L f +h(x) = 0, then it is a standard equilibrium point for the de-singularized sliding vector field, so that f ds (x) = 0. Proof. See Pagano and Ponce [010. Themathematicalexpressionofthe de-singularizedsliding vector field (10) is normally much easier to deal with than (7). Furthermore, the invariance of Σ under the flow determined by f ds allows us to reduce the dimension of the problem by one, to be usually done by projecting the dynamics in Σ onto one of the coordinate planes, see Section

3 E r L i L L q 1 Fig. 1. Circuit assumed for the DC-DC boost converter. Once assured the attractive character of Σ as, we can focus our attention on the f ds -dynamics on Σ, and specially on Σ as. Equivalently, we could start by analyzing such dynamics on Σ pas, which is less-dimensional by one. The usefulness of this methodology becomes clarified in next sections. 3. CONTROLLING THE BOOST CONVERTER The control of boost converters is a good exponent of the usefulness of bifurcation analysis in the design of non-smooth controllers. In Ponce and Pagano [009b an integral SMC was analyzed and some explanation for its possible chaotic behavior was given. Here, we continue the analysis of Pagano and Ponce [010 using the more realistic model that includes the effect of nonideal inductances. The basic circuit topology of the DC- DC boost converterisshownin Fig.1, whererstandsfora resistiveloadand E > 0 representsthevoltagesource.the voltage v out across R is the system output which should be driven to a regulated desired value v out > E. There are two switching elements in this circuit, to be represented by discrete variables. The value of q 1 {0,1} stands for the state of the MOSFET, and the variable q {0,1} corresponds to the diode state. The simultaneous conduction of the diode and the transistor is not possible: thus, the possible states for (q 1,q ) reduces to E 1 = (1,0), E 0 = (0,1), and E D = (0,0). Possible transitions are (E 1,E 0 ), (E 0,E 1 ), (E 0,E D ), and (E D,E 1 ). Here we will not study the last two possibilities related with the so called Discontinuous Conduction Mode (DCM), which leads to other possible scenarios, see for instance Ponce and Pagano [009b. The alternation of E 0 and E 1 structures leads to the Continuous Conduction Mode (CCM) of operation, with a non-null inductance current at any time. We take the inductor current i L and the capacitor voltage v C as state variables (the voltage output v out on the load R is given by v C ), and use the change of variables and parameters x = i L L E C, y = v C E, t = τ, a = 1 L LC R C, b = r L ar. Therefore we obtain the realistic dimensionless model q ẋ = (1 bx)q 1 +(1 bx y)q, ẏ = xq ay, C R (11) where q 1,q {0,1} (not simultaneously equal to 1; it is assumed that whenever one of the two discrete q-values is 1 the other is 0) and, due to the topology of the circuit, the states areconstrained to x 0 and y 0. The discrete variable q depends both on q 1 and the value of the states: it vanishes whenever q 1 = 1 and also when q 1 = 0 but x = 0 simultaneously with y > 1; otherwise, q = 1. We will endow the above circuit with a washout filter in order to achieve the usual control objectives, namely: (i) to regulate the desired value of the output voltage to y = y r > 1 in steady state; (ii) to ensure robustness under parameter variations (mainly, we must pay attention to the change of a produced by load changes of R); and (iii) to minimize as much as possible the transient response when a varies. A washout filter is a high-pass linear filter thatwashesoutsteady-stateinputswhilepassingtransient inputs (Lee and Abed [1991,Wangand Abed [1995)and seems more efficient than the integral SMC considered in Ponce and Pagano [009b. We suppose that the inductor current x can be filtered to get a new signal x F, and that the transfer function of the washout filter is G F (s) = X F(s) X(s) = s s+w = 1 w s+w, where w denotes the reciprocal of the filter constant and x F is the filter output. Thus we must add to the above model the differential equation ż = w(x z), where z is a new state that satisfies the output equation x F = x z. Now, for x = (x,y,z) R 3 we take the switching surface Σ = {x R 3 : h(x) = y y r + k(x z) = 0}, where y r is the desired normalized voltage and k is a control parametertobeadequatelyadjusted.weadoptthecontrol law u = 1 q 1, so that u = 0 if h(x) < 0, while u + = 1 whenh(x) > 0.Therefore,thecompleteCCMmodelofthe system with this SMC control law is built by combining the two vector fields f + (x) = [ 1 bx y x ay w(x z), f (x) = [ 1 bx ay, w(x z) according to the sign of h(x). Note that f(x) = f (x) and g(x) = [ y,x,0 T, so that L g h(x) = x ky. The equilibrium points for the vector fields f + and f (to be admissible or virtual, depending on their position with respect to Σ) are ( ) ( a e + = 1+ab, 1 1+ab, a 1, and e = 1+ab b,0, 1 ) b respectively.obviouslythe lastpoint cannotbeconsidered in the case b = 0. From 1 + ab > 1 and y r > 1, we have h(e + ) < 0, so that e + is a virtual equilibrium point. However h(e ) = y r < 0, and then e is a real or admissible stable equilibrium point, with negative eigenvalues b, a and w. In principle this is not good for our purposes but, since b is rather small, it is far from the desired operating point, and so the SMC strategy can lead to another attractor with a sufficiently big basin of attraction, as it will be seen below. 3.1 The sliding vector field and its desingularization Equation (7) shows that the sliding vector field is 1 f s (x) = x bx kwxy ay +kwyz k [ (b+w)x x+ay wxz, (1) x ky w(x z)(x ky) to be used only within the attractive set Σ as, where the two involved vector fields push orbits towards Σ. Here, 1395

4 Σ as is defined by the condition h(x) = 0, along with the inequalities L f h(x) = k[1 (b + w)x + wz ay > 0 and L f +h(x) = L f h(x)+x ky < 0, both implying that L g h(x) = x ky < 0. From its last component, equilibria for f s satisfy x = z, and then the condition h = 0 assures that y = y r as desired;furthermore,theirx-coordinatemustbe asolution of the quadratic bx x + ay r = 0. If b = 0 there is only one pseudo-equilibrium point at p = (ay r,y r,ay r ). When b > 0 there appear two pseudo-equilibrium points p (±) = ( 1±γ b,y r, 1±γ ) b where we have introduced the auxiliary value 0 < γ < 1, such that γ = 1 4aby r. The desired operating point in the non-ideal inductance case is p, which clearly satisfies p p as b 0. Note that for the existence of the operating point we must assume necessarily the design conditions ab = r L R < 1 4, and 1 < y r < 1 = 1 R. (13) 4ab r L Forinstance,areferencetypicalset ofcircuitparametersis E = 48V; V out = 96V; R = 40Ω; L = 1.4 mh; C = 10µF; and r L = Ω. This gives for the dimensionless boost model a = 0.958;b= 0.01;y r =.Then 1/ 4ab and γ = 1 4aby r so that γ and p ( ,, ). To study the steady state dynamics of the controlled system and its possible bifurcations, we will use Proposition 1 and start from (10) by writing the topologically Σ as - equivalent de-singularized version of (1), that is, f ds (x) = bx x+ay +kwy(x z) k [ bx x+ay +wx(x z), (14) w(x z)(x ky) which possesses the same equilibrium points than (1) but also some others out of Σ as satisfying the condition x ky = 0, to be considered later. 4. THE EQUIVALENT PLANAR VECTOR FIELD Theanalysisismucheasier,asstressedbefore,ifweproject the orbitsof (14)onto the plane z = 0, byconsideringonly the first two component of the vector field and eliminating z from the condition h(x) = 0. We arrive so to the planar vector field [ẋ [ = ẏ bx x+ay wy(y y r ) k ( bx x+ay ) +wx(y y r ), (15) to be significative only in Σ pas, projection of Σ as, namely {(x,y) : 0 < k(1 bx)+w(y y r ) ay < ky x}. (16) System (15) has not only the equilibrium points corresponding with the projections of those for (1), but also there appear new ones through the desingularization process, namely the origin and q d = (x d,y d ), where y d = (wy r k)/(w a bk ) and x d = ky d. The last point comes from the sliding singular point at the corner of the boundary of Σ as, as predicted by Proposition 1. Note that both new equilibria are located on the line x ky = 0. We will not analyze these two points in detail because they are out of Σ pas. It should be noticed however, that these additional equilibria help to understand the global dynamics in Σ pas, as shown below a HC H H sub HC a = Fig.. Bifurcation set for b = 0, w = 5, y r =. The dotted line a = is a reference value. The stability of the different equilibria of (15) can be deduced from the Jacobianmatrix. Since we have alwaysy = y r, we obtainthe determinantd(x,y r ) = w(1 bx)(ky r x), and trace T(x,y r ) = (w+b)x aky r 1. Standard local stability conditions will require (i) D(x,y r ) > 0 and (ii) T(x,y r ) < 0. Next, we do a bifurcation analysis by assuming all the parameters fixed excepting k and a. We first see the case b = 0, which is less involved. 4.1 Ideal inductance case (b = 0) Here the design conditions (13) are automatically fulfilled, and the desired operating point is (x,y) = (ay r,y r ). First, we check the conditions for the equilibrium to be in Σ pas given in (16), namely 0 < k ay r < ky r ay r = y r (k ay r ). Both inequalities hold whenever k > ay r, so that this is a necessary design condition. In fact, at k = ay r there appears a transcritical bifurcation where the two points (ay r,y r ) and (x d,y d ) collide. The pseudoequilibrium point p enters the region Σ as as a result of this bifurcation, after passing through the singular sliding point q d, the corner of Σ as. In our reduced differential system (15) however, this bifurcation is standard. Note that for ay r < k we already have D(ay r,y r ) > 0. Apart from ay r < k, the stability conditions also require k > k c = awy r 1 ay r, (17) thatcomesfromt(ay r,y r ) < 0.Inshort,thelocalstability of p asks for ay r < k, ay r (wy r k) < 1. A specific bifurcation set in the parameter plane (k,a) is depicted in Fig., showing the straight line k = ay r (labelled T, and drawn in red) that corresponds to D(ay r,y r) = 0 and the H sub curve corresponding to T(ay r,y r ) = 0. At the points of the H sub curve, it is rigorously shown in the Appendix that a subcritical Hopf bifurcation takes place. Thus, at the right of this curve, we have the (local) stability operating region, i.e. a stablepseudo-equilibriumpointsurroundedbyanunstable limit cycle. The H sub curve intersect the T-curve in two different non-tranversal Bogdanov-Takens co-dimensiontwo points, which organize almost all the bifurcations shown. The limit cycle dies in the right line of homoclinic connections HC which joins both codimension-two points. Such curve of homoclinic connection bifurcations can be obtained by numerical continuation software like AUTO. For a = 0.958(the horizontalline of Fig. ) the point p T T k 1396

5 (1.183,,1.183) becomes stable after a Hopf bifurcation for k > k c A value of k around 6 would be much better, since the basin of attraction bounded by the unstable limit cycle is bigger. We could take even greater values of k to the right of HC curve. In any case we should first analyze the effect in the dynamics of possible excursions out of Σ, by entering in the Discontinuous Current Mode, see Ponce and Pagano [009b. 4. Real inductance case (b > 0) a γ = 0 TB T SN (±) Now the design conditions (13) should be assured in order to get the possibility of pseudo-equilibria. The two equilibria p ± which are born for a = a SN = 1/(4byr ), that is for γ = 0, are equal for a = a SN to separate for a < a SN. At the critical value both have x = 1/(b) and after substituting in (16) we have 0 < k 1 4by r < ky r 1 b = y r ( k 1 ), 4by r so that the equilibria are born within Σ pas only if k > 1/(by r ). Thus we have a fold bifurcation of pseudoequilibria in Σ pas at a = a SN whenever k > 1/(by r ). We now check the possibility of entering or escaping the set Σ pas for each equilibrium point separately. For both points we have x(1 bx) = ay r, and since inequalities in (16) lead to 0 < ay r (ky r x)/x < ky r x, each projected pseudo-equilibrium point belongs to Σ pas if and only if ay r < x < ky r. The firstinequality isalwayssatisfied forboth points p (±), after simple algebraicmanipulations leading to the equivalentcondition1±γ < y r.regardingthesecondinequality, it should be noticed that when satisfied the equilibrium correspondingwithp haspositivedeterminant,beingthe determinant negative for p +. Thus the last point cannot be stable within Σ pas and will not be considered anymore. For the operating point p, easy computations show that it belongs to Σ pas for a < a SN whenever k > ay r or, if k < ay r when by r k k +ay r < 0, that is for parameter values to the right of the half-parabola in Fig. 3. At the points of this curve we have transcritical bifurcations: the operating point enters Σ pas through its corner q d. Recalling the condition for the trace, the local stability of p requires k > k c, where k c = (b+w)x 1 = w (b+w)γ. (18) ay r 4aby r For moderate values of a, this stability condition is again fulfilled through a subcritical Hopf bifurcation, as shown in the Appendix. Thus, when p becomes stable there is one unstablelimit cycle surroundingthe point in Σ as. This limit cycle defines within Σ as the basin of attraction of the pseudo-equilibrium point so that its stability is only local. Similar to the b = 0 case, the H sub curve intersect the T-curve in two different non-transversal Bogdanov-Takens co-dimension-two points. Asignificativebifurcation set in the parameterplane (k,a) is shown in Fig. 3. Only the bifurcation curves relevant for p are drawn. There the straight line (dashed line) γ = 0 corresponds with the saddle-node bifurcation (SN) at a = 1/(4by r ). The stabilityparameterregionliestothe 1H sub Fig.3.Bifurcationsetforp inthe parameterplane(k,a), for b = 0.01,w = 5,y r =. rightofthe H sub curve(in blue)andbelowthetranscritical bifurcation semi-parabola (in red). A qualitative reduction in the stability parameter region for the case b > 0, when comparing the two bifurcation sets of Figs. and 3, should be noticed. In particular, such region turns out to be bounded with respect to admissible values of the parameter a. In any case, a robust design requires to fix the operating point within the parameter stability region and sufficiently far from the bifurcation boundaries of this region. While the parameter a has a nominal value subjected to the possible variations that appear from external perturbations (load changes, mainly), the parameter k is to be chosen in advance by the designer. For this task, the above bifurcation sets constitute essential tools. k 5. CONCLUSION A useful methodology based on bifurcation theory to determine the stability and robustness of sliding mode controllers has been illustrated through a realistic problem. New concepts have been exploited; in particular, the de-singularized sliding vector field and its dimensional reduction via projection have been shown to be crucial for alleviate the computational effort for the analysis. The determination of the corresponding bifurcation sets leads to practical rules for choosing parameters in order to achieve a robust control design. In looking for both global stability conditions and performance in transient responses, we are not done: chattering avoiding devices, for instance, are needed. Anyway, the bifurcation analysis must not be underestimated since leads to a considerable insight about the dynamics of sliding mode controllersand how to modify it. REFERENCES A. Colombo, M. di Bernardo, E. Fossas, and M. R. Jeffrey. Teixeira singularities in 3D switched feedback control systems. Systems & Control Letters, submitted. F. B. Cunha, D.J. Pagano, and U. F. Moreno. Sliding bifurcations of equilibria in planar variable structure systems. IEEE Transactions on circuits and systems I: Fundamental Theory and Applications, 50 (8), ,

6 M. di Bernardo, D.J. Pagano and E. Ponce. Nonhyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach. International Journal of Bifurcations and Chaos, Vol. 18, No 5, , 008. M. di Bernardo, C. Budd, A.R. Champneys and P.S. Kowalczyk. Piecewise-smooth Dynamical Systems: Theory and Applications. Springer, Applied Mathematical Sciences 163, 008. A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, Dordrecht, The Netherlands. (Original in Russian, 1960). M. R. Jeffrey and A. Colombo. The two-fold singularity of discontinuous vector fields. SIAM J. Appl. Dyn. Syst. 8, , 009. Y. A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer-Verlag, 004. Y. A. Kuznetsov, S. Rinaldi, and A. Gragnani. Oneparameter bifurcations in planar Filippov systems. International Journal of Bifurcations and Chaos, 13 (8), , 003 H.C.LeeandE.H.Abed. Washoutfilterinthebifurcation control of high alpha flight dynamics. Proc. of the American Control Conference, vol. 01, pp , Boston, MA, D.J. Pagano and E. Ponce. Sliding Mode Controllers Design through Bifurcation Analysis. Preprints of 8th IFAC Symposium on Nonlinear Control Systems NOL- COS 010. University of Bologna, Italy, September 1-3, 010, E. Ponceand D.J. Pagano. Hopf Bifurcation in the Sliding Dynamics for a DCM Boost Converter under SMC strategy. IWDC 009, 15th International Workshop on Dynamics and Control, J. Rodellar and E. Reithmeier (Eds.), Barcelona, 009a. E. Ponce and D.J. Pagano. Chaos through Sliding Bifurcations in a Boost Converter under a SMC strategy. CHAOS09, Second IFAC meeting related to analysis and control of chaotic systems. London, June nd 4th, 009b. B. Sedghi. Control Design of Hybrid Systems via Dehybridization. Thesis No 859, EPFL, Lausanne, 003. H. Sira-Ramírez and R. Silva-Ortigoza. Control Design Techniques in Power Electronic Devices. Springer,006. V.I. Utkin. Sliding Modes in Control Optimization. Springer, 199. Wang, H. and E. H. Abed. Bifurcation control of a chaotic system. Automatica. vol. 31, pp , Appendix A. SLIDING HOPF BIFURCATIONS Here, we show that sliding Hopf bifurcations are always subcritical, that is, one unstable limit cycle bifurcates for k > k c, bounding the attraction basin of the corresponding pseudo-equilibrium point. For the notation and more details, see Section 3.5 of Kuznetsov [004. Recall that for b = 0 the pseudo-equilibrium point is ( x,ȳ) = (ay r,y r ) on the set Σ as and k c is given in (17), while for b > 0 the relevant pseudo-equilibrium is p with x = (1 γ)/(b) and the critical value of k c is given in (18). First, we apply in any case the change of variables x = x x, ỹ = y y r to put the equilibrium at the origin. The system can be expressed, dropping the tildes, as [ẋ [ [ γ (a w)yr x bx = + ẏ kγ w x akyr[ +(a w)y y wxy k(bx +ay, ) (A.1) where we separate the linear part at the new origin, from the nonlinear, quadratic part of the vector field, and from now on we assume that if b = 0 then γ = 1. Let us define as the bifurcation parameter α = k c k, so that α > 0 when k < k c. The eigenvalues for the linearization at the origin of system (A.1) are λ(α) = µ(α) ± iω(α), where µ(α) = ay r α and ω(α) = wγ(k c y r αy r x) a yr α. We see that µ(0) = 0 so that λ(0) = ±iω 0 = ±iω(0) with ω0 = wy r (k c ay r ), denoting a possible Hopf bifurcation at α = 0 (k = k c ). Since µ (0) = ay r > 0, the transversality condition for the eigenvalues is clearly satisfied. If A(α) is the linear part of system (A.1) in terms of the bifurcation parameter α, then its critical value is [ γ (a w)yr A(0) =. k c γ γ Now,weselectsomenormalizedright-andleft-eigenvectors q,p of A(0) for λ(0) = iω 0, so that A(0)q = iω 0 q, A(0) T p = iω 0 p and p,q = p T q = 1, where, stands for the standard scalar product in C. We choose [ [ 1 iγ +ω0 kc γ q =, p =, k c γω 0 ik c γ γ iω 0 and so p,q = 1. We now calculate the coefficient c 1 (0) = i (g 0 g 11 g ) ω g 0 + g 1 0. The sign of its real part determines the character of the Hopf bifurcation. Here g kl stands for the coefficient of the term Z k Zl in the scalar equation which results from the use of the complexified variable Z = k c γx+(γ +iω 0 )y. Our system is quadratic, so that g 1 = 0. We only need g 0 = p,b(q,q) = k c γb 1 (q,q)+(γ +iω 0 )B (q,q), g 11 = p,b(q, q) = k c γb 1 (q, q)+(γ +iω 0 )B (q, q), where B stands for the bilinear form associated to the quadratic terms of the system for the critical value k = k c. Computations give rise to B 1 (q,q) = q T [ b 0 0 (a w) [ B 1 (q, q) = q T b 0 0 (a w) w w ak c B (q, q) = q T [ bkc and B (q,q) = ak c ω0 +w γ iω 0 k c γω0 q = w a ω 0 +b (ω 0 +iγ) kcγ ω0, q = a w ω0 +b γ +ω0 kcγ ω0, q = b+ak c +w k c ω0 b k c γ b (ω 0 +iγ) k c γ ω0. Thus, after standard computations, we have for R(c 1 (0)) the expression 1 8k c ω4 0 γ3 ( (b+w)bω 4 0 +(b +3bw+w )ω 0γ + +(b+w)ak c ω 0 γ +(b+ak c +w)(1+k c )wγ4). Since all the coefficients are positive, we conclude that R(c 1 (0)) > 0, and the subcritical character of the Hopf bifurcation is shown. 1398