Hybrid Modelling and Control of a Class of Power Converters With Triangular-Carrier PWM Inputs

Received September 30, 2021, accepted October 28, 2021, date of publication November 8, 2021, date of current version November 16, 2021. Digital Object Identifier 10.1109/ACCESS.2021.3126433 Hybrid Modelling and Control of a Class of Power Converters With Triangular-Carrier PWM Inputs CAROLINA ALBEA 1, ANTONINO SFERLAZZA 2, (Member, IEEE), FABIO GÓMEZ-ESTERN 3, AND FRANCISCO GORDILLO 1 1Department of System Engineering and Automatic Control, Universidad de Sevilla, 41092 Seville, Spain 2Department of Engineering, University of Palermo, 90128 Palermo, Italy 3Department of Engineering, Universidad de Loyola Andalucía, Seville, 41704 Dos Hermanas, Spain Corresponding author: Carolina Albea ([email protected]) This work was supported in part by the Project PID2019-105890RJ-I00 and Project PID2019-109071RB-I00, in part by MCIN/ AEI /10.13039/501100011033/, in part by the ERDF A way of making Europe, in part by the Agence Nationale de la Recherche (ANR)–France under Grant ANR-18-CE40-0022-01, and in part by the ECSEL Joint Undertaking into the Project Frame of REACTION under Grant 783158. ABSTRACT In this paper, a new control design procedure for a class of power converters based on hybrid dynamical systems theory is presented. The continuous-time dynamics, as voltage and current signals, and discretetime dynamics, as the on-off state of the switches, are captured with a hybrid model. This model avoids the use of averaged and approximated models and includes the PWM as well as the sample-and-hold mechanism, commonly used in the industry. Then, another simplified hybrid system, whose trajectories match with the original one, is selected to design the controller and to analyse stability properties. Finally, an estimation of the chattering in steady state of the voltage and current signals is provided. The results are validated through simulation and experiments. INDEX TERMS Control of power converters, PWM, switched affine systems, hybrid dynamical systems, Lyapunov stability. I. INTRODUCTION Converter control has been widely studied by the electronics and control communities. Most of these studies deal with designing continuous-time control laws whose outputs are discretized by modulators. The most common modulator is the Pulse-Width Modulator (PWM) [1], [2]. Very often, this modulator is not considered in the converter control design and in the subsequent stability analysis. Instead, continuoustime approximate models, such as averaged models [3]–[5] are commonly used. These approaches have solved many practical problems and provided reasonably solid theoretical grounds, but these results might appear superficial when compared with the depth of analysis reached in other areas nowadays. Probably, the most relevant limitations are both the difficulty of quantifying the precision of the approximation resulting from the averaging procedure, and the fact that the properties of the control laws are only valid locally. In this way, it is known that PWM blocks have an effect on the output, [6], [7]. Mainly, the problem arises when the control The associate editor coordinating the review of this manuscript and approving it for publication was Zheng H. Zhu . designer ignores the discrete character of the signal, which is maintained constant for an elapsed time, possibly causing output jitter in steady state. Recently, the control community has devoted efforts to the study of new hybrid control techniques [8], such as sliding mode control [9] or model predictive control [10] applied to power converters with the possibility of considering the continuous-time dynamics (voltage and current signals) and discrete-time dynamics (the functioning mode of the switches). This class of hybrid systems is modelled by switched affine state-space equations, where the constant matrices change depending on the state of the switches. Other hybrid controllers are presented in [11], [12], where the unique control action is the selection of an operating mode among a finite set of possibilities. In the latter reference, the problem is formulated in terms of the control of switched systems, whose modes are described by affine differential equations. However, it is possible to show that the obtained switching rule can be interpreted as a sliding mode control law where the sliding surfaces are implicitly determined in terms of the state space variables (current, voltage) and of the selected operating point [13]. On the other hand, VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 151607 C. Albea et al.: Hybrid Modeling and Control of Class of Power Converters relevant results on Hybrid Dynamical System (HDS) theory [14]–[18] have been applied to power converters. It also is worth to mention the design of hybrid controllers with minimum dwell-time guarantees [19]. Moreover, in [20]–[22] DC-DC as well as DC-AC power converters were controlled using this approach. The main feature in these references is the implementation of an aperiodic sampled-data based control signal with arbitrarily fast switching, and the possible induction of a Zeno behaviour. This issue is solved in [23], [24], where suitable formalisms relying on controlled switches with control inputs updated in a periodic or aperiodic manner are presented, without considering PWM. A hybrid control using HDS theory was proposed in [25] for a DC-AC converter. However, with most of these variants, which do not include PWM in the controller, the state of the switching devices can only be changed at the sampling instants. Notice that when a PWM is used, the manipulated control input can change in any instant inside the sampling period, with the only constraint in the maximum number of commutations inside this time interval. When the use of PWM is avoided, it is necessary to increase the sampling frequency to preserve a suited performance. Thus, the consideration of PWM in power converters using HDS theory is of interest. In [26], under some assumptions, it is proved for general systems that the solutions of an averaged system are suited approximations of the original one composed of a PWM. They mention power converters as systems that benefit from this result. This paper proposes a new control law based on HDS models, avoiding the use of classical approximations derived from averaged systems and guaranteeing stability properties including non-linearities as PWM and sampling-and-hold mechanisms. A preliminary work regarding the direct design of controllers for power converters with PWM based on HDS is [27], where power converters with switched affine models are considered together with the PWM and sampling-and- hold mechanisms. Unlike this paper, •the PWM carrier considered thereby was a sawtooth signal. Although this requires simpler analysis, it is not the most usual carrier signal in PWM for power converters, since it is well known that triangular carriers yield better results in many applications, in terms of harmonic distortion [28], [29]. •The study using HDS theory of triangular carriers for PWM is more involved than the one for sawtooth signals, due to the fact that the number of commutations in a PWM interval is increased (it is almost doubled). As it will be shown here, it is not only necessary to enlarge the jump set for the hybrid model, but the number of state variables must also be increased. •A fictitious, simpler system is introduced in the present paper, and its behavior is proven to match the original system’s at the sampling instants. This result is used to design a control law with stability guarantee for this system and to extend its validity to the original one. •An estimation of the chattering in the steady-state signals is provided for the original hybrid model in the present work. Experimental results verify the validity of the proposed control loop. Moreover, these validations are extended with an external loop to guarantee voltage output regulation, as it is commonly done in this kind of converters [30], [31]. This paper is organized as follows. The problem statement is given in Section II. Then, the hybrid general model of triangular-carrier PWM-based converters is presented in Section III. The main result is presented in Section IV, and Section Vprovides a discussion about parameter tuning effects. Section VI and Section VII present simulated and experimental results, respectively. Finally, the paper closes with a conclusion section. Notation: Throughout the paper Ndenotes the set of natural numbers and Rthe set of real numbers, Rnthe n-dimensional Euclidean space and Rn×mthe set of all real n×mmatrices. The set of non-negative real numbers is denoted by R≥0.M0 (resp. M≺0) represents that Mis a symmetric positive (resp. negative) definite matrix. 0 and I are the zero matrix and identity matrix respectively, of suited dimension. The operator k·krepresents the Euclidean norm. He(M) is the Hermitian matrix of M, i.e. He(M)=(M+M>). Finally, satb a(φ) is the standard saturation function defined in R7→ [a,b]. II. PROBLEM STATEMENT Many switched power converters can be modelled as switched affine systems, as follows ˙z=Aσz+Bσ,(1) where z∈Rncontains the state variables, i.e. the continuoustime evolutions of the voltages and currents and σ∈ {0,1,2,...,N−1}is the control input that represents the switching between the modes of the converter. Finally, Aσand Bσare matrices of suitable dimensions. Model (1) covers many applications of power converters, such as the classical converters: buck, boost converter, quadratic boost converter, half bridge converter, boost inverter, etc. This set of power converters is modelled with two functioning modes, N=2, and this is the case considered here, while it is possible to extend the results to converters with N>2. Generally, one can find in the literature that these systems are governed by continuous-time control laws [7], [32]–[34], i.e., σ∈ {0,1}is modeled by continuous signals λ∈[0,1], obtained by using averaging approaches, and implemented in (1) by PWM, as depicted in Fig. 1, where κ(x) represents a continuous control law whose output is limited to the interval [0,1]. Furthermore, the control law is usually implemented in a digital device in discrete time by sampling the state of the converter z(t) periodically. Thus, the value of λin Fig. 1is constant during each sampling interval. The PWM mechanism with a triangular carrier is illustrated in Fig. 2. For the sake of simplicity, in this paper, it is assumed that the 151608 VOLUME 9, 2021 C. Albea et al.: Hybrid Modeling and Control of Class of Power Converters beginning of the triangular carrier period coincides with the time instants when the variables are measured, and the duty cycle is obtained from the controller equations. FIGURE 1. Feedback scheme. FIGURE 2. PWM mechanism. Top: Tp-periodic triangular carrier, s;λis the duty cycle, vis the slope of the carrier sand qis a binary signal, taking value 0 when sis depicted in red, and 1 when in blue. Bottom: output of the PWM block, σ. The control objective is to design a function λ=κ(x): Rn7→ Rsuch that the control signal σ(λs), modelled by a triangular carrier, ensures the convergence of zto a given operating point ze. Due to the sampling mechanism, asymptotic convergence to zeis not possible, and a chattering phenomenon is unavoidable. In any case, the desired operating point zemust satisfy the following assumption. Assumption 1: Given an operating point zethere exists a λ=λe∈[0,1] such that the following equation holds, 0=(A0+(A1−A0)λe)ze+B0+(B1−B0)λe.(2) This standard assumption for switched affine systems guarantees the existence of a switched signal for system (1), inducing an equilibrium in z=zein the generalized sense of Filippov [35]. This means that, in steady state, σis expected to be a periodic signal of period Tp, spending a time λTpin mode 1 and (1−λ)Tpin mode 0, corresponding to the convex combination of the right hand side of (2). Then, the time spent in each mode will be distributed in every sampling interval according to the used modulator. In this paper, it is assumed without loss of generality, that at t=0 a triangular carrier as well as a sampling period start. The error equation associated with (1) in each interval between sampling instants can be written as: ˙x=Aσx+Bσ,(3) where x:= z−zeand Bσ:= Bσ+Aσzesuch that Bλe=0, if Assumption 1is satisfied. Problem 1: Consider the switched system (1) with N=2 and a PWM with a triangular carrier, as shown in Fig. 2. Then, the goals here are •to model the closed-loop system considering its hybrid character, that is, the existence of both discrete-time and continuous-time signals, as well as the triangular-carrier PWM and sample-and-hold mechanism with a given periodic sampling time Tp. •To design a new control law for the duty cycle λ. •To achieve convergence of zto zeand to analyse stability properties for both hybrid systems. •To estimate the chattering of the voltages and currents in steady state. III. HYBRID DYNAMICAL MODEL In this section, the framework given in [14] about hybrid dynamical systems will be used to model the controlled system, considering continuous-time and discrete-time dynamics. Hence, the following hybrid dynamical model of the controlled switched system (3) is presented, considering a triangular carrier for the PWM mechanism, H:˙ ξ=f(ξ), ξ ∈C ξ+∈g(ξ), ξ ∈D,(4) being ξ=[x s v σqλ τ]>∈H, such that, H:= Rn× [0,1] ×{−2/Tp,2/Tp} ×{0,1}×{0,1}×[0,1] ×[0,Tp]. The maps fand gcapture the continuous-time and discretetime dynamics, respectively, and are defined as follows: f(ξ)=           Aσx+Bσ v 0 0 0 0 1           , g(ξ)=           x s v−2vq (2σ−1)q+(1 −σ) 1−q (1 −qσ)λ+qσsat1 0κ(x) (1 −qσ)τ           .(5) with ξ(0) =[x(0),0,2/Tp, σ(0),0, λ(0),0]. The following sets C1:= {ξ∈H:q=1,0≤s≤1} D1:= {ξ∈H:q=1,s≤0 or s≥1} C2:= {ξ∈H:q=0,sv ≤λv} D2:= {ξ∈H:q=0,sv ≥λv}, VOLUME 9, 2021 151609 C. Albea et al.: Hybrid Modeling and Control of Class of Power Converters define the so-called flow and jump sets C:= C1∪C2(6) D:= D1∪D2.(7) This hybrid scheme (4)–(7) gathers the complete dynamics of the system. Indeed, the continuous-time dynamics of x evolves according to the switching of σ. The selection of the latter is given by a triangular modulator defined by s,v,q and λ. It is easy to see that sis a continuous-time triangular carrier (the signal shown in Fig. 2); vdefines the slope of s; qis an artificial variable which takes values q:=            0 if (s< λ and v≥0 s≥λand v≤0 1 if (s> λ and v≥0 s≤λand v≤0. Variable λ∈[0,1] is the duty cycle, and κ(x) is the control law, to be defined, that computes the value of signal λ in every sampling time. Inside the sampling intervals, λis held constant. It is worth noting that the solutions to Hare generalized Krasovskii solutions, not only in the map f(ξ), but also in g(ξ). Solutions to H(f,G,C,D) are given on the well-known hybrid time domain: dom(ξ)⊂R≥0×N, such that, dom(ξ)= ¯ j−1 [ j=0 ([tj,tj+1],j),(8) for some sequence 0 =t0≤t1≤t2≤. . . ≤t¯ jwith ¯ jfinite (being a compact set) or infinite. In order to simplify the subsequent analysis, consider a second hybrid dynamical system, which is simpler than (4)–(7) and, as will be shown below, its trajectories match those of (4)–(7) at the sampling instants. Hp:(˙ ξp=fp(ξp), ξp∈Cp ξ+ p∈gp(ξp), ξp∈Dp,(9) where ξp=[xpλpτp]>∈Hp, such that, Hp:= Rn×[0,1]× [0,Tp]. The maps fpand gpcapture both the continuous-time and discrete-time dynamics and are defined as follows: fp(ξp)=  Aλxp+Bp 0 1 , gp(ξp)=  xp sat1 0κ(xp) 0 ,(10) where Aλand Bpcan be computed in different ways depending on the properties of the system. For this, consider a value of tcorresponding to a sampling instant (when the period of the triangular carrier starts), that is, q=1, s=0 in system (4)–(7). It is clear that at this moment system (4)–(7) jumps. Consider a value of ksuch that tkis equal to this time instant. By direct integration of the dynamics of xin Halong the triangular carrier period (Fig. 2), starting from an initial condition x(tk) yields1 x(tk+1)=eA1λ 2Tpx(tk)+(eA1λ 2Tp−I)A−1 1B1(11) x(tk+3)=eA0(1−λ)Tpx(tk+1)+(eA0(1−λ)Tp−I)A−1 0B0 (12) x(tk+Tp)=eA1λ 2Tpx(tk+2)+(eA1λ 2Tp−I)A−1 1B1,(13) being tk+1=tk+Tpλ 2and tk+3=tk+Tp(1 −λ 2). Then, the following assumption will explore different possibilities to compute Aλ(Bpwill be considered later), from (11)–(13) such that the following holds x(t+Tp)=eAλTpx(t)+(eAλTp−I)A−1 λBp.(14) Assumption 2: Using the following definition ϒ(λ):= eA1λ 2TpeA0(1−λ) 2TpeA0(1−λ) 2TpeA1λ 2Tp, which arises from rewriting (11)–(13) as x(tk+Tp)=ϒ(λ)x(tk)+(eAλTp−I)A−1 λBp, it will be assumed that one of the following conditions are verified by Tpand system matrices A0and A1. 1 The sampling period Tpis sufficiently small such that kϒ(λ)−Ik<1 for all λ∈[0,1]. 2 The matrices A0and A1are commutative, i.e. A0A1= A1A0. Under this condition, it is easy to prove that ϒ(λ)=e(A0+(A1−A0)λ)Tp. Now, for the definition of Aλ, there are two options. •Aλ:= log(ϒ(λ))/Tpif Assumption 2.1 holds, but Assumption 2.2 does not. The assumption guarantees the existence of the logarithmic matrix with real elements. •Aλ:= A0+(A1−A0)λif Assumption 2.2 holds. Remark 1: Note that in Assumption 2, the conditions would lead to exact results. However a less restrictive condition can provide an approximation of Aλ. Indeed, a firstorder approximation of the Taylor series expansion of ϒ(λ) with respect to Tparound Tp=0 yields ˜ ϒ(λ)≈e(A0+(A1−A0)λ)Tp=eAλTp. For the cases where these conditions are not satisfied, the search for an alternative to the matrix logarithm can be done using the Baker–Campbell–Hausdorff formula, but this will not be further investigated here. For the definition of Bpin (10), using Eqs. (11)–(13) gives, Bp:= (eAλTp−I)A−1 λ−1Bdλ=(eAλTp−I)−1AλBdλ =Aλ(eAλTp−I)−1Bdλ Bdλ:= eA1λ 2TpeA0(1−λ)Tp(eA1λ 2Tp−I)A−1 1B1 1This formulation is valid for λ∈(0,1) but can be extended to the cases λ=0 and λ=1 yielding the same results. 151610 VOLUME 9, 2021 C. Albea et al.: Hybrid Modeling and Control of Class of Power Converters +eA1λ 2Tp(eA0(1−λ)Tp−I)A−1 0B0 +(eA1λ 2Tp−I)A−1 1B1. For future uses, it is worth noting that, when Tp→0 (neglecting the second and higher order terms on Tp), this expression approaches to Bdλ→(B0+λ(B1−B0))Tp=BλTp,(15) where Bλhas been defined accordingly. The flow and jump sets, are now Cp:= {ξp∈Hp:τp∈[0,Tp]}(16) Dp:= {ξp∈Hp:τp=Tp}.(17) Note that system (9)–(10),(16)–(17), corresponds to a sampled system controlled with a sample-and-hold mechanism, and its relationship with system (4)–(7) is stated in the following proposition, which expresses that at the time instants when system Hpjumps, system Halso jumps and at these instants x=xp. For this, consider the pair (t,jp) such that system Hpjumps (subscript pin jhas been introduced in order to avoid confusion with the jumps of H). Proposition 1: Consider systems (4)–(7) and (9)–(10), (16)–(17), with the same initial condition, i.e., x(0,0) = xp(0,0) and with the same control law κ(x)=κ(xp), ∀x=xp. Moreover, consider that each subsystem is Hurwitz. Then, at the time instants when system (9)–(10) jumps, i.e., when ξp(t,jp)∈Dp, for a given jp, system (4)–(7) also jumps, i.e., there exists an integer jsuch that ξ(t,j)∈D. Furthermore, x(t,j)=xp(t,jp). Proof: We proceed by induction showing that if at a given time twhen system (9)–(10) jumps, x(t,j)=xp(t,jp) for certain values of jand jp, then at the following jump of this system, the state verifies that x(tj+Tp,j+4) =xp(tj+Tp,jp) (see Fig. 2). For this, integrating the dynamics of xpin (9)–(10) between two consecutive jumps and using Assumption 2the following equation is reached xp(t+Tp)=eAλTpxp(t)+(eAλTp−I)A−1 λBp.(18) From (14), and because at t=0 it holds that x(0,0) =xp(0,0). The proposition statement is proved by induction.  In the next section, a control law will be designed for system (9)–(10), (16)–(17) with stability guarantee. By Proposition 1, application of this control law to system(4)–(7) will inherit the stability property at sampling instants. The behaviour inside the time interval between sampling times will be analysed afterwards. In the sequel, possibly with abuse of notation, the symbols λand τ, instead of, λpand τpwill be used. Notice that the flow equation that governs the dynamics of xpin (9)–(10), (16)–(17), is ˙xp=Aλxp+Bp.(19) This equation plays a similar role to the one of averaged models used usually in power electronics. Indeed, the use of variable λinstead of σavoids the use of a discrete control signal. Nevertheless, the proposed approach presents important differences: •At sampling instants, the solution to (9)–(10), (16)–(17) matches exactly the solution to (4)–(7), as it has been stated in Proposition 1. In fact, the usual averaged model would be similar to (19) but with Aλ= A0+λ(A1−A0), instead of the logarithm definition above, and Bp=B0+λ(B1−B0), where the equality signs can be just an approximation depending on which case of Assumption 2 is satisfied. In the approximated case, some issues can appear, as has been reported in designs based in averaged modes [36], [37]. •Model (9)–(10),(16)–(17), takes care of the discretenature of the control action since the sample-and-hold mechanism is included in the model. Notice also that the non-hybrid model (18) (or its continuous version ˙xp=Aλxp+Bp) is not a ‘‘standard’’ affine problem since the transmission matrix eAλTp(or Aλ) depends on the control input λ. Now, rewriting the continuous and discrete-time dynamics of xpwill lead to a new Lyapunov function candidate. To do so, define 0q=hAqBq 0 0 i, such that during the flows: d dt xp 1=0qxp 1.(20) Hence, the following Lyapunov function candidate is considered V(xp, λ, σ, τ)=max W(xp, λ, σ, τ)−1,0,(21) where Wis a quadratic function of [xp1]>, defined as follows, W(xp, λ, σ, τ):= xp1Pσ(λ, τ)xp 1(22) with P0(λ, τ ):= e−0> 0τ¯ Pe−00τ P1(λ, τ ):= e−0> 1(λTp−τ)e−0> 0τ¯ Pe−00τe−01(λTp−τ) and ¯ P:= P0 0 0 , being P0 a symmetric matrix. We will see below that this Lyapunov function candidate enjoys nice properties. We are in position to define the compact attractor for which it is desired to establish uniform globally asymptotic stability (UGAS). This attractor set is A:= {ξp∈Hp:V(xp, λ, τ )=0}.(23) Note that if the solution evolves in the interior of Ain steady state, its associated xpis bounded. VOLUME 9, 2021 151611 C. Albea et al.: Hybrid Modeling and Control of Class of Power Converters The particular hybrid arcs starting from the origin after a jump are defined as follows: Eκ=ξp∈Hp:xp(τ, λ)=I0e0λ(λTp−τ)0 1,λ=sat1 0κ(0) . (24) Note that these arcs start from xp=0 and describe a solution of xin the time arcs, [tjp,tjp+1] with (tjp,jp)∈dom(ξ). IV. MAIN RESULT Inspired by [24], a controller is proposed for system (9)–(10), (16)–(17) providing stability guarantees of the compact attractor A. Theorem 1: Consider a λeassociated with the operating point xesuch that Assumption 1is satisfied and matrices P 0∈Rn,Q0∈Rnsuch that QPand MP−Q∈Rn satisfying A> 0P+PA0≺ −Q,(25) A> 1P+PA1≺ −Q.(26) Consider system (9)–(10), (16)–(17), with control law κ(xp)∈   λe1+x> pMxp 2B> 0Pxpif B> 0Pxp6= 0 [0,1] if B> 0Pxp=0, (27) Then, there exists a value T∗ p>0 such that for 0 <Tp< T∗ pthe following statements hold: (i) Ais UGAS. (ii) Eκis a subset of A. Proof: Hybrid system Hp(fp,gp,Cp,Dp) with control law (27) is well-posed, because it verifies: •Cpand Dpare closed sets in Hp. •fpis a continuous function, thus it is locally bounded and outer semi-continuous. Moreover, it is convex for each ξp∈Cp. •gpis outer semi-continuous and locally bounded. We will consider the proof item by item. Proof of (i): The proof of this item proceeds by applying [38, Theorem 1]. Note that the Lyapunov function candidate, V(xp, λ, τ ) (21) is continuous in Cp∪Dpand locally Lipschitz near each point in Cp\A. Moreover, V(xp, λ, τ) is strictly positive definite with respect to (Cp∪Dp)\A and radially unbounded. Likewise, it verifies, by definition, V(xp, λ, τ )=0, for all (xp, λ, τ) in A. The next step of the proof is to ensure that the time derivative of Valong flows outside of Ais non positive (or more precisely in this case, equal to zero). More formally, the objective is to show that h∇V(xp, λ, τ),f(xp, λ)i ≤ 0,∀(xp, λ, τ)∈Cp\A.(28) For any (xp, λ, τ)∈Cp\A, it is clear, from its definition, that V(x, λ, τ )=W(xp, λ, τ)−1 getting h∇V(xp, λ, τ),f(xp, λ)i =xp 1> ˙τ∂ ∂τ P(λ, τ)+˙ λ∂ ∂λP(λ, τ)  ×xp 1+2xp 1>P(λ, τ ) ˙xp 0 =xp 1>∂ ∂τ P(λ, τ)+P(λ, τ)He (0λ)xp 1=0. The last equality comes from ∂ ∂τ P(λ, τ)+P(λ, τ) He(0λ)=0. Next, let us analyse the second stability condition from [38, Theorem 1]. To do so, taking into account that the special structure of hybrid system (9)–(10) implies that the jumps occur periodically at the ordinary time instants t=jpTpfor jp∈N, the following notation will be adopted here according the hybrid time domain (8): xp,jp=xp(jpTp,jp), λjp= λ(jpTp,jp), τjp=τ(jpTp,jp) that corresponds to the variables right before the jump at t=jpTpand λ+ jp=λ(jpTp,jp+1), τ+ jp=τ(jpTp,jp+1) right after the same jump. In the same way, the definition 1V=V(x+ p,jp, λ+ jp, τ+ jp)−V(xp,jp, λjp, τjp) will be used. However, note that x+ p,jp=xp,jp,τ+ jp=0 and τjp=Tp. Now, ∀(xp, λ, τ)∈Dp\Athe following equations hold, 1V=W(xp,jp, λ+ jp,0) −W(xp,j, λjp,Tp) =xp,jp 1>¯ P−9> λjp¯ P9λjpxp,jp 1 =xp,jp+1 1>¯ P−9> λjp+1 ¯ P9λjp+1xp,jp+1 1 being 9λjp=e−00Tpe−01(λTp−Tp). Notice that the manipulable signal, λjp+1=λ+ jphas to be computed at t=jpTp. For this, it is convenient to write 1Vin terms of xp,jpinstead of xp,jp+1. Hence, from (20), the following relationship is obtained xp,jp+1 1=9λ+ jpxp,jp 1.(29) We have 1V=xp,jp 1>¯ 9> λjp¯ P¯ 9λjp−¯ Pxp,jp 1.(30) being ¯ 9λjp:= 9−1 λjp. We highlight that λ+ jp, which depends on xp,jpaccording to the definition of Hp, is associated with the value of xpat the jump instant. Indeed, (xp,jp, λ+ jp,0) refers here to the initial value in each hybrid arc. In the sequel, possibly with abuse of notation, λand xpwill be used to represent λ+ jpand xp,jp, respectively. From (30), it is not easy to verify that 1V<0. However, only small values of Tpare of interest, and hence, one just can analyse the following limit lim Tp→01V=lim Tp→0xp 1>¯ 9> λjp¯ P¯ 9λjp−¯ Pxp 1. 151612 VOLUME 9, 2021 C. Albea et al.: Hybrid Modeling and Control of Class of Power Converters When Tp→0, the following approximation can be used ¯ 9λjp≈I+Aλ+TpBλ+Tp 0 1 ,(31) with Bλ+:= B0+(B1−B0)λ+. Then, it holds that lim Tp→01V =lim Tp→0"xp 1#> ¯ P−e−0λ+Tp>¯ Pe−0λ+Tp"xp 1# =lim Tp→0"xp 1#> ¯ P−"P−He(PAλ+)Tp−PBλ+Tp −B> λ+PTp1#!"xp 1# =lim Tp→0x> pHe(PAλ+)xpTp+2xpB> λ+PxpTp=0,(32) which is achieved neglecting T2 pterms, both in (31) and in (32). On the other hand, in order to evaluate the behavior of lim Tp→01Vfor small Tpthe following computation holds, lim Tp→0 1V Tp =x> pHe(PAλ+)xp+2B> λ+Pxp =x> pHe(PAλ+)xp+2B> 0Pxp+2λ+(B1−B0)>Pxp =x> pHe(PAλ+)xp+21−λ+ λeB> 0Pxp.(33) The last step is reached from the next property B0+(B1−B0)λe=0⇒B1= −1−λe λe B0,(34) stemmed from Assumption 1and the error equation (3). It is worth to remind that λ+∈[0,1]. Now, consider that B> 0Pxp=0, then lim Tp→0 1V Tp=x> pHe(PAλ+)xp<−x> pQxp(35) The last condition stems from the fact that LMIs (25)–(26) are satisfied. Now, taking into account B> 0Pxp6= 0, there are three distinct cases: •0< λ+<1: Inserting (27) in (33), applying conditions LMI (25)–(26), MP−Qgiven in the Theorem statement and assuming that λ+is not saturated in (33), yields lim Tp→0 1V Tp =x> pHe(PAλ+)xp−x> pMxp <−x> p(Q+M)xp<−x> pPxp<0∀ξp∈Dp\A. •λ+=0: This case takes place when κ(xp)≤0, being (33) equal to lim Tp→0 1V Tp=x> pHe(PA0)xp+2B> 0Pxp.(36) Note that there are two possibilities, either M0 or QM0. First, let us consider M0. Here, the saturation in λ+=0 is reached if 2B> 0Pxp<0 (necessary for the argument of (27) to be negative or zero), being (36) negative ∀ξp∈Dp\Afrom the fact that condition (25) is satisfied. Finally, from applying PQ, (36) yields lim Tp→0 1V Tp <−x> pQxp<−x> pPxp. Secondly, if QM0, then λe 1+x> pMxp 2B> 0Pxp!≤0⇒ −x> pMxp≥2B> 0Pxp which implies lim Tp→0 1V Tp=x> pHe(PA0)xp+2B> 0Pxp ≤x> pHe(PA0)xp−x> pMxp <−x> p(Q+M)xp Consequently, from MP−Qthe following holds lim Tp→0 1V Tp <−x> pPxp<0∀ξp∈Dp\A. •λ+=1: In this case, corresponding to the case when κ(xp) is saturated in its upper bound, (33) yields lim Tp→0 1V Tp=x> pHe(PA1)xp+2B> 1Pxp.(37) Once again, two situations are particularized, M0 as well as QM0. The saturation of the expression of κ(xp) at λ+=1 with M0 can only happen because 2B> 0Pxp>0, which implies from the equilibrium equation (34) 2B> 1Pxp= −21−λe λe B> 0Pxp<0.(38) Therefore, (37) becomes lim Tp→0 1V Tp <−x> pQxp<−x> pPxp. Now, for the case QM0 and noting that λe 1+x> pMxp 2B> 0Pxp!≥1, which yields the following condition x> pMxp≤21−λe λe B> 0Pxp= −2B> 1Pxp. Then, lim Tp→0 1V Tp=x> pHe(PA0)xp+2B> 1Pxp −<x> p(Q+M)xp<−x> pPxp. VOLUME 9, 2021 151613 C. Albea et al.: Hybrid Modeling and Control of Class of Power Converters Hence, for a given M0 or QM0, it holds that lim Tp→0 1V Tp <−x> pPxp<0∀ξp∈Dp\A The last step is to prove that Ais an invariant set, i.e., g(A∩Dp)⊂A(remember that Wdoes not change when (xp, λ, τ )∈Cp∪Dp). To do so, remember that the following expression has been obtained W(xp, λ+,0) −W(xp, λ, Tp)<−x> pPxp. Moreover, note that W(xp, λ+,0) =x> pPxp. Then, some manipulations yield W(xp, λ+,0) <1 2W(xp, λ, Tp). Therefore, W(xp, λ+,0) is negative in the jumps for any (xp, λ, τ )∈A. Hence, if the solution to Hpreaches A, it will remain therein. Finally, applying the nonsmooth invariance principle given in [38], and using the well posedness result established at the beginning of the proof, leads to the conclusion that, for small enough values of Tp,Ais UGAS. Proof of (ii): We prove here that the particular solutions included in set Eκare in A. Remember that the set Eκconsists of the hybrid arcs that start at the origin xp=0. Then, for every point in Eκ W(ξp)= e0λτ"0 1#!> P(λ, τ ) e0λτ"0 1#! = e0λτ"0 1#!> e−0> λτP(λ, 0)e−0λτ e0λτ"0 1#! ="0 1#> P(λ, 0) "0 1#=0≤1, which holds for any (λ, τ)∈[0,1] ×[0,Tp].  Remark 2: The UGAS property guaranteed for system Hpimplies stability properties for system H, according to Proposition 1. Remark 3: It is worth noting that the particular case M=0 corresponds to an open-loop control κ(xp)=λe. Indeed, Theorem (1) proves the UGAS property of the attractor even with this particular control law. Nevertheless, the open-loop character of this case makes this control law unsuitable for practical applications. V. CONTROLLER PARAMETER TUNING OF THE SYSTEM PERFORMANCE Now, the question is how to provide performance-based criteria for the selection of Mand Q. These matrices can be chosen to improve performance, in terms of current peaks, or even, the chattering amplitude in steady state. To this goal, first a chattering estimation is obtained, and then a discussion about the selection of these matrices is provided. A. CHATTERING ESTIMATION It is interesting to have a measure of the amplitude of the chattering in steady state, i.e., once trajectories have entered in A. Theorem 1guarantees the convergence to kxpk ≤ ε(Tp) when Tpis small enough, but it is desirable to quantify the size of the attractor (related to the amplitude of the chattering) as a function of Tp. Property 1: Consider Assumption 1as well as Theorem 1 assumptions are satisfied and a given Tpsmall enough. Then, ˆε(Tp):= 2kBλekpM √pmTp kHe(PAλe)kpMTp−pm ,(39) is an approximation of the following upper bound kxss pk ≤ ˆε(Tp)∀xss p∈A. where xss pdenotes the steady state of xpof system (9)–(10),(16)–(17). Proof: First, it is easy to see that W(xp, λ, τ) is decreasing with respect to τ∈[0,Tp]. Moreover, consider Tp small enough and λesatisfying Assumption 1such that the following first-order approximation W(xp, λe,Tp)=xss p 1>e−0λeTp>¯ Pe−0λeTpxss p 1 ≈xss p 1>P−He(PAλe)Tp−PBλeTp −B> λePTp1xss p 1 is obtained applying (31) and neglecting T2 p. Now, if Theorem 1assumptions are satisfied, it holds that "xss p 1#>"P−He(PAλe)Tp−PBλeTp −B> λePTp1#"xss p 1#<1 ⇔0<xss p>(He(PAλe)Tp−P)xss p+2xss> pPBλeTp <xss p>(P−He(PAλe)Tp)xss p+2xss p>PBλeTp <(k1−He(Aλe)kpM pm Tp)kxss pk2 P+2kBλekpM √pm Tpkxss pkP ⇔ kxss pkP< 2kBλekpM √pmTp kHe(Aλe)kpM pmTp−1 ⇔ kxss pk< 2kBλekpM √pmTp kHe(Aλe)kpMTp−pm where pMand pmare the maximum and minimum eigenvalues of Pand kxss pkP=xss> pPxss p. It is worth noting, from Property 1, that if Tpdecreases, kxss pkalso decreases. Property 2: Consider that Assumption 1is satisfied, and a given Tpsmall enough such that Property 1is valid. Then, for 151614 VOLUME 9, 2021 C. Albea et al.: Hybrid Modeling and Control of Class of Power Converters a given positive parameter ε > 0, the state xin steady state, denoted as xss is limited by kxssk ≤ εe αTp 2(40) where α:= 2(aM+BM ε) such that aM:= 1 2max(a0,M,a1,M) with ai,Mthe maximum eigenvalue of |He(Ai)|and BM:= max(kB0k,kB1k). Moreover, ˆεdefined in (39) is an estimation of ε. Proof: Consider the following variable, χ:= kxssk2, i.e. the squared norm of xin steady state, and note that the dynamics of this variable is bounded from the definition of aMand BM, ˙χ=2xss>(Aσxss +Bσ)≤2aMχ+2√χBM.(41) Then, in order to find upper bounds on χin steady state, the worst cases bill be analyzed, i.e., the time subintervals inside the sampling period where kxssk>ˆε(assuming that Property 1provides a valid ˆε). Along those intervals, ˙χ≤2αχ with α:= aM+BM ˆε.(42) Now, it must be proved that the upper bound on the right hand side of the χdynamics directly implies the upper boundedness of the solution of (41) by the solution of the bounding equation (42). For this, let us define χ1(t)= χ(xss 0)e2αtas the solution to the bounding equation (42) and χ2(xss2) as the solution to the actual system (41). Then, if χ1(xss 0)=χ2(xss 0), d(χ1−χ2) dt =2αχ1−2xss 2>(Aσxss 2+Bσ)≥2α(χ1−χ2). This means that χ1being greater or equal than χ2at some time implies that it will remain so in the future. Note that the inequality comes from (41) and the fact that the focus lies on kxssk ˆε>1. If this does not hold, then ˆεis an even stricter bound than the one that Property 2suggests. In that case, the integration time of this proof would start when kxsskexits that bounding interval, leaving less time for escaping from it. Finally, χ1(t)≥χ2(t) implies that χ2(t)≤χ(xss 0)e2αTp⇒ kxssk ≤ ˆεe αTp 2. In that expression, the bounding exponential e2αthas only been allowed to evolve for a time period of Tp/2, as it is the maximum time distance from the interval edges 0,Tp, where the ˆεbound holds. It was also assumed χ(xss 0)=χ1(xss 0)= χ2(xss 0)≤ ˆε2, concluding the proof.  B. DISCUSSION Theorem 1establishes stability and performance properties for (4)–(7). Moreover, there are still degrees of freedom in the selection of matrices Mand Q, available for closed–loop performance tuning. 1) SELECTION OF Q On the one hand, for a selected matrix Q0∈Rn, the feasibility problem composed of (25)–(26) provides a matrix Pnecessary to compute the upper bound of kxss pkdefined in Property 1. Hence, the selection of Qcan be used to manage the chattering in steady state. 2) SELECTION OF M On the other hand, this tuning parameter adjusts the transient time, modifying the response time, voltage oscillations, current peak, among others. Indeed, if M0, the system response can be faster and/or can present voltage oscillations, with tendency to saturate the control signal. Conversely, if M≺0, the system time response can increase, diminishing the current peaks. VI. SIMULATIONS Some simulations have been performed to validate the results proposed here. For this, a boost converter has been selected, with the topology shown in Fig. 3. Taking the model given in (1)z=[iLvC]>,σ=0 when the switch Sis closed and σ=1 when this switch is open, A0=−R/L0 0−1/R0C0,A1=−R/L−1/L 1/C0−1/R0C0, B0=B1=Vin/L 0. The parameters inside these expressions are given in Table 1. FIGURE 3. Boost converter. TABLE 1. Boost converter parameters. The selected operating point is xe=8.4 100T, with its associated λe=0.76. The simulations are performed with Tp=10µs. The following matrices Q=8.35 0.01 0.01 8.33 ·103and P=416.40 −0.68 −0.68 17.47 satisfy the feasibility problem (25)–(26). Now, it is required to adjust matrix M. For this, some simulations have been performed with different choices of M. Fig. 4compares the state evolutions with VOLUME 9, 2021 151615