Hybrid control of power converters with affine models and pulse-width modulated inputs

1 Hybrid control of power converters with affine models and pulse-width modulated inputs Carolina Albea, Antonino Sferlazza, Francisco Gordillo, Fabio G´omez-Estern Abstract—In this paper, hybrid dynamical systems theory is applied to the analysis and control of switched converters with Pulse-Width Modulated (PWM) inputs. The system is described by a state-space model with continuous flows and discrete jumps, without averaged equations. The modulation effects are captured in full without using time-dependent signals, by enlarging the state vector to include the PWM waveform generation process. Furthermore, the sample-and-hold mechanism associated with the sampling frequency is also taken into account with this approach. A control law is proposed based on a Lyapunov function candidate. Furthermore, convergence sets and the steady state jitter, inherent to PWM-based controllers, are analyzed estimating limit sets for the augmented state. Consequently output chattering can be bounded. By using hybrid control approaches, the control designer gains a deeper understanding of the effect of modulation in the closed-loop dynamics, avoiding the problems associated with the use averaged models. Experimental results validate the proposed method. Index Terms—Converter control, PWM, hybrid dynamical system, Lyapunov function. I. Introduction In electronic power converters, the output of the controller is usually a discrete signal connected to the transistor gates with the aim at driving the output to the desired value. Usually, the design of controllers for this type of systems is based on averaged models that consider the control inputs as continuous signals [1–4]. In order to implement the resulting controller, the computed control signal is discretized using a modulator. The most common modulator is Pulse-Width Modulation [5, 6]. This approach may limit the accuracy of the signals in steady state [7] and even can fail to predict the system behavior [8]. Besides, it is usual to ignore the discrete-time nature of the control signals and the sample-and-hold mechanism. This work has been partially funded by the national Ministry of Higher Education, Research and Innovation under grants ANR-18- CE40-0022-01 (Project HISPALIS), by the national Ministry of Universities, Science and Innovation under grants PID2019-105890RJ- 100, PID2019-109071RB-I00 and DPI2016-75294-C2-1-R and by Junta de Andaluc´ıa Programa Operativo FEDER Andaluc´ıa 2014- 2020, US-1264655. This work has been conducted while C. Albea was with CNRS, LAAS, Universit´e de Toulouse, 7 avenue du colonel Roche, F-31400 Toulouse, France. Now, C. Albea is with Universidad de Sevilla, Department of System Engineering and Automatic Control, Camino de los Descubrimientos s/n, 41092, Seville, Spain (e-mail: calb[email protected]). A. Sferlazza is with University of Palermo, Department of Engineering, viale delle scienze Ed. 10, 90128 Palermo, Italy. E-mail: an[email protected]. F. Gordillo is with Universidad de Sevilla, Department of System Engineering and Automatic Control, Camino de los Descubrimientos s/n, 41092, Seville, Spain (e-mail: [email protected]). F. G´omez-Estern is with Universidad de Loyola Andaluc´ıa, Av. de las Universidades, s/n, 41704 Dos Hermanas, Seville, Spain (e-mail: [email protected]). Normally, PWM-based controllers [3, 4] are commonly based on a separation principle: a high–level control algorithm, usually based on a simple model, computes a continuous signal that is converted into a discrete input in the PWM block. In order for this separation principle to be tractable, the high-level controller is designed based on a model of the plant together with the modulator block that is simplified to keep the linear time–invariance properties. This implies averaging or approximating the exponential matrices by linear expressions, and generally neglecting the specifics of the PWM waveform. The approach works well provided that sampling rates are high. However, it is known that PWM blocks have an effect on the output, [4, 9]. PWM modulation of analog control signals limits in some ways the degrees of freedom of the control action, as signal values are discrete, and they can only be changed once or a few limited times along the sampling interval. This effect is often ignored, but at a micro scale the signal mismatch causes output jitter in steady state. Nevertheless, a modulator in the system model introduces the challenge of dealing with time-dependent signals and, consequently, non-autonomous dynamics. There exist control techniques that avoid the use of averaged models and deal directly with the discrete nature of signals, such as sliding mode control [10] or model predictive control [11]. Nevertheless, dealing directly with discrete input signals, avoiding the use of PWM gives rise to the loss of an important degree of freedom of the PWM mechanism. With PWM, the switches can be ordered at any instant inside the sampling interval (with a limitation in the total number of switches in this interval); with most of the techniques that avoid the use of averaged models, the control input can only be changed at the sampling instants. This makes necessary to increase the sampling frequency. Hybrid Dynamical System (HDS) theory can help in building autonomous state-space models of PWM-based switched converters, and also to design suitable controllers. Indeed, HDS theory is a mature framework for analyzing dynamical behaviors made of continuous dynamic fields (flows) and discrete changes in signals (jumps). Following notorious results from the literature [12, 13], it is easy to ascertain the conditions for an autonomous hybrid system to have proper solutions, to avoid conflictive behaviors such as Zeno dynamics, and to ensure stability and steady-state specifications. For relevant results that will be used here, see also [14–16] In this context, the authors in [17] proved that, for general systems under some assumptions, that the solutions of an averaged system are suited approximations of the original one, mentioning This work has been submitted to the IEEE for possible publication in IEEE Transactions on Circuits and Systems I: Regular Papers. Copyright may be transferred without notice, after which this version may no longer be accessible. 2 power converters as application. Nevertheless, these hybrid approaches suffer from the same drawback than other techniques mentioned above: the discrete control input can only be changed at the sampling instants. Switched power systems often appear in the hybrid framework as affine systems. This class of systems is described by standard state-space equations, where matrices Aand Bcan change depending on the state of the switches, see [18, 19]. There exist several results about controlling switched affine systems without considering PWM. For instance, robust and input-to-state stability has been analyzed in [20]. In [21], and more recently in [22, 23], this framework is presented and applied to DCDC converters. The main feature in these references is the implementation of an aperiodic sampled-data control signal with arbitrarily fast switching, making possible the appearance of Zeno behaviour. This practical difficulty has been overcome in [24]. This paper aims at designing a control law for power converters with affine models using hybrid dynamical systems theory. An advantage of this approach is to provide a rigorous state-space autonomous model that encompasses all elements of the converter architecture: power converter, PWM and sample-and-hold mechanism. Some results in this paper require sufficiently small sampling intervals, but this is not an a priori condition for modeling; it only arises when computing the convergence set, and the relation between chattering and the PWM-sampling period is made explicit. The novel idea developed here is focused on power converters with only two functioning modes, which are very common in power electronics [25–27]. A new controller is proposed to stabilize a limit cycle, based on a rigorous model, different from the classical averaged models used in the literature. The stability design is obtained at the cost of a harder analysis compared with usual controllers based on averaged models, but the controller implementation does not grow in complexity, and it can be readily adapted to other modulation techniques. Furthermore, an exact estimation is provided for the chattering in steadystate. Finally, experimental results show satisfactory performance of the proposed control loop. This paper is organized as follows. The problem statement is given in Section II. The hybrid general model of switched PWM-based converters is presented in Section III. In Section IV, the main result is presented. Section V and Section VI present simulation and experimental results, respectively. Finally, the paper closes with a conclusion section. Notation: Throughout the paper Ndenotes the set of the natural numbers,{0,1,2,3, ...}, and Rthe set of real numbers, Rnthe n-dimensional Euclidean space and Rn×m the 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 is a zero matrix of suited dimensions. He(M) is the hermitian matrix of M, i.e. (M+M>). Finally, satb a(φ) is the standard saturation function defined in R7→ [a, b]. II. Problem statement Many switched power systems made of linear components and switching devices can be modelled as switched affine systems, as follows ˙z=Aσz+Bσ,(1) where z∈Rnis the vector of physical state variables, i.e. voltages and currents and σ∈ {0,1,2, ..., N −1}is the control input that represents the functioning mode of the converter according to the switching state. Finally, Aσand Bσare matrices of suitable dimensions. Model (1) covers many applications of power converters, such as the buck converter, the boost converter, the quadratic boost converter, the half bridge converter, the boost inverter, etc. In this paper, only two functioning modes, N= 2, are considered. These systems are generally managed by continuoustime control laws, i.e., σ∈ {0,1}is modelled by continuous signals λ∈[0,1], obtained by using averaging approaches, and implemented in (1) by Pulse Width Modulation (PWM), as depicted in Fig. 1, where κ(x) (being x:= z−zeand zethe equilibrium point) represents a continuous control law whose output is limited to be in 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. 1 is constant during each sampling interval. The PWM mechanism with a sawtooth carrier is illustrated in Fig.2. ze − + xλσz PWM κ(x)_z=Aσz+Bσ Tp λs Fig. 1. Feedback scheme. The control objective is to design a function λ=κ(x) : Rn7→ Rsuch that the modulated control signal, σ(λ), 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 ze must satisfy the following assumption. Assumption 1: Consider that there exists a value for λ=λe∈[0,1] such that the following equation is hold, 0=(A0+ (A1−A0)λe)ze+B0+ (B1− B0)λe.(2) y From a practical point of view, this assumption is equivalent to suppose that the desired operating point is not beyond the physical constraints of the converter. In other words, this assumption guarantees the existence of a switched signal for system (1), inducing an equilibrium in z=zein the generalized sense of Krasovskii. 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)Tp 3 τ σ λ 1 0 1 0 t t0t1t2   Tp t3t4   Tp Fig. 2. PWM mechanism with sawtooth carrier. Top: Tp-periodic sawtooth carrier, τ, and λthe duty cycle. Bottom: output of the PWM block, σ. in mode 0, corresponding to the convex combination of the right hand of (2). Then, the time spent in each mode will be distributed in every time period according to the modulator shown in Fig 2. In this paper, it is assumed without loss of generality, that at t= 0 a sawtooth carrier as well as a sampling period start. The error equation associated with (1) in each sampling time tkfor all k∈Ncan be written as: ˙x=Aσx+Bσ,(3) where Bσ:= Bσ+Aσzesuch that Bλe= 0. Direct integration of the dynamics of xin (3) along the sawtooth carrier period (Fig. 2), starting from an initial condition x(t2k) at t=t2kyields x(t2k+1) = eA0λTpx(t2k)+(eA0λTp−I)A−1 0B0(4) x(t2k+2) = eA1(1−λ)Tpx(t2k+1)+(eA1(1−λ)Tp−I)A−1 1B1. (5) being t2k+2 −t2k=Tpfor all k∈N. The desired behavior in steady state corresponds to λ= λeand a limit cycle for x(see Fig. 3) considered in the following property. Property 1: Consider a couple (x, λ) = (xe, λe) associated to (3) that satisfies Assumption 1. If there exist two vectors xe,Tpand xe,λeTpfulfilling xe,λeTp=eA0λeTpxe,Tp+ (eA0λeTp−I)A−1 0B0 xe,Tp=eA1(1−λe)Tpxe,λeTp+ (eA1(1−λe)Tp−I)A−1 1B1, (6) then, there exists a limit cycle χ={(x∈Rn/∃τ∈[0, Tp) : x=xe(τ)} x1 x2 xe,Tp xe,λeTp xe(τ) Fig. 3. Example of limit cycle according to Property 1 for a case with n= 2. The red points and the blue trajectory represent points (6) and curve (7), respectively. in (3) defined by        xe(τ)= eA0λeτxe,Tp+Rτ 0eA0λe¯τB0d¯τ if 0 ≤τ < λeTp xe(τ)=eA1λe(τ−λeTp)xe,λeTp+Rτ λeTp eA1λe(¯τ−λeTp)B1d¯τ if λeTp≤τ < Tp (7) Proof. Curve (7) is a closed and isolated trajectory associated to (3). It is isolated because (xe, λe) and (6) form a linear system of two equations with unknowns xe,Tpand xe,λe,Tpand then, (7) is unique. Consequently, (7) is a limit cycle.  Then, we are in conditions of formulating the problem. Problem 1: Consider the switched system (1) with N= 2 and a PWM with a sawtooth carrier, as shown in Fig. 2. Then, the objectives 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 sample and hold mechanism with a given periodic sampling time Tp. •To design a control law for the duty cycle λ, without using averaged model. •To achieve convergence of x(t) to limit cycle (7) and to analyze stability properties for the hybrid system. III. Hybrid dynamical model In this section, we will use the framework given in [12] about hybrid dynamical systems to model the controlled system, considering continuous-time and discrete-time dynamics. Hence, we propose the following hybrid dynamical model of the controlled switched system (3), considering 4 a sawtooth carrier for the PWM mechanism, H:                            ˙x ˙ λ ˙σ ˙τ     =f(ξ), ξ ∈ C     x+ λ+ σ+ τ+     ∈g(ξ), ξ ∈ D, (8) where ξ= [x λ σ τ]>∈Hsuch that H:= {Rn× [0,1] × {0,1} × [0, Tp]}. The maps fand gcapture both the continuous-time and discrete-time dynamics and are defined as follows: f(ξ) =     Aσx+Bσ 0 0 1     , g(ξ) =     x σsat1 0κ(x) + (1 −σ)λ 1−σ (1 −σ)τ     , (9) The so-called flow and jump sets are C0:= {ξ∈H:τ∈[0, λTp], σ = 0} D0:= {ξ∈H:τ=λTp, σ = 0} C1:= {ξ∈H:τ∈[λTp, Tp], σ = 1} D1:= {ξ∈H:τ=Tp, σ = 1}, being C:= C0∪ C1(10) D:= D0∪ D1.(11) This hybrid scheme gathers the complete dynamics of the system. Indeed, the continuous-time dynamics of x evolves according the switching of σ. The selection of the last one is given by a sawtooth modulator defined by λ, τand Tp. Variable λ∈[0,1] is the duty cycle and τis a timer that defines a continuous-time sawtooth signal (the carrier shown in Fig. 2); Finally, κ(x) is the control law, to be defined, that computes the value of signal λin every sampling time, t2k. Inside the sampling intervals, λis held constant. Solutions to H(f, G, C,D) are given on the so-called hybrid time domain: dom(ξ)⊂R≥0×N, such that, dom(ξ) = ¯ j−1 [ j=0 ([tj, tj+1], j),(12) for some sequence 0 = t0≤t1≤t2≤... ≤t¯ jwith ¯ jfinite (being a compact set) or infinite. In the next section, a control law will be designed for system (8)–(11) with stability guarantee. Now, let us rewrite the continuous- and discrete-time dynamics of xto establish a Lyapunov function candidate. To do so, define Γσ=hAσBσ 01,n 0isuch that during the flows: d dt x 1= Γσx 1.(13) Now, the following Lyapunov function candidate is considered V(x, λ, σ, τ) = max W(x, λ, σ, τ)−x> cPxc,0,(14) where xc:= xe(τλeTp) and Wis a quadratic function of x, which is defined as follows, W(x, λ, σ, τ) := x 1> Pσ(λ, τ)x 1(15) with P0(λ, τ) := e−Γ> 0τ¯ Pe−Γ0τ P1(λ, τ) := eΓ> 1(λTp−τ)e−Γ> 0λTp¯ Pe−Γ0λTpeΓ1(λTp−τ) and ¯ P:= P0 0 0 where P0∈Rngiven in Property 1. We are in position to define the compact set for which it is desired to establish uniform globally asymptotic stability (UGAS). This desired attractor set is A:= {ξ∈H:V(x, λ, σ, τ)=0}.(16) It is worth noting that the Lyapunov function candidate is a single-valued function and continuous in ξ∈Hand enjoys nice properties. The idea behind of this Lyapunov function is to be constant during flows, to not change in the jumps ((j 2+λj)Tp, j) for j= 2k, k ∈Nand to get ∆V < 0 in each ((j+1 2)Tp, j) for j= 2k+ 1, k ∈N. This makes the candidate Lyapunov function to be not increasing (decreasing every two jumps), the state to converge to the set A, after each jump for j= 2k+ 1, k ∈N, and xto converge to the interior of x>Px ≤x> cPxcas jgoes to infinity. IV. Main result In this section, a controller is proposed for the hybrid system (8)–(11) providing uniform global asymptotic stability (UGAS) of the compact set A. Theorem 1: Consider there is a λeassociated with the operating point xesuch that Assumption 1 is satisfied, vectors xc:= xe,Tp∈Rnand xe(τ) given in Property 1 and matrices P, Q 0∈Rn×nsuch that QPand M≺Q−P∈Rn×nsatisfying A> 0P+PA0≺ −Q(17) A> 1P+PA1≺ −Q. (18) Moreover, consider system (8)–(11), with a control law verifying κ(x)∈(λe1−x>Mx−x> cMxc 2B> 1P x if B> 1Px 6= 0 [0,1] if B> 1Px = 0.(19) 5 Then, there exists a positive constant T∗ psuch that for 0< Tp< T∗ pthe following statements hold for H (i) Ais UGAS. (ii) set L:= {ξ∈H:x=xe(τ), λ =λe}(20) is in the interior of A. Proof. Hybrid system H(f, g, C,D) with control law (19) is well-posed, because it verifies: •Cand Dare closed sets in H. •fis a continuous function, thus it is locally bounded and outer semi-continuous. Moreover, it is convex for each ξ∈ C. •gis outer semi-continuous and locally bounded. It is worth noting that the maximal solutions to Hwith (19) are complete. We will consider the proof item by item. Proof of (i): The proof of this item proceeds applying [28, Theorem 1]. Note that the candidate Lyapunov function, V(x, λ, σ, τ) (14) is continuous in C ∪ Dpand locally Lipschitz near each point in C\A. Moreover, V(x, λ, σ, τ) is strictly positive definite in (C ∪ D)\A and radially unbounded. Likewise, it verifies, by definition V(x, λ, σ, τ) = 0, for all (x, λ, σ, τ) in A. The next step of the proof is to ensure that the derivative of Valong flows outside of Ais nonpositive (or more precisely in this case, equal to zero). More formally, the objective is to show h∇V(ξ), f(x, σ)i ≤ 0,∀(x, λ, σ, τ)∈ C\A.(21) For any (x, λ, σ, τ)∈ C\A, it is clear, from its definition, that V(x, λ, σ, τ) = W(x, λ, σ, τ)−x> cPxcgetting h∇V(x, λ, σ, τ), f(x, σ)i =x 1> ˙τ∂ ∂τ Pσ(λ, τ)+˙ λ∂ ∂λ Pσ(λ, τ) +˙σ∂ ∂σ Pσ(λ, τ)x 1 +2 x 1> Pσ(λ, τ) ˙x 0 =x 1> ∂ ∂τ Pσ(λ, τ) + Pσ(λ, τ)He (Γσ)x 1= 0. The last equality comes from ∂ ∂τ Pσ(λ, τ) + Pσ(λ, τ)He (Γλ) = 0. Let us proceed now to analyze the second stability condition from [28, Theorem 1]. To do so, we take into account the special structure of hybrid system (8)–(11) which implies that the jumps occur at the ordinary time instants either tj= (j 2+λj)Tpfor even jor tj=j+1 2Tpfor odd j. We adopt here the following notation according to the hybrid time domain (12): xj=x(tj, j), λj=λ(tj, j), τj=τ(tj, j) that correspond to the variables right before the jump at tj∈ {j 2+λj)Tp,j+1 2+λj)Tp}and x+ j= x(tj+1, j + 1), λ+ j=λ(tj+1, j + 1) and τ+ j=τ(tj+1, j + 1) right after the same jump. In the same way, we define ∆V=V(x+ j, λ+ j, τ+ j)−V(xj, λj, τj). Note that x+ j=xj, (λ+ j, λj, τ+ j, τj)∈ {(λj, λj, λjTp, λjTp),(λ+ j, λj,0, Tp)}. There are two cases where the solution is in D \ A: •(λ+ j, λj, τ+ j, τj)=(λj, λj, λjTp, λjTp): Here, we have ∆V=W(xj, λ, 1, λTp)−W(xj, λ, 0, Tp) =xj 1> e−Γ> 0λTp¯ P−¯ PPe−Γ0λTpxj 1= 0. •(λ+ j, λj, τ+ j, τj) = (λ+ j, λj,0, Tp): In this case, ∆V=W(xj, λ+ j,0,0) −W(xj, λj,1, Tp) =xj 1> ¯ P−Ψ(λj)>¯ PΨ(λj)xj 1 =x+ j 1> ¯ P−Ψ(λ+ j)>¯ PΨ(λ+ j)x+ j 1 where Ψ(λj) := eΓ1(λjTp−Tp)e−Γ0λjTp. Notice that the manipulable signal, λj+1 =λ+ jhas to be computed at tj=j+1 2Tp. For this, it is convenient to write ∆Vin terms of xjinstead of xj+1. Hence, from (13), we obtain the following relationship x+ j 1= Ψ(λ+ j)xj 1.(22) We have ∆V= xj 1> Ψ(λ+ j)> ¯ P−Ψ(λ+ j)>¯ PΨ(λ+ j)Ψ(λ+ j)xj 1 =xj 1>Ψ(λ+ j)>¯ PΨ(λ+ j)−¯ Pxj 1,(23) where Ψ(λ+ j) := Ψ(λ+ j)−1, which has been adopted for readability. We highlight that λ+ j, which depends on xj according to the definition of H, is associated with the value of xat the jump instant. Indeed, (xj, λ+ j,0,0) refers here to the initial value in each hybrid arc. In the sequel, with abuse of notation we will use λand xto represent λ+ jand xj, respectively. From (23), it is not easy to verify that ∆V < 0. However, we are interested in small values of Tp. Therefore, one just can analyse the following limit lim Tp→0∆V= lim Tp→0x 1>Ψ(λ+ j)>¯ PΨ(λ+ j)−¯ Px 1. As we are interested in the limit for Tp→0, the following approximation can be used ¯ Ψ(λ+ j)≈I+Aλ+TpBλ+Tp 01×n1,(24) with Aλ+:= A1+ (A0−A1)λ+and Bλ+:= B1+ (B0− B1)λ+. Then, it holds that lim Tp→0∆V= lim Tp→0x 1>¯ Ψ(λ+ j)>¯ P¯ Ψ(λ+ j)−¯ Px 1 = lim Tp→0x 1> He(PAλ+)Tp(PBλ+)Tp (B> λ+P)Tp0x 1 = lim Tp→02x>A> λ+PxTp+ 2B> λ+PxTp= 0,(25) which is achieved neglecting T2 pterms. 6 Moreover, to evaluate the behavior of ∆Vfor small Tp we can compute additionally, lim Tp→0 ∆V Tp ≤x>He(A> λ+P)x+ 2B> λ+Px =x>He(A> λ+P)x+2(B> 1+λ+(B0−B1) > )Pxn =x>He(A> λ+P)x+ 2 1−λ+ λeB> 1Px. (26) The last step is reached from the next property B1+ (B0−B1)λe= 0 ⇒B0=−1−λe λe B1,(27) stemmed from Assumption 1 and the error equation (3). Remember that λ+∈[0,1]. First, consider that B> 1Px = 0, then lim Tp→0 ∆V Tp ≤x>He(A> λ+P)x < −x>Qx < −x>Px (28) <−x>Px +x> cPxc<0 applying LMIs (17)–(18) and QP. Then, (28) is negative for all ξ∈ D1\A. Now, we take into account B> 1Px 6= 0 and we distinguish 3 cases: •0< λ+<1: Inserting (19) in (26), applying conditions LMI (17)–(18), M≺Q−Pand assuming that λ+is not saturated in (26), yields lim Tp→0 ∆V Tp ≤x>He(A> λ+P)x+x>Mx −x> cMxc <−x>(Q−M)x−x> cMxc <−x>Px +x> cPxc<0∀ξ∈ D1\A. •λ+= 0: This case takes place when κ(x)≤0 such that (26) becomes lim Tp→0 ∆V Tp =x>He(A> 0P)x+ 2B> 1Px. (29) There are two possibilities, either 0 MQor M 0. First, let us consider M0. Here, the saturation in λ+= 0 is reached if B> 1Px < 0 (necessary for the argument of (19) to be negative or zero), being (29) negative for all ξ∈ D1\A from the fact that condition (17) is satisfied. Secondly, if 0 MQ, then λe1−x>Mx −x> cMxc 2B> 1Px ≤0 ⇒2B> 1Px ≤x>Mx −x> cMxc which implies, from (17), lim Tp→0 ∆V Tp ≤x>He(A> 0P)x+x>Mx −x> cMxc <−x>(Q−M)x−x> cMxc. Consequently, from M≺Q−Pthe following holds lim Tp→0 ∆V Tp ≤ −x>Px +x> cPxc<0∀ξ∈ D1\A. •λ+= 1: In this case, corresponding to the case when κ(x) is saturated in its upper bound, being (26) equal to lim Tp→0 ∆V Tp =x>He(A> 1P)x+ 2B> 0Px. (30) Once again, two situations are particularized, 0  MQor as well as M0. The saturation of the expression of κ(x) at λ+= 1 with M0 can only happen because B> 1Px > 0, which implies from the equilibrium equation (27) that 2B> 0Px =−21−λe λe B> 1Px < 0.(31) Therefore, (30) is negative for all ξ∈ D1\A, if (18) is satisfied. Noting that 0 MQand λe1−x>Mx−x> cMxc 2B> 1Px ≥1, we obtain the following condition x>Mx −x> cMxc≥ −21−λe λe B> 1Px = 2B> 0Px. The last step comes from (27). Then, considering the same development than for the case 0 < λ+<1, we get lim Tp→0 ∆V Tp <−x>(Q−M)x−x> cMxc <−x>Px +x> cPxc. Hence, for κ(x) saturated in λ+= 1, we get lim Tp→0 ∆V Tp <0∀ξ∈ D1\A. Consequently, the solution jumps keep lim Tp→0 ∆V Tp<0 ∀(x, λ, σ, τ)∈ D1\A ⊂ D,which ensures that set Ais attractive for small enough values of Tp. The last step is to prove that Ais an invariant set, i.e., g(A ∩ D1)⊂ A (remember that Wdoes not change when (x, λ, σ, τ)∈ C∪D0). To do so, remember that we obtained W(x, λ+,0,0) −W(x, λ, 1, Tp)<−x>Px +x> cPxc. Moreover, note that W(x, λ+,0,0) = x>Px. Then, after some manipulations, we get W(x, λ+,0,0) −x> cPxc<1 2(W(x, λ, 1, Tp)−x> cPxc). Therefore, W(x, λ+,0,0)−x> cPxcis negative in the jumps for any (x, λ, σ, τ)∈ A. Hence, if the solution to Hreaches A, it will remain therein. Finally, applying the nonsmooth invariance principle given in [28], and using the well posedness result established at the beginning of the proof, we can conclude that, for small enough values of Tp,Ais UGAS. Proof of (ii): Now, we desire to prove that L⊂A. This is direct noting that V(ξ) is only updated in each x(t2k) of (4)–(5) that means when xis in A ∩ D1. Therefore, W(ξ) 7 takes the same value for each couple (x(t2k−1), x(t2k)), ensuring that xe(τλeTp) is inside A.  Remark 1: It is worth noting that the particular case M= 0 means an open-loop control κ(x) = λe. Indeed, Theorem (1) proves 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. y Remark 2: The maximum amplitude of the chattering in steady state for each variable of xis defined by max{xe,Tp, xe,λeTp}.y A. Adjusting M 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 current peaks and voltage oscillations, with tendency to saturate the control signal. Conversely, if 0 MQ, the system can reduces the oscillations, diminishing the current peaks. V. Simulations Some simulations are performed to validate the results proposed here. For this, we select a boost converter whose model matrices are [29]: A1=−R/L 0 0−1/R0C0, A2=−R/L −1/L 1/C0−1/R0C0, B1=B2=Vin/L 0. where the parameters are given in Table I. The state is TABLE I Boost converter parameters. COMPONENT NOMINAL VALUE Vin 24 V R11.5 mΩ L470 µH C20 µF R050 Ω z= [iLvC], being iLand vCthe inductor current and capacitor voltage, respectively. The selected operating point is ze=8.4 100T, which is associated with λe= 0.76. The simulations are performed with Tp= 10µs. The feasibility problem (17)–(18), provides Q=1.07 0.03 0.03 2.53·106, P =4.23 0 0 0.18·104. Now, we want to adjust matrix M. For this, we perform some simulations with different choices of M. Fig. 4 compares the state evolutions with M=−0.5Q,M= 0.1Q and M= 0. As mentioned in Section IV-A, if M≺0 the rise time decreases but the control signal has tendency to saturate. Conversely, choosing M0 the current peak is reduced and the control input is not saturated but the system dynamics become slower. Moreover, note that the particular case M= 0, provides λ+=λe, as mentioned in Remark 1. This control law yields a strong oscillating behavior in transient time with current peaks that can damage the circuit. Figure 5 shows a zoom of the state variables, the control input σand the Lyapunov function V(ξ) in transient time. It can be observed that the Lyapunov function is constant, excepts in the time instants t=kTpwith k∈Nwhen it jumps a step down. Moreover, the control input σevolves by switching between the functioning modes σ∈ {0,1}. Finally, the steady state of zis plotted in the state space in Fig. 6. Note that the state zevolves in a limit cycle between ze(tλeTp) and ze(tTp), which corresponds to the chattering behavior of the voltage and current. 0 0.005 0.01 0.015 0.02 0 50 100 150 0 0.005 0.01 0.015 0.02 0 5 10 15 20 0 0.005 0.01 0.015 0.02 0 0.5 1 Fig. 4. State and control input (λ+) evolutions, with M≺0, in green, M= 0, in red, and M0, in blue. The reference is in yellow. VI. Experimental setup A test setup was built to validate the proposed hybrid control scheme. It is composed of: •A boost converter whose electrical parameters are given in Table II. These parameters correspond to the ones used in the simulations of Sect. V. •An electronic card (model LEM LTS 15-NP) for the measurements of the inductor current and voltage 8 1 1.01 1.02 1.03 1.04 1.05 10-3 66 67 68 1 1.01 1.02 1.03 1.04 1.05 10-3 5.4 5.6 5.8 1 1.01 1.02 1.03 1.04 1.05 10-3 0 0.5 1 1 1.01 1.02 1.03 1.04 1.05 10-3 2.1 2.2 2.3 2.4 106 Fig. 5. Zoom of the state, control input (σ) and Lyapunov function evolutions, with M0 in transient time. 8.15 8.2 8.25 8.3 8.35 8.4 8.45 8.5 8.55 8.6 99.6 99.7 99.8 99.9 100 100.1 100.2 100.3 100.4 Fig. 6. Steady of the state zin blue, the operating point in red, the chattering extremes in green and yellow. TABLE II Circuit parameters values Component Value Model VIN 24V L470µH AGP4233-474ME rL11.5mΩ C120µF MKP1848C62090JP4 rC5mΩ R050Ω Diode C3D06060A switch C3M0065090D Driver 1EDI20N12AF sensor for the measurement of the output voltage. We built the voltage sensor by means of a resistor divider connected with operational amplifier in buffer configuration. •A dSPACE cardboard (DS1103) with a PowerPC 604e at 400 MHz and a fixed-point DSP TMS320F240. The complete control scheme was implemented in the dSPACE card by means of Matlab-Simulink. Fig. 7. Evolutions of the voltage, current and duty cycle in the start up. We selected for these tests M= 0.1Q, corresponding to the best results obtained in Sect V. Fig. 7 shows a start up transient from a initial condition equal to x0=0VIN to a reference operating point computed by imposing an output voltage equal to Vout = 100V. The voltage and the current signals present a smooth behavior, as performed in simulation (see Fig. 4). Likewise, the steady state operation is shown in Fig. 8. Finally, a perturbation in the load was introduced. The load R0was changed from R0= 50Ω to R0= 75Ω at 0.02s. The proposed algorithm does not provide a voltage output regulation, and an error in the voltage output is exhibited in steady state. However, if an external loop is added with 9 a PI control as in [30], the output voltage is regulated in its reference value, maintaining a good performance. Fig. 8. Evolutions of the voltage, current and duty cycle in the steady state. Fig. 9. Evolutions of the voltage, current and duty cycle without PI controller (in red) and with PI controller (in blue) for regulating the voltage output with a perturbation of R0. VII. Conclusion A hybrid model of switched power converters with PWM inputs and sample-and-hold mechanism is proposed here. The study is particularized for two functioning modes, which is a common power converter application. Moreover, a new control algorithm that chooses the value of the duty cycle at the beginning of each PWM-sampling interval is designed from a rigorous model. For this choice of controller, HDS stability analysis is done, proving convergence to a limit set that can be bounded based on the choice of the size of the sampling period. The hybrid analysis also provides a Lyapunov function for the closed-loop system. Finally, an upper and lower bound for the chattering peaks obtained by the signals is provided. Experimental results show satisfactory closed-loop performance. 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