Hybrid modelling and control of a class of power converters with triangular-carrier PWM inputs

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 discrete-time 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.


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, continuous-time approximate models, such as averaged models [3], [4], [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 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 equa-tions, 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, relevant results on Hybrid Dynamical System (HDS) theory [14], [15], [16], [17], [18] have been applied to power converters. It also is worth to mention the design of hybrid controllers with minimum dwelltime guarantees [19]. Moreover, in [20], [21], [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 V provides 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 N denotes the set of natural numbers and R the set of real numbers, R n the n-dimensional Euclidean space and R n×m the set of all real n × m matrices. The set of non-negative real numbers is denoted by R ≥0 . M 0 (resp. M ≺ 0) represents that M is a symmetric positive (resp. negative) definite matrix. 0 and I are the zero matrix and identity matrix respectively, of suited dimension. The operator · represents the Euclidean norm. He(M ) is the Hermitian matrix of M , i.e. He(M ) = (M + M ). Finally, sat b a (φ) is the standard saturation function defined in R → [a, b].

II. PROBLEM STATEMENT
Many switched power converters can be modelled as switched affine systems, as followṡ where z ∈ R n contains the state variables, i.e. the continuous-time 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, A σ and B σ are 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], [33], [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 λ in Fig. 1 is 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 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. The control objective is to design a function λ = κ(x) : R n → R such that the control signal σ(λ s ), modelled by a triangular carrier, ensures the convergence of z to a given operating point z e . Due to the sampling mechanism, asymptotic convergence to z e is not possible, and a chattering phenomenon is unavoidable. In any case, the desired operating point z e must satisfy the following assumption. Assumption 1: Given an operating point z e there exists a λ = λ e ∈ [0, 1] such that the following equation holds, This standard assumption for switched affine systems guarantees the existence of a switched signal for system (1), inducing an equilibrium in z = z e in the generalized sense of Filippov [35]. This means that, in steady state, σ is expected to be a periodic signal of period T p , spending a time λTp in mode 1 and (1 − λ)T p in 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: where x := z − z e and B σ := B σ + A σ z e such that B λe = 0, if Assumption 1 is 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 T p . • To design a new control law for the duty cycle λ. • To achieve convergence of z to z e and 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, . The maps f and g capture the continuous-time and discretetime dynamics, respectively, and are defined as follows: The following sets define the so-called flow and jump sets 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 s is a continuous-time triangular carrier (the signal shown in Fig. 2); v defines the slope of s; q is an artificial variable which takes values 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, λ is held constant. It is worth noting that the solutions to H are 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, for some sequence 0 = t 0 ≤ t 1 ≤ t 2 ≤ ... ≤ tj withj finite (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. where . The maps f p and g p capture both the continuous-time and discrete-time dynamics and are defined as follows: where A λ and B p can be computed in different ways depending on the properties of the system. For this, consider a value of t corresponding 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 k such that t k is equal to this time instant. By direct integration of the dynamics of x in H along the triangular carrier period (Fig. 2), starting from an initial condition x(t k ) yields 1 being t k+1 = t k + T p λ 2 and t k+3 = t k + T p (1 − λ 2 ). Then, the following assumption will explore different possibilities to compute A λ (B p will be considered later), from (11)- (13) such that the following holds Assumption 2: Using the following definition which arises from rewriting (11)-(13) as it will be assumed that one of the following conditions are verified by T p and system matrices A 0 and A 1 .
A 0 A 1 = A 1 A 0 . Under this condition, it is easy to prove that Now, for the definition of A λ , there are two options.
• A λ := log(Υ(λ))/T p if Assumption 2.1 holds, but Assumption 2.2 does not. The assumption guarantees the existence of the logarithmic matrix with real elements.
Remark 1: Note that in Assumption 2, the conditions would lead to exact results. However a less restrictive condition can provide an approximation of A λ . Indeed, a first-order approximation of the Taylor series expansion of Υ(λ) with respect to T p around T p = 0 yields 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 B p in (10), using Eqs. (11)-(13) gives, For future uses, it is worth noting that, when T p → 0 (neglecting the second and higher order terms on T p ), this expression approaches to where B λ has been defined accordingly. The flow and jump sets, are now 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 H p jumps, system H also jumps and at these instants x = x p . For this, consider the pair (t, j p ) such that system H p jumps (subscript p in j has 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) = x p (0, 0) and with the same control law κ(x) = κ(x p ), ∀x = x p . Moreover, consider that each subsystem is Hurwitz. Then, at the time instants when system (9)-(10) jumps, i.e., when ξ p (t, j p ) ∈ D p , for a given j p , system (4)-(7) also jumps, i.e., there exists an integer j such that Proof. We proceed by induction showing that if at a given time t when system (9)-(10) jumps, x(t, j) = x p (t, j p ) for certain values of j and j p , then at the following jump of this system, the state verifies that Fig. 2). For this, integrating the dynamics of x p in (9)-(10) between two consecutive jumps and using Assumption 2 the following equation is reached (14), and because at t = 0 it holds that x(0, 0) = x p (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 λ and τ , instead of, λ p and τ p will be used.
Notice that the flow equation that governs the dynamics of x p in (9)-(10), (16)- (17), iṡ This equation plays a similar role to the one of averaged models used usually in power electronics. Indeed, the use of variable λ instead of σ avoids 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 A λ = A 0 + λ(A 1 − A 0 ), instead of the logarithm definition above, and B p = B 0 + λ(B 1 − B 0 ), 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-andhold mechanism is included in the model. Notice also that the non-hybrid model (18) (or its continuous versionẋ p = A λ x p + B p ) is not a "standard" affine problem since the transmission matrix e A λ Tp (or A λ ) depends on the control input λ. Now, rewriting the continuous and discrete-time dynamics of x p will lead to a new Lyapunov function candidate. To do so, define Γ q = Aq Bq 0 0 , such that during the flows: d dt

VOLUME 4, 2016
Hence, the following Lyapunov function candidate is considered where W is a quadratic function of [x p 1] , defined as follows, with andP := [ P 0 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 Note that if the solution evolves in the interior of A in steady state, its associated x p is bounded.
The particular hybrid arcs starting from the origin after a jump are defined as follows: Note that these arcs start from x p = 0 and describe a solution of x in the time arcs, [t jp , t jp+1 ] with (t jp , j p ) ∈ 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 λ e associated with the operating point x e such that Assumption 1 is satisfied and matrices P 0 ∈ R n , Q 0 ∈ R n such that Q P and M P − Q ∈ R n satisfying Consider system (9)-(10), (16)- (17), with control law Then, there exists a value T * p > 0 such that for 0 < T p < T * p the following statements hold: Hybrid system H p (f p , g p , C p , D p ) with control law (27) is well-posed, because it verifies: • C p and D p are closed sets in H p . • f p is a continuous function, thus it is locally bounded and outer semi-continuous. Moreover, it is convex for each ξ p ∈ C p . • g p is 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 (x p , λ, τ ) (21) is continuous in C p ∪ D p and locally Lipschitz near each point in C p \A. Moreover, V (x p , λ, τ ) is strictly positive definite with respect to (C p ∪ D p )\A and radially unbounded. Likewise, it verifies, by definition, V (x p , λ, τ ) = 0, for all (x p , λ, τ ) in A.
The next step of the proof is to ensure that the time derivative of V along flows outside of A is non positive (or more precisely in this case, equal to zero). More formally, the objective is to show that The last equality comes from ∂ ∂τ P(λ, τ ) + P(λ, τ ) He(Γ λ ) = 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 = j p T p for j p ∈ N, the following notation will be adopted here according the hybrid time domain (8): x p,jp = x p (j p T p , j p ), λ jp = λ(j p T p , j p ), τ jp = τ (j p T p , j p ) that corresponds to the variables right before the jump at t = j p T p and λ + jp = λ(j p T p , j p + 1), τ + jp = τ (j p T p , j p + 1) right after the same jump. In the same way, the definition ∆V = V (x + p,jp , λ + jp , τ + jp )−V (x p,jp , λ jp , τ jp ) will be used. However, note that x + p,jp = x p,jp , τ + jp = 0 and τ jp = T p . Now, ∀(x p , λ, τ ) ∈ D p \A the following equations hold, Notice that the manipulable signal, λ jp+1 = λ + jp has to be computed at t = j p T p . For this, it is convenient to write ∆V in terms of x p,jp instead of x p,jp+1 . Hence, from (20), the following relationship is obtained We have beingΨ λj p := Ψ −1 λj p . We highlight that λ + jp , which depends on x p,jp according to the definition of H p , is associated with the value of x p at the jump instant. Indeed, (x p,jp , λ + jp , 0) refers here to the initial value in each hybrid arc. In the sequel, possibly with abuse of notation, λ and x p will be used to represent λ + jp and x p,jp , respectively.
From (30), it is not easy to verify that ∆V < 0. However, only small values of T p are of interest, and hence, one just can analyse the following limit When T p → 0, the following approximation can be usedΨ with B λ + := B 0 + (B 1 − B 0 )λ + . Then, it holds that which is achieved neglecting T 2 p terms, both in (31) and in (32).
On the other hand, in order to evaluate the behavior of lim Tp→0 ∆V for small T p the following computation holds, The last step is reached from the next property stemmed from Assumption 1 and the error equation (3).
It is worth to remind that λ + ∈ [0, 1]. Now, consider that B 0 P x p = 0, then The last condition stems from the fact that LMIs (25)- (26) are satisfied. Now, taking into account B 0 P x p = 0, there are three distinct cases: • 0 < λ + < 1: Inserting (27) in (33), applying conditions LMI (25)- (26), M P − Q given in the Theorem statement and assuming that λ + is not saturated in (33), yields • λ + = 0: This case takes place when κ(x p ) ≤ 0, being (33) 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 0 P x p < 0 (necessary for the argument of (27) to be negative or zero), being (36) negative ∀ξ p ∈ D p \A from the fact that condition (25) is satisfied. Finally, from applying P Q, (36) yields Secondly, if Q M 0, then Consequently, from M P − Q the following holds lim Tp→0 ∆V T p < −x p P x p < 0 ∀ξ p ∈ D p \A.
• λ + = 1: In this case, corresponding to the case when κ(x p ) is saturated in its upper bound, (33) yields Once again, two situations are particularized, M 0 as well as Q M 0. The saturation of the expression of κ(x p ) at λ + = 1 with M 0 can only VOLUME 4, 2016 happen because 2B 0 P x p > 0, which implies from the equilibrium equation (34) Therefore, (37) becomes Now, for the case Q M 0 and noting that which yields the following condition Then, Hence, for a given M 0 or Q M 0, it holds that The last step is to prove that A is an invariant set, i.e., g(A ∩ D p ) ⊂ A (remember that W does not change when (x p , λ, τ ) ∈ C p ∪ D p ). To do so, remember that the following expression has been obtained Moreover, note that W (x p , λ + , 0) = x p P x p . Then, some manipulations yield Therefore, W (x p , λ + , 0) is negative in the jumps for any (x p , λ, τ ) ∈ A. Hence, if the solution to H p reaches 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 T p , A is UGAS.

Remark 2:
The UGAS property guaranteed for system H p implies 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 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 M and 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 1 guarantees the convergence to x p ≤ ε(T p ) when T p is small enough, but it is desirable to quantify the size of the attractor (related to the amplitude of the chattering) as a function of T p . Property 1: Consider Assumption 1 as well as Theorem 1 assumptions are satisfied and a given T p small enough. Then,ε is an approximation of the following upper bound where x ss p denotes the steady state of x p of system (9)-(10),(16)- (17). Proof. First, it is easy to see that W (x p , λ, τ ) is decreasing with respect to τ ∈ [0, T p ]. Moreover, consider T p small enough and λ e satisfying Assumption 1 such that the following first-order approximation is obtained applying (31) and neglecting T 2 p . Now, if Theorem 1 assumptions are satisfied, it holds that where p M and p m are the maximum and minimum eigenvalues of P and x ss p P = x ss p P x ss p . It is worth noting, from Property 1, that if T p decreases, x ss p also decreases. Property 2: Consider that Assumption 1 is satisfied, and a given T p small enough such that Property 1 is valid. Then, for a given positive parameter ε > 0, the state x in steady state, denoted as x ss is limited by Then, in order to find upper bounds on χ in steady state, the worst cases bill be analyzed, i.e., the time subintervals inside the sampling period where x ss >ε (assuming that Property 1 provides a validε). Along those intervals, Now, it must be proved that the upper bound on the right hand side of the χ dynamics directly implies the upper boundedness of the solution of (41) by the solution of the bounding equation (42). For this, let us define χ 1 (t) = χ(x ss 0 )e 2αt as the solution to the bounding equation (42) and χ 2 (x ss 2 ) as the solution to the actual system (41). Then, if This means that χ 1 being greater or equal than χ 2 at 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 x ss ε > 1. If this does not hold, thenε is an even stricter bound than the one that Property 2 suggests. In that case, the integration time of this proof would start when x ss exits that bounding interval, leaving less time for escaping from it. Finally, χ 1 (t) ≥ χ 2 (t) implies that In that expression, the bounding exponential e 2αt has only been allowed to evolve for a time period of T p /2, as it is the maximum time distance from the interval edges 0, T p , where theε bound holds. It was also assumed χ(x ss 0 ) = χ 1 (x ss 0 ) = χ 2 (x ss 0 ) ≤ε 2 , concluding the proof.

B. DISCUSSION
Theorem 1 establishes stability and performance properties for (4)- (7). Moreover, there are still degrees of freedom in the selection of matrices M and Q, available for closed-loop performance tuning.

1) Selection of Q
On the one hand, for a selected matrix Q 0 ∈ R n , the feasibility problem composed of (25)-(26) provides a matrix P necessary to compute the upper bound of x ss p defined in Property 1. Hence, the selection of Q can 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 = [i L v C ] , σ = 0 when the switch S is closed and σ = 1 when this switch is open, The parameters inside these expressions are given in Table 1.   Fig. 4 compares the state evolutions with M = 0.5Q 0, M = 0 and M = −0.5Q ≺ 0. As mentioned in item 2, Section V-B, if M 0 the rise time is reduced, but the control signal will have a tendency to saturate, yielding a strong oscillating behaviour in transient time. In face of this, choosing M ≺ 0 reduces the current peak and the control input is not saturated, but the system dynamics become slower. Moreover, note that the particular case M = 0, provides λ + = λ e , as mentioned in Remark 3. Fig. 5 shows some evolutions with M ≺ 0. As M is more negative, the response time of the signal is slightly slower. Moreover, with M 0, as M is larger the signal control also tends to saturate, as shown in Fig. 6. Note that the control input does not saturate, but the high overshoot generated in the current signal can harm the converter.

VII. EXPERIMENTAL SETUP
A test setup was built to validate the proposed hybrid control scheme. Fig. 7 shows this experimental set up. It is composed of: • A boost converter whose electrical parameters are given in Table 2. • An electronic card with a current sensor (model LEM LTS 15-NP) for the measurements of the inductor current, and a voltage sensor for the measurement of the output voltage. We built the voltage sensor by means of a resistor divider connected with an operational amplifier in buffer configuration. • A dSPACE card (DS1103) that includes a Pow-erPC 604e at 400 MHz and a fixed-point DSP TMS320F240. The code of the control algorithm was generated by Matlab coder ® and it is automatically optimized for running in the dSPACE card.    We selected for these tests M = −0.5Q, after several simulation trials. Moreover, the switching frequency was the same than the one taken in the simulation section, 100kHz. Fig. 8 shows a startup transient from a initial condition equal to x 0 = 0 V in to a reference operating point computed by imposing an output voltage equal to V out = 100V. The voltage and current signals present a smooth behaviour without current peaks, as shown in simulation (see Fig. 4). Likewise, the steady-state operation is shown in Fig. 9. The measured errors of the voltage and current provide x ss = 3.01, whereas the computed estimated upper bound of (40) isε = 3.54 and εe αTp 2 = 9.77. In order to verify robustness of the proposed control system and to test its dynamic response in some different scenarios, three tests have been performed. In the first one, the input voltage is changed from 24V to 20V and the results are given in Fig. 10. This figure shows the ability of the system of ensuring an output voltage regulation even when an input voltage variation occurs. In a second test, the load R 0 was changed from R 0 = 50Ω to R 0 = 75Ω at t =0.02s (see Fig. 11). Note that here, differently from the previous test where the input voltage variation was easily measured and used in the computation of the new equilibrium point, the resistance value cannot be measured, and an error in the voltage output can be exhibited in steady state. This is due to the fact that the proposed algorithm does not guarantee an output voltage regulation when a load variation happens. However, if an external loop is added with a PI controller as in [39], the output voltage is regulated at its reference value, maintaining a suited performance as shown in Fig. 11. Finally, Fig. 12 shows the variable trajectories when the system suffers a perturbation due to a transition from no-load to the nominal one. The new test is more challenging because there is a 100% variation of the nominal load.

VIII. CONCLUSION
A hybrid model of switched power converters composed of two functioning modes, triangular-carrier PWM inputs and a sample-and-hold mechanism has been presented here. This study can be extended to other power converters with more than two functioning modes. The dynamic equations are then simplified to an equivalent system with trajectories that match those of the original one at the start of the sampling intervals, but with fewer jumps than the original one, and a reduced state vector. Moreover, a rigorous control law to select the value of the duty cycle at the beginning of each PWM-sampling interval has been proposed. Stability in closed loop has been established via hybrid dynamical systems theory. Finally, an estimation of the chattering peaks reached by the system state is provided. Experimental results show satisfactory closed-loop performance. PWMs with more than 2 modes will be considered in future works. Moreover, a stability analysis with an external control loop that controls the output voltage is also expected.