Ryan P C De Souza

Member, IEEE Zohra Kader
email: kader@laplace.univ-tlse.fr

Member, IEEE Stéphane Caux
email: caux@laplace.univ-tlse.fr

Switched affine systems with hysteresis-based switching control: application to power converters

Many power electronic converters can be modeled using the theoretical framework of switched affine systems, which considers the instantaneous dynamics of the converter instead of its averaged model. In this class of systems, a certain number of subsystems is present and a control law is designed to orchestrate the switching among them. When the switching law is state-dependent, the closed-loop system is stabilized at the desired equilibrium through sliding mode dynamics, leading to infinite-rate switching. In this paper, a control strategy is proposed using constant-width hysteresis as a way of bounding the switching frequency at a finite value when two subsystems are present. In addition, an upper bound on the hysteresis width is provided so that the system is not stabilized at the equilibrium of a subsystem. Finally, simulation results illustrating the application of the proposed method to power electronic converters are presented. The relation between the switching law parameters and the current ripple is discussed for a dc-dc Boost converter and a Buck converter.

I. INTRODUCTION

In the past decades, there has been a surge in the interest in the field of switched affine systems, which are characterized by a finite number of different possible modes (or subsystems), and at each time instant one of these dynamics is selected by a suitably designed control law [START_REF] Hetel | Local stabilization of switched affine systems[END_REF], [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF], [START_REF] Kader | Control and observation of switched affine systems[END_REF], [START_REF] Albea-Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF]. Moreover, the dynamics of each subsystem are given by an affine function of the state. One motivation for their study is that they are able to adequately model systems in a wide range of applications, including power electronic converters [START_REF] Albea-Sanchez | Robust hybrid control law for a boost inverter[END_REF], [START_REF] Deaecto | Switched affine systems control design with application to dc-dc converters[END_REF]. Indeed, by using the controlled switches present in a power converter, it is possible at each time instant to choose a different dynamical evolution dictated by the other elements of the circuit. This allows to bypass the need for a PWM module between the control signal and the switches.

However, the analysis of switched affine systems and their control remain challenging topics, due especially to the fact that each subsystem has its own equilibrium point, and the goal is to stabilize the system at a point that is not the equilibrium of any individual subsystem. Moreover, when the switching law is state-dependent, the closed-loop switched system is a discontinuous system that may present sliding modes [START_REF] Cortes | Discontinuous dynamical systems[END_REF].

In [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF], a state-dependent switching control law that globally stabilizes switched affine systems has been presented. The control law is based on a quadratic Lyapunov function and is obtained by solving a linear matrix inequality (LMI), for which efficient numerical methods are available. In general, the desired equilibrium point can only be reached by infinitely switching between different modes. This phenomenon is known as sliding motion and it is undesirable from a practical viewpoint since it leads to component wear and possibly even its destruction. In order to overcome this first drawback, this approach has been improved using different tools. In [START_REF] Albea-Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF], a variable-width hysteresis, which bounds the switching frequency both during transient behavior and in steady state has been used. In [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF], a sampled-data statedependent switching law has been proposed. Nevertheless, to the best of our knowledge, when applied to power converters the existing results do not provide an explicit relation between the controller parameters and the current and voltage ripples.

In the present paper, the switching strategy developed in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF] is applied to the control of power converters modeled as switched affine systems having two modes. Here, the issue of infinite-rate switching is overcome by means of a constant-width hysteresis. In addition, an upper bound on the hysteresis parameter is presented in order to avoid stabilization at the equilibrium point of a subsystem. The main advantage of the proposed technique is that the switching frequency in steady state can be easily chosen by the designer in order to respect constraints on the switching device or limit current and voltage ripples. This method presents an alternative to the ones proposed in [START_REF] Albea-Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF] for the case with two modes. Indeed, in [START_REF] Albea-Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF], the switching frequency can be large during transient behavior. Moreover, the choice of the switching frequency in steady state (which determines the current ripple) is not obvious. The method proposed in the current paper provides a way of addressing these shortcomings in applications where they may be undesirable. Another contribution of this paper regards the evolution of the switching frequency during transient behavior. Indeed, in some cases the frequency remains practically constant over the whole trajectory, which is desirable in practice since electromagnetic disturbances may be avoided this way.

This paper is organized as follows. In Section II, some preliminary notions are presented and the problem to be tackled is stated. In Section III, the ideal sliding dynamics associated with the method in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF] are briefly discussed. Then, in Section IV, the proposed hysteresis-based strategy is developed. Simulation results illustrating the technique are shown in Section V. Then, concluding remarks are presented in Section VI.

Notation: If v is a vector, v (j) denotes its j-th component. If M is a matrix, then M T denotes its transpose, and M ≺ 0 (resp. M 0) means that M is negative-definite (resp. positive-definite). The set formed of the real parts of the eigenvalues of M is denoted as Re (λ(M )). Given α ∈ [0, 1] and matrices M i , i = 1, 2, the convex combination of these matrices defined by α is denoted as M (α) := αM 1 + (1α)M 2 . Set R + is defined as R + := [0, ∞). The gradient of function s(x) is denoted as ∇s(x).

II. PRELIMINARIES

Consider switched affine systems expressed in the following form:

ż(t) = A σ(z(t)) z(t) + B σ(z(t)) , (1)
where z(t) ∈ R n denotes the system state at instant t, and σ : R n → {1, 2} is the state-dependent switching law, which selects at each time instant the active mode among the two available ones. Matrices A j ∈ R n×n and B j ∈ R n , j = 1, 2, with B 1 = B 2 , are the system parameters. The goal is to stabilize (1) at an admissible equilibrium point z ∈ R n (see, e.g. [START_REF] Albea-Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF]). It is convenient to define a coordinate shift given by x = z -z , so that stabilization can be analyzed with respect to the origin in x-coordinates. Using these new coordinates, the system is written as:

ẋ(t) = A σ(x(t)) x(t) + b σ(x(t)) , (2)
where b j := A j z + B j , j = 1, 2, and σ(x) := σ(z) is the switching function in x-coordinates. Since, by definition, the origin is an equilibrium point of (2), then, according to [START_REF] Hetel | Local stabilization of switched affine systems[END_REF], there exists α ∈ (0, 1) such that:

b(α ) = α b 1 + (1 -α )b 2 = 0. (3)
Consider the following assumption. Assumption 1: There exists P = P T 0 such that:

A(α ) T P + P A(α ) ≺ 0. (4)
It has been shown in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF] that, under Assumption 1, the switching law σ(x) ∈ arg min j∈{1,2}

x T P (A j x + b j )

(5) globally exponentially stabilizes system (2) to the origin.

A. Application to power converters

The switched system under study is composed of two modes. This is often the case of converters that have one controlled switch and operate in continuous current mode.

The focus in this paper is on DC/DC converters where the input voltage remains constant and the load is described by a constant resistance. These conditions, which are verified in a large number of applications, allow the converters addressed here to be modeled as switched affine systems. The dynamics of each mode are given by differential equations obtained by applying Kirchhoff's laws to the circuit according to whether the switch is closed or open.

The equilibrium point is chosen according to the desired operating point of the converter. This choice can be made, for instance, to impose a certain output voltage.

Remark 1: Equation ( 3) can be viewed as the dynamical equation of the state-averaged model (see [START_REF] Erickson | Fundamentals of power electronics[END_REF]) evaluated in steady state. By adopting the convention that mode 1 is the one where the switch is closed, then α directly provides the duty cycle in equilibrium.

With the desired equilibrium z and its correspondent value of α , matrix P is obtained by solving the LMI (4).

B. Problem statement

The downside of control law [START_REF] Albea-Sanchez | Robust hybrid control law for a boost inverter[END_REF] is that it only stabilizes the desired equilibrium point by switching infinitely fast between both modes. In this paper, the goal is to answer the following questions: how can the switching law (5) be adapted to control a power converter modeled with (1) while bounding the switching frequency at a finite value by using an hysteresis-based strategy? And how to do so in a way that makes it easy for the designer to specify the switching frequency in steady state (and hence the acceptable current ripple)?

The use of constant-width hysteresis may stabilize the system at the equilibrium of an individual mode, instead of the desired equilibrium z . This behavior is undesirable for two reasons: (i) a static error occurs; and (ii) switching does not occur as the system stabilizes at one particular mode, which may be a practical problem since converters are not designed to work under this condition. An additional question that the paper aims to answer is then: how can stabilization at the wrong equilibrium point be avoided?

III. CHARACTERIZATION OF THE SLIDING MOTION

Before tackling the problems at hand, the ideal dynamics of system (2) in sliding behavior are briefly discussed in this section.

According to [START_REF] Albea-Sanchez | Robust hybrid control law for a boost inverter[END_REF], switching only occurs when x T P (A 1 x+ b 1 ) = x T P (A 2 x + b 2 ). As a result, the state space is divided in two regions: one where mode 1 is active (R 1 ) and one where mode 2 is active (R 2 ). The surface separating both regions is switching surface R 1,2 := {x ∈ R n : s(x) = 0}, where:

s(x) := g(x) T P x, (6)
with g(x

) := (A 1 -A 2 )x + b 1 -b 2 . It is easy to see that R 1 = {x ∈ R n : s(x) < 0} and R 2 = {x ∈ R n : s(x) > 0}. If, for some t ≥ 0, x(t) ∈ R j , j ∈ {1, 2}
, then mode j is selected and the system evolves according to the dynamics of mode j. When the trajectory reaches R 1,2 , it may simply switch to the other mode and cross to the other side of R 1,2 . Alternatively, the trajectory may switch infinitely fast between both modes and slide on R 1,2 . The conditions for this to happen are discussed in the sequel. First, according to [START_REF] Hetel | Local stabilization of switched affine systems[END_REF], system (2) can be rewritten as:

ẋ = A(α )x + g(x)u σ(x) , (7)
where u 1 := 1 -α and u 2 := -α . Details on how these values were obtained can be found in [START_REF] Hetel | Local stabilization of switched affine systems[END_REF]. When the trajectory slides on R 1,2 , the dynamics in sliding mode are given by:

ẋ = A(α )x + g(x)ū(x), (8)
where ū(x) ∈ (u 2 , u 1 ) is the equivalent control signal [START_REF] Utkin | Sliding modes in control and optimization[END_REF] responsible for driving the trajectory along R 1,2 . When a sliding mode occurs on R 1,2 , the condition ṡ(x) = ∇s(x) T ẋ = 0 holds, with ẋ given by [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF]. Thus:

∇s(x) T ẋ = ∇s(x) T (A(α )x + g(x)ū(x)) = 0. (9)
The expression for ū(x) is then determined as:

ū(x) = - ∇s(x) T A(α )x ∇s(x) T g(x) , (10)
for every x ∈ R n such that ∇s(x) T g(x) = 0.

When a sliding motion occurs on x ∈ R 1,2 , then ū(x) calculated in [START_REF] Erickson | Fundamentals of power electronics[END_REF] lies inside the interval (u 2 , u 1 ). Therefore, the set S ⊂ R 1,2 , where sliding dynamics are actually possible, is given by:

S = {x ∈ R n : s(x) = 0, ū(x) ∈ (u 2 , u 1 )}. (11)

IV. CONSTANT-WIDTH HYSTERESIS

In order to prevent infinite-rate switching from arising as a consequence of sliding behavior, control law ( 5) is adapted in the sequel. Instead of switching as soon as s(x) = 0, switching is stopped until |s(x)| reaches a certain value h > 0, called the hysteresis parameter. Thus, the hysteresis-based switching law σ h : R n → {1, 2} is recursively defined as follows. Initially, at time t = 0, σ h (x(0)) = σ(x(0)) and, for all t > 0:

σ h (x(t)) = σ(x(t)), if |s(x(t))| ≥ h, lim τ →t -σ h (x(τ )), otherwise. , (12)
where σ(x) is given by [START_REF] Albea-Sanchez | Robust hybrid control law for a boost inverter[END_REF]. Since h > 0, the trajectory takes a non-zero time between any two successive switching instants, thus bounding the switching frequency at a finite value. The solution with hysteresis x(t) is obtained by replacing the switching law σ(x) with σ h (x) in the closedloop system [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF]. Note that the hysteresis width 2h is constant with respect to s(x), and not the state itself. As a consequence of hysteresis, the trajectory bounces between the surfaces s(x) = h and s(x) = -h around S. The hysteresis band H is then defined as

H := {x ∈ R n : |s(x)| ≤ h, ū(x) ∈ (u 2 , u 1 )}.
The following assumption is made.

Assumption 2: The hysteresis parameter is sufficiently small so that, if x ∈ H, then x belongs to a neighborhood of some y ∈ S where the linear approximation of s(y) by y is valid.

The main consequence of Assumption 2 is that the trajectory does not stray too far from S while in the hysteresis band. A sufficiently small value for h means that the resulting switching frequency is sufficiently high to consider that the dynamics of each mode is a constant vector while the mode is active when the trajectory is inside H.

Remark 2: Assumption 2 is analogous to the traditional claim made in power electronics that the switching frequency is much higher than the bandwidth of the system. This assumption is then commonly verified in practice, specially considering that the current ripples are chosen to be kept at a small level by the designer in practical applications. In light of Assumption 2, the time behavior of s(x) is approximated by a sequence of line segments, as shown in Fig. 1. The times spent in each mode change for each y ∈ S around which the trajectory x(t) evolves while in the hysteresis band. In Fig. 1, t j (y) denotes the time where mode j is active, j = 1, 2 , while the trajectory x(t) evolves in the vicinity of y ∈ S. By convention, ṡ(x) > 0 (resp. < 0) when mode 1 (resp. 2) is active.

Definition 1: Function f sw : S → R + associates with each y ∈ S the switching frequency as the trajectory x(t) evolves around y while x(t) ∈ H, i.e., f sw (y) = 1/(t 1 (y) + t 2 (y)).

Theorem 1: The switching frequency f sw (y) associated with y ∈ S is bounded and is given by the following expression:

f sw (y) = - 1 2h (A 1 y + b 1 ) T ∇s(y)∇s(y) T (A 2 y + b 2 ) ∇s(y) T g(y) . (13)
Proof: Under Assumption 2, if x ∈ H, there is a y ∈ S such that x ≈ y and thus s(x) ≈ s(y). From Fig. 1, the time derivative of s(y) can be written as follows:

ṡ(y) = 2h t1(y) , if mode 1 is active, -2h t2(y) , otherwise. , y ∈ S. (14)
From (2), ṡ(y) = ∇s(y) T (A j y + b j ) while mode j is active. Using (14), the following expressions are obtained:

t 1 (y) = 2h ∇s(y) T (A 1 y + b 1 ) , ( 15
) and t 2 (y) = - 2h ∇s(y) T (A 2 y + b 2 ) , (16)
for y ∈ S. As a consequence of the adopted convention for ṡ(x), t 1 (y) and t 2 (y) are always positive, implying that f sw (y) is also positive. From [START_REF] Deaecto | Switched affine systems control design with application to dc-dc converters[END_REF], the gradient of s(y) can be calculated as:

∇s(y) = [(A 1 -A 2 ) T P +P (A 1 -A 2 )]y +P (b 1 -b 2 ). ( 17
)
According to Definition 1, f sw (y) = 1/(t 1 (y) + t 2 (y)). Then, (13) is obtained from (15), ( 16) and (17). Note in [START_REF] Utkin | Sliding modes in control and optimization[END_REF] that S only contains points where ū(y) is defined, implying that ∇s(y) T g(y) = 0, ∀y ∈ S. Since, in addition, h > 0, f sw (y) is bounded. Equation (13) provides a way of estimating the switching frequency f sw (y) as the trajectory switches around a point y ∈ S. This means that it is possible to determine the possible values for the switching frequency during the transient behavior of the system. It is interesting to remark that the switching frequency is inversely proportional to the hysteresis parameter. As h → 0 (no hysteresis), the frequency tends to infinity, as expected.

Corollary 1: In steady state, the switching frequency in steady state f ss is given by:

f ss = 1 2h b T 1 P (b 1 -b 2 )(b 1 -b 2 ) T P b 2 (b 1 -b 2 ) T P (b 2 -b 1 ) . ( 18
)
Proof: As a consequence of Assumption 1, system (2) is asymptotically stabilized at the origin through infinite switching if hysteresis is not present. Because of Assumption 2, the solution with hysteresis evolves (in steady state) in a neighborhood of the origin y = 0 where the linear approximation is valid. Using Theorem 1, the expression for the steady-state frequency f ss is then obtained by setting f ss = f sw (0) in (13).

Using (18), it is possible to determine the parameter h for a desired switching frequency in steady state. Alternatively, h can be determined to ensure a certain ripple for currents or voltages in this regime. Indeed, around the origin, dynamics (2) can be approximated by ẋ = b j while mode j is active (due to Assumption 2). Integrating this expression while mode 1 is active yields:

X q = |b 1(q) |t 1 (0), ( 19
)
where X q is the steady-state peak-to-peak amplitude of the ripple in the switching frequency of the q-th state component, q = 1, . . . , n, which is usually an inductor current or a capacitor voltage. The absolute value of b 1(q) is taken in (19) so that the resulting amplitude is positive. The same result could be obtained by carrying out the integration while mode 2 is active: in this case, X q = |b 2(q) |t 2 (0). It can be shown using ( 3), ( 15) and ( 18), that t 1 (0) = α /f ss , reflecting the statement made in Remark 1 that α is the duty cycle in equilibrium. Equation ( 19) is then rewritten as:

X q = |b 1(q) |α f ss . ( 20
)
It is important to keep in mind that each individual mode j ∈ {1, 2} has its own equilibrium point x j , which is the solution of A j x j + b j = 0. If A j is Hurwitz, then x j attracts the system trajectory. By using switching law (5) in the ideal case without hysteresis, this is not a problem, because the trajectory always reaches S and then slides along S before ever stabilizing at the equilibrium point of any mode. In the case with hysteresis, if h is too high, then there is a risk that the trajectory will cross R 1,2 and stabilize at some x j before switching occurs again. Fig. 2 shows how this situation may arise in the case where n = 2.

In order to avoid the situation depicted in Fig. 2, the value of h must be constrained to be less than an upper bound

x (1) x (2) 0 s(x) = h s(x) = 0 s(x) = -h x j
Fig. 2. Example of a trajectory that is stabilized at x j , for some j ∈ {1, 2}, instead of the desired equilibrium x = 0.

denoted as h. To find h, note that x j must lie outside H for all j ∈ {1, 2} such that mode j is asymptotically stable. If mode j is not stable, then the system cannot stabilize at x j . Let J H denote the set of stable modes, i.e., J H := {j ∈ {1, 2} : max Re (λ(A j )) < 0}. Therefore, it must be ensured that |s(x j )| > h for all j ∈ J H . Or equivalently, the minimum value of |s(x j )| considering all j ∈ J H must be greater than h. If J H = ∅, then h must be chosen so that:

h < h = min j∈J H |s(x j )|. (21)
If, on the other hand, J H = ∅, then h does not need to be constrained with respect to the problem illustrated in Fig. 2. Usually, in practice, both modes are stable due at least to the parasitic resistances in the circuit. Thus, it may be considered that in general J H = {1, 2}.

It is important to recall here that h must be small anyway so that Assumption 2 holds. Nonetheless, (21) imposes an additional constraint which is specially useful when the duty cycle α is close to 0 or 1, since in this case the desired switching equilibrium is close to the equilibrium point of one of the modes.

V. SIMULATION RESULTS

In this section, the proposed constant hysteresis-based switching method is illustrated by simulations on MATLAB of two classical converter configurations: the Buck and the Boost converters.

A. Buck converter

The circuit corresponding to the Buck converter is shown in Fig. 3. Consider that the state vector z(t) is formed by the inductor current i(t) and the capacitor voltage v(t). Then, z(t) = [i(t) v(t)] T . Matrices A j and B j in (1), j = 1, 2, are given as follows:

A 1 = -R/L -1/L 1/C -1/(R o C) , B 1 = E/L 0 , (22)
A 2 = A 1 =: A, B 2 = 0. (23)
Matrices b 1 and b 2 in (2) are given by b 1 = (1-α )B 1 and b 2 = -α B 1 . The possible equilibrium points z = [i v ] are determined using (3). Thus, α = (Ri + v )/E and v = R o i . This implies that v can be written as: The steady-state current ripple is given by X 1 . Using (20), the following expression is obtained:

v = R o R + R o α E. ( 24
) - + E R L i(t) R o C + - v(t)
∆I := X 1 = α (1 -α )E Lf ss . ( 25
)
Suppose that the current ripple in steady state is allowed to be ∆I = 1A. From (25), the steady-state frequency is calculated as f ss = 49.9kHz. By using (18), the value of h is determined as h = 70212. From (21), h = 7.4 × 10 6 , meaning that indeed h < h.

Fig. 4 shows the trajectory in the simulation in zcoordinates, wherein the initial condition has been considered to be z(0) = [5 20] T . In Fig. 5, which shows current in steady-state operation, the ripple amplitude ∆I indeed equals 1A.

It is interesting to note that (13) provides a way of estimating the switching frequency along the line s(y) = 0. By writing y (1) as a function of y (2) using equation s(y) = 0, it is possible to write (13) only as a function of y [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF] . It can be written just as easily in z-coordinates, which is how Fig. is plotted. This figure should be interpreted as follows: if the state trajectory is passing around point y ∈ S while switching inside H with a voltage equal to z (2) , then Fig. 6 provides the estimation for the switching frequency at that point. In this particular case, it can be seen that the frequency remains practically constant along the switching line, showing how the constant hysteresis-based strategy can be advantageous in this example.

B. Boost converter

The circuit of the Boost converter, whose parameter values are the same as in Section V-A, is shown in Fig. 7. Matrices A j and B j , j = 1, 2, in this case, are given by:

A 1 = -R/L 0 0 -1/(R o C) , B 1 = E/L 0 , (26)
A 2 = -R/L -1/L 1/C -1/(R o C) , B 2 = B 1 . (27)
Vectors b 1 and b 2 in (2) are given by b

1 = [(E - Ri )/L -v /(R o C)] T and b 2 = [(E -v -Ri )/L (i - v /R o )/C] T .
Suppose that the converter is to be stabilized at v = 120V. Then, i = 3.1A and α = 0.22. It can be shown that these values are in accordance with (3). Matrix P has been

- + E R L i(t) R o C + - v(t)
∆I = α (E -Ri ) Lf ss , (28)
from which f ss = 40.9kHz. The corresponding value of h is h = 73748, which has been verified to be less than h = 4.3 × 10 6 . A simulation has been run with initial condition z(0) = [5 60] T . Fig. 8 shows that, in steady state, the current ripple is indeed equal to ∆I = 1A. In Fig. 9, the state trajectory is plotted. It can be seen that the system stabilizes at the desired equilibrium point z . Initially, x(0) ∈ R 1 and mode 1 is activated. Then, the trajectory reaches R 1,2 and, once s(x(t)) = h, mode 2 is activated but the trajectory does not start switching around R 1,2 . Instead, it ventures further into R 2 . This happens because x(t) has not reached H yet, since ū(x(t)) / ∈ (u 2 , u 1 ) at this point. It can be seen in Fig. 9 that the trajectory only enters H around z (2) = 70V, at which point switching between both modes around S starts. Fig. 10 shows the switching frequency along S. Even though it is not constant over S as in the case of the Buck converter, the method still provides an upper bound to the frequency during transient behavior: 67kHz, which occurs when the trajectory is switching in H and z (2) = 88V.

VI. CONCLUSIONS AND FUTURE WORKS

In this paper, a strategy to limit the switching frequency while using a state-dependent switching law for switched affine systems has been introduced. This strategy is based on a constant-width hysteresis band and it has been shown that the proposed method can be attractive from a design viewpoint with respect to its application to power converters. In fact, with the approach presented in this paper, one can determine the equilibrium point from the desired output voltage and then use the proposed expressions for calculating the hysteresis parameter from the desired switching frequency or current ripple in steady state. In addition, it is possible to determine all possible switching frequencies during transient behavior in the hysteresis band.

A future line of research is how to extend the results presented in this paper to more complex converter configurations, where more than one switch is present in the circuit. An adaptation of the method to the case where the load may change over time is also envisaged.

Fig. 1 .

 1 Fig. 1. Plot of s(x(t)) as a function of time during hysteresis.

Fig. 3 .

 3 Fig. 3. Electrical circuit of the Buck converter supplying a load Ro.

Fig. 4 .

 4 Fig. 4. Trajectory in the state space of the Buck converter.

6

 6

Fig. 5 .

 5 Fig. 5. Zoom on the current ripple in the Buck converter in steady state.

Fig. 6 .

 6 Fig. 6. Switching frequency in the Buck converter while the trajectory lies in the hysteresis band.

Fig. 7 .

 7 Fig. 7. Electrical circuit of the Boost converter supplying a load Ro.

Fig. 10 .

 10 Fig. 10. Switching frequency in the Boost converter while the trajectory lies in the hysteresis band.

  Fig. 8. Zoom on the current ripple in the Boost converter in steady state.Fig. 9. Trajectory in the state space of the Boost converter. determined to be P = [p 1 p 2 ] T , with p 1 = [0.66 0.97] T and p 2 = [0.97 3.34] T .Assume once again that the ∆I should be limited to 1A. Analogously to what has been done for the Buck converter, ∆I is expressed as:

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				4.9 4.91 4.92 4.93 4.94 4.95 4.96 4.97 4.98 4.99	5
									10 -3
		120
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		0	2	4	6	8	10	12	14	16	18	20
	10	12	14			16			18			20


Ryan P C De Souza

Member, IEEE Zohra Kader

Member, IEEE Stéphane Caux

Switching control applied to interconnected Boost converters: A comparison with hysteresis current control

In this paper, the stabilization of the voltage at the output of Boost converters is studied using a switching control method developed in the field of switched affine systems. This control law uses the instantaneous model instead of the nonlinear average model of the converter and has certain advantages compared to traditional approaches. Despite these advantages, the closed-loop system suffers from the occurrence of sliding modes, leading to unacceptable rises in the switching frequency. An adaptation of this control law is proposed in this paper to overcome this problem, and the adopted strategy is based on the introduction of hysteresis in the switching law. Moreover, the extension to Boost converters in parallel is also shown, with the attractive property of each converter only needing local measurements for global stabilization of the system. A comparison with the hysteresis current control method is conducted in simulations.

I. INTRODUCTION

Due to the objectives of mitigating the rise in greenhouse gas emissions in electricity generation, there has been a growing demand for renewable sources. Depending on the application, they can be integrated in a DC microgrid, which is a solution that is increasingly popular due to a number of reasons, including higher efficiency and a more natural interface with many types of renewable sources [START_REF] Dragičević | Dc microgrids-part I: A review of control strategies and stabilization techniques[END_REF].

In some scenarios, the energy sources connected to a microgrid produce a lower voltage compared with the voltage at the DC bus. This is often the case with photovoltaic panels [START_REF] Li | Review of nonisolated high-step-up dc/dc converters in photovoltaic grid-connected applications[END_REF]. In this case, Boost converters are suitable for interfacing the sources with the microgrid [START_REF] Fusheng | Power converters for dc microgridsmodelling and simulation[END_REF]. This raises the question of how to control each converter in order to stabilize the main bus at the desired voltage reference.

In order to tackle the challenge of global stabilization of the DC microgrid, a switching control law introduced in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF] is used in this paper. This control method is based on the instantaneous model of the converters, instead of the more traditional approach of using their (often nonlinear) average models. Moreover, the switching law can be deduced by solving a Linear Matrix Inequality (LMI), for which efficient solvers are available. Two advantages of this method are: (i) Pulse-Width Modulation (PWM) is not required to generate the gate signal to each switching device, since the control law can already provide it, and (ii) linearization of the model is not necessary (as it is normally the case when working with Boost converters, whose average models are nonlinear) and global stabilization can be theoretically ensured. The main issue with the control method in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF] is that it leads

The authors are with LAPLACE -Université de Toulouse, 118 route de Narbonne 31062 Toulouse cedex 9, France {pitanga,kader,caux}@laplace.univ-tlse.fr to sliding modes (with ideally infinite-rate switching), and thus the switching frequency rises to unacceptable values, damaging the switches.

The aim of this paper is to overcome the issue of infinite switching frequency by employing hysteresis. In addition, the proposed hysteresis-based switching method applies only local measurements, meaning that the resulting control law is decentralized. This can be interesting in practical situations where communication between different source units is impaired or non-existent.

This paper is organized as follows. In Section II, the instantaneous models of the two systems addressed here are presented. In Section III, the method proposed in this paper is introduced for both systems presented in Section II. Simulation results comparing the proposed method with hysteresis current control are shown in Section IV. Finally, conclusions are drawn in Section V.

Notation: If w is a vector, then

w (k) denoted its k-th component. If Q is a matrix, Q T is its transpose, and Q ≺ 0 (resp. Q 0)
means that Q is negative-definite (resp. positive-definite). The null matrix of q rows and r columns is denoted as 0 q×r . Given a positive integer N , the unit simplex ∆ N is defined as

∆ N := {δ ∈ [0, 1] N : N k=1 δ (k) = 1} and set I N is defined as I N := {1, 2, . . . , N }. Given matrices Q k , k = 1, 2, . . . , N , their convex combination defined by δ ∈ ∆ N is denoted as Q(δ) := N k=1 δ (k) Q k .

II. SYSTEM MODELS

In this section, the mathematical models of the two converter configurations addressed in this paper are presented. It is assumed that the converters always operate in continuous current mode. First, consider the equation describing the dynamics of a switched affine system [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF], [START_REF] Deaecto | Switched affine systems control design with application to dc-dc converters[END_REF], [START_REF] Albea-Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF]:

ż(t) = Aσz(t) + Bσ, (1)
where σ is a discrete variable taking values in the set I M , with M denoting the number of modes in the system. Vector z is the state vector, and matrices A σ and B σ , σ = 1, . . . , M , are the system parameters. At each time instant, the system evolves according to the dynamics dictated by one of the modes, and the system can always switch from one mode to another at any time.

A. Single Boost converter

Consider the idealized circuit of the Boost converter depicted in Fig. 1. Let i(t) be the current flowing in inductor L (i.e., the input current) and v(t) be the output voltage.

-

+ E L i(t) R o S C + - v(t)
Fig. 1: Boost converter supplying a load R o .

Since in this paper the instantaneous model is considered, the model is determined from the current and voltage laws applied to the circuit according to whether switch S is closed or not. By introducing the discrete variable σ ∈ I 2 , with σ = 1 corresponding to the situation where S is closed and σ = 2 to when S is open, the equation governing the system dynamics is given by (1), with state z(t) being equal to z(t) = [i(t) v(t)] T in this case and with:

A 1 = 0 0 0 -1/(RoC) , B 1 = E/L 0 , (2)
A 2 = 0 -1/L 1/C -1/(RoC) , B 2 = B 1 . (3)
Note that the number of modes is M = 2. Define i and v as the input current and the output voltage in equilibrium, respectively. The equilibrium points of (1) can be determined using the formalism of discontinuous systems (see, for instance, [START_REF] Kader | Control and observation of switched affine systems[END_REF]). Using this formalism, z = [i v ] T is an equilibrium of (1) if there exists δ ∈ ∆ M such that:

M σ=1 δ (σ) (Aσz + Bσ) = 0, (4)
Let α := δ (1) ∈ [0, 1]. Since δ (2) = 1 -α , then (4) can be written as:

α (A 1 z + B 1 ) + (1 -α ) (A 2 z + B 2 ) = 0. (5)
Note that ( 5) is exactly the same equation of the statespace average dynamics [START_REF] Erickson | Fundamentals of power electronics[END_REF] in equilibrium, wherein α corresponds to the duty cycle in steady state. Therefore, the equilibrium of the instantaneous model (1) can be characterized in the same way as in the case of the nonlinear average model. From [START_REF] Deaecto | Switched affine systems control design with application to dc-dc converters[END_REF], one can easily obtain the classical relationship between the input and steady-state output voltages:

v = E 1 -α . (6)
The inductor current reference i can be calculated as:

i = v (1 -α )Ro . ( 7
)
It is also useful to define the dynamics in equilibrium in mode σ as b σ := A σ z + B σ , σ = 1, 2.

B. Boost converters in parallel

In Fig. 2, the connection of two Boost converters feeding a load is shown. Each one is connected to the load through an LC filter modeled by capacitance C j and inductance L j , j ∈ {1, 2}. Note that the parasitic resistances associated with this inductor is represented by R j . Parameters L j and R j -

+ E1 L1 i1(t) L 1 i 1 (t) R 1 Ro S1 C1 + - v1(t) Co + - v(t) - + E2 L2 i2(t) L 2 i 2 (t) R 2 S2 C2 + - v2(t)
σ σ 1 σ 2 1 1 1 2 1 2 3 2 2 4 2 1
could also be used to model the inductance and the resistance of the cables connecting each converter to the DC bus, to which the load R o is connected. The capacitance of the bus is modeled by C o . The state vector, for this system, is given by z

(t) = [z 1 (t) T z 2 (t) T v(t)] T , where z j (t) = [i j (t) v j (t) i j (t)] T
represents the local state variables of the j-th converter, j = 1, 2. Note that the output current of each converter (i j ) has been included as a local variable.

Since two switches are present in the circuit, σ can take 4 different values, each one corresponding to a different combination of switches S 1 and S 2 . Thus, the number of modes is M = 4. Table I shows the state of each switch σ j , j = 1, 2, for each mode σ ∈ I 4 . The convention adopted here is that σ j = 1 when S j is closed and σ j = 2 when it is open.

Using the circuit laws once again for each different mode, the dynamics are dictated by [START_REF] Dragičević | Dc microgrids-part I: A review of control strategies and stabilization techniques[END_REF], and are written here as 1 :

ż =   ż1 ż2 v   =   g 1,σ 1 (z 1 , v) g 2,σ 2 (z 2 , v) v   = Aσz + Bσ, (8)
with g j,σj (z j , v) defined as the dynamics of the local variables of converter j, and matrices A σ and B σ , σ = 1, . . . , 4, are in this case given by:

A1 =   A1,1 03×3 03×3 A2,1 W U ao   , (9)
A2 =   A1,1 03×3 03×3 A2,2 W U ao   , (10)
A3 =   A1,2 03×3 03×3 A2,2 W U ao   , (11)
1 σ 1 and σ 2 are associated with mode σ according to Table I.

A4 =   A1,2 03×3 03×3 A2,1 W U ao   , (12)
and

B1 = B2 = B3 = B4 =      E1/L1 02×1 E2/L2 02×1 0      , (13)
where:

Aj,1 :=   0 0 0 0 0 -1/Cj 0 1/L j -R j /L j   , j = 1, 2, (14)
Aj,2 :=   0 -1/Lj 0 1/Cj 0 -1/Cj 0 1/L j -R j /L j   , j = 1, 2, (15)
W := 0 0 -1/L 1 0 0 -1/L 2 T , (16)
U := 0 0 1/Co 0 0 1/Co , (17)
and

ao := - 1 RoCo . (18)
By using once again the formalism of discontinuous systems, the equilibrium points z satisfy (4), with M = 4. Let α 1 := δ (1) + δ (2) and α 2 := δ (1) + δ (4) denote the duty cycles of converters 1 and 2, respectively. Then, the equilibrium points z satisfy the following relations:

v j = E j 1 -α j , j = 1, 2, (19)
i j = (1 -α j )i j , j = 1, 2, (20)
v j = v + R j i j , j = 1, 2, (21)
v = Ro(i 1 + i 2 ). (22)
There is a certain freedom with respect to how the load current is shared between the output currents of both converters. For the purposes of this paper, it is simply assumed that i 2 = κi 1 , where κ ≥ 0 is a design parameter. Thus, (22) becomes:

i 1 = v Ro(κ + 1) . ( 23
)
The dynamics in equilibrium for each mode of converter j is simply defined as b j,σj := g j,σj (z j , v ), σ j = 1, 2, j = 1, 2.

Remark 1: It can be seen that the modeling procedure for the converters in parallel is modular, in the sense that the parameter matrices can be algorithmically constructed from blocks corresponding to each module. This makes the modeling process scalable to a large number of converters in parallel, as in the case of a DC microgrid. Note also that more complex configurations for linear output filters can be considered through proper choices for matrices A j,1 , A j,2 , j = 1, 2, and W . This means that the control law discussed in the next two sections may be used in more complex scenarios.

III. HYSTERESIS-BASED SWITCHING CONTROL A. Ideal switching law for switched affine systems

The switching law presented in the sequel has been proposed in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF] for global stabilization of switched affine systems. Firstly, assume that the following LMI is feasible:

A(δ ) T P + P A(δ ) ≺ 0, (24)
for some P = P T 0. Vector δ is obtained according to the desired equilibrium point. For example, in the case of a single Boost converter, δ = [α 1-α ] T and α is the duty cycle determined from [START_REF] Deaecto | Switched affine systems control design with application to dc-dc converters[END_REF] to obtain the desired steady-state output voltage v .

With a matrix P satisfying (24), the following control law exponentially stabilizes system (1) at z [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF]:

σ(z) ∈ arg min m∈I M (z -z ) T P (Amz + Bm).
(25)

In the system models used in this paper, B 1 = B 2 = . . . = B M . With this in mind, (25) can be simply written as:

σ(z) ∈ arg min m∈I M (z -z ) T P Amz. ( 26
)
Since σ(z) only depends on z, it is said that ( 26) is a state-dependent control law. It is interesting to remark that the application of control law (26) does not require the system to be linearized around the desired equilibrium point. And yet, as shown in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF], the equilibrium point is globally exponentially stable. This contrasts with the use of more traditional methods, where linearization of the average model is employed and thus stability can only be ensured in a neighborhood of the equilibrium. Moreover, since (26) directly selects the mode that should be activated, a PWM module generating the gate signal for the switching device is not needed.

B. Adding hysteresis

The main drawback of the switching strategy described by ( 26) is that it leads to sliding modes [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF]. Thus, direct application of (26) causes the switching frequency to rise to unacceptable levels, possibly damaging the switching devices. The proposed method for overcoming this issue is based on an adaptation of control law (26) to incorporate hysteresis and the development of this method is carried out for the particular cases of the two systems presented in Section II. The newly obtained control shall be referred to as Hysteresis-Based Switching Control (HBSC).

1) Single Boost converter: In the case of one Boost converter, there are only two modes. Define s(z) := (zz ) T P Dz, where D := A 1 -A 2 . The application of (26) implies the following:

• σ(z) = 1, if s(z) < 0; • σ(z) = 2, if s(z) > 0; • σ(z) ∈ {1, 2} if s(z) = 0.
On the switching surface s(z) = 0, there are two possibilities for the evolution of the system trajectory: (i) the mode is switched from one to the other and the trajectory simply crosses the surface to the other side; (ii) a sliding mode occurs along the switching surface.

The HBSC can be described as follows: instead of switching as soon as the trajectory reaches the switching surface, switching is stopped while |s(z)| < h, where h > 0 is a design parameter. When |s(z)| ≥ h, then the mode is chosen according to (26). In the sequel, the link between h and the steady-state switching frequency is made clear.

Due to hysteresis, the behavior of the system in steady state is characterized by periodical oscillations in the current and voltage. Just as in traditional methods in power electronics, the system spends a certain time t 1 in mode 1 and a time t 2 in mode 2. The sum of t 1 and t 2 gives the switching period, whose inverse is the switching frequency f ss .

It is assumed that h is sufficiently small so that the time evolution of s(z) is piecewise linear. This assumption is analogous to the statement often made in power electronics that the switching frequency is very high with respect to the bandwidth of the system. Thus, in order to keep the current or voltage ripples at acceptable levels, this assumption often holds in practice. Since, during hysteresis, s(z) varies between -h and h (and thus the absolute variation of s(z) is 2h):

| ṡ(z)| = 2h tσ , σ = 1, 2, (27)
for z in a small neighborhood of z . The gradient of s(z) is given by:

∇s(z) = (D T P + P D)z -D T P z . (28)
In the neighborhood of z , ṡ(z) can be approximated by ṡ(z ) = ∇s(z ) T b σ while in mode σ. Since ∇s(z ) = P Dz , the following expressions are obtained from (27):

tσ = 2h |z T D T P bσ| , σ = 1, 2. (29)
From ( 29) and the fact that f ss is the inverse of the sum of t 1 and t 2 , the following expression is obtained:

fss = 1 2h |b T 1 P Dz z T D T P b 2 | |b T 1 P Dz | + |b T 2 P Dz | . ( 30
)
The acceptable current ripple ∆I is linked to the switching frequency by the same expression that is normally used in traditional methods:

∆I = α E Lfss . (31)
Thus, from (30) and ( 31), h can be determined to bound the current ripple at a value ∆I chosen by the designer.

2) Boost converters in parallel: In the case of the Boost converters in parallel, there are 4 modes instead of 2. This means that there are more than one switching surface, and each one may involve all state variables of the system. However, a procedure is proposed in the sequel to implement hysteresis separately in each converter, where only local measurements are used in the control law. As it will be seen, only one switching surface may be considered for each converter, and thus hysteresis may be implemented in a similar way as for a single converter. Firstly, rewrite matrices A σ and B σ , ∀σ ∈ I 4 , as follows:

Aσ = Ā 0 6×6 W U ao + Ã1,σ 1 A 1,σ 1 0 3×4 0 4×3 0 4×4 + +   0 3×3 0 3×3 0 3×1 0 3×3 A 2,σ 2 0 3×1 0 1×3 0 1×3 0   Ã2,σ 2 , ( 32
)
Bσ = B := B 1 , (33)
where the correspondence between σ and σ j , j = 1, 2, is given by Table I. The objective function in (26) can be written as (zz ) T P ( Āz+ B+ Ã1,σ1 z 1 + Ã2,σ2 z 2 ). Minimizing this expression is equivalent to simultaneously solving the minimization problems:

σ j (z) ∈ arg min m∈I 2 (z -z ) T P Ãj,m z, j = 1, 2, (34)
which directly give the values σ j corresponding to the state of each controlled switch S j . Once again, the mode σ can be recovered by association of each pair (σ 1 , σ 2 ) with each σ ∈ I 4 (see Table I). Now, consider (24) with P taking the following blockdiagonal form:

P =   P 1 0 3×3 0 3×1 0 3×3 P 2 0 3×1 0 1×3 0 1×3 p 3   , (35)
where P 1 , P 2 ∈ R 3×3 and p 3 ∈ R. This choice for the form of P has been made so that switching law (26) can be used in such a way that the control signal for each converter depends only on its local variables. The zeroes in P are there to cancel the dependence of the control law on variables of the other converter or the DC bus. The LMI (24) can then be solved for the variables P 1 , P 2 and p 3 . The solution of this problem is straightforward in commercial softwares such as MATLAB.

It can be shown that (34), with P taking the form (35), can be written as:

σ j (z j ) ∈ arg min m∈I 2 (z j -z j ) T P j A j,m z j , j = 1, 2. (36)
Note that control law (36) can directly provide the gate signal for each converter j and only requires local measurements from each one. Moreover, since (36) has been derived from (26) with a matrix P that is a solution of (24), then the whole closed-loop system is globally exponentially stable.

Similarly to what has been done for a single converter, define s j (z j ) := (z j -z j ) T P j D j z j , where D j := A j,1 -A j,2 , for j = 1, 2. The application of the ideal control law (36) for each converter j implies that:

• σ j (z j ) = 1, if s j (z j ) < 0; • σ j (z j ) = 2, if s j (z j ) > 0; • σ j (z j ) ∈ {1, 2}, if s j (z j ) = 0.
Therefore, the surface s j (z j ) = 0 can be recognized as the switching surface corresponding to each converter j, and it only depends on the local variables z j . Thus, by defining parameters h j > 0, j = 1, 2, hysteresis can be implemented for each converter in a similar fashion as it has been done for a single Boost converter in Section III-B.1. Under the assumption that h j is sufficiently small and developing the equations for | ṡj (z j )| as in Section III-B.1, f ss,j is determined as:

f ss,j = 1 2h j b T j,1 P j D j z j z j T D T j P j b j,2 |b T j,1 P j D j z j | + |b T j,2 P j D j z j | , j = 1, 2. (37)
The acceptable current ripple for each converter (∆I j ) is calculated in the same way as in (31). Thus:

∆Ij = α j Ej Ljfss,j , j = 1, 2. (38)
Remark 2: Applying control law (36) with hysteresis boils down to comparing the absolute value of s j (z j ) (whose expression is simply given by a matrix product) with h j , which can be advantageous in a real-time application. Moreover, P is computed offline.

IV. SIMULATION RESULTS

In this section, results from MATLAB simulations concerning the application of the methods developed in this paper are presented. In order to highlight their interest in the case of Boost converters, comparisons are made with current hysteresis control, briefly explained hereafter.

A. Current hysteresis control

Suppose that a certain current ripple ∆I is acceptable in the inductor of a single Boost converter. Then, Current Hysteresis Control (CHC) consists in the following:

• If the current i(t) is lower than i -∆I/2, then activate mode 1; • If the current i(t) is higher than i +∆I/2, then activate mode 2; • If i(t) is in the hysteresis band defined by |i(t) -i | ≤ ∆I/2, then keep the same mode as before. The use of CHC is widespread in the field of power electronics in general. Similarly to the technique discussed in this paper, CHC does not require a PWM module either. In the case of two converters, the procedure outlined above is carried out independently for each converter.

B. Single Boost converter

The parameters of the Boost converter represented in Fig. 1 II shows some performance indexes using both control techniques 2 . It can be seen that the HBSC 2 In this paper, the response time is considered to be the earliest time instant tr such that the voltage remains bounded within ±5% around v for all t ≥ tr. performed better in terms of current and voltage overshoot, as well as response time. Thus, the HBSC technique has brought improvements to the system response in transient behavior. Moreover, the steady-state behavior is the same using both methods. Indeed, it is shown in Fig. 3 that the current ripple is 5A in both cases. Fig. 5 shows the trajectory in state space using both methods.

C. Two Boost converters in parallel

The circuit parameters in Fig. 2 are the following:

E 1 = E 2 = 400V, L 1 = 10mH, L 2 = 8mH, L 1 = 1mH, L 2 =
and p 3 = 0.013. The acceptable current ripples are chosen to be ∆I 1 = 0.8A and ∆I 2 = 1.5A, corresponding to h 1 = 3.0 × 10 5 and h 2 = 5.2 × 10 5 , according to (38). The steadystate switching frequencies are f ss,1 = 17kHz and f ss,2 = 11.4kHz. Figs. 6 and 7 show the plots obtained for i 1 (t) and v(t), respectively. Current i 2 (t) is not shown due to lack of space, but its evolution is qualitatively similar to i 1 (t). Table III shows some performance indexes in this case. Note that the performance improvements seen in the case of a single converter have also been observed for the parallel association. Once again, the overshoots are smaller using the HBSC and the response time of the output voltage is faster.

V. CONCLUSIONS AND FUTURE WORKS

In this paper, a control law that is well known in the switched systems literature has been adapted to include an hysteretic behavior, thus preventing unacceptable rises of the switching frequency. This control law has been applied to Boost converters and to their parallel interconnection. It has been shown that the switching frequencies can be easily chosen through the choice of the hysteresis parameters. In simulations where a comparison is established with a classical method that does not require PWM modulation either, the method presented in this paper has performed better as far as the transient behavior is concerned. Indeed, it can be seen in Tables II and III that the response time and the peak values are more satisfactory using this method. The steady-state behavior has been shown to be similar to the more traditional method. In addition, the gate signal to be selected at the current time step is determined only from local measurements and from the signal that had been formerly selected. The next step in this research is to verify the results in practice on an experimental setup.

Fig. 2 :

 2 Fig. 2: Two Boost converters supplying load R o .

  are: E = 400V, L = 1mH, C = 10µF, and R o = 40Ω. The goal is to stabilize the voltage at v = 600V, which, from (5), corresponds to 1 -α = E/v = 2/3. From (7), i = 22.5A. The acceptable current ripple is taken to be ∆I = 5A. From (30) and (31), f ss = 27kHz and h = 19.8 × 10 7 . Using MATLAB, matrix P is determined from (24) as P = [ρ 1 ρ 2 ], with ρ 1 = [11.6 -0.002] T and ρ 2 = [-0.002 0.12] T . Figs. 3 to 5 show a comparison of the results obtained from each method. Table

Fig. 3 :Fig. 4 :

 34 Fig. 3: Evolution of input current i(t).

Fig. 5 :

 5 Fig. 5: System trajectory in the state space.

Fig. 7 :

 7 Fig. 7: Evolution of the DC bus voltage v(t).

TABLE I :

 I States of the switches in each mode σ ∈ {1, 2, 3, 4}.

TABLE II :

 II Performance indexes for the Boost converter.

							HBSC CHC
		Peak current (A)			37.6		44.6
		Peak output voltage (V)		625		687
		Response time (µs)		235		558
	20
	15
	10
	5
	0	0.5	1	1.5	2	2.5	3	3.5	4	4.5	5
											10 -3
	4			4.5					5
									10 -3

Fig. 6: Evolution of input current i 1 (t).

TABLE III :

 III Performance indexes for the parallel connection of two Boost converters.

				HBSC CHC
	Peak current in conv. 1 (A)	16.0	18.4
	Peak current in conv. 2 (A)	19.2	22.7
	Peak output voltage (V)	615	689
	Response time (µs)	848	1801
	0.6mH, C P 1 =	12.4 -0.010 -0.011 -0.004 -0.040 1.25 -0.004 0.013 -0.011	,	(39)
		10.2	-0.007 -0.028
	P 2 =	-0.028 -0.016 -0.007 0.019	0.75 -0.016	,

1 = C o = 10µF, C 2 = 15µF, R 1 = R 2 = 1Ω, and R o = 40Ω.

Here, the goal is also to stabilize the output voltage at v = 600V. Assume that the load current should be equally shared between both converters, leading to κ = 1. From (

19

)-(

23

), α 1 = α 2 = 0.34. A matrix P in the form (35) that solves (24) is obtained. P 1 P 2 and p 3 are given as: