# On the Discrete-Time Modeling of a DC-to-DC Power Converter and Control Design with Discrete-Time Sliding Modes

Authors: Jorge Rivera; Florentino Chavira; Alexander Loukianov
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This work presents a novel discrete-time modeling of a boost dc-to-dc power converter by means of the symplectic Euler method. Then, on the basis of this model, a discrete-time sliding mode regulator is designed in order to force the power converter to behave as a dc-to-ac power converter. Simulation and experimental results are carried on, where the great performance of the proposed methodology is verified.


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## Body

## 1. Introduction

Switched mode dc-to-dc converters as in [1, 2] are mainly used as constant current sources for LED, LED flashlights, industry lighting, mobile phones, and automotive applications as in [3], among other commercial devices. With respect to dc-to-ac power conversion using the traditional switched dc-to-dc power converter topologies has recently attracted the attention of researchers. So far, various control techniques, either linear or nonlinear, to regulate these converters have been proposed, such as I/O feedback linearization [4], linear designs [5], sliding mode control [6, 7], current-mode-control [8], artificial neural networks [9], fuzzy logic control [10], and passivity-based control [11], among others. Since dc-to-dc converters are absolutely nonlinear systems, for better performance, adopting nonlinear control methods is a good solution.It is worth mentioning that all of the previously cited works are designed in the continuous-time domain and have been digitally implemented. The recent advancements in digital microprocessor technology have rendered cheaper, simpler, and more flexible digital implementation of control algorithms designed in the continuous-time setting. The main problem is the degradation of the expected performance, due to the fact that the designing of the control law does not take into account the sampled dynamics of the system. This problem motivates the following possible solutions:(1)
the designing of the control law in the continuous-time setting and its implementation by means of zero-order holder;(2)
the exact sampling of the continuous-time system and the designing of the control law in the digital setting;(3)
the use of much simpler discretization methods, such as those due to Euler (explicit or implicit), and the designing of the control law in the digital setting.The first solution has the drawback of possible poor performance of the resulting sampled controller. For the second solution, the problem of exact sampling of continuous-time systems is not trivial. In fact, in general, a sampled closed representation of the sampled dynamics does not exist; whereas for linear systems, a sampled model in closed form can be easily obtained [12]; for nonlinear systems in general, the sampled data representations are given in the form of infinite series [13]. Hence, in practice, one uses truncated models of desired approximation [7]. The third solution [14] has the disadvantage that the accuracy of the resulting approximate discrete-time system decreases as the sampling period increases. But, there is an alternative to these approaches that is only valid for dynamical systems modeled by Euler-Lagrange methodology, the use of variational integrators as the symplectic Euler method. The advantages of this method are explained and compared with common Euler (explicit or implicit) methods in [15], and the application in robotics has been satisfactory developed in [16] and electric motors in [17].On the other hand, the sliding mode control [18] is a popular technique among control engineer practitioners due to the fact that introduces robustness to unknown bounded perturbations that belong to the control subspace; moreover, the residual dynamics under the sliding regime, that is, the sliding mode dynamics, can easily be stabilized with a proper choice of the sliding surface.  For an insightful treatment of the sliding mode technique, we refer the reader to the book entitled “Sliding mode control in electro-mechanical systems,” [19]. Electrical and electromechanical systems become vulnerable when output tracking signals present small oscillations of finite frequency known as chattering. The chattering problem is harmful because it leads to low control accuracy, high wear of moving mechanical parts, and high heat losses in power circuits. The chattering phenomenon can be caused by the deliberate use of classical sliding mode control technique. This control technique is characterized by a discontinuous control action with an ideal infinite frequency. When fast dynamics are neglected in the mathematical model such phenomenon can appear. Another situation responsible for chattering is due to implementation issues of the sliding mode control signal in digital devices operating with a finite sampling frequency, where the ideal infinite switching frequency of the control signal cannot be fully implemented, [20].Therefore, in this work one proposes to derive a novel sampled model for the boost converter by means of a variational integrator. For that reason, a Lagrangian formulation for the dc-to-dc boost converter must be first established in order to apply the symplectic Euler method. Then, on the basis of the obtained sampled model a discrete-time sliding mode controller is designed in order to force the dc-to-dc boost converter to behave as a dc-to-ac converter.The rest of this work is organized as follows: Section2 describes the discrete Lagrangian mechanics for conservative and nonconservative systems. Section 3 deals with the discrete-time modeling of the boost power converter by means of its Lagrangian formulation. The control algorithm is designed in Section 4 based on a discrete-time sliding mode regulator technique. Section 5 deals with the presentation of simulation and real-time results. Finally, some comments conclude the work.


## 2. Lagrangian Mechanics and Variational Integrators

In this section, the basic principles of Lagrangian mechanics and discrete Lagrangian mechanics are reviewed from a variational point of view.


### 2.1. Continuous-Time Lagrangian Mechanics

To define a mechanical system, letQ be a smooth manifold, called the configuration space, and let TQ be its tangent bundle, called the phase space. The Lagrangian is a function L:TQ→ℜ and is commonly defined as the kinetic energy K (usually, the only function of the velocity) minus the potential energy U of the system (usually, the only function of the state variable):
(1)L(q,q˙)=K(q˙)-U(q).The functional action is then introduced as the integral ofL along a path q(t) for time t∈[0,T]:
(2)S(q)=∫0T‍L(q,q˙)dt.With this definition, the main result of Lagrangian dynamics, Hamilton’s principle, can be expressed quite simply: this variational principle states that the correct path of motion of a dynamical system is such that its action has a stationary value; that is, the integral along the correct path has the same value as within first-order infinitesimal perturbations. As an integral principle this description encompasses the entire motion of a system between two fixed times (0and T in our setup). In more ways than one, this principle is very similar to a statement on the geometry of the path q(t): the action can be seen as the analog of a measure of curvature, and the path is such that this curvature is extremized (i.e., minimized or maximized).In order to determine the path which optimizes the action we take a variation. A variation of the pathq is written as δq and can be thought of as an infinitesimal perturbation to the path at each point, with the important property that the perturbation is null at the endpoints of the path. Computing variations of the action induced by variations δq of the path q(t) results in
(3)δS(q)=δ∫0T‍L(q(t),q˙(t))dtδS(q)=∫0T‍[∂L∂qδq+∂L∂q˙δq˙]dtδS(q)=∫0T‍[∂L∂q-ddt(∂L∂q˙)]δqdt+[∂L∂q˙δq]0T,
where integration by parts is used in the last equality. When the endpoints of q(t) are held fixed with respect to all variations δq(t) (i.e., δq(0)=δq(T)=0), the most right term in the previous equation vanishes. Therefore, the condition of stationary action for arbitrary variations ∂q with fixed endpoints stated in Hamilton’s principle directly indicates that the remaining integrand in the previous equation must be zero for all time t, yielding what is known as the Euler-Lagrange equation:
(4)∂L∂q-ddt(∂L∂q˙)=0.For a given Lagrangian, this formula will give the equations of motion of the system. Finally to account for nonconservative forces or dissipationF, the least action principle is modified as follows:
(5)δ∫0T‍L(q(t),q˙(t))dt+∫0T‍F(q(t),q˙(t))·δqdt=0,
this is known as the Lagrange-d’Alembert principle.


### 2.2. Discrete Mechanics and Variational Integrators

The idea of variational integrators was studied in [21, 22], among others, and a general theory was developed over the subsequent decade (see [23] for a comprehensive survey). The main idea when deriving discrete geometric mechanics is to handle the variational nature of mechanics and to preserve this variational structure in the discrete setting. In fact, very few integrators have a variational nature: the simpler Euler methods as the explicit and the implicit ones are not variational, and both are characterized by demanding small sampling periods in order to perform satisfactorily. Instead of simply approximating the Euler-Lagrange equation to first (or higher) order, one can directly discretize the variational principle behind it. That is, if one designs a discrete equivalent of the Lagrangian, then, a discrete Euler-Lagrange equation can be easily derived from it by paralleling the derivations followed in the continuous case. In essence, good numerical methods will come from discrete analogs to the Euler-Lagrange equation, that is, equations that truly derive from a variational principle.Suppose that we wish to discretize a mechanical system, whose continuous Lagrangian isL:TQ→ℜ, on the time interval t∈[0,T]. First, we partition the time interval by a finite number of points 0=t0,…,tN=T where k,N∈N; likewise, the path q:[0,T]→Q is replaced by a sequence of configurations q0,…,qN, where qk≈q(tk). Then the discrete Lagrangian is a function Ld:Q×Q→ℜ that approximates the Lagrangian L on each time interval [tk,tk+1], with h being the time interval between two samples h=tk+1-tk (chosen here to be constant for simplicity), so that
(6)Ld(qk,qk+1)≈∫tktk+1‍L(q,q˙)dt.
Now, the right-hand side integral can be approximated through a generalized midpoint quadrature rule; that is, by the length of the interval times the value of the integrand is evaluated somewhere between qk and qk+1 and with q˙ replaced by (qk+1-qk)/h:
(7)Ld(qk,qk+1,h)=hL((1-α)qk+αqk+1,qk+1-qkh),
where α∈[0,1] is parameter that generalizes the quadrature rule. For α=1/2, the quadrature is second-order accurate, while any other value leads to linear accuracy. Also, one can approximate the right-hand side integral through a generalized trapezoidal quadrature rule:
(8)Ld(qk,qk+1,h)=h(1-α)L(qk,qk+1-qkh)Ld(qk,qk+1,h)+hαL(qk+1,qk+1-qkh),
where again α∈[0,1] is parameter that generalizes the quadrature rule and for α=1/2, the quadrature is second-order accurate, meanwhile any other value leads to linear accuracy.Remark 1.
When the integrals are approximated by a generalized midpoint quadrature rule andα is fixed to 0 the variational integrator is known as the symplectic Euler method; when α is fixed to 1/2 the variational integrator is known as the midpoint Euler method. On the other hand, when the integrals are approximated by the generalized trapezoidal quadrature rule and α is fixed to 1/2 the variational integrator is known as the Störmer/Verlet method.Given the discrete Lagrangian, the discrete functional action becomes simply a sum:(9)Sd≔Sd({qi}i=0⋯N)=∑k=0N-1‍Ld(qk,qk+1)Sd≈∫ab‍L(q,q˙)dt=S(q).
Taking fixed-endpoint variations of this discrete action Sd, we obtain
(10)δSd=δ∑k=0N-1‍Ld(qk,qk+1)=∑k=0N-1‍D1Ld(qk,qk+1)·δqk+∑k=0N-1‍D2Ld(qk,qk+1)·δqk+1δSd=∑k=1N-1‍[D2Ld(qk-1,qk)+D1Ld(qk,qk+1)]·δqk+D1Ld(q0,q1)δq0+D2Ld(qN-1,qN)·δqN,
where D1L (resp., D2) denotes the partial derivative with respect to the first (resp., second) arguments of Ld. Using the fixed endpoint condition δq0=δqN=0, one gets
(11)δSd=∑k=1N-1‍[D1Ld(qk,qk+1)+D2Ld(qk-1,qk)]·δqk.
Setting this variation to be equal to 0 and noting that each δqk is arbitrary, we arrive at the discrete Euler-Lagrange (DEL) equation
(12)D1Ld(qk,qq+1)+D2Ld(qk-1,qk)=0.
Notice that this condition only involves three consecutive positions. Therefore, for two given successive positions qk and qk+1, (12) defines qk+2. Thus, these equations of motion are actually the algorithm for an integrator. And since the DEL equation derives from the extremization of a discrete action, such an algorithm enforces the variational aspect of the motion numerically.If we take initial conditions(q0,q1), then the discrete Euler-Lagrange equations define a recursive rule for calculating the sequence {qk}k=0N. Regarded in this way, they define a map (qk,qk+1)→(qk+1,qk+2) which we can think of as a one-step integrator for the system defined by the continuous Euler-Lagrange equations.We are primarily interested in discrete Lagrangian mechanics for deriving integrators. Any integrator which is the discrete Euler-Lagrange equation for some discrete Lagrangian is called a variational integrator. As we have seen previously, variational integrators can be implemented by taking two configurationsq0 and q1 of the system, which should approximate q0≈q(t0) and q1≈q(t0+h) and then solve the discrete Euler-Lagrange equation (12) for q2. This process can then be repeated to calculate an entire discrete trajectory. The map (qk-1,qk)→(qk,qk+1) defined by the discrete Euler-Lagrange equations is known as the discrete evolution map.In mechanics, the initial conditions are typically specified as a position and a velocity or momentum rather than two positions, hence it is beneficial to write (12) in a position-momentum form [24]. Therefore one defines the momentum at time tk to be
(13)pk≔D2Ld(qk-1,qk)=-D1Ld(qk,qq+1),
where the second equality holds due to (12). The position-momentum form of the variational integrator discussed previously is then given by
(14)pk=-D1Ld(qk,qk+1),(15)pk+1=D2Ld(qk,qk+1).For(qk,pk) known, (14) is an implicit equation whose solution gives qk+1. qk+1 is then substituted in (15) in order to find pk+1. This provides an update rule in phase space.Now we consider nonconservative systems, those with forcing and/or dissipation; in such case, the discrete action can be modified by adding the nonconservative force term and using the discrete Lagrange-d’Alembert principle as in [23]:
(16)δSd=δ∑k=0NLd(qk,qk+1)+∑k=0N‍[Fd-(qk,qk+1)·δqk+Fd+(qk-1,qk)·δqk+1]=0,
where Fd-(qk,qk+1) and Fd+(qk,qk+1) are discrete external forces acting, respectively, on the right of qk and on the left of qk+1. In other words, Fd-(qk,qk+1)·δqk+Fd+(qk-1,qk)·δqk+1 can be seen as a two-point quadrature approximation of the continuous forcing term, that is,
(17)Fd-(qk,qk+1)·δqk+Fd+(qk-1,qk)·δqk+1≈∫tkk+1‍F·δqdt.
It follows that the forced discrete Euler-Lagrange equation is as follows:
(18)D1Ld(qk,qq+1)+D2Ld(qk-1,qk)+Fd-(qk,qk+1)+Fd+(qk-1,qk)=0.
The forced discrete Euler-Lagrange equation can be expressed in a convenient, position-momentum form as follows:
(19)pk=-D1Ld(qk,qk+1)-Fd-(qk,qk+1),pk+1=D2Ld(qk,qk+1)+Fd+(qk,qk+1).
This variational treatment of energy decay, despite its simplicity, has also been proven to be superior to the usual time integration schemes that often add numerical viscosity to get stability [24].


## 2.1. Continuous-Time Lagrangian Mechanics

To define a mechanical system, letQ be a smooth manifold, called the configuration space, and let TQ be its tangent bundle, called the phase space. The Lagrangian is a function L:TQ→ℜ and is commonly defined as the kinetic energy K (usually, the only function of the velocity) minus the potential energy U of the system (usually, the only function of the state variable):
(1)L(q,q˙)=K(q˙)-U(q).The functional action is then introduced as the integral ofL along a path q(t) for time t∈[0,T]:
(2)S(q)=∫0T‍L(q,q˙)dt.With this definition, the main result of Lagrangian dynamics, Hamilton’s principle, can be expressed quite simply: this variational principle states that the correct path of motion of a dynamical system is such that its action has a stationary value; that is, the integral along the correct path has the same value as within first-order infinitesimal perturbations. As an integral principle this description encompasses the entire motion of a system between two fixed times (0and T in our setup). In more ways than one, this principle is very similar to a statement on the geometry of the path q(t): the action can be seen as the analog of a measure of curvature, and the path is such that this curvature is extremized (i.e., minimized or maximized).In order to determine the path which optimizes the action we take a variation. A variation of the pathq is written as δq and can be thought of as an infinitesimal perturbation to the path at each point, with the important property that the perturbation is null at the endpoints of the path. Computing variations of the action induced by variations δq of the path q(t) results in
(3)δS(q)=δ∫0T‍L(q(t),q˙(t))dtδS(q)=∫0T‍[∂L∂qδq+∂L∂q˙δq˙]dtδS(q)=∫0T‍[∂L∂q-ddt(∂L∂q˙)]δqdt+[∂L∂q˙δq]0T,
where integration by parts is used in the last equality. When the endpoints of q(t) are held fixed with respect to all variations δq(t) (i.e., δq(0)=δq(T)=0), the most right term in the previous equation vanishes. Therefore, the condition of stationary action for arbitrary variations ∂q with fixed endpoints stated in Hamilton’s principle directly indicates that the remaining integrand in the previous equation must be zero for all time t, yielding what is known as the Euler-Lagrange equation:
(4)∂L∂q-ddt(∂L∂q˙)=0.For a given Lagrangian, this formula will give the equations of motion of the system. Finally to account for nonconservative forces or dissipationF, the least action principle is modified as follows:
(5)δ∫0T‍L(q(t),q˙(t))dt+∫0T‍F(q(t),q˙(t))·δqdt=0,
this is known as the Lagrange-d’Alembert principle.


## 2.2. Discrete Mechanics and Variational Integrators

The idea of variational integrators was studied in [21, 22], among others, and a general theory was developed over the subsequent decade (see [23] for a comprehensive survey). The main idea when deriving discrete geometric mechanics is to handle the variational nature of mechanics and to preserve this variational structure in the discrete setting. In fact, very few integrators have a variational nature: the simpler Euler methods as the explicit and the implicit ones are not variational, and both are characterized by demanding small sampling periods in order to perform satisfactorily. Instead of simply approximating the Euler-Lagrange equation to first (or higher) order, one can directly discretize the variational principle behind it. That is, if one designs a discrete equivalent of the Lagrangian, then, a discrete Euler-Lagrange equation can be easily derived from it by paralleling the derivations followed in the continuous case. In essence, good numerical methods will come from discrete analogs to the Euler-Lagrange equation, that is, equations that truly derive from a variational principle.Suppose that we wish to discretize a mechanical system, whose continuous Lagrangian isL:TQ→ℜ, on the time interval t∈[0,T]. First, we partition the time interval by a finite number of points 0=t0,…,tN=T where k,N∈N; likewise, the path q:[0,T]→Q is replaced by a sequence of configurations q0,…,qN, where qk≈q(tk). Then the discrete Lagrangian is a function Ld:Q×Q→ℜ that approximates the Lagrangian L on each time interval [tk,tk+1], with h being the time interval between two samples h=tk+1-tk (chosen here to be constant for simplicity), so that
(6)Ld(qk,qk+1)≈∫tktk+1‍L(q,q˙)dt.
Now, the right-hand side integral can be approximated through a generalized midpoint quadrature rule; that is, by the length of the interval times the value of the integrand is evaluated somewhere between qk and qk+1 and with q˙ replaced by (qk+1-qk)/h:
(7)Ld(qk,qk+1,h)=hL((1-α)qk+αqk+1,qk+1-qkh),
where α∈[0,1] is parameter that generalizes the quadrature rule. For α=1/2, the quadrature is second-order accurate, while any other value leads to linear accuracy. Also, one can approximate the right-hand side integral through a generalized trapezoidal quadrature rule:
(8)Ld(qk,qk+1,h)=h(1-α)L(qk,qk+1-qkh)Ld(qk,qk+1,h)+hαL(qk+1,qk+1-qkh),
where again α∈[0,1] is parameter that generalizes the quadrature rule and for α=1/2, the quadrature is second-order accurate, meanwhile any other value leads to linear accuracy.Remark 1.
When the integrals are approximated by a generalized midpoint quadrature rule andα is fixed to 0 the variational integrator is known as the symplectic Euler method; when α is fixed to 1/2 the variational integrator is known as the midpoint Euler method. On the other hand, when the integrals are approximated by the generalized trapezoidal quadrature rule and α is fixed to 1/2 the variational integrator is known as the Störmer/Verlet method.Given the discrete Lagrangian, the discrete functional action becomes simply a sum:(9)Sd≔Sd({qi}i=0⋯N)=∑k=0N-1‍Ld(qk,qk+1)Sd≈∫ab‍L(q,q˙)dt=S(q).
Taking fixed-endpoint variations of this discrete action Sd, we obtain
(10)δSd=δ∑k=0N-1‍Ld(qk,qk+1)=∑k=0N-1‍D1Ld(qk,qk+1)·δqk+∑k=0N-1‍D2Ld(qk,qk+1)·δqk+1δSd=∑k=1N-1‍[D2Ld(qk-1,qk)+D1Ld(qk,qk+1)]·δqk+D1Ld(q0,q1)δq0+D2Ld(qN-1,qN)·δqN,
where D1L (resp., D2) denotes the partial derivative with respect to the first (resp., second) arguments of Ld. Using the fixed endpoint condition δq0=δqN=0, one gets
(11)δSd=∑k=1N-1‍[D1Ld(qk,qk+1)+D2Ld(qk-1,qk)]·δqk.
Setting this variation to be equal to 0 and noting that each δqk is arbitrary, we arrive at the discrete Euler-Lagrange (DEL) equation
(12)D1Ld(qk,qq+1)+D2Ld(qk-1,qk)=0.
Notice that this condition only involves three consecutive positions. Therefore, for two given successive positions qk and qk+1, (12) defines qk+2. Thus, these equations of motion are actually the algorithm for an integrator. And since the DEL equation derives from the extremization of a discrete action, such an algorithm enforces the variational aspect of the motion numerically.If we take initial conditions(q0,q1), then the discrete Euler-Lagrange equations define a recursive rule for calculating the sequence {qk}k=0N. Regarded in this way, they define a map (qk,qk+1)→(qk+1,qk+2) which we can think of as a one-step integrator for the system defined by the continuous Euler-Lagrange equations.We are primarily interested in discrete Lagrangian mechanics for deriving integrators. Any integrator which is the discrete Euler-Lagrange equation for some discrete Lagrangian is called a variational integrator. As we have seen previously, variational integrators can be implemented by taking two configurationsq0 and q1 of the system, which should approximate q0≈q(t0) and q1≈q(t0+h) and then solve the discrete Euler-Lagrange equation (12) for q2. This process can then be repeated to calculate an entire discrete trajectory. The map (qk-1,qk)→(qk,qk+1) defined by the discrete Euler-Lagrange equations is known as the discrete evolution map.In mechanics, the initial conditions are typically specified as a position and a velocity or momentum rather than two positions, hence it is beneficial to write (12) in a position-momentum form [24]. Therefore one defines the momentum at time tk to be
(13)pk≔D2Ld(qk-1,qk)=-D1Ld(qk,qq+1),
where the second equality holds due to (12). The position-momentum form of the variational integrator discussed previously is then given by
(14)pk=-D1Ld(qk,qk+1),(15)pk+1=D2Ld(qk,qk+1).For(qk,pk) known, (14) is an implicit equation whose solution gives qk+1. qk+1 is then substituted in (15) in order to find pk+1. This provides an update rule in phase space.Now we consider nonconservative systems, those with forcing and/or dissipation; in such case, the discrete action can be modified by adding the nonconservative force term and using the discrete Lagrange-d’Alembert principle as in [23]:
(16)δSd=δ∑k=0NLd(qk,qk+1)+∑k=0N‍[Fd-(qk,qk+1)·δqk+Fd+(qk-1,qk)·δqk+1]=0,
where Fd-(qk,qk+1) and Fd+(qk,qk+1) are discrete external forces acting, respectively, on the right of qk and on the left of qk+1. In other words, Fd-(qk,qk+1)·δqk+Fd+(qk-1,qk)·δqk+1 can be seen as a two-point quadrature approximation of the continuous forcing term, that is,
(17)Fd-(qk,qk+1)·δqk+Fd+(qk-1,qk)·δqk+1≈∫tkk+1‍F·δqdt.
It follows that the forced discrete Euler-Lagrange equation is as follows:
(18)D1Ld(qk,qq+1)+D2Ld(qk-1,qk)+Fd-(qk,qk+1)+Fd+(qk-1,qk)=0.
The forced discrete Euler-Lagrange equation can be expressed in a convenient, position-momentum form as follows:
(19)pk=-D1Ld(qk,qk+1)-Fd-(qk,qk+1),pk+1=D2Ld(qk,qk+1)+Fd+(qk,qk+1).
This variational treatment of energy decay, despite its simplicity, has also been proven to be superior to the usual time integration schemes that often add numerical viscosity to get stability [24].


## 3. Discrete-Time Modeling for a Boost Power Converter

In this section, a Lagrangian approach is used for deriving a novel discrete-time model for the boost power converter. The Lagrangian modeling is based on a suitable parametrization, in terms of the switch position parameter, of the EL functions describing each intervening system and subsequent application of the Lagrange formalism.


### 3.1. Discrete-Time Euler-Lagrange Modeling for a Boost Power Converter

Let us defineTu(q˙L) and Vu(qC) as the kinetic and potential energies of the circuit, respectively. One can denote by Fu(q˙) the Rayleigh dissipation function of the circuit and by Vu,nc(q) the nonconservative potential function. These quantities are taken from [25] where they readily are found to be(20)Tu(q˙L)=12Lq˙L2,Vu(qC)=12CqC2,Fu(q˙)=12R((1-u)q˙L-q˙C)2,Vu,nc(q)=-μTq,
where qL and qC are the circulating charge in the inductor and the electrical charge stored in the output capacitor, respectively, with q=(qL,qC)T; moreover, q˙Land q˙C are the corresponding currents with q˙=(q˙L,q˙C), the constant parameters are L as the inductance, C as the capacitance, and R as the resistance. The vector μ is defined as μ=(E,0)T, where E is the dc power supply value. Figure 1 shows an electric diagram of a boost power converter circuit.Figure 1
Boost converter circuit.Now, a nonconservative Lagrangian can be formulated as follows:(21)Lnc(q,q˙)=Lc(q,q˙)+∫0T‍F(q˙)dt-Vnc(q),
where Lc(q,q˙) is the conservative Lagrangian defined as usual:
(22)Lc(q,q˙)=Tu(q˙L)-Vu(qC)=12Lq˙L2-12CqC2.
Following  (7) one can obtain the discrete-time version of the conservative Lagrangian as
(23)Lc,d=h12L(qL,k+1-qL,kh)2-h12C((1-α)qC,k+αqC,k+1)2.
Moreover the discrete external forces Fd-(qk,qk+1) and Fd+(qk,qk+1) appearing in (19) are as follows:
(24)Fd-(qk,qk+1)=h(Qd(qk,qk+q)+μdT)(1-α),Fd+(qk,qk+1)=h(Qd(qk,qk+q)+μdT)(α),
where μd=(E,0)T and Qd are the sampled version of the external generalized forces of control input forces and dissipation forces, respectively. The vector of dissipation forces are obtained as
(25)Q=-∂Fu(q˙)∂q˙,
where its sampled version is as follows:
(26)Qd=(Qd,LQd,C)=∫tktk+1‍Qdt,
where again, the right-hand side integral can be approximated through a one-point quadrature, resulting as follows:
(27)Qd,L=-R((1-u)(qL,k+1-qL,kh))(1-u)Qd,L=+R(qC,k+1-qC,kh)(1-u),Qd,C=R((1-u)(qL,k+1-qL,kh))-R(qC,k+1-qC,kh).
After some algebraic steps and assigning α=0 (i.e., the symplectic Euler method), the updated rule for position shown in (19) yields to
(28)Pk=Aqk+1-Aqk+Bqk-V,
where Pk=(PL,k,PC,k)T and qk=(qL,k,qC,k)T; moreover,
(29)A=(Lh+R(1-u)2-R(1-u)-R(1-u)R),B=(000hC),V=(hE0).
Then, solving (28) for qk+1 results in
(30)qk+1=qk+A-1Pk-A-1Bqk+A-1V
and finally in scalar form:
(31)qL,k+1=qL,k-vkh2LCqC,k+hLPL,k+h2EL,qC,k+1=qC,k-(L+Rhvk2)hRLCqC,k+vkhLPL,k+vkh2EL,
where vk=1-uk.Now, the updated rule for momentum equation shown in (28) yields to
(32)PL,k+1=hL(qL,k+1-qL,k),PC,k+1=0.
Then, by replacing (30) in (32) it yields to
(33)PL,k+1=PL,k-vkhCqC,k+hE.
The variable qL,k can be omitted since it does not enter in the other equations. In the continuous-time model [25] for the boost power converter the first state variable is defined as the derivative of the circulating charge qL, that is, the input current represented as x1=q˙L. The second state variable is defined as the output voltage capacitor and it is represented as x2=qC/C where qC is the electrical charge stored in the output capacitor. Therefore, one proposes the following novel approximated discrete-time model for the boost power converter:
(34)x~1,k+1=x~1,k-vkhCx~2,k+hE,x~2,k+1=x~2,k-(L+Rhvk2)hRLCx~2,k+vkhLx~1,k+vkh2EL,x1,k=x~1,kL,x2,k=x~2,kC,
where x~1,k=PL,k,x~2,k=qC,k,x1,k≈x1(kh) is the sampled version of the input current, and x2,k≈x2(kh) is the sampled version of the output voltage capacitor.


## 3.1. Discrete-Time Euler-Lagrange Modeling for a Boost Power Converter

Let us defineTu(q˙L) and Vu(qC) as the kinetic and potential energies of the circuit, respectively. One can denote by Fu(q˙) the Rayleigh dissipation function of the circuit and by Vu,nc(q) the nonconservative potential function. These quantities are taken from [25] where they readily are found to be(20)Tu(q˙L)=12Lq˙L2,Vu(qC)=12CqC2,Fu(q˙)=12R((1-u)q˙L-q˙C)2,Vu,nc(q)=-μTq,
where qL and qC are the circulating charge in the inductor and the electrical charge stored in the output capacitor, respectively, with q=(qL,qC)T; moreover, q˙Land q˙C are the corresponding currents with q˙=(q˙L,q˙C), the constant parameters are L as the inductance, C as the capacitance, and R as the resistance. The vector μ is defined as μ=(E,0)T, where E is the dc power supply value. Figure 1 shows an electric diagram of a boost power converter circuit.Figure 1
Boost converter circuit.Now, a nonconservative Lagrangian can be formulated as follows:(21)Lnc(q,q˙)=Lc(q,q˙)+∫0T‍F(q˙)dt-Vnc(q),
where Lc(q,q˙) is the conservative Lagrangian defined as usual:
(22)Lc(q,q˙)=Tu(q˙L)-Vu(qC)=12Lq˙L2-12CqC2.
Following  (7) one can obtain the discrete-time version of the conservative Lagrangian as
(23)Lc,d=h12L(qL,k+1-qL,kh)2-h12C((1-α)qC,k+αqC,k+1)2.
Moreover the discrete external forces Fd-(qk,qk+1) and Fd+(qk,qk+1) appearing in (19) are as follows:
(24)Fd-(qk,qk+1)=h(Qd(qk,qk+q)+μdT)(1-α),Fd+(qk,qk+1)=h(Qd(qk,qk+q)+μdT)(α),
where μd=(E,0)T and Qd are the sampled version of the external generalized forces of control input forces and dissipation forces, respectively. The vector of dissipation forces are obtained as
(25)Q=-∂Fu(q˙)∂q˙,
where its sampled version is as follows:
(26)Qd=(Qd,LQd,C)=∫tktk+1‍Qdt,
where again, the right-hand side integral can be approximated through a one-point quadrature, resulting as follows:
(27)Qd,L=-R((1-u)(qL,k+1-qL,kh))(1-u)Qd,L=+R(qC,k+1-qC,kh)(1-u),Qd,C=R((1-u)(qL,k+1-qL,kh))-R(qC,k+1-qC,kh).
After some algebraic steps and assigning α=0 (i.e., the symplectic Euler method), the updated rule for position shown in (19) yields to
(28)Pk=Aqk+1-Aqk+Bqk-V,
where Pk=(PL,k,PC,k)T and qk=(qL,k,qC,k)T; moreover,
(29)A=(Lh+R(1-u)2-R(1-u)-R(1-u)R),B=(000hC),V=(hE0).
Then, solving (28) for qk+1 results in
(30)qk+1=qk+A-1Pk-A-1Bqk+A-1V
and finally in scalar form:
(31)qL,k+1=qL,k-vkh2LCqC,k+hLPL,k+h2EL,qC,k+1=qC,k-(L+Rhvk2)hRLCqC,k+vkhLPL,k+vkh2EL,
where vk=1-uk.Now, the updated rule for momentum equation shown in (28) yields to
(32)PL,k+1=hL(qL,k+1-qL,k),PC,k+1=0.
Then, by replacing (30) in (32) it yields to
(33)PL,k+1=PL,k-vkhCqC,k+hE.
The variable qL,k can be omitted since it does not enter in the other equations. In the continuous-time model [25] for the boost power converter the first state variable is defined as the derivative of the circulating charge qL, that is, the input current represented as x1=q˙L. The second state variable is defined as the output voltage capacitor and it is represented as x2=qC/C where qC is the electrical charge stored in the output capacitor. Therefore, one proposes the following novel approximated discrete-time model for the boost power converter:
(34)x~1,k+1=x~1,k-vkhCx~2,k+hE,x~2,k+1=x~2,k-(L+Rhvk2)hRLCx~2,k+vkhLx~1,k+vkh2EL,x1,k=x~1,kL,x2,k=x~2,kC,
where x~1,k=PL,k,x~2,k=qC,k,x1,k≈x1(kh) is the sampled version of the input current, and x2,k≈x2(kh) is the sampled version of the output voltage capacitor.


## 4. Discrete-Time Control for a Boost Power Converter

There are two options for controlling dc-to-dc power converters, the indirect and direct methods. The first method is characterized by controlling the inductor current where its reference signal is selected such that a desired capacitor voltage is obtained. This has the advantage that the system is minimum phase with respect to the inductor current variable, but the problem consists of guaranteeing the convergence of the output capacitor voltage to its reference signal. In the second method, the output capacitor voltage is directly controlled, where common converter topologies as the boost, buck-boost, and cúk result to be nonminimum phase systems with the main difficulty of stabilizing the unstable inductor current variable [25].Now, the control problem is formulated. The control problem consists of directly controlling the output capacitor voltage variablex~2,k to track a desired reference signal x~2r,k(ωk) and at the same time to stabilize the unstable inductor current variable.


### 4.1. Sliding Mode Regulator for a Boost Power Converter

Let us define the following tracking error:(35)ek=x~2,k-x~2r,k(ωk).
The reference signal is supposed to be generated by an autonomous exosystem given by
(36)ω1,k+1=cos(αh)ω1,k-sin(αh)ω2,k,ω2,k+1=sin(αh)ω1,k+cos(αh)ω2,k0,ω3,k+1=ω3,k,
with initial conditions ω1,0=ω2,0=a and ω3,0=b, with the following output
(37)x2r,k(ωk)=ω1,k+ω3,k.
Then, x2r,k(ωk) is a sinusoidal shape signal with amplitude of 2a, a frequency equal to α, and a bias value equal to b; moreover, ωk=(ω1,k,ω2,k,ω3,k)T. Finally one determines that x~2r,k(ωk)=π~2,k(ωk)=Cπ2,k(ωk).We define the steady state error as(38)zk=(z1,k,z2,k)T=x~k-π~k(ωk),
where x~k=(x~1,kx~2,k)T and π~k(ωk)=(π~1,kπ~2,k)T with π~1,k=Lπ1,k and π~2,k=Cπ2,k. Then, the dynamic equation for (38) can be obtained by taking one step ahead:
(39)z1,k+1=x~1,k-vkhCx~2,k+hE-π~1,k+1(ωk),z2,k+1=x~2,k-(L+Rhvk2)hRLCx~2,k+vkhLx~1,k,+vkh2EL-π~2,k+1(ωk).Proposition 2.
Consider the sliding mode function as(40)sk=z2,k+c1z1,k,
and the sliding mode controller:
(41)vk={veq,kif0<veq,k<1,12-12veq,k|veq,k|if0>veq,k>1,
where veq,k is the equivalent control calculated as usual from sk+1=0 and the design parameter c1 is chosen such that a11-a12c1=λ1, with |λ1|<1 and
(42)a1,j=∂f1,k∂x~j,k|(b2L/(RE),bC),
with j=1,2; moreover, f1,k is the right hand of the first equation in (39); then, the posed control problem is solved.Proof.
Conditionsk+1=0 results as follows:
(43)sk+1=x~2,k-(L+Rhvk2)hRLCx~2,k+vkhLx~1,k+vkh2EL-π~2,k+1(ωk)+c1(x~1,k-vkhCx~2,k+hE-π~1,k+1(ωk))=0.
After making some simplifications one obtains
(44)Sk+1=akveq,k2+bkveq,k+ck=0,
where
(45)ak=-h2LCx~2,k,bk=hLx~1,k+h2EL+c1hCx~2,k,ck=(1-hRC)x~2,k-π~2,k+1+c1(x~1,k+hE-π~1,k+1).
Equation (44) has been dealt in the work presented by [26], where the following equivalent control solution has been proposed:
(46)veq,k={-bk2ak±Δk2akifak>0,Δk≥0,-bk2akifak>0,Δk<0,-ckbkifak≤0,
where Δk=bk2-4akck. With this equivalent control, the sliding function is zeroed in one or more steps or tends asymptotically to zero. When 0>veq,k>1, the control action is saturated (41). In this case, for analysis purposes, the saturated control can be modified as follows:
(47)vk=v0veq,k|veq,k|,for|veq,k|>v0,
with v0=1/2; moreover, after some algebraic steps, the sliding function dynamics (44) can be represented in the following form:
(48)Sk+1=vk+bk2ak∓Δk2ak.
In order to reveal the structure of the sliding function dynamics, let us represent it as follows:
(49)Sk+1=Sk+vk+bk2ak∓Δk2ak-z2,k-c1z1,k,
and the equivalent control in the case of Δk≥0 is rewritten as follows:
(50)veq,k=-Sk-bk2ak±Δk2ak+z2,k+c1z1,k.
Now suppose that the control can vary within the fictitious bound of v0, therefore the available control resources are such that
(51)|bk2ak∓Δk2ak+z2,k+c1z1,k|<v0.
Substituting (50) in (47), and (47) in (49) yields to
(52)Sk+1=(Sk+bk2ak∓Δk2ak-z2,k-c1z1,k)×(1-v0|-Sk-bk/2ak±Δk/2ak+z2,k+c1z1,k|),
and by taking the absolute values
(53)|Sk+1|=|Sk+bk2ak∓Δk2ak-z2,k-c1z1,k|×(1-v0|-Sk-bk/2ak±Δk/2ak+z2,k+c1z1,k|)=|Sk+bk2ak∓Δk2ak-z2,k-c1z1,k|-v0.
At this point, one proposes the following candidate Lyapunov function:
(54)Vk=|Sk|,
where the increment of the Lyapunov function results as follows:
(55)ΔV=|Sk+1|-|Sk|ΔV=|Sk+bk2ak∓Δk2ak-z2,k-c1z1,k|-v0-|Sk|ΔV≤|Sk|+|bk2ak∓Δk2ak-z2,k-c1z1,k|-v0-|Sk|,
and by using condition (51), the increment of the Lyapunov function reduces to ΔV<0. Therefore, the sliding function (40) and the equivalent control (46) are decreasing monotonically, and after a finite number of steps, 0<veq,k<1 is achieved. Discrete-time sliding mode will take place from the next sampling point onwards.Remark 3.
Whenveq,k=-bk/2ak or veq,k=-ck/bk is chosen in (46), a similar Lyapunov analysis can be carried on, where the sliding function (40) can be demonstrated to decrease monotonically.After the sliding mode occurs, one hasz2,k=-c1z1,k (see (40)), and considering only the linear part of the first equation of (39) along with (46) as in [19], then, the motion of the linearized closed-loop system (sliding mode motion) will be governed by
(56)z1,k+1=(a11-a12c1)z1,k+ϕ1,s,k
with ϕ1,s,k as a function of higher order terms that vanish at the origin with their first derivative; then, one can assign a desired eigenvalue and solve for c1, that is, a11-a12c1=λ1, with |λ1|<1. With this choice it is easy to realize that the linear approximation of z1,k in the sliding regime (56) is asymptotically stable, and under the property of center manifold, one has a locally attractive manifold on (38), that is, z1,k→0⇒x~1,k=x~1r,k and z2,k→0⇒x~2,k=x~2r,k as k→∞.Remark 4.
Due to the complexity of the equivalent control (46), the sliding mode dynamics (56) must be determined for the three conditions presented in (46), where the design parameter c1 must stabilize the sliding mode dynamics for all cases.


### 4.2. Reference Signal Calculation for the Inductor Current

The smooth mappingsπ~1,k(ωk):W0→ℜ and π~2,k(ωk):W0→ℜ (where W0 is an open neighborhood of ωk=0), with π~1,0(0)=0 and π~2,0(0)=0, are such that the pair (π~1,k(ωk),π~2,k(ωk)) is the unique solution to the following difference equations:
(57)π~1,k+1(ωk)=π~1,k(ωk)-ckhCπ~2,k(ωk)+hE,(58)π~2,k+1(ωk)=π~2,k(ωk)-(L+Rhck2)hRLCπ~2,k(ωk)π~2,k+1(ωk)=+ckhLπ~1,k(ωk)+ckh2EL,(59)0=π~2,k(ωk)-x~2k(ωk).
From (37) and (59) one can determine π~2,k(ωk)=C(ω1,k+ω3,k). Then, one can calculate ck from (57) as follows:
(60)ck=Cπ~1,k(ωk)+hE-π~1,k+1(ωk)hπ~2,k(ωk),
and substituting it in (58) yields to
(61)π~2,k+1(ωk)=(1-hRC)π~2,k-CL(π~1,k+1-π~1,k-hE)2π~2,kπ~2,k+1(ωk)=-CL(π~1,k+hE)(π~1,k+1-π~1,k-hE)π~2,k.
The finding of a solution to this difference equation results in a difficult task that can be solved by proposing an approximated solution as in [27]. Thus, one proposes the following polynomial as an approximated solution for π~1,k(ωk):
(62)π~1,k(ωk)=a0+a1ω1,k3+a2ω1,k2+a3ω2,kω1,k2π~1,k(ωk)=+a4ω3,kω1,k2+a5ω1,kω2,k2+a6ω1,kω2,kπ~1,k(ωk)=+a7ω3,kω1,kω2,k+a8ω1,k+a9ω3,kω1,kπ~1,k(ωk)=+a10ω3,k2ω1,k+a11ω2,k3+a12ω2,k2π~1,k(ωk)=+a13ω3,kω2,k2+a14ω2,k+a15ω3,kω2,kπ~1,k(ωk)=+a16ω3,k2ω2,k+a17ω3,k+a18ω3,k2π~1,k(ωk)=+a19ω3,k3+O4(∥ωk∥1).
Multiplying (61) by π~2,k(ωk) and then substituting (62) in the resulting equation, one can find the values ai(i=0,…,19) if the coefficients of the same monomials appearing in both sides of such equation are equalized. In that case, the unique coefficients with values different from zero are
(63)a2=-Lcos(αh)(-cos(αh)h-RC+RCcos(αh))hER,a6=-Lsin(αh)(-2cos(αh)h-RC+2RCcos(αh))hER,a9=-L(RCcos(αh)-RC-2cos(αh)h)hER,a12=(RC-h)L(-1+cos2(αh))hER,a15=(RC-2h)Lsin(αh)hER,a18=LRE.
Note that the bias value for π~1,k(ωk) is the monomial with a18 as a coefficient, that is, b2L/(RE). This fact was used when linearizing the sliding mode dynamics.


## 4.1. Sliding Mode Regulator for a Boost Power Converter

Let us define the following tracking error:(35)ek=x~2,k-x~2r,k(ωk).
The reference signal is supposed to be generated by an autonomous exosystem given by
(36)ω1,k+1=cos(αh)ω1,k-sin(αh)ω2,k,ω2,k+1=sin(αh)ω1,k+cos(αh)ω2,k0,ω3,k+1=ω3,k,
with initial conditions ω1,0=ω2,0=a and ω3,0=b, with the following output
(37)x2r,k(ωk)=ω1,k+ω3,k.
Then, x2r,k(ωk) is a sinusoidal shape signal with amplitude of 2a, a frequency equal to α, and a bias value equal to b; moreover, ωk=(ω1,k,ω2,k,ω3,k)T. Finally one determines that x~2r,k(ωk)=π~2,k(ωk)=Cπ2,k(ωk).We define the steady state error as(38)zk=(z1,k,z2,k)T=x~k-π~k(ωk),
where x~k=(x~1,kx~2,k)T and π~k(ωk)=(π~1,kπ~2,k)T with π~1,k=Lπ1,k and π~2,k=Cπ2,k. Then, the dynamic equation for (38) can be obtained by taking one step ahead:
(39)z1,k+1=x~1,k-vkhCx~2,k+hE-π~1,k+1(ωk),z2,k+1=x~2,k-(L+Rhvk2)hRLCx~2,k+vkhLx~1,k,+vkh2EL-π~2,k+1(ωk).Proposition 2.
Consider the sliding mode function as(40)sk=z2,k+c1z1,k,
and the sliding mode controller:
(41)vk={veq,kif0<veq,k<1,12-12veq,k|veq,k|if0>veq,k>1,
where veq,k is the equivalent control calculated as usual from sk+1=0 and the design parameter c1 is chosen such that a11-a12c1=λ1, with |λ1|<1 and
(42)a1,j=∂f1,k∂x~j,k|(b2L/(RE),bC),
with j=1,2; moreover, f1,k is the right hand of the first equation in (39); then, the posed control problem is solved.Proof.
Conditionsk+1=0 results as follows:
(43)sk+1=x~2,k-(L+Rhvk2)hRLCx~2,k+vkhLx~1,k+vkh2EL-π~2,k+1(ωk)+c1(x~1,k-vkhCx~2,k+hE-π~1,k+1(ωk))=0.
After making some simplifications one obtains
(44)Sk+1=akveq,k2+bkveq,k+ck=0,
where
(45)ak=-h2LCx~2,k,bk=hLx~1,k+h2EL+c1hCx~2,k,ck=(1-hRC)x~2,k-π~2,k+1+c1(x~1,k+hE-π~1,k+1).
Equation (44) has been dealt in the work presented by [26], where the following equivalent control solution has been proposed:
(46)veq,k={-bk2ak±Δk2akifak>0,Δk≥0,-bk2akifak>0,Δk<0,-ckbkifak≤0,
where Δk=bk2-4akck. With this equivalent control, the sliding function is zeroed in one or more steps or tends asymptotically to zero. When 0>veq,k>1, the control action is saturated (41). In this case, for analysis purposes, the saturated control can be modified as follows:
(47)vk=v0veq,k|veq,k|,for|veq,k|>v0,
with v0=1/2; moreover, after some algebraic steps, the sliding function dynamics (44) can be represented in the following form:
(48)Sk+1=vk+bk2ak∓Δk2ak.
In order to reveal the structure of the sliding function dynamics, let us represent it as follows:
(49)Sk+1=Sk+vk+bk2ak∓Δk2ak-z2,k-c1z1,k,
and the equivalent control in the case of Δk≥0 is rewritten as follows:
(50)veq,k=-Sk-bk2ak±Δk2ak+z2,k+c1z1,k.
Now suppose that the control can vary within the fictitious bound of v0, therefore the available control resources are such that
(51)|bk2ak∓Δk2ak+z2,k+c1z1,k|<v0.
Substituting (50) in (47), and (47) in (49) yields to
(52)Sk+1=(Sk+bk2ak∓Δk2ak-z2,k-c1z1,k)×(1-v0|-Sk-bk/2ak±Δk/2ak+z2,k+c1z1,k|),
and by taking the absolute values
(53)|Sk+1|=|Sk+bk2ak∓Δk2ak-z2,k-c1z1,k|×(1-v0|-Sk-bk/2ak±Δk/2ak+z2,k+c1z1,k|)=|Sk+bk2ak∓Δk2ak-z2,k-c1z1,k|-v0.
At this point, one proposes the following candidate Lyapunov function:
(54)Vk=|Sk|,
where the increment of the Lyapunov function results as follows:
(55)ΔV=|Sk+1|-|Sk|ΔV=|Sk+bk2ak∓Δk2ak-z2,k-c1z1,k|-v0-|Sk|ΔV≤|Sk|+|bk2ak∓Δk2ak-z2,k-c1z1,k|-v0-|Sk|,
and by using condition (51), the increment of the Lyapunov function reduces to ΔV<0. Therefore, the sliding function (40) and the equivalent control (46) are decreasing monotonically, and after a finite number of steps, 0<veq,k<1 is achieved. Discrete-time sliding mode will take place from the next sampling point onwards.Remark 3.
Whenveq,k=-bk/2ak or veq,k=-ck/bk is chosen in (46), a similar Lyapunov analysis can be carried on, where the sliding function (40) can be demonstrated to decrease monotonically.After the sliding mode occurs, one hasz2,k=-c1z1,k (see (40)), and considering only the linear part of the first equation of (39) along with (46) as in [19], then, the motion of the linearized closed-loop system (sliding mode motion) will be governed by
(56)z1,k+1=(a11-a12c1)z1,k+ϕ1,s,k
with ϕ1,s,k as a function of higher order terms that vanish at the origin with their first derivative; then, one can assign a desired eigenvalue and solve for c1, that is, a11-a12c1=λ1, with |λ1|<1. With this choice it is easy to realize that the linear approximation of z1,k in the sliding regime (56) is asymptotically stable, and under the property of center manifold, one has a locally attractive manifold on (38), that is, z1,k→0⇒x~1,k=x~1r,k and z2,k→0⇒x~2,k=x~2r,k as k→∞.Remark 4.
Due to the complexity of the equivalent control (46), the sliding mode dynamics (56) must be determined for the three conditions presented in (46), where the design parameter c1 must stabilize the sliding mode dynamics for all cases.


## 4.2. Reference Signal Calculation for the Inductor Current

The smooth mappingsπ~1,k(ωk):W0→ℜ and π~2,k(ωk):W0→ℜ (where W0 is an open neighborhood of ωk=0), with π~1,0(0)=0 and π~2,0(0)=0, are such that the pair (π~1,k(ωk),π~2,k(ωk)) is the unique solution to the following difference equations:
(57)π~1,k+1(ωk)=π~1,k(ωk)-ckhCπ~2,k(ωk)+hE,(58)π~2,k+1(ωk)=π~2,k(ωk)-(L+Rhck2)hRLCπ~2,k(ωk)π~2,k+1(ωk)=+ckhLπ~1,k(ωk)+ckh2EL,(59)0=π~2,k(ωk)-x~2k(ωk).
From (37) and (59) one can determine π~2,k(ωk)=C(ω1,k+ω3,k). Then, one can calculate ck from (57) as follows:
(60)ck=Cπ~1,k(ωk)+hE-π~1,k+1(ωk)hπ~2,k(ωk),
and substituting it in (58) yields to
(61)π~2,k+1(ωk)=(1-hRC)π~2,k-CL(π~1,k+1-π~1,k-hE)2π~2,kπ~2,k+1(ωk)=-CL(π~1,k+hE)(π~1,k+1-π~1,k-hE)π~2,k.
The finding of a solution to this difference equation results in a difficult task that can be solved by proposing an approximated solution as in [27]. Thus, one proposes the following polynomial as an approximated solution for π~1,k(ωk):
(62)π~1,k(ωk)=a0+a1ω1,k3+a2ω1,k2+a3ω2,kω1,k2π~1,k(ωk)=+a4ω3,kω1,k2+a5ω1,kω2,k2+a6ω1,kω2,kπ~1,k(ωk)=+a7ω3,kω1,kω2,k+a8ω1,k+a9ω3,kω1,kπ~1,k(ωk)=+a10ω3,k2ω1,k+a11ω2,k3+a12ω2,k2π~1,k(ωk)=+a13ω3,kω2,k2+a14ω2,k+a15ω3,kω2,kπ~1,k(ωk)=+a16ω3,k2ω2,k+a17ω3,k+a18ω3,k2π~1,k(ωk)=+a19ω3,k3+O4(∥ωk∥1).
Multiplying (61) by π~2,k(ωk) and then substituting (62) in the resulting equation, one can find the values ai(i=0,…,19) if the coefficients of the same monomials appearing in both sides of such equation are equalized. In that case, the unique coefficients with values different from zero are
(63)a2=-Lcos(αh)(-cos(αh)h-RC+RCcos(αh))hER,a6=-Lsin(αh)(-2cos(αh)h-RC+2RCcos(αh))hER,a9=-L(RCcos(αh)-RC-2cos(αh)h)hER,a12=(RC-h)L(-1+cos2(αh))hER,a15=(RC-2h)Lsin(αh)hER,a18=LRE.
Note that the bias value for π~1,k(ωk) is the monomial with a18 as a coefficient, that is, b2L/(RE). This fact was used when linearizing the sliding mode dynamics.


## 5. Simulations and Experimental Results

### 5.1. Open-Loop Simulations

Open-loop simulations were carried out in order to verify the performance of the obtained discrete-time model. The following two figures show the comparison of the continuous-time model of the boost power converter simulated with a sampling time of 0.0001 s and the discrete-time model with a sampling time of 0.001 s. The switch input value was set tou=0. Figure 2 shows the performance of the inductor current, and Figure 3 shows the performance of the output capacitor voltage. In both figures can be appreciated that the discrete-time model it can accurately track the corresponding continuous-time signals.Figure 2
Open-loop current comparison, continuous-time model (dashed line), discrete-time model (solid line).Figure 3
Open-loop voltage comparison, continuous-time model (dashed line), discrete-time model (solid line).


### 5.2. Comparison of Two Control Schemes

The designed controller in [28] (see the appendix) is discretized by means of the explicit Euler method and compared with the controller here designed. It is worth mentioning that the discretization of continuous-time controllers via this method is a common practice among control engineer practitioners. Figure 4 shows the performance of the output capacitor voltage with a sample time of 10 μs. In this figure one observes that both controllers have a good tracking performance. Figure 5 shows the simulation with a sample time of 50 μs. In this figure one can observe that the proposed controller still performs well (Figure 5(b)); meanwhile, the sampled controller (Figure 5(a)) shows a difficulty for tracking the reference signal at lower peaks.Comparison of discrete-time controllers with a sample time of10 μs. Output capacitor voltage (continuous line), reference signal (dashed line). (a) Output capacitor voltage obtained with sampled controller. (c) Output capacitor voltage obtained with proposed controller. (b) and (d) are zoom views of same variables on the left.
(a)
(b)
(c)
(d)Comparison of discrete-time controllers with a sample time of50 μs. Output capacitor voltage (continuous line), reference signal (dashed line). (a) Output capacitor voltage obtained with sampled controller. (c) Output capacitor voltage obtained with proposed controller. (b) and (d) are zoom views of same variables on the left.
(a)
(b)
(c)
(d)Finally, Figure6 shows simulation results where the sample time is incremented to 100 μs. Here, one can appreciate the difference between the sampling of continuous-time controllers and the discrete-time control law design based on a good sampled model. There is a slight deviation in the output voltage capacitor for the proposed controller as can be seen in Figure 6(c), but, with the sampled controller, the output voltage capacitor signal is distorted (see Figure 6(a)). This is the intention of this simulation exercise, to show the degradation of the discretized controller with larger sampling periods and at the same time to show that the proposed controller can still perform better even with the larger sampling periods. It is worth mentioning that these discrete-time control laws were simulated with the continuous-time plant model.Comparison of discrete-time controllers with a sample time of100 μs. Output capacitor voltage (continuous line), reference signal (dashed line). (a) Output capacitor voltage obtained with sampled controller. (c) Output capacitor voltage obtained with proposed controller. (b) and (d) are zoom views of same variables on the left.
(a)
(b)
(c)
(d)


### 5.3. Simulations of the Proposed Control Algorithm

Closed-loop simulations were carried out in order to verify the performance of the proposed controller. The initial conditions for the exosystem (36) are set to a=7.071,b=60 in order to generate a sinusoidal shape signal with amplitude value of 10 and a bias value of 60. The parameters of the boost converter are L=0.098 H, C=10 μF, R=241 Ω, and E=40 V. The design parameter c1 in the sliding function (40) is fixed to 1 by means of a trial and error method. In the following, simulation results for the tracking of a sinusoidal signal with frequency of α=1 rad/s will be shown. Figure 7 shows the sliding function where the finite-time convergence can be appreciated.Figure 7
Sliding function withα=1 rad/s.Figure8 shows the minimum phase behavior of the inductor current that is commonly known to be a nonminimum phase variable.Figure 8
Inductor current tracking withα=1 rad/s, reference signal (dashed line), real signal (solid line).The output tracking result of the voltage capacitor can be appreciated in Figure9.Figure 9
Output tracking of the voltage capacitor withα=1 rad/s, reference signal (dashed line), real signal (solid line).Now, simulation results for the tracking of a sinusoidal signal with a frequency value ofα=5 rad/s will be shown. Figure 10 shows the sliding function where once again the finite-time convergence can be appreciated. Figures 11 and 12 show the good tracking performance of the inductor current and the voltage capacitor, respectively.Figure 10
Sliding function withα=5 rad/s.Figure 11
Inductor current tracking withα=5 rad/s, reference signal (dashed line), real signal (solid line).Figure 12
Output tracking of the voltage capacitor withα=5 rad/s, reference signal (dashed line), real signal (solid line).


### 5.4. Real-Time Experimentation

The experimental setup is shown in Figure13. Here, the VARIAC is a three-phase variable transformer fed from a three-phase voltage source. By rotating the knob of the VARIAC, the amplitude of the three-phase voltage source is regulated. These voltages are fed to the power module (Semikron) that incorporates a three-phase rectifier and a single switching transistor that is connected to the boost power converter elements; in that way, a regulated dc input voltage is available for the boost power converter. The control algorithm and PWM generation are programmed in Simulink and implemented with a DSP board (dSPACE 1104). This board comes along with a library that easily incorporates with Simulink.Figure 13
Experimental setup.Analog-to-digital converters included in the DSP board acquire the signals from the inductor current and capacitor voltage. Once the DSP executes the control algorithm in each sampling step, it generates one digital signal for switching the insulated-gate bipolar transistor (IGBT). This digital signal is TTL level and it is converted to a CMOS level of 15 V. This voltage level is the required one for switching on the IGBT. A block diagram of this setup is shown in Figure14.Figure 14
Block diagram of the experimental setup.The power converter has the same parameter values of the digitally simulated boost circuit in Section5.2. The inductor current and the output voltage capacitor are measured by means of hall-type sensors, as the HX 10-P and LV 25-P, respectively; both are manufactured by LEM. A first-order Butterworth low-pass filter having an 50 rad/s edge frequency was used for filtering the inductor current in order to attenuate the measurement noise. The sampling period has been fixed at h=70 μs. The dc input voltage E was open loop  (the switch position vk was set to 1) calibrated to a value of 20 V; then, the boost power converter was closed loop in order that the output voltage capacitor is forced to track a sinusoidal signal with an amplitude of 10 V and bias of 40 V. The real-time results for a frequency value of 1 rad/s are shown in Figures 15 and 16 and for a value of 5 rad/s are shown in Figures 17 and 18. In general, a good output tracking performance in both cases can be appreciated. The tracking of the inductor current shows in both cases big peak values of short duration that are admissible; moreover the first-order filter introduces the appreciated delay.Figure 15
Inductor current tracking withα=1 rad/s, reference signal (dashed line), real-time signal (solid line).Figure 16
Output tracking of the voltage capacitor withα=1 rad/s, reference signal (dashed line), real-time signal (solid line).Figure 17
Inductor current tracking withα=5 rad/s, reference signal (dashed line), real-time signal (solid line).Figure 18
Output tracking of the voltage capacitor withα=5 rad/s, reference signal (dashed line), real-time signal (solid line).Remark 5.
The tracking of the inductor is not so accurate due to the effect of the PWM switching which is correct since the system is not regulating the inductor current but the capacitor voltage.In order to better evaluate the control performance, we consider the precision errorPe and the chattering effect Ch. The former is defined as the relative error of the output variable and it is calculated as the difference between the reference value and the average steady-state control output, divided by the reference value
(64)Pe=|Sr-Vmc|Sr×100,
where Sr is the imposed reference and Vmc is the average of the output controlled variable in steady state. The amplitude of the signal characterizes the latter, and it is given by the difference between the amplitude of the output variable and the average in steady state of the reference signal and divided by the latter signal
(65)Ch=|Amc-Vmc|Vmc×100,
where Amc is the maximum amplitude of the deviations of controlled signals with respect to its average value Vmc. Table 1 summarizes the evaluation parameters for the two frequencies. One can observe that the proposed controller in closed loop yields to small percentiles in precision error and in chattering as well.Table 1
Frequency
P
e
C
h
1 rad/s
0.5
%
0.67
%
5 rad/s
0.5
%
0.54
%Remark 6.
The chattering problem can be significantly reduced with the incorporation of soft computing techniques. In particular, sliding mode control when combined with artificial neural networks can be useful to estimate the equivalent control and the magnitude of the switching control component using the conventional gradient learning algorithm. On the other hand, instead of using the well-defined sliding mode control functions, the control action can be derived from a fuzzy logic-based control table, compounded by fuzzy membership functions. In such a way, the convergence on the switching manifold and overshooting is softened, alleviating in this way the chattering problem. More directions can be found in [29].


## 5.1. Open-Loop Simulations

Open-loop simulations were carried out in order to verify the performance of the obtained discrete-time model. The following two figures show the comparison of the continuous-time model of the boost power converter simulated with a sampling time of 0.0001 s and the discrete-time model with a sampling time of 0.001 s. The switch input value was set tou=0. Figure 2 shows the performance of the inductor current, and Figure 3 shows the performance of the output capacitor voltage. In both figures can be appreciated that the discrete-time model it can accurately track the corresponding continuous-time signals.Figure 2
Open-loop current comparison, continuous-time model (dashed line), discrete-time model (solid line).Figure 3
Open-loop voltage comparison, continuous-time model (dashed line), discrete-time model (solid line).


## 5.2. Comparison of Two Control Schemes

The designed controller in [28] (see the appendix) is discretized by means of the explicit Euler method and compared with the controller here designed. It is worth mentioning that the discretization of continuous-time controllers via this method is a common practice among control engineer practitioners. Figure 4 shows the performance of the output capacitor voltage with a sample time of 10 μs. In this figure one observes that both controllers have a good tracking performance. Figure 5 shows the simulation with a sample time of 50 μs. In this figure one can observe that the proposed controller still performs well (Figure 5(b)); meanwhile, the sampled controller (Figure 5(a)) shows a difficulty for tracking the reference signal at lower peaks.Comparison of discrete-time controllers with a sample time of10 μs. Output capacitor voltage (continuous line), reference signal (dashed line). (a) Output capacitor voltage obtained with sampled controller. (c) Output capacitor voltage obtained with proposed controller. (b) and (d) are zoom views of same variables on the left.
(a)
(b)
(c)
(d)Comparison of discrete-time controllers with a sample time of50 μs. Output capacitor voltage (continuous line), reference signal (dashed line). (a) Output capacitor voltage obtained with sampled controller. (c) Output capacitor voltage obtained with proposed controller. (b) and (d) are zoom views of same variables on the left.
(a)
(b)
(c)
(d)Finally, Figure6 shows simulation results where the sample time is incremented to 100 μs. Here, one can appreciate the difference between the sampling of continuous-time controllers and the discrete-time control law design based on a good sampled model. There is a slight deviation in the output voltage capacitor for the proposed controller as can be seen in Figure 6(c), but, with the sampled controller, the output voltage capacitor signal is distorted (see Figure 6(a)). This is the intention of this simulation exercise, to show the degradation of the discretized controller with larger sampling periods and at the same time to show that the proposed controller can still perform better even with the larger sampling periods. It is worth mentioning that these discrete-time control laws were simulated with the continuous-time plant model.Comparison of discrete-time controllers with a sample time of100 μs. Output capacitor voltage (continuous line), reference signal (dashed line). (a) Output capacitor voltage obtained with sampled controller. (c) Output capacitor voltage obtained with proposed controller. (b) and (d) are zoom views of same variables on the left.
(a)
(b)
(c)
(d)


## 5.3. Simulations of the Proposed Control Algorithm

Closed-loop simulations were carried out in order to verify the performance of the proposed controller. The initial conditions for the exosystem (36) are set to a=7.071,b=60 in order to generate a sinusoidal shape signal with amplitude value of 10 and a bias value of 60. The parameters of the boost converter are L=0.098 H, C=10 μF, R=241 Ω, and E=40 V. The design parameter c1 in the sliding function (40) is fixed to 1 by means of a trial and error method. In the following, simulation results for the tracking of a sinusoidal signal with frequency of α=1 rad/s will be shown. Figure 7 shows the sliding function where the finite-time convergence can be appreciated.Figure 7
Sliding function withα=1 rad/s.Figure8 shows the minimum phase behavior of the inductor current that is commonly known to be a nonminimum phase variable.Figure 8
Inductor current tracking withα=1 rad/s, reference signal (dashed line), real signal (solid line).The output tracking result of the voltage capacitor can be appreciated in Figure9.Figure 9
Output tracking of the voltage capacitor withα=1 rad/s, reference signal (dashed line), real signal (solid line).Now, simulation results for the tracking of a sinusoidal signal with a frequency value ofα=5 rad/s will be shown. Figure 10 shows the sliding function where once again the finite-time convergence can be appreciated. Figures 11 and 12 show the good tracking performance of the inductor current and the voltage capacitor, respectively.Figure 10
Sliding function withα=5 rad/s.Figure 11
Inductor current tracking withα=5 rad/s, reference signal (dashed line), real signal (solid line).Figure 12
Output tracking of the voltage capacitor withα=5 rad/s, reference signal (dashed line), real signal (solid line).


## 5.4. Real-Time Experimentation

The experimental setup is shown in Figure13. Here, the VARIAC is a three-phase variable transformer fed from a three-phase voltage source. By rotating the knob of the VARIAC, the amplitude of the three-phase voltage source is regulated. These voltages are fed to the power module (Semikron) that incorporates a three-phase rectifier and a single switching transistor that is connected to the boost power converter elements; in that way, a regulated dc input voltage is available for the boost power converter. The control algorithm and PWM generation are programmed in Simulink and implemented with a DSP board (dSPACE 1104). This board comes along with a library that easily incorporates with Simulink.Figure 13
Experimental setup.Analog-to-digital converters included in the DSP board acquire the signals from the inductor current and capacitor voltage. Once the DSP executes the control algorithm in each sampling step, it generates one digital signal for switching the insulated-gate bipolar transistor (IGBT). This digital signal is TTL level and it is converted to a CMOS level of 15 V. This voltage level is the required one for switching on the IGBT. A block diagram of this setup is shown in Figure14.Figure 14
Block diagram of the experimental setup.The power converter has the same parameter values of the digitally simulated boost circuit in Section5.2. The inductor current and the output voltage capacitor are measured by means of hall-type sensors, as the HX 10-P and LV 25-P, respectively; both are manufactured by LEM. A first-order Butterworth low-pass filter having an 50 rad/s edge frequency was used for filtering the inductor current in order to attenuate the measurement noise. The sampling period has been fixed at h=70 μs. The dc input voltage E was open loop  (the switch position vk was set to 1) calibrated to a value of 20 V; then, the boost power converter was closed loop in order that the output voltage capacitor is forced to track a sinusoidal signal with an amplitude of 10 V and bias of 40 V. The real-time results for a frequency value of 1 rad/s are shown in Figures 15 and 16 and for a value of 5 rad/s are shown in Figures 17 and 18. In general, a good output tracking performance in both cases can be appreciated. The tracking of the inductor current shows in both cases big peak values of short duration that are admissible; moreover the first-order filter introduces the appreciated delay.Figure 15
Inductor current tracking withα=1 rad/s, reference signal (dashed line), real-time signal (solid line).Figure 16
Output tracking of the voltage capacitor withα=1 rad/s, reference signal (dashed line), real-time signal (solid line).Figure 17
Inductor current tracking withα=5 rad/s, reference signal (dashed line), real-time signal (solid line).Figure 18
Output tracking of the voltage capacitor withα=5 rad/s, reference signal (dashed line), real-time signal (solid line).Remark 5.
The tracking of the inductor is not so accurate due to the effect of the PWM switching which is correct since the system is not regulating the inductor current but the capacitor voltage.In order to better evaluate the control performance, we consider the precision errorPe and the chattering effect Ch. The former is defined as the relative error of the output variable and it is calculated as the difference between the reference value and the average steady-state control output, divided by the reference value
(64)Pe=|Sr-Vmc|Sr×100,
where Sr is the imposed reference and Vmc is the average of the output controlled variable in steady state. The amplitude of the signal characterizes the latter, and it is given by the difference between the amplitude of the output variable and the average in steady state of the reference signal and divided by the latter signal
(65)Ch=|Amc-Vmc|Vmc×100,
where Amc is the maximum amplitude of the deviations of controlled signals with respect to its average value Vmc. Table 1 summarizes the evaluation parameters for the two frequencies. One can observe that the proposed controller in closed loop yields to small percentiles in precision error and in chattering as well.Table 1
Frequency
P
e
C
h
1 rad/s
0.5
%
0.67
%
5 rad/s
0.5
%
0.54
%Remark 6.
The chattering problem can be significantly reduced with the incorporation of soft computing techniques. In particular, sliding mode control when combined with artificial neural networks can be useful to estimate the equivalent control and the magnitude of the switching control component using the conventional gradient learning algorithm. On the other hand, instead of using the well-defined sliding mode control functions, the control action can be derived from a fuzzy logic-based control table, compounded by fuzzy membership functions. In such a way, the convergence on the switching manifold and overshooting is softened, alleviating in this way the chattering problem. More directions can be found in [29].


## 6. Conclusions

A discrete-time sliding mode control scheme has been designed for the dc-to-ac boost power conversion problem. The contributions of this work are the following.(i)
A novel discrete-time model for the boost power converter circuit was obtained by means of a variational integrator scheme.(ii)
The discrete-time-sliding-mode-based controller can stabilize the nonminimum phase system with a proper choice of a sliding surface, resulting in the output tracking of the capacitor voltage and the stabilization of the inductor current.(iii)
The reference signal for the inductor current was proposed as a polynomial of order 3 that solves with a good accuracy the corresponding solution of the reference difference equation.Moreover, simulation and experimental results validate the good performance of the proposed dc-to-ac boost power conversion scheme. Some interesting issues, such as the robustness of the controller with respect to parameter variations, determination of the sampling period, are currently under study.



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*Source: 460281-2013-04-04.xml*