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.

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: Section 2 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, let be a smooth manifold, called the configuration space, and let be its tangent bundle, called the phase space. The Lagrangian is a function and is commonly defined as the kinetic energy (usually, the only function of the velocity) minus the potential energy of the system (usually, the only function of the state variable):

The functional action is then introduced as the integral of along a path for time :

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 (and in our setup). In more ways than one, this principle is very similar to a statement on the geometry of the path : 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 path is written as 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 of the path results in where integration by parts is used in the last equality. When the endpoints of are held fixed with respect to all variations (i.e., ), the most right term in the previous equation vanishes. Therefore, the condition of stationary action for arbitrary variations with fixed endpoints stated in Hamilton’s principle directly indicates that the remaining integrand in the previous equation must be zero for all time , yielding what is known as the Euler-Lagrange equation:

For a given Lagrangian, this formula will give the equations of motion of the system. Finally to account for nonconservative forces or dissipation , the least action principle is modified as follows: 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 is , on the time interval . First, we partition the time interval by a finite number of points where ; likewise, the path is replaced by a sequence of configurations , where . Then the discrete Lagrangian is a function that approximates the Lagrangian on each time interval , with being the time interval between two samples (chosen here to be constant for simplicity), so that 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 and and with replaced by : where is parameter that generalizes the quadrature rule. For , 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: where again is parameter that generalizes the quadrature rule and for , 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 and is fixed to the variational integrator is known as the symplectic Euler method; when is fixed to 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 is fixed to the variational integrator is known as the Störmer/Verlet method.

Given the discrete Lagrangian, the discrete functional action becomes simply a sum: Taking fixed-endpoint variations of this discrete action , we obtain where (resp., ) denotes the partial derivative with respect to the first (resp., second) arguments of . Using the fixed endpoint condition , one gets Setting this variation to be equal to and noting that each is arbitrary, we arrive at the discrete Euler-Lagrange (DEL) equation Notice that this condition only involves three consecutive positions. Therefore, for two given successive positions and , (12) defines . 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 , then the discrete Euler-Lagrange equations define a recursive rule for calculating the sequence . Regarded in this way, they define a map 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 configurations and of the system, which should approximate and and then solve the discrete Euler-Lagrange equation (12) for . This process can then be repeated to calculate an entire discrete trajectory. The map 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 to be where the second equality holds due to (12). The position-momentum form of the variational integrator discussed previously is then given by

For known, (14) is an implicit equation whose solution gives . is then substituted in (15) in order to find . 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]: where and are discrete external forces acting, respectively, on the right of and on the left of . In other words, can be seen as a two-point quadrature approximation of the continuous forcing term, that is, It follows that the forced discrete Euler-Lagrange equation is as follows: The forced discrete Euler-Lagrange equation can be expressed in a convenient, position-momentum form as follows: 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 define and as the kinetic and potential energies of the circuit, respectively. One can denote by the Rayleigh dissipation function of the circuit and by the nonconservative potential function. These quantities are taken from [25] where they readily are found to be where and are the circulating charge in the inductor and the electrical charge stored in the output capacitor, respectively, with ; moreover, and are the corresponding currents with , the constant parameters are as the inductance, as the capacitance, and as the resistance. The vector is defined as , where 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: where is the conservative Lagrangian defined as usual: Following (7) one can obtain the discrete-time version of the conservative Lagrangian as Moreover the discrete external forces and appearing in (19) are as follows: where and 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 where its sampled version is as follows: where again, the right-hand side integral can be approximated through a one-point quadrature, resulting as follows: After some algebraic steps and assigning (i.e., the symplectic Euler method), the updated rule for position shown in (19) yields to where and ; moreover, Then, solving (28) for results in and finally in scalar form: where .

Now, the updated rule for momentum equation shown in (28) yields to Then, by replacing (30) in (32) it yields to The variable 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 , that is, the input current represented as . The second state variable is defined as the output voltage capacitor and it is represented as where is the electrical charge stored in the output capacitor. Therefore, one proposes the following novel approximated discrete-time model for the boost power converter: where is the sampled version of the input current, and 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 variable to track a desired reference signal 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: The reference signal is supposed to be generated by an autonomous exosystem given by with initial conditions and , with the following output Then, is a sinusoidal shape signal with amplitude of , a frequency equal to , and a bias value equal to ; moreover, . Finally one determines that .

We define the steady state error as where and with and . Then, the dynamic equation for (38) can be obtained by taking one step ahead:

Proposition 2. Consider the sliding mode function as and the sliding mode controller: where is the equivalent control calculated as usual from and the design parameter is chosen such that , with and with ; moreover, is the right hand of the first equation in (39); then, the posed control problem is solved.

Proof. Condition results as follows: After making some simplifications one obtains where Equation (44) has been dealt in the work presented by [26], where the following equivalent control solution has been proposed: where . With this equivalent control, the sliding function is zeroed in one or more steps or tends asymptotically to zero. When , the control action is saturated (41). In this case, for analysis purposes, the saturated control can be modified as follows: with ; moreover, after some algebraic steps, the sliding function dynamics (44) can be represented in the following form: In order to reveal the structure of the sliding function dynamics, let us represent it as follows: and the equivalent control in the case of is rewritten as follows: Now suppose that the control can vary within the fictitious bound of , therefore the available control resources are such that Substituting (50) in (47), and (47) in (49) yields to and by taking the absolute values At this point, one proposes the following candidate Lyapunov function: where the increment of the Lyapunov function results as follows: and by using condition (51), the increment of the Lyapunov function reduces to . Therefore, the sliding function (40) and the equivalent control (46) are decreasing monotonically, and after a finite number of steps, is achieved. Discrete-time sliding mode will take place from the next sampling point onwards.