**1. Introduction**

It is well-known that the Boost converter can operate in three modes, named CCM, critical conduction mode (CRM), and DCM, related to the energy stored in the inductor. In CCM/DCM, the stored energy is strictly/non strictly greater than zero, and CRM stands for zero energy during an infinitesimal period. Many authors have considered the Boost converter's modeling and stabilization in CCM with a resistive load since no wide variations or constant power phenomena were expected at the load side. Some only mention that DCM operation should be avoided or treated with special care, and others have limited the DCM operation's study to its stationary behavior [1–6]. A brief survey of relevant DCM, CRM, and CCM models and controllers for the Boost converter is presented in the following.

The authors in [7] developed a low-output voltage Boost converter operating in DCM and designed a controller to stabilize the output by a steady-state analysis. Some authors analyzed the power quality in DCM operation for converters and rectifiers, for instance, in [8–13]. However, they based their analysis on steady-state behavior. Small signal linear models for DCM operation of the Boost converter, including power losses, were developed in [14]; however, the obtained transfer function includes many parameters that are not

**Citation:** Parada Salado, J.G.; Herrera Ramírez, C.A.; Soriano Sánchez, A.G.; Rodríguez Licea, M.A. Nonlinear Stabilization Controller for the Boost Converter with a Constant Power Load in Both Continuous and Discontinuous Conduction Modes. *Micromachines* **2021**, *12*, 522. https://doi.org/10.3390/mi12050522

Academic Editor: Young-Ho Cho

Received: 25 March 2021 Accepted: 1 May 2021 Published: 6 May 2021

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easy to acquire. In [15], a frequency control to operate the Boost converter in CRM was proposed. The authors based their study on a linear model of the Boost converter operating in DCM, obtained by a small-signal analysis. On the other hand, in [16], a numerical PSPICE implementation of an isolated full-bridge converter reproduced some CCM and DCM phenomena.

In [17], a nonlinear controller for DCM operation of the Boost converter was developed, whereby the authors obtained such controller from the discrete-time differenceapproximation equations and the steady-state solution of the system. The authors in [18] proposed a controller for the output current of the Boost converter operating in DCM, using a steady-state estimation of the dynamics.

In [19], the authors developed a conduction mode independent (CMI) model for the non-inverting Buck-Boost converter. However, the proposed model depends on the inductor's discharge period, which is an implicit function of the duty cycle and parameters and cannot be extended to the Boost converter. Regardless of the previous, a back-stepping controller for the Buck-Boost converter with a resistive load (non-CPL) was developed.

Some authors have stated unwanted phenomena during the transition between conduction modes. In [20,21], proofs of open-loop bifurcation phenomena in the DCM-CCM boundary were provided. Chaos and bifurcation behaviors for the boost converter were also demonstrated in [22]. Controllers for other converters operating in DCM or CCM with no CPL were developed in [23–25].

Furthermore, it has been reported that a CPL can destabilize the dynamics of the Boost converter in any conduction mode. Recently developed control strategies considered a CPL with single conduction modes or linearized models [26–33]. Unfortunately, it is not difficult to achieve DCM to CCM changes in a Boost converter, with both a resistive load or a CPL and in addition, with a CPL, the conduction mode depends on their power level and output voltage/current. Moreover, linearized models are accurate only in a small region containing the operating point.

It is worth mentioning the work in [34], the authors reported that the Boost converter operating in DCM with a CPL is stable during steady-state. However, the transient stage and CCM to CCM changes cannot be neglected for applications in which wide ranges of input/output voltage and a constant power load (CPL) can coexist.

The authors in [35] presented a current control for the Boost converter feeding a CPL and included a passive compensation (paralleled RC network). They based their analysis on small-signal models valid only in a small region containing the operating point.

From the above state of the art, one can find that the complicated scenario that includes CPLs, and conduction mode changes in the Boost converter, has not been formally studied. The models presented until now are small-signal or steady-state. Table 1 summarizes relevant proposals for converters operating with a CPL or different conduction modes; the checkmark symbol means that the research considers the characteristic. There is a gap for the Boost converter's modeling and control with a CPL, operating in both DCM and CCM. Unfortunately, in biomedicine, microelectromechanical, and other applications, wide CPL and input/output voltage variation ranges could induce changes in the conduction mode and stability properties.

Hence, the contributions of this paper are as follows. Precise Boost converter CMI models (CPL-switched, CPL non-switched, resistive load switched, and resistive load nonswitched) are presented. Through such models, it is shown that the Boost converter's operating point with a CPL is potentially unstable in open-loop; this has not been demonstrated from a CMI perspective to the authors' knowledge. Therefore, a nonlinear stabilizing controller is developed for the CMI model with a CPL. A stability analysis based on a common-Lyapunov function is provided, and numerical and experimental tests are presented to show the proposal's effectiveness. The controller does not depend on frequency control but the CPL power level and the output current, making it easy to implement.

This document is organized as follows. Section 2 presents the development of the mathematical models, and Section 3 its validation. Section 4 is devoted to instability

demonstration for the Boost converter with a CPL. In Section 5, the controller design is presented, and in Section 6, numerical and experimental tests of the closed-loop system are presented. Finally, some conclusions are presented in Section 7.


**Table 1.** Summary of the literature review.
