Sliding Mode Control Based on Internal Model for a Non-minimum phase Buck and Boost Converter

This work presents the application of different schemes to control a non-minimum phase Buck-Boost converter. Three control schemes are used. The first controller presented is a PI controller, the second one is Sliding Mode Control and the third one is a combination of two control schemes, Internal Model Control and Sliding Mode Control. The controllers are designed from a Right-Half Plane Zero (RHPZ) reduced order model. The RHPZ model is converted, using Taylor approximation, in a First Order Plus Dead Time (FOPDT) model and after that, the controllers are obtained. The performance of the SMC-IMC is compared against to a PI controller and a SMC. The simulation results show that SMC-IMC improves the converter response, reducing the chattering and presenting better robustness for load changes


Introduction
The Buck-Boost Converter (B-BC) is a DC / DC topology of power converters.DC-DC power converters are widely used in various applications such as in electrical power generation through photovoltaic systems, in DC energy storage systems, in DC power regulation systems, in photovoltaic power generation systems, aviation industry, aerospace technology and other fields (Rozanov, Ryvkin, Chavligin and Voronin, 2016).
A B-BC can be considered as a non-minimum phase system, which is characterized by the presence of a right-half plane (RHP) zero in the transfer function of system, when it receives the capacitance voltage as its output for feedback control.The ZRHP produces that the system operates in opposite direction during the initial transient response, affecting the dynamic response of the system.The non-minimum phase response of a Buck-Boost converter which works in Continuous Conduction Mode (CCM) is related to its inductance, load and duty ratio (Forouzesh M. et al 2017).Huang and Liu, (2016) present an analysis for non-minimum DC/DC converter.Internal model Control (IMC) has been used to eliminate problems originated by nonminimum phase behavior.Tarakanath, Pathwardan and Agarwal, (2014) present a study using IMC structure for Boost Converters.Other control structure is the sliding mode control (SMC).The design of SMC is attractive because it can deal with non-linear systems and exhibits robustness against modeling error and disturbances (Camacho and Lacruz, 2014).However, SMC can present a problem when it is used in the Buck-Boost causing chattering in the control action that is not ideal for the actuator, thus a hybrid approach (SMC-IMC) (Camacho, Smith and Moreno, 2003) can help in this sense.
This article shows the application of different control schemes to control a non-minimum phase Buck-Boost converter.Three control schemes are used.The first controller presented is a PI controller, the second one is Sliding Mode Control and the third one is a combination of two control schemes, Internal Model Control and Sliding Mode Control.The controllers are designed from a Right-Half Plane Zero (RHPZ) reduced order model.The RHPZ model is converted, using Taylor approximation, in a First Order Plus Dead Time (FOPDT) model and after that, the controllers are obtained.The performance of the SMC-IMC is compared against to a PI controller and a SMC.The simulation results show that SMC-IMC improves the converter response, reducing the chattering and presenting better robustness for load changes.

Converter Description
The topology of Buck-Boost converter is shown in Figure 1.This converter is a cascade connection of two basic converters: Buck converter and Boost converter.The notation used are: L is the inductance, C is the capacitance, R is the Resistance, V in is the input voltage, V o is the voltage in the load, V c is the voltage in the capacitor, i c is the current in the capacitor, V L is the voltage in the inductor.from 0 to 0.5 the converter works as Buck mode and from 0.5 to 1 it works as Boost mode.The variable to be controlled in the DC/DC converter is the output voltage in the load, the manipulated va56riable will be the duty cycle .The parameters used to analyze the Buck-Boost converter are shown in Table 1.

Modeling of converter
For modeling Buck-Boost converter two important conditions are considered.First, the converter works in continuous conduction mode and second is that the power semiconductor device work as an ideal switch, i.e. the semiconductor doesn't dissipate power.Therefore, the input power is the same that the output power in the converter.The mathematical analysis of converter is carried out considering the switches S1, S2 in two positions (On/Off).The following equations are obtained from the Buck-Boost analysis scheme (Figure 2).For the first case of analysis, switches S1 and S2 are in position 1, the converter response is described by Equation 1 and Equation 2. (1) (2) For the second case of analysis, switches S1 and S2 are in position 2, the converter response is described by Equation 3 and Equation 4: Combining Equation 1, Equation 2, Equation 3and Equation 4, is obtained the dynamic behavior of Buck-Boost converter described by Equation 5 and Equation 6.

Linear Model Approximation
Buck-Boost converter is approximated to a linear model to facilitate the design of controllers.To obtain a linear approximation of the system, a reaction curve is applied to the converter because it has a first order dynamic behavior.The method used to approximate the model is the "First Order Plus Dead Time" (FOPDT) proposed by Smith and Corripio, 1997.Equation 8 represents the linear approximation of system.To use the approaches proposed by Camacho, Rojas and García, (1999) and Camacho, Smith and Moreno (2003), let us . (8)

Internal Model Control (IMC)
IMC is a control scheme used in systems with non-minimum phase dynamic.The presence of delay and RHP zero in the system can originate non-invertible controllers which are not possible to implement.

Sliding Mode Control (SMC)
The idea of SMC is to define a surface with which the system can slide to its desired final value (Figure 4).The first step to design this controller is choose the sliding surface S(t) to define a global behavior for the system.The SMC control action U(t) consists of two additive parts (Equation 12).The first part is called discontinuous action U D (t) and it helps the system to reach the sliding surface.The second part is called continuous action Uc(t) and it helps the system to stay on the sliding surface (Dominguez et al, 2016).( 12) Sometimes, one problem that SMC presents is the chattering, this phenomenon consists in high-frequency oscillations around the value desired in the output of system.It happens when the discontinuous action reaches the sliding surface abruptly.Also, it causes that the control action damage to the actuator and appear unwanted dynamics in the system.Discontinuous control action U D (t) is based on Equation 13, the sign function is the discontinuous element. (13)

PI Controller
For tuning the PI controller (Equation 14), the FOPDT model of system (Equation 8) was used.The tuning method of Dahlin (Smith and Corripio, 1997) was considered.

SMC Controller
The method to design this controller was proposed by Camacho, Rojas and Garcia, (1999).It is a simple way of designing a SMC controller, and it can be generalized as a method of design for several processes.This last feature is the main difference with the original SMC (Tahri F., Tahri A. and Flazi, 2014), because the original procedure is based on a specific mathematic model of system and the design is exclusive for that system.For the design of SMC controller is used a linear approximation of the system FOPDT (Equation 8).
The design of SMC controller is shown in the work of Camacho, Garcia and Rojas, (1999).The sliding surface chosen in this work is presented in Equation 15. (15) The controller equation designed by Camacho is presented in Equation 16.
The parameters for S(t) and U(t) based on equations of Camacho are:

SMC-IMC Controller
For the design of SMC-IMC controller, FOPDT approach model (Equation 8) is used.One problem of FOPDT approximation is the delay, to solve this problem the delay is eliminated of the control loop through the internal model structure.Therefore, the model of system without delay (Equation 11) is used to design this controller.SMC is added to the internal model scheme to improve the robustness of the system (Camacho, Smith and Moreno, 2003).Figure 5 shows the structure of SMC-IMC proposed in this work.The condition that must be met for the system to follow the reference and remain stable on the sliding surface is shown in Equation 20: The continuous part U c (t) is designed below.The solution of the output of Equation 11 is presented in Equation 21.
Considering Equation 20, the sliding surface (Equation 17) is derived.Adding Equation 26and Equation 28 is obtained the control action (Equation 29). (29) The theory of Lyapunov is applied to analyze the stability of controller.Lyapunov's function (Equation 30) chose is the sliding surface (Equation 17).Lyapunov's condition is presented in Equation 31 .(36) The parameters of S(t) and U(t) were tuned according to Equation 25, 26 and 27.

Simulation Results
Using the parameters of Table 1 and the model FOPDT (Equation 8), is obtained the transfer function (Equation 37), it is a linear approximation of Buck-Boost converter.
As shown in Figure 6 the FOPDT approximation does not consider the inverse peak in the response of Buck-Boost converter

Set-point step change result
The variation of set-point is from 10 % with respect of the output as shown in the Figure 7 and 8.The control action of SMC present chattering phenomenon as shown in Figure 7. Therefore, this controller is not recommendable to implement in this Buck-Boost converter.Below, there are the graphs corresponding to the simulation of the PI and SMC-IMC controller applied to the Buck-Boost converter.Table 2 shows the performance indicators to analyze the operation of the controllers.

Load Disturbances Changes
The Load Disturbances applied to the converter are variables with the time (Figure 9).The variation of the resistive load is from 10 % to 60 % respect the initial load, the response of converter is shown in Figure 10.In Table 3 the parameters MPI %, Ts were obtained to the second disturbance, ISE parameter was obtained for all disturbances.ISE parameter of PI controller is very high because the system became unstable as shown in

Conclusions
Three controllers were evaluated to a non-minimum phase DC/DC converter.Classical methods usually have not good performance when they are used for non-minimum phase electrical systems.

Figure 3 .
Figure 3. Internal Model Control Scheme

Figure 4 .
Figure 4. Sliding and Reaching Mode

Figure 5 .
Figure 5. Sliding Mode control base on Internal Model (SMC-IMC) scheme Equation 19 represents the error of system considering it without delay.(19)

Figure 6 .
Figure 6.FOPDT approximation of Buck-boost converter applying a 5 % variation in the input

Figure 7 .
Figure 7. a) Response of system for a 10 % Setpoint change using SMC controller, b) Control action (a) (b)

Figure 8 .
Figure 8. a) Response of system for 10 % setpoint step change b) Control action (a) (b)

Table 1 .
Parameters of Buck-Boost Converter

Table 2 .
Controller Performance Indicators

Table 3 .
Controller Performance Indicators