# Experimental Investigation of an Adaptive Fuzzy-Neural Fast Terminal Synergetic Controller for Buck DC/DC Converters

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Related Work

#### 1.3. Novelty and Principal Contributions

- (i)
- It is the first time a FNN model has been developed based on FTSC for DC/DC buck converters;
- (ii)
- Unlike closely related work, the proposed AFN-FTSC is created by incorporating the macro-variable to tackle the chattering effects, decrease the time of convergence, simplify the expression of the controller, and ensure a fast transient reaction, low steady-state error, and high output voltage tracking accuracy;
- (iii)
- A Lyapunov stability theorem is rigorously used to demonstrate the overall stability of the system and to obtain the updated rules for the FNN weights.
- (iv)
- The removal of demand for an accurate model by using a FNN approximator to estimate an unknown buck DC/DC converter functions;
- (v)
- Experiments are carried out to demonstrate efficiency while assisting to achieve desired goals under a variety of operating situations.
- (vi)
- It simultaneously ensures the higher output voltage tracking accuracy, swift transient responses, and less impact by disturbances and uncertainties due to the use of adaptive time-varying reaching law.

#### 1.4. Organization of the Paper

## 2. Preliminaries

#### 2.1. Principle of Synergetic Control (sc)

- Invariant manifold is created in the state-space of a controllable system. On this attractor, we guarantee the organization of the preferred static and dynamic behavior of the controllable system. The design of the attractor is the indication of a directed self-organization principle.
- The most important premise in the theory of SC is the principle of compression–decompression of the phase flow of the controllable objects.
- The designer’s necessities are given in the form of an affine system that describes the preferred operating modes of the controlled systems.

^{n}, u ∈ R

^{m}, indicating the state-variable and control-variables, respectively. The f(.) indicates a continuous nonlinear function. The SC design structure starts with defining a macro-variable φ as a function of the state variables [19]:

_{0}(Figure 1). Since the time constant κ > 0, the macro-variable φ will decay exponentially with a speed determined by κ. As long as the system is stable, the smaller value of κ is faster than the macro-variable decays. When φ approaches zero, the system converges to the manifold and then functions on the manifold without interruption.

#### 2.2. Principle of Fnn Approximator

_{1}, x

_{2},…, x

_{n}]

^{T}and y = [y

_{1}, y

_{2},…, y

_{m}]

^{T}characterize the input and output parameter vectors of FNN. ${A}_{1}^{l}\text{}and\text{}{B}_{1}^{l}$ are the linguistic variables of the fuzzy sets, expressed by their membership function vectors μA

_{i}

^{l}(x

_{i}) and μB

_{i}

^{l}(y), respectively.

**Remark**

**1.**

^{2}). The total complexity of the AFN-FTSC algorithm is O(n × m) + O(m

^{2}).

## 3. Mathematical Model of Buck DC/DC Converter

_{in}with a DC load R

_{o}having its output voltage as V

_{o}. The buck DC/DC converter encompasses an adjustable switch Q, diode D, an inductor L with i

_{L}as the current flowing through it, a capacitor C with i

_{C}as the current flowing through it, and a load resistor R

_{0}where i

_{R}is the load current.

_{o}across R

_{o}, which is also the voltage across C. Similarly, i

_{L}will change as it flows through L. Since the switch will be in two modes, ON and OFF, there will be two stages.

_{1}and x

_{2}, respectively. Equation (18) represents the final model of the buck DC/DC converter, which is utilized to design the proposed composite controller as discussed in the section below:

## 4. Proposed Robust AFN-FTSC Design

_{1}, e

_{2}), which gives the desired value of converter output voltage V

_{o-ref}without the prior modeling information and is subject to the requirement that the voltage error converges to origin asymptotically in finite-time t

_{r}, i.e., Further analyses are presented to comprehensively analyze the stability and reachability of the proposed method. The determination of the switching control law is presented below.

#### 4.1. Determination of the Control Law

_{0}is the main task, the design process needs to start by defining the error for this output voltage.

- (1)
- Step 1:

_{1}) for V

_{0}can be expressed as:

_{0−ref}is the reference output voltage and its dynamic will be as:

- (2)
- Step 2:

- (3)
- Step 3:

- (4)
- Step 4:

_{r}for e

_{1}can be obtained by integrating Equation (20), which may be calculated as:

_{1}(0) is the initial value of e

_{1}(t). The time derivative of φ can be determined as:

_{FTSC}, guides to an enhanced reaching law (29):

#### 4.2. Stability Analysis

#### 4.3. Approximation of f(x) and g(x) Based on FNN

**Theorem**

**1.**

_{AFN-FTSC}is constructed as (32):

_{f}(x) and ϕ

_{g}(x) are the transfer functions from the input layer to the rule layer, w

_{f}and w

_{g}are the connection weights of FNN. By updating the network weights, the system uncertainties can be estimated adaptively.

**Theorem**

**2.**

_{1}and η

_{2}are arbitrary positive parameters.

**Assumption**

**1.**

^{n}={x∈ R

^{n}:∣x∣ ≤ K

_{x}< +∞}, with K

_{x}is a constant. The ideal FNN weights w

_{f}

^{∗}and w

_{g}

^{∗}are located in the convex area shown below:

_{f}and H

_{g}are designed parameters, and the radius Δ

_{f}and Δ

_{g}are limitations for w

_{f}and w

_{g}. The universal approximation theory states that there is an ideal FNN weights w

_{f}* and w

_{g}* satisfies:

**Assumption**

**2**

**(See**

**)**

**.**For each given real smooth variable f(x) and g(x) defined on a compact set x ∈ R

^{n}and for any arbitrary ε

_{f}> 0 and ε

_{g}> 0, there exists a FNN approximator and in the formula of (41) and (42) so that:

**Remark**

**2.**

_{f}(x) and ϕ

_{g}(x) can be scheduled adaptively according to the variety of x

_{1}and x

_{2}. Then, the minimum approximation error can be obtained as:

_{max}. The dynamic of the macro-variable is calculated by replacing (43) into (23):

^{∗}.

**Remark**

**3.**

_{1}, from (41), we recognize that the macro-variable is limited and every term in (42) is limited, hence $\left(\phi ,\dot{\phi}\right)\in {L}_{\infty},$ making use of the Barbalat lemma [26]. We can conclude that the tracking error converges to zero asymptotically, which confirms the stability condition of a closed-loop system. Consequently, the stable control performance of the buck DC/DC converter can be ensured without the necessity of system information. Figure 6 shows the overall schematic diagram of the developed AFN-FTSC algorithm.

## 5. Controller Performance Evaluation

#### 5.1. Selection of the User-Defined Parameters

#### 5.2. Experimental Validation

- Controller performance under variations in the load resistance,
- Controller performance under variations in the input supply voltage,
- Controller performance under start-up transient,
- Controller performance under variations in the output reference voltage,

**Case I:**Output voltage regulation with variations in the load resistance.

_{o}) and inductor current (i

_{L}), as illustrated in Figure 10. The output voltage quickly settles down to its reference value without affecting the steady-state behavior while the designed one is used. However, the overshoots are a bit higher with higher settling times, especially at the instant of changes, i.e., at the decrease in load resistance when the existing FTSC is used. Hence, Figure 10a provides an observation that the designed AFN-FTSC controller can stabilize the output voltage very quickly to its desired value without having any significant impact even with a large variation in the load resistance. Figure 10a also confirms the faster settling time with less transient with the AFN-FTSC controller during the post-disturbance operation while making its comparison with the FTSC controller.

**Case II**: Output voltage regulation with variations in the input supply voltage

**Case III:**Output voltage regulation with variations in the output reference voltage.

_{o}and i

_{L}) are depicted in Figure 12a,b. These responses clearly demonstrate the superior capability of the AFN-FTSC controller (Figure 12a) for tracking output reference voltage, as the existing FTSC controller fails to do so (Figure 12b).

**Case IV:**Output voltage regulation under nominal start-up in the reference voltage.

#### 5.3. Experimental Validation

## 6. Conclusions

- The designed AFN-FTSC properly tracks the reference value of the output voltage with the minimum overshoot and faster settling time—even variations in the input voltage, load resistance, and reference voltage.
- The steady-state is significantly low under any operating scenario.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Babes, B.; Boutaghane, A.; Hamouda, N.; Mezaache, M. Design of a robust voltage controller for a dc-dc buck converter using fractional-order terminal sliding mode control strategy. In Proceedings of the International Conference on Advanced Electrical Engineering (ICAEE), Algiers, Algeria, 19–21 November 2019. [Google Scholar]
- Amir, M.; Prajapati, A.K.; Refaat, S.S. Dynamic Performance Evaluation of Grid-Connected Hybrid Renewable Energy-Based Power Generation for Stability and Power Quality Enhancement in Smart Grid. Front. Energy Res.
**2022**, 10, 861282. [Google Scholar] [CrossRef] - Yan, Y.; Liu, J. Analysis of passivity-based sliding-mode control strategy in DC/DC converter. In Proceedings of the Chinese Control Conference, Harbin, China, 7–11 August 2006; pp. 171–174. [Google Scholar]
- Young, D.S.; Hen, T.-W.; Santi, E.; Monti, A. Synergetic control approach for induction motor speed control. In Proceedings of the Annual Conference of IEEE Industrial Electronics Society, Busan, Korea, 2–6 November 2004. IECON 2004. [Google Scholar]
- Dehri, K.; Nouri, A.S. A discrete repetitive adaptive sliding mode control for DC-DC buck converter. Inst. Mech. Eng. Part I J. Syst. Control. Eng.
**2021**, 235, 1698–1708. [Google Scholar] [CrossRef] - Chen, J.J.; Hwang, Y.S.; Lin, J.Y.; Ku, Y. A dead-beat-controlled fast-transient-response buck converter with active pseudo-current-sensing techniques. IEEE Trans. Very Large Scale Integr. (VLSI) Syst.
**2019**, 27, 1751–1759. [Google Scholar] [CrossRef] - Kumar, V.I.; Kapat, S. Mixed-signal hysteretic internal model control of buck converters for ultra-fast envelope tracking. In Proceedings of the IEEE Applied Power Electronics Conference and Exposition (APEC), Long Beach, CA, USA, 20–24 March 2016; pp. 3224–3230. [Google Scholar]
- Linares-Flores, J.; Hernandez Mendez, A.; Garcia-Rodriguez, C.; Sira-Ramirez, H. Robust nonlinear adaptive control of a boost converter via algebraic parameter identification. IEEE Trans. Ind. Electron.
**2014**, 61, 4105–4114. [Google Scholar] [CrossRef] - Xu, Q.; Yan, Y.; Zhang, C.; Dragicevic, T.; Blaabjerg, F. An offset-free composite model predictive control strategy for DC/DC buck converter feeding constant power loads. IEEE Trans. Power Electron.
**2020**, 35, 5331–5342. [Google Scholar] [CrossRef] - Hausberger, T.; Kugi, A.; Eder, A.; Kemmetmüller, W. High-speed nonlinear model predictive control of an interleaved switching DC/DC-converter. Control. Eng. Pract.
**2020**, 103, 104576. [Google Scholar] [CrossRef] - Albira, M.E.; Zohdy, M.A. Adaptive model predictive control for DC-DC power converters with parameters uncertainties. IEEE Access
**2021**, 9, 135121–135131. [Google Scholar] [CrossRef] - Hamouda, N.; Babes, B.; Boutaghane, A. Design and analysis of robust nonlinear synergetic controller for a PMDC motor driven wire-feeder system (WFS). In Lecture Notes in Electrical Engineering; Springer: Singapore, 2020; pp. 373–387. [Google Scholar]
- Babes, B.; Boutaghane, A.; Hamouda, N. Design and real-time implementation of an adaptive fast terminal synergetic controller based on dual RBF neural networks for voltage control of DC–DC step-down converter. Electr. Eng.
**2021**, 104, 945–957. [Google Scholar] [CrossRef] - Hamouda, N.; Babes, B. A DC/DC Buck converter voltage regulation using an adaptive fuzzy fast terminal synergetic control. In Lecture Notes in Electrical Engineering; Springer: Singapore, 2020; pp. 711–721. [Google Scholar]
- Hadjer, A.; Ameur, A.; Harmas, N.M. Adaptive non-singular terminal synergetic power system control using PSO. In Proceedings of the 8th International Conference on Modellin, Identification and Control (ICMIC-2016), Algiers, Algeria, 15–17 November 2016. [Google Scholar]
- Babes, B.; Boutaghane, A.; Hamouda, N.; Mezaache, M.; Kahla, S. A robust adaptive fuzzy fast terminal synergetic voltage control scheme for DC/DC buck converter. In Proceedings of the International Conference on Advanced Electrical Engineering (ICAEE), Algiers, Algeria, 19–21 November 2019. [Google Scholar]
- Wen, S.; Chen, M.Z.Q.; Zeng, Z.; Huang, T.; Li, C. Adaptive neural-fuzzy sliding-mode fault-tolerant control for uncertain nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst.
**2017**, 47, 2268–2278. [Google Scholar] [CrossRef] - Chen, Z.; Li, Z.; Chen, C.L.P. Adaptive neural control of uncertain MIMO nonlinear systems with state and input constraints. IEEE Trans. Neural Netw. Learn. Syst.
**2017**, 28, 1318–1330. [Google Scholar] [CrossRef] - Santi, E.; Monti, A.; Proddutur, D.; Li, K.; Dougal, R.A. Synergetic control for power electronics applications: A comparison with the sliding mode approach. J. Circuits Syst. Comput.
**2004**, 13, 737–760. [Google Scholar] [CrossRef] [Green Version] - Shahgholian, G. Power system stabilizer application for load frequency control in hydro-electric power plant. Int. J. Theor. Appl. Math.
**2017**, 3, 148. [Google Scholar] [CrossRef] [Green Version] - Rubaai, A.; Young, P. Hardware/software implementation of fuzzy-neural-network self-learning control methods for brushless DC motor drives. IEEE Trans. Ind. Appl.
**2016**, 52, 414–424. [Google Scholar] [CrossRef] - Nettari, Y.; Kurt, S. Design of a new non-singular robust control using synergetic theory for DC-DC buck converter. Electrica
**2018**, 18, 292–329. [Google Scholar] [CrossRef] [Green Version] - Zerroug, N.; Harmas, M.N.; Benaggoune, S.; Bouchama, Z.; Zehar, K. DSP-based implementation of fast terminal synergetic control for a DC–DC Buck converter. J. Frankl. Inst.
**2018**, 355, 2329–2343. [Google Scholar] [CrossRef] - Ullah, N.; Shaoping, W. High performance direct torque control of electrical aerodynamics load simulator using adaptive fuzzy backstepping control. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
**2015**, 229, 369–383. [Google Scholar] [CrossRef] - Sastry, S.; Bodson, M. Adaptive Control: Stability, Convergence, and Robustness; Englewood Cliffs, N.J., Ed.; Prentice-Hall: Hoboken, NJ, USA, 1989. [Google Scholar]
- Babes, B.; Mekhilef, S.; Boutaghane, A.; Rahmani, L. Fuzzy Approximation-Based Fractional-Order Nonsingular Terminal Sliding Mode Controller for DC–DC Buck Converters. IEEE Trans. Power Electron.
**2022**, 37, 2749–2760. [Google Scholar] [CrossRef] - Alanqar, A.; Durand, H.; Albalawi, F.; Cristofides, P.D. An economic model predictive control approach to integrated production management and process operation. AIChE J.
**2017**, 63, 1892–1906. [Google Scholar] [CrossRef]

**Figure 7.**The output voltage of the converter for different values of (p/q) with the designed controller.

**Figure 10.**Dynamic performance of the converter under load-resistance variations (V

_{o}: 10 V/div, i

_{L}: 0.25 A/div and time: 100 ms/div): (

**a**) AFN-FTSC controller, (

**b**) FTSC controller.

**Figure 11.**Dynamic performance of the converter under input supply voltage variations (V

_{o}: 10 V/div, i

_{L}: 0.5 A/div and time: 20 ms/div): (

**a**) AFN-FTSC controller, (

**b**) FTSC controller.

**Figure 12.**Dynamic performance of the converter under output reference voltage variations (V

_{o}: 10 V/div, i

_{L}: 0.5 A/div and time: 50 ms/div): (

**a**) AFN-FTSC controller, (

**b**) FTSC controller.

**Figure 13.**Dynamic performance of the converter under sinusoidal wave in the output voltage reference (V

_{o}: 10 V/div, i

_{L}: 0.5 A/div, and time: 1 s/div): (

**a**) AFN-FTSC controller, (

**b**) FTSC controller.

**Figure 14.**Dynamic responses of the converter with triangular-wave variations in the output voltage reference (V

_{o}: 10 V/div, i

_{L}: 0.25 A/div and time: 2 s/div): (

**a**) AFN-FTSC controller, (

**b**) FTSC controller.

**Figure 15.**Dynamic responses of the converter under nominal start-up in the output voltage reference (V

_{o}: 10 V/div, i

_{L}: 0.25 A/div and time: 20 ms/div): (

**a**) AFN-FTSC controller, (

**b**) FTSC controller.

p/q | 1.1 | 1.5 | 1.8 | 2.0 |

Voltage overshoot (V) | 15.85 | 0.13 | 0.11 | 0.07 |

Settling-time (ms) | 42.96 | 9.44 | 27.60 | 75.44 |

Regulator | Parameters | Gain Value |
---|---|---|

AFN-FTSC | κ | 0.005 |

a | 200 | |

b | 300 | |

p | 3 | |

q | 2 | |

µ_{1} | 500 | |

µ_{2} | 800 | |

FTSC | κ | 0.005 |

a | 200 | |

b | 300 | |

p | 3 | |

q | 2 |

Parameter | Value |

V_{in} | 100 V |

R_{o} | 40/80/40 Ω |

V_{o_red} | 20/30/50 V |

L | 7 mH |

C | 800 μF |

f_{s} | 20 kHz |

Controller | Peak Value (V) | Overshoot (%) | Undershoot (%) | Settling Time (ms) |
---|---|---|---|---|

AFN-FTSC | 0 | 0 | 6.66 | 2 |

FTSC | 8 | 16 | 40 | 20 |

Control Strategy | Peak Value (V) | Overshoot (%) | Undershoot (%) | Settling Time (ms) |
---|---|---|---|---|

AFN-FTSC | 3 | 10 | 0 | 5 |

FTSC | 11 | 36.66 | 0 | 28 |

**Table 6.**Peak value, overshoot, undershoot, and settling time with variations in the reference voltage.

Control Strategy | Peak Value (V) | Overshoot (%) | Undershoot (%) | Settling Time (ms) |
---|---|---|---|---|

AFN-FTSC | 0 | 0 | 0 | 2 |

FTSC | 2 | 6.66 | 0 | 40 |

Controller | Peak Value (V) | Overshoot (%) | Undershoot (%) | Settling Time (ms) |
---|---|---|---|---|

AFN-FTSC | 3 | 2 | 0 | 5 |

FTSC | 10 | 20 | 0 | 20 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Babes, B.; Hamouda, N.; Albalawi, F.; Aissa, O.; Ghoneim, S.S.M.; Abdelwahab, S.A.M.
Experimental Investigation of an Adaptive Fuzzy-Neural Fast Terminal Synergetic Controller for Buck DC/DC Converters. *Sustainability* **2022**, *14*, 7967.
https://doi.org/10.3390/su14137967

**AMA Style**

Babes B, Hamouda N, Albalawi F, Aissa O, Ghoneim SSM, Abdelwahab SAM.
Experimental Investigation of an Adaptive Fuzzy-Neural Fast Terminal Synergetic Controller for Buck DC/DC Converters. *Sustainability*. 2022; 14(13):7967.
https://doi.org/10.3390/su14137967

**Chicago/Turabian Style**

Babes, Badreddine, Noureddine Hamouda, Fahad Albalawi, Oualid Aissa, Sherif S. M. Ghoneim, and Saad A. Mohamed Abdelwahab.
2022. "Experimental Investigation of an Adaptive Fuzzy-Neural Fast Terminal Synergetic Controller for Buck DC/DC Converters" *Sustainability* 14, no. 13: 7967.
https://doi.org/10.3390/su14137967