Adaptive fuzzylogic state controller for DC–DC stepdown converter
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Abstract
This paper deals with the implementation of a control algorithm based on a state controller with an adaptation of the state controller gains by using the fuzzylogic approach. Adjustable state controller gains cause that the static error can be reduced arbitrarily for a system with variable parameters as is the DC–DC stepdown converter for power supply systems. Fuzzy sets are tuned offline by a genetic algorithm using fuzzylogic and a genetic toolbox in MATLAB. Fuzzification and defuzzification algorithms are implemented in real time within a Field Programmable Gate Array circuit. The whole control algorithm is performed within a sampling time of 5.33 \(\upmu \)s. Operation of the controller is verified experimentally.
Keywords
State controller Adaptive control Fuzzylogic, DC–DC converter1 Introduction
The state controller is not used extensively in the control of DC–DC converters due to its sensitivity to parameter changes, causing the steadystate error and output voltage overshoot. The FLCbased adaptation algorithm for state controllers’ gains can improve these properties. The FLC membership functions are adjusted offline by a genetic algorithm. The FLC algorithm is performed in real time inside the FPGA. This state controller assures the control of output voltage and its’ derivative, by using only one voltage sensor.
Section 2 introduces a set of state controller gains that are calculated as [26] suggests with respect to buck converter input voltage and load changes. The obtained sets define the search range of the parameters for the GA and present the physical limitations for the FLC membership functions. Section 3 provides an interaction between the FLC membership function and GA. The FLC system’s performance is adjusted to provide an appropriate state controller’s gain, which minimises the steadystate error and output voltage overshoot. Tuning can be done with a GA where the parameters of the fuzzy system’s membership functions’ sets evolve by means of selection, mutation and recombination. The whole system is verified by the experimental results in Sect. 4.
Control structures are tested on sophisticated hardware processing platforms with the ability to realise an accurate and fast dynamic response of converter which serve as a basis for research with the ability of exploitation to the required final product design. Applications requiring accurate and fast dynamic responses can nowadays be found in the diversity of data centres where with the digital content and computing services which makes them fast growing electricity demanding loads [27, 28].
2 The model of the system
2.1 State control scheme
2.2 The gain \(K_{r1}\) analysis
The limits’ combinations for the set of \(K_{r1_\varepsilon }\) evaluation
\(K_{r1_{\varepsilon }}\)  R  \(U_d\)  \(\varepsilon \) 

\(K_{r1_{\varepsilon 1}}\)  \(R_m\)  \(U_{d_{m}}\)  \(\varepsilon _m\) 
\(K_{r1_{\varepsilon 2}}\)  \(R_m\)  \(U_{d_{m}}\)  \(\varepsilon _M\) 
\(K_{r1_{\varepsilon 3}}\)  \(R_m\)  \(U_{d_{M}}\)  \(\varepsilon _m\) 
\(K_{r1_{\varepsilon 4}}\)  \(R_m\)  \(U_{d_{M}}\)  \(\varepsilon _M\) 
\(K_{r1_{\varepsilon 5}}\)  \(R_M\)  \(U_{d_{m}}\)  \(\varepsilon _m\) 
\(K_{r1_{\varepsilon 6}}\)  \(R_M\)  \(U_{d_{m}}\)  \(\varepsilon _M\) 
\(K_{r1_{\varepsilon 7}}\)  \(R_M\)  \(U_{d_{M}}\)  \(\varepsilon _m\) 
\(K_{r1_{\varepsilon 8}}\)  \(R_M\)  \(U_{d_{M}}\)  \(\varepsilon _M\) 
The limits’ combinations for the set of \(K_{r1_A}\) evaluation
\(K_{r1_{A}}\)  R  \(U_{d}\)  \(A_{ov}\) 

\(K_{r1_{A1}}\)  \(R_m\)  \(U_{d_{m}}\)  \(A_{ov_m}\) 
\(K_{r1_{A2}}\)  \(R_m\)  \(U_{d_{m}}\)  \(A_{ov_M}\) 
\(K_{r1_{A3}}\)  \(R_m\)  \(U_{d_{M}}\)  \(A_{ov_m}\) 
\(K_{r1_{A4}}\)  \(R_m\)  \(U_{d_{M}}\)  \(A_{ov_M}\) 
\(K_{r1_{A5}}\)  \(R_M\)  \(U_{d_{m}}\)  \(A_{ov_m}\) 
\(K_{r1_{A6}}\)  \(R_M\)  \(U_{d_{m}}\)  \(A_{ov_M}\) 
\(K_{r1_{A7}}\)  \(R_M\)  \(U_{d_{M}}\)  \(A_{ov_m}\) 
\(K_{r1_{A8}}\)  \(R_M\)  \(U_{d_{M}}\)  \(A_{ov_M}\) 
2.3 The gain \(K_{r2}\) analysis
The limits’ combinations for the set of \(K_{r2}\) evaluation
R  \(U_{d}\)  \(K_{r1}\) 

\(R_m\)  \(U_{d_{m}}\)  \(K_{r1_{m}}\) 
\(R_m\)  \(U_{d_{m}}\)  \(K_{r1_{M}}\) 
\(R_m\)  \(U_{d_{M}}\)  \(K_{r1_{m}}\) 
\(R_m\)  \(U_{d_{M}}\)  \(K_{r1_{M}}\) 
\(R_M\)  \(U_{d_{m}}\)  \(K_{r1_{m}}\) 
\(R_M\)  \(U_{d_{m}}\)  \(K_{r1_{M}}\) 
\(R_M\)  \(U_{d_{M}}\)  \(K_{r1_{m}}\) 
\(R_M\)  \(U_{d_{M}}\)  \(K_{r1_{M}}\) 
3 The FLCGA state controller
3.1 Fuzzylogic algorithm
Fuzzylogic rule tables
Fuzzy rules (\(K_{r1}\))  
\(u_0\)  \(NB_{1i}\)  \(NM_{1i}\)  \(NS_{1i}\)  \(ZE_{1i}\)  \(PS_{1i}\)  \(PM_{1i}\)  \(PB_{1i}\) 
out  \(NB_{o1}\)  \(NM_{o1}\)  \(NS_{o1}\)  \(ZE_{o1}\)  \(PS_{o1}\)  \(PM_{o1}\)  \(PB_{o1}\) 
Fuzzy rules (\(K_{r2}\))  
\(\varDelta u_0\)  \(NB_{2i}\)  \(NM_{2i}\)  \(NS_{2i}\)  \(ZE_{2i}\)  \(PS_{2i}\)  \(PM_{2i}\)  \(PB_{2i}\) 
out  \(NB_{2o}\)  \(NM_{2o}\)  \(NS_{2o}\)  \(ZE_{2o}\)  \(PS_{2o}\)  \(PM_{2o}\)  \(PB_{2o}\) 
Triangular membership functions were chosen in advance intentionally to use and apply some degrees of membership for the linguistic variable. As with the use of fast processing capability with the FPGA the computational delays have been significantly reduced in the scope of the time fame of switching period. Triangular membership functions that have been used are organised and optimised towards singleton membership function use, but with some degree of membership. With the use of singleton membership functions, additional computational time optimisation could be achieved although some accuracy would be lost with the digitalization which could lead to additional chattering in the output voltage. So no additional computational optimisation was taken into consideration.
3.2 GA tuning process for stateFLCgains (\(K_{r1}\), \(K_{r2}\))

The chromosome structure,

The fitness function on which parameter selection is determined and

The parameters for implementation of the genetic algorithms.
3.2.1 Definitions of chromosome coding
3.3 Offline chromosome calculation
The capability and feasibility of adjusting the state controller gains with a GA is demonstrated in this section. The search areas for the gains are defined by analysis of the feedback gains \(K_{r1}\) and \(K_{r2}\) done in Sect. 2. The condition (42) can be reached arbitrarily by using the appropriate fitness function. The GA technique is employed for tuning the membership functions of the FLC. This tuning approach employs the use of MATLAB mfiles and functions to manipulate the fuzzy membership boundaries, run the Simulinkbased simulation, check the resulting performance and modify the fuzzy system continuously a number of times during the search for adequate responses. The fitness function is evaluated by (41), where the weight factors are chosen equally (\(g_1=g_2=g_3=1/3\)). With the predefined dynamic parameters set as \(A_{ov}'=4\%\), \(t_r'= 100\,\upmu \)s, \(\varepsilon '=0\%\) the GA optimisation algorithm was run over 40 generations with each generation having a population size of 60. The average value of the fitness functions was calculated within every generation. Also, the best objective function was extracted from the set of 60 as the best objective function.
Performance metrics’ comparison between conventional and FLC adapted state controllers
Constant  FLC  FLC  

\(K_{r1}\), \(K_{r2}\)  \(A'_{\mathrm{ov}}\) = 4%  \(A'_{\mathrm{ov}}\) = 0%  
\(R=3.4~\Omega \)  
\(A_{ov}~(\%)\)  2.67  4.00  0.07 
\(t_r~(\upmu \)s)  120  120  172 
\(\varepsilon ~(\%)\)  2  0.1  0.006 
3.4 Stability
3.5 Comments for GA optimisation
The adaptation of gains \(K_{r1}\) and \(K_{r2}\) by using FLC was first tested by simulation. Figure 8 shows three cases of state controller verification. Figure 8a shows the voltage responses when nonadaptive gains were applied, and Fig. 8b, c shows the voltage responses when FLC was applied with the required overshoot for \(A_{ov}\le 4\%\) and \(A_{ov}\approx 0\%\).
The performance metrics of these results are shown in Table 5. It can be seen from such optimisation of the fuzzylogic controller that a desirable response performance can be obtained based on a predefined parameter set. When compared to the conventional state controller, the optimised FLC for 4% overshoot shows a comparable performance at the startup and a better performance at the load change. With further improvements, the FLC is evaluated for 0% overshoot and exhibited a better response at the startup than at the load change. The result of optimisation showed good performance in terms of achieving the desired overshoot value with small values of steadystate error, even by load changes, which is the main drawback of the conventional state controller.
4 Experimental results and discussion
Performance indicators for the FLCGA with adjustable \(K_{r1}\) and \(K_{r2}\) gains (\(A'_{ov}=0\%\)); experimental results (Fig. 9d)
\(R~(\Omega ) \)  \(U_d~(V)\)  \(A_{ov}~(\%)\)  \(t_r~(\mu s)\)  \(\varepsilon ~(\%)\) 

12.0  
6.8  0.48  108  \(\simeq \)0.00  
3.4  0.40  163  \(\simeq \)0.00 
Figure 9c shows the transient when the reference voltage was changed from 0 to 3.3 V and then to 5 V under noload condition (\(R \rightarrow \infty \)), and further when a load of \(6.8~\Omega \) was applied. According to the buckconverter’s operational modes, this mode represents the worst case operation. The oscillogram of the operational mode, during the startup and load change condition, is shown in Fig. 9d. When the reference voltage was changed from 0 to 5 V, the load was \(6.8~\Omega \). At a timeinstant of \(t=5\) ms, the load resistance changed to \(3.4~\Omega \). These results were used in order to measure the performance indicators (Table 6).
5 Conclusion
The results presented in this paper show the convenience of applying fuzzylogic control to DC–DC converter control as an alternative to conventional techniques. Genetic algorithms are a valuable tool for fuzzy controller tuning. An evolutionary strategy was applied to tune the input and output membership functions of a fuzzy controller. This algorithm was run under the MATLAB/genetic algorithm toolbox. The evaluated boundaries of the fuzzy sets function were, afterwards, implemented into the FPGA. The genetic tuning process was able to reduce the overshoots at the startup and the over/undershoots at the load change. The nonlinear gains (\(K_{r1}\) and \(K_{r2}\)) were calculated as the centre of gravity, of the chosen input and output membership functions, in real time. The FPGA unit spends 50–140 ns for this task. The presented process of control adaptation is demanding and requires additional skills to tune the process, but all the effort showed promising results by means of reducing the static and output voltage dynamic errors for the used state controller. The experimental set is open for testing other digital nonlinear controllers.
Footnotes
 1.
The parameters that will be used for further analysis are: \(R_{L}=0.2~\Omega \), \(L=68~\upmu \)H, \(C=220~\upmu \)F, \(U_0=5\) V, \(U_d=12\) V.
 2.
for nominal and open terminal load \(\omega _o\in [8.2\times 10^3~\mathrm{rad/s}, 9.2\times 10^3~\mathrm{rad/s}].\)
 3.
\(R\in [R_\mathrm{min},R_\mathrm{max}]=[3.4\,\Omega , \infty ].\)
 4.
\(\lambda _{Ts}=2\pi /T_s=1.12\times 10^6\, \mathrm{rad/s}.\)
 5.
\(R \in [R_m,~R_M)=[3.4~\Omega ,~\infty ).\)
 6.
\(U_d\in [U_{d_m},U_{d_M}]=\)[10.4, 14.4 V].
 7.
\(K_{r1}\in \left[ K_{r1_m}, K_{r1_M}\right] \doteq [0.9,~2.2]\).
 8.
\(K_{r2} \in [K_{r2_{m}},K_{r2_{M}}]\doteq [2.7\times 10^{5},~4.9\times 10^{5}].\)
 9.
Legend of membership functions: Negative (N) or Positive (P) set of Small (S), Big (B), Middle (M) and ZEro (ZE) membership functions.
 10.
\(K_{r1}\in [ K_{r1_m}, K_{r1_M} ]\doteq [0.5,~3.0]\).
 11.
\(K_{r2} \in [K_{r2_{m}},K_{r2_{M}}]\doteq [2.7\times 10^{5},~6.0\times 10^{5}].\)
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