Many-objective Adaptive Fuzzy With Sliding Mode Control for a Class of Switching Power Converters Using Global Optimization

Control of the power converters in renewable energy systems for stability and efficiency poses a technical challenge due to the intermittency of energy produced and inherent nonlinear dynamics. This paper presents a parametric optimization framework amid the synthesis of an adaptive fuzzy with sliding mode controller for a class of switching power converters suited for renewable energy systems. Four performance metrics essential to the practical needs are suggested. The potential design parameters of the controller are determined, and their influences on the performance metrics are studied and validated. A many-objective optimization problem is formulated accordingly, and a computational platform based on MATLAB/Simulink environment is established to solve the problem. Two multi-objective global search algorithms, i.e., particle swarm and bat optimization, are employed to obtain a set of Pareto optimal controllers, which noticeably enhance the performance metrics of the control system. An experimental platform with dSPACE controller board is utilized to further justify the simulation results. With those optimal controllers, the experimental results also demonstrate improvement of the performance.


I. INTRODUCTION
All high-performance applications require a regulated and stable power supply, which in turn demands an intermediate voltage regulator circuit for energy conversion of high quality. Renewable energy systems, such as photovoltaic (PV), wind, and battery storage, have been allocated to the power grids and electric vehicles [1]- [3]. The intermittency of the energy, i.e., randomly varying output, generated by those systems deems power electronic converters a vital part of the systems [4]. Power converters, such as buck, boost, flyback, push-pull, and full bridge converters, have been developed to interface those energy sources for stable and efficient power conditioning and control [5]. It is worth notice that power intermittency and load variation would also drag the power converter off its designated operating point, and stimulate the nonlinear dynamics of the components within the converter [6]- [8]. Therefore, control of power converters in renewable energy applications is a technically challenging task.
Feedback controllers have been adopted to enhance the performance of the power converter from the aspects of transient and steady state responses [9]. Depending on the situation of application, simple or complex control algorithm may be considered. Simple controllers such as proportionalintegral ones are easy to implement. However, they are usually designed for a certain operating point, thus is not resilient to input and parameter variation (i.e., lacks robustness). Robustness of these controllers have been improved for renewable energy applications, but achieving high performance is only possible when a thorough understanding of the converter in various operating points [10], [11] can be done. Control system design often comes across uncertainties. The uncertainties are due to disturbances, unknown parameters, nonlinear or unmodeled dynamics. To deal with nonlinear dynamics, feedback linearization is one common technique utilizing feedback to cancel part of or all the nonlinear terms. Adaptive feedback linearization can tackle both unknown parameters and nonlinear dynamics of certain structure (e.g., which can be linearly parameterized). For more generic types of uncertainties, a revision, also known as adaptive fuzzy control, has been shown to be effective [12]- [14]. Adaptive fuzzy control uses radial basis function network (or fuzzy inference network) to approximate uncertainties of unknown parameters and structure [15]- [17]. Due to the network output being a linear combination of the outputs from the hidden nodes (i.e., radial basis or fuzzy membership function), the approximated uncertainties become linearly parameterized. Sliding mode control is robust to external disturbances and nonlinearities of no prior knowledge [18]- [20]. However, either the existence of unmodelled dynamics or improper controller design may result in high-frequency switching, i.e., issue of chattering. High-order sliding mode controllers may attenuate chattering. However, chattering caused by the unmodelled dynamics cannot be eliminated.
A few parameters of the controller can be decided by following the design procedure of a control algorithm, mostly related to stability or steady state convergence (the first concern of control system design). Other parameters of the controller, which have connection with alternative performance indicators, are mainly customizable. Analytical relationship between those performance indices and the customizable parameters is usually complex or not clear. Therefore, determination of those parameters poses a challenge when performance requirements apart from steady state response are also critical [21]. Optimization algorithms have been applied to power converters pursuing better conversion efficiency, lower switching losses, better timedomain response, and other objectives. Relevant previous works can be separated into two categories. The first category focuses on the design of converter components, e.g., transformer turn ratio, values of capacities, and inductances [21]- [26]. The second category concentrates on the design of controller parameters [27]- [33]. A detailed comparison is provided in Table 1. As can be seen, most works effectively improve the performance of the power converters, but no particular one outranks the others. Note also that most works only deal with single objective optimization. Neither do they consider the robustness of the control system.
Optimization algorithms have been applied to parameter design of the sliding mode controllers, e.g., gain and sliding surface [34]. Instead of trial and error, optimization approach can effectively reduce the effort and time for acquiring feasible controller parameters. Works on single objective optimization, e.g., minimizing operation cost or reliability, have been reported [23], [26], [35]. When more than one performance indices are deemed essential, resorting to multior many-objective optimization is sensible. Many-objective optimization refers to multi-objective optimization problem containing large number of objectives, typically four or more. Two commonly used global optimization algorithms, suited for many-objective optimization, are particle swarm optimization and bat optimization [36]- [39]. Both are metaheuristic or swarm intelligence algorithms in terms of their mimicking collective behavior of ants, birds, fish, insects, or bats. Studies have shown that bat optimization algorithm solves constrained or unconstrained optimization problems with better robustness and efficiency. Not only can the algorithm increase the diversity of solutions in the population, it also has an automatic mechanism to balance exploration and exploitation during the search process. This paper proposes a parametric optimization framework amid the synthesis of a robust adaptive fuzzy controller for a class of switching power converters. The open-loop system is a phase-shift pulse width modulation (PSPWM) full bridge DC-DC power converter, which is of practical interest due to features such as wide-range voltage output, high efficiency, etc. A comprehensive mathematical and the corresponding numerical model for this converter has been established in [40] and will be adopted in subsequent optimization and simulation. The closed-loop controller is an integration of adaptive fuzzy and sliding mode controls, and possesses advantageous traits from both design paradigms. The controller encompasses a set of customizable or design parameters, which will be adopted by the optimization problem to be formulated. Besides steady state error of the output voltage, alternative performance metrics, i.e., voltage ripple, peak load current, and transient efficiency, are also considered. To begin with, the negative or positive influence of the design parameters on performance metrics is studied. Conflicting performance metrics are also clarified. Next, a many-objective optimization problem is formulated. Subsequently, two global optimization methods, i.e., particle swarm optimization and bat optimization, are employed to numerically solve the problem and identify a set of Pareto optimal controller. Both simulation and experiment will be performed to validate the effectiveness of those optimal controllers. In summary, the main contributions of this work are as follows:  A parametric optimization framework with multiple performance requirements is proposed, which is applicable to the synthesis of a robust adaptive fuzzy controller for a class of switching power converters.  A many-objective optimization problem is formulated.
Performance metrics of common practical needs are defined, and design parameters which influence the performance metrics are identified.  Both computational and experimental platforms are established to automate and facilitate the acquirement of Pareto optimal controllers and the validation of the respective performance.  The capability of the Pareto optimal controllers, in terms of the performance metrics, are justified both by simulation and experiment. The rest of the paper is organized into Section II-VII. The operation and the state-space model for a PSPWM full bridge DC-DC power converter is reviewed in Section II, followed by design and synthesis of the adaptive fuzzy with sliding mode controller. Section III introduces performance metrics incorporated for the power converter and defines them quantitatively. The parameters of the adaptive fuzzy with sliding mode controller and their effect on the four indicators are studied. Section IV formulates the corresponding manyobjective optimization problem. Two global optimization algorithms along with the concept of Pareto front for solving the problem are described in the context of this application. In Section V, a computing framework for parametric optimization of the controller is proposed. The control system with various sets of Pareto optimal parameters is numerically simulated, and the respective sets of performance metrics are compared. In Section VI, experimental setup is described and the results are demonstrated. Conclusion and future work are detailed in Section VII.

II. ADAPTIVE FUZZY WITH SLIDING MODE CONTROL SYSTEM
As illustrated in Figure 1, the overall system has the structure of a PSPWM full bridge DC-DC power converter and an adaptive fuzzy with sliding mode controller. This section will summarize the operation and first-principle modeling of the converters, which is followed by design and synthesis of the controller.
PSPWM full bridge DC-DC power converters draw practical interest due to characteristics like wide-range voltage output, high efficiency, etc. This category of converters provides voltage translation as well as isolation from the line voltage since the circuit topology includes a transformer. A typical PSPWM full bridge DC-DC power Proj( )   Figure 2. Specifically, the system dynamics of the full bridge power converter can be divided into eight (or ten) regions, labeling from one to eight, where four of them are in positive half cycle: trailing leg transition region from t 0 to t 1 , active region from t 1 to t 2 (this region actually contains one additional sub-region of duty cycle loss, i.e., t 1 to t 12 ), leading leg transition region from t 2 to t 3 , and passive region from t 3 to t 4 , and the other four regions are in negative half cycle, respectively.
A set of elaborate control-oriented state variable models comprising each operation interval were established [40]. The corresponding computational model is established using MATLAB/Simulink. The dynamics of the established model has been justified to be close to that of a conventional PSPWM full bridge DC-DC power converter in a laboratory environment. This computational model will be utilized for subsequent parametric optimization and verification.
From the perspective of control system design, utilization of elaborate computational model is advantageous and occasionally indispensable in various scenarios (e.g., reducing time and effort of design iteration, saving cost of experimentation). When it comes to controller synthesis, a sophisticated model is often too complex to be tackled with standard techniques. Hence, model reduction techniques are commonly employed to reduce the computational cost and storage requirement. The goal is to obtain a low dimensional model that encompasses the imperative dynamics of the sophisticated model. The 'neglected' dynamics can be addressed later on by adopting appropriate control paradigm.
Refer to the models established for the ten (including duty cycle loss) operation intervals [40]. Suppose that the extent of time is 1 for completing a cycle of operation (positive and negative half cycles). An averaging state-space model of the PSPWM full bridge DC-DC power converter can be formulated as where ( ) u t is the input voltage of the power converter ( ) The matrices A, B and C are defined in (3), where L and C are inductance and capacitance of the rectification filter, R is load, and n is turn ratio of the transformer.
Further simplification can be made by letting 1 Equation (5) can also be obtained by neglecting the transition intervals and taking only the dynamics of active and passive regions into consideration. Choose the state 1 i v , the input as d, and the direct output as v o . We obtain 1 1 By input-output linearization, the output differential model subject to disturbance is where the plant disturbance η is nonzero and assumed to be less than a positive constant d, i.e., ( , ) . In addition, ( ) g x is assumed to have a positive lower bound, i.e., . Define the tracking error as   , the control input can be specified as where 0 (with a design parameter). Let Substituting the control law given by (8) yields Thus, the sliding surface   is asymptotically attractive and the system restricted to the sliding surface can be made asymptotically stable with respect to the origin by an appropriate choice of the parameter 1 k .
Define the parametric error of adaptation as The adaptation law is specified as where f  and g  are positive design parameters. The projection operator is defined by The control law is specified as with selection of s k described in the proof of the subsequent theorem.
Theorem 1: The control law (21) along with parametric adaptation law (19). If slide u satisfies (22) with 0   , we have the following: It follows that the system is stable and ( ) t When s k is specified such that we show that (22) is satisfied. Indeed, considering (26) and rearranging terms gives Completing the square yields Therefore, (22) holds. 

III. PERFORMANCE METRICS AND DESIGN PARAMETERS
An understanding of the features and parameters is crucial for choosing applicable DC-DC power converters. Typical "static" parameters are input voltage range, output voltage range, and maximum required output current. Typical "dynamic" parameters are efficiency, output voltage ripples, and load transient regulation. There are also various practical aspects being regularly taken into consideration: EMI, size, input voltage ripple, operating temperature, output ripple frequency, failure rate, etc. When reviewing the features and parameters of converters, it is important to understand the different trade-offs between performance metrics. This helps determine realistic expectations for the converter that best fits the application. In this section, performance metrics incorporated for PSPWM full bridge DC-DC power converter are introduced and quantitatively defined. The potential design parameters of the adaptive fuzzy with sliding mode controller and their effect on the four indicators are studied.

A. PERFORMANCE METRICS
Four metrics are incorporated in this paper for performance evaluation of the PSPWM full bridge DC-DC power converter: root mean square error (RMSE), voltage ripple, switching peak load current, and transient conversion efficiency. Optimize these four performance indicators at the same time is challenging due to some of them being conflicting indices (as will be shown in subsequent study). In the following, we explain the practical meaning of the four metrics and provide their mathematical definition. The first metric is RMSE defined by For static reference voltage, voltage ripple can also be evaluated by The third metric is switching peak load current ,max

B. DESIGN PARAMETERS
Controller parameters directly related to stability or steady state convergence (the first priority of control system design) are mostly determined when following the algorithmic design procedure. Other controller parameters, which have connection with alternative performance indicators, are customizable. Analytical relationship between those performance indices and the customizable parameters is usually very complex or not clear. Therefore, determination of those customizable parameters poses a challenge when performance requirements apart from steady state response are also critical. In this and next subsection, we will investigate influence of various controller parameters on the performance metrics proposed previously. For this study, the number of fuzzy rules (i.e., M) is set to six. The Gaussian membership function l i F  in (14) can be expressed as where l c and l w are often called the center and RMS width (i.e., standard deviation) of the membership function, which can be adopted as design parameters if their impact on aforementioned performance metrics can be justified. A total of twenty-four design parameters are available if the center and width of the membership functions for both ( )  Table 2. It is concluded that both sets of parameters impact the performance metrics. Specifically, for the same range of center distribution, the RMSE, the voltage ripple, and the transient efficiency degrade as the width of the membership function increase, whereas the peak load current is barely affected. For the same width, the RMSE and the transient efficiency degrade as the range of center distribution widens, whereas the voltage ripple decreases and the peak load current is barely affected. Similar trend can be observed when varying the parameters in ( )  Table 3 and Table 4. It is concluded that both sets of parameters also impact the performance metrics. As shown in Table 3, as k 1 increases with other parameters fixed, both the RMSE and the voltage ripple reduce and the transient efficiency improves, but the peak load current deteriorates. When k 2 increases with other parameters fixed, the RMSE degrades gradually and the peak load current decreases whereas the voltage ripple and the transient efficiency are barely affected. As shown in Table 4, both f  and g  affect the performance metrics analogously.
When both parameters increase, the RMSE and the voltage ripple reduce gradually, the transient efficiency improves, and the peak load current is hardly affected.

IV. MANY-OBJECTIVE GLOBAL OPTIMIZATION
The key elements in formulating an optimization problem are selection of a set of decision variables (design parameters) and objective functions (performance metrics). It is important to only include in the formulation the decision variables that certainly influence the objective functions. It is also useful to understand whether the adopted objective functions are conflicting or consonant. In the previous section, we identify and define a set of performance metrics: RMSE, voltage ripple, switching peak load current, and transient efficiency.
We also determine three groups of decision variables: centers , respectively. This section will proceed with formulating a many-objective optimization problem, which is stated formally as ,max minimize ( ) RMSE , , , , where ( )   are lower and upper bounds for the decision variables. The bounds on the decision variables define a decision or search space for the problem. Setting reasonable bounds can prevent the optimization algorithms from settling on impractical solutions (e.g., large values, zeros, negative values, and so on). Note that the minus sign on f E indicates that this objective function is to be maximized. Two multiobjective algorithms, i.e., particle swarm optimization and bat optimization, are utilized to search the solutions. Both are swarm-based algorithms, by which they deploy a set of decision variables (i.e., particles or bats) into the search space and modify them based on certain rules. During the search process, the best-so-far decision variables will be tracked and recorded for subsequent determination of the optimal solution. Moreover, originally both algorithms are only applicable to single objective optimization. To extend their usage to many-objective optimization, both algorithms require modification and encompass the concept of Pareto solutions.

A. PARETO FRONT
Multi-objective function assigns to each decision variable a multi-objective vector function value in the objective function space. Instead of decision variable space, for multiobjective problems, we are usually more interested in the objective space and there is no natural ordering in this space. A solution 1 p dominates 2 p if the following two conditions are satisfied ( n  is the number of objectives)            (38) Hence, a solution is Pareto optimal, or nondominated, if there exists no other candidate solution that decreases some objectives without simultaneously increasing at least one other objective. The set of Pareto optimizers is called Pareto front. Computationally, the Pareto front is updated per iteration once non-dominated solutions are obtained during the search. Specifically, the newly acquired non-dominated solutions at each iteration are stored in an external archive of limited size. A scheme proposed in [41] may be utilized to decide whether a new solution enters the archive or a solution in the archive needs to be removed (i.e., update the Pareto front). This study incorporated the method proposed by [37], [38]. An adaptive grid is utilized to uniformly spread the nondominated solution along the Pareto front. The adaptive grid divides the objective space into hypercubes. Each hypercube is a bin that contains certain number of non-dominated solutions. When the external archive is full and a new nondominated solution arrives, a solution in the most clustered hypercube is randomly selected and removed from the archive. To select a global best solution from the current Pareto front, each non-empty hypercube is assigned with a probability inversely proportional to the number of nondominated solutions it holds. Then, a hypercube is chosen by employing roulette-wheel selection. From this hypercube, the global best may be selected randomly. Note that this will guide the search towards the area with less density of nondominated solutions. As for the personal best, at first it is equal to the initial position of each particle or bat. At each iteration, the new position of a particle or bat and its current personal best are compared. If one dominates the other, either the current personal best is kept or it is replaced by the new position. If neither of them is dominated by the other, a random selection is made.

B. PARTICAL SWARM OPTIMIZATION
Particle swarm optimization (PSO) is a population-based stochastic optimization technique which shares many similarities with evolutionary algorithms. PSO is initialized with a population of random solutions, i.e., particles. Each particle is configured with velocity, position, cognitive and social traits, and personal experience history of fitness values. Each particle flies through the search space, i.e., updates its position, based on three information: velocity inertia, best personal fitness, best group fitness. To apply PSO to the formulated many-objective optimization problem, the selection and adaptive grid schemes introduced previously are adopted. At the beginning of the optimization, the particles are randomly initialized in a given space satisfying (37). The fitness of each particle is evaluated based on the performance metrics defined previously. An initial set of nondominated solutions can be determined using the scheme proposed in [41] and stored in an external archive. The particles update their positions and velocities by  (39) where  is Schur product, (2) Initialize the position and the velocity of each particle. (3) Initialize an external archive of certain size for storing Pareto solutions. This archive also stores the personal best of each particle. (4) Evaluate the fitness of each particle by (36).  (1) Update the velocity and the position of each particle according to (39). (2) If any decision variable within the position of a particle exceeds the bounds described by (37), it will take the value of the corresponding lower or upper bound. (3) Evaluate the fitness of each particle by (36). (4) Update the non-dominated solutions in the external archive and perform adaptive grid scheme. (5) Select the global best from the archive. (6) Update the personal best of each particle. end while

C. BAT OPTIMIZATION
Bat optimization (BO) is inspired by the echolocation ability of microbats during foraging. The echolocation behavior is similar to an active sonar system which sends out loud sound pulses and listens to echoes. Utilizing this ability, the bats can proactively locate their prey, distinguish the type of the prey, and avoid obstacles in a complete darkness environment. BO is also initialized with a population of random solutions, i.e., bats. Each bat is configured with velocity, position, frequency of sound, loudness, and pulse emission rate. The selection and adaptive grid schemes introduced previously are also adopted to apply BO to the formulated many-objective optimization problem. The bats are randomly initialized in a given space, i.e. (37). The fitness of each bat is evaluated according to the performance metrics. An initial set of nondominated solutions can be determined using the scheme proposed in [41] and stored in an external archive. The bats update their positions and velocities by ,min ,max ,min 1 best , where  and  are constants typically in (0,1), the initial emission rate 0 i  is any value in [0,1], and the initial loudness 0 i A can be set in [1,2].  is analogous to the cooling factor in simulated annealing, which influences the convergence rate of the algorithm. The algorithm (including pseudo codes) is summarized as follows: initialization (k=0) (1) Determine the numbers of bats and iterations.
(2) Initialize the position, the velocity, the frequency, the loudness, and the pulse emission rate of each bat. end if end if end for (3) If any decision variable within the position of a particle exceeds the bounds described by (37), it will take the value of the corresponding lower or upper bound. Also, the sign of the respective decision variable within the velocity is reversed (i.e., positive to negative and vice versa). (4) Evaluate the fitness of each particle by (36). (5) Update the non-dominated solutions in the external archive and perform adaptive grid scheme.
(6) Select the global best from the archive. end while

V. PARAMETRIC OPTIMIZATION OF THE CONTROLLER
As described in Section II, design and synthesis of adaptive fuzzy with sliding model controller is based on a reducedorder model with specified uncertainty bounds. A computing framework is proposed in this section for parametric optimization of the controller. In order to have sophisticated dynamics in the output response for realistic evaluation of the performance metrics defined previously, a sophisticated model is required for the simulation framework. Specifically, the controller is connected with the elaborate computational model [40] (instead of the reduced-order model) in subsequent numerical simulation for analysis of the overall control system. Besides the simulation results having better practicability, this also reduces hardware design effort and saves experiment cost.

A. SIMULATION SETUP
A computational platform based on MATLAB/Simulink environment is established for parametric optimization of the controller (see Figure 3). The mathematical model of the PSPWM full bridge DC-DC power converter along with the adaptive fuzzy with sliding mode controller is realized and implemented in Simulink. The specification of a laboratory power converter to be used for subsequent experiment is detailed in Table 5. The many-objective optimization algorithm (PSO or BO) is implemented in MATLAB. For each iteration, an updated set of design parameters is generated by the optimization algorithm (in MATLAB), and given to the controller of the adaptive fuzzy with sliding mode control system for numerical simulation (in Simulink). After each simulation is completed, performance metrics are evaluated and provided to the optimization algorithm for further actions, e.g., updating the design parameters and the Pareto front. For synthesis of the adaptive fuzzy with sliding mode controller, apart from design parameters, there are other parameters which are relevant to stability, e.g., Lfi  are set to 50,000 and 10, respectively. The numbers of iteration, particles/bats, and the size of external archive are set to 50, 10, and 10, respectively. Other parameters are set to typical values provided in the literature. Note that several sophisticated dynamics, which might affect performance metrics, are included in the numerical model of PSPWM full bridge DC-DC power converter: PWM driver, phase-shift switching logic, and ZVS delay. In order to capture the above dynamics, the sampling frequency needs to be set to a much higher value than, for example, the PWM frequency (50 kHz). Therefore, a sampling frequency of 10 MHz is used for simulation. On a laboratory computer with Intel Core i7 (9700) 3.4 GHz CPU and 16GB RAM running MATLAB/Simulink R2020b, it takes approximately 20 minutes to complete a 2.5 seconds simulation for the adaptive fuzzy with sliding mode control system. Execution time of the optimization algorithm, performance metric evaluation, and Pareto front update also adds to the simulation time. Therefore, a full cycle of optimization run takes around seven days to complete. In the Conclusion section, methods or techniques for speeding up the simulation will be discussed.

B. PARETO SOLUTIONS
The many-objective optimization problem formulated by (36) and (37) is numerically solved based on the computational framework proposed previously. Both PSO and BO algorithms are utilized to produce separate Pareto fronts for comparative study. Recall that at each iteration the optimization algorithm will generate a set of feasible design parameters. Then the selection mechanism described previously is applied to extract candidate non-dominated or Pareto solutions from this set of design parameters. Note that the adopted selection mechanism can only approximate the 'true' Pareto front. The accuracy depends on factors such as number of particles, number of iterations, initial positions of particles, etc. A set of candidate Pareto solutions during BO is listed in Table 6. As can be verified that those solutions all satisfy (38) for the definition of non-dominated solutions. The Pareto fronts are acquired after the specified number of iterations. Since there are four performance metrics, it is not possible to demonstrate the Pareto fronts in fourdimensional space. Alternatively, "projected" twodimensional metric-versus-metric presentation is adopted and three representative results are shown in Figure 4. Specifically, Figure 4(a) shows the "projected" twodimensional Pareto fronts for RSME versus voltage ripple, Figure 4(b) shows the Pareto fronts for transient efficiency versus peak load current, and Figure 4(c) shows the Pareto front for RSME versus peak load current. Recall that the size of the external archive for storing Pareto solutions is set to ten. As can be seen, BO performs better than PSO in locating the Pareto fronts for RSME versus voltage ripple and RSME versus peak load current. Nevertheless, PSO performs better   Figure 5. The design parameters corresponding to Figure 5 . The corresponding fuzzy membership functions are given in Figure 6. The Pareto solutions acquired correspond to a set of controllers for the PSPWM full bridge DC-DC power converter. One characteristic of any two controllers from this set is that a gain in a performance metric from one controller to the other happens only because of a sacrifice in at least on other performance metric. This trade-off property raises a question concerning how a practitioner makes a final choice among the non-dominated controllers. If the decision maker has additional information regarding the preference of each performance metric, he/she may create a hyperplane (i.e., a single-objective function) by forming a weighted sum of the performance metrics with the weight indicating the importance. Locating the approximate tangent point of this hyperplane with the Pareto front will provide the decision maker with an optimal solution. An example is illustrated in Figure 4(a). Suppose that only trade-off between RMSE and voltage ripple needs to be made. The red curve in the figure is the Pareto front obtained using the MOBO algorithm. The

VI. EXPERIMENTAL RESULTS
An experimental platform is utilized to further justify the proposed parametric optimization scheme for controller synthesis. This platform, as shown in Figure 7, mainly consists of a PSPWM full bridge DC-DC power converter (as described in [42]) and a dSPACE DS1104 controller board (with peripherals and software) for rapid control prototyping. The power converter allows voltage output ranging from 0 V to 50 V. The controller board supports models created within Simulink environment and provides efficient method to develop and test new control strategies quickly without manual programming. More explicitly, to adopt the adaptive fuzzy with sliding mode control system developed previously (within Simulink environment) and implement it on the experimental platform, certain alterations are essential: (1) The model for the PSPWM full bridge DC-DC power converter is removed from the control system and replaced by a physical converter (see Figure 7). Some I/O blocks are added to the model serving as software and hardware interface; (2) Peripheral circuits (e.g., voltage and current sensors and transducers) connecting the power convert to the controller board are made; (3) The model for the adaptive fuzzy with sliding model controller requires certain modification (e.g., replacing fuzzy membership function with its numerical approximation) to ensure compatibility with the software for the controller board.
The current and voltage responses for the converter with a typical set of unoptimized controller parameters (formerly used for Figure 5(a)) are shown in Figure 8(a). The current and voltage responses corresponding to the Pareto solutions, which produce minimum or maximum performance metrics (as formerly shown in Figure 5(b)(c)), are shown in Figure  8(b)(c). As expected or predicted by the simulation results, the responses with optimized controller parameters demonstrate significant improvement when compared to those of the unoptimized case. Specifically, the output voltage in Figure 8(b) is closer to the reference output, (c) The Pareto optimal controller with minimum ripple voltage and minimum peak load current.  resulting in smaller RMSE. The transient efficiency is also maximized in this case. The voltage ripple in Figure 8(c) is less than that for the unoptimized case and the peak load current is also minimized. Overall, the experimental results suggest that the proposed parametric optimization scheme achieves 29.62, 25.00, 30.16, and 1.69 percentage of improvement in RMSE, ripple voltage, peak load current, and transient efficiency, respectively. Note that the amplitude of the peak load current for the power converter of the experimental platform is less than that for the simulation model. This may due to the snubber circuit composed by the parasitic parameters of switches or other electric components in the hardware-implemented power converter.

VII. CONCLUSION AND FUTURE WORK
A parametric optimization framework amid the synthesis of a robust adaptive fuzzy controller for a class of switching power converters applicable to renewable energy systems is presented in this paper. Four performance metrics essential to the practical needs of renewable energy application are suggested, and the corresponding many-objective optimization problem is formulated. MOPSO and MOBO are employed to numerically solve the problem and obtain a set of Pareto optimal controllers. Both simulation and experiment validate that those optimal controllers significantly improve the performance metrics of the control system.
Although the proposed optimization framework is only demonstrated for one set of operating point, i.e., input and output. To obtain the Pareto optimal controllers for other operating points, simply repeat the procedure described. Therefore, the decision maker will have numerous sets of optimal controllers with each corresponding to an operating point. Note that individual performance metric can have different emphasis with respect to different operating point, i.e., various set of weights can be designated to performance metrics for different operating points. Afterwards, the weighted sum method, e.g., Figure 4(a), can be used to choose the most appropriate controller for each operating point. It is also feasible to apply the proposed method to the scenario where parametric optimization is performed over a set of operating points, which will become an expansion of this work. However, it would be suspected that controllers obtained from such greedy strategy can have superior performance.
Another issue is regarding the lengthy off-line simulation time required utilizing the proposed computational platform for parametric optimization. Immediate solutions might be to upgrade the computing hardware or consider parallel computing using multiple computers (and merge the results). The major bottleneck of time is mainly due to the elaborate model incorporating the dynamics of the phase-shift PWM. Under the circumstance that the impact of the PWM (c) The Pareto optimal controller with minimum ripple voltage and minimum peak load current. dynamics on the performance metrics is negligible, the model can be further simplified, which should significantly reduce the time of simulation. Moreover, parts of the performance metrics adopted for this work, e.g., RMSE and efficiency, may be revised so that they are evaluated at each iteration or every few iterations. The controller parameters may be updated at the same pace. Therefore, machine learning algorithm such as reinforcement learning may be considered to perform on-line parametric optimization, which will require only one simulation run. Note, however, that not all performance metrics can be assessed immediately amid numerical simulation. This will be another future expansion of this work.