A buck converter is a step-down switching regulator. Buck convertors are being widely used in industrial applications that rely on regulated output voltage under fluctuating input voltage. A buck convertor works in the following modes: (a) current-controlled or (b) voltage-controlled mode. But these convertors manifest several nonlinearites because of the switching operation. Hence, in order to generate a quality output of the convertor, the design of a controller becomes crucial. In this paper, the synthesis of a QFT-based robust controller and prefilter has been carried out for an uncertain buck converter with varying input voltage and varying load. The controller synthesis problem has been posed as an optimization problem, and metaheuristic algorithms have been used for obtaining the optimal gains for the QFT controller and prefilter. By doing this, the QFT synthesis can be carried out in a single step instead of following the sequential classical QFT process on Nichols charts and the need for the generation of templates and bounds has be eliminated. The designed 2-degree-of-freedom QFT control system offers a robust behavior and efficiently handles the parametric uncertainties. The robustness of the designed controller has been confirmed through simulation results for large input voltage and load fluctuations.
Buck converters are center to various diverse applications that require a tunable/fixed DC supply form a fixed/tunable DC supply such as aerospace, instrumentation, medical appliances, and computers. [
In the literature, several classical control methods have been implemented for the linearized buck convertor models but under the influence of uncertainties the performance of the convertors degrades [
In 1960s, Issac Horowitz introduced Quantitative Feedback Theory (QFT). QFT has 2-degree-of-freedom (2-DoF) controller architecture, viz., (a) controller
This paper introduces an automated single-step QFT controller synthesis technique for buck convertors using metaheuristic algorithms. The desired QFT bounds and performance objectives for the voltage mode control in the buck convertor have been expressed in terms of design objectives and constraints. The QFT controller system design problem has been expressed as an optimization problem. This eliminates the generation of the templates and bounds which else are required for the manual loop shaping. The designed control system offers a robust response over a range of parametric uncertainty both in frequency and time domain and also offers performance robustness for large input voltage variations. The work has also been compared with classical controller synthesis methods like Ziegler–Nichols, internal mode control (IMC), and the QFT controller proposed by Ibarra et al. [
The paper has been split into subsequent sections: in Section
Often, modeled plant dynamics have a lot of assumptions, and the operation of the plant and due to the aging of the instruments overtime lead to the deviation of the plant’s nominal dynamics that were used while controller design. This makes it difficult to assure quality control over time. To address this issue, several control theories like
In 1960s, Issac Horowitz introduced a frequency-domain controller design technique of Quantitative Feedback Theory (QFT) [
QFT has found application is several diverse engineering applications [
Recently, many researchers have emphasised on the automatic synthesis of QFT controllers. Gera and Horowitz [
Metaheuristic algorithms are now been widely used for the design of control systems. QFT control synthesis cannot be accomplished by using conventional gradient-based optimization algorithms. Several evolutionary algorithms have been used for the synthesis of the QFT controllers. Gracia-Sanz et al. [
Still the applications of such algorithms in electrical engineering particularly in power electronics are very limited. Olalla et al. [
For the buck convertor, QFT has been employed for tackling the parametric uncertainty. In [
The dynamics of the DC-DC convertor has been by small-signal state-space averaging to obtain a set of time-invariant equations [
Conventional buck convertor.
A buck convertor primarily operates in two configurations: (a) continuous current mode (CCM) and (b) discontinuous current mode (DCM). In this work, the continuous current mode (CCM) mode has been considered for the controller synthesis. For a duty cycle, equation (
Values of the elements used in the DC-DC buck convertor.
Parameter | Symbol | Value |
---|---|---|
Input voltage |
|
24 V |
Inductor |
|
300 |
Capacitor |
|
220 |
Load |
|
12 Ω |
PWM period |
|
10 |
PWM duty cycle |
|
1 |
tON switch resistance |
|
0.01 Ω |
Inductor resistance |
|
16.3 mΩ |
Capacitor resistance |
|
0.305 Ω |
Quantitative Feedback Theory (QFT) is a frequency-domain controller synthesis methodology introduced by Issac Horowitz in 1960s. QFT is based on Bode’s famous gain-phase integrals and has a 2-DoF controller configuration as shown in Figure
2-Degree-of-freedom QFT control configuration.
In 2010, Yang proposed a metaheuristic algorithm of the bat algorithm (BA), which was established upon the echolocation behavior of the bats [
When prey is identified by a bat, the loudness
The values of loudness and pulse emission rates are modified only when improved new solutions are obtained.
Flower pollination algorithm is inspired by the occurrence of pollination that occurs in flowering plants [
Following four steps in pollination characterize the flower pollination algorithm [ For the pollination to occur from one flower to other, the pollinators perform Lévy flights and are regarded as global pollination Self-pollination is regarded as local pollination The more fertile the flower is, the greater the chances of the reproduction probability are The rate of global and local pollination is controlled by a switching probability,
In global pollination, pollens are transmitted via pollinators, and the fertility of the flower secures pollination and the selection of the most fertile flower. Mathematically, it is represented as
Mathematically, the flower constancy in self-pollination and is given as follows:
Predominantly, flower pollination happens both at local and global extent. In case of nearby flowers, local pollination is dominant. So, switching probability
Artificial bee colony (ABC) algorithm has been introduced by Karaboga and Basturk in 2005 and is based on the intelligent behavior of honeybees [
In the initialization phase, each bee in the population is assigned with a random location
In this stage, the employer bee phase will perform search for the food
In this phase, the onlooker bees dance in the waiting area in the hive to share information regarding the employer bees. Based upon the probability of food and distance from the hive, the onlooker bees make their decision and are given mathematically as follows:
Based upon the information shared, the onlooker bee will search the neighborhoods and calculate its fitness. Comparing the fitness of the current position with the previous one, the onlooker bees choose the new position.
After a certain iterations/searches, if an employed bee does not change its position, it becomes a scout. Scout bees are limited to one in a current cycle and performs search for the new food sources. When a new food location is found, it stores that in its memory till maximum number of cycles has been reached.
In 2008, Simon introduced a biogeography-based optimization algorithm [
The probability
Two important operators: (a) migration and (b) mutation, govern the BBO algorithm. A habitat/island with higher HSI is regarded as more suitable for living, and the lower HSI means that it is not suitable. Migration is an adaptive activity. Probability
In 2000, Geem and Loganathan introduced a population-based algorithm which is based on the principles of the extemporization process in jazz instruments [ Initialization of the randomly generated harmony search memory ( In this step, a now improved result In this step, the HM is updated. The fitness of the new solutions is evaluated, it returns a better value than the worst in the HM, and the worst one is replaced by the new solution. If not, the new solution is discarded. Repeat steps 2 and 3 till the stopping criterion is reached.
The harmony search algorithm has many operators like those in evolutionary algorithms, but harmony search differs from all as it offers single search memory for the solution to evolve. This also boosts the convergence speed of the algorithm.
Differential evolution (DE) [
A mutation vector is generated for each target vector
The trial vector
In selection, any individual from the population can form the parent despite of its fitness. After mutation and crossover, the competency of the child is assessed and equated with the competency of the parent and the individual with a better competency value is chosen.
Imperialist colony algorithm is inspired by the imperialist competition [
Both the power of the imperialist and its colonies together form the strength of the nation. The empire that fails to adhere with the competition becomes extinct. The competition mainly strengthens the empire and decreases the power of the weak nations and makes them extinct. The competition amongst empires is a way to converge them towards a single powerful empire in the world and with all the other countries as its colonies.
In 2006, Mehrabian introduced the invasive weed optimization (IWO) algorithm [
In this phase, a population of weeds is randomly generated.
In this phase, only a few plants in the population produce seeds and this depends on the fitness of the plant. The plant, which is least fit, will produce lesser seeds, while the fittest one will produce the most number of seeds, and this relation is linear.
In this phase, a random dispersion of the seeds is carried out such that the seeds remain nearer the parent. As the generations pass by, the standard deviation
If a weed plant fails to produce seeds, it will become extinct; otherwise, they would take over the world. So, competition limits the number of plants in the colony. As the generations pass, it is desired that the fitter plant reproduce more than the unfit ones. When a maximum number of weeds
In 2011, motivated by the process of teaching and learning, Rao et al. introduced teaching-learning-based optimization (TLBO) [
In the 1st phase of TLBO, the students learn form the teacher. The teacher is regarded as the elite being and shares his expertise with the students to increase their knowledge (the mean result). Initially, a random population is generated, and the individual with the minimum fitness value is chosen as a teacher (for minimization problems), and this information is shared with the students to increase their mean scores from
Suppose
Teaching factor is limited to 1 or 2 only and is chosen by
The new solution
In the learning phase, the pupils learn from mutual interaction. The interactions are random and happen if and only if the grade of one student is larger than other. For two learners
The new solution is accepted only when it minimizes or maximizes the objective function. As the teaching-learning process progresses, the level of knowledge of learners increases towards to that of the teacher and the algorithm converges towards a solution.
Ant colony optimization is a probabilistic metaheuristic algorithm inspired by the behavior of ants for finding the optimal path from their colony to the food source [
The global ant system updates the pheromone trail, and all the ants in the colony have to share the information of their journeys and the deposition of the pheromone. Mathematically, it is given as
QFT controller is primarily implemented to mitigate the consequences of parametric variations of the uncertain dynamics of the plant. The synthesis of the controller
For the closed loop system to assure robust stability, the minimization of the maximum magnitude of the closed-loop frequency response of the closed-loop system at each design frequency is desired. Mathematically, it is given as
Equation (
Tracking ratios guide the shaping of the open-loop transmission, such that a set of time- and frequency-domain specifications is satisfied. The upper and lower tracking ratios are declared at the starting of the design process. Mathematically, it is given as in the following equation:
Upper
Minimization of
The designed system must be immune to external disturbances. So, minimization of the sensitivity ensures that and is given mathematically as
The QFT controller and prefilter must satisfy the design specifications of robust stability, tracking performance, and sensitivity. The QFT controller synthesis problem has been expressed as an optimization problem which offers a templates-and-bounds-free approach for designing optimal QFT controllers within very less time and also naïve loop-shaping experience. In this paper, a standard PID controller and a fixed structure prefilter have been chosen and are given by equations (
Algorithms mentioned in Section
In this paper, the voltage mode-controlled DC-DC buck convertor is considered. The parameters of the physical components in Table
The optimal QFT controller and prefilter obtained from the automated synthesis are given by
To compare the designed QFT-based controller and prefilter, the results have been compared with several classical controller synthesis methodologies.
Figures
Compared step response of the closed-loop system (the nominal case).
Compared frequency response of the closed-loop system (the nominal case).
Time-domain performance indices.
Performance indices | Rise time | Settling time | Overshoot percentage (%) |
---|---|---|---|
BA | 5.983 × 10−04 | 0.0011 | 0 |
FPA | 5.989 × 10−04 | 0.0011 | 0 |
ABC | 5.231 × 10−04 | 9.315 × 10−04 | 0 |
BBO | 8.994 × 10−04 | 0.0016 | 0 |
HS | 5.795 × 10−04 | 0.0010 | 0 |
DE | 7.531 × 10−04 | 0.0013 | 0 |
ICA | 8.593 × 10−04 | 0.0015 | 0 |
IWO | 5.223 × 10−04 | 0.9045 | 0 |
TLBO | 6.048 × 10−04 | 0.0011 | 0 |
ACO | 0.0017 | 0.0030 | 0 |
ZN | 1.83 × 10−04 | 1.021 | 51.3 |
Ibarra | 6.368 × 10−04 | 0.0011 | 0 |
IMC | 5.72 × 10−04 | 0.001 | 0 |
So, to verify the robustness of the controller to parametric uncertainties, an uncertain buck convertor is considered as
The compared closed-loop step and frequency response of the parametrically uncertain system are shown in Figures
Compared time-domain worst case response of the closed-loop system: (a) BA; (b) FPA; (c) ABC; (d) BBO; (e) HS; (f) DE; (g) ICA; (h) IWO; (i) TBLO; (j) ACO; (k) Ibarra et al; (l) Zeigler–Nichols; (m) IMC.
Compared frequency-domain worst case response of the closed-loop system: (a) BA; (b) FPA; (c) ABC; (d) BBO; (e) HS; (f) DE; (g) ICA; (h) IWO; (i) TBLO; (j) ACO; (k) Ibarra et al.; (l) Zeigler Nichols; (m) IMC.
A buck convertor with varying input voltage has been designed in SIMULINK to test the efficacy of the designed QFT controllers and prefilters. For the ideal response, the buck convertor must maintain a fixed output voltage despite of the fluctuation in the input voltage. In this case, a fixed load of 50 Ω has been considered, while the input voltage has been varied from 20–28 V for a fixed output of 12 V. The plot for load voltage for variable input voltages has been shown in Figure
Plot for (a) output load voltage and the variable input voltage and (b) output current and voltage for varying input voltage case using flower pollination algorithm-tuned QFT controller.
Convertor performance parameters (variable input voltage).
Parameters | Voltage ripple (Δ |
Current ripple (Δ |
---|---|---|
BA | 1.167 | 1.042 |
FPA | 0.833 | 0.833 |
ABC | 0.833 | 1.041 |
BBO | 1.083 | 1.375 |
HS | 1.167 | 1.25 |
DE | 0.833 | 1.041 |
ICA | 1.041 | 1.041 |
IWO | 1.083 | 1.375 |
TLBO | 1.041 | 1.041 |
ACO | 1.041 | 1.041 |
ZN | 0.477 | 0.333 |
IMC | 0.375 | 0.479 |
Ibarra | 0.67 | 0.67 |
Plot for output current and voltage ripple for varying input voltage case using flower pollination algorithm-tuned QFT controller.
The plot for load voltage for variable input voltages and the plot for load voltage and load current are shown in Figures
Output voltage across load under variable input voltage for ZN-PID controller.
Output current and voltage for varying input voltage for ZN-PID controller.
Plot for (a) output load voltage and the variable input voltage and (b) output current and voltage for varying input voltage case using the IMC controller.
Plot for (a) output voltage across load under variable input voltage and (b) output current and voltage for varying input voltage using the QFT controlled proposed by Ibarra.
To check the efficacy of the designed robust QFT-based control scheme, the buck convertor with varying load has been considered. Here, the input voltage has been fixed at 48 V, for obtaining a fixed output voltage of 24 V. The variation in the resistive output load has been considered and has been varied form 2 Ω to 57 Ω. Figure
Plot for load current and voltage for varying load using the flower pollination algorithm-tuned QFT controller.
Plot for output current and voltage ripple for varying load case using flower pollination algorithm-tuned QFT controller.
Convertor performance parameters (variable load).
Parameters | Voltage ripple (Δ |
Current ripple (Δ |
---|---|---|
BA | 1.458 | 1.271 |
FPA | 1.0417 | 1.0294 |
BBO | 1.146 | 1.102 |
HS | 1.146 | 1.101 |
DE | 1.25 | 1.203 |
ICA | 1.146 | 1.101 |
IWO | 1.25 | 1.203 |
TLBO | 1.25 | 1.186 |
ACO | 1.146 | 1.102 |
Ibarra | 1.36 | 1.416 |
IMC | 0.724 | 0.692 |
ZN | 0.267 | 0.294 |
Figures
Plot for load current and voltage for varying load using Ziegler–Nichols-tuned PID controller.
Plot for load current and voltage for varying load using the IMC-based controller.
Figure
Plot load current and load voltage for varying load using the QFT controller proposed by L Ibarra.
In DC-DC convertors, nonlinear behavior due to switching operations and parametric uncertainties due to continuous operations make it hard to yield quality output overtime. In this paper, the automatic synthesis of the QFT controller and prefilter has been carried out using metaheuristic algorithms. The design process has been posed as an optimization problem, which eliminates the need of generation of templates and bounds and eases the design process. The flower pollination algorithm-based designed controller satisfies the design requirement both in the time and frequency domains and offers better performance than other algorithms. At the end, the designed controller has been implemented for a Simulink model of the DC-DC converter for two different cases of varying input voltage. The designed controller significantly reduces the voltage and current ripples and thus offering a quality voltage and current characteristics.
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.