Abstract
In this paper, a discrete state-feedback controller is applied to control the current in power LEDs by using a buck converter. Sufficient conditions for the existence of the controller are given in terms of linear matrix inequalities (LMIs) with D-stability theory. In order to consider the specific closed-loop dynamics of the closed-loop LED driver, the genetic algorithm (GA) technique is applied, as a complement of the D-stability theory. Therefore, the proposed GA technique is used to search offline the optimal closed-loop eigenvalues to have a desired response of the closed-loop system, subject to a proposed cost function equation. The main contribution of this work is the design and the experimental validation of both the state-feedback controller and the GA to achieve the desired closed-loop dynamics of the LED-driver system. Finally, simulations and experimental results are done in order to illustrate the effectiveness of the proposed methodology.
1. Introduction
Power LEDs are the most efficient light source in the market, and also it has a long useful life. Power LEDs are used in several lighting systems, such as residential, industrial, enterprise, and street lighting applications. The LED driver uses a dc-dc converter to maintain a constant LED current because it can be damaged if an overcurrent circulates over them. These days, power LED-driver systems work by using a power converter like dc-dc converter, which is used to obtain a constant LED current because power LEDs are very sensitive to variations of this variable. In an extreme case, these variations can destroy them. Therefore, a LED driver based on a dc-dc converter is required for the control of the LED current because the LEDs have a high sensitivity to voltage variations. This phenomenon causes the current to have a great increment in its value when the voltage has small variations [1–11].
A LED driver based on a buck converter with a constant on-time control loop was presented in [2], and a good performance in the current control of the LEDs is achieved. A digital primary-side controller for a flyback light-emitting diode driver with a conventional constant on-time control was presented in [9], and good line regulation with good LED current regulation is achieved.
One of the problems in control theory is the output feedback stabilization problem. Using this theory, it is necessary to find static output feedback, applied to a given system, so that the closed-loop system has some desirable characteristics [12]. Various approaches have been used to study the stability of the state-feedback control [13]. It has been shown that an extremely wide array of feedback controller designs may be reduced to the problem of finding a feasible point under a linear matrix inequality (LMI) constraint [14]. Recent works of the theory of LMIs have been used for discrete feedback controller design in [15–18]. Most of the above works present an iterative algorithm in which a set of equations or a set of LMI problems are repeated until certain convergence criteria are met.
D-stability theory can be applied in the LMI design to select a specific region of the pole-placement closed-loop system [14, 17, 19, 20]. Nevertheless, this approach can be difficult to design. An alternative to designing an optimal discrete state-feedback controller could be the combination of machine learning techniques and the LMI design [21]. More specifically, evolutionary algorithms are an important category of machine learning techniques that are based on the evolutionary ideas of natural selection and genetics [21]. There are approaches reported in the literature that combines the use of LMI design and a genetic algorithm (GA) to design a robust controller, e.g., robust controller design procedure applied to a grid-connected converter by combining LMIs and genetic algorithm is proposed in [22]. Such a strategy allows to overcome the control design, ensuring a good tradeoff between performance and robustness against uncertain grid parameters. Experimental results are presented to demonstrate the viability of the proposed approach. In [23], two robust decentralized control design methods for load frequency control are proposed by using LMIs and GA techniques. GA optimization is used to tune the control parameters of a PI controller. The proposed controllers are tested in simulation to demonstrate their robust performance. An LMI condition of nonlinear systems by GA-based adaptive fuzzy sliding mode controller is discussed in [24]. Numerical simulations are presented to demonstrate that the control methodology can rapidly and efficiently control a nonlinear system. To our knowledge, there are few studies on designing discrete state-feedback controllers with genetic algorithms to have the desired characteristic of a closed-loop LED driver, so the study of such problems remains important and challenging and is of both practical and theoretical importance. The buck converter is used as a LED driver because it has been used with satisfactory results in LED lighting applications [6]. This converter is a minimum phase system, and due to this property, it is easy to control it [2].
The simulation results are obtained by using the LED model presented in [10]. It consists of a constant voltage source in series with a resistor with a constant value. The performance of the power LEDs is well represented by this model, and good simulation results are presented in [10, 11].
This document is organized as follows: In Section 2, the power stage is presented. The system modeling is shown in Section 3 and the control stage is shown in Section 4. Section 5 presents the simulation results, and in Section 6 experimental results are shown. Finally, the conclusions are presented in Section 7.
2. Power Stage
Figure 1 shows the schematic diagram of the power stage that is based on a dc-dc converter named buck converter that is used as a LED driver where the objective is to maintain a constant current in the power LEDs. For this purpose, closed-loop control by sensing the LED current is commonly designed. Buck converter is considered a minimum phase system that allows easy control of the LED-driver system. The buck converter uses a MOSFET transistor M, a diode D, an inductor L, and a capacitor C.
3. System Modelling
The LED model presented in [25] is used to simulate the LED-driver system. It consists of a constant voltage source in series with a resistor with a constant value that is shown in Figure 2. This model is used because is simple, and due to this the model of the LED-driver system can be easily calculated.
Figure 3 shows the schematic diagram of the LED-driver system with the model proposed in [25]. The LED model presented in Figure 2 is used to represent the power LED performance. An average state-space model of the dc-dc converter is calculated because it facilitates the design of the control stage.
Thee LED panel LMT-P12Y-77-N LED from the SiLED manufacturer is used, and its model parameters are and .
4. Analysis and Design of the Control Stage
The LED-driver system is nonlinear system. The state-space averaging modeling method is used instead of the state-space switching model because it facilitates the design of the control stage. Based on the aftermentioned words, and by using the schematic diagram of the LED-driver system (Figure 3) the nonlinear state-space averaged system model can be written as follows:withwhere denotes the state vector, represented by the inductor current and the capacitor voltage, receptively, with . The output vector is given by . For tracking control design, it is considered that the output is only , but in the experimental circuit both states can be measured and they will be used in the stabilizing control stage, as illustrated in Figure 4. To control the LED driver, the dynamic system (1) is discretized by using the Euler method. Then the following system is obtained:withwhere denotes the sample time.
The objective is to design a control law with a state variable feedback controller such that steady-state response tends to the desired inductor current reference. To reach the desired time-varying values , an integrator comparator is considered. Then, the dynamics of the error are represented bywhere , and is an auxiliary variable for integral control design. Then, systems (3) and (5) are rewritten in the augmented system form , as follows:with
By assuming that the pair is controllable, the control law is obtained:where and are the state feedback gains matrices to be calculated and is the pseudoinverse of the matrix . The term is proposed to eliminate the effect of the matrix in the controller design (see Figure 4). Now, the problem is reduced to determine optimal values of the controller gains. By synthesizing (8) in system (6), assuming that , gives the closed-loop system
The following result gives sufficient LMIs conditions to guarantee the global asymptotic convergence of the tracking controller:
Theorem 1. Given system (6), and the comparator integrator control law (9), the closed-loop error system (5) is asymptotically stable if there exists a symmetric matrix and a matrix such thatThe controller gain matrices are computed by
Proof. Let us choose the following candidate quadratic Lyapunov function aswith a positive define matrix and consider that (11) is monotonically decreasing, i.e.,for all Since condition (12) is ensured ifholds, is stabilizing gain if there exists a matrix satisfying (13). Now, by multiplying on the left- and right-hand sides of the left-hand term of (13), the stability conditions (13) can be rewritten asApplying the Schur complement to (14), the matrix inequality is equivalent toNote that due to the quantity , the inequality expression (15) is not an LMI. To express it with LMI conditions, the following change of variables can be performed: , and with the property of congruence with full rank matrix (15) is equivalent to the LMI in Theorem 1. Finally, this completes the proof.
Lemma 1. To avoid slow dynamics, pole assignment can be considered aswhere is the disk stability with all eigenvalues of located inside the unit circle, as defined in [14, 19]. The area associated with the decay rate consists of a circle centered at the origin, with its radius being the parameter that is defined by the desired decay rate.
Then, by considering the previous Lemma, the following Corollary of Theorem 1 is obtained.
Corollary 1. Given system (6), and the comparator integrator control law (9), the close-loop error system (5) is asymptotically stable if there exists a positive scalar , a symmetric matrix and a matrix such thatThe controller gain matrices are computed by
The successfully obtained control vector guarantees the closed-loop system stability. However, the positive scalar in (17) is selected without considering any dynamics of the controlled system. Therefore, the proposed genetic algorithm (GA) technique is used to search offline the optimal scalar value of , to have the desired response of the closed-loop system subjected to a proposed cost function equation.
Evolutionary algorithms are an important category of machine learning techniques that are based on the evolutionary ideas of natural selection and genetics. A population of individuals, called a generation, compete at a given task with a defined cost function. GA is based on the propagation of generations of individuals by selection through fitness. Each individual represents a possible solution within a search space. After the initial generation is populated with individuals, each is evaluated and assigned a fitness based on their performance on the cost function metric [21]. In this paper, the Global Optimization Toolbox [26] was used to implement the GA optimization.
The performance of the given control law is judged based on the value of the following cost function:where the matrix Q weighs the cost of deviations of the error and it is positive semidefinite. The goal is to develop a control strategy , with an optimal selection of in (17), to minimize the cost function ensuring a good performance of the closed-loop system (9). The offline procedure for the selection of optimal using the genetic algorithm technique is shown in Figure 5.
5. Simulation and Experimental Results
5.1. Simulation Results
In this section, simulation implementations of the proposed algorithm are presented. Simulation of the control algorithm implemented in the STM32F4 discovery development board applied to the LED-driver system was carried out in the program MATLAB/Simulink. The computer used to find the optimal controller gain has the following characteristics: MSI Prestige 15 A10SC, Intel Core i7 10-th generation @ 1.10 GHz, and 16 GB of RAM. Table 1 shows the constant parameters of the LED-driver system.
The open-loop operation points are and with a constant input , which represents a pulse width modulation (PWM) with a duty cycle of 50%. In the STM32F4 discovery development board, the PWM generation is configured with a counter period of 800. In other words, 50% of the duty cycle is equivalent to selecting a pulse bits value of 400. By considering previous remarks, the matrix can be rewritten in (3) as
The sample time is calculated by considering (1) and the step response (Figure 6). The continuous transfer function of the open-loop LED-driver system is
Note that the overall system is a second-order system; however, the inductor current and the capacitor voltage present a first-order open-loop dynamics that we have used to find the sample time for practical implementation.
From (19), the constant time of the system is with a system bandwidth of According to the sampling theorem [27], a suitable sampling frequency is chosen as
The corresponding sampling time is selected as , with .
Figure 7 shows the GA applied to the closed-loop system (9) to select the optimal value of in (14) by minimizing the cost function (18). The defined parameters for the GA are a maximum generation of and each generation with a population size of individuals.
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(b)
The controller gains for (8) are obtained by solving the LMI of Corollary 1, using Yalmip Toolbox [28]. By using the procedure for selection of optimal with a genetic algorithm, presented in Figures 5 and 6, the following is obtained
To test the effectiveness of the proposed method, four scenarios are considered, depicted in Table 2. In each scenario, initial conditions of the LED-driver system (3) are established in .
In Figures 8 and 9, the simulation of the LED-driver system performance for each scenario is presented. The top and middle figures show both the controlled inductor current and the capacitor voltage, respectively, while the bottom figures show the control input (duty cycle) applied to the system.
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(b)
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Scenarios 1 and 2 are displayed in Figure 8 and scenarios 3 and 4 are presented in Figure 9. Also, these figures show the good performance of the tracking controller with a small error between the reference and the inductor current, also with a small overshoot, and a small settling time. Additionally, Figures 8 and 9 show how the proposed control system can track the inductor current, even with a pulse reference.
5.2. Experimental Results
The feedback controller was implemented in the STM32F4 discovery development board with an STM32F429 as the processor. ARM controller was chosen because it is lightweight, small in dimension, easy to be operated, and low cost. Also, this development board can handle signal processing, is easy to be programmed, and has a friendly user interface. The STM32F4 discovery development board has a 6.5 × 11.8 cm dimension, with 32 bit ARM Cortes-M4 RISC CPU and it has a high-speed embedded memory with 2 MB flash memory and 256 KB RAM.
The schematic diagram implemented physically is presented in Figure 10.
Experimental results are shown in Figures 11–14. Figures 11 and 12 show experimental results under stationary state conditions. Figure 11 shows the experimental results when the LED current is 310 mA, and Figure 12 shows experimental results when the current is 50 mA.
Figures 13 and 14 show experimental results under a current pulse test (dynamical conditions), and the step current test is made with a pulse control signal. Figure 13 shows the current pulse test when the LED current changes from 150 mA to 310 mA, and Figure 14 shows a current pulse test when the LED current changes from 50 mA to 310 mA. A good dynamic response is observed in these tests with good reference pursuit obtained, and also the current error is near zero because an integrator is used for this purpose.
Figure 15 shows a picture of the prototype and the equipment used in the experiments.
6. Conclusions
A discrete state-feedback controller design by combining D-stability theory and GA technique was proposed and validated experimentally. The dynamic model of the LED driver is presented. Furthermore, tracking controllers were designed using a feedback integrator. Sufficient conditions for the existence of the controller are given in terms of linear matrix inequalities (LMIs) with D-stability theory. The proposed GA technique is used to search offline the optimal closed-loop eigenvalues to have a desired response of the closed-loop LED driver, subjected to a proposed cost function equation. Finally, simulations and experimental validation was carried out. The results of the proposed methodology demonstrate that the discrete controller designed with GA can ensure a good performance of the tracking controller with a small error between the reference and the inductor current, also with a small overshoot, and a small settling time.
Data Availability
The simulations and experimental data used to support the findings of this study are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.