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This work analyzes the response of the electricity market to varied renewable and nonrenewable installed capacity scenarios while taking into account the variability of renewables due to seasonality and El Niño-Southern Oscillation (ENSO) episodes. A hybrid system dynamics/dynamic systems (SD/DS) model was developed by first deriving an SD hypothesis and stock-flow structure from the Colombian electricity supply and demand dynamics. The model’s dynamic behavior was then transformed into a Simulink model and analyzed using the DS tools of bifurcation and control theory to provide deeper insights into the system, both from a Colombian perspective and from the perspective of other market scenarios. Applying the developed hybrid model to the Colombian electricity market provided a detailed description of its dynamics under a broad range of permanent (fossil fuel) and variable (renewable) installed capacity scenarios, including a number of counterintuitive insights. Greater shares of permanent capacity were found to guarantee the security of supply and system robustness in the short-term (2021–2029), whereas greater shares of variable capacity make the system more vulnerable to increased prices and blackouts, especially in the long-term (2040–2050). These critical situations can be avoided only if additional capacity from either conventional or non-conventional generation is quickly installed. Overall, the methodology proposed for building the hybrid SD/DS model was found to provide deeper insights and a broader spectrum of analysis than traditional SD model analysis, and thus can be exploited by policy makers to suggest improvements in their respective market structures.
José Morcillo; Fabiola Angulo; Carlos Franco. Simulation and Analysis of Renewable and Nonrenewable Capacity Scenarios under Hybrid Modeling: A Case Study. Mathematics 2021, 9, 1560 .
AMA StyleJosé Morcillo, Fabiola Angulo, Carlos Franco. Simulation and Analysis of Renewable and Nonrenewable Capacity Scenarios under Hybrid Modeling: A Case Study. Mathematics. 2021; 9 (13):1560.
Chicago/Turabian StyleJosé Morcillo; Fabiola Angulo; Carlos Franco. 2021. "Simulation and Analysis of Renewable and Nonrenewable Capacity Scenarios under Hybrid Modeling: A Case Study." Mathematics 9, no. 13: 1560.
The design of robust and reliable power converters is fundamental in the incorporation of novel power systems. In this paper, we perform a detailed theoretical analysis of a synchronous ZETA converter controlled via peak-current with ramp compensation. The controller is designed to guarantee a stable Period 1 orbit with low steady state error at different values of input and reference voltages. The stability of the desired Period 1 orbit of the converter is studied in terms of the Floquet multipliers of the solution. We show that the control strategy is stable over a wide range of parameters, and it only loses stability: (i) when extreme values of the duty cycle are required; and (ii) when input and reference voltages are comparable but small. We also show by means of bifurcation diagrams and Lyapunov exponents that the Period 1 orbit loses stability through a period doubling mechanism and transits to chaos when the duty cycle saturates. We finally present numerical experiments to show that the ramp compensation control is robust to a large set of perturbations.
David Angulo-García; Fabiola Angulo; Juan-Guillermo Muñoz. DC-DC Zeta Power Converter: Ramp Compensation Control Design and Stability Analysis. Applied Sciences 2021, 11, 5946 .
AMA StyleDavid Angulo-García, Fabiola Angulo, Juan-Guillermo Muñoz. DC-DC Zeta Power Converter: Ramp Compensation Control Design and Stability Analysis. Applied Sciences. 2021; 11 (13):5946.
Chicago/Turabian StyleDavid Angulo-García; Fabiola Angulo; Juan-Guillermo Muñoz. 2021. "DC-DC Zeta Power Converter: Ramp Compensation Control Design and Stability Analysis." Applied Sciences 11, no. 13: 5946.
The boost-flyback converter is a DC-DC step-up power converter with a wide range of technological applications. In this paper, we analyze the boost-flyback dynamics when controlled via a modified Zero-Average-Dynamics control technique, hereby named Zero-Average-Surface (ZAS). While using the ZAS strategy, it is possible to calculate the duty cycle at each PWM cycle that guarantees a desired stable period-1 solution, by forcing the system to evolve in such way that a function that is constructed with strategical combination of the states over the PWM period has a zero average. We show, by means of bifurcation diagrams, that the period-1 orbit coexists with a stable period-2 orbit with a saturated duty cycle. While using linear stability analysis, we demonstrate that the period-1 orbit is stable over a wide range of parameters and it loses stability at high gains and low loads via a period doubling bifurcation. Finally, we show that, under the right choice of parameters, the period-1 orbit controller with ZAS strategy satisfactorily rejects a wide range of disturbances.
Juan-Guillermo Muñoz; Fabiola Angulo; David Angulo-Garcia. Zero Average Surface Controlled Boost-Flyback Converter. Energies 2020, 14, 57 .
AMA StyleJuan-Guillermo Muñoz, Fabiola Angulo, David Angulo-Garcia. Zero Average Surface Controlled Boost-Flyback Converter. Energies. 2020; 14 (1):57.
Chicago/Turabian StyleJuan-Guillermo Muñoz; Fabiola Angulo; David Angulo-Garcia. 2020. "Zero Average Surface Controlled Boost-Flyback Converter." Energies 14, no. 1: 57.
In this paper, we study the time optimal control problem in a DC-DC buck converter in the underdamped oscillatory regime. In particular, we derive analytic expressions for the admissible regions in the state space, satisfying the condition that every point within the region is reachable in optimal time with a single switching action. We then make use of the general result to establish the minimum and maximum variation allowed to the load in two predefined design set-ups that fulfills the time optimal single switching criteria. Finally, we make use of numerical simulations to show the performance of the proposed control under changes in the reference voltage and load resistance.
Ilya Dikariev; Fabiola Angulo; David Angulo-Garcia. Single-Switching Reachable Operation Points in a DC-DC Buck Converter: An Approximation from Time Optimal Control. Micromachines 2020, 11, 834 .
AMA StyleIlya Dikariev, Fabiola Angulo, David Angulo-Garcia. Single-Switching Reachable Operation Points in a DC-DC Buck Converter: An Approximation from Time Optimal Control. Micromachines. 2020; 11 (9):834.
Chicago/Turabian StyleIlya Dikariev; Fabiola Angulo; David Angulo-Garcia. 2020. "Single-Switching Reachable Operation Points in a DC-DC Buck Converter: An Approximation from Time Optimal Control." Micromachines 11, no. 9: 834.
In this paper, we present a method to control a boost flyback converter using a hysteresis band, which is designed using information of the magnetization current and a proportional integral control action of the error. The stability of the periodic orbit is proven via the monodromy matrix, and the robustness, after the first tests of the disturbance rejection, is proven using bifurcation diagrams. When the period-1 orbit losses the stability, it disappears and gives rise to other period-1 orbit with different topological sequence. In all cases that we tested, the system always presented a good performance in a period-1 orbit with very low error.
Juan-Guillermo Muñoz; Fabiola Angulo; David Angulo-Garcia. Designing a hysteresis band in a boost flyback converter. Mechanical Systems and Signal Processing 2020, 147, 107080 .
AMA StyleJuan-Guillermo Muñoz, Fabiola Angulo, David Angulo-Garcia. Designing a hysteresis band in a boost flyback converter. Mechanical Systems and Signal Processing. 2020; 147 ():107080.
Chicago/Turabian StyleJuan-Guillermo Muñoz; Fabiola Angulo; David Angulo-Garcia. 2020. "Designing a hysteresis band in a boost flyback converter." Mechanical Systems and Signal Processing 147, no. : 107080.
In this paper, the variations in hydropower generation are addressed considering the seasonality and ENSO (El Niño-Southern Oscillation) episodes. The dynamic hypothesis and the stock-flow structure of the Colombian electricity market were analyzed. Moreover, its dynamic behavior was analyzed by using Dynamic Systems tools aimed at providing deep insight into the system. The MATLAB/Simulink model was used to evaluate the Colombian electricity market. Since we combine System Dynamics and Dynamic Systems, this methodology provides a novel insight and a deeper analysis compared with System Dynamics models and can be easily implemented by policymakers to suggest improvements in regulation or market structures. We also provide a detailed description of the Colombian electricity market dynamics under a broad range of demand growth rate scenarios inspired by the bifurcation and control theory of Dynamic Systems.
José D. Morcillo; Fabiola Angulo; Carlos J. Franco. Analyzing the Hydroelectricity Variability on Power Markets from a System Dynamics and Dynamic Systems Perspective: Seasonality and ENSO Phenomenon. Energies 2020, 13, 2381 .
AMA StyleJosé D. Morcillo, Fabiola Angulo, Carlos J. Franco. Analyzing the Hydroelectricity Variability on Power Markets from a System Dynamics and Dynamic Systems Perspective: Seasonality and ENSO Phenomenon. Energies. 2020; 13 (9):2381.
Chicago/Turabian StyleJosé D. Morcillo; Fabiola Angulo; Carlos J. Franco. 2020. "Analyzing the Hydroelectricity Variability on Power Markets from a System Dynamics and Dynamic Systems Perspective: Seasonality and ENSO Phenomenon." Energies 13, no. 9: 2381.
We analyze a generalized form of the Fujikawas growth model which involves an adaptation function that enhances the representation of the lag phase. This model is autonomous, and combines a power law term, a saturation term and an adaptation function that suppresses the growth rate during initial period corresponding to the lag phase. The properties of the adaptation function are determined, and the proposed model is examined separately for the regular measure and the logarithmic measure, including: Convergence and boundedness properties; population at the inflection point; conditions for the existence of the inflection point and lag phase; effect of model parameters on the existence of the inflection point and lag phase; population size of the inflection point under limiting values of the model parameters; and parameter values that lead to inflection point located at the mean value of the curve. Different combinations of model parameters lead to different possibilities for the existence of the inflection point and the lag phase. It was noticed that the power law term has a strong effect on the representation of the exponential growth phase, whereas the adaptation function has a strong effect on the representation of the lag phase. The lag phase duration depends on the exponent parameter of the adaptation function, and its dependence with respect to the power law parameter is low. Also, an approach is proposed for the analytical determination of the lag time, based on the application of the classical approach to a simplified model. Ascertained lag time values were obtained, what confirms the assumptions. At last, the model is applied to experimental data.
Alejandro Rincón; Fabiola Angulo; Fredy E. Hoyos. Analysis of a generalized Fujikawa’s growth model. Mathematical Biosciences and Engineering 2020, 17, 2103 -2137.
AMA StyleAlejandro Rincón, Fabiola Angulo, Fredy E. Hoyos. Analysis of a generalized Fujikawa’s growth model. Mathematical Biosciences and Engineering. 2020; 17 (2):2103-2137.
Chicago/Turabian StyleAlejandro Rincón; Fabiola Angulo; Fredy E. Hoyos. 2020. "Analysis of a generalized Fujikawa’s growth model." Mathematical Biosciences and Engineering 17, no. 2: 2103-2137.
Controlling switched systems is a difficult task, even when dealing with piecewise linear systems (CPWLs), which consist of a set of linear differential equations and a set of switching conditions. This difficulty is largely due to the loss of linearity in the entire system, and it is necessary to solve differential and algebraic equations to determine the solution. In this paper, a new method to tune the parameters of the controllers applied to switched systems is derived using information from the saltation matrix, particularly its induced norm. First, the parameters are tuned using classical methods, and then, after analyzing the norm of the saltation matrix, a new set of parameters that guarantees the stability of the period-1 orbit is obtained. The method is validated using analytical solutions for two different systems (boost and boost-flyback power converters) and is also experimentally validated for the boost-flyback power converter.
Juan-Guillermo Muñoz; Arnold Perez; Fabiola Angulo. Enhancing the Stability of the Switched Systems Using the Saltation Matrix. International Journal of Structural Stability and Dynamics 2019, 19, 1 .
AMA StyleJuan-Guillermo Muñoz, Arnold Perez, Fabiola Angulo. Enhancing the Stability of the Switched Systems Using the Saltation Matrix. International Journal of Structural Stability and Dynamics. 2019; 19 (5):1.
Chicago/Turabian StyleJuan-Guillermo Muñoz; Arnold Perez; Fabiola Angulo. 2019. "Enhancing the Stability of the Switched Systems Using the Saltation Matrix." International Journal of Structural Stability and Dynamics 19, no. 5: 1.
This paper presents three homotopic methods to determine the suitable topology to be simulated in DC-DC power electronic converters. One of the proposed methods is based on Newton and hyperspheres tracking, by using the canonical piecewise-linear model of Chua-Kang to represent the characteristic curve of the diodes. The others two homotopic methods are based on fixed point and Newton with uniform variation of the homotopic parameter, by using hybrid diode models. Numerical simulations via MATLAB were compared with the results obtained from PLECS, and a good agreement was found between both simulation alternatives.
Guillermo Leon Gallo-Hernandez; Hector Vazquez-Leal; Victor Manuel Jimenez-Fernandez; Gustavo A. Osorio; Fabiola Angulo; Jose Alfredo Martinez-Melchor. Homotopic Approach for the Simulation of DC-DC Power Electronic Converters. ANZIAM Journal 2018, 60, 25 -58.
AMA StyleGuillermo Leon Gallo-Hernandez, Hector Vazquez-Leal, Victor Manuel Jimenez-Fernandez, Gustavo A. Osorio, Fabiola Angulo, Jose Alfredo Martinez-Melchor. Homotopic Approach for the Simulation of DC-DC Power Electronic Converters. ANZIAM Journal. 2018; 60 ():25-58.
Chicago/Turabian StyleGuillermo Leon Gallo-Hernandez; Hector Vazquez-Leal; Victor Manuel Jimenez-Fernandez; Gustavo A. Osorio; Fabiola Angulo; Jose Alfredo Martinez-Melchor. 2018. "Homotopic Approach for the Simulation of DC-DC Power Electronic Converters." ANZIAM Journal 60, no. : 25-58.
Reliable and robust control of power converters is a key issue in the performance of numerous technological devices. In this paper we show a design technique for the control of a DC-DC buck converter with a switching technique that guarantees both good performance and global stability. We show that making use of the contraction theorem in the Jordan canonical form of the buck converter, it is possible to find a switching surface that guarantees stability but it is incapable of rejecting load perturbations. To overcome this, we expand the system to include the dynamics of the voltage error and we demonstrate that the same design procedure is not only able to stabilize the system to the desired operation point but also to reject load, input voltage, and reference voltage perturbations.
David Angulo-Garcia; Fabiola Angulo; Gustavo Osorio; Gerard Olivar. Control of a DC-DC Buck Converter through Contraction Techniques. Energies 2018, 11, 3086 .
AMA StyleDavid Angulo-Garcia, Fabiola Angulo, Gustavo Osorio, Gerard Olivar. Control of a DC-DC Buck Converter through Contraction Techniques. Energies. 2018; 11 (11):3086.
Chicago/Turabian StyleDavid Angulo-Garcia; Fabiola Angulo; Gustavo Osorio; Gerard Olivar. 2018. "Control of a DC-DC Buck Converter through Contraction Techniques." Energies 11, no. 11: 3086.
Peak current-mode control is widely used in power converters and involves the use of an external compensation ramp to suppress undesired behaviors and to enhance the stability range of the Period-1 orbit. A boost converter uses an analytical expression to find a compensation ramp; however, other more complex converters do not use such an expression, and the corresponding compensation ramp must be computed using complex mechanisms. A boost-flyback converter is a power converter with coupled inductors. In addition to its high efficiency and high voltage gains, this converter reduces voltage stress acting on semiconductor devices and thus offers many benefits as a converter. This paper presents an analytical expression for computing the value of a compensation ramp for a peak current-mode controlled boost-flyback converter using its simplified model. Formula results are compared to analytical results based on a monodromy matrix with numerical results using bifurcations diagrams and with experimental results using a lab prototype of 100 W.
Juan-Guillermo Muñoz; Guillermo Gallo; Fabiola Angulo; Gustavo Osorio. Slope Compensation Design for a Peak Current-Mode Controlled Boost-Flyback Converter. Energies 2018, 11, 3000 .
AMA StyleJuan-Guillermo Muñoz, Guillermo Gallo, Fabiola Angulo, Gustavo Osorio. Slope Compensation Design for a Peak Current-Mode Controlled Boost-Flyback Converter. Energies. 2018; 11 (11):3000.
Chicago/Turabian StyleJuan-Guillermo Muñoz; Guillermo Gallo; Fabiola Angulo; Gustavo Osorio. 2018. "Slope Compensation Design for a Peak Current-Mode Controlled Boost-Flyback Converter." Energies 11, no. 11: 3000.
Power converters with coupled inductors are very promising due to the high efficiency and high voltage gain. Apart from the aforementioned advantages, the boost-flyback converter reduces the voltage stress on the semiconductors. However, to obtain good performance with high voltage gains, the controller must include two control loops (current and voltage), and a compensation ramp. One of the most used control techniques for power converters is the peak current-mode control with compensation ramp. However, in the case of a boost-flyback converter there is no mathematical expression in the literature, to compute the slope of the compensation ramp. In this paper, a formula to compute the slope of the compensation ramp is proposed in such a way that a stable period-1 orbit is obtained. This formula is based on the values of the circuit parameters, such as inductances, capacitances, input voltage, switching frequency and includes some assumptions related to internal resistances, output voltages, and some other electrical properties related with the physical construction of the circuit. The formula is verified numerically using the saltation matrix and experimentally using a test circuit.
Juan-Guillermo Muñoz; Guillermo Gallo; Fabiola Angulo; Gustavo Osorio. Slope Compensation Design for a Peak Current-Mode Controlled Boost-Flyback Converter. 2018, 1 .
AMA StyleJuan-Guillermo Muñoz, Guillermo Gallo, Fabiola Angulo, Gustavo Osorio. Slope Compensation Design for a Peak Current-Mode Controlled Boost-Flyback Converter. . 2018; ():1.
Chicago/Turabian StyleJuan-Guillermo Muñoz; Guillermo Gallo; Fabiola Angulo; Gustavo Osorio. 2018. "Slope Compensation Design for a Peak Current-Mode Controlled Boost-Flyback Converter." , no. : 1.
Viktor Avrutin; José D. Morcillo; Zhanybai T. Zhusubaliyev; Fabiola Angulo. Bubbling in a power electronic inverter: Onset, development and detection. Chaos, Solitons & Fractals 2017, 104, 135 -152.
AMA StyleViktor Avrutin, José D. Morcillo, Zhanybai T. Zhusubaliyev, Fabiola Angulo. Bubbling in a power electronic inverter: Onset, development and detection. Chaos, Solitons & Fractals. 2017; 104 ():135-152.
Chicago/Turabian StyleViktor Avrutin; José D. Morcillo; Zhanybai T. Zhusubaliyev; Fabiola Angulo. 2017. "Bubbling in a power electronic inverter: Onset, development and detection." Chaos, Solitons & Fractals 104, no. : 135-152.
In this work we have presented a methodology for the hybrid simulation of DC-DC power electronic converters. This methodology only requires a description of the circuit through the incidence matrix and a control rule to generate all the needed circuit topologies in the simulation. We have performed a comparative analysis of the simulations performed with the proposed methodology and a commercial software like PLECS.
Guillermo Gallo; Juan-Guillermo Muñoz; Fabiola Angulo; Gustavo Osorio. Automatic hybrid simulation of DC-DC power electronic converters. 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC) 2017, 1 -6.
AMA StyleGuillermo Gallo, Juan-Guillermo Muñoz, Fabiola Angulo, Gustavo Osorio. Automatic hybrid simulation of DC-DC power electronic converters. 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC). 2017; ():1-6.
Chicago/Turabian StyleGuillermo Gallo; Juan-Guillermo Muñoz; Fabiola Angulo; Gustavo Osorio. 2017. "Automatic hybrid simulation of DC-DC power electronic converters." 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC) , no. : 1-6.
In this work we have simulated and physically implemented a boost-flyback power converter with an average current control mode which presents two sliding dynamics. We have tested four applications to perform the simulations: NGSPICE, PSIM, PLECS, MATLAB/SIMULINK/SIMSCAPE. We have developed a program that detects the sliding dynamics and solves a Filippov field in order to avoid blockages and long simulation times. We compared our program with PSIM and obtained speed ups around 119 times. Finally, a discussion about pros and cons is presented.
Cristian Galindo; Guillermo Gallop; Juan-Guillermo Muñoz; Gustavo Osorio; Fabiola Angulo. Effects of sliding dynamics and filippov fields in the simulation of a boost-flyback power converter with an average current control mode. 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC) 2017, 1 -5.
AMA StyleCristian Galindo, Guillermo Gallop, Juan-Guillermo Muñoz, Gustavo Osorio, Fabiola Angulo. Effects of sliding dynamics and filippov fields in the simulation of a boost-flyback power converter with an average current control mode. 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC). 2017; ():1-5.
Chicago/Turabian StyleCristian Galindo; Guillermo Gallop; Juan-Guillermo Muñoz; Gustavo Osorio; Fabiola Angulo. 2017. "Effects of sliding dynamics and filippov fields in the simulation of a boost-flyback power converter with an average current control mode." 2017 IEEE 3rd Colombian Conference on Automatic Control (CCAC) , no. : 1-5.
Jose D. Morcillo; Carlos J. Franco; Fabiola Angulo. Delays in electricity market models. Energy Strategy Reviews 2017, 16, 24 -32.
AMA StyleJose D. Morcillo, Carlos J. Franco, Fabiola Angulo. Delays in electricity market models. Energy Strategy Reviews. 2017; 16 ():24-32.
Chicago/Turabian StyleJose D. Morcillo; Carlos J. Franco; Fabiola Angulo. 2017. "Delays in electricity market models." Energy Strategy Reviews 16, no. : 24-32.
In many practical applications power Converters are designed to work in a desired operation point depending on the needs of the appliance. The ability of the system to work in that point is related to a correct dynamic analysis that eventually leads to de design of a proper control strategy. Linear model approximations are an option to design control loops, but using this kind of models the nonlinear dynamics are neglected. In this paper we propose to study a boost-flyback converter by means of the numerical analysis of the converter hybrid model. The bifurcations diagrams show that the converter exhibits coexistence of solutions when varying the coupling coefficient of the inductors. Four different coexistence scenarios are reported, including period-1, period-2 and chaotic solutions. Results for two cases of study are experimentally validated.
Juan-Guillermo Muñoz; Guillermo Gallo; Fabiola Angulo; Gustavo Osorio. Coexistence of solutions in a boost-flyback converter with current mode control. 2017 IEEE 8th Latin American Symposium on Circuits & Systems (LASCAS) 2017, 1 -4.
AMA StyleJuan-Guillermo Muñoz, Guillermo Gallo, Fabiola Angulo, Gustavo Osorio. Coexistence of solutions in a boost-flyback converter with current mode control. 2017 IEEE 8th Latin American Symposium on Circuits & Systems (LASCAS). 2017; ():1-4.
Chicago/Turabian StyleJuan-Guillermo Muñoz; Guillermo Gallo; Fabiola Angulo; Gustavo Osorio. 2017. "Coexistence of solutions in a boost-flyback converter with current mode control." 2017 IEEE 8th Latin American Symposium on Circuits & Systems (LASCAS) , no. : 1-4.
High voltage gain power converters are very important in photovoltaic applications mainly due to the low output voltage of photovoltaic arrays. This kind of power converters includes three or more semiconductor devices and four or more energy storage elements, making the dynamical analysis of the controlled system more difficult. In this paper, the boost-flyback power converter is controlled by peak-current mode with compensation ramp. The closed-loop analysis is performed to guarantee operation conditions such that a period-1 orbit is attained. The converter is considered as a piecewise linear system, and the closed-loop stability is determined by using the monodromy matrix, obtained by the composition of the saltation matrixes with the solutions of the dynamical equations in the linear intervals. The largest eigenvalue of the monodromy matrix gives the stability of the period-1 orbit, and a deep analysis using bifurcation diagrams let us reach a conclusion about the loss of the stability, which is experimentally verified. To avoid overcompensation effects, the minimum value required by the compensation ramp is obtained, and the minimum and maximum values of the load resistance are found too. The system has a good transient response under disturbances in the load and in the input voltage.
Juan-Guillermo Muñoz; Guillermo Gallo; Gustavo Osorio; Fabiola Angulo. Performance Analysis of a Peak-Current Mode Control with Compensation Ramp for a Boost-Flyback Power Converter. Journal of Control Science and Engineering 2016, 2016, 1 -14.
AMA StyleJuan-Guillermo Muñoz, Guillermo Gallo, Gustavo Osorio, Fabiola Angulo. Performance Analysis of a Peak-Current Mode Control with Compensation Ramp for a Boost-Flyback Power Converter. Journal of Control Science and Engineering. 2016; 2016 ():1-14.
Chicago/Turabian StyleJuan-Guillermo Muñoz; Guillermo Gallo; Gustavo Osorio; Fabiola Angulo. 2016. "Performance Analysis of a Peak-Current Mode Control with Compensation Ramp for a Boost-Flyback Power Converter." Journal of Control Science and Engineering 2016, no. : 1-14.
The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps. 1. IntroductionComplex bifurcation scenarios have been observed in nonlinear dynamic systems from virtually all areas of science, including a broad range of natural sciences, mechanical and electrical engineering, and economics and other areas of the social sciences [1–3]. Theoretical and applied researches have explained various bifurcation scenarios [4–6], and analytical, numerical, and experimental works have contributed to unraveling the complexity inherent to chaotic motion [3].Coupled chaotic maps are a set of special discrete-time dynamical systems that can describe chemical, epidemiological, physiological, biological, or engineering systems [7]. Interesting nonlinear phenomena have been reported in them. For example, in [8, 9], coupled logistic maps were analyzed and new scenarios for transition to chaos were found via the creation and destruction of multilayered tori. In those papers, novel routes to chaos were described, and authors found that, depending on the coupling constant value, the system approaches different periodic attractors.Control methods of coexisting attractors in multistability scenarios have been widely studied in the last decades. In [10], periodic signals were replaced by chaotic ones in order to eliminate multiple domains of attraction. In [11, 12], the influence of noise on preference and dominance of attractors in multistable systems was studied. In [13, 14], multistability was controlled using small periodic modulation of a system parameter. In [15], the basins of attraction (BA) were controlled using harmonic and stochastic modulation. In [16], an impulsive force was used in order to perturb one attractor of the system and to change its response such that the system response evolved to another attractor. In [17], control of multistability scenarios based on the selection of a particular attractor by periodic external modulation was presented. A complete report of control of multistability can be found in [18].In this paper, we prove a different technique to control domains of attraction in multistability scenarios which is called Fixed-Point Inducting Controller (FPIC). The FPIC is a feed-forward controller that forces the system solution to evolve to an existing desired attractor which cannot be always reached for the uncontrolled case because of initial conditions. This technique was initially proposed in [19] and successfully applied in [20, 21]. In particular, in [21], experiments showed the good performance of the controlled system, where the FPIC was used as a second control loop. However, in all previous works published until now, FPIC has been only used to stabilize unstable periodic orbits.The main contributions of this paper lie in numerical analysis and control design areas. The development of a novel methodology to compute and analyze the basins of attraction reinforced the numerical analysis. The methodology consists in decomposing the map of the system in submaps, where corresponds to the order of the periodic orbit to be controlled. To have the BA, the long-time response terms of this submap are depicted according to key colours. Apart from help to the control design, this way to compute the basin of attraction can be seen as the basins of attraction of period-1 orbits which are computed every sixth or fifth iteration for linear and nonlinear coupling, respectively, and the control goal can be thought of as the control of period-1 orbit. On the other hand, the methodology to design the controller based on FPIC technique belongs to control theory. When the proposed controller is used, the system is forced to evolve to a known periodic orbit that exists in the uncontrolled map. Moreover, by using the information obtained from bifurcation diagrams and basins of attraction, it is possible to compute the minimum control effort required to stabilize the target orbit in the defined region. In particular, the coexistence of periodic solutions in coupled logistic maps [8, 9, 22] is controlled by widening the basin of attraction of a period- orbit that coexists with another one, and the minimum control effort is computed aided by the unstable period-2 orbit for the linear coupling and by unstable period-1 orbits for the nonlinear coupling case.The paper is organized as follows. In Section 2, the coupled logistic maps are presented. The coexistence of period-6 and period-5 orbits is identified. In Section 3, the methodology to compute the basins of attraction is explained, and the procedure is applied to the linear and nonlinear coupling maps. In Section 4, the methodology to compute the controller is presented and applied to the considered systems and the minimum control effort required to guarantee monostabilization of the periodic orbit is computed. In Section 5, a brief discussion of the results is presented, and finally, in Section 6, the conclusions are given.2. Coupled Logistic MapsCoupled chaotic maps provide a source of bifurcation scenarios with nonlocal phenomena and coexistence of attractors. To design the proposed controller, two maps with different coupling mechanisms are chosen. For linear coupling map, there are two period-6 orbits that coexist in a range of the parameter set. Similarly, for nonlinear coupling case, two period-5 orbits coexist.2.1. Coupled Logistic Maps: Linear CouplingThis system is described bywhere is the coupling constant and , , is the parameter associated with the nonlinear part of (1). Since the system is symmetrical, there is an invariant line and the restriction of the 2D map to reduces it to a 1D map (the logistic map). The dynamics on this invariant set help us to study the dynamics and bifurcations of the 2D system. Moreover, symmetrical trajectories are generated by symmetrical initial conditions leading to symmetrical basins of attraction [22]. Novel routes to chaos through torus-breakdown mechanism of this system were reported in [9] and different dynamics were characterized in the parameter region given by .Figure 1 shows the dynamic behavior of (1) when is varied and two symmetrical initial conditions are considered. Note that a pair of coexisting period-6 orbits are identified in these diagrams for , which makes the main difference between Figures 1(a) and 1(b). The dynamical behavior in the rest of the interval is very similar and other differences cannot be easily seen. Table 1 shows the specific values of the two coexisting period-6 orbits. It can be observed that the solutions are mutually symmetrical in the sense that .Table 1: Two coexisting period-6 orbits for the linear coupling maps when and . For the sake of simplicity, the periodic orbit values are presented with 8 decimal digits.Figure 1: Bifurcation diagrams of (1) with and two symmetrical initial conditions.2.2. Coupled Logistic Maps: Nonlinear CouplingThis system is defined bywhere and were defined before, , and . Figure 2 shows an interesting coexistence scenario for this system. Two mutually symmetrical period-5 cycles coexist in it when and .Figure 2: Bifurcation diagrams of (2) with and two symmetrical initial conditions.3. Methodology to Compute the Basins of AttractionIn this section, a methodology to compute the basins of attraction (BA) is developed. By using this methodology, it is possible to find the minimum control effort required to stabilize a desired periodic orbit.3.1. Computation of the Basins of Attractions in Linear Coupling MapsThe BA corresponds to the set of initial conditions whose long-time response (LTR) approaches the attractor. The first and second period-6 orbits were defined in Table 1. All numerical results associated with the second period-6 orbit are differentiated from the first one with an upper bar. In order to compute the BA and after applying FPIC control, we associate with each point of the first period-6 orbit a section that is given by , such as what is illustrated in Figure 3. A similar procedure is developed to compute the BA of the second period-6 orbit, whose sections are defined as .Figure 3: Colour key to compute the BA.To compute the LTR terms, we start from any initial condition inside the considered region ( and ) and compute its successive iterations until the solution approaches the attractor. The general map to describe the evolution of the system is , and it is given by where is high enough. The map is decomposed in six submaps . Each is formed by the set of points given by , where and . If the LTR terms of lie inside some (or when the BA for the second period-6 orbit is computed), it is coloured according to the colour key displayed in Figure 3. If the LTR terms of lie outside these regions, then it is gray coloured. This procedure is repeated for a lot of initial conditions.The BA of the first period-6 orbit (second period-6 orbit) can be computed and depicted only using () submaps for a fixed , and its diagram
John Alexander Taborda; Fabiola Angulo. Computing and Controlling Basins of Attraction in Multistability Scenarios. Mathematical Problems in Engineering 2015, 2015, 1 -13.
AMA StyleJohn Alexander Taborda, Fabiola Angulo. Computing and Controlling Basins of Attraction in Multistability Scenarios. Mathematical Problems in Engineering. 2015; 2015 ():1-13.
Chicago/Turabian StyleJohn Alexander Taborda; Fabiola Angulo. 2015. "Computing and Controlling Basins of Attraction in Multistability Scenarios." Mathematical Problems in Engineering 2015, no. : 1-13.
A. Rincón; D. Piarpuzán; F. Angulo. A new adaptive controller for bio-reactors with unknown kinetics and biomass concentration: Guarantees for the boundedness and convergence properties. Mathematics and Computers in Simulation 2015, 112, 1 -13.
AMA StyleA. Rincón, D. Piarpuzán, F. Angulo. A new adaptive controller for bio-reactors with unknown kinetics and biomass concentration: Guarantees for the boundedness and convergence properties. Mathematics and Computers in Simulation. 2015; 112 ():1-13.
Chicago/Turabian StyleA. Rincón; D. Piarpuzán; F. Angulo. 2015. "A new adaptive controller for bio-reactors with unknown kinetics and biomass concentration: Guarantees for the boundedness and convergence properties." Mathematics and Computers in Simulation 112, no. : 1-13.