- Research
- Open Access
FPGA implementation of adaptive sliding mode control and genetically optimized PID control for fractional-order induction motor system with uncertain load
- Karthikeyan Rajagopal^{1}Email authorView ORCID ID profile,
- Guessas Laarem^{2},
- Anitha Karthikeyan^{1} and
- Ashokkumar Srinivasan^{1}
https://doi.org/10.1186/s13662-017-1341-9
© The Author(s) 2017
- Received: 22 May 2017
- Accepted: 27 August 2017
- Published: 7 September 2017
Abstract
In this paper, we investigate the control of 4-D nonautonomous fractional-order uncertain model of a PI speed-regulated current-driven induction motor (FOIM) using a fractional-order adaptive sliding mode controller (FOASMC). First, we derive a dimensionless fractional-order model of the induction motor from the well-known integer -model of the induction motor. Various dynamic properties of the fractional-order induction motor, such as stability of the equilibrium points, Lyapunov exponents, bifurcation, and bicoherence, are investigated. An adaptive controller is derived to suppress the chaotic oscillations of the fractional-order model of the induction motor. Numerical simulations of the adaptive chaos suppression methodology are depicted for the fractional-order uncertain model of the induction motor to validate the analytical results of this work. A genetically optimized fractional-order PID (FOPID) controller is also derived to stabilize the states of the FOIM system. FPGA implementation of the proposed FOASMC is also presented to show that the proposed controller is hardware realizable.
Keywords
- induction motor
- chaos suppression
- fractional order
- Lyapunov stability
- FPGA
MSC
- 34H10
- 26A33
- 34A08
1 Introduction
Electric motors consume approximately 65% to 70% of the electric energy [1]. Industry and household applications depend mostly on alternating current (AC) electric motors. It is well known that 90% of these AC electric motors are induction motors. A majority of modern devices use induction motors, and the motor drive operates under various loads. A rapid change of load allows one to increase the productivity of the motor, but at the same time it may lead to various undesirable effects such as motor stopping, vibration, damage, or failure of the device itself. So the investigation of induction motor operation under sudden changes of load becomes a critical issue. Mathematical models of induction motor with various rotors and analysis of their stability and oscillations were studied by Solovyeva [2]. The control of an induction motor is a very complicated research problem due to highly nonlinear characteristics, coupling, and time varying dynamics [3].
Fractional-order calculus developed from ordinary calculus is a generalization of the integration and differentiation to the noninteger-(fractional-)order generalization operator \({}_{a}D_{t}^{q}\) in which a and t are limits and q is the order of the operator. This notation is used for both the fractional derivatives and fractional integrals in a single expression [4]. Two general fractional-order integral/differential operations are commonly discussed, viz. Caputo and Riemann-Liouville (R-L) fractional operators. Physically, the R-L fractional operator has an initial value problem [5]. Thus, the Caputo fractional operator is more practical than the R-L fractional one.
The benefits of using fractional-order models of real dynamical objects and processes of applications appear in various fields of science and technology [6]. The synchronization of chaotic systems has been implemented in many engineering applications with integer-order derivatives [7–11]. However, only a few works have been reported on the synchronization of fractional-order chaotic systems, since the proof of stability of fractional-order systems is more complex than the chaotic systems with integer-order derivatives [6]. Synchronization of chaotic systems deals with asymptotically synchronizing the state trajectories of a pair of chaotic systems called the master and slave systems. Many control techniques have been developed for the chaos synchronization of integer-order chaotic systems such as active control [12–16], adaptive control [17–21], sliding mode control [22–25], backstepping control [26–28], fuzzy control [29, 30], and so on.
A bifurcation diagram shows the long-term qualitative changes (equilibria or periodic orbits) of a system as a function of a bifurcation parameters of the system. The complete dynamics of the system with variation of parameters can be studied with the help of bifurcation diagrams [31, 32]. Nonlinear dynamical system undergoes abrupt qualitative changes when crossing bifurcation points [33]. For a more exhaustive qualitative analysis of a nonlinear dynamic system, it is compulsory to identify both singularities of the parameter plane and singularities of the phase plane [34, 35].
The stability of fractional-order systems using Lyapunov stability theory has been investigated in the literature [36, 37]. A fractional-order controller to stabilize the unstable fixed points of an unstable open-loop system was proposed by Tavazoei and Haeri [38]. A delayed feedback control (DFC) based on the act-and-wait concept for nonlinear dynamical systems was proposed by Konishi et al. [39], who reduce the dynamics of DFC systems to that of discrete-time systems.
With the LabVIEW simulation module, we can investigate the dynamic behavior of complex engineering systems. An experimental study of the fractional-order proportional derivative (FO-PD) controllers using LabVIEW was investigated by Jin et al. [40]. Digital implementation of a 3-D chaotic system with three quadratic nonlinearities using LabVIEW was studied by Vaidyanathan et al. [41]. The control and synchronization of an induction motor system was investigated by Chen et al. [42]. Fractional-order systems were investigated by many methods in the literature [43–47]. Many chaotic systems with hidden attractors were investigated by Jafari et al. [48–53]. Analysis of chaotic systems with multistability helps researchers in nonlinear controller design to modify the algorithms with reference to the parameter selections [54, 55]. Asymptotic stability of Caputo fractional derivatives are recently investigated [56]. Some recent works discussed the fractional-order applications in financial models [57].
This paper is organized as follows. In Section 2, we derive a fractional-order model of the induction motor system [3, 42]. In Section 3, we investigate the dynamic properties of the fractional-order induction motor system (FOIMS). In Section 4, we derive fractional-order controllers for suppressing the chaotic oscillations of the FOIMS. In Section 5, we derive the stability of the controllers, and Section 6, we numerically check the results. In Section 6, we derive PID control [58] optimized by genetic optimization algorithm [59–61]. In Section 7, we implement adaptive sliding mode controllers in FPGA [55] using Xilinx vivado tools to show that the controller is hardware realizable.
2 Fractional-order current driven induction motor
Theoretically, fractional-order differential equations use infinite memory. When we wish to numerically calculate or simulate fractional-order equations, we have to use finite-memory principle, where L is the memory length, and h is the time sampling as \(N(t) = \min \{ [ \frac{t}{h} ], [ \frac{L}{h} ] \},b_{j} = ( 1 - \frac{a + \alpha}{j} )b_{j - 1}\).
3 Dynamical analysis of the fractional-order system
In this section, we analyze the fractional-order induction motor system (6) for various properties of chaotic behavior like equilibrium points, Lyapunov exponents, bifurcation, and bicoherence.
3.1 Equilibria points and Lyapunov exponents
3.2 Bifurcation and bicoherence
4 Stability analysis of FOIM system
Commensurate Order: For a commensurate FOIM system of order q, the system is stable and exhibits chaotic oscillations if \(\vert \arg (\operatorname{eig}(J_{E})) \vert = \vert \arg (\lambda_{i}) \vert > \frac{q\pi}{ 2}\), where \(J_{E}\) is the Jacobian matrix at the equilibrium E, and \(\lambda_{i}\) are the eigenvalues of the FOIM system, where \(i = 1,2,3,4\). As seen from the FOIM system, the eigenvalues should remain in the unstable region, and the necessary condition for the FOIM system to be stable is \(q > \frac{2}{\pi} \tan^{ - 1} ( \frac{ \vert \operatorname{Im} \lambda \vert }{\operatorname{Re} \lambda} )\). As the eigenvalues of the system are \(\lambda_{1,2} = 1.303 \pm 29.080i, \lambda_{3} = - 18.26\), and \(\lambda_{4} = - 13.82\), it is clearly seen that \(\lambda_{1,2}\) is a complex pair of eigenvalues and remains in the unstable region contributing to the existence of chaotic oscillations.
Incommensurate Order: The necessary condition for the FOIM system to exhibit chaotic oscillations in the incommensurate case is \(\frac{\pi}{ 2M} - \min_{i} ( \vert \arg (\lambda i) \vert ) > 0\), where M is the LCM of the fractional orders. If \(q_{x} = 0.9,q_{y} = 0.9,q_{z} = 0.8,q_{w} = 0.8\), then \(M = 10\). The characteristic equation of the system evaluated at the equilibrium is \(\det (\operatorname{diag}[\lambda^{Mq_{x}},\lambda^{Mq_{y}},\lambda^{Mq_{z}},\lambda^{Mq_{w}}] - J_{E}) = 0\), and by substituting the values of M and the fractional orders, \(\det (\operatorname{diag}[\lambda^{9},\lambda^{9},\lambda^{8},\lambda^{8}] - J_{E}) = 0\), the characteristic equation is \(\lambda^{26} + 15.22\lambda^{25} + 33.57\lambda^{18} + 62.85\lambda^{17} + 189.29\lambda^{16} + \lambda^{12} + 19.34\lambda^{11} + 857.47\lambda^{10} + 16{,}699\lambda^{9} + 447.95\lambda^{8} + \lambda^{4} + 29.46\lambda^{3} + 1{,}016\lambda^{2} + 26{,}515\lambda + 213{,}680\). The approximated solution of the characteristic equation is \(\lambda_{26} = - 1.3159\), its argument is zero, which is the minimum argument, and hence the necessary stability condition becomes \(\frac{\pi}{ 20} - 0 > 0\), that is, \(0.1571 > 0\). Hence, the FOIM system is stable, and chaos exists in the incommensurate system.
5 Chaos suppression of the fractional-order system using adaptive sliding mode control (ASMC)
Thus, it is clear that stability calculations with (22) are very difficult. So, we use modified Lyapunov stability theory as given in [47, 54, 55].
Let \(e ( t )\) be a continuous and differentiable function.
As \(\rho_{i}\) and \(\eta_{i}\) are positive for \(i = 1,2,3,4\), the Lyapunov first derivative (28) is a negative definite function. This establishes that the closed-loop control system is asymptotically stable for all initial conditions.
6 Numerical simulations of ASMC
The fractional-order induction motor (FOIM) given by equation (18) with the robust adaptive sliding mode controller (26) is implemented in LabVIEW for numerical analysis and validation.
The initial conditions are chosen as in (12), and the parameter values are chosen as in (13). The fractional orders of system (14) are chosen as in (7). The controller gains are taken to be \(k_{1} = 10, k_{2} = 10, k_{3} = 10, k_{4} = 10\).
7 Chaos suppression using genetically optimized fractional-order PID controllers
Fractional-order PID controllers described by using fractional calculus are the most common and useful algorithms in control system engineering. In most cases, feedback loops are controlled using PID Algorithms, designed to correct error(s) between instant value(s) in a system and chosen set point values. Design of FOPID controller using Genetic Algorithms (GAs), which are a stochastic global search method that mimics the process of natural evolution. It is one of the methods used for optimization successfully applied in [59–61].
A genetic algorithm has to be initialized before the algorithm can proceed. The Initialization of the population size, variable bounds, and the evaluation objective functions are required to evaluate the best gain values of FOPID controller for the system. An objective function can created to find a FOPID controller that gives a minimal error. The error functions such as sum absolute error (SAE) are used as objective functions in this work.
MATLAB is used for numerical simulation with the following options:
Variable bound matrix of the proportional, integral, and derivative gains is [−0.001, 0.001], but for the states (\(x_{3},x_{4}\)), the values are multiplied by hundred and ten for the difference of variation range.
FOPID controller gain values optimized with GA
FOPID controller | \(\boldsymbol {K_{P}}\) | \(\boldsymbol {K_{I}}\) | \(\boldsymbol {K_{D}}\) |
---|---|---|---|
\(u_{x_{1}}\) | 0.0023 | 0.0128 | 0.0076 |
\(u_{x_{2}}\) | −0.0028 | −0.0110 | 0.0052 |
\(u_{x_{3}}\) | 1.29 | 0.98 | 2.02 |
\(u_{x_{4}}\) | 0.065 | 0.106 | 0.045 |
Many real dynamic systems are better characterized using a noninteger-order dynamic model based on fractional calculus or differentiation or integration of noninteger order. Therefore fractional-order PID controllers are the future of nonlinear control theory.
8 FPGA implementation of the FOASMC
9 Conclusion
This paper investigates the control of dimensionless nonautonomous fractional-order uncertain load torque model of an induction motor via an adaptive control technique. First, the dimensionless fractional-order model of the induction motor is derived from the integer-order model discussed in the literature using the Caputo-Riemann-Liouville fractional derivatives. To study the effects of variation of parameters on the fractional-order system performance, we have investigated the bifurcation analysis of a fractional-order system with respect to the load torque. It is also shown that the fractional-order induction motor is not only prone to instability due to Hopf bifurcation, but it also exhibits limit cycles and chaos due to bifurcation other than Hopf bifurcation, which is shown by the bicoherence plots. This bispectrum analysis helps us in choosing the appropriate parameters for the proper work of the motor. As understood from the dynamic analysis of the fractional-order system, it is seen that chaos oscillations are exhibited for a particular selection of parameters. To suppress such chaotic oscillations, we have derived an adaptive control technique assuming that the operating load torque parameters of the fractional-order induction motor system are unknown. Numerical results are shown to illustrate the adaptive controller derived in this work.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Machowski, J, Bialek, JW, Bumby, JR: Power System Dynamics: Stability and Control. Wiley, New Jersey (2008) Google Scholar
- Solovyeva, E: Mathematical Models and Stability Analysis of Induction Motors Under Sudden Changes of Load. Jyväskylä Studies in Computing, vol. 182. University of Jyväskylä, Jyväskylä (2013) Google Scholar
- Chang, YH, Wu, CI, Lin, HW, Chen, HC, Chang, CW: Fractional order integral sliding mode flux observer for direct field oriented induction machines. Int. J. Innov. Comput. Inf. Control 8(7A), 4851-4868 (2012) Google Scholar
- Tavazoei, MS, Haeri, M: A note on the stability of fractional order systems. Math. Comput. Simul. 70(5), 1566-1579 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Cao, YG, Li, Y, Ren, W, Chen, YQ: Distributed coordination of networked fractional order systems. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40(2), 362-370 (2010) View ArticleGoogle Scholar
- Podlubny, I: Fractional order systems and \(\mathrm{PI}^{\lambda}\mathrm{D}^{\mu}\) controllers. IEEE Trans. Autom. Control 44(1), 208-213 (1999) MathSciNetView ArticleGoogle Scholar
- Azar, AT, Vaidyanathan, S: Chaos Modeling and Control Systems Design. Springer, Berlin (2015) View ArticleMATHGoogle Scholar
- Azar, AT, Vaidyanathan, S: Advances in Chaos Theory and Intelligent Control. Springer, Berlin (2016) View ArticleMATHGoogle Scholar
- Vaidyanathan, S, Volos, C: Advances and Applications in Nonlinear Control Systems. Springer, Berlin (2016) View ArticleMATHGoogle Scholar
- Vaidyanathan, S, Volos, C: Advances and Applications in Chaotic Systems. Springer, Berlin (2016) View ArticleMATHGoogle Scholar
- Nategh, M, Baleanu, D, Valinejad, MR: On a discrete chaos induction via an aperiodic kicks pattern. J. Comput. Nonlinear Dyn. 12(4), 041008 (2017) View ArticleGoogle Scholar
- Yang, J, Zhao, L: Bifurcation analysis and chaos control of the modified Chua’s circuit system. Chaos Solitons Fractals 77, 332-339 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Sundarapandian, V, Karthikeyan, R: Hybrid synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems via active control. J. Eng. Appl. Sci. 7(3), 254-264 (2012) Google Scholar
- Karthikeyan, R, Sundarapandian, V: Hybrid chaos synchronization of four-scroll systems via active control. J. Electr. Eng. 65(2), 97-103 (2014) Google Scholar
- Vaidyanathan, S, Azar, AT, Rajagopal, K, Alexander, P: Design and SPICE implementation of a 12-term novel hyperchaotic system and its synchronisation via active control. Int. J. Model. Identif. Control 23(3), 267-277 (2015) View ArticleGoogle Scholar
- Vaidyanathan, S: Active control design for the anti-synchronization of Lotka-Volterra biological systems with four competitive species. Int. J. PharmTech Res. 8(7), 58-70 (2015) MathSciNetGoogle Scholar
- Pehlivan, I, Moroz, IM, Vaidyanathan, S: Analysis, synchronization and circuit design of a novel butterfly attractor. J. Sound Vib. 333(20), 5077-5096 (2014) View ArticleGoogle Scholar
- Vaidyanathan, S, Rajagopal, K: Analysis, control, synchronization and LabVIEW implementation of a seven-term novel chaotic system. Int. J. Control Theory Appl. 9(1), 151-174 (2016) Google Scholar
- Vaidyanathan, S: Synchronization of Tokamak systems with symmetric and magnetically confined plasma via adaptive control. Int. J. ChemTech Res. 8(6), 818-827 (2015) Google Scholar
- Vaidyanathan, S: Anti-synchronization of Rikitake two-disk dynamo chaotic systems via adaptive control method. Int. J. ChemTech Res. 8(9), 393-405 (2015) MathSciNetGoogle Scholar
- Vaidyanathan, S, Rajagopal, K: Adaptive control, synchronization and LabVIEW implementation of Rucklidge chaotic system for nonlinear double convection. Int. J. Control Theory Appl. 9(1), 175-197 (2016) Google Scholar
- Yau, HT: Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Solitons Fractals 22(2), 341-347 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Vaidyanathan, S, Boulkroune, A: A novel hyperchaotic system with two quadratic nonlinearities, its analysis and synchronization via integral sliding mode control. Int. J. Control Theory Appl. 9(1), 321-337 (2016) Google Scholar
- Vaidyanathan, S, Sampath, S, Azar, AT: Global chaos synchronization of identical chaotic systems via novel sliding mode control method and its application to Zhu system. Int. J. Model. Identif. Control 23(1), 92-100 (2015) View ArticleGoogle Scholar
- Sampath, S, Vaidyanathan, S: Hybrid synchronization of identical chaotic systems via novel sliding control method with application to Sampath four-scroll chaotic system. Int. J. Control Theory Appl. 9(1), 221-235 (2016) Google Scholar
- Wang, C, Ge, SS: Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos Solitons Fractals 12(7), 1199-1206 (2001) View ArticleMATHGoogle Scholar
- Vaidyanathan, S: A novel hyperchaotic hyperjerk system with two nonlinearities, its analysis, adaptive control and synchronization via backstepping control method. Int. J. Control Theory Appl. 9(1), 257-278 (2016) Google Scholar
- Vaidyanathan, S, Rasappan, S: Global chaos synchronization of n-scroll Chua circuit and Lur’e system using backstepping control design with recursive feedback. Arab. J. Sci. Eng. 39(4), 3351-3364 (2014) View ArticleGoogle Scholar
- Senouci, A, Boukabou, A: Fuzzy modeling, stabilization and synchronization of multi-scroll chaotic systems. Optik 127(13), 5351-5358 (2016) View ArticleGoogle Scholar
- Vaidyanathan, S, Azar, AT: Takagi-Sugeno fuzzy logic controller for Liu-Chen four-scroll chaotic system. Int. J. Intell. Eng. Inform. 4(2), 135-150 (2016) Google Scholar
- Revel, G, Leon, AE, Alonso, DM, Moiola, JL: Multi-parameter bifurcation analysis of subsynchronous interactions in DFIG-based wind farms. Electr. Power Syst. Res. 140, 643-652 (2016) View ArticleGoogle Scholar
- Zarei, A, Tavakoli, S: Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system. Appl. Math. Comput. 291, 323-339 (2016) MathSciNetGoogle Scholar
- Kuznetsov, YA: Elements of Applied Bifurcation Theory. Springer, Berlin (1995) View ArticleMATHGoogle Scholar
- Jabli, N, Khammari, H, Mimouni, MF, Dhifaoui, R: Bifurcation and chaos phenomena appearing in induction motor under variation of PI controller parameters. WSEAS Trans. Syst. 9(7), 784-793 (2010) MATHGoogle Scholar
- Sundarapandian, V, Pehlivan, I: Analysis, control, synchronization and circuit design of a novel chaotic system. Math. Comput. Model. 55(7-8), 1904-1915 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Li, Y, Chen, Y, Podlubny, I: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59(5), 1810-1821 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Gallegos, JA, Duarte-Mermoud, MA: On the Lyapunov theory for fractional order systems. Appl. Math. Comput. 287, 161-170 (2016) MathSciNetMATHGoogle Scholar
- Tavazoei, MS, Haeri, M: Chaos control via a simple fractional order controller. Phys. Lett. A 372, 798-807 (2008) View ArticleMATHGoogle Scholar
- Konishi, K, Kokame, H, Hara, N: Delayed feedback control based on the act-and-wait concept. Nonlinear Dyn. 63, 513-519 (2011) View ArticleGoogle Scholar
- Jin, Y, Chen, YQ, Xue, D: Time-constant robust analysis of a fractional order [proportional derivative] controller. IET Control Theory Appl. 5(1), 164-172 (2011) View ArticleGoogle Scholar
- Vaidyanathan, S, Rajagopal, K, Volos, C, Kyprianidis, IM, Stouboulos, IN: Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system with three quadratic nonlinearities and its digital implementation in LabVIEW. J. Eng. Sci. Technol. Rev. 8(2), 130-141 (2015) Google Scholar
- Chen, D, Shi, P, Ma, X: Control and synchronization of chaos in an induction motor system. Int. J. Innov. Comput. Inf. Control 8(10B), 7237-7248 (2012) Google Scholar
- Li, H, Liao, X, Luo, M: A novel non-equilibrium fractional order chaotic system and its complete synchronization by circuit implementation. Nonlinear Dyn. 68(1), 137-149 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Petras, I: A note on the fractional order Chua’s system. Chaos Solitons Fractals 38(1), 140-147 (2008) View ArticleGoogle Scholar
- Katugampola, UN: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6(4), 1-15 (2014) MathSciNetMATHGoogle Scholar
- Herzallah, MAE: Notes on some fractional calculus operators and their properties. J. Fract. Calc. Appl. 5(19), 1-10 (2014) MathSciNetGoogle Scholar
- Rajagopal, K, Vaidyanathan, S, Karthikeyan, A, Duraisamy, P: Dynamic analysis and chaos suppression in a fractional order brushless DC motor. Electr. Eng. (2016). doi:10.1007/s00202-016-0444-8 Google Scholar
- Jafari, S, Sprott, JC: Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57, 79-84 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Jafari, S, Sprott, JC, Golpayegani, SMRH: Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A 377(9), 699-702 (2013) MathSciNetView ArticleGoogle Scholar
- Pham, VT, Volos, C, Jafari, S, Wang, X, Vaidyanathan, S: Hidden hyperchaotic attractor in a novel simple memristive neural network. Optoelectron. Adv. Mater., Rapid Commun. 8(11-12), 1157-1163 (2014) Google Scholar
- Pham, VT, Volos, C, Jafari, S, Wei, Z, Wang, X: Constructing a novel no-equilibrium chaotic system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 24(5), Article ID 1450073 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Jafari, S, Sprott, JC, Nazarimehr, F: Recent new examples of hidden attractors. Eur. Phys. J. Spec. Top. 224(8), 1469-1476 (2015) View ArticleGoogle Scholar
- Sprott, JC, Jafari, S, Pham, VT, Hosseini, ZS: A chaotic system with a single unstable node. Phys. Lett. A 379(36), 2030-2036 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Rajagopal, K, Laarem, G, Karthikeyan, A, Srinivasan, A, Adam, G: Fractional order memristor no equilibrium chaotic system with its adaptive sliding mode synchronization and genetically optimized fractional order PID synchronization. Complexity 2017, Article ID 1892618 (2017) MathSciNetMATHGoogle Scholar
- Rajagopal, K, Karthikeyan, A, Srinivasan, A: FPGA implementation of novel fractional order chaotic system with two equilibriums and no equilibrium and its adaptive sliding mode synchronization. Nonlinear Dyn. 87(4), 2281-2304 (2017) View ArticleGoogle Scholar
- Baleanua, D, Wu, G, Zeng, S: Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals (2017). doi:10.1016/j.chaos.2017.02.007 MathSciNetGoogle Scholar
- Jajarmi, A, Hajipour, M, Baleanu, D: New aspects of the adaptive synchronization and hyperchaos suppression of a financial model. Chaos Solitons Fractals 99, 285-296 (2017) MathSciNetView ArticleGoogle Scholar
- Astrom, KJ, Hagglund, T: PID Controllers: Theory, Design and Tuning. Research Triangle Park, North Carolina (1995) Google Scholar
- Goldberg, DE: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading (1989) MATHGoogle Scholar
- Wang, Q, Spronck, P, Tracht, R: An overview of genetic algorithms applied to control engineering problems. In: Proceedings of the Second International Conference on Machine Learning and Cybernetics, Xi’an, 3-5 Nov. 2003 (2003). doi:10.1109/ICMLC.2003.1259761 Google Scholar
- Chen, Z, Yuan, X, Ji, B, Wang, P, Tian, H: Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II. Energy Convers. Manag. 84, 390-404 (2014) View ArticleGoogle Scholar
- Pezeshki, C: Bispectral analysis of systems possessing chaotic motions. J. Sound Vib. 137(3), 357-368 (1990) MathSciNetView ArticleMATHGoogle Scholar
- Chandran, V, Elgar, S, Pezeshki, C: Bispectral and trispectral characterization of transition to chaos in the Duffing oscillator. Int. J. Bifurc. Chaos Appl. Sci. Eng. 3(3), 551-557 (1993) MathSciNetView ArticleMATHGoogle Scholar