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Research Papers

Fractional PD-IλDμ Error Manifolds for Robust Tracking Control of Robotic Manipulators

[+] Author and Article Information
Aldo Jonathan Muñoz-Vázquez

CONACYT,
School of Engineering,
Autonomous University of Chihuahua,
Campus II,
Chihuahua 31100, Chihuahua, Mexico
e-mail: aldo.munoz.vazquez@gmail.com

Vicente Parra-Vega

Robotics and Advanced Manufacturing,
Center for Research and Advanced Studies,
Saltillo 25900, Coahuila, Mexico
e-mail: vparra@cinvestav.mx

Anand Sánchez-Orta

Robotics and Advanced Manufacturing,
Center for Research and Advanced Studies,
Saltillo 25900, Coahuila, Mexico
e-mail: anand.sanchez@cinvestav.mx

Gerardo Romero-Galván

Electrical and Electronic Engineering
Autonomous University of Tamaulipas
Reynosa-Rodhe 88779, Tamaulipas, Mexico
e-mail: gromero@docentes.uat.edu.mx

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received September 25, 2017; final manuscript received September 23, 2018; published online November 8, 2018. Assoc. Editor: Srinivasa M. Salapaka.

J. Dyn. Sys., Meas., Control 141(3), 031006 (Nov 08, 2018) (6 pages) Paper No: DS-17-1485; doi: 10.1115/1.4041605 History: Received September 25, 2017; Revised September 23, 2018

Linear proportional-integral-derivative (PID) controller stands for the most widespread technique in industrial applications due to its simple structure and easy tuning rules. Recently, considering fractional orders λ and μ, there has been studied the fractional-order PIλDμ (FPID) controller to provide salient advantages in comparison to the conventional integer-order PID, such as, a more flexible structure and a preciser performance. In addition, proportional and derivative (PD) and PID error manifolds have been classically proposed; however, there remains the question on how FPID-like error manifolds perform for the control of nonlinear plants, such as robots. In this paper, this problem is addressed by proposing a PD-IλDμ error manifold for novel vector saturated control. The stability analysis shows convergence into a small vicinity of the origin, wherein, such hybrid combination of integer- and fractional-order error manifolds provides further insights into the closed-loop response of the nonlinear plant. Simulations studies are carried out to illustrate the feasibility of the proposed scheme.

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References

Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Rahimian, M. A. , and Tavazoei, M. S. , 2013, “ Optimal Tuning for Fractional-Order Controllers: An Integer-Order Approximating Filter Approach,” ASME J. Dyn. Syst. Meas. Contr., 135(2), p. 021017. [CrossRef]
Liebst, B. S. , and Torvik, P. J. , 2007, “ Asymptotic Approximations for Systems Incorporating Fractional Derivative Damping,” ASME J. Dyn. Syst. Meas. Contr., 118(3), pp. 572–579. [CrossRef]
Barbosa, R. , Tenreiro-Machado, J. , and Ferreira, I. , 2004, “ Tuning of PID Controllers Based on Bode's Ideal Transfer Function,” Nonlinear Dyn., 38(1–4), pp. 305–321. [CrossRef]
Dadras, S. , Dadras, S. , and Momeni, H. R. , 2017, “ Linear Matrix Inequality Based Fractional Integral Sliding-Mode Control of Uncertain Fractional-Order Nonlinear Systems,” ASME J. Dyn. Syst. Meas. Contr., 139(11), p. 111003. [CrossRef]
Tavazoei, M. S. , and Haeri, M. , 2010, “ Stabilization of Unstable Fixed Points of Fractional-Order Systems by Fractional-Order Linear Controllers and Its Applications in Suppression of Chaotic Oscillations,” ASME J. Dyn. Syst. Meas. Contr., 132(2), p. 021008. [CrossRef]
Muñoz-Vázquez, A. J. , Parra-Vega, V. , and Sánchez-Orta, A. , 2014, “ Free-Model Fractional-Order Absolutely Continuous Sliding Mode Control for Euler-Lagrange Systems,” 53rd IEEE Conference on Decision and Control, Los Angeles, CA, Dec. 15–17, pp. 6933–6938.
Muñoz-Vázquez, A. J. , Parra-Vega, V. , and Sánchez-Orta, A. , 2015, “ Control of Constrained Robot Manipulators Based on Fractional Order Error Manifolds,” 11th IFAC Symposium on Robot Control, Salvador, Brazil, Aug. 26–28, pp. 131–126.
Podlubny, I. , 1999, “ Fractional-Order Systems and P I λ D μ-Controllers,” IEEE Trans. Autom. Contr., 44(1), pp. 208–214. [CrossRef]
Tavakoli-Kakhki, M. , and Haeri, M. , 2011, “ Temperature Control of a Cutting Process Using Fractional Order Proportional-Integral-Derivative Controller,” ASME J. Dyn. Syst. Meas. Contr., 133(5), p. 051014. [CrossRef]
Badri, V. , and Tavazoei, M. S. , 2016, “ Simultaneous Compensation of the Gain, Phase, and Phase-Slope,” ASME J. Dyn. Syst. Meas. Contr., 138(12), p. 121002. [CrossRef]
Li, H. , Luo, Y. , and Chen, Y. Q. , 2000, “ A Fractional Order Proportional and Derivative (FOPD) Motion Controller: Tuning Rule and Experiments,” IEEE Trans. Contr. Syst. Technol., 18(2), pp. 516–520. [CrossRef]
Yang, H. , Jiang, Y. , and Yin, S. , 2018, “ Fault-Tolerant Control of Time-Delay Markov Jump Systems With It ô Stochastic Process and Output Disturbance Based on Sliding Mode Observer,” IEEE Trans. Ind. Inf. (in press).
Yin, S. , Yang, H. , and Kaynak, O. , 2017, “ Sliding Mode Observer-Based FTC for Markovian Jump Systems With Actuator and Sensor Faults,” IEEE Trans. Autom. Contr., 62(7), pp. 3551–3558. [CrossRef]
Samko, S. , Khilbas, A. , and Marichev, O. , 1993, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, Switzerland.
Muñoz-Vázquez, A. J. , Parra-Vega, V. , and Sánchez-Orta, A. , 2016, “ Uniformly Continuous Differintegral Sliding Mode Control of Nonlinear Systems Subject to Hölder Disturbances,” Automatica, 66, pp. 179–184. [CrossRef]
Parra-Vega, V. , Arimoto, S. , Liu, Y. H. , Hirzinger, G. , and Akella, P. , 2003, “ Dynamic Sliding PID Control for Tracking of Robots Manipulators: Theory and Experiments,” IEEE Trans. Rob. Autom., 19(6), pp. 967–976. [CrossRef]
Matignon, D. , 1996, “ Stability Results for Fractional Differential Equations With Applications to Control Processing,” Multiconference on Computation Engineering in Systems Applications, pp. 963–968.
Gutman, S. , 1979, “ Uncertain Dynamic Systems: A Lyapunov Min-Max Approach,” IEEE Trans. Autom. Contr., 24(3), pp. 437–449. [CrossRef]
Chen, Y. Q. , Petráš, I. , and Xue, D. , 2009, “ Fractional Order Control—A Tutorial,” IEEE American Control Conference, St. Louis, MO, June 10–12, pp. 1397–1411.
Oustaloup, A. , Mathieu, B. , and Lanusse, P. , 1995, “ The CRONE Control of Resonant Plants: Application to a Flexible Transmission,” Eur. J. Contr., 1(2), pp. 113–121. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Bode's diagrams of transfer function Gi(s). Notice that in this particular example, the fractional response is in between of phase and gain to the integer order cases.

Grahic Jump Location
Fig. 2

Regulation: bended tracking errors q̃(t). Joint 1 in solid line, joint 2 in dot dashed line: (a) t versus q̃(t): λ = 0, μ = 0, (b) t versus q̃(t): λ = 0.5, μ = 0.25, and (c) t versus q̃(t): λ = 1, μ = 1.

Grahic Jump Location
Fig. 3

Regulation: error variable S(t) and control torque τ(t): (a) t versus S(t): λ = 0.5, μ = 0.25 and (b) t versus τ(t): λ = 0.5, μ = 0.25

Grahic Jump Location
Fig. 4

Tracking: bended tracking errors q̃(t). Joint 1 in solid line, joint 2 in dot dashed line: (a) t versus q̃(t): λ = 0, μ = 0, (b) t versus q̃(t): λ = 0.6, μ = 0.5, and (c) t versus q̃(t): λ = 1, μ = 1.

Grahic Jump Location
Fig. 5

Tracking: error manifold S(t) and control torque τ(t): (a) t versus S(t): λ = 0.6 μ = 0.5 and (b) t versus τ(t): λ = 0.6, μ = 0.5

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