Abstract
This paper proposes \(H_{2}\)/\(H_{\infty }\) full state-feedback synthesis for robot manipulators, using uncertain polytopic linear parameter-varying (LPV) system modeling, with pole placement constraints to assign the poles of closed-loop system in a desired linear matrix inequality (LMI) region. The desired state trajectory of the system is used for generating an uncertain polytopic model of the system applying usual Lagrangian equations. The control gain matrix is derived by solving a set of LMIs to design a robust pole placement controller such that a prescribed mixed \(H_{2}\)/\(H_{\infty }\) performance is fulfilled and the response of the manipulator has a proper damping ratio. A sufficient condition is proposed to guarantee the asymptotic stability of the closed-loop uncertain polytopic LPV system against the uncertainties on the vertices. The proposed scheme is applied to controller synthesis of a two-degree-of-freedom manipulator trajectory-tracking problem. The simulation results show the effectiveness of the proposed controller.
Similar content being viewed by others
References
Ali, H. S., Boutat-Baddas, L., Becis-Aubry, Y., & Darouach, M. (2006). H-infinity control of a SCARA robot using polytopic LPV approach. In 14th Mediterranean Conference on Control and Automation (pp. 1–5).
Amato, F., Garofalo, F., Glielmo, L., & Pironti, A. (1995). Robust and quadratic stability via polytopic set covering. International Journal of Robust and Nonlinear Control, 5(8), 745–756.
Cai, G., Hu, C., Yin, B., He, H., & Han, X. (2014). Gain-scheduled H2 controller synthesis for continuous-time polytopic LPV systems. Mathematical Problems in Engineering, 2014, 1–14.
Chilali, M., & Gahinet, P. (1996). H\(\infty \) design with pole placement constraints: An LMI approach. IEEE Transactions on Automatic Control, 41(3), 358–367.
Hashemi, S. M., Abbas, H. S., & Werner, H. (2009). LPV modelling and control of a 2-DOF robotic manipulator using PCA-based parameter set mapping. In Proceedings of the 48th IEEE Conference on Decision and Control (pp. 7418–7423).
Hashemi, S. M., Abbas, H. S., & Werner, H. (2012). Low-complexity linear parameter-varying modeling and control of a robotic manipulator. Control Engineering Practice, 20(3), 248–257.
Hoffmann, C., Hashemi, S. M., Abbas, H. S., & Werner, H. (2013). Benchmark problem—nonlinear control of a 3-DOF robotic manipulator. In Proceedings of the 52nd IEEE Conference on Decision and Control (CDC) (pp. 5534–5539).
Hoffmann, C., & Werner, H. (2014a). A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations. IEEE Transactions on Control Systems Technology, 23(2), 416–433.
Hoffmann, C., & Werner, H. (2014b). Complexity of implementation and synthesis in linear parameter-varying control. In Proceedings of the 19th IFAC World Congress (pp. 11749–11760).
Kelly, R., Davila, V. S., & Loria, A. (2005). Control of robot manipulators in joint space. London: Springer.
Kozlowski, K. (2006). Robot motion and control, recent developments. London: Springer.
Ordóñez-Hurtado, R. H., & Duarte-Mermoud, M. A. (2012). Finding common quadratic Lyapunov functions for switched linear systems using particle swarm optimisation. International Journal of Control, 85(1), 12–25.
Ramos, S. D., Domingos, A. C. J., & Vazquez Silva, E. (2014). An algorithm to verify asymptotic stability conditions of a certain family of systems of differential equations. Applied Mathematical Sciences, 8(31), 1509–1520.
Rugh, W. J., & Shamma, J. S. (2000). Research on gain scheduling. Automatica, 36(10), 1401–1425.
Shamma, J. S. (2012). An overview of LPV systems. In J. Mohammadpour & C. W. Scherer (Eds.), Control of linear parameter varying systems with applications (pp. 3–26). New York: Springer.
Spong, M. W., Hutchinson, S., & Vidyasagar, M. (2006). Robot modeling and control. Hoboken: Wiley.
Stilwell, D. J., & Rugh, W. J. (1999). Interpolation of observer state feedback controllers for gain scheduling. IEEE Transactions on Automatic Control, 44(6), 1225–1229.
Stilwell, D. J., & Rugh, W. J. (2000). Stability preserving interpolation methods for the synthesis of gain scheduled controllers. Automatica, 36(5), 665–671.
Sun, Z. (2004). A robust stabilizing law for switched linear systems. International Journal of Control, 77(4), 389–398.
Tong, Y., Zhang, L., Shi, P., & Wang, C. (2013). A common linear copositive Lyapunov function for switched positive linear systems with commutable subsystems. International Journal of Systems Science, 44(11), 1994–2003.
Xiang, W., & Xiao, J. (2013). Finite-time stability and stabilisation for switched linear systems. International Journal of Systems Science, 44(2), 384–400.
Xie, W. (2012). Multi-objective H2/L2 performance controller synthesis for LPV systems. Asian Journal of Control, 14(5), 1273–1281.
Xuping, X., & Antsaklis, P. J. (2000). Stabilization of second-order LTI switched systems. International Journal of Control, 73(14), 1261–1279.
Yanga, Y., Xianga, C., & Leea, T. H. (2012). Sufficient and necessary conditions for the stability of second-order switched linear systems under arbitrary switching. International Journal of Control, 85(12), 1977–1995.
Yu, Z., Chen, H., & Woo, P. (2002). Gain scheduled LPV H-infinity control based on LMI approach for a robotic manipulator. Journal of Robotic Systems, 19(12), 585–593.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abolhasani Jabali, M.B., Kazemi, M.H. A New Polytopic Modeling with Uncertain Vertices and Robust Control of Robot Manipulators. J Control Autom Electr Syst 28, 349–357 (2017). https://doi.org/10.1007/s40313-017-0309-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40313-017-0309-z