Parallel Distributed Compensation Based Stabilization of A 3-DOF RC Helicopter: A Tensor Product Transformation Based Approach
Péter Baranyi*, Péter Korondi**, and Kazuo Tanaka***
*Computer and Automation Research Institute of the Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary
**Department of Automation and Applied Informatics, Budapest University of Technology and Economics, Hungary
***Department of Mechanical Engineering and Intelligent Systems, The University of Electro-Communications, Tokyo 182-8585, Japan
This paper presents a control solution for the stabilization of the 3-DOF RC helicopter via the combination of the TP (Tensor Product) model transformation and the PDC (Parallel Distributed Compensation) control design framework. First we recall the nonlinear model of the RC helicopter and its simplified version, in order, to facilitate control design. Then we execute the TP model transformation on the simplified model to yield its TP model representation, that is a kind of polytopic model with specific characteristics, whereupon the PDC framework can immediately be applied. The control design considers practical control specifications such as: good speed of response and physical constrain on the control effort to avoid actuator saturations. We guarantee these specifications by LMI (Linear Matrix Inequality) conditions developed under the PDC frameworks. Further, we avoid the discrepancies, introduced via the simplification of the model, by applying LMI conditions specialized for robust control. By simultaneously solving these LMI conditions, we render a stabilizing nonlinear controller that achieves good speed of response with small control effort without actuator saturations. Simulation results are included to validate the control design. It will be pointed out that the resulting controller is equivalent with the controller successfully applied in the real control experiment of the helicopter presented in a recent paper.
The main conclusion of this paper is, that the proposed design process is systematic, non-heuristic and straightforward, the stability proof of the resulting controller is tractable via the feasibility test of LMIs and, hence, exact. The whole design procedure is automatically computed via commercialized mathematical tools (MATLAB LMI Toolbox) without analytical interaction. The computational time is about minutes on a regular PC.
-  B. K. Bose, “Neural network applications in power electronics and motor drives–an introduction and perspective,” IEEE Transaction on Industrial Electronics, Vol.54, No.1, pp. 14-33, February 2007.
-  M. Cirrincione, M. Pucci, G. Cirrincione, and G.-A. Capolino, “Sensorless control of induction machines by a new neural algorithm: The TLS EXIN neuron,” IEEE Transaction on Industrial Electronics, Vol.54, No.1, pp. 127-149, February 2007.
-  R. -J. Wai and C. -C. Chu, “Robust petri fuzzy-neural-network control for linear induction motor drive,” IEEE Transaction on Industrial Electronics, Vol.54, No.1, pp. 177-189, February 2007.
-  M. N. Uddin and M. A. Rahman, “High-speed control of ipmsm drives using improved fuzzy logic algorithms,” IEEE Transaction on Industrial Electronics, Vol.54, No.1, pp. 190-199, February 2007.
-  T. Pajchrowski and K. Zawirski, “Application of artificial neural network to robust speed control of servodrive,” IEEE Transaction on Industrial Electronics, Vol.54, No.1, pp. 200-207, February 2007.
-  X. le Wei, J. Wang, and Z. xuan Yang, “Robust smooth-trajectory control of nonlinear servo systems based on neural networks,” IEEE Transaction on Industrial Electronics, Vol.54, No.1, pp. 208-217, February 2007.
-  A. Rubaai, M. Castro-Sitiriche, M. Garuba, and L. Burge, “Implementation of artificial neural network-based tracking controller for high-performance stepper motor drives,” IEEE Transaction on Industrial Electronics, Vol.54, No.1, pp. 218-227, February 2007.
-  P. Baranyi, “ TP model transformation as a way to LMI based controller design,” IEEE Transaction on Industrial Electronics, Vol.51, No.2, pp. 387-400, April 2004.
-  K. Tanaka and H. O. Wang, “Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach,” John Wiley & Sons, Inc., 2001.
-  Special Session, “TP model transformation in non-linear control,” in IEEE International Conference on Fuzzy Systems (FUZZ-IEEE'04), July 25-29 2004, pp. 1063-1085.
-  P. Baranyi, “Tensor-product model-based control of two-dimensional aeroelastic system,” Journal of Guidance, Control, and Dynamics, Vol.29, No.2, pp. 391-400, May-June 2005.
-  P. Baranyi, P. Zoltán, “Output feedback control of two-dimensional aeroelastic system,” Journal of Guidance Control and Dynamics, Vol.29, No.3, pp. 762-767, May-June 2005.
-  K. Tanaka, H. Ohtake, and H. O. Wang, “A practical design approach to stabilization of a 3-DOF RC helicopter,” IEEE Transaction on Control Systems Technology, Vol.12, No.1, pp. 315-325, January 2004.
-  L. D. Lathauwer, B. D. Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM Journal on Matrix Analysis and Applications, Vol.21, No.4, pp. 1253-1278, 2000.
-  D. Tikk, P. Baranyi, R. J. Patton, and J. Tar, “Approximation properties of TP model forms and its consequences to TPDC design framework,” Asian Journal of Control, Vol.9, No.3, pp. 221-231, 2007.
-  P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali, “LMI Control Toolbox User's Guide,” The MathWorks, Inc., 1995.
-  M.-L. Tomescu, S. Preitl, R.-E. Precup, and J. K. Tar, “Stability Analysis Method for Fuzzy Control Systems Dedicated Controlling Nonlinear Processes,” in Acta Polytechnica Hungarica, Vol.4, No.3, pp. 127-141 (ISSN 1785-8860), 2007.
-  C. Arino and A. Sala, “Relaxed LMI conditions for closed-loop fuzzy systems with tensor-product structure,” in Engineering Applications of Artificial Intelligence, Vol.20, pp. 1036-1046, 2007.
-  F. Kolonic, A. Poljugan, and I. Petrovic, “Tensor Product Model Transformation-based Controller Design for Gantry Crane Control System – An Application Approach,” in Acta Polytechnica Hungarica, Vol.3, No.4, pp. 95-112, 2006.
-  O. Pages and A.E. Hajjaji, “Two fuzzy Multiple Reference Model Tracking Control Designs with an Application to Vehicle Lateral Dynamics Control,” European Control Conference on Decision and Control (CDC-ECC '05), pp. 3267-3272, ISBN: 0-7803-9567-0, 12-15 Dec. 2005.
-  G. Hancke and Á. Szeghegyi: “Nonlinear Control via TP Model Transformation: the TORA System Example,” 2nd Slovakian-Hungarian Joint Symposium on Applied Machine Intelligence (SAMI 2004), Herl'any, Slovakia, pp. 333-340, 16-17 January, 2004.
-  R.-E. Precup, S. Preitl, J. Fodor, I.-B. Ursache, P.A. Clep, and S. Kilyeni, “Experimental validation of iterative feedback tuning solutions for inverted pendulum crane mode control,” proc. of Conf. on Human Systems Interaction (HSI 2008), Article no.4581496, pp. 536-541.
-  R.-E. Precup, S- Preitl, “PI-Fuzzy controllers for integral plants to ensure robust stability,” Information Sciences, Vol.177, pp. 4410-4429, 2007.
-  R.-E. Precup, Z. Preitl and E.M. Petriu, “Delta domain design of low-cost fuzzy controlled servosystems,” proc. of IEEE Int. Symp. on Intelligent Signal Processing (WISP 2007), Article no:4447588. pp. 125-130.
-  R.-E. Precup, W. S. Lee, M. Venkata, and Zs. Preitl, “Linear and fuzzy control solutions for tape drives,” in Electrical Engineering (Archiv für Elektrotechnik), Vol.90, No.5, pp. 361-377, (ISSN 0948-7921), May 2008.