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2013, 3(1): 127-143. doi: 10.3934/naco.2013.3.127

## Sensitivity based trajectory following control damping methods

 1 McCoy School of Engineering, Midwestern State University, 3410 Taft Blvd., Wichita Falls, TX 76308, United States

Received  December 2011 Revised  November 2012 Published  January 2013

Synthesis of trajectory following optimization methods with Lyapunov optimizing control techniques create continuous controls suitable for systems with switching surfaces. Fundamental contributions include the optimization structure from which the control law is derived and mitigation of undesirable system behavior. It is assumed the analyst is concerned with minimization of control effort and a positive definite function of the state. Disturbance rejection and stability are primary control objectives. Given these priorities, a cost functional with optimal control inspired structure due to the use of minimum cost descent control is constructed. It contains an analyst defined state dependent cost function. Another term is included which is a function of the control effort modified appropriately to reflect that sufficient control is needed to drive the state to a switching surface. This derivation establishes optimization rationale for disturbance rejection and switching surface placement. The control effort term provides for mitigation of finite time interval switching control; a continuous time analog to discontinuous chatter. A different perspective on chatter elimination via sensitivity analysis is provided and inspires the final control damping" algorithm.
Citation: Dale McDonald. Sensitivity based trajectory following control damping methods. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 127-143. doi: 10.3934/naco.2013.3.127
##### References:
 [1] M. Ahmad and J. Osman, Robust sliding mode control for robot manipulator tracking problem using a proportional-integral switching surface, Proc. of the Student Conference on Research and Development, Putrajaya, Malaysia, (2003), 29-35. [2] M. Chen, Y. Hwang and M. Tomizuka, A state dependent boundary layer design for sliding mode control, IEEE Trans. Aut. Cont., 47 (2002), 1677-1681. doi: 10.1109/TAC.2002.803534. [3] M. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Aut., ac-26 (1981), 1139-1134. [4] R. Figliola and D. Beasely, "Theory and Design for Mechanical Measurements," 4th Edition, Wiley, New York, 2006. [5] B. Goh, Algorithms for unconstrained optimization via control theory , Journal of Optimization Theory and Applications, 92 (1997), 581-604. doi: 10.1023/A:1022607507153. [6] W. Grantham and T. Vincent, "Modern Control Systems Analysis and Design," Wiley, New York, 1993. [7] W. Grantham, Trajectory following optimization by gradient transformation differential equations, Proc. 42nd IEEE Conf. on Decision and Control, Maui, HI, Dec. 9-12, 003. [8] W. Grantham, Some necessary conditions for steepest descent controllability, Proceedings of the 1st American Controls Conference, Alexandria, VA, 1982. [9] D. McDonald and W. Grantham, Singular perturbation trajectory following methods for min-max differential games, in "Advances in Dynamic Game Theory and Applications" (eds. S. Jorgensen, T. Vincent, and M. Quincampoix), Birkhauser, Boston, 2006, 659-678. [10] T. Vincent and W. Grantham, Trajectory following methods in control system design, Journal of Global Optimization, 23 (2002), 267-282. doi: 10.1023/A:1016530713343. [11] T. Vincent, B. Goh and K. Teo, Trajectory-following algorithms for min-max optimization problems, Journal of Optimization Theory and Application, 75 (1992), 501-519. doi: 10.1007/BF00940489. [12] T. Vincent and W. Grantham, "Nonlinear and Optimal Control Systems," Wiley, New York, 1997.

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##### References:
 [1] M. Ahmad and J. Osman, Robust sliding mode control for robot manipulator tracking problem using a proportional-integral switching surface, Proc. of the Student Conference on Research and Development, Putrajaya, Malaysia, (2003), 29-35. [2] M. Chen, Y. Hwang and M. Tomizuka, A state dependent boundary layer design for sliding mode control, IEEE Trans. Aut. Cont., 47 (2002), 1677-1681. doi: 10.1109/TAC.2002.803534. [3] M. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Aut., ac-26 (1981), 1139-1134. [4] R. Figliola and D. Beasely, "Theory and Design for Mechanical Measurements," 4th Edition, Wiley, New York, 2006. [5] B. Goh, Algorithms for unconstrained optimization via control theory , Journal of Optimization Theory and Applications, 92 (1997), 581-604. doi: 10.1023/A:1022607507153. [6] W. Grantham and T. Vincent, "Modern Control Systems Analysis and Design," Wiley, New York, 1993. [7] W. Grantham, Trajectory following optimization by gradient transformation differential equations, Proc. 42nd IEEE Conf. on Decision and Control, Maui, HI, Dec. 9-12, 003. [8] W. Grantham, Some necessary conditions for steepest descent controllability, Proceedings of the 1st American Controls Conference, Alexandria, VA, 1982. [9] D. McDonald and W. Grantham, Singular perturbation trajectory following methods for min-max differential games, in "Advances in Dynamic Game Theory and Applications" (eds. S. Jorgensen, T. Vincent, and M. Quincampoix), Birkhauser, Boston, 2006, 659-678. [10] T. Vincent and W. Grantham, Trajectory following methods in control system design, Journal of Global Optimization, 23 (2002), 267-282. doi: 10.1023/A:1016530713343. [11] T. Vincent, B. Goh and K. Teo, Trajectory-following algorithms for min-max optimization problems, Journal of Optimization Theory and Application, 75 (1992), 501-519. doi: 10.1007/BF00940489. [12] T. Vincent and W. Grantham, "Nonlinear and Optimal Control Systems," Wiley, New York, 1997.
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