Caratheodory-π solution
A Carathéodory-π solution is a generalized solution to an ordinary differential equation. The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory.[1] Its practicality was demonstrated in 2008 by Ross et al.[2] in a laboratory implementation of the concept. The concept is most useful for implementing feedback controls, particularly those generated by an application of Ross' pseudospectral optimal control theory.[3]
Mathematical background
A Carathéodory-π solution addresses the fundamental problem of defining a solution to a differential equation,
- [math]\displaystyle{ \dot x = g(x,t) }[/math]
when g(x,t) is not differentiable with respect to x. Such problems arise quite naturally[4] in defining the meaning of a solution to a controlled differential equation,
- [math]\displaystyle{ \dot x = f(x,u) }[/math]
when the control, u, is given by a feedback law,
- [math]\displaystyle{ u = k(x,t) }[/math]
where the function k(x,t) may be non-smooth with respect to x. Non-smooth feedback controls arise quite often in the study of optimal feedback controls and have been the subject of extensive study going back to the 1960s.[5]
Ross' concept
An ordinary differential equation,
- [math]\displaystyle{ \dot x = g(x,t) }[/math]
is equivalent to a controlled differential equation,
- [math]\displaystyle{ \dot x = u }[/math]
with feedback control, [math]\displaystyle{ u = g(x,t) }[/math]. Then, given an initial value problem, Ross partitions the time interval [math]\displaystyle{ [0, \infty) }[/math] to a grid, [math]\displaystyle{ \pi = \{t_i\}_{i\ge 0} }[/math] with [math]\displaystyle{ t_i \to \infty \text{ as } i \to \infty }[/math]. From [math]\displaystyle{ t_0 }[/math] to [math]\displaystyle{ t_1 }[/math], generate a control trajectory,
- [math]\displaystyle{ u(t) = g(x_0, t), \quad x(t_0) = x_0, \quad t_0 \le t \le t_1 }[/math]
to the controlled differential equation,
- [math]\displaystyle{ \dot x = u(t), \quad x(t_0) = x_0 }[/math]
A Carathéodory solution exists for the above equation because [math]\displaystyle{ t \mapsto u }[/math] has discontinuities at most in t, the independent variable. At [math]\displaystyle{ t = t_1 }[/math], set [math]\displaystyle{ x_1 = x(t_1) }[/math] and restart the system with [math]\displaystyle{ u(t) = g(x_1, t) }[/math],
- [math]\displaystyle{ \dot x(t) = u(t), \quad x(t_1) = x_1, \quad t_1 \le t \le t_2 }[/math]
Continuing in this manner, the Carathéodory segments are stitched together to form a Carathéodory-π solution.
Engineering applications
A Carathéodory-π solution can be applied towards the practical stabilization of a control system.[6][7] It has been used to stabilize an inverted pendulum,[6] control and optimize the motion of robots,[7][8] slew and control the NPSAT1 spacecraft[3] and produce guidance commands for low-thrust space missions.[2]
See also
References
- ↑ Biles, D. C., and Binding, P. A., “On Carathéodory’s Conditions for the Initial Value Problem," Proceedings of the American Mathematical Society, Vol. 125, No. 5, May 1997, pp. 1371–1376.
- ↑ 2.0 2.1 Ross, I. M., Sekhavat, P., Fleming, A. and Gong, Q., "Optimal Feedback Control: Foundations, Examples and Experimental Results for a New Approach," Journal of Guidance, Control and Dynamics, Vol. 31, No. 2, pp. 307–321, 2008.
- ↑ 3.0 3.1 Ross, I. M. and Karpenko, M. "A Review of Pseudospectral Optimal Control: From Theory to Flight," Annual Reviews in Control, Vol.36, No.2, pp. 182–197, 2012.
- ↑ Clarke, F. H., Ledyaev, Y. S., Stern, R. J., and Wolenski, P. R., Nonsmooth Analysis and Control Theory, Springer–Verlag, New York, 1998.
- ↑ Pontryagin, L. S., Boltyanskii, V. G., Gramkrelidze, R. V., and Mishchenko, E. F., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.
- ↑ 6.0 6.1 Ross, I. M., Gong, Q., Fahroo, F. and Kang, W., "Practical Stabilization Through Real-Time Optimal Control," 2006 American Control Conference, Minneapolis, MN, June 14-16 2006.
- ↑ 7.0 7.1 Martin, S. C., Hillier, N. and Corke, P., "Practical Application of Pseudospectral Optimization to Robot Path Planning," Proceedings of the 2010 Australasian Conference on Robotics and Automation, Brisbane, Australia, December 1-3, 2010.
- ↑ Björkenstam, S., Gleeson, D., Bohlin, R. "Energy Efficient and Collision Free Motion of Industrial Robots using Optimal Control," Proceedings of the 9th IEEE International Conference on Automation Science and Engineering (CASE 2013), Madison, Wisconsin, August, 2013
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