Humberto Gonzalez
ABSTRACT
One of the oldest problems in the study of dynamical systems is the calculation of an optimal control. Though the determination of a numerical solution for the general non-convex optimal control problem for hybrid systems has been pursued relentlessly to date, it has proven difficult, since it demands nominal mode scheduling. In this paper, we calculate a numerical solution to the optimal control problem for a constrained switched nonlinear dynamical system with a running and final cost. The control parameter has a discrete component, the sequence of modes, and two continuous components, the duration of each mode and the continuous input while in each mode. To overcome the complexity posed by the discrete optimization problem, we propose a bi-level hierarchical optimization algorithm: at the higher level, the algorithm updates the mode sequence by using a single-mode variation technique, and at the lower level, the algorithm considers a fixed mode sequence and minimizes the cost functional over the continuous components. Numerical examples detail the potential of our proposed methodology.
- H. Axelsson, Y. Wardi, M. Egerstedt, and E. Verriest. Gradient Descent Approach to Optimal Mode Scheduling in Hybrid Dynamical Systems. Journal of Optimization Theory and Applications, 136(2):167--186, 2008.Google Scholar
Digital Library
- A. Bemporad, F. Borrelli, and M. Morari. Piecewise Linear Optimal Controllers for Hybrid Systems. In American Control Conference, 2000. Proceedings of the 2000, volume 2, 2000.Google Scholar
- D. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific Belmont, MA, 1995. Google ScholarDigital Library
- F. Borrelli, M. Baotić, A. Bemporad, and M. Morari. Dynamic Programming for Constrained Optimal Control of Discrete-Time Linear Hybrid Systems. Automatica, pages 1709--1721, June 2005. Google ScholarDigital Library
- M. Branicky, V. Borkar, and S. Mitter. A Unified Framework for Hybrid Control: Model and Optimal Control theory. IEEE Transactions on Automatic Control, 43(1):31--45, 1998.Google ScholarCross Ref
- R. Brockett. Stabilization of Motor Networks. In Decision and Control, 1995., Proceedings of the 34th IEEE Conference on, volume 2, 1995.Google ScholarCross Ref
- M. Egerstedt and Y. Wardi. Multi-Process Control Using Queuing Theory. In IEEE Conference on Decision and Control, volume 2, pages 1991--1996.IEEE; 1998, 2002.Google Scholar
- M. Egerstedt, Y. Wardi, and H. Axelsson. Transition-Time Optimization for Switched-Mode Dynamical Systems. IEEE Transactions on Automatic Control, 51(1):110--115, 2006.Google ScholarCross Ref
- M. Egerstedt, Y. Wardi, and F. Delmotte. Optimal Control of Switching Times in Switched Dynamical Systems. In Proceedings 42nd IEEE Conference on Decision and Control, pages 2138--2143, Dec. 2003.Google ScholarCross Ref
- R. Ghosh and C. Tomlin. Hybrid System Models of Biological Cell Network Signaling and Differentiation. In Proceedings of the Annual Allerton Conference on Communication Control and Computing, volume 39, pages 942--951, 2001.Google Scholar
- A. Giua, C. Seatzu, and C. Van Der Mee. Optimal Control of Switched Autonomous Linear systems. In IEEE Conference on Decision and Control, volume 3, pages 2472--2477. IEEE; 1998, 2001.Google Scholar
- H. Gonzalez, R. Vasudevan, M. Kamgarpour, S. S. Sastry, R. Bajcsy, and C. Tomlin. A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems: Appendix. Technical Report UCB/EECS-2010-9, EECS Department, University of California, Berkeley, Jan 2010.Google ScholarDigital Library
- S. Hedlund and A. Rantzer. Optimal Control of Hybrid Systems. In Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, Arizona, Dec. 1999.Google ScholarCross Ref
- G. Hoffmann, H. Huang, S. Waslander, and C. Tomlin. Quadrotor Helicopter Flight Dynamics and Control: Theory and Experiment. In Proc. of the AIAA Guidance, Navigation, and Control Conference. Citeseer, 2007.Google ScholarCross Ref
- B. Lincoln and A. Rantzer. Optimizing Linear System Switching. In IEEE Conference on Decision and Control, volume 3, pages 2063--2068. IEEE; 1998, 2001.Google Scholar
- D. Mayne and E. Polak. First-Order Strong Variation Algorithms for Optimal Control. Journal of Optimization Theory and Applications, 16(3):277--301, 1975.Google ScholarDigital Library
- E. Polak. Optimization: Algorithms and Consistent Approximations. Springer, 1997. Google ScholarDigital Library
- L. Pontryagin, V. Boltyanskii, R. Gamkrelidze, and E. Mishchenko. The Mathematical Theory of Optimal Processes. Interscience New York, 1962.Google Scholar
- A. Rantzer and M. Johansson. Piecewise Linear Quadratic Optimal Control. IEEE Transactions on Automatic Control, 45(4):629--637, 2000.Google ScholarCross Ref
- H. Rehbinder and M. Sanfridson. Scheduling of a Limited Communication Channel for Optimal Control. Automatica, 40(3):491--500, 2004. Google ScholarDigital Library
- M. Rinehart, M. Dahleh, D. Reed, and I. Kolmanovsky. Suboptimal Control of Switched Systems with an Application to the Disc Engine. IEEE Transactions on Control Systems Technology, 16(2):189--201, 2008.Google ScholarCross Ref
- M. Shaikh and P. Caines. On the Optimal Control of Hybrid Systems: Optimization of Trajectories, Switching Times, and Location Schedules. Lecture Notes in Computer Science, 2623:466--481, 2003. Google ScholarDigital Library
- R. Vinter. Optimal control, Systems & Control: Foundations & Applications, 2000.Google Scholar
- G. Walsh, H. Ye, L. Bushnell, et al. Stability Analysis of Networked Control Systems. IEEE Transactions on Control Systems Technology, 10(3):438--446, 2002.Google ScholarCross Ref
- X. Xu and P. Antsaklis. Optimal Control of Switched Systems via Nonlinear Optimization Based on Direct Differentiation of Value Functions. International Journal of Control, 75, 16(17):1406--1426, 2002.Google ScholarCross Ref
- X. Xu and P. Antsaklis. Results and Perspective on Computational Methods for Optimal Control of Switched Systems. Lecture Notes in Computer Science, 2623:540--556, 2003. Google ScholarDigital Library
- W. Zhang, A. Abate, and J. Hu. Stabilization of Discrete-Time Switched Linear Systems: A Control--Lyapunov Function Approach. In R. Majumdar and P. Tabuada, editors, Hybrid Systems: Computation and Control, volume 5469 of Lecture Notes in Computer Science, pages 411--425. Springer, 2009. Google ScholarDigital Library
Index Terms
- A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems
Recommendations
Optimal control of switched systems and its parallel optimization algorithm
In this paper, a class of optimal switching control problems is considered in which the mode sequence of active subsystems and the number of mode switchings are not pre-specified, and both the switching sequence and the control inputs are to be chosen ...
Minimizing control variation in nonlinear optimal control
In any real system, changing the control signal from one value to another will usually cause wear and tear on the system's actuators. Thus, when designing a control law, it is important to consider not just predicted system performance, but also the ...
Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems
A pseudospectral method for generating optimal trajectories of linear and nonlinear constrained dynamic systems is proposed. The method consists of representing the solution of the optimal control problem by an mth degree interpolating polynomial, using ...
Comments