# 最优控制

## 通用方法

${\displaystyle J[{\textbf {x}}(\cdot ),{\textbf {u}}(\cdot ),t_{0},t_{f}]:=E\,[\,{\textbf {x}}(t_{0}),t_{0},{\textbf {x}}(t_{f}),t_{f}\,]+\int \limits _{t_{0}}^{t_{f}}F\,[\,{\textbf {x}}(t),{\textbf {u}}(t),t\,]\,\operatorname {d} t}$

${\displaystyle {\dot {\textbf {x}}}(t)={\textbf {f}}\,[\,{\textbf {x}}(t),{\textbf {u}}(t),t\,],}$

${\displaystyle {\textbf {h}}\,[\,{\textbf {x}}(t),{\textbf {u}}(t),t\,]\leq {\textbf {0}},}$

${\displaystyle {\textbf {e}}\,[\,{\textbf {x}}(t_{0}),t_{0},{\textbf {x}}(t_{f}),t_{f}\,]=0}$

## LQ控制器

${\displaystyle J={\tfrac {1}{2}}\mathbf {x} ^{\mathsf {T}}(t_{f})\mathbf {S} _{f}\mathbf {x} (t_{f})+{\tfrac {1}{2}}\int _{t_{0}}\limits ^{t_{f}}[\,\mathbf {x} ^{\mathsf {T}}(t)\mathbf {Q} (t)\mathbf {x} (t)+\mathbf {u} ^{\mathsf {T}}(t)\mathbf {R} (t)\mathbf {u} (t)\,]\,\operatorname {d} t}$

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),}$

${\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}$

${\displaystyle J={\tfrac {1}{2}}\int \limits _{0}^{\infty }[\,\mathbf {x} ^{\mathsf {T}}(t)\mathbf {Q} \mathbf {x} (t)+\mathbf {u} ^{\mathsf {T}}(t)\mathbf {R} \mathbf {u} (t)\,]\,\operatorname {d} t}$

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t),}$

${\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}$

${\displaystyle \mathbf {u} (t)=-\mathbf {K} (t)\mathbf {x} (t)}$

${\displaystyle \mathbf {K} (t)=\mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} (t),}$

${\displaystyle \mathbf {S} (t)}$ 是微分Riccati方程的解，微分Riccati方程如下：

${\displaystyle {\dot {\mathbf {S} }}(t)=-\mathbf {S} (t)\mathbf {A} -\mathbf {A} ^{\mathsf {T}}\mathbf {S} (t)+\mathbf {S} (t)\mathbf {B} \mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} (t)-\mathbf {Q} }$

${\displaystyle \mathbf {S} (t_{f})=\mathbf {S} _{f}}$

${\displaystyle \mathbf {0} =-\mathbf {S} \mathbf {A} -\mathbf {A} ^{\mathsf {T}}\mathbf {S} +\mathbf {S} \mathbf {B} \mathbf {R} ^{-1}\mathbf {B} ^{\mathsf {T}}\mathbf {S} -\mathbf {Q} }$

## 最佳控制的數值方法

${\displaystyle {\begin{array}{lcl}{\dot {\textbf {x}}}&=&\partial H/\partial {\boldsymbol {\lambda }}\\{\dot {\boldsymbol {\lambda }}}&=&-\partial H/\partial {\textbf {x}}\end{array}}}$

${\displaystyle H=F+{\boldsymbol {\lambda }}^{\mathsf {T}}{\textbf {f}}-{\boldsymbol {\mu }}^{\mathsf {T}}{\textbf {h}}}$

${\displaystyle F(\mathbf {z} )\,}$

${\displaystyle {\begin{array}{lcl}\mathbf {g} (\mathbf {z} )&=&\mathbf {0} \\\mathbf {h} (\mathbf {z} )&\leq &\mathbf {0} \end{array}}}$

## 參考資料

1. Ross, Isaac. A primer on Pontryagin's principle in optimal control. San Francisco: Collegiate Publishers. 2015. ISBN 978-0-9843571-0-9. OCLC 625106088.
2. ^ Luenberger, David G. Optimal Control. Introduction to Dynamic Systems. New York: John Wiley & Sons. 1979: 393–435. ISBN 0-471-02594-1. 已忽略未知参数|url-access= (帮助)
3. ^ Kamien, Morton I. Dynamic Optimization : the Calculus of Variations and Optimal Control in Economics and Management.. Dover Publications. 2013. ISBN 978-1-306-39299-0. OCLC 869522905.
4. ^ Ross, I. M.; Proulx, R. J.; Karpenko, M. An Optimal Control Theory for the Traveling Salesman Problem and Its Variants. 2020-05-06. [math.OC].
5. ^ Ross, Isaac M.; Karpenko, Mark; Proulx, Ronald J. A Nonsmooth Calculus for Solving Some Graph-Theoretic Control Problems**This research was sponsored by the U.S. Navy.. IFAC-PapersOnLine. 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016. 2016-01-01, 49 (18): 462–467. ISSN 2405-8963. （英语）.
6. ^ Sargent, R. W. H. Optimal Control. Journal of Computational and Applied Mathematics. 2000, 124 (1–2): 361–371. Bibcode:2000JCoAM.124..361S. .
7. ^ Bryson, A. E. Optimal Control—1950 to 1985. IEEE Control Systems. 1996, 16 (3): 26–33. doi:10.1109/37.506395.
8. ^ Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control. Collegiate Publishers. 2009. ISBN 978-0-9843571-0-9.
9. ^ Kalman, Rudolf. A new approach to linear filtering and prediction problems. Transactions of the ASME, Journal of Basic Engineering, 82:34–45, 1960
10. ^ Oberle, H. J. and Grimm, W., "BNDSCO-A Program for the Numerical Solution of Optimal Control Problems," Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, 1989
11. ^ Ross, I. M.; Karpenko, M. A Review of Pseudospectral Optimal Control: From Theory to Flight. Annual Reviews in Control. 2012, 36 (2): 182–197. doi:10.1016/j.arcontrol.2012.09.002.
12. ^ Betts, J. T. Practical Methods for Optimal Control Using Nonlinear Programming 2nd. Philadelphia, Pennsylvania: SIAM Press. 2010. ISBN 978-0-89871-688-7.
13. ^ Gill, P. E., Murray, W. M., and Saunders, M. A., User's Manual for SNOPT Version 7: Software for Large-Scale Nonlinear Programming, University of California, San Diego Report, 24 April 2007
14. ^ von Stryk, O., User's Guide for DIRCOL (version 2.1): A Direct Collocation Method for the Numerical Solution of Optimal Control Problems, Fachgebiet Simulation und Systemoptimierung (SIM), Technische Universität Darmstadt (2000, Version of November 1999).
15. ^ Betts, J.T. and Huffman, W. P., Sparse Optimal Control Software, SOCS, Boeing Information and Support Services, Seattle, Washington, July 1997
16. ^ Hargraves, C. R.; Paris, S. W. Direct Trajectory Optimization Using Nonlinear Programming and Collocation. Journal of Guidance, Control, and Dynamics. 1987, 10 (4): 338–342. Bibcode:1987JGCD...10..338H. doi:10.2514/3.20223.
17. ^ Gath, P.F., Well, K.H., "Trajectory Optimization Using a Combination of Direct Multiple Shooting and Collocation", AIAA 2001–4047, AIAA Guidance, Navigation, and Control Conference, Montréal, Québec, Canada, 6–9 August 2001
18. ^ Vasile M., Bernelli-Zazzera F., Fornasari N., Masarati P., "Design of Interplanetary and Lunar Missions Combining Low-Thrust and Gravity Assists", Final Report of the ESA/ESOC Study Contract No. 14126/00/D/CS, September 2002
19. ^ Izzo, Dario. "PyGMO and PyKEP: open source tools for massively parallel optimization in astrodynamics (the case of interplanetary trajectory optimization)." Proceed. Fifth International Conf. Astrodynam. Tools and Techniques, ICATT. 2012.
20. ^ RIOTS 互联网档案馆存檔，存档日期16 July 2011., based on Schwartz, Adam. Theory and Implementation of Methods based on Runge–Kutta Integration for Solving Optimal Control Problems (Ph.D.). University of California at Berkeley. 1996. OCLC 35140322.
21. ^ Ross, I. M., Enhancements to the DIDO Optimal Control Toolbox, arXiv 2020. https://arxiv.org/abs/2004.13112
22. ^ Williams, P., User's Guide to DIRECT, Version 2.00, Melbourne, Australia, 2008
23. ^ FALCON.m, described in Rieck, M., Bittner, M., Grüter, B., Diepolder, J., and Piprek, P., FALCON.m - User Guide, Institute of Flight System Dynamics, Technical University of Munich, October 2019
24. ^ GPOPS 互联网档案馆存檔，存档日期24 July 2011., described in Rao, A. V., Benson, D. A., Huntington, G. T., Francolin, C., Darby, C. L., and Patterson, M. A., User's Manual for GPOPS: A MATLAB Package for Dynamic Optimization Using the Gauss Pseudospectral Method, University of Florida Report, August 2008.
25. ^ Rutquist, P. and Edvall, M. M, PROPT – MATLAB Optimal Control Software," 1260 S.E. Bishop Blvd Ste E, Pullman, WA 99163, USA: Tomlab Optimization, Inc.
26. ^ E. Polak, On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems Math. Prog. 62 pp. 385–415 (1993).
27. ^ Ross, I M. A Roadmap for Optimal Control: The Right Way to Commute. Annals of the New York Academy of Sciences. 2005-12-01, 1065 (1): 210–231. Bibcode:2005NYASA1065..210R. ISSN 0077-8923. PMID 16510411. S2CID 7625851. doi:10.1196/annals.1370.015.
28. ^ Fahroo, Fariba; Ross, I. Michael. Convergence of the Costates Does Not Imply Convergence of the Control. Journal of Guidance, Control, and Dynamics. September 2008, 31 (5): 1492–1497. Bibcode:2008JGCD...31.1492F. ISSN 0731-5090. doi:10.2514/1.37331.
29. ^ RIOTS

## 延伸閱讀

• Bertsekas, D. P. Dynamic Programming and Optimal Control. Belmont: Athena. 1995. ISBN 1-886529-11-6.
• Bryson, A. E.; Ho, Y.-C. Applied Optimal Control: Optimization, Estimation and Control Revised. New York: John Wiley and Sons. 1975. ISBN 0-470-11481-9.
• Fleming, W. H.; Rishel, R. W. Deterministic and Stochastic Optimal Control. New York: Springer. 1975. ISBN 0-387-90155-8.
• Kamien, M. I.; Schwartz, N. L. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management Second. New York: Elsevier. 1991. ISBN 0-444-01609-0.
• Kirk, D. E. Optimal Control Theory: An Introduction. Englewood Cliffs: Prentice-Hall. 1970. ISBN 0-13-638098-0.
• Ross, I. M. (2015). A Primer on Pontryagin's Principle in Optimal Control. Collegiate Publishers. ISBN 978-0-9843571-0-9.
• Stengel, R. F. Optimal Control and Estimation. New York: Dover (Courier). 1994. ISBN 0-486-68200-5.