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Adaptation in Stochastic Dynamic Systems—Survey and New Results III: Robust LQ Regulator Modification

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The paper is intended to provide algorithmic and computational support for solving the frequently encountered linear-quadratic regulator (LQR) problems based on receding-horizon control methodology which is most applicable for adaptive and predictive control where Riccati iterations rather than solution of Algebraic Riccati Equations are needed. By extending the most efficient computational methods of LQG estimation to the LQR problems, some new algorithms are formulated and rigorously substantiated to prevent Riccati iterations divergence when cycled in computer implementation. Specifically developed for robust LQR implementation are the two-stage Riccati scalarized iteration algorithms belonging to one of three classes: 1) Potter style (square-root), 2) Bierman style (

*LDL*), and 3) Kailath style (array) algorithms. They are based on scalarization, factorization and orthogonalization techniques, which allow more reliable LQR computations. Algorithmic templates offer customization flexibility, together with the utmost brevity, to both users and application programmers, and to ensure the independence of a specific computer language.^{T}Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

I. V. Semushin, "Adaptation in Stochastic Dynamic Systems—Survey and New Results III: Robust LQ Regulator Modification,"

*International Journal of Communications, Network and System Sciences*, Vol. 5 No. 9A, 2012, pp. 609-623. doi: 10.4236/ijcns.2012.529071.

[1] | M. G. Singh, Ed., “Systems and Control Encyclopedia: Theory, Technology, Applications,” Pergamon Press, Inc., Elmsford, 1986. |

[2] | J. E. Gibson, “Nonlinear Automatic Control: Chapter 11,” McGrow Hill, New York, 1962. |

[3] | I. V. Semushin, “Adaptation in Stochastic Dynamic Systems—Survey and New Results I,” International Journal of Communications, Network and System Sciences, Vol. 4, No. 1, 2011, pp. 17-23. doi:10.4236/ijcns.2011.41002 |

[4] | I. V. Semushin, “Adaptation in Stochastic Dynamic Systems—Survey and New Results II,” International Journal of Communications, Network and System Sciences, Vol. 4, No. 4, 2011, pp. 266-285. doi:10.4236/ijcns.2011.44032 |

[5] | I. V. Semushin and S. A. Ponyrko, “On the Choice of Start-Stop Algorithm while Minimizing the Square Mean Performance Index,” Autometria, Siberian Division of the USSR Academy of Sciences, No. 2, 1973, pp. 68-74. |

[6] | T. L. Lai, “Sequential Changepoint Detection in Quality Control and Dynamical Systems,” Journal of the Royal Statistical Society, Series B (Methodological), Vol. 57, No. 4, 1995, pp. 613-658. |

[7] | T. L. Lai and H. P. Xing, “Sequential Change-Point Detection When the Pre- and Post-Change Parameters Are Unknown,” Technical Report No. 2009-5, Department of Statistics, Stanford University, Stanford, 2009. |

[8] | D. G. Lainiotis, “Partitioning: A Unified Framework for Adaptive Systems, I. Estimation,” Proceedings of the IEEE, Vol. 64, No. 8, 1976, pp. 1126-1143. doi:10.1109/PROC.1976.10284 |

[9] | D. G. Lainiotis, “Partitioning: A Unified Framework for Adaptive Systems, II. Control,” Proceedings of the IEEE, Vol. 64, No. 8, 1976, pp. 1182-1197. doi:10.1109/PROC.1976.10289 |

[10] | M. Athans and C.-B. Chang, “Adaptive Estimation and Parameter Identification Using Multiple Model Estimation Algorithm,” Technical Note, Lincoln Lab, MIT, Lexington, 1976. |

[11] | G. C. Goodwin and K. S. Sin, “Adaptive Filtering Prediction and Control,” Prentice Hall, Englewood Cliffs, 1984. |

[12] | H. W. Sorenson, “Kalman Filtering: Theory and Application,” IEEE Press, New York, 1985. |

[13] | J. K. Uhlmann, “Algorithms for Multiple Target Tracking,” American Scientist, Vol. 80, No. 2, 1992, pp. 128- 141. |

[14] | B. Carew and P. Belanger, “Identification of Optimaum Filter Steady-State Gain for Systems with Unknown Noise Covariances,” IEEE Transactions on Automatic Control, Vol. 18, No. 6, 1973, pp. 582-587. doi:10.1109/TAC.1973.1100420 |

[15] | L. Ljung, “System Identification—Theory for the User,” Prentice Hall, Englewood Cliffs, 1987. |

[16] | P. E. Caines, “Linear Stochastic Systems,” John Wiley & Sons, Inc., New York, 1987. |

[17] | H. Kaufman and D. Beaulier, “Adaptive Parameter Identification,” IEEE Transactions on Automatic Control, Vol. 17, No. 5, 1972, pp. 729-731. doi:10.1109/TAC.1972.1100111 |

[18] | T. Yoshimura and T. Soeda, “A Technique for Compensating the Filter Performance by Fictitious Noise,” ASME Journal of Dynanmic Systems, Measurement, and Control, Vol. 100, 1978. |

[19] | H. S. Zhang, D. D. Zhang, W. Wang and L. H. Xie, “Robust Filtering by Fictitious Noises,” Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, 9-12 December 2003, pp. 1280-1284. |

[20] | I. V. Semoushin, “Identifying Parameters of Linear Stochastic Differential Equations from Incomplete Noisy Measurements,” In: Y.-C. Hon, M. Yamamoto, J. Cheng and J.-Y. Lee, Eds., Recent Developments in Theories & Numerics, World Scientific, 2003, pp. 281-290. |

[21] | C. Barrios, H. Himberg, Y. Motai and A. Sadek, “Multiple Model Framework of Adaptive Extended Kalman Filtering for Predicting Vehicle Location,” Proceedings of the 2006 IEEE Intelligent Transportation Systems Conference, Toronto, 17-20 September 2006, pp. 1053-1059. |

[22] | D. Rupp, G. Ducard, E. Shafai and H. P. Geering, “Extended Multiple Model Adaptive Estimation for the Detection of Sensor and Actuator Faults,” Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, 12-15 December 2005, pp. 3079-3084. |

[23] | I. Semoushin, J. Tsyganova and M. Kulikova, “Fault Point Detection with the Bank of Competitive Kalman Filters,” Lecture Notes in Computer Science, Vol. 2658, Part 2, 2003, pp. 417-426. |

[24] | P. Eide and P. Maybeck, “An MMAE Failure Detection System for the F-16,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 32, No. 3, 1996, pp. 1125-1136. doi:10.1109/7.532271 |

[25] | H. S. Witsenhausen, “Separation of Estimation and Control for Discrete Time Systems,” Proceedings of the IEEE, Vol. 59, No. 11, 1971, pp. 1557-1566. doi:10.1109/PROC.1971.8488 |

[26] | E. Mosca, “Optimal, Predictive and Adaptive Control,” Prentice Hall, Englewood Cliffs, 1994. |

[27] | S. E. Dushin, N. S. Zotov, D. Kh. Imaev, N. N. Kuzmin and V. B. Yakovlev, “Theory of Automatic Control,” 3rd Edition, V. B. Yakovlev, Ed., Vysshaya Shkola, Moscow, 2009. |

[28] | A. G. Aleksandrov, “Optimal and Adaptive Systems,” Vysshaya Shkola, Moscow, 1989. |

[29] | P. Lancaster and L. Rodman, “Algebraic Riccati Equations,” Oxford University Press, Inc., New York, 1995. |

[30] | J. C. Willems, “Least Squares Stationary Optimal Control and the Algebraic Riccati Equation,” IEEE Transactions on Automatic Control, Vol. 16, No. 6, 1971, pp. 621-634. doi:10.1109/TAC.1971.1099831 |

[31] | V. Ionescu, C. Oara, M. D. Weiss and M. Weiss, “Generalized Riccati Theory and Robust Control. A Popov Function Approach,” John Wiley & Sons, Inc., New York, 1999. |

[32] | B. C. Kuo, “Digital Control Systems,” Holt, Rinehart and Winston, Inc., New York, 1980. |

[33] | “Riccati Solvers at ScienceDirect: 485 Articles Found,” 2012. http://www.sciencedirect.com/ |

[34] | A. J. Laub, “A Schur Method for Solving Algebraic Riccati Equations,” IEEE Transactions on Automatic Control, Vol. 24, No. 6, 1979, pp. 913-921. doi:10.1109/TAC.1979.1102178 |

[35] | T. Pappas, A. J. Laub and N. R. Sandell Jr., “On the Numerical Solution of the Discrete-time Algebraic Riccati Equation,” IEEE Transactions on Automatic Control, Vol. 25, No. 4, 1980, pp. 631-641. doi:10.1109/TAC.1980.1102434 |

[36] | W. F. Arnold III and A. J. Laub, “Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations,” Proceedings of the IEEE, Vol. 72, No. 12, 1984, pp. 1746-1754. doi:10.1109/PROC.1984.13083 |

[37] | F. A. Aliyev, B. A. Bordyug and B. V. Larin, “Orthogonal Projections and Solution of Algebraic Riccati Equations,” USSR Computational Mathematics and Mathematical Physics, Vol. 29, No. 3, 1989, pp. 104-108. doi:10.1016/0041-5553(89)90154-7 |

[38] | P. G. Kaminski, A. E. Bryson Jr. and S. F. Schmidt, “Discrete Square Root Filtering: A Survey of Current Techniques,” IEEE Transactions on Automatic Control, Vol. 16, No. 6, 1971, pp. 727-736. doi:10.1109/TAC.1971.1099816 |

[39] | D. G. Lainiotis, “Discrete Riccati Equation Solutions: Partitioned Algorithms,” IEEE Transactions on Automatic Control, Vol. 20, No. 4, 1975, pp. 555-556. doi:10.1109/TAC.1975.1101010 |

[40] | G. J. Bierman, “Factorization Methods for Discrete Sequential Estimation,” Academic Press, New York, San Francisco, London, 1977. |

[41] | D. G. Lainiotis, N. D. Assimakis and S. K. Katsikas, “A New Computationally Effective Algorithm for Solving the Discrete Riccati Equation,” Journal of Mathematical Analysis and Applications, Vol. 186, No. 3, 1994, pp. 868-895. doi:10.1006/jmaa.1994.1338 |

[42] | N. D. Assimakis, D. G. Lainiotis, S. K. Katsikas and F. L. Sanida, “A Survey of Recursive Algorithms for the Solution of the Discrete Time Riccati Equation,” Nonlinear Analysis, Theory, Methods & Applications, Vol. 30, No. 4, 1997, pp. 2409-2420. |

[43] | T. Kailath, A. H. Sayed and B. Hassibi, “Linear Estimation,” Prentice Hall, Englewood Cliffs, 2000. |

[44] | N. Assimakis, S. Roulis and D. Lainiotis, “Recursive Solutions of the Discrete Time Riccati Equation,” Neural, Parallel & Scientific Computations, Vol. 11. No. 3, 2003, pp. 343-350. |

[45] | “New Features in Maple 15: Algebraic Riccati Equation Solvers,” 2012. http://www.maplesoft.com/products/maple/newfeatures/algebraicriccati.aspx |

[46] | “Wolfram Mathematica 8: Control System Professional Documentation. 10.3 Riccati Equations,” 2012. http://reference.wolfram.com/legacy/applications/control/OptimalControlSystemsDesign/10.3.html |

[47] | “MATLAB Toolboxes of SLICOT (Subroutine Library in Systems and Control Theory),” 2012. http://www.slicot.org/index.php?site=home |

[48] | “SLICOT Basic Systems and Control Toolbox,” 2012. http://www.slicot.org/index.php?site=slbasic |

[49] | P. Benner and V. Sima, “Solving Algebraic Riccati Equations with SLICOT,” CD-ROM Proceedings of the 11th Mediterranean Conference on Control and Automation MED’03,” Rhodes, 18-20 June 2003, Invited Session IV01, Paper IV01-01. |

[50] | V. Sima, “Computational Experience in Solving Algebraic Riccati Equations,” Proceedings of the 44th IEEE Conference on Decision and Control, and European Control Conference 2005,” Seville, 12-15 December 2005, pp. 7982-7987. |

[51] | V. Sima, “Algorithms for Linear-Quadratic Optimization (Vol. 200 of Series Pure and Applied Mathematics: A series of Monographs and Textbooks),” Chapman and Hall/CRC, London, 1996. |

[52] | V. Armstrong, “Updated Discrete Algebraic Riccati Equation Solver in Python,” 2010. http://jeff.rainbow-100.com/?p=141 |

[53] | W. H. Kwon, Y. S. Moon and S. C. Ahn, “Bounds in Algebraic Riccati and Lyapunov Equations: A Survey and Some New Results,” International Journal of Control, Vol. 64, No. 3, 1996, pp. 377-389. doi:10.1080/00207179608921634 |

[54] | A. Bunse-Gerstner, “Computational Solution of the Algebraic Riccati Equation,” 2012. http://www.math.uni-bremen.de/zetem/numerik/Published/Report982.ps.gz |

[55] | H. Dai and Z.-Z. Bai. “On Eigenvalue Bounds and Iteration Methods for Discrete Algebraic Riccati Equations,” Journal of Computational Mathematics, Vol. 29, No. 3, 2011, pp. 341-366. http://www.jcm.ac.cn/EN/10.4208/jcm.1010-m3258 http://www.jcm.ac.cn/EN/Y2011/V29/I3/341 |

[56] | D. L. Kleinman, “On an Iterative Technique for Riccati Equation Computations,” IEEE Transactions on Automatic Control, Vol. 13, No. 1, 1968, pp. 114-115. doi:10.1109/TAC.1968.1098829 |

[57] | S. J. Hammarling, “Newton’s Method for Solving Algebraic Riccati Equation,” NPL Report, DITC 12/82, 1982. |

[58] | G. Kreisselmeier, “Stabilization of Linear Systems by Constant Output Feedback Using the Riccati Equation,” IEEE Transactions on Automatic Control, Vol. 20, No. 4, 1975, pp. 556-557. doi:10.1109/TAC.1975.1101013 |

[59] | T. Zheng, Ed., “Advanced Model Predictive Control,” InTech, Rijeka, 2011. |

[60] | I. V. Semushin, “Computational Methods of Algebra and Estimation,” UlSTU, Ulyanovsk, 2011. http://venec.ulstu.ru/lib/disk/2012/Semuwin.pdf |

[61] | P. S. Maybeck, “Stochastic Models, Estimation, and Control, Vol. 3,” Academic Press, Inc., New York, London, Paris, San Diego, San Francisco, S?o Paulo, Sydney, Tokyo, Toronto, 1982. |

[62] | P. Comon, “Independent Component Analysis, a New Concept?” Signal Processing, Vol. 36. No. 3, 1994, pp. 287-314, Special Issue on Higher-Order Statistics. |

[63] | J. E. Potter and R. G. Stern, “Statistical Filtering of Space Navigation Measurements,” Proceedings of 1963 AIAA Guidance and Control Conference, New York, 12-14 August 1963, 19 pp. |

[64] | I. V. Semushin, Ed., “Adaptive Systems of Filtering, Control and Fault Detection,” USU, Ulyanovsk, 2011. |

[65] | P. G. Park and T. Kailath, “New Square-Root Algorithms for Kalman Filtering,” IEEE Transactions on Automatic Control, Vol. 40, No. 5, 1995, pp. 895-899. doi:10.1109/9.384225 |

[66] | P. S. Maybeck, “Stochastic Models, Estimation, and Control, Vol. 1,” Academic Press, Inc., New York, San Francisco, London, 1979, Chapter 6. |

[67] | D. Song, J. T. Qi, J. D. Han and G. J. Liu, “Predictive Control for Active Model and its Applications on Unmanned Helicopters,” InTech, Rijeka, 2011. doi:10.5772/17716 |

[68] | A. Fakharian, T. Gustafsson and M. Mehrfam, “Adaptive Kalman Filtering Based Navigation: An IMU/GPS Integration Approach,” Proceedings of the 8th IEEE International Conference on Networking, Sensing and Control, Delft, 11-13 April 2011, pp. 181-185. |

[69] | D. Chu, M. Forbes, J. Backstrom, C. Gheorghe and S. Chu, “Model Predictive Control and Optimization for Papermaking Processes,” InTech, Rijeka, 2011. |

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