## Max-Plus Methods for Nonlinear Control and EstimationThe central focus of this book is the control of continuous-time/continuous-space nonlinear systems. Using new techniques that employ the max-plus algebra, the author addresses several classes of nonlinear control problems, including nonlinear optimal control problems and nonlinear robust/H-infinity control and estimation problems. Several numerical techniques are employed, including a max-plus eigenvector approach and an approach that avoids the curse-of-dimensionality. Well-known dynamic programming arguments show there is a direct relationship between the solution of a control problem and the solution of a corresponding Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE). The max-plus-based methods examined in this monograph belong to an entirely new class of numerical methods for the solution of nonlinear control problems and their associated HJB PDEs; they are not equivalent to either of the more commonly used finite element or characteristic approaches. The potential advantages of the max-plus-based approaches lie in the fact that solution operators for nonlinear HJB problems are linear over the max-plus algebra, and this linearity is exploited in the construction of algorithms. The book will be of interest to applied mathematicians, engineers, and graduate students interested in the control of nonlinear systems through the implementation of recently developed numerical methods. Researchers and practitioners tangentially interested in this area will also find a readable, concise discussion of the subject through a careful selection of specific chapters and sections. Basic knowledge of control theory for systems with dynamics governed by differential equations is required. |

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### Contents

Preface | vii |

Introduction | 1 |

11 Some Control and Estimation Problems | 3 |

12 Concepts of MaxPlus Methods | 6 |

MaxPlus Analysis | 11 |

21 Spaces of Semiconvex Functions | 13 |

22 Bases | 15 |

23 TwoParameter Families | 21 |

61 Constituent Problems | 130 |

62 Operating on the Transformed Operators | 133 |

63 The HJB PDE Limit Problems | 134 |

64 A Simple Example | 138 |

CurseofDimensionalityFree Method | 143 |

71 DP for the Constituent and Originating Problems | 146 |

72 MaxPlus Spaces and Dual Operators | 150 |

73 Discrete Time Approximation | 158 |

24 Dual Spaces and Reflexivity | 22 |

Dynamic Programming and Viscosity Solutions | 31 |

31 Dynamic Programming Principle | 32 |

32 Viscosity Solutions | 42 |

MaxPlus Eigenvector Method for the Infinite TimeHorizon Problem | 57 |

41 Existence and Uniqueness | 58 |

42 MaxPlus Linearity of the Semigroup | 60 |

43 Semiconvexity and a MaxPlus Basis | 66 |

44 The Eigenvector Equation | 70 |

45 The Power Method | 72 |

Initial Notes | 83 |

47 Outline of Algorithm | 84 |

481 A Game Problem | 93 |

49 An Example | 95 |

MaxPlus Eigenvector Method Error Analysis | 97 |

51 Allowable Errors in Computation of B | 98 |

52 Convergence and Truncation Errors | 107 |

521 Convergence | 108 |

522 Truncation Error Estimate | 110 |

53 Errors in the Approximation of B | 119 |

531 A Method for Computing B | 122 |

54 Error Summary | 124 |

55 Example of Convergence Rate | 126 |

A Semigroup Construction Method | 129 |

74 The Algorithm | 164 |

75 Practical Issues | 170 |

752 Initialization | 171 |

77 More General Quadratic Constituents | 175 |

78 Future Directions | 180 |

Finite TimeHorizon Application Nonlinear Filtering | 183 |

81 Semiconvexity | 187 |

82 MaxPlus Propagation | 192 |

Mixed jLooL2 Criteria | 197 |

92 Dynamic Programming | 200 |

922 Dynamic Programming Equations | 204 |

93 MaxPlus Representations and Semiconvexity | 205 |

94 MaxPlus Numerical Methods | 209 |

941 Nonuniqueness for the MaxPlus Affine Equation | 211 |

942 The Affine Power Method | 212 |

Miscellaneous Proofs | 217 |

A02 Proof of Theorem 313 | 218 |

A03 Proof of Lemma 315 | 220 |

A04 Sketch of Proof of Theorem 727 | 222 |

A05 Sketch of Proof of Lemma 731 | 224 |

A06 Existence of RobustHoo Estimator and a Disturbance Bound | 228 |

233 | |

239 | |

### Other editions - View all

Max-Plus Methods for Nonlinear Control and Estimation William M. McEneaney No preview available - 2011 |

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### References to this book

Controlled Markov Processes and Viscosity Solutions Wendell H. Fleming,Halil Mete Soner No preview available - 2006 |