## Nonlinear H2/H-Infinity Constrained Feedback Control: A Practical Design Approach Using Neural Networks (Google eBook)The authors present algorithms for H2 and H-infinity design for nonlinear systems which provide solution techniques which can be implemented in real systems; neural networks are used to solve the nonlinear control design equations. Constraints on the control actuator inputs are dealt with. Results are proven to give confidence and performance guarantees. The algorithms can be used to obtain practical controllers. Nearly optimal applications to constrained-state and mimimum-time problems are discussed as is discrete-time design for digital controllers. 'Nonlinear H2/H-infinity Constrained Feedback Control' is of importance to control designers working in a variety of industrial systems. Case studies are given and the design of nonlinear control systems of the same caliber as those obtained in recent years using linear optimal and bounded-norm designs is explained. The book will also be of interest to academics and graduate students in control systems as a complete foundation for H2 and H-infinity design. |

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### Contents

Preliminaries and Introduction | 1 |

112 Discretetime Nonlinear Systems | 2 |

12 Stability of Nonlinear Systems | 3 |

121 Lyapunov Stabiltity of Continuoustime Nonlinear Systems | 4 |

13 Dissipativity of Nonlinear Systems | 8 |

131 Dissipativity of Continuoustime Nonlinear Systems | 9 |

132 Dissipativity of Discretetime Nonlinear Systems | 12 |

14 Optimal Control of Nonlinear Systems | 14 |

36 Policy Iterations Without Solving the LEVu | 75 |

37 Bibliographical Notes | 76 |

Policy Iterations and Nonlinear H Constrained State Feedback Control | 77 |

42 Policy Iterations and the Nonlinear Bounded Real Lemma | 78 |

43 L2gain of Nonlinear Control Systems with Input Saturation | 83 |

44 The HJI Equation and the Saddle Point | 86 |

45 Solving the HJI Equation Using Policy Iterations | 90 |

Nearly H Optimal Neural Network Control for ConstrainedInput Systems | 95 |

142 Discretetime HJB Equation | 17 |

15 Policy Iterations and Optimal Control | 18 |

151 Policy Iterations and H2 Optimal Control | 19 |

152 Policy Iterations and the Bounded Real Lemma | 21 |

16 Zerosum Games of Nonlinear Systems | 23 |

162 Linear Quadratic Zerosum Games and H0 Optimal Control | 25 |

163 Discretetime HJI Equation | 26 |

17 Neural Networks and Function Approximation | 28 |

172 Function Approximation Theorems | 30 |

18 Bibliographical Notes | 31 |

Policy Iterations and Nonlinear H2 Constrained State Feedback Control | 32 |

22 Optimal Regulation of Systems with Actuator Saturation | 34 |

23 Policy Iterations for ConstrainedInput Systems | 37 |

24 Nonquadratic Performance Functionals for Minimumtime and Constrained States Control | 41 |

25 Bibliographical Notes | 42 |

Nearly H2 Optimal Neural Network Control for ConstrainedInput Systems | 43 |

32 Convergence of the Method of Least Squares to the Solution of the LEVu | 45 |

33 Convergence of the Method of Least Squares to the Solution of the HJB Equation | 52 |

Introducing a Mesh in Rn | 54 |

35 Numerical Examples | 56 |

352 Nonlinear Oscillator with Constrained Input | 62 |

353 Constrained State Linear System | 65 |

354 Minimumtime Control | 68 |

355 Parabolic Tracker | 71 |

51 Neural Network Representation of Policies | 96 |

52 Stability and Convergence of Least Squares Neural Network Policy Iterations | 100 |

The Nonlinear Benchmark Problem | 104 |

54 Bibliographical Notes | 113 |

Taylor Series Approach to Solving HJI Equation | 115 |

62 Power Series Solution of HJI Equation | 118 |

63 Explicit Expression for Hk | 126 |

64 The Disturbance Attenuation of RTAC System | 135 |

65 Bibliographical Notes | 146 |

An Algorithm to Solve Discrete HJI Equations Arising from Discrete Nonlinear H Control Problems | 147 |

72 Taylor Series Solution of Discrete HamiltonJacobiIsaacs Equation | 151 |

73 Disturbance Attenuation of Discretized RTAC System | 164 |

74 Computer Simulation | 172 |

H Static Output Feedback | 176 |

82 Intermediate Mathematical Analysis | 178 |

83 Coupled HJ Equations for H Static Output Feedback Control | 182 |

84 Existence of Static Output Feedback Game Theoretic Solution | 185 |

85 Iterative Solution Algorithm | 187 |

86 H Static Output Feedback Design for F16 Normal Acceleration Regulator | 188 |

87 Bibliographical Notes | 192 |

193 | |

201 | |

### Common terms and phrases

2L gain activation functions actuator algebraic Riccati equation algorithm approximate assumption asymptotically stable available storage Bounded Real Lemma chapter closed-loop system constrained constrained-input continuous-time control input control law convergence derived discrete discrete-time disturbance attenuation dynamical system exists feedback control Figure H control Hamilton–Jacobi equation Hamilton–Jacobi–Isaacs Hamiltonian HJI equation initial condition initial stabilizing control L2-gain least squares linear systems linear time-invariant systems Lyapunov equation Lyapunov function matrix method minimum-time Nearly Optimal Controller neural network nonlinear control nonlinear H nonlinear systems nonquadratic performance functional Note optimal control problem policy iterations positive definite Proof quadratic saddle point saturated control Section shown Sobolev space static output feedback Taylor series Taylor series solution techniques Theorem third-order controller trajectories TT TT uniform convergence uniformly value function vector zero-sum game