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Chapter 8

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  1. Chapter 8 Nonlinear Programming with Constraints Chapter 8

  2. Chapter 8

  3. Chapter 8

  4. Methods for Solving NLP Problems Chapter 8

  5. ; see Fig. E 8.1a Chapter 8

  6. Chapter 8

  7. Chapter 8

  8. Chapter 8

  9. Chapter 8

  10. Chapter 8

  11. Chapter 8

  12. Chapter 8 Note that there are n + m equations in the n + m unknowns x and λ

  13. Chapter 8

  14. Chapter 8

  15. Chapter 8

  16. Chapter 8

  17. By the Lagrange multiplier method. Solution: The Lagrange function is Chapter 8 The necessary conditions for a stationary point are

  18. Chapter 8

  19. Chapter 8

  20. Chapter 8

  21. Penalty functions for handling equality constraints Chapter 8

  22. Chapter 8

  23. for handling inequality constraints Chapter 8 Note g must be >0 ; r 0

  24. Chapter 8 The logarithmic barrier function formulation for m constraints is

  25. Chapter 8

  26. Chapter 8

  27. Chapter 8

  28. Chapter 8

  29. Use xc = 2 yc = 2 for linearization Chapter 8 (step bounds)

  30. Chapter 8

  31. Chapter 8

  32. Chapter 8

  33. Quadratic Programming (QP) Chapter 8

  34. 8.3 QUADRATIC PROGRAMMING Chapter 8

  35. Use of Quadratic Programming to Design Multivariable Controllers(Model Predictive Control) • Targets (set points) selected by real-time optimization software based on current operating and economic conditions • Minimize square of deviations between predicted future outputs and specific reference trajectory to new targets using QP • Framework handles multiple input, multiple output (MIMO) control problems with constraints on manipulated and controlled variables. Dynamics obtained from transfer function model. Chapter 8

  36. Successive Quadratic Programming • Considered by some to be the best general nonlinear programming algorithm • Repetitively approximates nonlinear objective function with quadratic function and nonlinear constraints with linear constraints • Uses line search rather than QP step for each iteration • Inequality constrained Quadratic Programming (IQP) keeps all inequality constraints • Equality constrained Quadratic Programming (EQP) only keeps equality constraints by utilizing and active set strategy • SQP is an Infeasible Path method Chapter 8

  37. Chapter 8

  38. Chapter 8 solve for

  39. Generalized Reduced Gradient (GRG) Chapter 8

  40. Chapter 8

  41. Chapter 8

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  50. Chapter 8