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Adaptive Neural Network Control of Nonlinear Systems

Adaptive Neural Network Control of Nonlinear Systems

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Adaptive Neural Network Control of Nonlinear Systems

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  1. Adaptive Neural Network Control of Nonlinear Systems S. Sam GeDepartment of Electrical & Computer Engineering National University of Singapore Singapore 117576 E-mail: elegesz@nus.edu.sg http://vlab.ee.nus.edu.sg/~sge

  2. Work Place Inaugural IEEE Multi-conference on Systems & Control 16th IEEE Conference on Control Applications (CCA) 22nd IEEE International Symposium on Intelligent Control F. L. Lewis, Sponsorship Chair S. Jagannathan, ISIC Program Chair T. Parisini, Conference Editorial Board Chair

  3. Work Place

  4. Content • Introduction • System Descriptions • Neural Network Approximation • State-Feedback Control for SISO • Output-Feedback Control for SISO • Simulation Study • Conclusion

  5. 1. Introduction • Neural network control has gone through • the pioneering works, • the pains against the skeptics and doubts, and • the graceful acceptance, and maturity • as a powerful tool for control of nonlinear systems. Thanks to the many distinguished individuals and their families: Narendra, Levin, Lewis, Calise, Polycarpou, Hovakimyan, Jagannathan, Slotine, Ge, …

  6. 1 Intelligent Control The most intelligent system in nature! Info. Feedback Decision Making Real-time Control

  7. 1 Adaptive NN Control Adaptation & Learning Control Law Plant Info. feedback Cycling or driving，we never thinking of the so-called mathematical models！

  8. 1 Adaptive NN Control

  9. 1 Adaptive NN Control

  10. 1 Adaptive NN Control

  11. 1 Adaptive NN Control

  12. 1. Adaptive NN Control • System Modeling is usually more difficult than control system design • Model based control though rigorous, it depends too much on model building。 • Before 90s：Off-line NN Training • After 90s：Combining adaptive control, and NN parametrization, on-line adaptive NN control is investigated.

  13. Continuous to Discrete Owing to different analytical tools used, results in continuous time are not necessarily hold in discrete time.

  14. 2 System Descriptions Discrete-time SISO system where

  15. 2.1 Assumptions

  16. 2.2 System Descriptions Discrete-time MIMO system

  17. 2.4 System Properties We have the following observations: • The inputs are in triangular form. • There are both inputs and states interconnections. • The system cannot be expressed as • (k+1)=F((k))+G((k))U(k) • which makes the feedback linearization method • not applicable.

  18. 3 Neural Network Approximation In control engineering, different types of neural networks, including LPNN (RBF, HONN) and MLNN, have been used as function approximators over a compact set. LPNN:

  19. 3 Neural Network Approximator For clarity of analysis, consider HONNs where

  20. 3 Neural Network Approximation • The particular choice of NN is used for analysis only, similar results can be obtained for (extended to) • other linearly parametrized networks, • radial basis function networks, polynomials, splines functions, fuzzy systems, • and, the multiple layer neural networks (Nonlinear). • Different choices affect performance though.

  21. Part I

  22. 4 State-Feedback Control The non-causality is one of the main problems for strict-feedback nonlinear system through backstepping in discrete time. The following issues are in order • Non-causal Problem, • System Transformation, • Desired Control, • Stable Control System Design

  23. 4.1 Non-causal Problem Consider the discrete SISO system given Direct application of backstepping, the following ideal fictitious controls are in order:

  24. 4.1 n-step Ahead Predictor

  25. 4.2 System Transformation Re-examining the system

  26. 4.2 System Transformation

  27. 4.2 System Transformation

  28. 4.2 System Transformation Through the coordinate transformation, we have

  29. 4.2 Desired Virtual Controls The desired (virtual) controls are given by:

  30. 4.4 Adaptive Neural Control The desired controls are functions of unknown functions, thus are not feasible. As such NN control is called upon to construct a feasible controller. Let us consider the fictitious controls and the control as:

  31. 4.4 Adaptive Neural Control The errors are defined as Neural network weight update laws are

  32. 4.5 Stability Analysis

  33. 4.5 Stability Analysis

  34. 4.5 Stability Analysis

  35. 4.5 Stability Analysis

  36. 4.5 Stability Analysis

  37. 4.5 Stability Analysis

  38. Before Part II

  39. After Part II

  40. 5 Output-Feedback Control For output feedback control, the strict-feedback form is transformed into a cascade form. For equivalent transformation of coordinates, it is important to ensure that the transformation map is diffeomorphism. The following issues will be highlighted: • Coordinate Transformation • Diffeomorphism • Cascade Form • Control Design

  41. 5.1 Coordinate Transformation

  42. 5.1 Coordinate Transformation

  43. 5.2 Diffeomorphism

  44. 5.2 Diffeomorphism

  45. 5.2 Diffeomorphism

  46. 5.2 Diffeomorphism

  47. 5.3 Cascade Form

  48. 5.3 Cascade Form

  49. 5.3 Cascade Form

  50. 5.3 Cascade Form

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