Lu LIU and Jie HUANG Department of Mechanics & Automation Engineering

1. Applied Control and Computing Laboratory Global Robust Output Regulation of Lower Triangular Systems with Unknown High-Frequency Gain Sign Lu LIU and Jie HUANG Department of Mechanics & Automation Engineering The Chinese University of Hong Kong 9 December, 2006 2006 Systems Workshop on Autonomous Networks

2. Outline • Introduction • Problem Formulation • Main Result • An Example • Conclusion Applied Control and Computing Laboratory

3. 1. Introduction Applied Control and Computing Laboratory

4. Background • The global robust output regulationproblem for nonlinear systems in lower triangular form is considered with various solvability conditions. • A basic assumption is that the sign of the high-frequency gain, i.e., the control direction, is known. • The knowledge of the high-frequency gain sign makes control design much more tractable. Applied Control and Computing Laboratory

5. Objective • Solve the global robust output regulation problem for nonlinear systems in lower triangular form without knowing the high-frequency gain sign. • Remark: Nonlinear systems in lower triangular form is an important class of nonlinear systems, and many systems can be converted into lower triangular form by coordinate transformation. Applied Control and Computing Laboratory

6. Related Work • When the high-frequency gain sign is known: • The global robust output regulation problem (GRORP) for nonlinear systems in lower triangular form has been solved by using the robust control method. • When the high-frequency gain sign is unknown: • The same problem has been rarely considered in the existing literature. Applied Control and Computing Laboratory

7. Proposed Approach • Approach: • Integrate the robust control approach and the adaptive control approach to develop a Lyapunov direct method. Applied Control and Computing Laboratory

8. 2. Problem Formulation Applied Control and Computing Laboratory

9. Problem Formulation • Nonlinear systems in lower triangular form: • Remark: The high-frequency gain sign, i.e., the sign of , is unknown. Applied Control and Computing Laboratory

10. Problem Formulation • Global Robust Output Regulation Problem (GRORP): Design a control law such that, for all bounded exogenous signal and any uncertain parameter , the trajectories of the closed-loop system starting from all initial states are bounded, and the tracking error e converges to zero asymptotically. Applied Control and Computing Laboratory

11. Remark • Output regulation problem is more challenging than stabilization and the conventional tracking and disturbance rejection problem. • Requires more than stabilization. • The class of reference signals and disturbances are generated by some autonomous differential equation. Applied Control and Computing Laboratory

12. Remark • A general framework has been established to convert the output regulation problem for a nonlinear system into a stabilization problem for an appropriately augmented system (Huang and Chen, 2004). • The GRORP for the original system can be converted into a GRSP for an augmented system composed of the original plant and the internal model • The solvability of the GRSP for the augmented system implies the solvability of the GRORP for the original system. Applied Control and Computing Laboratory

13. Solvability of the Problem • Using the existing framework, the GRORP for the original plant can be converted into the GRSP for the augmented system: • Remark: • The augmented system is not in the lower triangular form as system (1). Some standard assumptions are needed to solve the stabilization problem for the augmented system. Applied Control and Computing Laboratory

14. Standard Assumptions Applied Control and Computing Laboratory

15. 3. Main Result Applied Control and Computing Laboratory

16. Control Strategy • When the sign of is known, the existing result gives: • When the sign of is unknown, we propose: Applied Control and Computing Laboratory

17. Remark • Apply change supply rate technique to handle the dynamic uncertainty, and Nussbaum gain technique for the unknown high-frequency gain sign. • is introduced to estimate the unknown control coefficient b(w). • N(k) is a type of dynamically generated gain which oscillates to ensure that both positive and negative control directions are tried (Nussbaum, 1983). Applied Control and Computing Laboratory

18. Main Theorem • Theorem: Applied Control and Computing Laboratory

19. Idea of the Proof • Use a recursive approach to design virtual control • Define Applied Control and Computing Laboratory

20. Idea of the Proof Applied Control and Computing Laboratory

21. Outline of the Proof • By appropriately selecting the design functions, we obtain: • then applying a lemma by Ye and Jiang and Barbalat’s lemma gives, Applied Control and Computing Laboratory

22. 4. An Example Applied Control and Computing Laboratory

23. An Example • Plant: • Exosystem: Applied Control and Computing Laboratory

24. Controller: Applied Control and Computing Laboratory

25. Simulation Applied Control and Computing Laboratory

26. Simulation Applied Control and Computing Laboratory

27. Simulation Applied Control and Computing Laboratory

28. 5. Conclusion Applied Control and Computing Laboratory

29. Conclusion • Solved the GRORP for nonlinear systems in lower triangular form without knowing the high-frequency gain sign. • Obtained the control law by integrating the robust control method and the adaptive control method. Applied Control and Computing Laboratory

30. Thank you! Applied Control and Computing Laboratory