Methods of finite-time control and observation
0372-4007-01
Prof. Arie Levant

Preliminary detailed syllabus:

1. Control theory and dynamic systems. Stability notions and Lyapunov theory overview. Controllability and observability. Normal form of controllable linear autonomous system. Ackermann's formula.

2. Lie derivative. Feedback linearization notion. Relative degree of non-linear control system. Scalar and vector relative degrees. Zero dynamics. Minimum-phase feature. Backstepping. Uncertainty conditions and robustness problem.

3. Sliding modes (SMs) motivation. Differential equations with discontinuous right-hand sides: the Filippov theory. Filippov differential inclusions (FDIs). Theorems on solution existence, continuous dependence on the right-hand side and initial conditions. 

4. SM control introduction:  Equivalent control method. Invariance to disturbances  and matching conditions. Chattering effect and sliding mode regularization. Higher-order sliding mode: sliding order and sliding accuracy. First-order sliding modes (1-SMs). Single-input single-output (SISO) and multi-input multi-output (MIMO) cases. MIMO unit 1-SM control.  

5. The Anosov theorem. Asymptotically stable 2-SM. 2-SM: Twisting, super-twisting and terminal controllers. 1st-order differentiator. 

6. Homogeneity approach to control design. System homogeneity degree. Homogeneous Filippov differential inclusions (FDIs). Existence of homogeneous Lyapunov functions. Infinite-time, finite-time (FT) and fixed-time convergence. Continuity properties of the convergence time functions.
 
7. Sampled and delayed homogeneous FDIs. Accuracy in the presence of noises and disturbances. Computer simulation methods: implicit and explicit Euler schemes. Their comparison. Regions of convergence and FT attractors.

8. Homogeneous SISO control. Stabilization of uncertain systems in fixed or finite time, or exponentially. Homogeneity in bilimit. General recursive homogeneous SISO control design. Quasi-continuous homogeneous SM control. 

9. Output-feedback problem. Differentiation problem. High gain observers. Inequalities by Landau-Kolmogorov and asymptotically optimal differentiation.  Asymptotically optimal differentiation.

10. Homogeneous, FT stable differentiators. Filtering duifferentiators. Hybrid (bihomogeneous) differentiators.

11. Discrete homogeneous systems. Homogeneous discrete differentiation. Computer realization and its accuracy restrictions. Real-time-programming notion. 

12. Quasi-continuous MIMO SM control. Integral SMs, chattering reduction. Lyapunov methods of homogeneous-control design.

13. Output-feedback homogeneous SM control in the presence of disturbances. Systems with actuators. Practical relative degree notion. Bifurcation of relative degree. Car control example. Blood glucose control. 

Main References

Books:
1. Shtessel Y., Edwards C., Fridman L., Levant A., Sliding Mode Control and Observation. Control Engineering series, Birkhauser Basel, US, 2013
2. Filippov, A.F., Differential Equations with Discontinuous Right-hand Side. Kluwer, Dordrecht, the Netherlands, 1988, 2001
3. Slotine, J.-J. E. and Li W., Applied Nonlinear Control, 1991, Prentice-Hall, Inc., London.
4. Khalil, H. Nonlinear Systems, 3rd ed., Prentice Hall, 2001. 
5. Isidori, A., 1995, Nonlinear Control Systems, (New York: Springer Verlag).
6. Bacciotti, A., & Rosier, L., 2001, Liapunov functions and stability in control theory. Lecture notes in control and inf. sc. 267, Springer Verlag, London. Also the 2nd edition, 2005

Papers in alphabetic order:
1. Cruz-Zavala, E. and Moreno, J.A., Lyapunov approach to higher-order sliding mode design. In "Recent Trends in Sliding Mode Control",  pp. 3-28, 2016, Institution of Engineering and Technology IET
2. Hanan, A., Jbara, A., Levant A., New homogeneous controllers and differentiators. Studies in Systems, Decision and Control, Springer, 271, pp. 3-28, 2020
3. Hanan, A., Jbara, A., Levant A., Homogeneous Output-Feedback Control, to be presented on line at the International Federation of Automatic Control (IFAC) World Congress, July 12-17, 2020
4. Levant A., Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, 76 (9/10), 924-941, special issue on Sliding-Mode Control, 2003
5. Levant A., Non-Lyapunov Homogeneous SISO Control Design, IEEE Conference on Decision and Control, 2017
6. Levant A., Sliding Mode Control: Finite-Time Observation and Regulation. In Baillieul J., Samad T. (eds), Encyclopedia of Systems and Control. Springer, London, 2020, available online
7. Levant A., Efimov D.,Polyakov A.,Perruquetti W., Stability and Robustness of Homogeneous Differential Inclusions, IEEE Conference on Decision and Control, 2016
8. Levant A., Livne M., Weighted homogeneity and robustness of sliding mode control. Automatica, 72(10), 186-193, 2016
9. Levant, A., Livne, M., & Yu, X. Sliding-mode-based differentiation and its application. In Proc. of the 20th IFAC World Congress, Toulouse, July 9-14, France, 2017
10. Levant, A., Livne, M., Globally convergent differentiators with variable gains, International journal of Control, 91(9), 1994–2008, 2018
11. Levant, A., Livne, M., Robust exact filtering differentiators, European Journal of Control, available online, 2020
12. Livne M., Levant A., Proper discretization of homogeneous differentiators. Automatica, 50(8), 2007-2014, 2014