All practical control systems are subject to operational constraints such as restricted dimensions
and limited control capacity. In many situations these constraints are purposely imposed and are
intended to be made as tight as possible in order to reduce energy consumption, to minimize the
utilization of resources, or to merely reduce the size of a certain device. Magnetic suspension
systems, e.g., a fluid pump as depicted below, are typical examples with severe constraints:
extremely small clearance, low power consumption and small size.  

Project Description: Our project aims to explore the best performances of control systems with
limited control capacity and resources. Since constrained control systems are intrinsically nonlinear,
a simple linear controller is unlikely to make full utilization of the available resources and we have to
resort to nonlinear control strategies, possibly involving switching and hybrid structures. This project
will develop systematic approaches to the construction of nonlinear controllers by using non-quadratic Lyapunov functions whose level sets reflect the structure of the constraints better than ellipsoids
generated by quadratic functions, which are usually used for the design of linear controllers. To bridge
the gap between theory and practice, this work emphasizes the realizablility of the controller and
numerical tractability of the design algorithms.

Approaches: Control design via quadratic Lyapunov functions can be numerically implemented
via the Linear Matrix Inequality (LMI) based methods. We seek to use functions which are constructed
from several quadratic Lyapunov functions – Composite Quadratic Lyapunov Functions (CQLF).
These functions are potentially numerically tractable while providing more flexibility and degree
of design freedom for achieving better performances.  Three types of CQLF have been
constructed and exploited in our recent works and they have shown great potential in
performance improvement for uncertain systems, switched systems and constrained control systems.   

Our works:

1. T. Hu, A. R. Teel and L. Zaccarian, ``Stability and performance for saturated systems 
   via quadratic and non-quadratic Lyapunov functions," IEEE Transactions on Automatic 
   Control, 51(11), pp.~1770-1786, 2006. 

2. T. Hu and Z. Lin, Control Systems with Actuator Saturation: Analysis and Design,
   Birkhauser, Boston, xvi, 392 p, July, 2001. (ISBN: 0-8176-4219-6).

3. T. Hu, R. Goebel, A.R. Teel and Z. Lin, ``Conjugate Lyapunov functions for saturated 
   linear systems," Automatica, 41(11), pp.~1949-1956, 2005.  

4. T. Hu, A. R. Teel and L. Zaccarian, ``Regional anti-windup compensation for linear 
   systems with input saturation," The 2005 American Control Conference, pp.3397-3402, 2005.  

5. T. Hu and Z. Lin,``Output regulation of linear systems with bounded continuous feedback,'' 
   IEEE Transactions on Automatic control, Vol.49, No.11, pp.1941-1953, 2004. 

6. H. Fang, Z. Lin and T. Hu, ``Analysis and control design of linear systems in the presence 
   of actuator saturation and L2-disturbances,'' Automatica, Vol.40, July, pp.1229-1238, 2004.

7. T. Hu and Z. Lin, ``Controlled invariance of ellipsoids: linear vs nonlinear feedback," 
    Systems & Control Letters, Vol.53, pp.203-210, 2004. 

8. T. Hu, Z. Lin and Y. Shamash, ``On maximizing the convergence rate for linear systems 
   with input saturation," IEEE Transactions on Automatic Control, Vol.48, No.7, 
   pp.1249-1253, 2003. 

9. T. Hu and Z. Lin, ``On the tightness of a recent set invariance condition under 
   actuator saturation," Systems & Control Letters, Vol.49, No.5, pp.389-399, 2003. 

10. T. Hu and Z. Lin, ``Composite quadratic Lyapunov functions for constrained control systems," 
    IEEE Transactions on Automatic Control, Vol.48, No.3, pp.440-450, March 2003.

11. T. Hu and Z. Lin,``Output regulation of general discrete-time linear systems with 
    saturation nonlinearities,'' Int. J. of Robust and Nonlinear Control, Vol.12, No.13, 
    pp.1129-1143, 2002. 

12. T. Hu, D. Miller and L. Qiu, ``Null controllable region of LTI discrete-time systems 
    with input saturation,'' Automatica,  Vol.38, No.11, pp.2009-2013, 2002. 

13. T. Hu and Z. Lin, ``On improving performances with continuous feedback laws," 
    IEEE Transactions on Automatic Control, Vol.47, No.9, pp.1570-1575, 2002. 

14. T. Hu, Z. Lin and L. Qiu, ``An explicit description of the null controllable regions of
    linear systems with saturating actuators," Systems & Control Letters, Vol.47, No.1, 
    pp.65-78, 2002.

15. T. Hu and Z. Lin, ``On semi-global stabilizability of anti-stable systems by saturated 
    linear feedback," IEEE Transactions on Automatic Control, Vol.47, No.7, pp.1193-1198, 2002.

16. T. Hu and Z. Lin, ``Exact characterization of invariant ellipsoids for linear systems 
    with saturating actuators,'' IEEE Transactions on Automatic Control, Vol.47, No.1, 
    pp.164-169, 2002.

17. T. Hu, Z. Lin and B. M. Chen, ``An analysis and design method for linear systems subject 
    to actuator saturation and disturbance," Automatica, Vol.38, No.2, pp.351-359, 2002. 

18. T. Hu, Z. Lin and B. M. Chen, ``Analysis and design for linear discrete-time systems 
    subject to actuator saturation," Systems & Control Letters, Vol.45, No.2,
    pp.97-112, 2002. 

19. Y. Y. Cao, Z. Lin and T. Hu,``Stability analysis of linear time-delay systems subject to 
    actuator saturation,'' IEEE Transactions on Circuits and Systems - Part I: Fundamental 
    Theory and Applications, Vol.49, No.2, pp.233-240, 2002.

20. T. Hu, Z. Lin and Y. Shamash, ``Semi-global stabilization with guaranteed regional perfor-
    mance of linear systems subject to actuator saturation,'' Systems & Control Letters, 
    Vol.43, No.3, pp.203-210, 2001.

21. Z. Lin and T. Hu, ``Semi-global stabilization of linear system subject to output saturation,"
    Systems & Control Letters, Vol.43, No.3, pp.211-217, 2001. 

22. T. Hu and Z. Lin, ``A complete stability analysis of planar discrete-time linear systems 
    under saturation," IEEE Transactions on Circuits and Systems - Part I: Fundamental Theory
    and Applications, Vol.48, No.6, pp.710-725, 2001. 

23. T. Hu, Z. Lin and L. Qiu, ``Stabilization of exponentially unstable linear systems with 
    saturating actuators," IEEE Transactions on Automatic Control, Vol.46, No.6, 
    pp.973-979, 2001. 

24. T. Hu and Z. Lin, ``Practical stabilization of exponentially unstable linear systems 
    subject  to actuator saturation nonlinearity and disturbance,'' Int. J. of Robust and 
    Nonlinear Control, Vol.11, pp.555-588, 2001. 

25. T. Hu and Z. Lin, ``On enlarging the basin of attraction for linear systems under saturated 
    linear feedback,'' Systems & Control Letters, Vol.40, No.1, pp.59-69, 2000. 

26. T. Hu and Z. Lin, ``A complete stability analysis of planar linear systems under saturation,'' 
    IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol.47, No.4, 
    pp.498-512, 2000.