(569d) Constructing Constrained Control Lyapunov Functions for Control Affine Nonlinear Systems

Summary

Systems exhibiting nonlinearity, constraints, and uncertainty are ubiquitous in practice and can pose significant challenges. Previous results have shown that control designs which do not acknowledge these features can under-perform more advanced techniques or even result in closed-loop instability [1]. In many applications, particularly those systems with input constraints, control designs are sought that guarantee stabilization to the origin from the largest possible set of initial conditions. This set has been termed the null controllable region (NCR) [2].

Constrained Control Lyapunov Functions (CCLF) are Lyapunov functions designed so that the control laws which guarantee their decay are admissible and stabilizing for the constrained system with a region of attraction equal to the NCR [3]. Clearly, control designs using CLFs that are not CCLFs only guarantee stabilization within subsets of the NCR [4]. There currently do not exist results that identify the NCR of a general nonlinear system or give controls laws for its stabilization everywhere within the NCR.

Motivated by the above, in this work we consider the problem of determining the NCR of control-affine nonlinear unstable systems with constrained controls.

Our construction of the nonlinear NCR is a generalization of the results for linear systems [1], where it was shown that for single-input linear systems of arbitrary dimension, the NCR of the system was covered by extremal trajectories of the reverse-time system, i.e. its reachable set. It was shown that these trajectories are induced solely by all bang-bang controls with a known number of switches. To determine the NCR for nonlinear systems, we will exploit the fact that time-optimal trajectories traverse the boundary of the system’s reachable set [5, 6].

We will then show that the NCR is a CCLF and that decay of the trajectory into successive concentric sub-NCRs, corresponding to proportionally smaller input requirements, is sufficient to result in stabilization to the origin. To illustrate the proposed approach, we will employ the CCLF in a model predictive controller that guarantees stability everywhere by constraining the control action to those which force the trajectory into successive sub-shells of the NCR.

References

[1] T. Hu, Z. L. and Qiu, L. (2001). Stabilization of exponentially unstable linear systems with saturating actuators. IEEE Transactions on Automatic Control, 46(6), 973-979.

[2] T. Hu, Z. L. and Qiu, L. (2002). An explicit description of null controllable regions of linear systems with saturating actuators. Systems & Control Letters, 47, 65-78.

[3] Mahmood, M. and Mhaskar, P. (2014). Constrained control Lyapunov function based model predictive control design. Int. J. Robust Nonlinear Control, 24, 374-388.

[4] Mahmood, M. and Mhaskar, P. (2008). Enhanced stability regions for model predictive control of nonlinear process systems. Proceedings of the American Control Conference,

1133-1138.

[5] Lewis, A. (2006). The Maximum Principle of Pontryagin in Control and in Optimal Control. Department of Mathematics and Statistics, Queen's University, Kingston, Canada.

[6] Shattler, H. and Ledzewicz, U. (2012). Geometric Optimal Control. Springer, New York.

[7] Sussmann, H. (1987). The structure of time optimal trajectories for single input systems in the plane. SIAM Journal of Control and Optimization, 25(2), 433-465.