(390e) Constrained Control Lyapunov Functions: Design and Application for Nonlinear Systems

Constrained Control Lyapunov Function: Design and application for nonlinear systems
Systems exhibiting nonlinearity, constraints, and uncertainty are ubiquitous in practice, from chemical reactors to flight control systems. 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) [1, 2].

There currently do not exist results that identify the NCR of a general nonlinear system or give controls laws for stabilization everywhere within the NCR. Motivated by this, in this work we consider the problems of constructing the NCR for control-affine input-constrained nonlinear systems and determining a control law which guarantees stabilization from everywhere within the NCR.

It is known that the NCR is described by time-optimal trajectories of the reverse-time system [7]. We use the fact that reachable points in reverse-time are stabilizing for the nominal system. Our construction reinforces the special relationship that time-optimal controls have with a system's reachable set: that they steer to and then traverse the boundary of the NCR [5].
In [2], it was shown that the NCR for linear systems is covered by certain bang-bang trajectories of the reverse-time system. Our construction of the NCR for nonlinear systems can be seen as a generalization of this result. In that work, the boundary of the NCR is formed by a collection of alternating bang-bang arcs with arbitrary switching times, whereas in this paper, the switching times of the reverse-time bang-bang arcs are indicated by the evolution of a certain â??switching functionâ?? ODE [6].
The notion of Constrained Control Lyapunov Functions (CCLF) was recently defined as 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].

Notice that that decay of the trajectory into successive sub-NCRs, corresponding to the same dynamics but successively smaller input constraints, is sufficient to result in stabilization to the origin. Thus, by recognizing that such a collection of sub-shells of the NCR is a CCLF, we implement a control law that guarantees stabilization from everywhere in the NCR. We employ this CCLF in a model predictive controller that constrains the control action to those which force the trajectory into successive sub-shells of the NCR. This result will be illustrated with an example.

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.