(12h) The Stabilization of Input-Constrained Nonlinear Systems with Maximal Region of Attraction

All physical control systems exhibit input constraints: all real motors have finite torque and valves can only open so wide. However, many control system fail to incorporate this limitation into their design, resulting in potential loss of stability. Regardless of the control law, input constraints often limit the set of initial conditions from where stabilization is achievable to the so- called null controllable region (NCR).

For linear control systems, it was shown in [2] that the boundary of the NCR is formed from certain trajectories of the time-reversed system. These trajectories can be shown to be solutions to a time-optimal control problem [3]. There currently do not exist any methods to determine the NCR for general nonlinear systems.

In this work, we will introduce a new, computationally efficient algorithm to determine the NCR for any nonlinear system. The method consists of expanding a quadrative Lyapunov-function based estimate of the NCR with successive estimations of the NCR. By simulating trajectories just outside estimated NCR, the set of ‘stable’ states grows by adding states that can reach the current region. This can be seen as exploiting the positive invariance property of the NCR.

We next use knowledge of the NCR to design a stabilizing control law. This design is the first to be demonstrated which can guarantee stabilization from everywhere in the NCR (it is impossible to stabilize the system outside of the NCR) for nonlinear systems, and is similar to the ‘Constrained control Lyapunov function’-based design which was first reported in [5] for linear systems.

The design can be summarized as driving the states away from the boundary of the NCR. Before online implementation, we compute several NCR “shells” corresponding to various input constraints. Then, we implement a model predictive controller that drives the states into successively smaller shells, which results in asymptotic stabilization to the origin. The versatility and efficacy of our results are illustrated with several linear and nonlinear examples in various dimensions.

References

[1] P. Mhaskar, N. El-Farra, P. Christofides (2006). Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control. Systems & Control Letters, 55, 650-659.

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

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