Null Controllable Regions Research Articles

A trajectory‐based method for constructing null controllable regions

SummaryThe problem of determining a system's set of stabilizable states, the null controllable region (NCR), is intricately related to the problem of determining control Lyapunov functions. In this paper, we address the problem of construction of the NCR for control‐affine nonlinear systems with input constraints. To this end, we explain how the boundary of the NCR is covered by time‐optimal trajectories. To construct the NCR, we employ an algorithm based on Pontryagin's minimum principle, which integrates optimal trajectories of a special smooth system in reverse time from the boundary of an initial NCR estimate. We illustrate this algorithm with linear and nonlinear system examples.

Utilizing null controllable regions to stabilize input-constrained nonlinear systems

In this paper, we present a method for control of input-constrained nonlinear systems that offers guaranteed stabilization from the entire null controllable region (NCR). The controller achieves stabilization by using a constrained control Lyapunov function (CCLF) based on this NCR. Prior to online implementation, the level sets the CCLF are constructed using an iterative algorithm. The algorithm works by using an invariance principle to expand an initial quadratic Lyapunov function-based region of attraction. The level sets of this CCLF are then utilized in the control calculations, and in particular, an MPC is formulated that requires the system to go to lower level sets of the CCLF. The proposed MPC thus achieves stabilization from the entire NCR. The proposed approach is first corroborated against existing results for linear systems using two- and three-dimensional linear systems examples. Subsequently, the implementation is shown for two and three dimensional nonlinear systems.

Null controllable region of delta operator systems subject to actuator saturation

ABSTRACTIn this paper, we give exact description of null controllable regions for delta operator systems subject to actuator saturation. The null controllable region is in terms of a set of extremal trajectories of anti-stable subsystems. For the delta operator system with real eigenvalues or complex eigenvalues, the description is simplified to an explicit formula which is used to characterise the boundary of a null controllable region. The relations of null controllable regions are shown separately for continuous-time systems, discrete-time systems and delta operator systems. Two numerical examples are given to illustrate the effectiveness of the proposed techniques on null controllable regions.

Computation of null controllable regions for antistable linear systems subject to input saturation: an iterative approach

In this study an alternative characterisation of the null controllable region for linear systems subject to input saturation with strictly positive real eigenvalues is proposed. First an outer estimate of the null controllable region is obtained and then an iterated convexification technique is developed.

An explicit description of null controllable regions of linear systems with saturating actuators

We give simple exact descriptions of the null controllable regions for general linear systems with saturating actuators. The description is in terms of a set of extremal trajectories of the anti-stable subsystem. For lower order systems or systems with only real eigenvalues, this description is further simplified to result in explicit formulae for the boundaries of the null controllable regions.