Adaptive Horizon Model Predictive Control and Al'brekht's Method

A standard way of finding a feedback law that stabilizes a control system to an operating point is to recast the problem as an infinite horizon optimal control problem. If the optimal cost and the optmal feedback can be found on a large domain around the operating point then a Lyapunov argument can be used to verify the asymptotic stability of the closed loop dynamics. The problem with this approach is that is usually very difficult to find the optimal cost and the optmal feedback on a large domain for nonlinear problems with or without constraints. Hence the increasing interest in Model Predictive Control (MPC). In standard MPC a finite horizon optimal control problem is solved in real time but just at the current state, the first control action is implimented, the system evolves one time step and the process is repeated. A terminal cost and terminal feedback found by Al'brekht's methoddefined in a neighborhood of the operating point is used to shorten the horizon and thereby make the nonlinear programs easier to solve because they have less decision variables. Adaptive Horizon Model Predictive Control (AHMPC) is a scheme for varying the horizon length of Model Predictive Control (MPC) as needed. Its goal is to achieve stabilization with horizons as small as possible so that MPC methods can be used on faster and/or more complicated dynamic processes.Comment: arXiv admin note: text overlap with arXiv:1602.0861

Further Results on Active Magnetic Bearing Control with Input Saturation

We study the low-bias stabilization of active magnetic bearings (AMBs) subject to voltage saturation based on a recently proposed model for the AMB switching mode of operation. Using a forwarding-like approach, we construct a stabilizing controller of arbitrarily small amplitude and a control-Lyapunov function for the AMB dynamics. We illustrate our construction using a numerical example.Comment: 9 pages, 2 figures. IEEE Transactions on Control Systems Technology, accepted for publication in January 200

Constrained Finite Receding Horizon Linear Quadratic Control

Issues of feasibility, stability and performance are considered for a finite horizon formulation of receding horizon control (RHC) for linear systems under mixed linear state and control constraints. It is shown that for a sufficiently long horizon, a receding horizon policy will remain feasible and result in stability, even when no end constraint is imposed. In addition, offline finite horizon calculations can be used to determine not only a stabilizing horizon length, but guaranteed performance bounds for the receding horizon policy. These calculations are demonstrated on two examples

Unconstrained receding-horizon control of nonlinear systems

It is well known that unconstrained infinite-horizon optimal control may be used to construct a stabilizing controller for a nonlinear system. We show that similar stabilization results may be achieved using unconstrained finite horizon optimal control. The key idea is to approximate the tail of the infinite horizon cost-to-go using, as terminal cost, an appropriate control Lyapunov function. Roughly speaking, the terminal control Lyapunov function (CLF) should provide an (incremental) upper bound on the cost. In this fashion, important stability characteristics may be retained without the use of terminal constraints such as those employed by a number of other researchers. The absence of constraints allows a significant speedup in computation. Furthermore, it is shown that in order to guarantee stability, it suffices to satisfy an improvement property, thereby relaxing the requirement that truly optimal trajectories be found. We provide a complete analysis of the stability and region of attraction/operation properties of receding horizon control strategies that utilize finite horizon approximations in the proposed class. It is shown that the guaranteed region of operation contains that of the CLF controller and may be made as large as desired by increasing the optimization horizon (restricted, of course, to the infinite horizon domain). Moreover, it is easily seen that both CLF and infinite-horizon optimal control approaches are limiting cases of our receding horizon strategy. The key results are illustrated using a familiar example, the inverted pendulum, where significant improvements in guaranteed region of operation and cost are noted

Global stabilization of linear systems with bounds on the feedback and its successive derivatives

We address the global stabilization of linear time-invariant (LTI) systems when the magnitude of the control input and its successive time derivatives, up to an order $p\in\mathbb N$, are bounded by prescribed values. We propose a static state feedback that solves this problem for any admissible LTI systems, namely for stabilizable systems whose internal dynamics has no eigenvalue with positive real part. This generalizes previous work done for single-input chains of integrators and rotating dynamics

Sparse and Constrained Stochastic Predictive Control for Networked Systems

This article presents a novel class of control policies for networked control of Lyapunov-stable linear systems with bounded inputs. The control channel is assumed to have i.i.d. Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. Our proposed class of policies is affine in the past dropouts and saturated values of the past disturbances. We further consider a regularization term in a quadratic performance index to promote sparsity in control. We demonstrate how to augment the underlying optimization problem with a constant negative drift constraint to ensure mean-square boundedness of the closed-loop states, yielding a convex quadratic program to be solved periodically online. The states of the closed-loop plant under the receding horizon implementation of the proposed class of policies are mean square bounded for any positive bound on the control and any non-zero probability of successful transmission

Thermal quench effects on ferroelectric domain walls

Using piezoresponse force microscopy on epitaxial ferroelectric thin films, we have measured the evolution of domain wall roughening as a result of heat-quench cycles up to 735C, with the effective roughness exponent \zeta\ changing from 0.25 to 0.5. We discuss two possible mechanisms for the observed \zeta\ increase: a quench from a thermal 1-dimensional configuration, and from a locally-equilibrated pinned configuration with a crossover from a 2- to 1-dimensional regime. We find that the post-quench spatial structure of the metastable states, qualitatively consistent with the existence of a growing dynamical length scale whose ultra slow evolution is primarily controlled by the defect configuration and heating process parameters, makes the second scenario more plausible. This interpretation suggests that pinning is relevant in a wide range of temperatures, and in particular, that purely thermal domain wall configurations might not be observable in this glassy system. We also demonstrate the crucial effects of oxygen vacancies in stabilizing domain structures.Comment: 17 pages (preprint), 4 figure