Gene Grimm

Examples when nonlinear model predictive control is nonrobust

We consider nominal robustness of model predictive control for discrete-time nonlinear systems. We show, by examples, that when the optimization problem involves state constraints, or terminal constraints coupled with short optimization horizons, the asymptotic stability of the closed loop may have absolutely no robustness. That is to say, it is possible for arbitrarily small disturbances to keep the closed loop strictly inside the interior of the feasibility region of the optimization problem and, at the same time, far from the desired set point. This phenomenon does not occur when using model predictive control for linear systems with convex constraint sets. We emphasize that a necessary condition for the absence of nominal robustness in nonlinear model predictive control is that the value function and feedback law are discontinuous at some point(s) in the interior of the feasibility region.

Nominally robust model predictive control with state constraints

We present robust stabilization results for constrained, discrete-time, nonlinear systems using a finite-horizon model predictive control (MPC) algorithm that does not require any particular properties for the terminal cost. We introduce a property that characterizes the robustness properties of the MPC optimization problem. Assuming the system has this property (for which we give sufficient conditions), we make two further key assumptions. These are that the value function is bounded by a K/sub /spl infin// function of a state measure (related to the distance of the state to some target set) and that this measure is detectable from the stage cost used in the MPC algorithm. We show that these assumptions lead to stability that is robust to sufficiently small disturbances and measurement noise. While in general the results are semiglobal practical, when the detectability and upper bound assumptions are satisfied with linear K/sub /spl infin// functions, the stability and robustness is global with respect to the feasible set. We discuss algorithms employing terminal equality or inequality constraints. We provide two examples, one involving a terminal equality constraint and the other involving a nonrobustness-inducing state constraint.

Model predictive control when a local control Lyapunov function is not available

This paper presents closed-loop stability results for the control of unconstrained nonlinear systems using the model predictive control methodology with semidefinite costs. The results do not require the use of a local control Lyapunov function as the terminal cost. The key assumptions are that the value function is bounded by a K/sub /spl infin// function of some measure of the state and that this measure is detectable through the stage cost. Sufficient conditions to yield semiglobal practical (and global) MPC stability results are given. In each case, a minimum horizon (uniform for global results) is determined for which the MPC method will result in the stabilization of a desired set.

Case Studies Using Linear Matrix Inequalities for Optimal Anti-Windup Synthesis**

In this paper, we present several case studies illustrating the synthesis of static and dynamic optimal finite L2 gain anti-windup compensation using linear matrix inequalities, the theoretical underpinnings having been described in [6]. The resulting performance is also compared to the well-known IMC-based anti-windup construction both through comparison of the finite L2 gain and through simulations.

The l/sub 2/ anti-windup problem for discrete-time linear systems: definition and solutions

The anti-windup problem for discrete-time linear systems is formalized in terms of the l/sub 2/ norm of the deviation of the actual response of the system with saturation and anti-windup compensation from the (ideal) unconstrained response. We show that, paralleling continuous-time results, the problem is globally solvable if and only if the plant is non exponentially unstable and it is robustly globally solvable if and only if the plant is exponentially stable. We provide a constructive solution whenever the problem is solvable. Also offered is a high-performance global solution for exponentially stable plants based on receding horizon control. Illustrative simulations are included.

Results on linear LMI-based external anti-windup design

We study linear anti-windup augmentation for linear control systems with saturated linear plants in a special case when the anti-windup compensator can only modify the input and output of the windup-prone linear controller. We also measure the arising performance in terms of the finite L/sub 2/ gain from selected exogenous inputs to selected performance outputs. Our main results are a system theoretic feasibility characterization for fixed order anti-windup design and a linear matrix inequality (LMI) formulation for optimal static and plant-order anti-windup design. Interpretations of lower bounds on the achievable performance are also given. The effectiveness of the design procedure is demonstrated on a simulation example.

Examples of zero robustness in constrained model predictive control

Nominal robustness of model predictive control for nonlinear systems is considered. It is shown, by examples, that when the optimization problem involves state constraints, or terminal constraints coupled with short optimization horizons, the asymptotic stability of the closed loop may have absolutely no robustness. Namely, it is possible for arbitrarily small disturbances to keep the closed loop strictly inside the interior of the feasibility region of the optimization problem and, at the same time, far from the desired set point. This phenomenon does not occur when using model predictive control for linear systems with convex constraint sets. It is emphasized that a necessary condition for the absence of nominal robustness in nonlinear model predictive control is that the value function and feedback law are discontinuous at some point(s) in the interior of the feasibility region.

Robust LMI-based linear anti-windup design: optimizing the unconstrained response recovery

Fixed order anti-windup augmentation is addressed by considering the dynamics of the mismatch between the constrained and unconstrained responses. The method utilizes LMIs, and the result is similar to other recent LMI-based anti-windup synthesis methods. Robustness is directly addressed-which is not done in previous LMI work. An optimal LMI-based synthesis procedure is provided for static and plant-order linear anti-windup augmentation and the performance of the resulting design strategy is shown via a simulation example.

Experimental results in optimal linear anti-windup compensation

The optimal anti-windup synthesis proposed by G. Grimm et al. (2001) is demonstrated on an experimental mechanical system. A windup-prone controller is first shown to induce severe performance degradation when saturation is hit. According to a linearized model of the mechanical system, static anti-windup compensation is infeasible, hence dynamic anti-windup of order equal to that of the plant is shown to induce performance recovery both in simulation with the linearized and nonlinear model, and in the experimental runs on the physical system.