Lars Grüne

Nonlinear Model Predictive Control

In this chapter, we introduce the nonlinear model predictive control algorithm in a rigorous way. We start by defining a basic NMPC algorithm for constant reference and continue by formalizing state and control constraints. Viability (or weak forward invariance) of the set of state constraints is introduced and the consequences for the admissibility of the NMPC-feedback law are discussed. After having introduced NMPC in a special setting, we describe various extensions of the basic algorithm, considering time varying reference solutions, terminal constraints, and costs and additional weights. Finally, we investigate the optimal control problem corresponding to this generalized setting and prove several properties, most notably the dynamic programming principle.

Economic receding horizon control without terminal constraints

We consider a receding horizon control scheme without terminal constraints in which the stage cost is defined by economic criteria, i.e., not necessarily linked to a stabilization or tracking problem. We analyze the performance of the resulting receding horizon controller with a particular focus on the case of optimal steady states for the corresponding averaged infinite horizon problem. Using a turnpike property and suitable controllability properties we prove near optimal performance of the controller and convergence of the closed loop solution to a neighborhood of the optimal steady state. Two examples illustrate our findings numerically and show how to verify the imposed assumptions.

On the Infinite Horizon Performance of Receding Horizon Controllers

Receding horizon control is a well established approach for control of systems with constraints and nonlinearities. Optimization over an infinite time-horizon, which is often computationally intractable, is therein replaced by a sequence of finite horizon problems. This paper provides a method to quantify the performance degradation that comes with this approximation. Results are provided for problems both with and without terminal costs and constraints and for both exactly and practically asymptotically stabilizable systems.

Analysis and Design of Unconstrained Nonlinear MPC Schemes for Finite and Infinite Dimensional Systems

We present a technique for computing stability and performance bounds for unconstrained nonlinear model predictive control (MPC) schemes. The technique relies on controllability properties of the system under consideration, and the computation can be formulated as an optimization problem whose complexity is independent of the state space dimension. Based on the insight obtained from the numerical solution of this problem, we derive design guidelines for nonlinear MPC schemes which guarantee stability of the closed loop for small optimization horizons. These guidelines are illustrated by a finite and an infinite dimensional example.

Homogeneous State Feedback Stabilization of Homogenous Systems

We show that for any asymptotically controllable homogeneous system in euclidean space (not necessarily Lipschitz at the origin) there exists a homogeneous control Lyapunov function and a homogeneous, possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system. If the system satisfies the usual local Lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate. If the system satisfies a global Lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate. The control Lyapunov function and the feedback are based on the Lyapunov exponents of a suitable auxiliary system and admit a numerical approximation.

Optimization-Based Stabilization of Sampled-Data Nonlinear Systems via Their Approximate Discrete-Time Models

We present results on numerical regulator design for sampled-data nonlinear plants via their approximate discrete-time plant models. The regulator design is based on an approximate discrete-time plant model and is carried out either via an infinite horizon optimization problem or via a finite horizon with terminal cost optimization problem. In both cases, we discuss situations when the sampling period T and the integration period h used in obtaining the approximate discrete-time plant model are the same or they are independent of each other. We show that, using this approach, practical and/or semiglobal stability of the exact discrete-time model is achieved under appropriate conditions.

Analysis of Unconstrained Nonlinear MPC Schemes with Time Varying Control Horizon

For nonlinear discrete time systems satisfying a controllability condition, we present a stability condition for model predictive control without stabilizing terminal constraints or costs. The condition is given in terms of an analytical formula which can be employed in order to determine a prediction horizon length for which asymptotic stability or a performance guarantee is ensured. Based on this formula a sensitivity analysis with respect to the prediction and the possibly time varying control horizon is carried out.

Asymptotic stability equals exponential stability, and ISS equals finite energy gain — if you twist your eyes

In this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables. The same is shown for input-to-state stability and input-to-state exponential stability, and for input-to-state exponential stability and a nonlinear H∞ estimate.

Input-to-state dynamical stability and its Lyapunov function characterization

We present a new variant of the input-to-state stability (ISS) property which is based on using a one-dimensional dynamical system for building the class /spl Kscr//spl Lscr/ function for the decay estimate and for describing the influence of the perturbation. We show the relation to the original ISS formulation and describe characterizations by means of suitable Lyapunov functions. As applications, we derive quantitative results on stability margins for nonlinear systems and a quantitative version of a small gain theorem for nonlinear systems.

A Generalization of Zubov's Method to Perturbed Systems

A generalization of Zubovs theorem on representing the domain of attraction via the solution of a suitable partial differential equation is presented for the case of perturbed systems with a singular fixed point. For the construction it is necessary to consider solutions in the viscosity sense. As a consequence, maximal robust Lyapunov functions can be characterized as viscosity solutions.