A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control

Abstract

Abstract. An efficient Newton-type scheme for the approximate on-line solution of optimization problems as they occur in optimal feedback control is presented. The scheme allows a fast reaction to disturbances by delivering approximations of the exact optimal feedback control which are iteratively refined during the runtime of the controlled process. The contractivity of this real-time iteration scheme is proven, and a bound on the loss of optimality-compared with the theoretical optimal solution-is given. The robustness and excellent real-time performance of the method is demonstrated in a numerical experiment, the control of an unstable system, namely, an airborne kite that shall fly loops. 1. Introduction. Feedback control based on the real-time optimization of nonlinear dynamic process models, also referred to as nonlinear model predictive control (NMPC), has attracted increasing attention over the past decade, particularly in chemical engineering [4,27,1,28]. Based on the current system state, feedback is provided by an online optimization of the predicted system behavior, using the mathematical model. The first part of the optimized control trajectory is implemented at the real system, and a sampling time later the optimization procedure is repeated. Among the advantages of this approach are the flexibility provided in formulating the objective and in modeling the process using ordinary or partial differential equations (ODEs or PDEs), the capability of directly handling equality and inequality constraints, and the possibility of treating large disturbances quickly.One important precondition, however, is the availability of reliable and efficient numerical optimal control algorithms. One particularly successful algorithm that is designed to achieve this aim, the recently developed real-time iteration scheme, will be the focus of this paper. In the literature, several suggestions have been made on how to adapt off-line optimal control algorithms for use in on-line optimization. For an overview and comparison of important approaches, see, e.g., Binder et al. [6]. We particularly mention here the "Newton-type control algorithm" proposed by Li and Biegler [32] and de Oliveira and Biegler [15] and the "feasibility-perturbed SQP" approach to NMPC by Tenny, Wright, and Rawlings [38]. Both approaches keep even intermediate optimization iterates feasible. This is in contrast to the simultaneous dynamic optimization methods, as the collocation method proposed in Biegler [5] or the direct multiple shooting method in Bock et al. [7] and in Santos [37], which allow

A Feedback Optimal Control Algorithm with Optimal Measurement Time Points

“…for given and fixed controls u : T → U and weights w ω i ∈ W. As the measurement times t i may be a priori unknown, we will in our analysis in Section 3.1 also look at the continuous analogue to (4). This is given by min…”

“…As a fast feedback of the controller is important in many applications, clever approaches doing most of the necessary calculations before a new measurement arrives have been proposed in the literature. The most important numerical concepts comprise real-time iterations [4,5], multi-level iterations [6], parallel multi-level iterations [7], an exploitation of the KKT structures [8,9], adaptive control [10], automatic code export [11,12], and usage of parametric QPs [13,14]. For a benchmark problem, the continuously stirred tank reactor of [15], a speedup of approximately 150,000 has been achieved comparing the 60 seconds per iteration reported in 1997 [16] and the 400 microseconds per iteration reported in 2011 by [17].…”

Prevention of Emergency Voltage Collapses in Electric Power Networks using Hybrid Predictive Control

“…To deal with control and optimization of systems modeled by DAEs, several strategies have been proposed, such as simultaneous strategies [6,9,21], multiple shooting strategies [11,12], and direct search methods [43]. The particular structure of power network models (see Section 3.3.4) can especially be used advantageously to set up tractable models for model-based control approaches, such as model predictive control (MPC) [31].…”

Efficient robust optimization for robust control with constraints

“…2.4] or Differential Dynamic Programming [33] technique in which the state feedback equation x + = Ax + Bu is explicit in every stage, compared to O(N 3 ) time for the more compact formulation in which states are eliminated from the system. More direct motivation for our work comes from [8,16,44,49,54], which describe efficient implementations of optimization methods for solving optimal control problems with state and control constraints, though without disturbances.…”