**Part 1**

**New Theoretical Frontier** 

6 Frontiers of Model Predictive Control

Yuzgec U.; Becerikli Y.; Turker M. (2006). Nonlinear predictive control of a drying process

ISSN 0019-0578.

using genetic algorithms. *ISA Transactions*, Vol.45, No.4, October, 2006, pp.589-602,

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**1**

*University of Ulm*

*Germany*

**A Real-Time Gradient Method for Nonlinear Model Predictive Control**

*Institute of Measurement, Control and Microtechnology*

Model predictive control (MPC) is a modern control scheme that relies on the solution of an optimal control problem (OCP) on a receding horizon. MPC schemes have been developed in various formulations (regarding continuous/discrete-time systems, finite/infinite horizon length, terminal set/equality constraints, etc.). Comprehensive overviews and references on MPC can, for instance, be found in Diehl et al. (2009); Grüne & Pannek (2011); Kothare &

Although the methodology of MPC is naturally suited to handle constraints and multiple-input systems, the iterative solution of the underlying OCP is in general computationally expensive. An intuitive approach to reducing the computational load is to solve the OCP approximately, for instance, by using a fixed number of iterations in each sampling step. In the next MPC step, the previous solution can be used for a warm-start of the optimization algorithm in order to successively reduce the suboptimality of the predicted trajectories. This incremental strategy differs from the "optimal" MPC case where the

There exist various suboptimal and real-time approaches in the literature with different kinds of terminal constraints and demands on the optimization algorithm (Cannon & Kouvaritakis, 2002; DeHaan & Guay, 2007; Diehl et al., 2005; Graichen & Kugi, 2010; Lee et al., 2002; Michalska & Mayne, 1993; Ohtsuka, 2004; Scokaert et al., 1999). In particular, the approaches of Ohtsuka (2004) and Diehl et al. (2005) are related to the MPC scheme presented in this chapter. In Ohtsuka (2004), an algorithm is developed that traces the solution of the discretized optimality conditions over the single sampling steps. The real-time iteration scheme presented by Diehl et al. (2005) uses a Newton scheme together with terminal constraints in order to compute an approximate solution that is refined in each sampling step. Suboptimal MPC schemes require special attention regarding their convergence and stability properties. This is particularly important if an MPC formulation without terminal constraints is used in order to minimize the computational complexity and to allow for a real-time implementation for very fast dynamical systems. In this context, a suboptimal MPC approach without terminal constraints was investigated in Graichen & Kugi (2010). Starting from the assumption that an optimization algorithm with a linear rate of convergence exists, it is

Morari (2000); Mayne et al. (2000); Rawlings & Mayne (2009).

(numerically exact) OCP solution is assumed to be known.

**1. Introduction**

Knut Graichen and Bartosz Käpernick
