Slovakia

Recently, a regional MPC approach has been proposed that exploits the piecewise affine structure of the optimal solution (without computing the entire explicit solution before). Here, regional refers to the idea of using the affine feedback law that is optimal in a vicinity of the current state of operation, and therefore provides the optimal input signal without requiring to solve a QP. In the present paper, we apply the idea of regional MPC to min-max MPC problems. We show that the new robust approach can significantly reduce the number of QPs to be solved within min-max MPC resulting in a reduced overall computational effort. Moreover, we compare the performance of the new approach to an existing robust regional MPC approach using a numerical example with varying horizon. Finally, we provide a rule for choosing a suitable robust regional MPC approach based on the horizon.

Nonlinear Model Predictive Control Conference 2021

Reducing the computational effort of min-max model predictive control with regional feedback laws

model-based control

robust control

predictive control

quadratic programming

constraints

The IFAC Conference on Nonlinear Model Predictive Control (NMPC 2021) aims to bring together researchers interested in and work in the MPC field from both academia and industry. This allows us to reflect and establish the current state-of-the-art and focus on the future development of the MPC field towards relevant directions.

Model predictive control (MPC) is indisputably one of the advanced control techniques that have significantly affected control engineering practice with thousands of controllers implemented in various fields, spanning from the process industry to automotive and robotics. This is due to its unique ability to handle constraints, take preview information about disturbances and setpoint changes into account, and optimize performance. Furthermore, MPC allows the consideration of various types of dynamics, from linear to nonlinear, discrete to continuous, as well as an online adaptation of models and parameters. The research community on MPC is very active, with hundreds of papers published every year. Over the past decades, many theoretical, as well as implementation results, have been achieved.

### **All content presented at NMPC2021, including videos, papers and slides will be available only to registered (regular/student) participants. The content can be accessed until August 31st, 2021.**



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The IFAC Conference on Nonlinear Model Predictive Control (NMPC 2021) aims to bring together researchers interested in and work in the MPC field from both academia and industry. This allows us to reflect and establish the current state-of-the-art and focus on the future development of the MPC field towards relevant directions.

technical paper

Nonlinear Model Predictive Control (NMPC) formulations through quasi-Linear Parameter Varying (qLPV) embeddings have been brought to focus in recent literature. The qLPV realisation of the nonlinear dynamics yields linear predictions at each sampling instant. Thereby, these control strategies offer online implementation with numerical toughness of Sequential Quadratic Programs (SQPs), which can be solved much faster than the Nonlinear Programs (NPs) generated with “regular” NMPC design. The general lines of such methods are as follows: (i) The qLPV embedding is formulated with state-dependent scheduling parameters; (ii) recursive extrapolation procedures are used to estimate the values of these parameters along the prediction horizon; (iii) these estimates are used to compute linear predictions, which are used to compute the control law through a constrained optimisation procedure. This presentation details the overall concept of these novel NMPC

NMPC through qLPV Embedding: A Tutorial Review of Different Approaches

We show that exact solutions to nonlinear MPC problems for a special class of bilinear systems can be obtained by solving a finite number of convex linear MPC problems. The underlying reformulation is realized through exact (input-state) linearization and supported by special biquadratic stage costs. While such stage costs are unusual, we show that they preserve beneficial properties such as positive definiteness and that they allow for meaningful tuning. We further show that the proposed reformulation reveals some interesting insights on the structureof the optimal control law and we illustrate our approach with a numerical example.

Exact Solution to a Special Class of Nonlinear MPC Problems

The use of derivative-based solvers to compute solutions to optimal control problems with non-differentiable cost or dynamics often requires reformulations or relaxations that complicate the implementation or increase computational complexity. We present an initial framework for using the derivative-free Mesh Adaptive Direct Search (MADS) algorithm to solve Nonlinear Model Predictive Control problems with non-differentiable features without the need for reformulation. The MADS algorithm performs a structured search of the input space by simulating selected system trajectories and computing the subsequent cost value. We propose handling the path constraints and the Lagrange cost term by augmenting the system dynamics with additional states to compute the violation and cost value alongside the state trajectories, eliminating the need for reconstructing the state trajectories in a separate phase. We demonstrate the practicality of this framework by solving a robust rocket control problem, where the objective is to reach a target altitude as close as possible, given a system with uncertain parameters. This example uses a non-differentiable cost function and simulates two different system trajectories simultaneously, with each system having its own free final time.

Towards a Framework for Nonlinear Predictive Control using Derivative-Free Optimization

We study a property of dynamic optimization (DO) problems (as those encountered in model predictive control and moving horizon estimation) that is known as exponential decay of sensitivity (EDS). This property indicates that the sensitivity of the solution at stage $i$ against a data perturbation at stage $j$ decays exponentially with $|i-j|$. Building upon our previous results, we show that EDS holds under uniform boundedness of the Lagrangian Hessian, a uniform second order sufficiency condition (uSOSC), and a uniform linear independence constraint qualification (uLICQ). Furthermore, we prove that uSOSC and uLICQ can be obtained under uniform controllability and observability. Hence, we have that uniform controllability and observability imply EDS. These results provide insights into how perturbations propagate along the horizon and enable the development of approximation and solution schemes. We illustrate the developments with numerical examples.

Controllability and Observability Imply Exponential Decay of Sensitivity in Dynamic Optimization

In this paper we present a distributed second-order augmented Lagrangian method

A Distributed Second-Order Augmented Lagrangian Method for Distributed Model Predictive Control

This paper investigates how various caching strategies can reduce the computational effort of the active set method (ASM) applied to solve constrained model predictive control problems with quadratic objective function and linear constraints. Specifically, we show that during closed-loop operation, the active set method often re-visits the same combination of active constraints while searching for optimal control inputs by factoring Karush-Kuhn-Tucker (KKT) systems. By storing the factors of the corresponding KKT system in a cache, these repetitive

Streamlining Active Set Method in MPC Using Cache Memory

Objective of the vehicle thermal management system is to achieve precise coolant temperature control that ensures operation of the combustion engine in the region with the maximum thermal efficiency. In this study it is achieved by a combined effect of coolant temperature control algorithm that is operating an advanced electronic valve with one input and three outputs that balances distribution of the coolant flow

Advanced Coolant Temperature Control Study with Integrated Thermal Management System Valve

We address the problem of operating up- and down-time restricted components with activation thresholds using linear predictive control instead of hybrid predictive control. Accounting for operational requirements of this kind usually requires optimal control problems with discrete variables, which are computationally demanding. As a workaround, run-time requirements are often ignored in the control design and enforced independently from the optimal input value. We present conditions and an algorithm that ensures minimum run-time requirements in linear MPC. The control performance for a heating application is compared to linear MPC with rule-based post-processing and hybrid MPC. The results show that our approach ensures the up-and down-time requirements at a high control performance and is significantly faster than hybrid MPC.

MPC for Heating Systems with Minimum up- and Down-Time Requirements

This presentation gives an introduction to an application of MPC to an industrial process. The aim is to control the temperature of a substrate that is transported through a heating zone with changing speed. The heat is applied using hot air. The talk gives an overview on the modeling approach, the MPC setup and some experimental results.

Modeling and Temperature Control of a Moving Substrate

In recent years, theoretical results for model predictive control (MPC) have been expanded to address discrete actuators (decisions) and high-level planning and scheduling problems. The application of MPC-style methods to scheduling problems has been driven, in part, by the robustness afforded by feedback. The ability of MPC, and feedback methods in general, to reject small persistent disturbances is well-recognized. In many planning and scheduling applications, however, we must also consider an additional class of discrete and infrequent disturbances, such as breakdowns and unplanned maintenance. In this paper, we establish that nominal MPC is robust, in a stochastic context, to this class of discrete and infrequent disturbances. We illustrate these results with a nonlinear blending example.

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Reducing the computational effort of min-max model predictive control with regional feedback laws

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NMPC through qLPV Embedding: A Tutorial Review of Different Approaches

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