Slovakia

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

Nonlinear Model Predictive Control Conference 2021

Streamlining Active Set Method in MPC Using Cache Memory

model predictive control

active set method

cache replacement policies

cache memory

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

The analysis of convergence and stability conditions for Non-linear Model Predictive

Towards Necessary and Sufficient Stability Conditions for NMPC

The stability analysis of NMPC schemes has seen significant progress in recent years, which includes schemes with and without terminal ingredients such as penalties and constraints. In the context of economic MPC, turnpike properties, which are closely related to dissipativity properties of the underlying optimal control problem, have enabled novel insights on stability conditions. In this talk, we show that turnpike properties naturally enable to bound the required stabilizing horizon length in MPC. The main idea is to define the turnpike with respect to a level-set of the terminal penalty. This way we derive a bound on the stabilizing horizon which guarantees that a terminal constraint is satisfied without being explicitly stated in the underlying optimal control problem. A numerical example indicates that if the terminal set is not too small, the horizon bound is not overly conservative.

First Results on Turnpike Bounds for Stabilizing Horizons in NMPC

Predictive functional control (PFC) is the simplest model-based algorithm, equipped with the attributes of a fully fledged predictive controller but at the cost and complexity threshold of a standard PID regulator. It has proven benefits in controlling stable SISO dynamic systems, but similarly to its competitor PID, it loses efficacy when a challenging application is introduced. In this paper, we present a modified PFC approach, especially tailored for open-loop unstable processes, using pre-stabilisation to efficiently control the undesirable dynamics at hand. This is essentially a two-stage design scheme with implications for PFC tuning and constraint handling. The proposal, nevertheless, is straightforward and intuitive, and provides improved closed-loop control in the presence of external perturbations against the standard PFC, and significantly better performance overall compared to the common PID algorithm, as demonstrated in a numerical case-study.

Pre-Stabilised Predictive Functional Control for Open-Loop Unstable Dynamic Systems

Stabilizing conditions for nonlinear predictive control typically rely on standard Lyapunov functions and thus require a monotonically decreasing cost function. These conditions cannot certify stability of predictive controllers in the presence of non-monotonic cost functions. In this paper we develop new dissipativity-based stabilizing conditions for nonlinear predictive control that allow for non-monotonic cost functions. Firstly, we establish that dissipation inequalities with a cyclically negative supply imply asymptotic stability. Secondly, we show that closed-loop trajectories generated by predictive control satisfy a fundamental dissipation inequality. This enables dissipativity-based stabilizing conditions that do not require a special terminal cost and apply to both model-based and data-driven predictive control algorithms.

A Dissipativity-Based Framework for Analyzing Stability of Predictive Controllers

We accelerate nonlinear model predictive control with an approach that successively detects and removes inactive constraints from the optimal control problem. In every time step and for every constraint, the cost function value is compared to a bound that can be calculated offline. If the current cost function value drops below one of these bounds, the corresponding constraint can be removed. We

Accelerating Nonlinear Model Predictive Control by Constraint Removal

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

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Streamlining Active Set Method in MPC Using Cache Memory

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