arXiv subjects

Daniel Limon

Publications and source records attributed to Daniel Limon.

29 records · Page 2Linked to original sources

arXiv2022

Restart of accelerated first order methods with linear convergence under a quadratic functional growth condition

Accelerated first order methods, also called fast gradient methods, are popular optimization methods in the field of convex optimization. However, they are prone to suffer from oscillatory behaviour that slows their convergence when medium to high accuracy is desired. In order to address this, restart schemes have been proposed in the literature, which seek to improve the practical convergence by suppressing the oscillatory behaviour. This paper presents a restart scheme for accelerated first order methods for which we show linear convergence under the satisfaction of a quadratic functional growth condition, thus encompassing a broad class of non-necessarily strongly convex optimization problems. Moreover, the worst-case convergence rate is comparable to the one obtained using a (generally non-implementable) optimal fixed-rate restart strategy. We compare the proposed algorithm with other restart schemes by applying them to a model predictive control case study.

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arXiv2021

Implementation of model predictive control for tracking in embedded systems using a sparse extended ADMM algorithm

This article presents a sparse, low-memory footprint optimization algorithm for the implementation of the model predictive control (MPC) for tracking formulation in embedded systems. This MPC formulation has several advantages over standard MPC formulations, such as an increased domain of attraction and guaranteed recursive feasibility even in the event of a sudden reference change. However, this comes at the expense of the addition of a small amount of decision variables to the MPC's optimization problem that complicates the structure of its matrices. We propose a sparse optimization algorithm, based on an extension of the alternating direction method of multipliers, that exploits the structure of this particular MPC formulation. We describe the controller formulation and detail how its structure is exploited by means of the aforementioned optimization algorithm. We show closed-loop simulations comparing the proposed solver against other solvers and approaches from the literature.

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arXiv2021

Oracle-based economic predictive control

This paper presents an economic model predictive controller, under the assumption that the only measurable signal of the plant is the economic cost to be minimized. In order to forecast the evolution of this economic cost for a given input trajectory, a prediction model with a NARX structure, the so-called oracle, is proposed. Sufficient conditions to ensure the existence of such oracle are studied, proving that it can be derived for a general nonlinear system if the economic cost function is a Morse function. Based on this oracle, economic model predictive controllers are proposed, and their stability is demonstrated in nominal conditions under a standard dissipativity assumption. The viability of these controllers in practical settings (where the oracle may provide imperfect predictions for generic inputs) is proven by means of input-to-state stability. These properties have been illustrated in a case study based on a continuously stirred tank reactor.

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arXiv2021

Suboptimal multirate MPC for five-level inverters

The application of multilevel converters to renewable energy systems is a growing topic due to their advantages in energy efficiency. Regarding its control, model predictive control (MPC) has become very appealing due to its natural consideration of discrete inputs, its optimization capability, and the present-day availability of powerful processing hardware. The main drawback of MPC compared to other control techniques in this field is that the control input is held constant during the sampling period, and it is usually difficult or even impossible to reduce this sampling period because of hardware limitations. For this reason, a multirate MPC algorithm is proposed, which allows to change the control input several times within the sampling period. The optimization problem is simplified and made suboptimal to substantially decrease computational burden. This approach is tested in simulation on a three-phase, five-level diode-clamped converter (DCC) operating in inverted mode with a three-phase resistive load. Results show significant reduction in harmonic distortion at the cost of an increase in the number of commutations with respect to a standard MPC operating at the same sampling period.

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arXiv2021

Real-time implementation of MPC for tracking in embedded systems: Application to a two-wheeled inverted pendulum

This article presents the real-time implementation of the model predictive control for tracking formulation to control a two-wheeled inverted pendulum robot. This formulation offers several advantages over standard MPC formulations at the expense of the addition of a small number of decision variables, which complicates the inner structure of the matrices of the optimization problem. We implement a sparse solver, based on an extension of the alternating direction method of multipliers, in the system's embedded hardware. The results indicate that the solver is suitable for controlling a real system with sample times in the range of milliseconds using current, readily-available hardware.

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arXiv2021

Harmonic based model predictive control for set-point tracking

This paper presents a novel model predictive control (MPC) formulation for set-point tracking. Stabilizing predictive controllers based on terminal ingredients may exhibit stability and feasibility issues in the event of a reference change for small to moderate prediction horizons. In the MPC for tracking formulation, these issues are solved by the addition of an artificial equilibrium point as a new decision variable, providing a significantly enlarged domain of attraction and guaranteeing recursive feasibility for any reference change. However, it may suffer from performance issues if the prediction horizon is not large enough. This paper presents an extension of this formulation where a harmonic artificial reference is used in place of the equilibrium point. The proposed formulation achieves even greater domains of attraction and can significantly outperform other MPC formulations when the prediction horizon is small. We prove the asymptotic stability and recursive feasibility of the proposed controller, as well as provide guidelines for the design of its main ingredients. Finally, we highlight its advantages with a case study of a ball and plate system.

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arXiv2020

Online learning-based Model Predictive Control with Gaussian Process Models and Stability Guarantees

Model predictive control allows to provide high performance and safety guarantees in the form of constraint satisfaction. These properties, however, can be satisfied only if the underlying model, used for prediction, of the controlled process is sufficiently accurate. One way to address this challenge is by data-driven and machine learning approaches, such as Gaussian processes, that allow to refine the model online during operation. We present a combination of an output feedback model predictive control scheme and a Gaussian process-based prediction model that is capable of efficient online learning. To this end, the concept of evolving Gaussian processes is combined with recursive posterior prediction updates. The presented approach guarantees recursive constraint satisfaction and input-to-state stability with respect to the model-plant mismatch. Simulation studies underline that the Gaussian process prediction model can be successfully and efficiently learned online. The resulting computational load is significantly reduced via the combination of the recursive update procedure and by limiting the number of training data points while maintaining good performance.

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arXiv2020

Nonlinear MPC for Tracking for a Class of Non-Convex Admissible Output Sets

This paper presents an extension to the nonlinear Model Predictive Control for Tracking scheme able to guarantee convergence even in cases of non-convex output admissible sets. This is achieved by incorporating a convexifying homeomorphism in the optimization problem, allowing it to be solved in the convex space. A novel class of non-convex sets is also defined for which a systematic procedure to construct a convexifying homeomorphism is provided. This homeomorphism is then embedded in the Model Predictive Control optimization problem in such a way that the homeomorphism is no longer required in closed form. Finally, the effectiveness of the proposed method is showcased through an illustrative example.

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arXiv2020

Reference Dependent Invariant Sets: Sum of Squares Based Computation and Applications in Constrained Control

The goal of this paper is to present a systematic method to compute reference dependent positively invariant sets for systems subject to constraints. To this end, we first characterize these sets as level sets of reference dependent Lyapunov functions. Based on this characterization and using Sum of Squares (SOS) theory, we provide a polynomial certificate for the existence of such sets. Subsequently, through some algebraic manipulations, we express this certificate in terms of a Semi-Definite Programming (SDP) problem which maximizes the size of the resulting reference dependent invariant sets. We then present the results of implementing the proposed method to an example system and propose some variations of the proposed method that may help in reducing the numerical issues of the method. Finally, the proposed method is employed in the Model Predictive Control (MPC) for Tracking scheme to compute the terminal set, and in the Explicit Reference Governor (ERG) scheme to compute the so-called Dynamic Safety Margin (DSM). The effectiveness of the proposed method in each of the schemes is demonstrated through a simulation study.

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arXiv2020

Elimination of Redundant Polynomial Constraints and Its Use in Constrained Control

The reduction of constraints to obtain minimal representations of sets is a very common problem in many engineering applications. While well-established methodologies exist for the case of linear constraints, the problem of how to detect redundant non-linear constraints is an open problem. In this paper we present a novel methodology based on Sum of Squares for the elimination of redundant polynomial constraints. The paper also presents some relevant applications of the presented method to constrained control problems. In particular, we show how the proposed method can be used in the Model Predictive Control and in the Reference Governor frameworks to reduce the computational burden of the online algorithms. Furthermore, this method can also be used to eliminate the terminal constraints in MPC in a simple way that is independent from the cost function.

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arXiv2019

Restart FISTA with Global Linear Convergence

Fast Iterative Shrinking-Threshold Algorithm (FISTA) is a popular fast gradient descent method (FGM) in the field of large scale convex optimization problems. However, it can exhibit undesirable periodic oscillatory behaviour in some applications that slows its convergence. Restart schemes seek to improve the convergence of FGM algorithms by suppressing the oscillatory behaviour. Recently, a restart scheme for FGM has been proposed that provides linear convergence for non strongly convex optimization problems that satisfy a quadratic functional growth condition. However, the proposed algorithm requires prior knowledge of the optimal value of the objective function or of the quadratic functional growth parameter. In this paper we present a restart scheme for FISTA algorithm, with global linear convergence, for non strongly convex optimization problems that satisfy the quadratic growth condition without requiring the aforementioned values. We present some numerical simulations that suggest that the proposed approach outperforms other restart FISTA schemes.

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