Application Of Nonlinear Model Predictive Control Research Articles

Tracking Control of Unforced and Forced Equilibrium Positions of the Pendubot System: A Nonlinear MHE and MPC Approach

This paper presents a unified control scheme of the Pendubot based on nonlinear model predictive control (NMPC) and nonlinear moving horizon estimation (NMHE) with the objective of point-to-point tracking its unstable unforced and ultimately forced equilibrium positions. In order to implement it on this fast, underactuated mechatronic system, we employ the Gauss–Newton real-time iteration scheme tailored to obtain the efficient solution of the underlying nonlinear optimization problems via sequential quadratic programming. The control performance is experimentally assessed on a real-world laboratory setup featuring an execution timing analysis and hints how to further improve the computational efficiency of the proposed nonlinear estimation control scheme. Even nowadays, the number of practical NMPC applications in the millisecond range is still rather limited, and the presented NMHE-based NMPC of the Pendubot thus also represents a unique case study for control practitioners.

Passivity based nonlinear model predictive control (PNMPC) of multi-robot systems for space applications.

In the past 2 decades, there has been increasing interest in autonomous multi-robot systems for space use. They can assemble space structures and provide services for other space assets. The utmost significance lies in the performance, stability, and robustness of these space operations. By considering system dynamics and constraints, the Model Predictive Control (MPC) framework optimizes performance. Unlike other methods, standard MPC can offer greater robustness due to its receding horizon nature. However, current literature on MPC application to space robotics primarily focuses on linear models, which is not suitable for highly non-linear multi-robot systems. Although Nonlinear MPC (NMPC) shows promise for free-floating space manipulators, current NMPC applications are limited to unconstrained non-linear systems and do not guarantee closed-loop stability. This paper introduces a novel approach to NMPC using the concept of passivity to multi-robot systems for space applications. By utilizing a passivity-based state constraint and a terminal storage function, the proposed PNMPC scheme ensures closed-loop stability and a superior performance. Therefore, this approach offers an alternative method to the control Lyapunov function for control of non-linear multi-robot space systems and applications, as stability and passivity exhibit a close relationship. Finally, this paper demonstrates that the benefits of passivity-based concepts and NMPC can be combined into a single NMPC scheme that maintains the advantages of each, including closed-loop stability through passivity and good performance through one-line optimization in NMPC.

PANTR: A Proximal Algorithm With Trust-Region Updates for Nonconvex Constrained Optimization

This work presents PANTR, an efficient solver for nonconvex constrained optimization problems, that is well-suited as an inner solver for an augmented Lagrangian method. The proposed scheme combines forward-backward iterations with solutions to trust-region subproblems: the former ensures global convergence, whereas the latter enables fast update directions. We discuss how the algorithm is able to exploit exact Hessian information of the smooth objective term through a linear Newton approximation, while benefiting from the structure of box-constraints or ℓ1-regularization. An open-source C++ implementation of PANTR is made available as part of the NLP solver library ALPAQA. Finally, the effectiveness of the proposed method is demonstrated in nonlinear model predictive control applications.

Observer-based fast nonlinear MPC for multi-DOF maglev positioning system: Theory and experiment

This paper presents an observer-based fast nonlinear model predictive control (NMPC) scheme for translation control of magnetically levitated (maglev) positioning system subject to input saturation. The motivation lies in the improvement of transient characteristics and control performance for positioning systems. The nonlinear dynamical translation model of the maglev positioning system is derived that does not affect the rotation dynamics with special current conditions. The disturbance estimation, obtained by nonlinear disturbance observer, is introduced in the state receding prediction to compensate the errors caused by disturbances and uncertainties. To reduce the computational burden, the stability of the proposed NMPC is established without using any stability-related terminal costs or constraints, and only the short prediction horizon is required for real-time feasibility. The online optimization algorithm underlying the NMPC scheme takes the process constraints into account, and solves the optimal control problem using a parallel structure at each iteration. Comparative experiments are carried out on the positioner to validate the proposed controller has the outperformance in transient/steady-state trajectory tracking, frequency characteristics and robustness against disturbances. The proposed scheme also provides a guidance for the application of NMPC in industrial mechatronic system with fast dynamics.

Experimental operation of a solar‐driven climate system with thermal energy storages using mixed‐integer nonlinear model predictive control

AbstractThis work presents the results of experimental operation of a solar‐driven climate system using mixed‐integer nonlinear model predictive control (MPC). The system is installed in a university building and consists of two solar thermal collector fields, an adsorption cooling machine with different operation modes, a stratified hot water storage with multiple inlets and outlets as well as a cold water storage. The system and the applied modeling approach is described and a parallelized algorithm for mixed‐integer nonlinear MPC and a corresponding implementation for the system are presented. Finally, we show and discuss the results of experimental operation of the system and highlight the advantages of the mixed‐integer nonlinear MPC application.

A roadmap for in silico development and evaluation of industrial NMPC applications: A practical case study

Linear Model Predictive Control (MPC) can be considered as the state of the art advanced process control technology in model-based automation of continuous chemical processes. For (semi-)batch processes, that often present a strongly nonlinear (or even unstable) behavior in combination with fast dynamics, Nonlinear Model Predictive Control (NMPC) is a more suited technology. However, online applications of NMPC have a hard time to penetrate in industry despite methodological developments, tools and examples in academia. In this paper, we propose a roadmap to argue against the intrinsic reasons and practical limitations that slow down the practical online applications of NMPC. This roadmap is applied to an existing semi-batch plant as a practical case study. The results have shown that the NMPC algorithm can provide an improved control, namely a better tracking of the main process variables, a reduction in the reaction time, and robustness with respect to model-plant mismatch and disturbances.

FPGA Implementation Framework for Low Latency Nonlinear Model Predictive Control

Embedded implementation of real-time Nonlinear Model Predictive Control (NMPC) is extremely challenging and complex. This paper presents a framework for implementation of NMPC on Field Programmable Gate Array (FPGA). We show the step-by-step procedure of FPGA implementation framework design of NMPC for a case study of 2D-crane system. In the implementation, we used GRAMPC software to construct NMPC and subsequently generate an FPGA specific low-level C/C++ code of the optimization solver. Generated C/C++ code is optimized for memory, speed, and resource utilization by the customized approach of applying pipelining and directives using Xilinx Vivado HLS toolchain. The NMPC is implemented on a Xilinx’s ZYNQ-7000 SoC ZC706 FPGA board. The detailed analysis of the controller computational complexity in terms of memory, resource utilization, clock, and power consumption is presented. The performance of implemented NMPC is verified through Hardware-in-the-Loop (HIL) co-simulation using system generator tool. The presented results show the feasibility of FPGA-based GRAMPC framework for ultra-fast applications of NMPC.

An Adaptive Partial Sensitivity Updating Scheme for Fast Nonlinear Model Predictive Control

In recent years, efficient optimization algorithms for Nonlinear Model Predictive Control (NMPC) have been proposed, that significantly reduce the on-line computational time. In particular, direct multiple shooting and Sequential Quadratic Programming (SQP) are used to efficiently solve Nonlinear Programming (NLP) problems arising from continuous-time NMPC applications. One of the computationally demanding steps for on-line optimization is the computation of sensitivities of the nonlinear dynamics at every sampling instant, especially for systems of large dimensions, strong stiffness, and when using long prediction horizons. In this paper, within the algorithmic framework of the Real-Time Iteration (RTI) scheme based on multiple shooting, an inexact sensitivity updating scheme is proposed, that performs a partial update of the Jacobian of the constraints in the NLP. Such update is triggered by using a Curvature-like Measure of Nonlinearity (CMoN), so that only sensitivities exhibiting highly nonlinear behaviour are updated, thus adapting to system operating conditions and possibly reducing the computational burden. An advanced tuning strategy for the updating scheme is provided to automatically determine the number of sensitivities being updated, with a guaranteed bounded error on the Quadratic Programming (QP) solution. Numerical and control performance of the scheme is evaluated by means of two simulation examples performed on a dedicated implementation. Local convergence analysis is also presented and a tunable convergence rate is proven, when applied to the SQP method.

Demonstration of non-linear model predictive control of post-combustion CO2 capture processes

Nonlinear model predictive control applications have been deployed on two large pilot plants for post combustion CO2 capture. The control objective is formulated in such a way that the CO2 capture ratio is controlled at a desired value, while the reboiler duty is formulated as an unreachable maximum constraint. With a correct tuning, it is demonstrated that the controllers automatically compensate for disturbances in flue gas rates and compositions to obtain the desired capture ratio while the reboiler duty is minimized. The applications are able to minimize the transient periods between two different capture rates with the use of minimum reboiler duty.

Recent Advances in Quadratic Programming Algorithms for Nonlinear Model Predictive Control

Over the past decades, the advantages of optimization-based control techniques over conventional controllers inspired developments that enabled the use of model predictive control (MPC) in applications with very high sampling rates. Since at the heart of most linear and nonlinear MPC controllers resides a quadratic programming (QP) solver, the implementation of efficient algorithms that exploit the underlying problem structure drew the attention of many researchers and the progress in the field has been remarkable. The aim of this paper is to summarize the main algorithmic advances in the field and to provide a consistent benchmark between a selection of software tools that have been recently developed. The code that was used for the simulations is publicly available for readers that wish to reproduce the results or test the benchmarked solvers on their own nonlinear MPC applications.

Data-driven model reduction-based nonlinear MPC for large-scale distributed parameter systems

Model predictive control (MPC) has been effectively applied in process industries since the 1990s. Models in the form of closed equation sets are normally needed for MPC, but it is often difficult to obtain such formulations for large nonlinear systems. To extend nonlinear MPC (NMPC) application to nonlinear distributed parameter systems (DPS) with unknown dynamics, a data-driven model reduction-based approach is followed. The proper orthogonal decomposition (POD) method is first applied off-line to compute a set of basis functions. Then a series of artificial neural networks (ANNs) are trained to effectively compute POD time coefficients. NMPC, using sequential quadratic programming is then applied. The novelty of our methodology lies in the application of POD's highly efficient linear decomposition for the consequent conversion of any distributed multi-dimensional space-state model to a reduced 1-dimensional model, dependent only on time, which can be handled effectively as a black-box through ANNs. Hence we construct a paradigm, which allows the application of NMPC to complex nonlinear high-dimensional systems, even input/output systems, handled by black-box solvers, with significant computational efficiency. This paradigm combines elements of gain scheduling, NMPC, model reduction and ANN for effective control of nonlinear DPS. The stabilization/destabilization of a tubular reactor with recycle is used as an illustrative example to demonstrate the efficiency of our methodology. Case studies with inequality constraints are also presented.

Cone Crusher Model Identification Using Block-Oriented Systems with Orthonormal Basis Functions

In this paper, block-oriented systems with linear parts based on Laguerre functions is used to approximation of a cone crusher dynamics. Adaptive recursive least squares algorithm is used to identification of Laguerre model. Various structures of Hammerstein, Wiener, Hammerstein-Wiener models are tested and the MATLAB simulation results are compared. The mean square error is used for models validation. It has been found that Hammerstein-Wiener with orthonormal basis functions improves the quality of approximation plant dynamics. The mean square error for this model is 11% on average throughout the considered range of the external disturbances amplitude. The analysis also showed that Wiener model cannot provide sufficient approximation accuracy of the cone crusher dynamics. During the process it is unstable due to the high sensitivity to disturbances on the output. The Hammerstein-Wiener model will be used to the design nonlinear model predictive control application.

An off-line NMPC strategy for continuous-time nonlinear systems using an extended modal series method

This paper presents a new off-line nonlinear model predictive control (NMPC) approach for continuous-time affine-input nonlinear systems. In this approach, the NMPC-related nonlinear two-point boundary value problem derived from the Pontryagin’s maximum principle is solved by the extended modal series method. The resulting suboptimal control law explicitly depends on the initial conditions and is updated by replacing the initial conditions with the new state measurements in future sampling instants. Therefore, there is no need to repeat the recursive online optimization process in each sampling instant. Since the applicability of NMPC is generally restricted by computational burden of the online optimization, we propose an NMPC scheme, which not only reduces the online computational burden significantly, but also can be applied to fast dynamic systems with short prediction horizons. An efficient algorithm is presented which approximates the order of the modal series such that feasibility of the optimization problem is guaranteed. Closed-loop stability of the proposed NMPC approach is shown using the off-line terminal region calculations suggested in quasi-infinite horizon NMPC scheme. The applicability and effectiveness of the proposed approach are illustrated by two numerical examples.

Cone Crusher Model Identification Using Block-Oriented Systems with Orthonormal Basis Functions

In this paper, block-oriented systems with linear parts based on Laguerre functions is used to approximation of a cone crusher dynamics. Adaptive recursive least squares algorithm is used to identification of Laguerre model. Various structures of Hammerstein, Wiener, Hammerstein-Wiener models are tested and the MATLAB simulation results are compared. The mean square error is used for models validation.It has been found that Hammerstein-Wiener with orthonormal basis functions improves the quality of approximation plant dynamics. The mean square error for this model is 11% on average throughout the considered range of the external disturbances amplitude. The analysis also showed that Wiener model cannot provide sufficient approximation accuracy of the cone crusher dynamics. During the process it is unstable due to the high sensitivity to disturbances on the output.The Hammerstein-Wiener model will be used to the design nonlinear model predictive control application.

Nonlinear Model Predictive Control of PEM Fuel Cell Systems for Generation of Exhaust Gas with Low Oxygen Content

Polymer electrolyte membrane (PEM) fuel cells are highly efficient energy converters and provide electrical energy, cathode exhaust gas with low oxygen concentration and water. For aircraft applications, PEM fuel cell systems are investigated as replacement for auxiliary power units (APU) that are currently used for electrical power generation. The system investigated is intended for generation of oxygen depleted cathode exhaust air (ODA), which must have a low oxygen concentration. A challenging task is controlling the fuel cell system for this product and simultaneously keeping fuel cell stack degradation, voltage losses and stack damage as low as possible as well as keeping the system within operational limitations such as bounds and gradients on control parameters. This constrained control task is attacked by a nonlinear model predictive control (NMPC) strategy. This paper demonstrates the applicability of NMPC for a fuel cell system. The prediction horizon has a significant influence on control performance and computational cost. Results of Matlab® simulations are shown.

Mixed–-Level Iteration Schemes for Nonlinear Model Predictive Control

In general, numerical schemes for nonlinear model predictive control (NMPC) require the (approximate) solution of a nonlinear program in each sample for feedback generation. Thus, the application of NMPC to processes that need fast feedback poses a major computational challenge. Recently, new multi–level iteration schemes have been proposed, extending the well–known idea of real–time iterations. These algorithms take into account different time scales inherent in the dynamic model by updating the data of the feedback–generating quadratic program (QP), i.e., Hessians and Jacobians, gradients, and constraint residuals, on different levels. In this contribution we consider new mixed–level updates of the QP data, which interval–wise apply different update levels. In particular, we apply higher–level updates more frequently on the first intervals of the control horizon, given their importance in the context of model predictive control in general. Targeting at modern computers with multi–core processing units, we describe an efficient parallel implementation of the mixed–level iteration approach and apply it to a benchmark problem from automotive engineering.

Set membership approximation of discontinuous nonlinear model predictive control laws

In this paper, the use of Set Membership (SM) methods is investigated, in order to derive off-line an approximation of a discontinuous nonlinear model predictive control (NMPC) law. The approximating function can then be evaluated on-line, instead of solving the nonlinear program embedded in the NMPC scheme. This way, a significant decrease of computational times may be obtained, thus allowing the application of NMPC also to systems with “fast” dynamics. It is shown that the knowledge of the discontinuities is needed to achieve an approximated controller with arbitrarily small approximation error. By exploiting such a knowledge, SM techniques already developed for the case of continuous NMPC laws are generalized in order to approximate discontinuous ones. The stability of the origin of the closed loop system with the approximated control law is analyzed, and a numerical example is employed to illustrate the features of the proposed approach.

Stability of Economically-Oriented NMPC with Periodic Constraint

AbstractIn smart grid environment, it is economically attractive to manage the load with respect prices and demands that could change periodically. Thus it is desired to operate some plants and system applications in a cyclic fashion. Model predictive control (MPC) and nonlinear MPC (NMPC) are widely accepted advanced control tools in process industry, due to their advantages of easily handling constraints and multi-input-multi-output systems. Nevertheless, the objective of the majority of NMPC applications is to track a predefined set point. This work proposes an economically-oriented NMPC formulation with periodic constraints that directly deal with systems that exhibit cyclic steady state behavior. The periodic constraint ensures the system converges to a cyclic steady state. In addition, nominal stability of the proposed NMPC formulation is analyzed, where we introduce a transformed system with the origin as the steady state. Hence, a Lyapunov function can be established at the steady state for the transformed system that is asymptotically stable at the origin. Consequently, asymptotic stability of the original system at the cyclic steady state can be inferred from the stability of the transformed system.

Nonlinear model predictive control of a pH neutralization process based on Wiener–Laguerre model

In this paper, Laguerre filters and simple polynomials are used respectively as linear and nonlinear parts of a Wiener structure. The obtained model structure is the so-called Wiener–Laguerre model. This model is used to evaluate identification of a pH neutralization process. Then the model is used in a nonlinear model predictive control framework based on the sequential quadratic programming (SQP) algorithm. Various orders of Laguerre filters and nonlinear polynomials are tested, and the results are compared for the validation of these models. Validation results for various orders suggest that in order to have a good trade-off between simplicity of the model and its corresponding fitness, a second order nonlinear polynomial along with two Laguerre filters may be selected. The fitness of this model according to variance account for (VAF) criterion is 92.32%, which is completely acceptable for nonlinear model predictive control applications. Then the identified Wiener–Laguerre model is used for nonlinear model predictive control and the results are compared with model predictive control in which just Wiener model was used for identification. It is shown that the use of the Wiener–Laguerre structure improves the quality of modeling together with the rate of convergence of SQP in a reasonable time. Furthermore, these results are also compared with the performance of a linear model predictive controller based on Laguerre model to provide a fair comparison between linear and nonlinear systems. © 2008 Elsevier B.V. All rights reserved.

Fast reduced multiple shooting methods for nonlinear model predictive control

Nonlinear model predictive control (NMPC) has become an appealing control concept for chemical processes because it can directly take into account the multivariable character, nonlinearities and constraints on inputs and states. While linear MPC is frequently applied in the process industries, practical applications of NMPC are rare. One reason is the relatively large computation time still needed to solve the nonlinear constrained optimization problems inherent to NMPC schemes. For faster controller response, the use of an extended partially reduced SQP method within the direct multiple shooting framework is proposed. This method is implemented in the code MSOPT which allows for an accelerated calculation of the directional derivatives and thereby saves computation time. Furthermore, this method is adapted to the real-time iteration scheme, which leads to a tremendous reduction of the time needed to provide new controls. The new NMPC variant is compared with a previously introduced scheme based on the optimization code MUSCOD-II, where the extended reduction is not in use. Numerical results are presented for both NMPC schemes and a decentralized PI controller in closed loop with a simulation model of a highly nonlinear thermally coupled distillation column which separates a ternary mixture.