Real-time Nonlinear Model Predictive Control Research Articles

A Fast NMPC Approach based on Bounded-Variable Nonlinear Least Squares

Abstract In this paper, we present an approach for real-time nonlinear model predictive control (NMPC) of constrained multivariable dynamical systems described by nonlinear difference equations. The NMPC problem is formulated by means of a quadratic penalty function as an always feasible, sparse nonlinear least-squares problem subject to box constraints on the decision variables. This formulation is exploited by the proposed fast solution algorithm, which is based on the Gauss-Newton method and bounded-variable least squares (BVLS). Linear time-invariant and linear time-varying model predictive control based on BVLS are special cases of the proposed NMPC framework. The proposed approach and its benefits are demonstrated through a typical numerical example in simulation.

Real-Time Nonlinear Model Predictive Control of a Battery–Supercapacitor Hybrid Energy Storage System in Electric Vehicles

A nonlinear model predictive control (NMPC) method has been presented as the energy management strategy of a battery–supercapacitor (SC) hybrid energy storage system (H-ESS) in a Toyota Rav4EV. For the first time, the NMPC has been shown to be real-time implementable for these fast systems. The performance of the proposed controller has been demonstrated against a linear model predictive control (LMPC) and a rule-based control (RBC) strategy. The NMPC shows to outperform the RBC even with no prior knowledge of the future trip available. The NMPC also shows performance improvement over the LMPC by compensating for the error accompanied by linearization in LMPCs. Hardware-in-the-loop (HiL) testing has been performed to demonstrate the NMPC capability for real-time implementation in a battery–SC H-ESS. Upon carefully choosing the prediction horizon and control horizon size, as well as the maximum number of iterations, the turn-around time for the control update is shown to fall far below the necessary sampling time of 10 ms in vehicle control.

An algorithm for gradient-based dynamic optimization of UV flash processes

This paper presents a novel single-shooting algorithm for gradient-based solution of optimal control problems with vapor–liquid equilibrium constraints. Such optimal control problems are important in several engineering applications, for instance in control of distillation columns, in certain two-phase flow problems, and in operation of oil reservoirs. The single-shooting algorithm uses an adjoint method for the computation of gradients. Furthermore, the algorithm uses either a simultaneous or a nested approach for the numerical solution of the dynamic vapor–liquid equilibrium model equations. Two numerical examples illustrate that the simultaneous approach is faster than the nested approach and that the efficiency of the underlying thermodynamic computations is important for the overall performance of the single-shooting algorithm. We compare the performance of different optimization software as well as the performance of different compilers in a Linux operating system. These tests indicate that real-time nonlinear model predictive control of UV flash processes is computationally feasible.

Thermal Management in Plug-In Hybrid Electric Vehicles: A Real-Time Nonlinear Model Predictive Control Implementation

A real-time nonlinear model predictive control (NMPC) for the thermal management (TM) of the electrical components cooling circuit in a Plug-In Hybrid Electric Vehicle (PHEV) is presented. The electrical components are highly temperature sensitive and, therefore, working out of the ranges recommended by the manufacturer can lead to their premature aging or even failure. Consequently, the goals for an accurate and efficient TM are to keep the main component, the Li-ion battery, within optimal working temperatures, and to consume the minimum possible electrical energy through the cooling circuit actuators. This multi-objective requirement is formulated as a finite-horizon optimal control problem (OCP) that includes a multi-objective cost function, several constraints, and a prediction model especially suitable for optimization. The associated NMPC is performed on real time by the optimization package MUSCOD-II and is validated in three different repeatable test-drives driven with a PHEV. Starting from identical conditions, each cycle is driven once being the cooling circuit controlled with NMPC and once with a conventional approach based on a finite-state machine. Compared to the conventional strategy, the NMPC proposed here results in a more accurate and healthier temperature performance, and at the same time, leads to reductions in the electrical consumption up to 8%.

Implicit Integrators for Linear Dynamics Coupled to a Nonlinear Static Feedback and Application to Wind Turbine Control

Efficient integration schemes with sensitivity propagation are crucial for deploying real-time Nonlinear Model Predictive Control on systems described by continuous time dynamics. Implicit integration schemes are preferred when stiff modes are present in the model equations, or when the equations are implicit. We consider here a class of models, where the dynamics are linear, but coupled to a general static nonlinear feedback function. We propose a collocation-based implicit integration scheme where a lifting-condensing approach is used to exploit this specific structure to reduce the size of the linear algebra underlying the integrator. This technique yields a significant reduction in the computational complexity of performing the system integration and sensitivity analysis, when the static nonlinearity is of much smaller dimensions than the complete dynamics. The proposed method is illustrated on a complex wind turbine model, resulting in a significant gain of computational time in the linear algebra, and an overall gain of computational time of a factor 2.

Real-Time Nonlinear Model Predictive Control of a Transport–Reaction System

Two real-time nonlinear model predictive control (NMPC) algorithms for a transport–reaction system are designed. The system is modeled by a hyperbolic partial differential equation and discretized by means of a two-time-level semi-implicit semi-Lagrangian scheme. For the resulting lumped-parameter system, a constrained optimal control problem is formulated and state constraints are implemented in the form of barrier functions. The NMPC algorithms perform a single step or several steps of an iterative solution routine of the optimal control problem at every sampling point. With this suboptimal solution strategy, a fixed maximum evaluation time and execution in real time are guaranteed. An analysis of the nominal stability is provided for one NMPC scheme. The robustness of the controllers is evaluated for an example problem, where a nonisothermal plug-flow reactor with irreversible exothermic reactions is considered. The control objectives are to limit the maximum reactor temperature (avoid hot spots) and to maximize the process output.

A Reactive Walking Pattern Generator Based on Nonlinear Model Predictive Control

The contribution of this work is to show that real-time nonlinear model predictive control (NMPC) can be implemented on position controlled humanoid robots. Following the idea of “walking without thinking,” we propose a walking pattern generator that takes into account simultaneously the position and orientation of the feet. A requirement for an application in real-world scenarios is the avoidance of obstacles. Therefore, this letter shows an extension of the pattern generator that directly considers the avoidance of convex obstacles. The algorithm uses the whole-body dynamics to correct the center of mass trajectory of the underlying simplified model. The pattern generator runs in real-time on the embedded hardware of the humanoid robot HRP-2 and experiments demonstrate the increase in performance with the correction.

Power Control for Efficient Operation of a PEM Fuel Cell System by Nonlinear Model Predictive Control

Efficiency of polymer electrolyte membrane (PEM) fuel cells depends directly on anode and cathode gas pressures, and stack temperature. For an efficient chemical reaction and a safe operation for both stationary and dynamic loads all values must fall within specific ranges. This paper deals with realtime nonlinear model predictive control (NMPC) of electrical power for a PEM fuel cell system. The NMPC is split into three nonlinear optimization problems to calculate the optimal stack current and the supply of anode and cathode with hydrogen and oxygen. For real-time application the predictive control algorithm is matched to the sampling time of the system. This NMPC handles all kinds of dynamic load changes, while maximizing power efficiency without a steady state error. Modeling of relevant anode and cathode site fuel cell peripherals is shown. Experimental results on a test bench with a 4.4kW PEM fuel cell are presented.

Block Condensing for Fast Nonlinear MPC with the Dual Newton Strategy

A recently proposed quadratic programming (QP) solver dedicated to bandstructured optimal control problems as they may arise in the field of model predictive control (MPC) is the dual Newton strategy, as implemented in the publicly available software qpDUNES. Such a structure-exploiting solver is often preferable over the use of condensing for problems with long horizons. In this paper, we propose an effcient and exible algorithm for real-time nonlinear MPC which tries to combine the advantages of the two approaches by means of a block condensing technique. Using a nontrivial benchmarking problem, we show that this approach can improve the performance of the dual Newton strategy by a factor of 10 in terms of total computation times while reducing the delay between state estimate and feedback even further. Additionally, an open-source implementation of the resulting scheme is provided as part of the ACADO code generation tool.

Real-Time NMPC for Semi-Automated Highway Driving of Long Heavy Vehicle Combinations

This work presents a Nonlinear Model Predictive Control (NMPC) approach to realtime trajectory generation for highway driving with long truck combinations. In order to consider all relevant information for the road and the surrounding traffic, we formulate a fininite-horizon optimal control problem (OCP) that incorporates a prediction model in spatial coordinates for the vehicle and the surrounding traffic. This allows road properties (such as curvature) to appear as known variables in the prediction horizon. The objective function of the resulting constrained nonlinear least-squares problem provides a trade-off between tracking performance, driver comfort, and keeping a comfortable distance from fellow road users. The OCP is solved with a direct multiple shooting solution method implemented in a real-time iteration scheme using ACADO code generation and compared with a feedback scheme that solves the entire nonlinear program in each time step with the interior point solver IPOPT. To illustrate the efficacy and the real-time implementability of the methodology, simulation results are presented for a high way merging scenario of a long heavy vehicle combination.

Real-time Nonlinear Model Predictive Path-Following Control of a Laboratory Tower Crane

A path-following controller is developed and applied to a laboratory tower crane. The control task is to move a load along a predefined geometric path. The time evolution along the path is not fixed but left as a degree of freedom to be determined by the controller. In order to be able to account for system constraints, a model predictive control scheme is adopted with special focus on real-time feasibility with small sampling times. The resulting controller is applied to a laboratory-scale tower crane and validated through simulation studies and measurement results.

A novel real-time non-linear wavelet-based model predictive controller for a coupled tank system

This article presents the design, simulation and real-time implementation of a constrained non-linear model predictive controller for a coupled tank system. A novel wavelet-based function neural network model and a genetic algorithm online non-linear real-time optimisation approach were used in the non-linear model predictive controller strategy. A coupled tank system, which resembles operations in many chemical processes, is complex and has inherent non-linearity, and hence, controlling such system is a challenging task. Particularly important is low-level control where often instability and oscillatory responses are observed. This article designs a wavelet neural network with high predicting precision and time–frequency localisation characteristics for an online prediction model in the non-linear model predictive controller to show the effectiveness of this approach in controlling the liquid at low level. To speed up the training process, a fast global search stochastic non-linear conjugate wavelet gradient algorithm is initially used to train the wavelet neural network structure before the genetic algorithm optimisation technique is utilised to tune adaptively the wavelet neural network parameters. The non-linear model predictive controller algorithm is tested for both approaches: first, in a simulation using identified models, and second, in a real-time practical application to a single-input single-output system coupled tank system. The results show an excellent control performance with respect to mean square error and average control energy values obtained.

Nonlinear model-predictive control with hysteresis compensation of an electro-pneumatic clutch for truck applications

This contribution presents two real-time capable nonlinear model-predictive control (NMPC) approaches for an electro-pneumatic clutch for heavy trucks: a centralized control approach and a cascaded one. A clutch is necessary at start-up or during gear shifts to connect or disconnect the combustion engine and the gear box. This automated actuator disburdens the driver and provides the necessary actuation force according to the large torque typically transmitted through the drive train. The force characteristic of the clutch, however, is subject to hysteresis, which is described by a generalized Bouc–Wen model and used for a feedforward hysteresis compensation in the control algorithm. The proposed NMPC-algorithm involves (i) a minimization of the difference between the desired and predicted state vector at the end of the prediction horizon and (ii) flatness-based techniques to compute desired trajectories for the complete state vector as well as the control input. The optimal control is given by an additional, minimum-norm control input that minimizes the difference between the predicted state vector and the desired one at the end of the prediction horizon. Thereby, the computation effort of the NMPC approaches can be kept relatively small, and a real-time evaluation becomes possible. A reduced-order observer estimates an effective pressure in the clutch that also accounts for an uncertain disturbance force. Thereby, a disturbance compensation and a high tracking accuracy is achievable for the piston position as controlled variable. The efficiency of the two proposed control structures is emphasized by experimental results from a dedicated test rig.

Auto Generation of Implicit Integrators for Embedded NMPC with Microsecond Sampling Times

Algorithms for fast real-time Nonlinear Model Predictive Control (NMPC) for mechatronic systems face several challenges. They need to respect tight real-time constraints and need to run on embedded control hardware with limited computing power and memory. A combination of efficient online algorithms and code generation of explicit integrators was shown to be able to overcome these hurdles. This paper generalizes the idea of code generation to Implicit Runge-Kutta (IRK) methods with efficient sensitivity generation. It is shown that they often outperform existing auto-generated Explicit Runge-Kutta (ERK) methods. Moreover, the new methods allow to treat Differential Algebraic Equation (DAE) systems by NMPC with microsecond sampling times.

An auto-generated real-time iteration algorithm for nonlinear MPC in the microsecond range

In this paper, we present an automatic C-code generation strategy for real-time nonlinear model predictive control (NMPC), which is designed for applications with kilohertz sample rates. The corresponding code export module has been implemented within the software package ACADO Toolkit. It is capable of exporting fixed step-size integrators together with their sensitivities as well as a real-time Gauss–Newton method. Here, we employ the symbolic representation of optimal control problems in ACADO in order to auto-generate plain C-code which is optimized for final production. The exported code has been tested for model predictive control scenarios comprising constrained nonlinear dynamic systems with four states and a control horizon of ten samples. The numerical simulations show a promising performance of the exported code being able to provide feedback in much less than a millisecond.

Real-Time Optimization and Nonlinear Model Predictive Control of Processes Governed by Differential-Algebraic Equations

Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting method has been known for a while as a fast off-line method for optimization problems in ODE and later in DAE. Some factors crucial for its fast performance are briefly reviewed. The direct multiple shooting approach has been successfully adapted to the specific requirements of real-time optimization. Special strategies have been developed to effectively minimize the on-line computational effort, in which the progress of the optimization iterations is nested with the progress of the process. They use precalculated information as far as possible (e.g. Hessians, gradients and QP presolves for iterated reference trajectories) to minimize response time in case of perturbations. In typical real-time problems they have proven much faster than fast off-line strategies. Compared with an optimal feedback control computable upper bounds for the loss of optimality can be established that are small in practice. Numerical results for the Nonlinear Model Predictive Control (NMPC) of a high-purity distillation column subject to parameter disturbances are presented. # 2002 Published by Elsevier Science Ltd.