Nonlinear Model Predictive Control Algorithm Research Articles

Fast Sensitivity-Based Nonlinear Economic Model Predictive Control with Degenerate NLP

We present a fast sensitivity-based nonlinear model predictive control (NMPC) algorithm, that can handle non-unique multipliers in the discretized dynamic optimization problem. Non-unique multipliers may arise, for example when path constraints are active for longer periods of the prediction horizon. This is a common situation in economic model predictive control. In such cases, the optimal nonlinear programming (NLP) solution often satisfies the Mangasarian-Fromovitz constraint qualification (MFCQ), which implies non-unique, but bounded multipliers. Consequently, any sensitivity-based fast NMPC scheme must allow for discontinuous jumps in the multipliers. In this paper, we apply a sensitivity-based path-following algorithm that allows multiplier jumps within the advance-step NMPC (asNMPC) framework. The path-following method consists of a corrector and a predictor step, which are computed by solving a system of linear equations, and a quadratic programming problem, respectively, and a multiplier jump step determined by the solution of a linear program. We demonstrate the proposed method on an economic NMPC case study with a CSTR.

Efficient Partial Condensing Algorithms for Nonlinear Model Predictive Control with Partial Sensitivity Update

Abstract In Nonlinear Model Predictive Control(NMPC), an optimal control problem (OCP) is solved repeatedly at every sampling instant. To satisfy the real-time restriction, modern methods tend to convert the OCP into structured Nonlinear Programming problems (NLP), which are approximately solved on-line. Real-Time Iteration is one of the promising NMPC algorithms that solves the NLP using a single Sequential Quadratic Programming (SQP) iteration. To solve Quadratic Programming (QP) problems efficiently, recently proposed partial condensing techniques reformulate large and sparse QP problems into smaller but still sparse ones. In this paper, we propose two tailored partial condensing algorithms by combining partial sensitivity updating schemes, that are recently proposed to update part of the sensitivities of dynamics. We show that such a combination can significantly reduce the computational time for partial condensing and for the full RTI step.

A New Lagrange-Newton-Krylov Solver for PDE-constrained Nonlinear Model Predictive Control

Abstract Real-time optimization of systems governed by partial differential equations (PDEs) presents significant computational challenges to nonlinear model predictive control (NMPC). The large-scale nature of PDEs often limits the use of standard nested black-box optimizers that require repeated forward simulations and expensive gradient computations. Hence, to ensure online solutions at relevant time-scales, large-scale NMPC algorithms typically require powerful, customized PDE-constrained optimization solvers. To this end, this paper proposes a new Lagrange-Newton-Krylov (LNK) method that targets the class of time-dependent nonlinear diffusion-reaction systems arising from chemical processes. The LNK solver combines a high-order spectral Petrov-Galerkin (SPG) method with a new, parallel preconditioner tailored for the large-scale saddle-point systems that form subproblems of Sequential Quadratic Programming (SQP) methods. To establish proof-of-concept, a case study uses a simple parallel MATLAB implementation of the preconditioner with 10 cores. As a step towards real-time control, the results demonstrate that large-scale diffusion-reaction optimization problems with more than 106 unknowns can be solved efficiently in less than a minute.

Control of diesel engines with electrically assisted turbocharging through an extended state observer based nonlinear MPC

A two-layer controller for a diesel engine equipped with an electrically assisted turbocharger is proposed. A previously identified control-oriented plant model is used for control design. A GT-SUITE based high fidelity engine model is used as the plant. The high-level controller is designed based on the active disturbance rejection control method and uses an extended state observer for added robustness. The high-level controller tracks the boost pressure and intake manifold oxygen concentration. The low-level controller seeks to deliver the desired compressor power through an optimal affine split between the exhaust turbine power and the electric power. The optimal power split is determined by an optimal vane position for the variable geometry turbine. The corresponding optimization problem is solved using a nonlinear model predictive control algorithm that is adapted for fast numerical solution. Controller validation is conducted on the GT-SUITE engine model over the FTP-75 cycle. Results confirm the effectiveness the proposed control design and also illustrate engine performance benefits of an assisted turbocharging system, in terms of transient response (~70% reduction in accumulated boost pressure tracking error) and fuel economy (~4.4% reduction in brake specific fuel consumption) relative to a conventionally boosted system.

A Real-Time Path-Planning Algorithm based on Receding Horizon Techniques

In this article we present a real-time path-planning algorithm that can be used to generate optimal and feasible paths for any kind of unmanned vehicle (UV). The proposed algorithm is based on the use of a simplified particle vehicle (PV) model, which includes the basic dynamics and constraints of the UV, and an iterated non-linear model predictive control (NMPC) technique that computes the optimal velocity vector (magnitude and orientation angles) that allows the PV to move toward desired targets. The computed paths are guaranteed to be feasible for any UV because: i) the PV is configured with similar characteristics (dynamics and physical constraints) as the UV, and ii) the feasibility of the optimization problem is guaranteed by the use of the iterated NMPC algorithm. As demonstration of the capabilities of the proposed path-planning algorithm, we explore several simulation examples in different scenarios. We consider the existence of static and dynamic obstacles and a follower condition.

Nox conversion efficiency optimization based on NSGA-II and state-feedback nonlinear model predictive control of selective catalytic reduction system in diesel engine

The selective catalytic reduction (SCR) technology is a promising diesel after-treatment technology due to its high NOx reduction potential. A nonlinear model predictive control (NMPC) approach was proposed based on the three-state SCR model here for control and optimization of SCR system to maximize NOx reduction while minimizing NH3 slip and urea consumption; moreover, the ammonia coverage ratio was designed to asa control target, whose desired range was obtained by means of the singular perturbation method, that is to say, the upper limit of ammonia coverage ratio was derived by 10ppm NH3 slip while the lower limit was calculated by maximum NOx conversion efficiency, which was obtained through a novel model based non-dominated sorting genetic algorithm (NSGA-II) according to an optimum tradeoff between NOx conversion and NH3 slip at each engine operating point. The state-feedback NMPC algorithm was applied to ensure the estimated ammonia coverage ratio falling in the desired range so that the SCR performance can be optimized over entire engine operating. The standard European steady-state cycle (ESC) was performed for data generation from a diesel engine equipped with a commercial SCR and to verify the effectiveness of the designed control strategy. Simulation results over the entire ESC indicated that the NOx conversion efficiency may be optimal in various engine operating modes and the NMPC controller ensures the most part of the cycle may conform to the NOx emissions requirements; moreover, the downstream NH3 concentration was less than the limit; especially, the whole mean NH3 slip was 9.7ppm which is basically close to the given limit (10ppm); furthermore, the ammonia storage capacity falls rapidly along with growth of the temperature but the NMPC still worked well for prevention of any unacceptably high NH3 slip. The experimental verification was carried out and the results showed that the designed strategy was effective and acceptable compared with the simulated results.

NMPC using Pontryagin’s Minimum Principle-Application to a two-phase semi-batch hydroformylation reactor under uncertainty

Nonlinear model predictive control (NMPC) is an important tool for the real-time optimization of batch and semi-batch processes. Direct methods are often the methods of choice to solve the corresponding optimal control problems, in particular for large-scale problems. However, the matrix factorizations associated with large prediction horizons can be computationally demanding. In contrast, indirect methods can be competitive for smaller-scale problems. Furthermore, the interplay between states and co-states in the context of Pontryagin’s Minimum Principle (PMP) might turn out to be computationally quite efficient.This work proposes to use an indirect solution technique in the context of shrinking-horizon NMPC. In particular, the technique deals with path constraints via indirect adjoining, which allows meeting active path constraints explicitly at each iteration. Uncertainties are handled by the introduction of time-varying backoff terms for the path constraints. The resulting NMPC algorithm is applied to a two-phase semi-batch reactor for the hydroformylation of 1-dodecene in the presence of uncertainty, and its performance is compared to that of NMPC that uses a direct simultaneous optimization method. The results show that the proposed algorithm (i) can enforce feasible operation for different uncertainty realizations both within batch or from batch to batch, and (ii) is significantly faster than direct simultaneous NMPC, especially at the beginning of the batch. In addition, a modification of the PMP-based NMPC scheme is proposed to enforce active constraints via tracking.

A design technique for fast sampled-data nonlinear model predictive control with convergence and stability results

ABSTRACTIn this study, a sampled-data nonlinear model predictive control scheme is developed. The control algorithm uses a prediction horizon with variable length, a terminal constraint set, and a feedback controller defined on this set. Following a suboptimal solution strategy, a defined number of steps of an iterative optimisation routine improve the current input trajectory at each sampling point. The value of the objective function monotonically decreases and the state converges to a target set. A discrete-time formulation of the algorithm and a discrete-time design model ensure high computational efficiency and avoid an ad hoc quasi-continuous implementation. This design technique for a fast sampled-data nonlinear model predictive control algorithm is the main contribution of the paper. Based on a benchmark control problem, the performance of the developed control algorithm is assessed against state-of-the-art nonlinear model predictive control methods available in the literature. This assessment demonstrates that the developed control algorithm stabilises the system with very low computational effort. Hence, the algorithm is suitable for real-time control of fast dynamical systems.

Towards Parallelizable Sampling–based Nonlinear Model Predictive Control

This paper proposes a new sampling–based nonlinear model predictive control (MPC) algorithm, with a bound on complexity quadratic in the prediction horizon N and linear in the number of samples. The idea of the proposed algorithm is to use the sequence of predicted inputs from the previous time step as a warm start, and to iteratively update this sequence by changing its elements one by one, starting from the last predicted input and ending with the first predicted input. This strategy, which resembles the dynamic programming principle, allows for parallelization up to a certain level and yields a suboptimal nonlinear MPC algorithm with guaranteed recursive feasibility, stability and improved cost function at every iteration, which is suitable for real–time implementation. The complexity of the algorithm per each time step in the prediction horizon depends only on the horizon, the number of samples and parallel threads, and it is independent of the measured system state. Comparisons with the fmincon nonlinear optimization solver on a benchmark example indicates that, as the simulation time progresses, the proposed algorithm converges rapidly to the “optimal” solution, even when using a small number of samples.

Optimal lane change motion of intelligent vehicles based on extended adaptive pseudo-spectral method under uncertain vehicle mass

Lane change operation is widely used in numerous traffic scenarios; to minimize lane change time and avoid collisions with other vehicles required by intelligent vehicles, an optimal lane change motion is proposed for the shortest time free lane change and emergency obstacle avoidance lane change. First, two optimal lane change motion models are constructed based on a 3-degree-of-freedom dynamical vehicle model. Because of uncertain parameters in the vehicle model, the optimal lane change motion problem is an uncertain optimal control problem. Subsequently, targeted at nonlinearity and uncertain parameters, an extended adaptive pseudo-spectral method is also presented on the basis of approximate and numerical integration of parameter space. Finally, solving the optimal lane change motion problem, optimization results, control results based on the nonlinear model predictive control algorithm, and experimental results are shown under circumstances of certain vehicle mass and uncertain vehicle mass. As for uncertain parameters, optimization results are featured with robustness. Thus, uncertain parameters need not be measured, and optimization calculation need not be carried out in real time. The algorithm put forward here could be adopted to solve an uncertain optimal control problem. The optimal lane change motion can be extended to apply in more complex operations like merging, double-lane change, entering/exiting highways, or overtaking another vehicle.

Electrical Power Distribution System Reconfiguration: Case Study of a Real-life Grid in Croatia

This paper describes application of a nonlinear model predictive control algorithm to the problem of dynamic reconfiguration of an electrical power distribution system with distributed generation and storage. Power distribution systems usually operate in a radial topology despite being physically built as interconnected meshed networks. The meshed structure of the network allows one to modify the network topology by changing status of the line switches (open/closed). The goal of the control algorithm is to find the optimal radial network topology and the optimal power references for the controllable generators and energy storage units that will minimize cumulative active power losses while satisfying all system constraints. Validation of the developed algorithm is conducted on a case study of a real-life distribution grid in Croatia. The realistic simulations show that large loss reductions are feasible (more than 13%), i.e. that the developed control algorithm can contribute to significant savings for the grid operator.

Optimal Predictive Control for Path Following of a Full Drive-by-Wire Vehicle at Varying Speeds

The current research of the global chassis control problem for the full drive-by-wire vehicle focuses on the control allocation (CA) of the four-wheel-distributed traction/braking/steering systems. However, the path following performance and the handling stability of the vehicle can be enhanced a step further by automatically adjusting the vehicle speed to the optimal value. The optimal solution for the combined longitudinal and lateral motion control (MC) problem is given. First, a new variable step-size spatial transformation method is proposed and utilized in the prediction model to derive the dynamics of the vehicle with respect to the road, such that the tracking errors can be explicitly obtained over the prediction horizon at varying speeds. Second, a nonlinear model predictive control (NMPC) algorithm is introduced to handle the nonlinear coupling between any two directions of the vehicular planar motion and computes the sequence of the optimal motion states for following the desired path. Third, a hierarchical control structure is proposed to separate the motion controller into a NMPC based path planner and a terminal sliding mode control (TSMC) based path follower. As revealed through off-line simulations, the hierarchical methodology brings nearly 1700% improvement in computational efficiency without loss of control performance. Finally, the control algorithm is verified through a hardware in-the-loop simulation system. Double-lane-change (DLC) test results show that by using the optimal predictive controller, the root-mean-square (RMS) values of the lateral deviations and the orientation errors can be reduced by 41% and 30%, respectively, comparing to those by the optimal preview acceleration (OPA) driver model with the non-preview speed-tracking method. Additionally, the average vehicle speed is increased by 0.26 km/h with the peak sideslip angle suppressed to 1.9°. This research proposes a novel motion controller, which provides the full drive-by-wire vehicle with better lane-keeping and collision-avoidance capabilities during autonomous driving.

Identification and nonlinear model predictive control of MIMO Hammerstein system with constraints

This work is concerned with identification and nonlinear predictive control method for MIMO Hammerstein systems with constraints. Firstly, an identification method based on steady-state responses and sub-model method is introduced to MIMO Hammerstein system. A modified version of artificial bee colony algorithm is proposed to improve the prediction ability of Hammerstein model. Next, a computationally efficient nonlinear model predictive control algorithm (MGPC) is developed to deal with constrained problem of MIMO system. The identification process and performance of MGPC are shown. Numerical results about a polymerization reactor validate the effectiveness of the proposed method and the comparisons show that MGPC has a better performance than QDMC and basic GPC.

Nonlinear predictive control of dynamic systems represented by Wiener–Hammerstein models

This paper is concerned with computationally efficient nonlinear model predictive control (MPC) of dynamic systems described by cascade Wiener–Hammerstein models. The Wiener–Hammerstein structure consists of a nonlinear steady-state block sandwiched by two linear dynamic ones. Two nonlinear MPC algorithms are discussed in details. In the first case the model is successively linearised on-line for the current operating conditions, whereas in the second case the predicted output trajectory of the system is linearised along the trajectory of the future control scenario. Linearisation makes it possible to obtain quadratic optimisation MPC problems. In order to illustrate efficiency of the discussed nonlinear MPC algorithms, a heat exchanger represented by the Wiener–Hammerstein model is considered in simulations. The process is nonlinear, and a classical MPC strategy with linear process description does not lead to good control result. The discussed MPC algorithms with on-line linearisation are compared in terms of control quality and computational efficiency with the fully fledged nonlinear MPC approach with on-line nonlinear optimisation.

Optimal Energy Management for Stable Operation of an Islanded Microgrid

This paper presents a methodology on the design of an optimal predictive control scheme applied to an islanded microgrid. The controller manages the batteries energy and performs a centralized load shedding strategy to balance the load and generation within the microgrid, and to keep the stability of the voltage magnitude. A nonlinear model predictive control (NMPC) algorithm is used for processing a data set composed of the batteries state of charge, the distributed energy resources (DERs) active power generation, and the forecasted load. The NMPC identifies upcoming active power unbalances and initiates automated load shedding over noncritical loads. The control strategy is tested in a medium voltage distribution system with DERs. This control strategy is assisted by a distribution monitoring system, which performs real-time monitoring of the active power generated by the DERs and the current load demand at each node of the microgrid. Significant performance improvement is achieved with the use of this control strategy over tested cases without its use. The balance between the power generated by the DERs and the load demand is maintained, while the voltage magnitude is kept within the maximum variation margin of $\pm 5\%$ recommended by the standard ANSI C84.1-1989.

A graph search and neural network approach to adaptive nonlinear model predictive control

Systems with a priori unknown and time-varying dynamic behavior pose a significant challenge in the field of Nonlinear Model Predictive Control (NMPC). When both the identification of the nonlinear system and the optimization of control inputs are done robustly and efficiently, NMPC may be applied to control such systems. This paper considers stable systems and presents a novel method for adaptive NMPC, called Adaptive Sampling Based Model Predictive Control (Adaptive SBMPC), that combines a radial basis function neural network identification algorithm with a nonlinear optimization method based on graph search. Unlike other NMPC methods, it does not rely on linearizing the system or gradient based optimization. Instead, it discretizes the input space to the model via pseudo-random sampling and feeds the sampled inputs through the nonlinear model, producing a searchable graph. For this discretization, an optimal path is found using Lifelong Planning A⁎, an efficient graph search method. Adaptive SBMPC is used in simulation to identify and control a simple plant with clearly visualized nonlinear behavior. In these simulations, both fixed and time-varying dynamic systems are considered. Results are compared with an adaptive version of Neural GPC, an existing NMPC algorithm based on Newton–Raphson optimization and a back propagation neural network model. When the cost function exhibits many local minima, Adaptive SBMPC is successful in finding a low-cost solution that appears close globally optimal while Neural GPC converges to a solution that is only locally optimal. This paper presents the method, soundness and completeness theory, and two simulated NMPC examples. The first is a transparent single-input single-output example, and the second considers a more complex power plant combustion process with two inputs and three outputs.

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.

Robust Constrained Learning-based NMPC enabling reliable mobile robot path tracking

This paper presents a Robust Constrained Learning-based Nonlinear Model Predictive Control (RC-LB-NMPC) algorithm for path-tracking in off-road terrain. For mobile robots, constraints may represent solid obstacles or localization limits. As a result, constraint satisfaction is required for safety. Constraint satisfaction is typically guaranteed through the use of accurate, a priori models or robust control. However, accurate models are generally not available for off-road operation. Furthermore, robust controllers are often conservative, since model uncertainty is not updated online. In this work our goal is to use learning to generate low-uncertainty, non-parametric models in situ. Based on these models, the predictive controller computes both linear and angular velocities in real-time, such that the robot drives at or near its capabilities while respecting path and localization constraints. Localization for the controller is provided by an on-board, vision-based mapping and navigation system enabling operation in large-scale, off-road environments. The paper presents experimental results, including over 5 km of travel by a 900 kg skid-steered robot at speeds of up to 2.0 m/s. The result is a robust, learning controller that provides safe, conservative control during initial trials when model uncertainty is high and converges to high-performance, optimal control during later trials when model uncertainty is reduced with experience.

Modelling and predictive control of a neutralisation reactor using sparse support vector machine Wiener models

This paper has two objectives: (a) it describes the problem of finding a precise and uncomplicated model of a neutralisation process, (b) it details development of a nonlinear Model Predictive Control (MPC) algorithm for the plant. The model has a cascade Wiener structure, i.e. a linear dynamic part is followed by a nonlinear steady-state one. A Least-Squares Support Vector Machine (LS-SVM) approximator is used as the steady-state part. Although the LS-SVM has excellent approximation abilities and it may be found easily, it suffers from a huge number of parameters. Two pruning methods of the LS-SVM Wiener model are described and compared with a classical pruning algorithm. The described pruning methods make it possible to remove as much as 70% of support vectors without any significant deterioration of model accuracy. Next, the pruned model is used in a computationally efficient MPC algorithm in which a linear approximation of the predicted output trajectory is successively found on-line and used for prediction. The control profile is calculated on-line from a quadratic optimisation problem. It is demonstrated that the described MPC algorithm with on-line linearisation based on the pruned LS-SVM Wiener model gives practically the same trajectories as those obtained in the computationally complex MPC approach based on the full model with on-line nonlinear optimisation repeated at each sampling instant.

Non-linear model predictive control schemes with application on a 2 link vertical robot manipulator

The industrial requirements for controllers able to perform tasks in the presence of plant nonlinearities are growing. In addition, an increase in industrial computation power is allowing the implementation of more complex control algorithms in the fast processing industry. In this investigation three different nonlinear model predictive control algorithms are tested and evaluated in simulation and experimentally. The methodologies are adaptive nonlinear model predictive control (nMPC), PID based nMPC (PIDnMPC), and a novel simplified nMPC (SnMPC). These are tested in simulation with an inverted pendulum, a Van der Pol oscillator, and a planar 2-link vertical robotic arm. The controllers are tested experimentally using a fabricated planar 2-link vertical robotic arm apparatus. A comparison of the different algorithms is made with special attention to trajectory tracking, computational complexity and transient response dynamics.