Fast Nonlinear Model Predictive Control Research Articles

A new fast nonlinear model predictive control of parallel manipulators: Design and experiments

High-speed parallel manipulators are characterized by fast sampling rates and challenges owing to the presence of constraints, high nonlinearities, uncertainties, and fast dynamics. The development of a reliable nonlinear model predictive control (NMPC) approach for this type of robotic systems is challenging because of the complex dynamics that involves high computational burden. To address this control problem, this paper proposes a new fast NMPC strategy applied to high-speed parallel robots. Based on (i) a parameterization technique, (ii) a fast gradient solver, (iii) a proportional integral derivative (PID) control term, (iv) a nonlinear dynamic model, and (v) an artificial neural network (ANN) model, two fast NMPC frameworks were developed to reduce the computational cost of the classical NMPC and address online implementation. In this study, we focus on improving the performance of the standard predictive control strategy outside of the range of its classical applicability by developing a fast NMPC controller for high-speed parallel kinematic manipulators (PKMs). Numerical simulations and real-time experimental results were presented and discussed to validate the relevance of the proposed controllers. Experiments were conducted on a four-degree-of-freedom (4-DOF) PKM called VELOCE developed in our laboratory. The results show that the proposed solution outperforms the original scheme in terms of real-time tracking performance.

Fast nonlinear model predictive planner and control for an unmanned ground vehicle in the presence of disturbances and dynamic obstacles

This paper presents a solution for the tracking control problem, for an unmanned ground vehicle (UGV), under the presence of skid-slip and external disturbances in an environment with static and moving obstacles. To achieve the proposed task, we have used a path-planner which is based on fast nonlinear model predictive control (NMPC); the planner generates feasible trajectories for the kinematic and dynamic controllers to drive the vehicle safely to the goal location. Additionally, the NMPC deals with dynamic and static obstacles in the environment. A kinematic controller (KC) is designed using evolutionary programming (EP), which tunes the gains of the KC. The velocity commands, generated by KC, are then fed to a dynamic controller, which jointly operates with a nonlinear disturbance observer (NDO) to prevent the effects of perturbations. Furthermore, pseudo priority queues (PPQ) based Dijkstra algorithm is combined with NMPC to propose optimal path to perform map-based practical simulation. Finally, simulation based experiments are performed to verify the technique. Results suggest that the proposed method can accurately work, in real-time under limited processing resources.

Fast nonlinear model predictive controller using parallel PSO based on divide and conquer approach

The high computationally expensive nature of nonlinear optimisation algorithms leads to limitations of its use in a real-time era. So, this is the main challenge against researchers to develop a fast algorithm that is used in real-time computations. This paper proposes a fast nonlinear model predictive control algorithm that utilises a parallel particle swarm optimisation with synchronous and asynchronous methods inclusion, that handles nonlinear optimisation problems with constraints. The additional divide and conquer approach of the proposed algorithm improves the speed of computation and disturbance rejection capability which proves its efficacy in real-time applications. A highly nonlinear fast dynamic real-time inverted pendulum system with hybrid embedded hardware platform (ARM + FPGA) is used to validate the performance of this algorithm under constraints. The solution presented in the paper is computationally feasible for smaller sampling times i.e. in miliseconds and it gives promising results with synchronous and asynchonous parallel PSO compared to the state-of-art PSO algorithm.

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.

Fast approximate learning-based multistage nonlinear model predictive control using Gaussian processes and deep neural networks

Scenario-based model predictive control (MPC) methods introduce recourse into optimal control and can thus reduce the conservativeness inherent to open-loop robust MPC. However, the uncertainty scenarios are often generated offline using worst-case uncertainty bounds quantified a priori, limiting the potential gains in control performance. This paper presents a learning-based multistage MPC (msMPC) for systems with hard-to-model dynamics and time-varying plant-model mismatch. Gaussian Processes (GP) are used to learn state- and input-dependent plant-model mismatch in real-time and accordingly adapt the scenario tree online. Due to the increased computational complexity associated with incorporating the GP predictions into the optimal control problem, the learning-based msMPC (LB-msMPC) law is approximated by a deep neural network (DNN) that is cheap-to-evaluate online and has a small memory footprint, which makes it suitable for embedded applications. In addition, we present a novel algorithm for training the DNN-based controller that uses a GP description of the plant-model mismatch to generate closed-loop simulation data, which ensures the LB-msMPC law is evaluated in regions of the state space most relevant to closed-loop operation. The proposed LB-msMPC strategy is demonstrated on a cold atmospheric plasma jet with applications in (bio)materials processing. The simulation results indicate the promise of the approximate LB-msMPC strategy for control of hard-to-model systems with fast dynamics on millisecond timescales.

Fast hybrid dual mode NMPC for a parallel double inverted pendulum with experimental validation

This study presents a novel fast non-linear model predictive control approach for a parallel double inverted pendulum. The approach uses dual-mode closed-loop predictions to obtain numerically robust optimal solutions. Moreover, it uses the real-time iteration (RTI) scheme to reduce the computational burden and achieve real-time performance. Furthermore, two main modifications are proposed which significantly improve the performance of the RTI scheme in the presence of large disturbances, namely; additional energy-based costs, and a hybrid switching scheme. Finally, the approach uses a non-standard discretised model combined with an online system identification scheme to address parameter uncertainty, and with an extended Kalman filter for state-estimation. The resulting performance is validated through both simulations and experimental results.

Quasi-translations for fast hybrid nonlinear model predictive control

We present the quasi-translation (QT) strategy for mixed-integer model predictive control (MPC) problems. This strategy handles the complexity introduced by discrete controls in a simple manner that takes advantage of the sequential nature of MPC optimization problems. Some general criteria is provided guiding their application. The QT strategy can be used to balance controller performance, chatter and turnaround time. We also introduce an approach to handle hybrid models with state-dependent switches for use with a symbolic formulation of the MPC problem that integrates seamlessly with the QT strategy.We investigate the QT strategy for a variety of hybrid nonlinear MPC problems including the classic benchmark Lotka–Volterra fishing problem, a diesel airpath control problem and longitudinal vehicle control. Results show the QT strategy consistently yields a reduction in turnaround times with excellent or minimally degraded performance as compared to competing strategies.

Toward Safe Dose Delivery in Plasma Medicine using Projected Neural Network-based Fast Approximate NMPC

Atmospheric pressure plasma jets (APPJs) are increasingly used for biomedical applications. Reproducible and effective operation of APPJs hinges on controlling the nonlinear effects of plasma on a target substrate in the face of intrinsic variabilities of the plasma as well as exogenous disturbances. This paper presents a low-memory fast approximate nonlinear model predictive control (NMPC) strategy for an APPJ with prototypical applications in plasma medicine. The NMPC objective is to regulate the delivery of the cumulative thermal effects of plasma to a substrate, while adhering to constraints pertaining to a patient’s safety and comfort. Deep neural networks are used to approximate the implicit NMPC law with a cheap-to-evaluate explicit control law that has low memory requirements. Robust constraint satisfaction is guaranteed by projecting the output of the neural network onto a set that ensures the state stays within an appropriately defined invariant set. Closed-loop simulations and real-time control experiments indicate that the proposed approximate NMPC strategy is effective in handling nonlinear control costs at fast sampling times, while guaranteeing satisfaction of safety-critical system constraints. This work takes a crucial step toward fast NMPC of safety-critical plasma applications using resource-limited embedded control hardware.

Real-Time Model Predictive Control of Rigid Body Motion via Discretization Using the Cayley Map

This paper presents a fast nonlinear model predictive control method for rigid body dynamical systems such as spacecraft or aerial vehicles on the special Euclidean group SE(3). The focus of this research is on the real-time execution of the optimal control on a low-cost embedded computer. The position and orientation of a rigid body are expressed as a 6-dimensional vector via the Cayley map for SE(3). Based on the representation, the motion can be exactly discretized on the algebra se(3). As a result, coarse sampling intervals can be selected to reduce the number of prediction steps. Furthermore, the recursive discretization technique is applied to effectively reduce the decision variables of the nonlinear optimization problem by eliminating states from them. Simulation results on a Raspberry Pi single-board computer are given to prove that the present model predictive control method is feasible in real time. The effectiveness of the present controller is further verified by an experiment using a fully actuated hexarotor unmanned aerial vehicle.

Fast economic nonlinear model predictive control strategy of Organic Rankine Cycle for waste heat recovery: Simulation-based studies

Effective control for Organic Rankine Cycle (ORC) systems is required to ensure safety of each component and attain satisfactory performance in spite of the waste heat sources varying in a broad range. To maximally recover the waste heat and to handle the multivariate constraints during the ORC transient operation, in this paper an economic nonlinear model predictive controller (EMPC) using the net power output as an objective is designed. To fast obtain a solution of EMPC in practical applications, the computation of the gradient of the EMPC objective is simplified and the quasi-sequential method is employed for the online dynamic optimization of EMPC. Unlike the conventional nonlinear model predictive control (MPC) scheme, the results in a case study show that the proposed EMPC can quickly improve the net power output of the ORC system during the operation while still satisfying the load tracking requirements.

Fast nonlinear model predictive control of a chemical reactor: a random shooting approach

Abstract This paper presents a fast way of implementing nonlinear model predictive control (NMPC) using the random shooting approach. Instead of calculating the optimal control sequence by solving the NMPC problem as a nonlinear programming (NLP) problem, which is time consuming, a sub-optimal, but feasible, sequence of control inputs is determined randomly. To minimize the induced sub-optimality, numerous random control sequences are selected and the one that yields the smallest cost is selected. By means of a motivating case study we demonstrate that the random shooting-based approach is superior, from a computational point of view, to state-of-the-art NLP solvers, and features a low level of sub-optimality. The case study involves a continuous stirred tank reactor where a fast multi-component chemical reaction takes place.

A Fast Nonlinear Model Predictive Control Method Based on Discrete Mechanics

Abstract Nonlinear Model Predictive Control (NMPC) is an advanced control technique that often relies on a computationally demanding optimization problem and numerical integration algorithms. This paper proposes and investigates a novel method with less computational effort to improve the efficiency of NMPC using a formulation based on discrete mechanics (DM). In contrast to classical NMPC formulations, the proposed method merges the two stages, for solving both the initial value problem (IVP) for prediction as well as the nonlinear programming problem (NLP) for optimization, into a single stage for solving an optimal boundary value problem (BVP) using NLP techniques. By exploiting the structural features of DM a symbolic solution set of the equations of motion are derived offline on each discretization node along the whole optimization horizon. Within an NLP, the optimal solution is efficiently obtained online under the consideration of the boundary constraints. As a benchmark, the widely used NMPC formulation based on direct multiple shooting (MS) method is served to assess the convergence and the excellent real-time performance of this method. The closed-loop performance is demonstrated by the swing-up of an unstable numerical experiment.

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.

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.

Modified C/GMRES Algorithm for Fast Nonlinear Model Predictive Tracking Control of AUVs

This brief presents a nonlinear model predictive control (NMPC) method for the trajectory tracking problem of an autonomous underwater vehicle (AUV). By augmenting the desired output trajectory to a reference dynamical system, the tracking task can fit into the standard NMPC framework, which effectively avoids possible numerical difficulties in the following fast NMPC implementation. To relieve the conflict between short sampling period and high demand of online calculation, Ohtsuka’s continuation/generalized minimal residual (C/GMRES) algorithm is investigated. In order to handle the realistic constraints on the AUV thrusters, we incorporate the log barrier functions into the cost function and modify the C/GMRES algorithm. Several different reference trajectories are tested using the identified dynamic model of the Saab SeaEye Falcon open-frame ROV/AUV, which demonstrate the effectiveness and efficiency of the proposed fast algorithm for the AUV tracking control.

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.

Receding horizon control of a 3 DOF helicopter using online estimation of aerodynamic parameters

This study presents a numerical implementation of fast nonlinear model predictive control (NMPC) and nonlinear moving horizon estimation (NMHE) for the trajectory tracking problem of a 3 degree of freedom (DOF) helicopter. The motivation behind using the NMPC instead of its linear counterpart is that the helicopter is operated over nonlinear regions. Moreover, this system has cross-couplings that make the control of the system even more complicated. What is more, according to our simulation scenario, the system has a time-varying dynamical model because it has time-varying parameters which are estimated online using NMHE and the extended Kalman filter (EKF) throughout the control. Although NMHE is computationally more demanding, its capability of incorporating the constraints encourages us to utilize NMHE rather than EKF. Two reference trajectories, namely, sinusoidal and square-like, are tracked, and owing to the better learning capability of NMHE over EKF, the NMPC-NMHE closed-loop control framework is able to track both reference signals with more accuracy than the NMPC-EKF control framework, even under parameter uncertainties. Thanks to the ACADO toolkit, the combined average execution time is 4 milliseconds, demonstrating the potential of the proposed framework for real-time aerospace applications using relatively cheaper processors.

Sensitivity-Based Economic NMPC with a Path-Following Approach

We present a sensitivity-based predictor-corrector path-following algorithm for fast nonlinear model predictive control (NMPC) and demonstrate it on a large case study with an economic cost function. The path-following method is applied within the advanced-step NMPC framework to obtain fast and accurate approximate solutions of the NMPC problem. In our approach, we solve a sequence of quadratic programs to trace the optimal NMPC solution along a parameter change. A distinguishing feature of the path-following algorithm in this paper is that the strongly-active inequality constraints are included as equality constraints in the quadratic programs, while the weakly-active constraints are left as inequalities. This leads to close tracking of the optimal solution. The approach is applied to an economic NMPC case study consisting of a process with a reactor, a distillation column and a recycler. We compare the path-following NMPC solution with an ideal NMPC solution, which is obtained by solving the full nonlinear programming problem. Our simulations show that the proposed algorithm effectively traces the exact solution.

Fast Nonlinear Model Predictive Control on FPGA Using Particle Swarm Optimization

Nonlinear model predictive control (NMPC) requires a repeated online solution of a nonlinear optimal control problem. The computation load remains the main challenge for the real-time practical application of the NMPC technique, particularly for fast systems. This paper presents a fast NMPC algorithm implemented on a field-programmable gate array (FPGA) that employs a particle swarm optimization (PSO) algorithm to handle nonlinear optimization. The FPGA is used to explore the possibilities of parallel architecture for the substantial acceleration of NMPC. PSO is employed to achieve real-time operation due to its naturally parallel capabilities. The proposed FPGA-based NMPC-PSO controller consists of a random-number generator, a fixed-point arithmetic, a PSO solver, and a universal asynchronous receiver/transmitter communication interface. Then, this controller is applied to an engine idle speed control problem and demonstrated with an FPGA-in-the-loop testbench. The experimental results indicate that the NMPC-on-FPGA-chip strategy has good computational performance and achieves satisfactory control performance.

Distributed nonlinear model predictive control of an autonomous tractor–trailer system

This paper addresses the trajectory tracking problem of an autonomous tractor–trailer system by using a fast distributed nonlinear model predictive control algorithm in combination with nonlinear moving horizon estimation for the state and parameter estimation in which constraints on the inputs and the states can be incorporated. The proposed control algorithm is capable of driving the tractor–trailer system to any desired trajectory ensuring high control accuracy and robustness against environmental disturbances.