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

https://doi.org/10.1016/j.rcim.2016.02.003Get rights and content

Highlights

  • Three non-linear MPC Algorithms are developed to control fast systems, nMPC, PIDnMPC and SnMPC.

  • The algorithms are tested in simulation using three non-linear models.

  • The controller features are discussed and future directions of study are suggested.

Abstract

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.

Introduction

Since the 1970s MPC has gained popularity in the process industry and academia as a robust control form able to handle problem features such as constraints, disturbances, and complex modeling. The importance of control algorithms capable of negotiating nonlinear systems in current and future automation industries is undeniable. A number of controller methodologies have been developed to address this class of problems. These nonlinear schematics often present themselves as adaptations of classic control algorithms such as PID control and model predictive control (MPC) [20], [11], [18], [7].

In many cases different adaptations of MPC for nonlinear systems are designed for a class of problems, or to emphasize a control objective [4], [13], [21]. Other examples of advanced nonlinear MPC techniques incorporate tools such as artificial neural networks, and global optimization methods [6], [3]. The trade-off of these advanced nonlinear MPC methods is complexity and longer computer processing time [23]. This can be mitigated by efficient nonlinear MPC algorithm formulations, and becomes relevant when considering the more recent extension of MPC to applications, beyond the process industry, into electro-mechanical systems [16], [15], [10]. Examples of nonlinear MPC control formulations applied to fast systems include robotics, active vibration control, and unmanned aerial vehicles [12], [24], [17], [14], [8].

Other forms of state dependent MPC that address nonlinear dynamics have been formulated [1], [19]. While these schemes work well, they are complex in formulation and cannot be readily applied to fast response systems, such as robotic manipulators, that involve kinematic solutions within the control sampling instant. In addition, these methodologies did not address the challenge of tracking irregular reference trajectories that are prevalent in robotic systems. This study focuses on developing nonlinear formulations that can be applied to the general field of MPC with respect to fast response systems to address some of these challenges.

The development and the application of three different formulations of nonlinear adaptive MPC are considered here. The first, nMPC, is based on standard nMPC methods. The second is a hybrid form of MPC that includes a PID component inside the nMPC control loop (PIDnMPC). The first two presented approaches can be applied to most adaptive variations of MPC schemes such as generalized predictive control (GPC), M-shifted MPC, and extended predictive control (EPC) [2], [9]. The third methodology is a novel simplified version of nonlinear MPC (SnMPC). The performance of these nMPC formulations are evaluated with respect to fast processes in industry. Knowledge of underlying characteristics of these nonlinear methodologies can be helpful in selecting the proper control type for a given application. The algorithms designed here are tested first in simulation on standard nonlinear plant examples and then with a planar two link vertical robot manipulator. The simulation results are validated with experiments conducted using the modeled robot manipulator. Features of the nonlinear controllers are summarized and suggestions for future research directions are made.

Section snippets

Model predictive control

The standard Model Predictive control algorithm rests on the optimization of control moves, u, in order to minimize future errors. The optimization is done over a prediction horizon, N using a model in order to construct predicted outcomes of plant actuations y^. The MPC control algorithm considered here is dynamic matrix control (DMC) which is characterized by a dynamic matrix model A:A=[a100a2a10aNaN1aNnu]The dynamic matrix is constructed with a normalized open loop response vector, a

Simulation models

The three nonlinear MPC controllers were tested in simulation on three different problems: the inverted pendulum problem, a Van der Pol oscillator, and a vertical robot manipulator. The simulation models and parameters used are described here.

Simulation results

In the following section, simulation results obtained by virtually implementing the three control algorithms on the described models are presented. The results will outline important performance features expected from the different control types. These should be considered when selecting nonlinear MPC platforms. For comparison, computation times of one control step are included in Table 4. Tracking capacity of the controllers was measured in mean squared error (MSE) with results found in Table 5

Conclusions

The extension of MPC to electro-mechanical systems is driven by faster industrial processing capability and an underlining expectation to continuously improve product performance. To this effect, it is necessary to consider nonlinear formulations of MPC while keeping in mind problem requirements such as controller accuracy and computational complexity. In this investigation, three formulations of nMPC were evaluated. Features of the different nMPC schematic were outlined. Traditional nMPC was

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