Theoretical analysis of unconstrained nonlinear model predictive control

Model predictive control (MPC) is a well-established controller design strategy for linear process models. Because many chemical and biological processes exhibit significant nonlinear behaviour, several MPC techniques based on nonlinear process models have recently been proposed. The most significant difference between these techniques is the computational approach used to solve the nonlinear model predictive control (NMPC) optimization problem. Consequently, analysis of NMPC techniques is often connected to the computational approach employed. In this paper, a theoretical analysis of unconstrained NMPC is presented that is independent of the computational approach. A nonlinear discrete-time, state-space model is used to predict the effects of future inputs on future process outputs. It is shown that model inverse, pole-placement, and steady-state controllers can be obtained by suitable selection of the control and prediction horizons. Moreover, the NMPC optimization problem can be modified to yield nonlinear internal model control (NIMC). The computational requirements of NIMC are considerably less than NMPC, but the NIMC approach is currently restricted to nonlinear models with well-defined and stable inverses. The NIMC controller is shown to provide superior servo and regulatory performance to a linear IMC controller for a continuous stirred tank reactor.     

Computation delay compensation for real time implementation of robust model predictive control

The implementation of model predictive control (MPC) requires to solve an optimization problem online. The computation time, often not negligible especially for nonlinear MPC (NMPC), introduces a delay in the feedback loop. Moreover, it impedes fast sampling rate setting for the controller to react to uncertainties quickly. In this paper, a dual time scale control scheme is proposed for linear/nonlinear systems with external disturbances. A pre-compensator works at fast sampling rate to suppress uncertainty, while the outer MPC controller updates the open loop input sequence at a slower rate. The computation delay is explicitly considered and compensated in the MPC design. Four robust MPC algorithms for linear/nonlinear systems in the literature are adopted and tailored for the proposed control scheme. The recursive feasibility and stability are rigorously analysed. Three simulation examples are provided to validate the proposed approaches.     

Real-time nonlinear MPC and MHE for a large-scale mechatronic application

Progress in optimization algorithms and in computational hardware made deployment of Nonlinear Model Predictive Control (NMPC) and Moving Horizon Estimation (MHE) possible to mechatronic applications. This paper aims to assess the computational performance of NMPC and MHE for rotational start-up of Airborne Wind Energy systems. The capabilities offered by an automatic code generation tool are experimentally verified on a real physical system, using a model comprising 27 states and 4 inputs at a sampling frequency of 25 Hz. The results show the feedback times less than 5 ms for the NMPC with more than 1500 variables.     

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.     

Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty

Model predictive control (MPC) has become one of the most popular control techniques in the process industry mainly because of its ability to deal with multiple-input–multiple-output plants and with constraints. However, in the presence of model uncertainties and disturbances its performance can deteriorate. Therefore, the development of robust MPC techniques has been widely discussed during the last years, but they were rarely, if at all, applied in practice due to the conservativeness or the computational complexity of the approaches. In this paper, we present multi-stage NMPC as a promising robust non-conservative nonlinear model predictive control scheme. The approach is based on the representation of the evolution of the uncertainty by a scenario tree, and leads to a non-conservative robust control of the uncertain plant because the adaptation of future inputs to new information is taken into account. Simulation results show that multi-stage NMPC outperforms standard and min–max NMPC under the presence of uncertainties for a semi-batch polymerization benchmark problem. In addition, the advantages of the approach are illustrated for the case where only noisy measurements are available and the unmeasured states and the uncertainties have to be estimated using an observer. It is shown that better performance can be achieved than by estimating the unknown parameters online and adapting the plant model.     

High-order Volterra Model Predictive Control and its application to a nonlinear polymerisation process

Model Predictive Control (MPC) has recently found wide acceptance in the process industry, but existing design and implementation methods are restricted to linear process models. A chemical process, however, involves severe nonlinearity which cannot be ignored in practice. This paper aims to solve this nonlinear control problem by extending MPC to accommodate nonlinear models. It develops an analytical framework for nonlinear model predictive control (NMPC). It also offers a third-order Volterra series based nonparametric nonlinear modelling technique for NMPC design, which relieves practising engineers from the need for deriving a physical-principles based model first. An on-line realisation technique for implementing NMPC is then developed and applied to a Mitsubishi Chemicals polymerisation reaction process. Results show that this nonlinear MPC technique is feasible and very effective. It considerably outperforms linear and low-order Volterra model based methods. The advantages of the developed approach lie not only in control performance superior to existing NMPC methods, but also in eliminating the need for converting an analytical model and then convert it to a Volterra model obtainable only up to the second order.     

Extension of efficient predictive control to the nonlinear case

The combined use of the closed‐loop paradigm, an augmented autonomous state space formulation, partial invariance, local affine difference inclusion, and polytopic invariance are deployed in this paper to propose an NMPC algorithm which, unlike earlier algorithms that have to tackle online a nonlinear non‐convex optimization problem, requires the solution of a simple QP. The proposed algorithm is shown to outperform earlier algorithms in respect of size of region of attraction and online computational load. Conversely, for comparable computational loads, the proposed algorithm outperforms earlier algorithms in terms of optimality of dynamic performance. Copyright © 2005 John Wiley & Sons, Ltd.     

不确定系统的鲁棒与随机模型预测控制算法比较研究

近几十年来,不确定系统模型预测控制的理论和应用得到了飞速发展.本文简要地回顾了不确定系统中鲁棒模型预测控制和随机模型预测控制的发展历史,总结了它们的相关应用,并较为细致地分析了线性不确定系统模型预测控制的各种主要算法.通过总结各种算法的通用模型、运作方式、问题规模,以及它们保证递归可行性、稳定性的方法,分析了部分算法可行域间的关系,揭示了各种算法的主要特点、适用场合和未来可发展方向,并通过仿真实例直观地分析了各种算法的性能和可靠性.     

Latent‐variable Nonlinear Model Predictive Control Strategy for a pH Neutralization Process

Linear model predictive control (MPC) is a widely‐used control strategy in chemical processes. Its extension to nonlinear MPC (NMPC) has drawn increasing attention since many process systems are inherently nonlinear. When implementing the NMPC based on a nonlinear predictive model, a nonlinear dynamic optimization problem must be calculated. For the sake of solving this optimization problem efficiently, a latent‐variable dynamic optimization approach is proposed. Two kinds of constraint formulations, original variable constraint and Hotelling T² statistic constraint, are also discussed. The proposed method is illustrated in a pH neutralization process. The results demonstrate that the latent‐variable dynamic optimization based the NMPC strategy is efficient and has good control performance.     

Nonlinear Model Predictive Control of Electrical Submersible Pumps based on Echo State Networks

Employed for artificial lifting in oil well production, Electrical Submersible Pumps (ESP) can be operated with Model Predictive Control (MPC) to drive an optimal production, while ensuring a safe operation and respecting system constraints. Due to the nonlinear dynamics of ESPs, Echo State Networks (ESNs), a recurrent neural network with fast training, are employed for efficient system identification of unknown dynamic systems. Besides the synthesis of highly accurate prediction models, this work contributes by designing two Nonlinear MPC (NMPC) strategies for the control of an ESP-lifted oil well: a standard Single-Shooting NMPC that embeds the ESN model completely, and the Practical Nonlinear Model Predictive Controller (PNMPC) that approximates the NMPC through fast trajectory-linearization of the ESN model. Another contribution is the implementation of an error correction filter to reject disturbances and counter modeling errors in both NMPC strategies. Finally, in computational experiments, both ESN-based NMPC strategies performed well in controlling simulated ESP-lifted oil wells when the model of the plant is unknown. However, PNMPC was more efficient and induced a similar performance to standard NMPC.     

Multirate sliding mode disturbance compensation for model predictive control

In this paper, a novel hierarchical multirate control scheme for nonlinear discrete‐time systems is presented, consisting of a robust nonlinear model predictive controller (NMPC) and a multirate sliding mode disturbance compensator (MSMDC). The proposed MSMDC acts at a faster rate than the NMPC in order to keep the system as close as possible to the nominal trajectory predicted by NMPC despite model uncertainties and external disturbances. The a priori disturbance compensation turns out to be very useful in order to improve the robustness of the NMPC controller. A dynamic input allocation between MSMDC and NMPC allows to maximize the benefits of the proposed scheme that unites the advantages of sliding mode control (strong reduction of matched disturbances, low computational burden) to those of NMPC (optimality, constraints handling). Sufficient conditions required to guarantee input‐to‐state stability and constraints satisfaction by the overall scheme are also provided. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear model predictive control for hydrobatics: Experiments with an underactuated AUV

Hydrobatic autonomous underwater vehicles (AUVs) can be efficient in range and speed, as well as agile in maneuvering. They can be beneficial in scenarios such as obstacle avoidance, inspections, docking, and under-ice operations. However, such AUVs are underactuated systems—this means exploiting the system dynamics is key to achieving elegant hydrobatic maneuvers with minimum controls. This paper explores the use of model predictive control (MPC) techniques to control underactuated AUVs in hydrobatic maneuvers and presents new simulation and experimental results with the small and hydrobatic SAM AUV. Simulations are performed using nonlinear model predictive control (NMPC) on the full AUV system to provide optimal control policies for several hydrobatic maneuvers in Matlab/Simulink. For implementation on AUV hardware in robot operating system, a linear time varying MPC (LTV-MPC) is derived from the nonlinear model to enable real-time control. In simulations, NMPC and LTV-MPC shows promising results to offer much more efficient control strategies than what can be obtained with PID and linear quadratic regulator based controllers in terms of rise-time, overshoot, steady-state error, and robustness. The LTV-MPC shows satisfactory real-time performance in experimental validation. The paper further also demonstrates experimentally that LTV-MPC can be run real-time on the AUV in performing hydrobatic maneouvers.     

A nonlinear model predictive control system based on Wiener piecewise linear models

In this paper a nonlinear model predictive control (NMPC) based on a Wiener model with a piecewise linear gain is presented. This approach retains all the interested properties of the classical linear model predictive control (MPC) and keeps computations easy to solve due to the canonical structure of the nonlinear gain. Some guidelines for the identification of the nominal model as well as the uncertainty bounds are discussed, and two examples that show the possibility of application of this control scheme to real life problems are presented.     

An alternative use of the Riccati recursion for efficient optimization

In optimization routines used for on-line Model Predictive Control (MPC), linear systems of equations are solved in each iteration. This is true both for Active Set (AS) solvers as well as for Interior Point (IP) solvers, and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main computational effort is spent while solving these linear systems of equations, and hence, it is of great interest to solve them efficiently. In high performance solvers for MPC, this is performed using Riccati recursions or generic sparsity exploiting algorithms. To be able to get this performance gain, the problem has to be formulated in a sparse way which introduces more variables. The alternative is to use a smaller formulation where the objective function Hessian is dense. In this work, it is shown that it is possible to exploit the structure also when using the dense formulation. More specifically, it is shown that it is possible to efficiently compute a standard Cholesky factorization for the dense formulation. This results in a computational complexity that grows quadratically in the prediction horizon length instead of cubically as for the generic Cholesky factorization.     

Design of observer based nonlinear model predictive controller for a continuous stirred tank reactor

Model predictive control (MPC) schemes are now widely used in process industries for the control of key unit operations. Linear model predictive control (LMPC) schemes which make use of linear dynamic model for prediction, limit their applicability to a narrow range of operation (or) to systems which exhibit mildly nonlinear dynamics.

In this paper, a nonlinear observer based model predictive controller (NMPC) for nonlinear system has been proposed. An approach to design NMPC based on fuzzy Kalman filter (FKF) and augmented state fuzzy Kalman filter (ASFKF) has been presented. The efficacy of the proposed NMPC schemes have been demonstrated by conducting simulation studies on the continuous stirred tank reactor (CSTR). The analysis of the extensive dynamic simulation studies revealed that, the NMPC schemes formulated produces satisfactory performance for both servo and regulatory problems. Simulation results also include an inferential control case, where the reactor concentration is not measured but estimated from temperature measurement and used in the NMPC based on FKF and ASFKF formulations.     

Linear programming and model predictive control

The practicality of model predictive control (MPC) is partially limited by the ability to solve optimization problems in real time. This requirement limits the viability of MPC as a control strategy for large scale processes. One strategy for improving the computational performance is to formulate MPC using a linear program. While the linear programming formulation seems appealing from a numerical standpoint, the controller does not necessarily yield good closed-loop performance. In this work, we explore MPC with an l₁ performance criterion. We demonstrate how the non-smoothness of the objective function may yield either dead-beat or idle control performance.     

A COMPARATIVE STUDY ON COMPUTATIONAL SCHEMES FOR NONLINEAR MODEL PREDICTIVE CONTROL

This paper briefly reviews development of nonlinear model predictive control (NMPC) schemes for finite horizon prediction and basic computational algorithms that can solve the stable real‐time implementation of NMPC in space state form with state and input constraints. In order to ensure stability within a finite prediction horizon, most NMPC schemes use a terminal region constraint at the end of the prediction horizon — a particular NMPC scheme using a terminal region constraint, namely quasi‐infinite horizon, that guarantees asymptotic closed‐loop stability with input constraints is presented. However, when nonlinear processes have both input and state constraints, difficulty arises from failure to satisfy the state constraints due to constraints on input. Therefore, a new NMPC scheme without a terminal region constraint is developed using soften state constraints. A brief comparative simulation study of two NMPC schemes: quasi‐infinite horizon and soften state constraints is done via simple nonlinear examples to demonstrate the ability of the soften state constraints scheme. Finally, some features of future research from this study are discussed.     

Lock-free dynamic hash tables with open addressing

We present an efficient lock-free algorithm for parallel accessible hash tables with open addressing, which promises more robust performance and reliability than conventional lock-based implementations. "Lock-free" means that it is guaranteed that always at least one process completes its operation within a bounded number of steps. For a single processor architecture our solution is as efficient as sequential hash tables. On a multiprocessor architecture this is also the case when all processors have comparable speeds. The algorithm allows processors that have widely different speeds or come to a halt. It can easily be implemented using C-like languages and requires on average only constant time for insertion, deletion or accessing of elements. The algorithm allows the hash tables to grow and shrink when needed. Lock-free algorithms are hard to design correctly, even when apparently straightforward. Ensuring the correctness of the design at the earliest possible stage is a major challenge in any responsible system development. In view of the complexity of the algorithm, we turned to the interactive theorem prover PVS for mechanical support. We employ standard deductive verification techniques to prove around 200 invariance properties of our algorithm, and describe how this is achieved with the theorem prover PVS. CR Subject Classification (1991): D.1 Programming techniques AMS Subject Classification (1991): 68Q22 Distributed algorithms, 68P20 Information storage and retrieval Received: 13 April 2004, Accepted: 27 October 2004, Published online: 25 November 2004     

Controlling the level of sparsity in MPC

In optimization algorithms used for on-line Model Predictive Control (MPC), linear systems of equations are often solved in each iteration. This is true both for Active Set methods as well as for Interior Point methods, and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main computational effort is spent while solving these linear systems of equations, and hence, it is of greatest interest to solve them efficiently. Classically, the optimization problem has been formulated in either of two ways. One leading to a sparse linear system of equations involving relatively many variables to compute in each iteration and another one leading to a dense linear system of equations involving relatively few variables. In this work, it is shown that it is possible not only to consider these two distinct choices of formulations. Instead it is shown that it is possible to create an entire family of formulations with different levels of sparsity and number of variables, and that this extra degree of freedom can be exploited to obtain even better performance with the software and hardware at hand. This result also provides a better answer to a recurring question in MPC; should the sparse or dense formulation be used.     

Multiplexed model predictive control

This paper proposes a form of MPC in which the control variables are moved asynchronously. This contrasts with most MIMO control schemes, which assume that all variables are updated simultaneously. MPC outperforms other control strategies through its ability to deal with constraints. This requires on-line optimization, hence computational complexity can become an issue when applying MPC to complex systems with fast response times. The Multiplexed MPC (MMPC) scheme described in this paper solves the MPC problem for each subsystem sequentially, and updates subsystem controls as soon as the solution is available, thus distributing the control moves over a complete update cycle. The resulting computational speed-up allows faster response to disturbances, which may result in improved performance, despite finding sub-optimal solutions to the original problem. This paper describes nominal and robust MMPC, states some stability results, and demonstrates the effectiveness of MMPC through two examples.