On-line Nonlinear Optimisation Research Articles

Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems

Abstract Motion planning and controller design are challenging tasks for highly coupled and nonlinear dynamical systems such as autonomous vehicles and robotic applications. Nonlinear model predictive control (NMPC) is an emerging technique in which sampling-based methods are used to synthesize the control and trajectories for complex systems. In this study, we have developed the sampling-based motion planning algorithm with NMPC through Bayesian estimation to solve the online nonlinear constrained optimization problem. In the literature, different filtration techniques have been applied to extract knowledge of states in the presence of noise. Due to the detrimental effects of linearization, the Kalman filter with NMPC only achieves modest effectiveness. Moving horizon estimation (MHE), on the other hand, frequently relies on simplifying assumptions and lacks an effective recursive construction. Additionally, it adds another optimization challenge to the regulation problem that has to be solved online. To address this problem, particle filtering is implemented for Bayesian filtering in nonlinear and highly coupled dynamical systems. It is a sequential Monte Carlo method that involves representing the posterior distribution of the state of the system using a set of weighted particles that are propagated through time using a recursive algorithm. For nonlinear and strongly coupled dynamical systems, the novel sampling-based NMPC technique is effective and simple to use. The efficiency of the suggested method has been assessed using simulated studies.

Nonlinear and infinite gain scheduling neural predictive control of the outlet temperature in a parabolic trough solar field: A comparative study

Solar thermal plants have high nonlinearities and non-manipulated energy source which make their control task a very challenging work. Linear controllers cannot cope with undesirable deviations of the outlet temperature over all the operation range of the dynamics of this type of plants. Moreover, nonlinear predictive control relying on online nonlinear optimization have the drawback of time consuming and numerical calculus issues. In this paper, an infinite gain scheduling neural predictive control is designed and applied to control the temperature in a distributed parabolic trough solar collector field. The performance of both tracking and disturbance rejection of the proposed controller is compared to those of four nonlinear predictive control variants: Two unconstrained neural predictive control using the Levenberg–Marquardt and the Broyden–Fletcher–Goldfarb–Shanno algorithms, and two constrained nonlinear predictive control using interior point algorithm, one is based on a neural network model and the other one is based on a first principal model. The superiority of the proposed control strategy is well demonstrated through simulation results.

Computationally Simple Nonlinear MPC Algorithm for Vehicle Obstacle Avoidance With Minimization of Fuel Utilization

This work is concerned with Model Predictive Control (MPC) algorithm for vehicle obstacle avoidance. The second objective of the algorithm is on-line minimization of fuel utilization. At first, the rudimentary nonlinear MPC optimization problem is formulated. Next, the constraints related to the predicted process state variables are formulated as soft ones to guarantee computational safety. Furthermore, in order to obtain a computationally simple procedure, the process dynamics and the fuel utilization model are linearized on-line and used for prediction in MPC. It leads to a quadratic programming MPC task, the necessity of nonlinear optimization performed in real-time is eliminated. In order to stress advantages of the discussed computationally uncomplicated MPC method it is compared with the basic scheme with on-line nonlinear optimization in terms of control quality and computational time. Additionally, effectiveness of the MPC algorithm is discussed in presence of modeling errors and measurement noise. Finally, additional constraints imposed on the rate of change of the manipulated variables are considered.

State–space approach to adaptive fuzzy modeling for financial investment

This paper proposes a new adaptive learning framework for fuzzy system under dynamically changing environment. Especially, a state–space model with filtering algorithm, traditionally used for the estimation of unobservable state variables, is applied to online non-linear optimization problems by reinterpreting control variables and objective function as state variables and observation model, respectively. Our proposed methodology substantially improves the flexibility of the objective function, which enables to construct the adaptive fuzzy system achieving arbitrarily designed user’s objective. In addition, time-series structure is actively introduced into the parameter transition, whose proper modeling is expected to enhance the performance. Particularly, the introduction of mean-reversion process makes it possible to adaptively learn model parameters around specific predetermined levels obtained by existing learning methodologies. As an application of adaptive learning fuzzy system for financial investment, the current work focuses on the construction of the target return replication portfolio. Concretely, the target return is specified as zero floored market index considering investor’s great demand to construct a portfolio with restricted downside risk. The validity of our framework is shown by out-of-sample numerical experiments with the data of well-known high liquid instruments such as S&P500 and TOPIX, which indicates the robustness and reliability of our proposed method in practice.

Wiener structures for modeling and nonlinear predictive control of proton exchange membrane fuel cell

The proton exchange membrane (PEM) fuel cell is a nonlinear dynamic system which cannot be precisely described and controlled using a linear model. This work has two objectives: (a) it discusses model selection for the PEM and (b) it develops two nonlinear computationally efficient model predictive control (MPC) algorithms for the PEM. Three Wiener model types of different orders of dynamics and complexity of the nonlinear steady-state block are compared. The model consisting of three dynamic blocks and a neural network with five hidden nodes is chosen. To obtain simple MPC quadratic optimization problems, a linear approximation of the model or a linear approximation of the predicted trajectory is repeatedly found. The first MPC scheme gives very good control accuracy, whereas the second MPC scheme leads to the same trajectories as those possible in the “ideal” MPC scheme with full online nonlinear optimization.

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.

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.

Nonlinear predictive control for Hammerstein–Wiener systems

This paper discusses a nonlinear Model Predictive Control (MPC) algorithm for multiple-input multiple-output dynamic systems represented by cascade Hammerstein–Wiener models. The block-oriented Hammerstein–Wiener model, which consists of a linear dynamic block embedded between two nonlinear steady-state blocks, may be successfully used to describe numerous processes. A direct application of such a model for prediction in MPC results in a nonlinear optimisation problem which must be solved at each sampling instant on-line. To reduce the computational burden, a linear approximation of the predicted system trajectory linearised along the future control scenario is successively found on-line and used for prediction. Thanks to linearisation, the presented algorithm needs only quadratic optimisation, time-consuming and difficult on-line nonlinear optimisation is not necessary. In contrast to some control approaches for cascade models, the presented algorithm does not need inverse of the steady-state blocks of the model. For two benchmark systems, it is demonstrated that the algorithm gives control accuracy very similar to that obtained in the MPC approach with nonlinear optimisation while performance of linear MPC and MPC with simplified linearisation is much worse.

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.

On improving accuracy of computationally efficient nonlinear predictive control based on neural models

For nonlinear processes the classical model predictive control (MPC) algorithm, in which a linear model is used, usually does not give satisfactory closed-loop performance. In such nonlinear cases a suboptimal MPC strategy is typically used in which the nonlinear model is successively linearised on-line for the current operating point and, thanks to linearisation, the control policy is calculated from a quadratic programming problem. Although the suboptimal MPC algorithm frequently gives good results, for some nonlinear processes it would be beneficial to further improve control accuracy. This paper details a computationally efficient nonlinear MPC algorithm in which a neural model is linearised on-line along the predicted trajectory in an iterative way. The algorithm needs solving on-line only a series of quadratic programming problems. Advantages of the discussed algorithm are demonstrated in the control system of a high-purity ethylene–ethane distillation column for which the classical linear MPC algorithm does not work and the classical suboptimal MPC algorithm is slow. It is shown that the discussed algorithm can give practically the same control accuracy as the algorithm with on-line nonlinear optimisation and, at the same time, the algorithm is significantly less computationally demanding.

Online set-point optimisation cooperating with predictive control of a yeast fermentation process: A neural network approach

Online set-point optimisation which cooperates with model predictive control (MPC) and its application to a yeast fermentation process are described. A computationally efficient multilayer control system structure with adaptive steady-state target optimisation (ASSTO) and a suboptimal MPC algorithm are presented in which two neural models of the process are used. For set-point optimisation, a steady-state neural model is linearised online and the set-point is calculated from a linear programming problem. For MPC, a dynamic neural model is linearised online and the control policy is calculated from a quadratic programming problem. In consequence of linearisation of neural models, the necessity of online nonlinear optimisation is eliminated. Results obtained in the proposed structure are comparable with those achieved in a computationally demanding structure with nonlinear optimisation used for set-point optimisation and MPC.

On-line set-point optimisation and predictive control using neural Hammerstein models

This paper discusses a computationally efficient approach to set-point optimisation which cooperates with predictive control and its application to a multivariable neutralisation reactor. In the presented system structure a neural Hammerstein model of the process is used. For set-point optimisation, a linearisation of the steady-state model derived from the neural Hammerstein model is calculated on-line. As a result, the set-point is determined from a linear programming problem. For predictive control, a linear approximation of the neural Hammerstein model is calculated on-line and the control policy is determined from a quadratic programming problem. Thanks to linearisation, the necessity of on-line nonlinear optimisation is eliminated. This article emphasises advantages of neural Hammerstein models: accuracy, a limited number of parameters and a simple structure. Thanks to using such models, model transformations can be carried out very efficiently on-line. It is demonstrated that results obtained in the presented structure are very close to those achieved in a computationally demanding structure with on-line nonlinear optimisation. It is also shown that for the considered neutralisation reactor the classical system structure in which for control and set-point optimisation linear models are used gives numerically wrong results.

Clarification and Achievement of Theoretical Limitation in Vehicle Dynamics Integrated Management

In this article, vehicle dynamics integrated control algorithm using an on-line non-linear optimization method is proposed for 4-wheel-distributed steering and 4-wheel-distributed traction/ braking systems. The proposed distribution algorithm calculates the magnitude and direction of tire forces, which satisfy constraints corresponding to the target resultant force and moment of the vehicle motion and which minimizes maximum μ rate (=tire force/friction circle) of each tire. Convexity of this problem is shown, so that the global optimality of the convergent solution of the recursive algorithm is guaranteed. This implies that theoretical limited performance of vehicle dynamics integrated control is clarified. The proposed algorithm is based on SQP (Sequential Quadratic Programming) and steepest gradient algorithm. Calculation speed performance of the proposed algorithm is shown in comparison with primal-dual interior-point method, which is representative optimization method. Furthermore, the effect of this vehicle dynamics control is demonstrated by simulation and experiment to compare with various vehicle dynamics integrated control methods.

Clarification and Achievement of Theoretical Limitation in Vehicle Dynamics Integrated Management

In this article, vehicle dynamics integrated control algorithm using an on-line non-linear optimization method is proposed for 4-wheel-distributed steering and 4-wheel-distributed traction/ braking systems. The proposed distribution algorithm calculates the magnitude and direction of tire forces, which satisfy constraints corresponding to the target resultant force and moment of the vehicle motion and which minimizes maximum μ rate (=tire force/friction circle) of each tire. Convexity of this problem is shown, so that the global optimality of the convergent solution of the recursive algorithm is guaranteed. This implies that theoretical limited performance of vehicle dynamics integrated control is clarified. The proposed algorithm is based on SQP (Sequential Quadratic Programming) and steepest gradient algorithm. Calculation speed performance of the proposed algorithm is shown in comparison with primal-dual interior-point method, which is representative optimization method. Furthermore, the effect of this vehicle dynamics control is demonstrated by simulation and experiment to compare with various vehicle dynamics integrated control methods.

Predictive control of a class of bilinear systems based on global off-line models

A new multi-step adaptive predictive control algorithm for a class of bilinear systems is presented. The structure of the bilinear system is converted into a simple linear model by using nonlinear support vector machine (SVM) dynamic approximation with analytical control law derived. The method does not need on-line parameters estimation because the system’s internal model has been transformed into an off-line global model. Compared with other traditional methods, this control law reduces on-line parameter estimating burden. In addition, its overall linear behavior treating method allows an analytical control law available and avoids on-line nonlinear optimization. Simulation results are presented in the article to illustrate the efficiency of the method.

Vehicle dynamics integrated control for four-wheel-distributed steering and four-wheel-distributed traction/braking systems

In this article, vehicle dynamics integrated control algorithm using an on-line non-linear optimization method is proposed for 4-wheel-distributed steering and 4-wheel-distributed traction/braking systems. The proposed distribution algorithm minimizes work load of each tire, which is controlled to become the same value. The global optimality of the convergent solution of the recursive algorithm can be proved by extension to convex problems. This implies that theoretical limited performance of vehicle dynamics integrated control is clarified. Furthermore, the effect of this vehicle dynamics control for the 4-wheel-distributed steering and 4-wheel-distributed traction/braking systems is demonstrated by simulation to compare with the combination of the various actuators.

A FORMULATION OF NONLINEAR MODEL PREDICTIVE CONTROL USING AUTOMATIC DIFFERENTIATION

The computational burden, which obstacles Nonlinear Model Predictive Control techniques to be widely adopted, is mainly associated with the requirement to solve a set of nonlinear differentiation equations and a nonlinear dynamic optimisation problem in real-time online. In this work, an efficient algorithm has been developed to alleviate the computational burden. The new approach uses the automatic differentiation techniques to solve the set of nonlinear differentiation equations, at the same time to produce the differential sensitivity of the solution against input variables. Using the differential sensitivity, the gradient of the cost function against control moves is accurately obtained so that the online nonlinear dynamic optimisation can be efficiently solved. The new algorithm has been applied to an evaporation process with satisfactory results to cope with setpoint changes, unmeasured disturbances and process-model mismatches.

Fuzzy model predictive control

A fuzzy model predictive control (FMPC) approach is introduced to design a control system for a highly nonlinear process. In this approach, a process system is described by a fuzzy convolution model that consists of a number of quasi-linear fuzzy implications. In controller design, prediction errors and control energy are minimized through a two-layered iterative optimization process. At the lower layer, optimal local control policies are identified to minimize prediction errors in each subsystem. A near optimum is then identified through coordinating the subsystems to reach an overall minimum prediction error at the upper layer. The two-layered computing scheme avoids extensive online nonlinear optimization and permits the design of a controller based on linear control theory. The efficacy of the FMPC approach is demonstrated through three examples.