The design of dynamic ship positioning control systems using stochastic optimal control theory

The design of dynamic ship positioning control systems and their associated wave filters is considered. The control problems and the basic components in the system are discussed first. The model of the vessel is then described, since this plays an important role in the design of the Kalman filter which is used to estimate the low frequency motions of the vessel. The model involves both high and low frequency subsystems. The total linearized model for the vessel is represented by state equations, and simulation results are given to illustrate the performance of the Kalman filter. A comparison with a notch filter is also presented. The design of the optimal control system using LQG stochastic control results is considered. The operation of the system under various operating conditions is illustrated. Schemes involving the extended Kalman filter are then described. These enable more realistic non-linear models of the plant to be employed within the filter. This is shown to be unnecessary in the case of the low frequency ship model. However, the extended filter may be useful to track varying wave conditions (using parameter estimation) as represented in the high frequency model. Two extended filtering schemes are presented for this case.     

Learning-based parametrized model predictive control for trajectory tracking

This article is concerned with the tracking of nonequilibrium motions with model predictive control (MPC). It proposes to parametrize input and state trajectories of a dynamic system with basis functions to alleviate the computational burden in MPC. As a result of the parametrization, an optimization problem with fewer variables is obtained, and the memory requirements for storing the reference trajectories are reduced. The article also discusses the generation of feasible reference trajectories that account for the system's dynamics, as well as input and state constraints. In order to cope with repeatable disturbances, which may stem from unmodeled dynamics for example, an iterative learning procedure is included. The approach relies on a Kalman filter that identifies the repeatable disturbances based on previous trials. These are then included in the system's model available to the model predictive controller, which compensates them in subsequent trials. The proposed approach is evaluated on a quadcopter, whose task is to balance a pole, while flying a predefined trajectory.     

Parametric uncertainty and disturbance attenuation in the suboptimal control of a non‐linear electrochemical process

The optimal control of the hydrogen evolution reactions is attempted for the regulation and change of set‐point problems, taking into account that model parameters are uncertain and I/O signals are corrupted by noise. Bilinear approximations are constructed, and their dimension eventually increased to meet accuracy requirements with respect to the trajectories of the original plant. The current approximate model is used to evaluate the optimal feedback through integration of the Hamiltonian equations. The initial value for the costate is found by solving a state‐dependent algebraic Riccati equation, and the resulting control is then suboptimal for the electrochemical process. The bilinear model allows for an optimal Kalman–Bucy filter application to reduce external noise. The filtered output is reprocessed through a non‐linear observer in order to obtain a state‐estimation as independent as possible from the bilinear model. Uncertainties on parameters are attenuated through an adaptive control strategy that exploits sensitivity functions in a novel fashion. The whole approach to this control problem can be applied to a fairly general class of non‐linear continuous systems subject to analogous stochastic perturbations. All calculations can be handled on‐line by standard ordinary differential equations integration software. Copyright © 2007 John Wiley & Sons, Ltd.     

Inverse optimal controller based on extended Kalman filter for discrete‐time nonlinear systems

In this study, we present an inverse optimal control approach based on extended Kalman filter (EKF) algorithm to solve the optimal control problem of discrete‐time affine nonlinear systems. The main aim of inverse optimal control is to circumvent the tedious task of solving the Hamilton‐Jacobi‐Bellman equation that results from the classical solution of a nonlinear optimal control problem. Here, the inverse optimal controller is based on defining an appropriate quadratic control Lyapunov function (CLF) where the parameters of this candidate CLF were estimated by adopting the EKF equations. The root mean square error of the system states is used as the observed error in the case of classical EKF algorithm application, whereas, here, the EKF tries to eliminate the same root mean square error defined over the parameters by generating a CLF matrix with appropriate elements. The performance and the applicability of the proposed scheme is illustrated through both simulations performed on a nonlinear system model and a real‐time laboratory experiment. Simulation study demonstrate the effectiveness of the proposed method in comparison with 2 other inverse control approaches. Finally, the proposed controller is implemented on a professional control board to stabilize a DC‐DC boost converter and minimize a meaningful cost function. The experimental results show the applicability and effectiveness of the proposed EKF‐based inverse optimal control even in real‐time control systems with a very short time constant.     

A combined state and state estimate feedback solution to the ship positioning control problem

A stochastic linear quadratic optimal control problem is considered in which some of the plant states may be measured without a measurement noise component. This set of states are assumed to be associated with the plant inputs and force transducers. The optimal controller is shown to include state feedback from this part of the system. The states which cannot be measured are assumed to be combined in noisy output signal. The optimal controller corresponding to this second subsystem is shown to include a Kalman filter and state-estimate feedback. The combination of state and state-estimate feedback has the advantage that the dimension of the Kalman filter is equal to that of the second subsystem mentioned above. In the conventional solution to this problem, no states are assumed measurable, and the dimension of the Kalman filter is equal to the dimension of the complete system. In many industrial control problems, the combined control law enables a significant reduction in the dimension of the filter to be achieved. The technique has been proposed for use in dynamic ship positioning control systems, and this problem is discussed.     

A fractional Fokker–Planck control framework for subdiffusion processes

An efficient framework for the optimal control of the probability density function of a subdiffusion process is presented. This framework is based on a fractional Fokker–Planck equation that governs the time evolution of the PDF of the subdiffusion process and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open‐loop optimality systems in a model predictive control strategy. The resulting optimality system with fractional evolution operators is discretized by a suitable scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with numerical experiments. Copyright © 2015 John Wiley & Sons, Ltd.     

Design and application of nonlinear model‐based tracking control schemes employing DEKF estimation

This paper deals with the design and application of nonlinear model‐based control schemes for stable and nonlinear benchmark industrial processes. The primary control objective is to facilitate set‐point (constant/time‐varying) tracking in the presence of external disturbances, process noise, measurement noise, parametric uncertainty, and model mismatch. We first propose a “noninferential‐type” model‐based control scheme which involves a finite‐dimensional, nonlinear, and deterministic process model to generate the model states. Secondly, an “inferential‐type” model‐based control scheme has been introduced particularly to take into account the stochastic uncertainties such as process noise and measurement noise. The second scheme exploits the dual extended Kalman filter for estimating the immeasurable states and the process parameters through which disturbance is injected. Unlike fixed‐parameter controllers, the proposed schemes update the controller gains at each step depending on the real‐time process gains. In order to demonstrate the usefulness of the proposed closed‐loop tracking control schemes, two exhaustive case studies have been carried out on the CSTR and Van de Vusse reactor processes, which are considered to be benchmark industrial processes due to highly nonlinear and unpredictable behaviour and due to nonminimum phase property. Finally, the performance of the proposed schemes are compared with an EKF‐based adaptive PI control framework and the simulation results reveal that the transient performance of the proposed schemes are better than that of the aforementioned PI technique especially in perturbed condition (ie, in presence of model mismatch and measurement noise).     

A reduced‐order stochastic observer approach to optimal state estimation with noise‐free measurements

In a continuous‐time Kalman filter, it is required that the measurement noise covariance be non‐singular. If the measurements are noise‐free, then this condition does not hold and, in practice, the measurement data are differentiated to define a derived measurement function to build what is known as Deyst filter. It is proposed here that a reduced‐order observer be used in deriving the linear minimum‐variance filter to construct state estimates based on the original measurement data with no need for differentiation. This filter is of dimension (n?p) where n and p are the state and measurement vector dimensions, respectively. In this work, we consider both the finite‐time and infinite‐time results. The set of all assignable estimation error covariances are characterized and the set of all estimator gains are parametrized in addition to the linear minimum variance optimal results. The conditions for the existence of the optimal steady‐state filter are obtained in terms of the system theoretic properties of the original signal model. A simple example is included to illustrate the effectiveness of the proposed technique. Copyright © 2001 John Wiley & Sons, Ltd.     

A new optimal linear quadratic observer‐based tracker under input constraint for the unknown system with a direct feed‐through term

This paper develops a new optimal linear quadratic observer‐based tracker with input constraint for the linear unknown system with a direct transmission term from input to measured output. The off‐line observer/Kalman filter identification method is used to determine the linear sampled‐data model with a direct feed‐through term. On the basis of this model, a high‐gain optimal linear quadratic analog observer‐based tracker is proposed, so that it can effectively induce a high quality performance on the state estimation and the trajectory tracking design. Besides, the prediction‐based digital redesign method is utilized to obtain a relatively low‐gain and implementable observer and digital tracker from the theoretically well‐designed high‐gain analog observer and tracker for the linear system with a direct transmission term from input to output. To reduce the magnitude of control input, which is caused by the high‐gain property to fit the requirement of the input constraint, the modified linear quadratic analog tracker is proposed. Thus, the control input can be compressed effectively without losing the original high performance of tracking much. Copyright © 2014 John Wiley & Sons, Ltd.     

A multiobjective optimization approach for linear quadratic Gaussian/loop transfer recovery design

This article bestows the linear quadratic Gaussian (LQG)/Loop Transfer Recovery (LTR) optimal controller design for a perturbed linear system having insufficient information about systems states through a multiobjective optimization approach. A Kalman filter observer is required to estimate the unknown states at the output from the noisy data. However, the main downside of the LQG controller's is that its robustness cannot be guaranteed because it consists of linear quadratic regulator (LQR) and Kalman observer, and due to observer incorporation within the LQR framework results in loss of robustness which is undesirable. Therefore, it is necessary to recover the robustness by tuning the controller which further plays havoc with system performance and control effort for certain plants. The present work addresses the investigation of the trade-off between multiobjective indexes (formulated on the basis of robustness, optimal control, and performances) through three multiobjective optimization algorithms as NSGA-II, multiobjective simulated annealing and multiobjective particle swarm optimization. The tuned parameters meet the competitive multiobjective performance indexes that are verified through simulation results. The Pareto front with multiple solutions helps to design a robust controller depending on the weightage given to the respective performance indexes. Simulation results reveal that the proposed multiobjective control strategy helps in recovering the characteristics of LQG/LTR.     

A predictive control framework for 3‐phase induction motors modeled in natural variables

This paper presents a nonlinear control approach for 3‐phase induction motors. The proposed structure combines a 3‐phase predictive controller with an integrative reference filter. The predictive controller is designed based on an induction motor model established in natural variables (without using transformations), which is a nonlinear and time‐variant one. This model enables the controller to work independently with the supply voltages, considering unbalanced situations. A dynamic evaluation of the state equation coefficients is used to perform the process variables prediction, thereby executing a point‐to‐point linearization. The conversion of the rotation speed and stator flux modulus reference values is realized by a integrative 3‐phase referrer, which acts as a reference filter, expressing the references as 3‐phase signals and acting as an integrator to eliminate steady‐state errors. Also, a constraint feature is implemented, to reduce the currents. Simulation results satisfactorily show the proposed control architecture characteristics for various reference values and for motor operation as a brake and with load variation.     

Flexible spacecraft attitude measurement and control scheme incorporating state estimation

The design of an attitude measurement and control scheme for a flexible space vehicle is presented. A proposed spacecraft configuration is used to study the performance of an inertial-optical measurement system incorporating a Kalman filter to obtain the best estimates in the presence of differing noise models. These estimates are used to correct for gyroscope drift and also provide the information necessary to implement optimal attitude control with active damping of the flexural motion.     

Optimization of time-variable-parameter model for data-based soft sensor of industrial debutanizer

The enhancement of modern process control methods has caused the popularity of soft sensors in online quality prediction. It is significant to consider the reduction of model complexity, the performance increment, and decrement of input variables in soft sensor design, simultaneously. The aim of this paper is designing and applying a new data-based soft sensor with minimum input variables for the enhancement of product quality estimation. Time-varying-parameter model by employing the Kalman filter and fixed interval smoothing algorithms has been developed to determine the dynamic transfer function and parameters setting based on time. A novel hybrid method with a dynamic autoregressive exogenous variable model and genetic algorithm has been presented for both state identification and parameter prediction. The combinatorial optimization problem has constructed based on a selection of input variables and an evaluation of Akaike information criterion as a fitness function. An industrial debutanizer column has been used for soft sensor performance validation. The result has indicated that the final soft sensor model in comparison to other presented soft sensing methods for this case has less complexity, fewer input variables, more robust and higher predictive performance. Due to fewer input variables, rapid convergence, and low complexity of this model, it can be efficient in industrial processes control, time-saving, and improvement of quality prediction.     

A multiobjective optimization approach to filter tuning applied to coupled hyperbolic PDEs describing gas flow dynamics

In this work, we introduce a multiobjective optimization approach that seeks the optimal process noise statistics in the extended Kalman filter (EKF). The bi‐objective Mesh Adaptive Direct Search (Bi MADS) algorithm was used to minimize a performance index based on state estimate errors. The EKF estimated the gas flow dynamics in a pipeline system. Simulations were conducted with outflow boundary conditions for the flow model that contain gradual changes and discontinuities. To ensure shock‐capturing properties, the model was approximated with a semidiscrete finite volume scheme using Roe's SUPERBEE limiter. The knee point in the Pareto front was based on normal boundary intersection approach and selected to compute the flow estimates. Numerical experiments demonstrated that Bi MADS is suitable for tuning the EKF and, compared to the normalized weighted sum method and nondominated sorting genetic algorithm, it showed to be superior in terms of computation time and most effective in finding Pareto optimal solutions.     

Minimizing control energy in a class of bounded‐control linear‐quadratic regulator problems

Minimal‐control‐energy strategies are substantiated and illustrated for linear‐quadratic problems with penalized endpoints and no state‐trajectory cost, when bounds in control values are imposed. The optimal solution for a given process with restricted controls, starting at a known initial state, is shown to coincide with the saturated solution to the unrestricted problem that has the same coefficients but starts at a generally different initial state. This result reduces the searching span for the solution: from the infinite‐dimensional set of admissible control trajectories to the finite‐dimensional Euclidean space of initial conditions. An efficient real‐time scheme is proposed here to approximate (eventually to find) the optimal control strategy, based on the detection of the appropriate initial state while avoiding as much as possible the generation and evaluation of state and control trajectories. Numerical (including model predictive control) simulations are provided, compared, and checked against the analytical solution to ‘the cheapest stop of a train’ problem in its pure‐upper‐bounded brake, flexible‐endpoint setting. Copyright © 2013 John Wiley & Sons, Ltd.     

A novel hybrid model predictive control design with application to a quadrotor helicopter

In this paper, a novel control design strategy based on a hybrid model predictive control in combination with fuzzy logic control is presented for a quadrotor helicopter system. In the proposed scheme, a 2‐part control structure is used. In the first part, a linear model predictive controller with receding horizon design strategy is combined with a nonlinear model predictive controller, which is applied as the main controller. In the second part, a 2‐level fuzzy logic controller is utilized to assist the first controller when the error exceeds a predefined value. The proposed nonlinear predictive control method utilizes a novel approach in which a prediction of the future outputs is used in the modeling stage. Using this simple technique, the problem can be solved using linear methods and, thereby, due to considerable reduction in the computational cost, it will be applicable for the systems with fast dynamics. Moreover, the fuzzy logic controller is used as a supervisor to adjust a proportional‐integral‐derivative controller to enhance the system performance by decreasing the tracking error. The proposed scheme is applied to a model of quadrotor system such that the difference between the predicted output of the system and the reference value is minimized while there are some constraints on inputs and outputs of the nonlinear quadrotor system. Simulation results demonstrate the efficiency of the proposed control scheme for the quadrotor system model.     

Approximate parallel controllers for discrete stochastic weakly coupled linear systems

The global Kalman filter of linear weakly coupled discrete systems is exactly decomposed into separate reduced-order local filters via the use of a decoupling transformation. The approximate parallel controllers, up to an arbitrary degree of accuracy, are derived by approximating coefficients of the optimal control law. The proposed method allows parallel processing of information and reduces both off-line and on-line computational requirements. A real-world example demonstrates the efficiency of the proposed method.     

Jump linear quadratic Gaussian problem for a class of nonhomogeneous Markov jump linear systems

This paper concerns with the jump linear quadratic Gaussian problem for a class of nonhomogeneous Markov jump linear systems (MJLSs) in the presence of process and observation noise. By assuming that mode transition rate matrices (MTRMs) are piecewise homogeneous whose variation is subjected to a high‐level Markov process, two Markov processes are proposed to model the characteristics of nonhomogeneous MJLSs: the variation of system mode is governed by a low‐level Markov process, while the variation of MTRM is governed by a high‐level one. Based on this model, a mode‐MTRM‐based optimal filter is firstly given where filter gain can be obtained via coupled Riccati equations. Secondly, we extend the separation principle of the linear quadratic problem to the nonhomogeneous MJLSs case. An optimal controller is then designed to minimize the quadratic system cost. Finally, a potential application in solar boiler system is given to illustrate the developed theoretical methods. Copyright © 2016 John Wiley & Sons, Ltd.     

New methods for the estimation of Takagi–Sugeno model based extended Kalman filter and its applications to optimal control for nonlinear systems

This paper describes new approaches to improve the local and global approximation (matching) and modeling capability of Takagi–Sugeno (T‐S) fuzzy model. The main aim is obtaining high function approximation accuracy and fast convergence. The main problem encountered is that T‐S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the application of the T‐S method because this type of membership function has been widely used during the last 2 decades in the stability, controller design of fuzzy systems and is popular in industrial control applications. The approach developed here can be considered as a generalized version of T‐S identification method with optimized performance in approximating nonlinear functions. We propose a noniterative method through weighting of parameters approach and an iterative algorithm by applying the extended Kalman filter, based on the same idea of parameters’ weighting. We show that the Kalman filter is an effective tool in the identification of T‐S fuzzy model. A fuzzy controller based linear quadratic regulator is proposed in order to show the effectiveness of the estimation method developed here in control applications. An illustrative example of an inverted pendulum is chosen to evaluate the robustness and remarkable performance of the proposed method locally and globally in comparison with the original T‐S model. Simulation results indicate the potential, simplicity, and generality of the algorithm. An illustrative example is chosen to evaluate the robustness. In this paper, we prove that these algorithms converge very fast, thereby making them very practical to use. Copyright © 2011 John Wiley & Sons, Ltd.     

Low‐order multi‐rate linear time‐invariant decentralized trackers using the new observer‐based sub‐optimal method for unknown sampled‐data nonlinear time‐delay system with closed‐loop decoupling

This paper presents the low‐order multi‐rate linear time‐invariant decentralized trackers using the new observer‐based sub‐optimal method for a class of unknown sampled‐data nonlinear time‐delay system with closed‐loop decoupling. For the unknown sampled‐data nonlinear time‐delay system, we assume that the inner time delay is clearly known. Under this prerequisite, the appropriate (low‐) order decentralized linear observer for the unknown sampled‐data nonlinear system is determined by the off‐line observer/Kalman filter identification (OKID) method with artificial delay input and actual delay output. Then, the above observer has been further improved based on the proposed new observer‐based sub‐optimal approach. Sequentially, the decentralized tracker with the high gain property is proposed, so that the closed‐loop system has the decoupling property. The proposed approach constructs complete mathematics method including the concept of optimal control theory and state‐matching digital redesign technique and is quite useful for the complicated interconnected large‐scale sampled‐data nonlinear time‐delay system with unknown system equation. Copyright © 2010 John Wiley & Sons, Ltd.