Recursive Prediction Error Method Research Articles

Neural Networks for Optimal Control

Two neural networks are trained to act as an observer and a controller, respectively, to control anon-linear, multi-variable process. The process model is the well-known Innovation State Space model. Firstly, the observer network is trained with a Recursive Prediction Error Method using a Gauss-Newton search direction to minimize the the prediction error. Next, the trained observer network is applied in a closed-loop simulation to train another neural network, the controller. During this training an optimal control cost function is minimized using a recursive, off-line, backward training method, similar to the Back Propagation Through Time (BPTT) method. Finally, a practical, non-linear, noisy and multi-variable example confirms, that the model and the training methods are a promising technique to control non-linear processes, which are difficult to model.

Functional Series Modelling Approach to Time Varying Zero Tracking Algorithms

In many situations, the process representation in terms of poles and zeros may provide a more appropriate description than other widely used representations. This paper deals with time-varying process. A weighted Functional Series Modelling for system's poles and zeros is proposed. The approach is based on the Gauss-Newton recursive prediction error method with an exact expression for the gradient.

Identification of rudder—yaw and rudder—roll steering models by using recursive prediction error techniques

AbstractLinear and non‐linear recursive prediction error (RPE) methods have been employed for the purpose of identification of the dynamics of ship rudder—yaw and —roll motions. The evaluations of the goodness of model structures have been carried out by comparing simulated outputs from the identified model with real measurements. In the linear case Nomoto's first‐order model has been initially considered for the transfer functions between rudder—yaw and —sway, with a second‐order model for rudder—roll with coupling effects considered only between sway—yaw and —roll. Identification results have shown that such a model is not appropriate. Changing the model to second‐order between rudder—sway and —yaw and further considering the coupling effects between roll angle—sway and —yaw shows that a significant improvement is achieved, while the coupling effect between yaw and roll appears negligible. In the nonlinear case effects due to various non‐linear terms have been studied with the non‐linear RPE algorithm.

Recursive prediction error methods for online estimation in nonlinear state-space models

Several recursive algorithms for online, combined state and parameter estimation in nonlinear state-space models are discussed in this paper. Well-known algorithms such as the extended Kalman filter and alternative formulations of the recursive prediction error method are included as well as a new method based on a line-search strategy. A comparison of the algorithms illustrates that they are very similar although the differences can be important to the online tracking capabilities and robustness. Simulation experiments on a simple nonlinear process show that the performance under certain conditions can be improved by including a line-search strategy. >

Real-Time Neural Network Control - An IMC Approach

An internal model control (IMC) structure provides a practical framework to nonlinear control design. Current nonlinear IMC design procedures are restricted to open-loop stable systems with stable inverse. In this paper, a novel IMC design method is presented for time-discrete, nonlinear single-input single-output systems. The design method is based on a nonlinear parametric optimal control law which is realized by a perceptron network. The parameters of the network are obtained by minimizing the cost function by the recursive prediction error method (RPEM).Also, a parameter projection is employed to restrict the parameters of the controller into the stable region of the parameter space. This procedure relaxes the assumption about the stability of the model inverse. The properties of the resulting controller are studied in real-time experiments with a laboratory heating process. The results demonstrate that the proposed design method yields good performance characteristics and robust control

An ordinary differential equation technique for continuous-time parameter estimation

An ordinary differential equation technique is developed via averaging theory and weak convergence theory to analyze the asymptotic behavior of continuous-time recursive stochastic parameter estimators. This technique is an extension of L. Ljung's (1977) work in discrete time. Using this technique, the following results are obtained for various continuous-time parameter estimators. The recursive prediction error method, with probability one, converges to a minimum of the likelihood function. The same is true of the gradient method. The extended Kalman filter fails, with probability one, to converge to the true values of the parameters in a system whose state noise covariance is unknown. An example of the extended least squares algorithm is analyzed in detail. Analytic bounds are obtained for the asymptotic rate of convergence of all three estimators applied to this example. >

A convergent approximation of the continuous-time optimal parameter estimator

Continuous-time linear stochastic systems that are bilinear in the state and parameters are considered. A specific approximation to the optimal nonlinear filter used as a recursive parameter estimator is derived by retaining third-order moments and using a Gaussian approximation for higher order moments. With probability one, the specific approximation is proved to converge to a minimum of the likelihood function. The proof uses the ordinary differential equation technique and requires that the trajectories of the slow system be bounded on finite time intervals and that the fixed parameter fast system by asymptotically stable. The fixed parameter fast system is proved to be asymptotically stable if the parameter update gain is small enough. Essentially, the specific approximation is asympotically equivalent to the recursive prediction error method, thus inheriting its asymptotic rate of convergence. A numerical simulation for a simple example indicates that the specific approximation has better transient response than other commonly used convergent parameter estimators. >

Properties of the Neural Network Internal Model Controller

Nonlinear Internal Model Controller (IMC), realized with a multilayer perceptron neural network, is studied in this paper. The neural networks for the process model and the resulting controller are identified using the Recursive Prediction Error (RPE) method with the applied gradient calculation procedure. Novel stability analysis and stability projection methods are also introduced. Both simulation and laboratory processes are used in testing the control performance. Results indicate that the IMC control structure provides robust performance and is clearly a good alternative for controlling nonlinear plants. The real-time experiment addresses the very important question of implementing and gaining practical experience with neural network controllers.

Application of Neural Networks to Rudder Roll Stabilization

Abstract A number of research studies have been carried out, during the last two decades, devoted to rudder roll stabilization of surface ships. A possible solution has been recently proposed, which is based on adaptive control. According to such approach, the unknown system parameters are recursively estimated by an RPE (Recursive Prediction Error) method for a linear or nonlinear rudder-yaw and rudder-roll dynamical model, while an LQG (Linear Quadratic Gaussian) optimal control strategy is subsequently applied in order to minimize a preset cost function associated to the dynamical process. It should be pointed out, however, that in the presence of a relatively high number of unknown parameters, RPE method can give rise to time consuming numerical algorithms with a too low convergence rate. This paper outlines a possible alternative use of artificial neural network as an efficient recursive parameter estimation tool. For this purpose, a particular type of neural network called CINN (Competitively Inhibited Neural Network) is considered, which explicitely takes into account competition among neurons. The competition process is used for achieving adaptive features which are used to improve the efficiency of the estimation algorithm in terms of convergence rate and robustness. The performances of the resultant neural network based adaptive controller applied to the rudder roll stabilization problem are illustrated through a number of simulation examples.

Properties of the neural network internal model controller

Nonlinear Internal Model Controller (IMC), realized with a multilayer perceptron neural network, is studied in this paper. The neural networks for the process model and the resulting controller are identified using the Recursive Prediction Error (RPE) method with the applied gradient calculation procedure. Novel stability analysis and stability projection methods are also introduced. Both simulation and laboratory processes are used in testing the control performance. Results indicate that the IMC control structure provides robust performance and is clearly a good alternative for controlling nonlinear plants. The real-time experiment addresses the very important question of implementing and gaining practical experience with neural network controllers.

System Identification of Remotely Operated Vehicle Dynamics

Abstract Models for Remotely Operated Underwater Vehicles (ROVs) are difficult to derive because their dynamics are strongly coupled, highly nonlinear and vary according to the vehicle’s operating configuration. In addition, conventional modeling techniques require the use of expensive, specialized testing equipment. An alternative procedure using System Identification (SI) to process data gathered during simple free-running trials can generate fast, inexpensive and accurate ROV models. Three SI algorithms, Least Squares (LS), Extended Least Squares (ELS) and the Recursive Prediction Error Method (RPEM), are tested on simulated ROV data and compared for accuracy and economy. The vehicle studied in this paper, the UMEL Seapup, can be represented accurately by a linear model at the low speeds where accurate maneuvering control is most important and gain scheduling can be used to switch models at higher speeds.

Identification of errors-in-variables models with Faurre type realization algorithms

Identification of multiple input output discrete time linear dynamic systems operating in open or closed loop are considered in the time invariant case. Two methods have been used for such a purpose: the recursive prediction error method on the input-output data and the successive Gram-Schmidt orthogonalization of the spectral density function of the joint input-output variable.

Adaptive Modeling, Identification, and control of dynamic structural Systems. I: Theory

A concise review of the fheory of adaptive modeling, identification, and control of dynamic structural systems based on discrete‐time recordings is presented. Adaptive methods have four major advantages over the classical methods: (1) Removal of the noise from the signal is done over the whole frequency band; (2) time‐varying characteristics of systems can be tracked; (3) systems with unknown characteristics can be controlled; and (4) a small segment of the data is needed during the computations. Included in the paper are the discrete‐time representation of single‐input single‐output (SISO) systems, models for SISO systems with noise, the concept of stochastic approximation, recursive prediction error method (RPEM) for system identification, and the adaptive control. Guidelines for model selection and model validation and the computational aspects of the method are also discussed in the paper. The present paper is the first of two companion papers. The theory given in the paper is limited to that which is...

Estimation of the State and Parameters of a Bioprocess Using the Recursive Prediction Error Method

This paper presents an alternative to the Extended Kalman filter for the estimation of the state and parameters of a bioprocess. This method called Recursive Prediction Error Method has been developped by Ljung [83]and can be applied on a model of any degree of complexity as EKF, but stability and convergences properties can be analyzed more easily. From a practical point of view, another advantage is that the tuning parameters of the algorithm have a simpler meaning, and then made the algorithm easier to use.

Identification of a class of nonlinear state-space models using RPE techniques

The RPE (recursive prediction error) method in state-space form is developed in the nonlinear systems and extended to include the exact form of a nonlinearity, thus enabling structure preservation for certain classes of nonlinear systems. Both the discrete and the continuous-discrete versions of the algorithm in an innovations model are investigated, and a nonlinear simulation example shows a quite convincing performance of the filter as combined parameter and state estimator. >

Surface accuracy control in end milling using adaptive state tracking

A geometric adaptive control system for the end milling process is developed based on adaptive state tracking. The control specification is given by an external reference trajectory, which represents the desired geometry of the finished workpiece. The machining system is represented by a state-space model which includes the flexibilities of both the cutter and the workpiece. The Recursive Prediction Error (RPE) method is used for joint state and parameter estimation of the state innovations model and the necessary adaptive state tracking control law is derived. In this study, on-line assessment of workpiece geometry is considered not realizable in practice, and only the cutting force is assumed measurable. The desired control is achieved by manipulating the position command for the servo loop so that the actual tool position follows the desired workpiece geometry. Simulations are obtained for the proposed geometric adaptive control system and the results are shown to be quite satisfactory.

State-space Self-tuning Control with Application to End Milling

An algorithm for self-tuning control of an immeasurable physical state of linear system is introduced. The control objective is specified by time-varying external reference signals The suggested algorithm can be considered as a concatenation of the Recursive Prediction Error (RPE) method for joint state and parameter estimation of state innovations model and state feedback control law with pole-zero placement.The practical usefulness of the algorithm is demonstrated by application to surface accuracy control in end milling process.

Towards a Globally Convergent Approximation of Optimal On-line Parameter Estimation

A possibly globally convergent approximation to the optimal continuous time non linear filter for n pole parameters and n states only is found by computing the first, second, and third order moment equations. Closure of the moment equations results from approximating fourth order moments by the Gaussian formula involving second order moments. Five necessary conditions for stability are given. The unique convergence point of the algorithm is shown to be the true pole parameter value in the scalar case. The new algorithm is asymptotically equivalent to the non-stationary version of the recursive prediction error method. The error incurred by the Gaussian approximation is shown to be small in several examples of the discrete time optimal nonlinear filter.

Adaptive control of a manipulator when the mass of the load is unknown

Abstract In this paper an adaptive controller for a robotic manipulator is presented. In the control system, the computed torque technique is used to linearize and decouple the dynamic model. The computed torque technique requires the complete dynamic model to be known. Normally, this model is known except maybe for a few parameters, e.g. load inertia and center of gravity and friction co-efficients. In this paper, the unknown parameters, which all represent physical quantities, are estimated with a recursive prediction error method. The most recent parameter estimate is then used in the computed torque technique. The control system is applied to the positioning part of an industrial robot in a simulation experiment where the unknown parameter is the mass of the load. The controller performs well, and the mass of the load can even be estimated when the manipulator is accelerating and the control deviation is large.

LQG Feedback and Feedforward Self-Tuning Control

Abstract A new LQG self-tuning control algorithm which includes feedforward action is derived using a very general polynomial equation system model. The algorithm is derived for use in advanced or complex industrial applications where the simple ARMAX model (used successfully in process control applications) is not adequate. The main contribution of the paper is the derivation of an identification algorithm (based on the recursive prediction-error method) for estimation of the parameters of the general system model. Calculation of the control law requires spectral factorization and the solution of diophantine equations. Although the full algorithm is quite complex some optional simplifications are suggested which make the computations more straightforward.