Abstract — Model identification and measurement acquisition is always to some degree uncertain. Therefore, a framework for Nonlinear Model Predictive Control (NMPC) is proposed that explicitly considers the noise influence on nonlinear dynamic systems with continuous state spaces and a finite set of control inputs in order to significantly increase the control quality. Integral parts of NMPC are the prediction of system states over a finite horizon as well as the problem specific modeling of reward functions. For achieving an efficient and also accurate state prediction, the introduced framework uses transition densities approximated by means of axis-aligned Gaussian mixtures. The representation power of Gaussian mixtures is also used to model versatile reward functions. Thus, together with the prediction technique a closed-form calculation of the optimization problems arising from NMPC is possible. Additionally, an efficient algorithm for calculating an approximate value function of the corresponding optimal control problem employing dynamic programming is presented. Thus, the value function can be calculated off-line, which reduces the on-line computational burden significantly and also permits the use of long optimization horizons. The capabilities of the framework and especially the benefits that can be gained by incorporating the noise in the controller are illustrated by the example of a two-wheeled differential-drive mobile robot following a given path. I.

Abstract: In this paper, a framework for Nonlinear Model Predictive Control (NMPC) that explicitly incorporates the noise influence on systems with continuous state spaces is introduced. By the incorporation of noise, which results from uncertainties during model identification and the measurement process, the quality of control can be significantly increased. Since NMPC requires the prediction of system states over a certain horizon, an efficient state prediction technique for nonlinear noise-affected systems is required. This is achieved by using transition densities approximated by axis-aligned Gaussian mixtures together with methods to reduce the computational burden. A versatile cost function representation also employing Gaussian mixtures provides an increased freedom of modeling. Combining the prediction technique with this value function representation allows closed-form calculation of the necessary optimization problems arising from NMPC. The capabilities of the framework and especially the benefits that can be gained by considering the noise in the controller are illustrated by the example of a mobile robot following a given path. NOTATION x variable x random variable x vector-valued random variable ˜f x (x) probability density function of x f x (x) approximate of ˜f x (x) N (x − µ;σ 2) Gaussian density with mean µ and standard deviation σ Ex{x} expected value of x Jk(x k) value function Vk(x k,u k) input dependent value function gn(x n,u n) cost function k time index n time index of prediction horizon 1

Abstract — A novel online-computation approach to optimal control of nonlinear, noise-affected systems with continuous state and control spaces is presented. In the proposed algorithm, system noise is explicitly incorporated into the control decision. This leads to superior results compared to state-of-the-art nonlinear controllers that neglect this influence. The solution of an optimal nonlinear controller for a corresponding deterministic system is employed to find a meaningful state space restriction. This restriction is obtained by means of approximate state prediction using the noisy system equation. Within this constrained state space, an optimal closed-loop solution for a finite decisionmaking horizon (prediction horizon) is determined within an adaptively restricted optimization space. Interleaving stochastic dynamic programming and value function approximation yields a solution to the considered optimal control problem. The enhanced performance of the proposed discrete-time controller is illustrated by means of a scalar example system. Nonlinear model predictive control is applied to address approximate treatment of infinite-horizon problems by the finite-horizon controller. I.

This paper presents a filter approach for estimating the state of nonlinear dynamic systems based on recursive approximation of posterior densities by means of Dirac mixture functions. The filter consists of a prediction step and a filter step. The approximation approach is based on a systematic minimization of a distance measure and is hence optimal and deterministic. In contrast to non-deterministic methods we are able to determine the optimal number of components in the Dirac mixture. A further benefit of the proposed approach is the consideration of measurements during the approximation process in order to avoid parameter degradation. NOTATION k xk yk

Abstract — Filtering or measurement updating for nonlinear stochastic dynamic systems requires approximate calculations, since an exact solution is impossible to obtain in general. We propose a Gaussian mixture approximation of the conditional density, which allows performing measurement updating in closed form. The conditional density is a probabilistic representation of the nonlinear system and depends on the random variable of the measurement given the system state. Unlike the likelihood, the conditional density is independent of actual measurements, which permits determining its approximation off-line. By treating the approximation task as an optimization problem, we use progressive processing to achieve high quality results. Once having calculated the conditional density, the likelihood can be determined on-line, which, in turn, offers an efficient approximate filter step. As result, a Gaussian mixture representation of the posterior density is obtained. The exponential growth of Gaussian mixture components resulting from repeated filtering is avoided implicitly by the prediction step using the proposed techniques. I.

For several tasks in sensor networks, such as localization, information fusion, or sensor scheduling, Bayesian estimation is of paramount importance. Due to the limited computational and memory resources of the nodes in a sensor network, evaluation of the prediction step of the Bayesian estimator has to be performed very efficiently. An exact and closed-form representation of the predicted probability density function of the system state is typically impossible to obtain, since exactly solving the prediction step for nonlinear discrete-time dynamic systems in closed form is unfeasible. Assuming additive noise, we propose an accurate approximation of the predicted density, that can be calculated efficiently by optimally approximating the transition density using a hybrid density. A hybrid density consists of two different density types: Dirac delta functions that cover the domain of the current density of the system state, and another density type, e.g. Gaussian densities, that cover the domain of the predicted density. The freely selectable, second density type of the hybrid density depends strongly on the noise affecting the nonlinear system. So, the proposed approximation framework for nonlinear prediction is not restricted to a specific noise density. It further allows an analytical evaluation of the Chapman-Kolmogorov prediction equation and can be interpreted as a deterministic sampling estimation approach. In contrast to methods using random sampling like particle filters, a dramatic reduction in the number of components and a subsequent decrease in computation time for approximating the predicted density is gained.

In this paper, a framework for Nonlinear Model Predictive Control (NMPC) that explicitly incorporates the noise influence on systems with continuous state spaces is introduced. By the incorporation of noise, which results from uncertainties during model identification and the measurement process, the quality of control can be significantly increased. Since NMPC requires the prediction of system states over a certain horizon, an efficient state prediction technique for nonlinear noise-affected systems is required. This is achieved by using transition densities approximated by axis-aligned Gaussian mixtures together with methods to reduce the computational burden. A versatile cost function representation also employing Gaussian mixtures provides an increased freedom of modeling. Combining the prediction technique with this value function representation allows closed-form calculation of the necessary optimization problems arising from NMPC. The capabilities of the framework and especially the benefits that can be gained by considering the noise in the controller are illustrated by the example of a mobile robot following a given path.

Abstract — A novel online-computation approach to optimal control of nonlinear, noise-affected systems with continuous state and control spaces is presented. In the proposed algorithm, system noise is explicitly incorporated into the control decision. This leads to superior results compared to state-of-the-art nonlinear controllers that neglect this influence. The solution of an optimal nonlinear controller for a corresponding deterministic system is employed to find a meaningful state space restriction. This restriction is obtained by means of approximate state prediction using the noisy system equation. Within this constrained state space, an optimal closed-loop solution for a finite decisionmaking horizon (prediction horizon) is determined within an adaptively restricted optimization space. Interleaving stochastic dynamic programming and value function approximation yields a solution to the considered optimal control problem. The enhanced performance of the proposed discrete-time controller is illustrated by means of a scalar example system. Nonlinear model predictive control is applied to address approximate treatment of infinite-horizon problems by the finite-horizon controller. I.

Abstract: Many information fusion tasks involve the processing of Gaussian mixtures with simple underlying shape, but many components. This paper addresses the problem of reducing the number of components, allowing for faster density processing. The proposed approach is based on identifying components irrelevant for the overall density’s shape by means of the curvature of the density’s surface. The key idea is to minimize an upper bound of the curvature while maintaining a low global reduction error by optimizing the weights of the original Gaussian mixture only. The mixture is reduced by assigning zero weights to reducible components. The main advantages are an alleviation of the model selection problem, as the number of components is chosen by the algorithm automatically, the derivation of simple curvature-based penalty terms, and an easy, efficient implementation. A series of experiments shows the approach to provide a good trade-off between quality and sparsity. 1