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11
A ClosedForm Model Predictive Control Framework for Nonlinear Noise Corrupted Systems
 in Proceedings of the 4th International Conference on Informatics in Control, Automation, and Robotics (ICINCO 2007
, 2007
"... 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 p ..."
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Cited by 5 (5 self)
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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 noiseaffected systems is required. This is achieved by using transition densities approximated by axisaligned 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 closedform 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 vectorvalued 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
A State Estimator for Nonlinear Stochastic Systems Based on Dirac Mixture Approximations
 in Proceedings of the 4th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2007
, 2007
"... 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 min ..."
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Cited by 4 (3 self)
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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 nondeterministic 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
Efficient Control of Nonlinear Noise–Corrupted Systems Using a Novel Model Predictive Control Framework
 In Proceedings of the 2007 American Control Conference (ACC 2007
"... 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 ..."
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Cited by 4 (4 self)
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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 axisaligned Gaussian mixtures. The representation power of Gaussian mixtures is also used to model versatile reward functions. Thus, together with the prediction technique a closedform 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 offline, which reduces the online 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 twowheeled differentialdrive mobile robot following a given path. I.
Efficient Nonlinear Measurement Updating based on Gaussian Mixture Approximation of Conditional Densities
 In Proceedings of the American Control Conference (ACC
, 2007
"... 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 ..."
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Cited by 4 (3 self)
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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 offline. 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 online, 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.
OnlineComputation Approach to Optimal Control of NoiseAffected Nonlinear Systems with Continuous State and Control Spaces
 In Proceedings of the European Control Conference (ECC 2007
, 2007
"... Abstract — A novel onlinecomputation approach to optimal control of nonlinear, noiseaffected 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 ..."
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Cited by 3 (3 self)
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Abstract — A novel onlinecomputation approach to optimal control of nonlinear, noiseaffected 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 stateoftheart 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 closedloop 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 discretetime controller is illustrated by means of a scalar example system. Nonlinear model predictive control is applied to address approximate treatment of infinitehorizon problems by the finitehorizon controller. I.
Hybrid Transition Density Approximation for Efficient Recursive Prediction of Nonlinear Dynamic Systems
 in International Conference on Information Processing in Sensor Networks (IPSN
, 2007
"... 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 ha ..."
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Cited by 3 (3 self)
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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 closedform representation of the predicted probability density function of the system state is typically impossible to obtain, since exactly solving the prediction step for nonlinear discretetime 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 ChapmanKolmogorov 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.
www.informatik2011.de Superficial Gaussian Mixture Reduction
"... 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 componen ..."
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Cited by 1 (1 self)
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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 curvaturebased penalty terms, and an easy, efficient implementation. A series of experiments shows the approach to provide a good tradeoff between quality and sparsity. 1
SupportVector Conditional Density Estimation for Nonlinear Filtering
"... Abstract – A nonparametric conditional density estimation algorithm for nonlinear stochastic dynamic systems is proposed. The contributions are a novel support vector regression for estimating conditional densities, modeled by Gaussian mixture densities, and an algorithm based on crossvalidation f ..."
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Cited by 1 (1 self)
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Abstract – A nonparametric conditional density estimation algorithm for nonlinear stochastic dynamic systems is proposed. The contributions are a novel support vector regression for estimating conditional densities, modeled by Gaussian mixture densities, and an algorithm based on crossvalidation for automatically determining hyperparameters for the regression. The conditional densities are employed with a modified axisaligned Gaussian mixture filter. The experimental validation shows the high quality of the conditional densities and good accuracy of the proposed filter.
NOTATION
"... 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, th ..."
Abstract
 Add to MetaCart
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 noiseaffected systems is required. This is achieved by using transition densities approximated by axisaligned 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 closedform 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.
Nonlinear Systems with Continuous State and Control Spaces
"... Abstract — A novel onlinecomputation approach to optimal control of nonlinear, noiseaffected 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 ..."
Abstract
 Add to MetaCart
Abstract — A novel onlinecomputation approach to optimal control of nonlinear, noiseaffected 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 stateoftheart 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 closedloop 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 discretetime controller is illustrated by means of a scalar example system. Nonlinear model predictive control is applied to address approximate treatment of infinitehorizon problems by the finitehorizon controller. I.