Fully probabilistic control for stochastic nonlinear control systems with input dependent noise

Robust controllers for nonlinear stochastic systems with functional uncertainties can be consistently designed using probabilistic control methods. In this paper a generalised probabilistic controller design for the minimisation of the Kullback-Leibler divergence between the actual joint probability density function (pdf) of the closed loop control system, and an ideal joint pdf is presented emphasising how the uncertainty can be systematically incorporated in the absence of reliable systems models. To achieve this objective all probabilistic models of the system are estimated from process data using mixture density networks (MDNs) where all the parameters of the estimated pdfs are taken to be state and control input dependent. Based on this dependency of the density parameters on the input values, explicit formulations to the construction of optimal generalised probabilistic controllers are obtained through the techniques of dynamic programming and adaptive critic methods. Using the proposed generalised probabilistic controller, the conditional joint pdfs can be made to follow the ideal ones. A simulation example is used to demonstrate the implementation of the algorithm and encouraging results are obtained.

A probabilistic indirect adaptive control for systems with input-dependent noise

A probabilistic indirect adaptive controller is proposed for the general nonlinear multivariate class of discrete time system. The proposed probabilistic framework incorporates input–dependent noise prediction parameters in the derivation of the optimal control law. Moreover, because noise can be nonstationary in practice, the proposed adaptive control algorithm provides an elegant method for estimating and tracking the noise. For illustration purposes, the developed method is applied to the affine class of nonlinear multivariate discrete time systems and the desired result is obtained: the optimal control law is determined by solving a cubic equation and the distribution of the tracking error is shown to be Gaussian with zero mean. The efficiency of the proposed scheme is demonstrated numerically through the simulation of an affine nonlinear system.

Instantaneous phase of the autocorrelation and robustness to signal-dependent noise

The phase of an analytic signal constructed from the autocorrelation function of a signal contains significant information about the shape of the signal. Using Bedrosian's (1963) theorem for the Hilbert transform it is proved that this phase is robust to multiplicative noise if the signal is baseband and the spectra of the signal and the noise do not overlap. Higher-order spectral features are interpreted in this context and shown to extract nonlinear phase information while retaining robustness. The significance of the result is that prior knowledge of the spectra is not required.

A common stochastic accumulator with effector-dependent noise can explain eye-hand coordination

The computational architecture that enables the flexible coupling between otherwise independent eye and hand effector systems is not understood. By using a drift diffusion framework, in which variability of the reaction time (RT) distribution scales with mean RT, we tested the ability of a common stochastic accumulator to explain eye-hand coordination. Using a combination of behavior, computational modeling and electromyography, we show how a single stochastic accumulator to threshold, followed by noisy effector-dependent delays, explains eye-hand RT distributions and their correlation, while an alternate independent, interactive eye and hand accumulator model does not. Interestingly, the common accumulator model did not explain the RT distributions of the same subjects when they made eye and hand movements in isolation. Taken together, these data suggest that a dedicated circuit underlies coordinated eye-hand planning.

Generalized Partial Response Equalization and Data-Dependent Noise Predictive Signal Detection Over Media Models for TDMR

Two-dimensional magnetic recording (2-D TDMR) is an emerging technology that aims to achieve areal densities as high as 10 Tb/in(2) using sophisticated 2-D signal-processing algorithms. High areal densities are achieved by reducing the size of a bit to the order of the size of magnetic grains, resulting in severe 2-D intersymbol interference (ISI). Jitter noise due to irregular grain positions on the magnetic medium is more pronounced at these areal densities. Therefore, a viable read-channel architecture for TDMR requires 2-D signal-detection algorithms that can mitigate 2-D ISI and combat noise comprising jitter and electronic components. Partial response maximum likelihood (PRML) detection scheme allows controlled ISI as seen by the detector. With the controlled and reduced span of 2-D ISI, the PRML scheme overcomes practical difficulties such as Nyquist rate signaling required for full response 2-D equalization. As in the case of 1-D magnetic recording, jitter noise can be handled using a data-dependent noise-prediction (DDNP) filter bank within a 2-D signal-detection engine. The contributions of this paper are threefold: 1) we empirically study the jitter noise characteristics in TDMR as a function of grain density using a Voronoi-based granular media model; 2) we develop a 2-D DDNP algorithm to handle the media noise seen in TDMR; and 3) we also develop techniques to design 2-D separable and nonseparable targets for generalized partial response equalization for TDMR. This can be used along with a 2-D signal-detection algorithm. The DDNP algorithm is observed to give a 2.5 dB gain in SNR over uncoded data compared with the noise predictive maximum likelihood detection for the same choice of channel model parameters to achieve a channel bit density of 1.3 Tb/in(2) with media grain center-to-center distance of 10 nm. The DDNP algorithm is observed to give similar to 10% gain in areal density near 5 grains/bit. The proposed signal-processing framework can broadly scale to various TDMR realizations and areal density points.

TOPS (Task optimization in the presence of signal-dependent noise) model

This paper proposes a movement trajectory planning model, which is a maximum task achievement model in which signal-dependent noise is added to the movement command. In the proposed model, two optimization criteria are combined, maximum task achievement and minimum energy consumption. The proposed model has the feature that the end-point boundary conditions for position, velocity, and acceleration need not be prespecified. Consequently, the method can be applied not only to the simple point-to-point movement, but to any task. In the method in this paper, the hand trajectory is derived by a psychophysical experiment and a numerical experiment for the case in which the target is not stationary, but is a moving region. It is shown that the trajectory predicted from the minimum jerk model or the minimum torque change model differs considerably from the results of the psychophysical experiment. But the trajectory predicted from the maximum task achievement model shows good qualitative agreement with the hand trajectory obtained from the psychophysical experiment. © 2004 Wiley Periodicals, Inc.

Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises

In this paper, we consider the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises under two criteria. The first one is an unconstrained mean-variance trade-off performance criterion along the time, and the second one is a minimum variance criterion along the time with constraints on the expected output. We present explicit conditions for the existence of an optimal control strategy for the problems, generalizing previous results in the literature. We conclude the paper by presenting a numerical example of a multi-period portfolio selection problem with regime switching in which it is desired to minimize the sum of the variances of the portfolio along the time under the restriction of keeping the expected value of the portfolio greater than some minimum values specified by the investor. (C) 2011 Elsevier Ltd. All rights reserved.

Regression with input-dependent noise: A Gaussian process treatment

Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance.

Regression with input-dependent noise: A Bayesian treatment

In most treatments of the regression problem it is assumed that the distribution of target data can be described by a deterministic function of the inputs, together with additive Gaussian noise having constant variance. The use of maximum likelihood to train such models then corresponds to the minimization of a sum-of-squares error function. In many applications a more realistic model would allow the noise variance itself to depend on the input variables. However, the use of maximum likelihood to train such models would give highly biased results. In this paper we show how a Bayesian treatment can allow for an input-dependent variance while overcoming the bias of maximum likelihood.

Enhanced information transmission with signal dependent noise in an array of LIF neurons

Sensory cells usually transmit information to afferent neurons via chemical synapses, in which the level of noise is dependent on an applied stimulus. Taking into account such dependence, we model a sensory system as an array of LIF neurons with a common signal. We show that information transmission is enhanced by a nonzero level of noise. Moreover, we demonstrate a phenomenon similar to suprathreshold stochastic resonance with additive noise. We remark that many properties of information transmission found for the LIF neurons was predicted by us before with simple binary units [Phys. Rev. E 75, 021121 (2007)]. This confirmation of our predictions allows us to point out identical roots of the phenomena found in the simple threshold systems and more complex LIF neurons.

Optimal design for correlated processes with input-dependent noise

Optimal design for parameter estimation in Gaussian process regression models with input-dependent noise is examined. The motivation stems from the area of computer experiments, where computationally demanding simulators are approximated using Gaussian process emulators to act as statistical surrogates. In the case of stochastic simulators, which produce a random output for a given set of model inputs, repeated evaluations are useful, supporting the use of replicate observations in the experimental design. The findings are also applicable to the wider context of experimental design for Gaussian process regression and kriging. Designs are proposed with the aim of minimising the variance of the Gaussian process parameter estimates. A heteroscedastic Gaussian process model is presented which allows for an experimental design technique based on an extension of Fisher information to heteroscedastic models. It is empirically shown that the error of the approximation of the parameter variance by the inverse of the Fisher information is reduced as the number of replicated points is increased. Through a series of simulation experiments on both synthetic data and a systems biology stochastic simulator, optimal designs with replicate observations are shown to outperform space-filling designs both with and without replicate observations. Guidance is provided on best practice for optimal experimental design for stochastic response models. © 2013 Elsevier Inc. All rights reserved.

Enhanced information transmission with signal-dependent noise in an array of nonlinear elements

We have investigated information transmission in an array of threshold units that have signal-dependent noise and a common input signal. We demonstrate a phenomenon similar to stochastic resonance and suprathreshold stochastic resonance with additive noise and show that information transmission can be enhanced by a nonzero level of noise. By comparing system performance to one with additive noise we also demonstrate that the information transmission of weak signals is significantly better with signal-dependent noise. Indeed, information rates are not compromised even for arbitrary small input signals. Furthermore, by an appropriate selection of parameters, we observe that the information can be made to be (almost) independent of the level of the noise, thus providing a robust method of transmitting information in the presence of noise. These result could imply that the ability of hair cells to code and transmit sensory information in biological sensory systems is not limited by the level of signal-dependent noise. © 2007 The American Physical Society.

Impedance control reduces instability that arises from motor noise.

There is ample evidence that humans are able to control the endpoint impedance of their arms in response to active destabilizing force fields. However, such fields are uncommon in daily life. Here, we examine whether the CNS selectively controls the endpoint impedance of the arm in the absence of active force fields but in the presence of instability arising from task geometry and signal-dependent noise (SDN) in the neuromuscular system. Subjects were required to generate forces, in two orthogonal directions, onto four differently curved rigid objects simulated by a robotic manipulandum. The endpoint stiffness of the limb was estimated for each object curvature. With increasing curvature, the endpoint stiffness increased mainly parallel to the object surface and to a lesser extent in the orthogonal direction. Therefore, the orientation of the stiffness ellipses did not orient to the direction of instability. Simulations showed that the observed stiffness geometries and their pattern of change with instability are the result of a tradeoff between maximizing the mechanical stability and minimizing the destabilizing effects of SDN. Therefore, it would have been suboptimal to align the stiffness ellipse in the direction of instability. The time course of the changes in stiffness geometry suggests that modulation takes place both within and across trials. Our results show that an increase in stiffness relative to the increase in noise can be sufficient to reduce kinematic variability, thereby allowing stiffness control to improve stability in natural tasks.