Lenka Pavelková

Projection-based Bayesian recursive estimation of ARX model with uniform innovations

Abstract Autoregressive model with exogenous inputs (ARX) is a widely-used black-box type model underlying adaptive predictors and controllers. Its innovations, stochastic unobserved stimulus of the model, are white, zero mean with time-invariant variance. Mostly, the innovations are assumed to be normal. It induces least squares as the adequate estimation procedure. The light tails of the normal distribution allow one to accept the unbounded support as a reasonable approximate description of bounded physical quantities. In some cases, however, this approximation is too crude or does not fit subsequent processing, for instance, robust control design. Then, techniques that deal with unknown-but-bounded equation errors are used. More often than not these techniques give up a stochastic interpretation of innovations and develop estimation algorithms of a min-max type. The paper assumes bounded innovations but stays within the standard Bayesian estimation framework by assuming uniformly distributed innovations. The posterior probability density function (pdf) is first described and approximated by a pdf with a fixed-dimensional statistic. Consequently, the estimation can run in real time. Moreover, its limited memory allows for tracking time-varying parameters. In this manner, an alternative to popular forgetting techniques is also obtained. The paper provides a complete algorithmic solution and illustrates its behavior.

Fuel Consumption Optimization: Early Experiments*

Abstract The paper deals with a problem of fuel consumption optimization. Solutions existing in this field are mainly based on the various conceptual approaches such as hybrid and electric vehicles. However, it leads to high initial cost of a vehicle. The approach presented in this paper aims at conventional vehicles and is based on recursive algorithms of system identification and adaptive quadratic optimal control under Bayesian methodology. Experiments with real data measured on a driven vehicle are provided.

SERVO PROBLEM WITHIN FUEL CONSUMPTION OPTIMIZATION

The presented paper deals with a problem of fuel consumption optimization. Today’s automotive industry solves this problem mainly via various conceptual approaches (hybrid and electric vehicles). However, it leads to high initial cost of a vehicle. This paper focuses on fuel economy for conventional vehicles. For this aim, recursive algorithms of adaptive optimal quadratic control under Bayesian methodology are used. A stochastic servo problem, including setpoint tracking, is a part of the considered adaptive control design. In this paper, fuel consumption and speed of a driven vehicle are the controlled variables, where the first one is to be optimized and the second one is pushed to track its set-point. This set-point is a recommended roaddependent speed. Experiments with real data measured on a driven vehicle are provided.

Double Optimization of Fuel Consumption and Speed Tracking

Abstract This paper presents automatic fuel consumption optimization with simultaneous keeping the recommended vehicles speed. These tasks are closely related since a simple minimization of fuel consumption leads to stopping a vehicle. The proposed “double” optimization is performed online using combination of two controllers. The first of them is based on fully probabilistic design (FPD) under Bayesian methodology. It optimizes the “driver-vehicle” closed loop with the aim to save fuel and keep the recommended speed, using externally given setpoints. Optimized values serve as setpoints for PID controller, which provides necessary setpoint tracking. Research is performed in collaboration with Skoda auto ( www.skoda-auto.com ).

Approximate Bayesian Recursive Estimation of Linear Model with Uniform Noise

Abstract Recursive estimation forms core of adaptive prediction and control. Dynamic exponential family is the only but narrow class of parametric models that allows exact Bayesian estimation. The paper provides an approximate estimation of important autoregressive model with exogenous variables (ARX) and uniform noise. This model reflects well physical nature of modelled system: majority of signals, noise and estimated parameters are bounded. Unlike former solutions, the paper proposes an algorithm that provides a full (approximate) posterior probability density function (pdf) of unknown parameters. Behaviour of the designed algorithm is illustrated by simulations.

Parameter Estimation With Partial Forgetting Method

Abstract The paper proposes a new estimating algorithm for linear parameter varying systems with slowly time-varying parameters when the rate of change of individual parameters is different. It introduces a true probability density function, describing ideally the behaviour of parameters. However, as it is unknown, we search for its best approximation. A convex combination of point estimates, defined by individual hypotheses about the true probability density function, is then approximated by a single density. That serves as the best available description of parameters’ behaviour and it is therefore suitable e.g. for prediction purposes.

Bayesian filtering with discrete-valued state

The paper deals with estimation of a state with discrete values. The proposed estimation technique is evolved as an application of Bayesian filtering to a state-space model with discrete distribution. The example of filtering is shown with Bernoulli distributions. The considered problem is one of the items aiming at filtering with mixed continuous and discrete state. Illustrative experiments demonstrate the filtering with discrete simulated data from the traffic control area, which is a potential application domain of the research.

ESTIMATION OF APPROXIMATE MARKOV CHAINS

Abstract Controlled Markov chains represent important universal class of models distinguished by their ability to describe non-linear stochastic dynamic systems. However, their use is restricted: the demands on the amount of data as well as on computer power required for the estimation blow up exponentially with the dimension of their state. Recently proposed Bayesian infonnation pooling has been recognized as a promising tool for approaching the dimensionality problem. Essentially, Markov chains with low-dimensional states are estimated independently and pooled into a global description of the system. The partial models are naturally approximate and this fact should be taken into account by the estimation procedure used which has to be adaptive. Within the Bayesian framework, an analogy of exponential and/or restricted forgetting can be used. In this paper, an attempt to improve the estimation of the approximate Markov chain is presented. It is based on the same pooling idea, now applied to a single model understood as a collection of models labelled by the measured states.