## Osion kuvaus

• ### Yleinen

• ##### Statistics and stochastics

• Gaussian Random Variables; pdf, Mean and Covariance;   Joint and Conditional Random Variables
• Fundamental Equations of Linear Estimation
• Stochastic processes; Correlation; White noise process
• Random Sequences, Markov Processes; Markov  Sequences and Markov Chains

• #### Basic concepts in estimation

Random and nonrandom parameters
Definitions of estimates
• ML Maximum Likelihood
• MAP Maximum A Posteriori
• LS Least Squares
• MMSE Minimum Mean square Error
Measures of quality of estimates
• Unbiased, variance, consistency,the Cramer-Rao lower bound, Fisher information, Efficiency

#### Linear Estimation in Static Systems

• Minimum Mean Square Error (MMSE)
• MMSE estimation of Gaussian random vectors
• Linear MMSE estimator for arbitrarily distributed random vectors
• LS estimation of unknown constant vectors from linear observations, batch form, recursive form.
• Apply the LS technique

• ##### LINEAR DYNAMIC SYSTEMS WITH RANDOM INPUTS

• Gaussian pdf, mean and covariance
• Stochastic sequences, Markov property
• Discrete-time linear stochastic dynamic systems. Prediction, propagation of mean and covariance
• Continuous-time linear stochastic dynamic systems. Propagation of mean and covariance

##### STATE ESTIMATION IN DISCRETE TIME LINEAR DYNAMIC SYSTEMS = Kalman Filter

The estimation of the state vector of a stochastic linear dynamic system is considered.
The state estimator for discrete-time linear dynamic systems driven by white noise — the (discrete-time) Kalman filter — is introduced.
Estimation of Gaussian random vectors.

For linear systems, white noise Gaussian processes
Linear equations used for state prediction,  prediction of the
measurement and for measurement update.
Exact propagation and measurement  update equations for a priori
and a posteriori covariances
All pdfs stay exactly Gaussian, no need for approximations

• ##### COMPUTATIONAL ASPECTS OF ESTIMATION; Information Filter

Implementation of Linear Estimation
The Information filter implementation of Kalman filter

• carries out the recursive computation of the inverse of the covariance matrix.
• an alternative to the “standard“ Kalman filter formulation and is less demanding computationally for systems with dimension of
the measurement vector larger than that of the state.
• has the advantage that it allows the start-up of the estimation without an initial estimate,  i.e., with a noniformative prior
• Suitable for distributed estimation for example in swarm robotics

##### THE CONTINUOUS-TIME LINEAR STATE ESTIMATION FILTER
The linear minimum mean square error (LMMSE) filter for this continuous time problem, known as the Kalman-Bucy filter
The duality of the LMMSE estimation with the linear-quadratic (LQ) control problem is discussed. These two problems have their
solutions determined by the same Riccati equation

• ##### STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS

• Present the optimal discrete-time estimator.
• Discuss the difficulty in implementing it in practice due to memory  requirements and computational requirements.
• Discuss numerical implementation of the optimal discrete-time estimator.
• Derive the suboptimal filter known as the extended Kalman Filter

##### THE EXTENDED KALMAN FILTER

• Very limited feasibility of the implementation of the optimal filter, the functional recursion, suboptimal algorithms are of interest.
• The recursive calculation of the sufficient statistic consisting of the  conditional mean and variance in the linear-Gaussian case is the simplest possible state estimation filter.
• As indicated earlier, in the case of a linear system with non-Gaussian random variables the same simple recursion yields an approximate mean and variance.
• A framework similar is desirable for a nonlinear system. Such an estimator, called the extended Kalman filter (EKF), can be obtained by a series expansion of the nonlinear dynamics and of the measurement equations.

• ##### EKF based Localization of ePolaris ATV

• ePolaris. Conventions.
• Kinematic model of ePolaris, Odometry.
• Different EKF -implemention of Sensor Fusion with Odometry
• Reference from Novatel SPAN with most accurate correction signals

• EKF
• GPS
• Odometry

##### Particle Filter
▪ Represent belief by random samples
▪ Estimation of non-Gaussian, nonlinear processes
▪ Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter

• ##### USE OF EKF FOR SIMULTANEOUS STATE AND PARAMETER ESTIMATION

The EKF can be used to estimate simultaneously
• the base state and
• the unknown parameters of a system.
This is accomplished by stacking them into an augmented state and
carrying out the series expansions of the EKF for this augmented state.

##### MULTIPLE MODELS

The multiple model (MM) algorithms; Hybrid systems — the system behaves according to one of a finite number of models, it is in one of several modes (operating regimes),  both discrete (structure/parameters) and continuous uncertainties
1. The static MM algorithm —for fixed (nonswitching) models
2. The optimal dynamic MM algorithm — for switching models —Markov chain, two suboptimal approaches: Generalized pseudo-Bayesian (GPB) ; Interacting multiple model (IMM)

• WIENER -NN MODEL

Linear system: Orthogonal expansion of impulse response
Reduced Wiener representation
’Feedforward’ Wiener-NN-model
Wiener-NN with feedback model
Comparison with nonlinear ‘regression’ models
Identification of NOE-type models
Case: Tricoderma fungi process producing enzyme
Conclusions

• ##### COLLECTION OF EQUATIONS

You can use the attached Equation collection in the exam. You don't need to have this with you in the exam. It is given in the exam.

##### CHAPTERS IN BAR-SHALON ET AL

The lecture slides and exercise material cover the whole area. You don't need to have the book Yaakov Bar-Shalom, Xiao-Rong Li & T. Kirubarajan: Estimation with applications to tracking and navigation (2001. But if you have one, the chapters and sections of the book, listed in the second attachment,  have been studied.