ELECE8107  Stochastic models, estimation and control D, Lecture, 5.9.20238.12.2023
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Topic outline


Lecture 1, Sep 5, 2023: Course Arrangements, Statistics and Stochastics Folder
Course Arrangements
Statistics and Stochastics
• Gradient, Jacobian and Hessian; Eigenvalues,
Eigenvectors, and Quadratic Forms
• 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 ChainsEquation collection, can be used in Exam

Lecture 2, Sep 14, 2023: Basic concepts in estimation Folder
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 CramerRao
lower bound, Fisher information, Efficiency 

Lecture 3, Sep 19, 2023: Linear Estimation in Static System Folder
Linear Estimation in Static System
• 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 
Lectrure 4, Sep 26, 2023: LINEAR DYNAMIC SYSTEMS WITH RANDOM INPUTS, STATE ESTIMATION IN DISCRETE TIME LINEAR DYNAMIC SYSTEMS, KALMAN FILTER Folder
LINEAR DYNAMIC SYSTEMS WITH RANDOM INPUTS
• Gaussian pdf, mean and covariance
• Stochastic sequences, Markov property
• Discretetime linear stochastic dynamic systems, Prediction, propagation of mean and covariance
• Continuoustime linear stochastic dynamic systems, Propagation of mean and covarianceSTATE ESTIMATION IN DISCRETE TIME LINEAR DYNAMIC SYSTEMS
 The estimation of the state vector of a stochastic linear dynamic system is considered.
 The state estimator for discretetime linear dynamic systems driven by white noise — the (discretetime) Kalman filter — is introduced.
 Estimation of Gaussian random vectors.
KALMAN FILTER
 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

Lecture 5, Oct 3 2023: COMPUTATIONAL ASPECTS OF ESTIMATION, Implementation of Linear Estimation, Information Filter, THE CONTINUOUSTIME LINEAR STATE ESTIMATION FILTER Folder
COMPUTATIONAL ASPECTS OF ESTIMATION
Implementation of Linear Estimation
Information 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 startup of the estimation without an initial estimate, i.e., with a noniformative prior
• Suitable for distributed estimation for example in swarm roboticsTHE CONTINUOUSTIME LINEAR STATE ESTIMATION FILTER
The linear minimum mean square error (LMMSE) filter for this continuous time problem, known as the KalmanBucy filter
The duality of the LMMSE estimation with the linearquadratic (LQ)
control problem is discussed. These two problems have their
solutions determined by the same Riccati equation 

Lecture 6, Oct 24, 2023: STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS; EXTENDED KALMAN FILTER Folder
STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS
• Present the optimal discretetime estimator.
• Discuss the difficulty in implementing it in practice due to memory requirements and computational requirements.
• Discuss numerical implementation of the optimal discretetime estimator.
• Derive the suboptimal filter known as the extended Kalman FilterEXTENDED 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 linearGaussian case is the
simplest possible state estimation filter.
• As indicated earlier, in the case of a linear system with nonGaussian 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. 
Lecture 7, Nov 7, 2023: Particle Filter; Example for EKF (Honfa ATV) Folder
Particle Filter
▪ Represent belief by random samples
▪ Estimation of nonGaussian, nonlinear processes
▪ Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter
▪ Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]
▪ Computer vision: [Isard and Blake 96, 98]
▪ Dynamic Bayesian Networks: [Kanazawa et al., 95]d 
Lecture 8, Nov 14: MULTIPLE MODELS AND ADAPTIVE ESTIMATION Folder
MULTIPLE MODELS AND ADAPTIVE ESTIMATION
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 uncertainties1. The static MM algorithm —for fixed (nonswitching) models
2. The optimal dynamic MM algorithm — for switching models —Markov chain, two
suboptimal approaches: Generalized pseudoBayesian (GPB) ; Interacting multiple model (IMM) 
Lecture 9, Nov 21, 2023: DUALITY BETWEEN ESTIMATION AND CONTROL; WIENERNN MODEL Folder
DUALITY BETWEEN ESTIMATION AND CONTROL
• Rudolf Kalman invented optimal state feedback control
• He invented the Kalman Filter, too.
• In Kalman’s paper with Bucy, it is shown that the problem of the optimal estimation is dual to the optimal control problem
• Here, the duality between estimation and control is shown using discrete formulationsWIENER NN MODEL AND ITS USE IN MODELING OF BIOTECHNICAL PROCESSES
 Linear system: Orthogonal expansion of impulse response
 Reduced Wiener representation
 ’Feedforward’ WienerNNmodel
 WienerNN with feedback model
 Comparison with nonlinear ‘regression’ models
 Identification of NOEtype models
 Case: Tricoderma fungi process producing enzyme