ELEC-E8107 - Stochastic models, estimation and control D, Lecture, 5.9.2023-8.12.2023

Topic outline

• General
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    Lecture 1, Sep 5, 2023: Course Arrangements, Statistics and Stochastics Folder

    Not available unless: You belong to H01 (SISU)

     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 Chains

    Equation collection, can be used in Exam
    

    
    

    
    

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    Exercise Session 1 Folder
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    Lecture 2, Sep 14, 2023: Basic concepts in estimation Folder

    Not available unless: You belong to H01 (SISU)

    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
    

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    Homework 1 Assignment

    Materials for homework 1. The deadline is 29-09-2023. 

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    Lecture 3, Sep 19, 2023: Linear Estimation in Static System Folder

    Not available unless: You belong to H01 (SISU)

    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
    

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    Exercise Session 2 Folder
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    Lectrure 4, Sep 26, 2023: LINEAR DYNAMIC SYSTEMS WITH RANDOM INPUTS, STATE ESTIMATION IN DISCRETE TIME LINEAR DYNAMIC SYSTEMS, KALMAN FILTER Folder

    Not available unless: You belong to H01 (SISU)

    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

    • 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.

    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

    
    
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    Exercise Seesion 3 Folder
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    Lecture 5, Oct 3 2023: COMPUTATIONAL ASPECTS OF ESTIMATION, Implementation of Linear Estimation, Information Filter, THE CONTINUOUS-TIME LINEAR STATE ESTIMATION FILTER Folder

    Not available unless: You belong to H01 (SISU)

    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 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
    

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    Homework 2 Assignment

    Materials for Homework 2. The deadline is 02.11.2023.
    

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    Lecture 6, Oct 24, 2023: STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS; EXTENDED KALMAN FILTER Folder

    Not available unless: You belong to H01 (SISU)

    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
    

    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.
    

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    Exercise Session 4 Folder
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    Lecture 7, Nov 7, 2023: Particle Filter; Example for EKF (Honfa ATV) Folder

    Not available unless: You belong to H01 (SISU)

    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
    ▪ Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]
    ▪ Computer vision: [Isard and Blake 96, 98]
    ▪ Dynamic Bayesian Networks: [Kanazawa et al., 95]d
    

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    Lecture 8, Nov 14: MULTIPLE MODELS AND ADAPTIVE ESTIMATION Folder

    Not available unless: You belong to H01 (SISU)

    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 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)
    

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    Lecture 9, Nov 21, 2023: DUALITY BETWEEN ESTIMATION AND CONTROL; WIENER-NN MODEL Folder

    Not available unless: You belong to H01 (SISU)

    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 formulations

    WIENER -NN MODEL AND ITS USE IN MODELING OF BIOTECHNICAL PROCESSES
    

    • 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
      

    
    

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    Exercise Session 5 Folder