ELEC-E8107 - Stochastic models, estimation and control D, Lecture, 5.9.2023-8.12.2023
This course space end date is set to 08.12.2023 Search Courses: ELEC-E8107
Översikt
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Lecture 1, Sep 5, 2023: Course Arrangements, Statistics and Stochastics Mapp
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
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Lecture 2, Sep 14, 2023: Basic concepts in estimation Mapp
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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Lecture 3, Sep 19, 2023: Linear Estimation in Static System Mapp
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 Mapp
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 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 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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Lecture 5, Oct 3 2023: COMPUTATIONAL ASPECTS OF ESTIMATION, Implementation of Linear Estimation, Information Filter, THE CONTINUOUS-TIME LINEAR STATE ESTIMATION FILTER Mapp
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 roboticsTHE 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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Lecture 6, Oct 24, 2023: STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS; EXTENDED KALMAN FILTER Mapp
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 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 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. -
Lecture 7, Nov 7, 2023: Particle Filter; Example for EKF (Honfa ATV) Mapp
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 -
Lecture 8, Nov 14: MULTIPLE MODELS AND ADAPTIVE ESTIMATION Mapp
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 pseudo-Bayesian (GPB) ; Interacting multiple model (IMM) -
Lecture 9, Nov 21, 2023: DUALITY BETWEEN ESTIMATION AND CONTROL; WIENER-NN MODEL Mapp
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’ Wiener-NN-model
- Wiener-NN with feedback model
- Comparison with nonlinear ‘regression’ models
- Identification of NOE-type models
- Case: Tricoderma fungi process producing enzyme