Computer Science, Electrical Engineering
SMR013 Optimal Control 6.0 ECTS credits
DENNA SIDA FINNS OCKSÅ PÅ SVENSKA
General information about studying at Luleå University
TIMEPERIOD: Lp III
LANGUAGE: English/Swedish
EXAMINER
A Medvedev Univ lekt
PREREQUISITES
Background courses: Control Theory SMR017, System Identification SMR012
The course includes 44 hours of lectures and 16 hours of computer aided exercises.
COURSE AIM
This course covers the basic theory of the control of linear discrete-time systems which are subject to stochastic disturbances. The main objective is to provide students with the theoretical knowledge that comprise the core of the ''modern" (state-space) control science, with much emphasis placed in the practical aspects of analysis and design.
The main treated topics are the Linear Quadratic Control and Kalman filtering.
To meet the demands of digital implementation, the computational issues of optimal control are considered. The complexity arises when the principles of the classical optimal control theory are reduced to practice, for then the basic notions of robust control have to be be addressed in the course.
Four obligatory computer exercises are devised to facilitate understanding of the theoretical material and provide basic skills in optimal control system design using MATLAB Control System Toolbox.
CONTENTS
Introduction. Optimality and Robustness in Dynamic Systems (1 hour)
A. Estimation Theory and the Kalman Filter (10 lectures= 20 hours)
Summary: This chapter is devoted to a systematic derivation of the Kalman Filter algorithm with necessary preliminaries on Stochastic Processes.
Numerical issues related to Lyapunov and Riccati equations are considered.
1. Some concepts from probability theory (2 hours)
Probability density function, conditional probability, independent and uncorrelated stochastic variables.
2. Minimum variance estimation (3 hours)
Estimators, optimal estimators, orthogonality and projection.
3. Some properties of normally distributed random variables. (3 hours)
Independence of normally distributed random variables, covariance of independent variables, conditional distributions.
4. Linear estimators. (2 hours)
5. The Kalman Filter. (6 hours)
Sufficient statistic, Riccati equation, stationary gain,the innovations form, optimal prediction, correlated disturbances. Reduced-order estimators . Asymptotic Stability of the Kalman filter.
6. Numerical procedures in Estimation Theory (4 hours).
Solution of the matrix Lyapunov equation.Direct solution, direct iteration, transform method.
Solution of the stationary Riccati equation. Eigenvector method, Hewer's method.
Lab 1. Numerical procedures for solving matrix equations (4 hours)
Lab 2. Kalman filter design (4 hours)
B . Optimal Control.
Summary: This chapter addresses Linear Quadratic Control Problem both in state-space and input-output frameworks to shed more light on the optimal controller properties, its structure and possible implementations.
Introduction.
1. The linear quadratic gaussian control problem.
State-space approach .(6 lectures= 12 hours)
Quadratic loss function, admissible control strategy.
1.1 Statement of the problem.
1.2 Dynamic programming
The Principle of Optimality. Sequential decision process, optimal strategy, Bellman's equation
1.3 Solution of the optimal control problem.(9 hours)
Complete state information. Incomplete state information. Separation principle.
Stationary optimal control laws.
1.4. Robust linear quadratic control (1 hour)
2. The linear quadratic gaussian control problem.
Input-output approach (6 lectures= 12 hours)
2.1. Problem formulation. Process and disturbance models.
2.2. Optimal prediction.
2.3. Minimum variance control.
Stable and unstable inverses, pole-placement analogy.
2.4. LQG-control
Spectral factorization. Design procedure and computational issues.
2.5. Comparisons with the State-space approach.
Lab 3. LQG-controller design. State-space approach. (4 hours)
Lab 4. LQG-controller design. Input-output approach. (4 hours)
TEACHING
Lectures, laboratory tests and computer exercises.
EXAMINATION
COURSE GRADE SCALE: U, 3, 4, 5
ITEMS AND CREDITS
Laboratory work 2.2 ECTS
Written exam 3.7 ECTS
COURSE LITERATURE
Åström K.J., Wittenmark B.: Computer Controlled Systems, 2nd ed., Prentice Hall 1990.
REMARKS
Last modified 97-03-05
Further information: Alexander Medvedev, Tel. 0920-91302
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