Optimal Control
- Workload:
- 1 ECTS
- Purpose:
- The students should acquire knowledge and skills in optimal and predictive control and experience with formulation and solution of control problems including a quadratic performance criterion possibly with constraints in control inputs and outputs
- Rationale:
- Parameterization of a controller for a physical system and choice of controller parameters is often a compromise between conflicting goals. Optimal control is a systematic way to obtain a controller which balance tight control with reasonable use of resources available through the manipulated variable. Optimal control also is background for extensions to other modern methods like predictive control and robust control and should be familiar to a control engineer at high level
- Objectives:
- The student should be able to apply formulation of control problems using models of disturbances and references combined with a quadratic performance function and know the use of integral states to eliminate steady state errors. The student should be able to apply observers as a tool to asses the state of a system where measurement are incomplete and noisy. The students should know stability properties of optimal controllers and how observers affect this. Students should know quadratic programming as a method to solve predictive control problems with constraints.
- Contents:
-
- Linear model, Quadratic performance, dynamic programming, Riccati equation
- Formulation of optimal control problems with references and disturbances
- Elimination of steady state errors
- Use of observer, LQG control
- Stability properties of optimal controller
- Solving predictive control with constraints using quadratic programming
- Prerequisites:
- Analog and digital control, Stochastic systems
Literature:
Lecture notes: Preliminary version (This is subject to change, chapters for each lecture will also be placed at the lecture plan.
The lecture notes make up a revised version of Ole Sørensen's notes August 1995, in Danish: Optimal regulering
Lecture Plan
- Lecture 1:
- Introduction to optimal control and quadratic cost functions. Dynamic programming and the Riccati equation. Time varying Linear quadratic control.
Litterature: Lecture notes Chapters 1, 2 and 3.
Exercise: Exercise 1 Solution: SolutionEx1
Description of watermixing exercise: Watermixing exercise
Files:: simulate.m, constants.m ,plotdata.m
Solutionfiles:lopg1.m,contmodel.m,discmodel.m,weight1.m,riccatitimevary.m,
Slides: pdf
- Lecture 2:
- Stationary LQ Controllers. Control including references and disturbances
Litterature: Lecture notes Chapters 4-7 Chapters 4-7 in separate file
Exercise: Exercise 2
Solution:Solution exercise 2
Files: riccaticontrol.m, servo.m
Slides: pdf
- Lecture 3:
- Stochastic control, use of observer, LQG control.
Litterature: Lecture notes chapters 9 and 10
Exercise: Exercise 3
Solution:Solution exercise 3
Slides: pdf
- Lecture 4:
- Disturbance rejection and controllers with integral action. Stability of LQ controlled systems
Litterature: Lecture notes chapter 8, 11, and note on disturbance rejection and integral action pdf
Exercise: Exercise 4
Slides: pdf
- Lecture 5:
- Solving predictive control with constraints using quadratic programming
Litterature: Notes refering to chapter 3 of Maciejowski: Predictive control with constraints: notes pdf
Slides: pdf
Supplementary Litterature:
- Karl. J. Åström & Björn Wittenmark: Computer Controlled
Systems, Prentice Hall Int.
- Gene F. Franklin, J.David Powel, Michael Workman: Digital Control of Dynamic
Systems.
- J.M.Maciejowski: Predictive Control with Constraints, Prentice Hall, 2002