Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 1: 1 September 2009

Information Instructor: Agung Julius Office hours: JEC 6044 Mon,Wed 2 – 3pm Textbook: S. Skogestad & I. Postlethwaite, Multivariable Feedback Control 2 nd ed, Wiley. Additional reading: J. Doyle, B. Francis, A. Tannenbaum, Feedback Control Theory, Macmillan. Downloadable from Bruce Francis’ website (google) Online contents: (Notes, HW sets) RPI LMS (grades)

Prerequisite(s) The course is for graduate or advanced undergraduate students with working knowledge in differential calculus, linear algebra, complex numbers, and classical linear systems/control theory. Attendance background?

Grading Homeworks (5 sets) = 40% 2 x Exams = 30% + 30% Homework sets are due one week after handout. Late submissions will get point deduction (no later than 1 week). Exams are take home tests. Will include control design type of task.

Other issues Exchanging ideas is allowed for solving the homework sets, but not copying. No collaboration is allowed for exams! You will need MATLAB. An installer with campus license is available from  software Beware: Need to be connected to RPI network, use VPN client from outside.  networking

Course outline Introduction Classical Feedback Control Loop Shaping MIMO Control Performance Limitations Disturbance and Robustness Controller Design Model Reduction

Linear Time Invariant Systems  Linearity Linearity: Time invariance Time invariance:

Linear Differential Systems zero With zero initial conditions. Why? Two ways to describe the systems: Time domain Frequency domain Laplace transform inverse transform

Example (Spring Mass System) M

Control Problem M

Feedforward vs feedback Feedforward: use an inverse model of the plant to compute the control input. Generally not a good idea! Why?

Feedforward vs feedback Feedback: use output measurement to compute control input. How to design a good controller? What is a good controller? PlantController

Performance limitation Performance criteria: stability, speed of response, overshoot, disturbance rejection, etc. Can we always attain any desired performance using feedback control? NO Short answer: NO. Why?

Disturbance and Robustness PlantController disturbance Design a controller that works, despite the presence of disturbances.

Disturbance and robustness How to best model the disturbance M

Disturbance and robustness How to best model the disturbance M Square wheels!!!

Robustness issue Suppose that we know how to design a good controller if we know the plant (and disturbance) model. IF It is still a very big IF ! In practice, we don’t know the model precisely. There’s always uncertainty, modeling error, parameter variation, etc. Challenge: Challenge: design a good controller, even though we don’t know the plant model. Is it possible? How?

Model reduction complexity inessentials Reduce the complexity of the mathematical model, by throwing out the inessentials. High order system

Fictitious application Low order system NOT a class project!!

Model reduction complexity inessentials Reduce the complexity of the mathematical model, by throwing out the inessentials. how much We need to know how much detail is lost.

Multivariable? Input and output variables are multidimensional, i.e. vectors instead of scalars. Consequences: Different algebraic rules Quantities have directions, in addition to magnitudes. Controller topology can be important. (which output influences which input?) How do we generalize SISO results to MIMO?