Multivariable Control Introduction John T. Wencats-fs.rpi.edu/~wenj/ECSE646F06/lec01.pdf · 2006-08-21 · JTW-MV01 RPI ECSE 6460 Multivariable Control. Course Objective This course

Multivariable Control

Lecture 01

Introduction

John T. Wen

August 29, 2006

JTW-MV01 RPI ECSE 6460 Multivariable Control.

Outline

• Course outline

• An industrial control example

• Review of SISO controller design

Ref: Text Ch. 1 and Ch. 2.

August 29, 2006Copyrighted by John T. Wen Page 1


Course Objective

This course covers the tools and methods for theanalysis and synthesisof linear multivari-

ablecontrol systems. The emphasis is on contemporary control systemdesign, connection

betweenfrequency domain and state space methods, systematic consideration ofmodel

uncertainty and closed loop performance, andconvex analysis and designmethods.



Two Types of Problems

Analysis: Given a controller, determine if desired properties are satisfiedin the presence

of admissible noises, disturbances, and model uncertainties.

Synthesis: Design a controller so that the desired properties are satisfied.

Other considerations: mechanical design and sensor/actuator selection/placement to make

control problem easier: see inverted pendulum video.



Format

• Lectures (T/F 8:30-9:50, W used for make-up classes only)

• In-class activities and Homework (15%)

• Exam (30%, 35%)

• Project (20%)

Knowledge of MATLAB (>7.0) is expected! We will use the MATLABrobust control

toolbox extensively.

Send me an email to get on thecourse mailing list (you’ll need to use your RCS ID and

password to log in).



Text

Essentials of Robust Controlby Kemin Zhou (with John C. Doyle)

On the web: J.C. Doyle, B.A. Francis, A.R.Tannenbaum,Feedback Control Theory,

Macmillan, 1992. (Nice introduction ofH∞ design for SISO systems).

Additional references on reserve in library.



What is Multivariable Control?

Multivariable control considers feedback control systems with multiple input and output

variables (multi-input/multi-output, or MIMO, systems).

More broadly, multivariable control means systematically addressing modeling, uncer-

tainty, performance in control system design.

– Classical Control(frequency domain techniques): Feedback amplifier (Black ’27),

Nyquist criterion (Nyquist ’32), Bode integral formula, gain/phasemargin (Bode

’38), Servomechanism (Hazen ’34, James, Nichols, Phillips ’47)

– Modern Control(state space methods): Stochastic control (Wiener ’49), Nonlinear

control (Popov ’61), Optimal control (Bellman ’57, Pontryagin ’58, Kalman ’60),

Stability theory (Kalman ’60), Geometric approach (Wonham ’79)

– Robust Control(combination of frequency domain and state space):H∞ Optimization

(Zames ’81), Robustness of MIMO systems, Robust performance,µSynthesis (Doyle

’82), Convex programming approach with linear matrix inequality (LMI) (Boyd ’91)



What You Might Have Learned So Far

Classical Control:(Control Systems Eng., DTS)

– Stabilization: PID, lead/lag, additional compensation

– Performance: closed loop poles (loop gain bandwidth)

– Command Following: zero steady state error (high DC loop gain)

– Disturbance Rejection: large loop gain in spectrum ofd

– Trajectory Tracking: large loop gain in spectrum ofydes

– Robustness: gain/phase margins



What You Might Have Learned So Far (Cont.)

Modern Control(SAT, Optimal Control)

x = Ax+Bu, y = Cx+Du

– Stabilization: Observer based control (observer + full state feedback with either pole

placement or LQR)

– Performance: quadratic optimization index

– Command Following: Integral control

– Disturbance Rejection: ??

– Robustness: ??



An Example from Electronic Manufacturing

The wafer on top of thex-y linear stage needs to follow the specified trajectory precisely

(about 50nm position tracking error) on air bearing for laser processing of wafers.



Example (Cont.)

MIMO Control problem: 3 inputs (Fx1, Fx2, Fy), 2 outputs (x-y positions of the wafer

carriage).



Example (Cont.)

Nonlinearity: Dynamic nonlinearity – inertia is a function of carriagey position; geomet-

ric nonlinearity – interferometer measurement of position.



Example (Cont.)

SISO approach:

• Combine the twoFxi inputs into a singleFx (with moment arms used as weightings).

• Treat the system as two decoupled SISO systems (Fx to x interferometer output,Fy to

y interferometer output) with nonlinearities as model uncertainties.

• Tune SISO controller to achieve certain robustness margins (gain/phase margins).

What about the model?

There is no analytic model, just small input amplitude frequency responses at selected

operating points.



Multivariable Approach

• Develop a MIMO LTI design model through either

– analytic modeling with measured or identified model parameters, and then lin-

earize about operating point;

– model identification from measured time or frequency domain data collected at

operating point.

• Characterize model uncertainty: parametric (model parameter uncertainty) vs. non-

parametric (frequency response uncertainty)

• Translate control objectives (trajectory following, disturbancerejection) to perfor-

mance metrics. Select (frequency dependent) weightings for the metrics.

• Design controller to achieve stability while minimizing weighting performance met-

rics in the presence of model uncertainty.

• Test controller on the nonlinear model (or evaluate robustnesson the experimental

frequency responses) and then the actual system.

• Iterate (adjust model, re-tune weightings).



Before Next Class

• Send me an email to get on the course mailing list. Please tell me a bit of your

background (Master or Ph.D., Department, research area, preparation so far, etc.)

• Check out the course web site.

• Read Chapters 1 and 2.

• Start on Homework #1.


Documents

Multivariable Control Introduction John T. Wencats-fs.rpi.edu/~wenj/ECSE646F06/lec01.pdf · 2006-08-21 · JTW-MV01 RPI ECSE 6460 Multivariable Control. Course Objective This course