Department of Mechanical EngineeringEDC2002

A Graphical User Interface for Computer-aided Robust Control System Design

J.F. Whidborne, S.J. King P. Pangalos, Y.H. Zweiri

Department of Mechanical EngineeringEDC2002

Introduction

• Graphical classical control design tools (Bode, Nyquist etc) developed before advent of efficient numerical computation - good qualitative information

• Early quantitative methods (linear-quadratic optimal control) developed in 1950’s before availability of graphical input and output devices

• Multivariable computer-based graphical methods (inverse Nyquist array, characteristic locus array) do not exploit numerical capabilities of modern digital computer & suffer from curse of dimensionality

• Modern frequency-based approaches, (H, ) exploit graphical &

numerical potential of modern computers - suffer curse of dimensionality less - but lack of supporting GUI-based tools.

Department of Mechanical EngineeringEDC2002

McFarlane & Glover’s Loop Shaping Design Procedure (LSDP)

• modern H-optimization approach (H-norm is max magnitude of frequency response)

• multivariable (many inputs and outputs)• robust (stability guaranteed in the face of plant

perturbations & uncertainty)• based on concepts from classical Bode plot methods

- graphical frequency domain method• number of graphical plots required is max(n,m)+n+m

(Inverse Nyquist Array requires nxm)

Department of Mechanical EngineeringEDC2002

LSDP - Step 1

• augment plant G with weighting functions W1 and W2

G(s) W2(s)W1(s)

Augmented Plant Gs(s)

Department of Mechanical EngineeringEDC2002

WW11 and and WW2 2 chosen so weighted plant has “good” shapechosen so weighted plant has “good” shape

high gain at low

freq

Low gain at high freq

Singular values close at cross over

Roll-off < 20 dB/dec

max sing. value

min sing. value

freq

Sin

gula

r va

lues

of G

s (d

B)

Department of Mechanical EngineeringEDC2002

LSDP - Step 2

G(s) W2(s)W1(s)

Ks(s)

optimalcontroller

• check design index - if > 5 return to step 1

• synthesize H-optimal controller to robustly stabilize

shaped plant

Department of Mechanical EngineeringEDC2002

G(s)

W1(s) Ks(s) W2(s)

LSDP - Step 3

• Final controller K(s) = W1(s) Ks(s) W2(s)

K(s)

Department of Mechanical EngineeringEDC2002

LSDPTOOL - A Graphical User Interface MATLAB© Toolbox

• Features– main GUI for designing weighting functions

W1 and W2

– GUI for input and editing model G(s)– window for displaying design index and

step responses– full MATLAB© help system– load, save, print options

Department of Mechanical EngineeringEDC2002

Case Study - A Maglev System

• magnetic levitation of a ball bearing

controller

detector lightz

• open loop unstable

• electromagnet current, i, varied by controller

• vertical displacement of ball, z, measured by light

emitter & detector

F

mg

i

Department of Mechanical EngineeringEDC2002

Maglev Controller Design

• Small deviations of system from equilibrium gives linearised system state description

where

• LSDPTOOL used to design controller

Department of Mechanical EngineeringEDC2002

Department of Mechanical EngineeringEDC2002

Department of Mechanical EngineeringEDC2002

Department of Mechanical EngineeringEDC2002

Department of Mechanical EngineeringEDC2002

Department of Mechanical EngineeringEDC2002

Department of Mechanical EngineeringEDC2002

• Design index value = 4.23 indicates a good design

• Weighting functions:

• System simulated in SIMULINK©

Controller Design

Department of Mechanical EngineeringEDC2002

Step Responses (Non-linear Model)

Department of Mechanical EngineeringEDC2002

Maglev Laboratory Rig

Department of Mechanical EngineeringEDC2002

Toolbox Availability

• Available on WWW at

http://www.eee.kcl.ac.uk/mecheng/jfw/lsdptool.html

• or through MATLAB CENTRAL on Mathworks

WWW site at

http://www.mathworks.com/