A method is presented that can often reduce the number of scheduling parameters for gain-scheduled controller implementation by transformation of the system representation using parameter-dependent dimensional transformations. In some cases, the reduction in parameter dependence is so significant that a linear parameter-varying system can be transformed to an equivalent linear time invariant (LTI) system, and a simple example of this is given. A general analysis of the parameter-dependent dimensional transformation using a matrix-based approach is then presented. It is shown that, while some transformations simplify gain scheduling, others may increase the number of scheduling parameters. This work explores the mathematical conditions causing an increase or decrease in varying parameters resulting from a given transformation, thereby allowing one to seek transformations that most reduce the number of gain-scheduled parameters in the controller synthesis step.

Issue Section:
Technical Briefs
Keywords:
control system synthesis, controllers, gain control, linear systems, scheduling, gain scheduling, pi theorem, dimensionless parameters, dimensional transformation
Topics:
Control equipment, Gain scheduling
1.
Shimomura
,
T.
, 2003, “
Hybrid Control of Gain-Scheduling and Switching: A Design Example of Aircraft Control
,”
Proceedings of the American Control Conference
,
Denver, CO
, Jun. 4–6, pp.
4639
–
4644
.
2.
Rugh
,
W. J.
, 1991, “
Analytical Framework for Gain Scheduling
,”
IEEE Control Syst. Mag.
0272-1708,
11
, pp.
79
–
84
.
3.
Bruzelius
,
F.
, 2002, “
LPV-Based Gain Scheduling, An H-Infinity-LMI Aproach
,”
Signals and Systems
,
Chalmers University of Technology
,
Goteborg
, pp.
112
.
4.
Choi
,
D. J.
, and
Park
,
P. G.
, 2001, “
Guaranteed Cost LPV Output-Feedback Controller Design for Nonlinear Systems
,”
Proceedings of the 2001 IEEE International Symposium on Industrial Electronics
,
Pusan, Korea
, Jun. 12–16, pp.
1198
–
1203
.
5.
Lee
,
C. H.
, and
Chung
,
M. J.
, 2001, “
Gain-Scheduled State Feedback Control Design Technique for Flight Vehicles
,”
IEEE Trans. Aerosp. Electron. Syst.
0018-9251,
37
(
1
), pp.
173
–
182
.
6.
Leith
,
D. J.
, and
Leithead
,
W. E.
, 2000, “
On Formulating Nonlinear Dynamics in LPV Form
,”
Proceedings of the 39th IEEE Conference on Decision and Control
,
Sydney, Australia
, Dec., pp.
3526
–
3527
.
7.
Fedigan
,
S. J.
,
Knospe
,
C. R.
, and
Williams
,
R. D.
, 1998, “
Gain-Scheduled Control for Substructure Properties
,”
Proceedings of the 1998 American Control Conference
, Jun. 24–26, pp.
1205
–
1209
.
8.
Wang
,
F.
, and
Balakrishnan
,
V.
, 2002, “
Improved Stability Analysis and Gain-Scheduled Controller Synthesis for Parameter-Dependent Systems
,”
IEEE Trans. Autom. Control
0018-9286,
47
, pp.
720
–
734
.
Crossref
Search ADS
 
9.
Corriga
,
G.
,
Guia
,
A.
, and
Usai
,
G.
, 1998, “
An Implicit Gain-Scheduling Controller for Cranes
,”
IEEE Trans. Control Syst. Technol.
1063-6536,
6
(
1
), pp.
15
–
20
.
Crossref
Search ADS
 
10.
Tavakoli
,
S.
, and
Tavakoli
,
M.
, 2003, “
Optimal Tuning of PID Controllers for First Order Plus Time Delay Models Using Dimensional Analysis
,”
The Fourth IEEE International Conference on Control and Automation
,
Canada
, pp.
942
–
946
.
11.
Astrom
,
K. J.
,
Hang
,
C. C.
,
Persson
,
P.
, and
Ho
,
W. K.
, 1992, “
Toward Intelligent PID Control
,”
Automatica
0005-1098,
28
, pp.
1
–
9
.
Crossref
Search ADS
 
12.
Brennan
,
S.
, and
Alleyne
,
A.
, 2001, “
Robust Scalable Vehicle Control via Non-Dimensional Vehicle Dynamics
,”
Veh. Syst. Dyn.
0042-3114,
36
(
4–5
), pp.
255
–
278
.
13.
Brennan
,
S.
, and
Alleyne
,
A.
, 2001, “
Using a Scale Testbed: Controller Design and Evaluation
,”
IEEE Control Syst. Mag.
0272-1708,
21
(
3
), pp.
15
–
26
.
14.
Brennan
,
S.
, and
Alleyne
,
A.
, 2005, “
Dimensionless Robust Control With Application to Vehicles
,”
IEEE Trans. Control Syst. Technol.
1063-6536,
13
(
4
), pp.
624
–
630
.
Crossref
Search ADS
 
15.
Ghanekar
,
M.
,
Wang
,
D. W. L.
, and
Heppler
,
G. R.
, 1997, “
Scaling Laws for Linear Controllers of Flexible Link Manipulators Characterized by Nondimensional Groups
,”
IEEE Trans. Rob. Autom.
1042-296X,
13
(
1
), pp.
117
–
127
.
Crossref
Search ADS
 
16.
Fox
,
R. W.
, and
McDonald
,
A. T.
, 1992,
Introduction to Fluid Mechanics
, 4th ed.,
Wiley
,
New York
.
17.
Bridgman
,
P. W.
, 1943,
Dimensional Analysis
, 3rd printing, revised ed.,
Yale University Press
,
New Haven
.
18.
Szirtes
,
T.
, 1998,
Applied Dimensional Analysis and Modeling
,
McGraw-Hill
,
New York
.
19.
Buckingham
,
E.
, 1914, “
On Physically Similar Systems Illustrations of the Use of Dimensional Equations
,”
Phys. Rev.
0031-899X,
4
(
4
), pp.
345
–
376
.
Crossref
Search ADS
 
20.
Hailu
,
H.
, and
Brennan
,
S.
, 2005, “
Use of Dimensional Analysis to Reduce the Parametric Space for Gain-Scheduling
,”
Proceedings of the 2005 American Control Conference
,
Portland, OR
, Jun. 8–10.
21.
Franklin
,
G. F.
,
Powell
,
J. D.
, and
Emami-Naeini
,
A.
, 1994,
Feedback Control of Dynamic Systems
, 3rd ed.,
Addison-Wesley
,
Reading, MA
.
Copyright © 2008
 by American Society of Mechanical Engineers
You do not currently have access to this content.