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MODEL ERROR CONCEPTS AND
COMPENSATION
Proceedings of the IF AC Workshop, Boston, USA,
17-18 June 1985
Edited by
R. E. SKELTON
School of Aeronautics £sf Astronautics, Purdue University, Indiana, USA
and
D. H. OWENS
University of Strathclyde, Glasgow, UK
Published for the
INTERNATIONAL FEDERATION OF AUTOMATIC CONTROL
by
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Copyright © 1986 IFAC
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in
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First edition 1986
British Library Cataloguing in Publication Data
Model error concepts and compensation : proceedings of the IFAC workshop, Boston, USA,
17-18 June 1985.—(An IFAC publication)
1. Control theory
I. Skelton, R. E. II. Owens, D. H.
III. International Federation of Automatic Control
629.8'312 QA402.3
ISBN 0-08-032575-0
These proceedings were reproduced by means of the photo-offset process using the manuscripts supplied by the
authors of the different papers. The manuscripts have been typed using different typewriters and typefaces. The
lay-out, figures and tables of some papers did not agree completely with the standard requirements: consequently
the reproduction does not display complete uniformity. To ensure rapid publication this discrepancy could not be
changed: nor could the English be checked comphtely. Therefore, the readers are asked to excuse any deficiencies
of this publication which may be due to the above mentioned reasons.
The Editors
Printed in Great Britain by A. Wheaton &f Co. Ltd., Exeter
IFAC WORKSHOP ON MODEL ERROR CONCEPTS AND
COMPENSATION
Organized by
The Organizing Committee, with the aid of the Secretariat of
the 1985 American Control Conference
Sponsored by
International Federation of Automatic Control (IFAC)
The Mathematics of Control Committee
The Theory Committee
Co-sponsored by
The IEEE Control Systems Society
The American Automatic Control Council
International Programme Committee
D. H. Owens, UK (Chairman)
G. Zames, Canada
G. Stein, USA
J. Ackermann, FRG
J. Cruz, USA
G. Goodwin, Australia
H. Kwakernaak, Netherlands
National Organizing Committee
D. H. Owens
R. E. Skelton
Copyright © IF AC Model Error Concepts
and Compensation, Boston, USA, 1985
WORKSHOP EDITORIAL
D. H. Owens* and R. E. Skelton**
^Department of Mathematics, University of Strathclyde, Glasgow
**School of Aeronautics and Astronautics, Purdue University, USA
The following text represents the proceedings This'indistinguishability will be retained under
of a Workshop held in Boston, Massachusetts in June closed-loop control conditions provided that the
1985. The idea of holding the Workshop arose during closed-loop bandwidth does not contain the
general discussion between the editors at another characteristic frequencies of the parasitic term.
IFAC.Workshop on 'Singular perturbations and For example, under negative proportional output
robustness of control systems*held in Ohrid in feedback of gain K, G can be confidently used as
Yugoslavia in 1981. The basic idea behind the
a model for the plant G for the purposes of closed-
discussion became the theme of the Workshop and is
loop predictions of stability and performance if,
most easily stated in the deceptively simple phrase-
roughly speaking,
the modelling problem and the control design
problem are not independent! The phrase is -1
K + 1 « (4)
deceptively simple in the sense that its acceptance
leads one into a minefield of conceptual, technical
and computational problems that are only just being The interesting thing about this relationship is
recognized by the control community. The phrase that it tells us that the usefulness of the model
also contains the elements of controversy as it G as a predictive device for controller design
implicitly asserts that the systems modeller cannot depends both on the open-loop modelling error
'sensibly' proceed in a systematic and scientific (represented by ε) and on the final designed
way to produce a model for the purpose of controller control K. More precisely, success depends upon
design without taking direct account of the the relationship between the modelling errors and
structure of the control scheme to be designed the control design - we return to the theme of the
(e.g. feedback, feedforward, decoupling, optimal, Workshop: The modelling and control problems are
adaptive...), the changes in performance to be not independent.
achieved by the scheme (e.g. bandwidth,...) and
assessing the magnitude and effect of modelling Finally, we can comment on the consequences of
errors on model-based predictions. To illustrate violating the above requirements as represented by
this simple idea, consider the following examples (4). Take, for illustrative purposes a situation
chosen to highlight some well-known and some not where the designer inadvertently chooses
so well-known pitfalls in the modelling and design
exercize: K » 1/4ε. (5)
Ex 1 Small open-loop modelling errors can lead to Using his model G he will predict a stable
large closed-loop prediction errors
closed-loop system with fast response
characteristics, no overshoot or oscillation and
Consider the single-input/single-output
small steady state errors. To a large extent he
system described by the transfer function
will be correct as the real plant G will be stable
and have small steady-state errors. However, his
G(s) = (l+s)(l+es)a (1) time response/performance predictions will be
greatly in error as the implemented controller
where ε is small, strictly positive real parameter will cause severely underdamped oscillations in
representing (say) fast, 'parasitic' sensor or the plant G!
actuator dynamics. A natural modelling assertion
is to claim that the effect of the parasitic term Clearly the conclusions obtained from the example
on open-loop input/output behaviour can be apply more widely and it is quite possible for
neglected and the system modelled by the small open-loop modelling errors to lead to
approximate model predictions of closed-loop stability when, in
reality, the real plant will be unstable.
1
(2)
GA(S> " (1+S) Ex 2 Large open-loop modelling errors do not
preclude small closed-loop prediction errors
if 0 < e « 1. This idea can be quantified more
precisely by noting that errors in the unit step
There is a natural tendency in modelling to try
response Y of G as modelled by the unit step
to achieve as accurate a model as possible. Infact,
response Y of G satisfy (for 0 < ε < 1 and a = 1)
example 1 may provide some motivation for this
goal. It is clear however that it is not necessary
2ε6 t > 0 (3)
|Y(t) - YA(0| 1-ε to have an accurate model for control design - if
the control scheme has the right characteristics
(remember the theme of the workshop^), To
i.e. the open-loop properties of plant G and model
illustrate this fact, consider a scalar process
G. (as measured by a commonly accepted simulation
with transfer function
data comparison of step characteristics) are
virtually in distinguishable if ε is small! G(s) " sTT (6)
X D. H. Owens and R. E. Skelton
modelled by the integrator For p < — the actual closed-loop system is
unstable, whereas the predicted behaviour is
GA(S> " Ί (7) approaching its maximal accuracy. Fig. 1
illustrates that the region of predicted maximal
for the purposes of negative output proportional accuracy (smallest value of V predicted by the
feedback control of gain K. It is perhaps not
controller design model G ) corresponds to the
neccesary to point out that G^ is not a good model
of the open-loop properties of G. It is worst closed-loop performance for the actual
interesting however to evaluate in what sense G system. Moreover, over the whole range of
positive p the predicted closed-loop performance
is a good model for predicting closed-loop
is an improvement (i.e. generates a smaller value
performance of the plant G!
of V ) over the open-loop performance, whereas the
Stability: G is stable under gains K > 0 and actual closed-loop performance is always worse
than the open-loop performance.
it is easily seen that a choice of gain K based
uuppoonn GG aanndd tthhee rreeqquuiirreedd cclloosseedd--loop performance Now consider the "absurd" design model
will lead to stabilization of G.
1
GA lies
Performance: For a gain choice of K > 0 G will
predict closed-loop performance with some degree instead of G.. This model is "absurd" to the
A
of error. If Y (resp. Y ) is the closed-loop unit
extent that the open-loop step response error is
step response of G (resp. G ) in the presence of not neccessarily small. All we can say is that
the gain K, then it is trivially verified that
|Y(t) - YA(t)| < 1, t > 0.
|Y(t) - YA(t)| l+K' t > 0 (8)
This is to be compared with the smaller error
i.e. the fidelity of the model G for predicting oobbttaaiinneedd wwiitthh GGA 4 ,, see (3). The optimal control
K in this case is
closed-loop step response of the plant G depends
upon the controller K. The closed-loop properties K = (-1 Jw >
of the plant G and model G can be made arbitrarily +
close by choosing gains (closed-loop bandwidth) to
be sufficiently large, despite the fact that open- which is precisely the same K as for model G !
loop modelling errors are extremely large! In
Hence, Fig. 1 applies for this model as well!
this example the effects of model errors are
Great persuasion abounds in the model reduction
reduced by large gains K. However, in general,
literature for retaining the "dominant" mode G ,
large gains are not the solution. The point of the
t A
example is that the modelling errors need not be but the optimal controller for G (the fast mode)
made small, they need to be made appropriate to
yields the same closed-loop performance as the
the chosen controller.
optimal controller for G ! Hence no reliable
Ex 3 Modelling errors can lead to erroneous statement about the quality of a reduced model can
optimality predictions be made independently of knowledge of the
controller, (i.e. the Workshop theme).
Reconsider example 1. Let K be optimal for G
with respect to the performance criterion Now consider Ex. 2. Let K be optimal for GA with
A
(Y2 + pu2)dt. Then respect to the performance criterion
(Y2A + pu2)dt.
J0 Then,
K (i - /Ϊ+Ί7Ρ)
K =
and the closed-loop performance of G is described /P
and the actual closed-loop performance of G is
described in Fig. 2
A\
V Predicted performance based on G.
Actual performance
of plant G
.Actual peformance for
2(ε+1) x ^ r Predicted performance based 2(ε+1) plant G
yg model G.
\
\
\
dt
-^:l·
Fig 1. Mean-squared behaviour of input Fig 2.
and output.
Workshop Editorial XI
In this case, the regions of maximal accuracy are (Note: (1) and (2) are inevitably interconnected
in agreement but the predictions of performance at and represent the crucial decisions that relate the
low levels of control are arbitrarily far apart. modelling stage to the design stage - successful
Hence, once again the fidelity of the model resolution of these problems are the key to
approximation depends upon the control. successful design...)
These examples and many others too numerous to (3) the construction of techniques for robust
mention clearly justify the validity of the theme stability assessment and performance
of the Workshop. They also indicate the complexity deterioration using available error data
of the problem as the conclusions of the examples and formal rules for controller redesign to
are, in many ways, counter-intuitive or, at best, minimize sensitivity to the modelling error.
not generally known. For the purposes of approaching
the problem, it is useful to look at the modelling The papers and discussion of the Workshop
and control design problems as part of an focus on the general area outlined above while
integrated procedure as illustrated in Fig. 3 where pursuing a specific area of systems and control
plant data or a plant idealization is used interest. The general reader will find something
to interest him or her as consideration is given to
(a) to construct a model of plant dynamics and apparently diverse topics such as approximation,
model reduction, process control, adaptation,
(b) to characterize in an explicit quantitative
large-scale systems, optimal control, robust
manner errors or mismatch between plant
stabilization and many others where the underlying
and model open-loop behaviour.
theme of the Workshop plays a role. The editors
have resisted the temptation of attempting to guide
Normal design routes can then be used to iteratively
the reader through the contributions, leaving the
choose a controller for the model, but before
choice of reading material to his or her discretion.
implementation, the effect of modelling errors on
We do hope however that the generality of the
closed-loop model predictions must be assessed for
underlying theme is recognized despite the clear
acceptability. If the errors are not predicted to
difference in technical background required for
degrade performance by an unacceptable amount the
each topic and that the theme will play a role in
design is completed. Otherwise the designer must
future developments of a more unified approach to
look back to either
control design that avoids the following myths:
(i) improve his model to reduce uncertainty in MYTH 1: the control problem begins after a
closed-loop predictions model is available, (i.e. the modelling
and control problems are separable).
MYTH 2: Control theory exists which can
(ii) redesign the controller to reduce the
accommodate all errors in the model.
affects of the given modelling errors to: an
acceptable level. MYTH 3: The model and control can be optimized
separately.
There are a large number of technical problems
MYTH 4: The model and control can be optimized
implicit in the conceptual design procedure
simultaneously.
centered on:
MYTH 5: All important linear control problems
(1) The choice of the simplest form of the plant have been solved.
model that represent dominant plant
(Note the MYTHs 3 and 4 imply an iterative process
characteristics of importance in the sort
to modelling and control design).
of controlled situation envisaged.
(2) explicit characterization of modelling Finally our thanks go out to all contributors
errors in numerical form and the to the proceedings and all participants in the
classification of error structures that are Workshop. Without their invaluable help, neither
compatible with given design objectives. the event itself nor the text following would have
been possible.
Plant Data
Error Characterization
Performance
Specifications
-p> Assess effects of modelling errors NOT
OK
on design predictions
I OK
Fig. 3 An Integrated Design
STOP
Procedure
Copyright © IF AC Model Error Concepts PLENARY SESSION
and Compensation, Boston, USA, 1985
MULTI-MODEL APPROACHES TO ROBUST
CONTROL SYSTEM DESIGN
J. Ackermann
DFVLR-Institut für Dynamik der Flugsysteme, D—8031 Oberpfaffenhof en,
Federal Republic of Germany
Abstract. Consider structured uncertainties, i.e. explicit uncertainty bounds
for physical parameters in a plant model of known structure. A finite number of
typical admissible plant parameter values is used to define a multi-model problem.
The problem is to find a fixed gain controller, such that the closed loop has its
eigenvalues in a specified region Γ in the complex plane for each plant model.
Some results on existence and required order of the simultaneous stabilizer (or
Γ-stabilizer) are reviewed. In a practical approach the structure of the controller
is assumed, and admissible sets of controller parameters can be determined by al
gebraic and graphic methods. A track-guided bus is used as an example.
Keywords. Robust control; Simultaneous stabilization of a family of plant models;
Pole region assignment.
INTRODUCTION parameters θ in a given admissible
set Ω.
Most plant models used for controller de
sign are uncertain. Even if an exact model A linear dynamic controller structure may
is available, it may be so complicated that be assumed in form of a state-space model
it must be approximated by a simpler, but or transfer function. In order to satisfy
uncertain, design model. For example, non requirements ii) it may contain an internal
linear models may be linearized for small model or filter in the loop and a prefilter
deviations from an operating condition. for shaping the reference responses. In
Then the linear model depends on this un this paper the design of the compensator
certain operating condition. Also physical part for Γ-stabilization is studied. The
parameters of the plant and its environment free design parameters in the controller
may be uncertain. Suppose the linearized structure are combined to a controller-
plant can be described by a state space parameter vector k'.
model
We call this a fixed-gain controller if k
x = Α(Θ)χ + B(0)u is constant. In a gain scheduling controller,
_ some components of fr or related external
(1) variables are measured and used for adjust
y_ = Cx
ment of k = k(C)) . In an adaptive controller,
where Q_ is the vector of uncertain plant k is a functional k(t) = F (υ(τ) ,y_(x)) ,
parameters. Assume the state variables in τ < t.
x are chosen such that the output matrix C^
Hoes not depend on Θ. Equivalently the Example: Track-guided bus(Christ, Darenberg,
plant may be described by its transfer Panik, Weidemann, 1977), (Ackermann, Turk,
function 1982) .
G(s,0) = C[sX-A(0)]"1B(0) (2) A bus is guided by the field generated by
a wire in the street. Fig. 1 shows the case
It is desired to find a linear feedback where the nominal track is a straight guide
control law, such that the closed loop line .
i) is nicely stable. In the case when Q_
is constant this means that the
closed-loop system has rapidly decay
ing and well-damped modes, i.e. all
dominant eigenvalues must be located
in a specified region Γ of the complex
plane.
ii) has required disturbance compensation,
filtering and tracking properties,
iii) requires only admissible actuator sig Fig. 1 Track-guided bus
nal magnitudes, i.e. |u| < u , öR, δρ = rear and front displacement
of the bus from the guide
iv) has the properties i), ii) and iii) line ,
for all constant values of the plant 3 = steering angle.
1
2 J. Ackermann
The measured variables are d , d and 3,
A feasible state vector is c D
thus F R
x = [d a d a er (3) 10 0 0 0
F F R R
y_ = Cx 0 0 10 0 (8)
A good controller will keep all state vari 0 0 0 0 1
ables small, thus the controller design
may be performed with the linearized model Important design specifications are
i) For passenger comfort and safety the
0 1 0 0 0 "o lateral motion should be smooth, i.e.
a lower bound for the damping of the
a a a a a 0
closed-loop eigenvalues should be
0 2 1 0 2 2 0 2 3 1 2 4 0 2 5 X + 0 specified.
a a a a a 0
ii) For good tracking (e.g. when entering
4 5
b into a curve in the guideline) an upper
5 . . 5. bound for the real part of the eigen
(4) values may be required.
bs = 4.7s ■1' is the time constant of the iii) The closed-loop bandwidth should be
power steering. The coefficients a^ are below unmodelled high-frequency effects
of the form ij from suspension, structure, motor vi
brations etc.
for j = 1, 3, 5
ij m These specifications lead to a Γ-stability
(5) region as shown in Fig. 3.
aij 2- + 3 μ for j = 2, 4
1J mv ij Jv
bandwidth
The coefficients a.. and β.. depend only on
the bus geometry and are known. The vari
able plant parameters are s-plane
m = mass
J = moment of inertia
v = velocity
μ = road adhesion coefficient.
For uniform passenger distribution, J is
determined by m and the plant parameter
vector is
Θ = (6)
The parameter ranges are:
πι . (empty bus) < m < in (full bus) Fig. 3 A region of Γ-stability. The hyper
mmv v 7 J maxv J bola boundary guarantees minimum
vm ·m ^ v < v (7) damping and minimum negative real
part of the eigenvalues. The band
width circle completes the boundary
μ . (wet road) < μ <, 1 (dry road)
3Γ.
Note that the system is not controllable
for v = 0, therefore a minimum speed v . PROBLEM FORMULATION
is assumed for the controller design.
Assume that the uncertain vector of plant
Fig. 2 illustrates the parameter ranges of parameters @_ is constant and in an admissi
m and v. ble region Ω, Also assume a fixed gain line
ar controller structure
m l u(s) = C(s,k)y_(s) + r(s) (9)
A B
full with the free controller parameters k. Then
the closed-loop characteristic polynomial
Ω has the form
empty ID P(s,0,k)=po(0,k)+Pi (0,k)s+.. .+pn_1 (G^lOs^+s11
I 1I 1 = [s-s (0,k)][s-s (0,k)]...[s-s(0,k)]
n
vmax
(10)
Fig. 2 Plant-parameter plane with coordi
A typical robustness problem is: Find the
nates v and m. Admissible region f
set of all k such that s,(0,k)er, i=1,2...n,
for all Q£ti. An appropriate space is the ge-
Multi-model Approaches 3
neral parameter space with the coordinates iii) Select a good
of Θ_ and k. In this space a Γ-stability re
gion can Fe determined. It must then be k ε κ (14)
a Γ
analyzed, for which values of k this admis
sible region contains Ω. An algebraic for in consideration of additional design
mulation of this problem is: requirements (Ackermann, 1985)
• Map Γ by a rational mapping onto the • small ||k|| in order to reduce |u|,
left half plane of a new complex vari
able w. The Hurwitz conditions for the • a safety margin for k away from the
characteristic polynomial in w yield a boundaries of Kp for the case of im
set of some linear and some nonlinear in
plementation inaccuracies (e.g.
equalities in Q_ and k (Gutman, Jury,
quantization),
1981), (Barnett, Scraton 1982),
(Sondergeld, 1983) .
• robustness with respect to sensor
failures,
• The linear Hurwitz conditions (all coef
ficients positive) describe the convex
• gain reduction margins.
hull of the stability region and there
fore give a "tight" necessary condition
(Fam, Meditch, 1978) . iv) It can be checked whether the admissi
ble parameter range Ω was well repre
sented by the chosen Θ_.€Ω, see step
• A sufficient condition can be obtained by
imbedding the problem into one of inter i). For this purpose fix k = k , see
val polynomials with independent coeffi step iii) and determine tEe Γ- stabi
cients. Only four specific polynomials lity region Ω„ in £3-space and check
must be tested for the Hurwitz property whether
(Kharitonov, 1978) .
Ω C Ω (15)
Γ
In order to make the problem tractable, we
break it down into subproblems in lower This is illustrated by Fig. 5.
dimensional spaces. This multi-model ap
proach consists of the following steps:
i) discretize Θ_. Select representative ©2 ^
values <3-€Ω, j = 1,2...J. This gives / B E / \
a family^of plant models ©2max· / \
— - ]
Ω /
A. = Α(Θ..) , B. = HQ.) (11)
02min l V D /
or in transfer function notation
Gj(s) CCsI-Aj)"1^ (12) ^Imin ^1max θ,
ii) For each j find the Γ-stability region Fig. Γ-stability was assured for the
K„. in k-space. The intersection points A, B, C and D by a multi-
model formulation. In the situ
ation depicted the multi-model
approach may be repeated for
cr = Π K rj (13) A, B, C, D and E.
j-1 Assume that Ω was first represented by
the model family A, B, C, D and a si
is the set of simultaneous Γ-stabiliz- multaneous Γ-stabilizer k was found.
ers. Fig. 4 illustrates this for two —a
plant models (A , b ), (A , b ) of Then the Γ-stability region·Ω„ in the
(3-plane must contain A, B, C, D by
second order and a state-feedback con
construction. Stability regions may be
troller u = - [k k ]x + r.
non-convex, therefore Ω^ does not ne
cessarily contain all admissible op
erating conditions in Ω. In the situ
ation of Fig. 5 the design may be re
peated with the additional plant model
for the operating condition E.
EXISTENCE OF SIMULTANEOUS STABILIZERS
Only few resu lts are aval iable on the ex-
istence and dynamic order of simultaneous
stabilizers or r-stabili zers for a family
- plane K-space of plant mode ls (Ay b.) , j = 1 , 2...J. Some
of these resu lts are brie fly reviewed in
this section For simplic ity the discussion
Fig. kGKr places all eigenvalues of is restricted to the sing le-input case with
(A -b k') into Γ,
state feedbac k and Γ = 1e ft half plane.
places all eigenvalues of Fig. 6 shows the general compensator struc-
— 1 2 (A -b k') into Γ. ture.
The intersection K„ - Κ Π K de
Γ r
scribes the set of simuΓl it aneTosu s Γ-
stabilizers for the two plant models.
