Guaranteed Estimates, Adaptation and Robustness in Control Systems

195 Pages·1992·6.054 MB·English

by  Vsevolod M. Kuntsevich, Michail M. Lychak (auth.), V. M. Kuntzevich, M. M. Lychak (eds.)

INTRODUCTION
The majority of real control plants feature considerable
uncertainty regarding both the properties (parameters) of these plants
and disturbances affecting them. In many cases, the a priori estimates
of parameters are rather rough thus preventing the obtaining of
practically reasonable solutions of problems of analysis and synthesis
of respective control systems. This clrcumstance leads to the need to
use one or other parametric identification procedure which can be
realized either at the preliminary identification stage l.e. prior to
the realization of the control process itself (what Is obviously
admissible only for the class of stable controlled plants), or within
the framework of adaptive control systems in which the both processes:
the studying (identification) and the control proper are carried out
simultaneously. Thls approach Is required specifically in solving the
problem of control of unstable plants.
It might be well to point out that the majority of scientists
studying control processes under uncertainty conditions use stochastic
(probablllstlc) uncertainty models until the present time. They do it
not because the processes of uncertainty actually have a stochastic
nature. In many cases thls Is the result of a certain inertia of
thought, the action of an established stereotype.
The assumption oZ stochastic nature of uncertainty can not be
accepted at least in two cases: )I when the volume of a priori
experimental data on the nature of the uncertain factors is so small
that it does not allow the conclusion on the availability of stable
2
statistical characteristics, 2) when It Is known a priori that the
uncertainty basically can not be considered to be produced by some
probablllstlc mechanism. As correctly noted by R.Kalman: "It would be
a great untruth to claim that ... the whole uncertainty arises by
virtue of the mechanism of statistical choice. The nature does not
conform to the rules of traditional probability".
It Is thls reasoning that has given rlse ot the advent In the last
decade of the method of obtaining ~ranteed estimates of the vectors
of parameters or (and) the state of the processes being controlled.
The first chapter of the book calling the attention of the reader Is
devoted to the above range of problems.
Another important problem to which Chapter 2 of the present book
is dedicated is the problem of constructing adaptive control systems
in which (in view of one or another reason) It Is necessary to combine
in time both the process of improvement of the accuracy of the
estimates of the controlled plant parameters (and in some cases, of
the guaranteed estimates of its state vector inaccessible for a direct
measurement) and the process of the plant control proper, The
simultaneous realization of these two processes has a significant
mutual effect on the both processes.
The presence of uncertainty as regards the non-controllable
disturbances (noise) acting upon the controlled plant results in
incorrectness of the problems of analysis and synthesis of systems
controlling such plants even given ideal estimates of their
parameters. To eliminate this uncertainty, a galm approach to the
control synthesis problem statement is used i.e. the assumption of
3
"Ill-intentions" of the nature( of the environment generating these
disturbances) is introduced. I.e., the nature tends (within the limits
of specified restrictions) to maximize the performance index which the
system designer seeks to minimize by selecting control. Thus, the
problem of control synthesis is formulated as a minimax problem and
some results in solving this problems with different statements of
control synthesis problems are presented In Chapter 2 .
The result in solvlog the identification problem under real
conditions (irrespective of the identification algorithms used in thls
case and irrespective whether this solution Is obtained within the
framework of adaptive control systems or at a stage of preliminary
(independent) identification) Is always the obtaining of only some of
parameter estimates. Thls takes place already even if the solution is
sought for on a final time interval and in the presence of one or
other disturbance (noise). In the event that these estimates are in
some sense reasonably close to the true values of the parameters, It
is allowable to identify these estimates wlth the true values of the
parameters. When thls takes place, no special problems arise, as a
rule, wlth the analysis and synthesis of control systems ~or such
plants. However, when the control system designer has no reasons to
neglect the errors in solving the identification problems, then the
situation is substantially complicated since the presence of such
"residual" irremovable in principle uncertainty concerning the
characteristics of the controlled plants is equivalent to the
necessity to solve the problems of analysis and synthesis not for one
specific plant but for a complete class of such plants, i.e. it leads
us to the necessity to solve the problem of the robustness of control
systems in general and of adaptive control systems in particular. Some
4
problem statements and solutions of the control system robustness
problem are set forth in the last tklrd chapter.
This book si based on results of a research conducted by the
authors at the V.M. Glushkov Institute of Cybernetics of the
Ukrainian Academy of Sciences which have been rather widely discussed
by the authors with their colleagues not only at the Institute. No
doubt, the results of eht discussions have been not only beneficial
for the improvement of the style of their statement, but have had a
positive impact on the direction of the studies themselves.
We should like gratefully acknowledge the support which was
rendered to the authors by Professor .M Thoma during preparation of
thls book for publication.
Vsevolod M. Kuntsevich
Michall .M Lychak
Klev, ~ay, 1991.
CHAPTER I
G U ~ ESTIMATES OF PARA~ER AND STATE VECTORS
"Throw off the whole
of the impossible and
then what remains is
the truth".
ConanDoyle
1.1.Guaranteed Estimates of Linear Systems Parameters
Automatic control of any plant assumes that the control system
designer has a more or less adequate mathematical model of the plant
which is specified In many cases only up to a parameter vector.
Therefore, the first problem which faces the control system designer
is the problem of refinement of the a priori estimates of this
parameter vector, i.e. the parameter identification problem. By virtue
of a number of reasons mentioned already In the Introduction, we shall
consider below only nonstochastic method of its solution which wlll
prevent us from using stochastic methods for its solution described
comprehensively in a number of papers which are now already classical
(e.g., ref. to [I]-[7]). In the present chapter, the case will be
consldered when the parametric identification process can be carried
out independently on the process of control proper and comes before to
it in time. Thus, let us consider first the problem of determining
the guaranteed estimates of the parameter vector for the simplest
class of control plants, namely of linear plants without memory wlth
the state described in discrete time by the equation
Yn = ]]TUn + fn ' n = 1,2,..., (I)
where n U is 1-dlmenslonal control vector ("input" ' of the plant), L
is i -dimensional vector of constant but unknown parameters, fn is
scalar uncontrollable disturbance (noise), Yn is the measured scalar
("output"of the plant).
Let us assume that only a priori estimate
L c ~o ' (2)
is known concerning parameter vector L , where 9o is a given
bounded convex set.
For disturbance fn " given is also only its a priori estimate
fn C ~ , V n)O , (3)
where [ is a preset bounded convex set determined in the following
way:
= ( fl [fI ~ a = const ~ . (~:)
It is assumed that vector U is formed generally wlth due regard for
n
constraints
U c~, (5)
n
where ~ is a specified set in control space.
Such a priori information practically predetermines the use of the
set identification procedure which enables a guaranteed estimate of
parameter vector L to be obtained at each n-th step in form of its
belonging to some set
n
Let us consider flrsl the solution of the passive identification
problem when ~put signals n U are assumed to be knownbut they are
generated in the way unknown to the investigator. Solution of this
problem should be based both on the ~e of a priori information
(equation (11) and estimates )2( and (3)) and on the use of a
posteriori information (the resets of measurements of values n U and
y~)-
Let us construct a recurrent procedure for the refinement of
estimates of vector L . Assume that estimate L at the (n+1)-th
step is known in the form:
,r" c ~ . (6)
wlth n=1 . The role o£ such estimate plays estimate (2).
Once the values Un+ I and Yn+1 are measured, we obtain from
)I( the estimate of vector L in the form:
rv
L ¢ ~n+l = { :L LI+n~U + fn+1 - Yn+1 = 0 .} )7(
From two conslstent estimates )6( and (7), we obtaln a new,
generally
improved, estimate 9n+I . which ls described by the set evolutlon
equatlon [8]-[11]:
(8)
L E ~n+1 = ~n+l n 9n ' n = 0,1,2 .... , •
It should be emphasized here, that unlike the tzadltlonal
Identlflcatlon problem solutlon methods whlch allow to obtain only
approximate "point" estimates of parameters, estimate )6( is a
guaranteed one in the sense that true values of parameters Sought for
are known to belong to a set 9n+I When this takes place, all
elements (points) o5 the set ~n+1 are equal In rights since there
are no preferences between the elements of the set.
An important property of the sequence of estimates )8( is that
these estimates are non-deterlorating, i.e.
1+n~ ~ ~ (9)
even through non-informative changes are posslble wlth some n when no
rv
improvements of the estimate take place. In the latter case ~n ~n+1
and ~n+1 = ~n " Clearly, the number of non-informative measurements
grows wlth a su~flclently good accuracy and hence the obtained
estimates may appear hereinafter to be non-improving wlth su~flclently
large n . Only In some special cases, sets ~n wlth large n can
contract (degenerate) to a s/ngle-point set containing only one point
$
corresponding to the true value of ,T Of the plant parameter vector.
2Z
';+nI
0
Fig. .i Intersection of convex polyhedra
Another important property of the sequence of estimates )8( ls as
follows: when a priori estimate ~o is a convex polyhedron, then
all following estimates ~n are also convex polyhedra (ref. to Fig. )I
since the class of convex polyhedra is closed with respect to the
intersection operation.
Hereinafter, we shall assume that ~o is a convex polyhedron
without stipulating this each time separately.
Let us now dwell on the definition of the necessary and sufflclent
condltlons with fulfillment of which the sequence of estimates
n
degenerates into a single-point set. We consider first the degenerate
case in whlch set f is empty and which is realized wlth a = .O
Then we obtain from )I( at A = 0 that this equation describes
some ~fpersurface in space (L} for each n. Let us note that when
ddrecting vectors n U of each of the hypersurfaces are linearly-
independent on each other then , as a result of the execution of each
of the set intersectlon operations, we obtain set ~n+1 with
dlmenslonallty smaller by one than the preceding one. Thus, after the
(l+1)-fold executlon of operation (8), we obtain set ~n+1 In the
form of a slngle-point set which contains only one point .L
From the algebraic point of view, this result can be interpreted
as follows. Assume that there are (i+I) observations of form )I( wlth
f = 0 , i.e. assume that n takes values from n = I to n = 1
n
Let us introduce the following designations:
lY -JTE"
1T J-
v~
1Y = (lO)
1 Z =
lY
Then the considered system of observation equations can be written as
ZIL = YI " (11)
OI
If the condition of linear independence of vectors U is met, l.e.
n
that of linear independence of rows of matrix Z~ , then det I ~ 0 Z
and we obtain from I( ) I
,T = z~yz - T.. (12)
whlch Is the required result.
Let us consider now a more interesting case when r ~ O , i.e.
a ~ .O Let us assume that there are two linearly-dependent
observations among the whole set of observations (I), l.e.
"s = cu~ , (is)
where o Is a known constant, such that
I fs I ~ I ~k I =~ , (14)
sign t = -sign k t . (15)
The fulfillment of conditions (14), (15) is in essence equivalent to
the existence o5 one summary observation free from noise. Indeed, when
condition (13) is met, we have
= Yk- (167
kTUC L = Ys + fk " )71(
From here we obtain that wlth fulfillment of (14), (15)
kY sY
(1 + c)U~.T, = + ' (18)
as we wished to prove.
It is obvious that the problem is how to single out the pair of
observations (16), (17) sought for from the whole sequence of
observation of type (I). Let us show that the application of procedure
)8( allows to solve this problem.