Sampling with arbitrary choice of the sampling instants

Automatica, VoI. 9, pp. 117-124. Pergamon Press, 1973. Printed in Great Britain.

Sampling with Arbitrary Choice of the Sampling Instants* Echantillonnage avec Choix Arbitraire des Instants d'l~chantillonage Representativstatistik mit beliebiger wahl der momente der Representativstatistik K B a H T O B a H H e r i o B p e M e r l r t c n p o I , I3BO3IbI-IbIM O V 6 O p O M MOMeI-ITOB I
The solution of problems of control, observation of the initial state, parameter estimation and related topics can be optimized by using the freedom in the choice of the instants of sampling. Summary--For linear multivariable control systems the question of observing the state by means of sampling with arbitrary but fixed choice of the sampling instants is discussed. The freedom due to this arbitrary choice is used to derive criteria for a choice of the sampling instants which is as advantageous as possible as far as the propagation of measuring errors is concerned. The extended principle of duality is formulated and the dual problem of control by means of step functions is covered. Further topics mentioned are e.g. the identification problem and the influence of uncertainties of parameters.

sampling instants in this generalized case, too. But intervals can be specified, so that complete observability is preserved, if the sampling instants are taken out of such an interval. By the extended principle of duality there is also the possibility to make use of these results in the problem of (optimal) control. A further interesting application of these ideas is e.g. the identification problem.

THE CONCEPT OF p-OBSERVABILITY INTRODUCTION

In this paper multivariable control systems shall be discussed, which can be described by the following equations:

IN a previous work GILCrmlST [4] discussed for systems with scalar output the problem of observing the initial state by means of n samples of the output, taken at instants which need not being equidistant. Later on [8, 9], these ideas have been treated in a generalized form, a short survey of the main results will be given in the following. The problem of observing the state by means of sampling, results in the task of solving a system of linear equations. It will be shown how the freedom in the choice of the observation instants can be used to obtain a rather well conditioned system of linear equations for the unknown initial state. Thus, the propagation of measuring and/or rounding errors can be reduced to a minimum. As known previously for the case of sampling with constant sampling period, there is a - - r a t h e r weak--restriction for the choice of the

2 =Ax+Bu y = C x + Du.

(1)

Here xeR", u~R" and yeR S denote the state-, control-, and output vector resp., the matrices A, B, C and D are of appropriate order and constant. Furtheron the concepts "completely observable" and "completely controllable" shall be used as defined by KALMAN in [5]. For the main criteria the reader is referred to [6] or [12]. Just the fact shall be mentioned, that the observability problem can be discussed without loss of generality when putting u = 0 . N o w the main definitions shall be formulated:

* Received 3 February 1972; revised 10 July 1972. The original version of this paper was presented at the 2nd IFAC Symposium on Multivariable Technical Control Systems which was held in DUsseldorf, Germany during October, 1971. It was recommended for publication in revised form by Associate Editor K. J. Astr/Sm. t Dozent Dr. techn., Hochschulassistent am I. Institut fiir Mathematik der Technischen Hochschule Wien, Karlsplatz 13, A-1040 Wien, Austria.

Definition 1 System (1) is called completely p-observable, if there exist p numbers 0~
1 18

INGE TROCH

This concept can be interpreted as sampling with variable sampling interval, whereas the following leads to a control built of step functions.

of (6) has rank n. This matrix F can be rewritten (comp. ( 3 ) ) as product: ¢"

Definition 2

u(t)=u(Tk)

cq(T0CA ~

[ Cexp(AT1)

System (1) is called completely p-controllable, if there exist p numbers O< T~ < . . . < Tp, such that for every 2eR", (1) can be directed to x=O (i.e. x(Tp) = O)by means of a control of the form (T o= O) for

Tk<<.t
m-I

k=0

F= m-I

C exp(ATm)

~k(r,.)C A ~ k=0

p-1. (2)

In most cases the number p will be the smallest number with the properties required in Definitions 1 or 2, resp. But this is not necessary, since every number q>p will satisfy these definitions ifp does. Yet, the smallest number which can be used is of great importance as shall be pointed out when discussing the concepts of observability- and controllability-index. In the following this minimum number will be preferred and used. First the question of p-observability will be discussed at some length, the dual results for the control problem shall be given later. If the minimal polynomial of the matrix A is of degree m and the fundamental solution matrix belonging to (1) is represented by

CA

011 • • . Ol,n 1

i

(1)ml

.

.

.

O)mm

CA"--~ [ C

CA =:O.Q

CA

m-

1

with m-

l

exp(At)= ~,

C~k(OA~

(3)

k=0

01~ = Diag{~k- I ( T i ) . . . . .

then the following is valid [8]:

Direct calculations show, that

Suppose that system (1) is completely observable in the sense of Kalman. Then (l) is completely m-observable too, if the sampling instants Tk fulfill the condition = Det {ai(Tk) } + 0 (i=0,

t(Ti) } = e k - l(Ti) 'I

(S rOWS).

Theorem 1

~:

~/,--

1. . . . .

m-l'k=l,2

.....

m). (4)

Proof (outline). As mentioned before, it is sufficient to restrict to the case u = 0 . Then the solution of system (1) starting in 2 can be described by x(t,.2)=-exp (At)2

(5)

and it has to be shown,that m numbers Tiaccording Definition 1 exist such that the system of linear equations C . exp(AT i) .2=y(Ti) i = 1, 2 . . . . .

m

(6)

can be solved uniquely for every 2c-R". This is possible if and only if the matrix F o f t h e coefficients

Det{Ou} =~'(i, j = 1, 2 . . . . .

m).

Therefore and because of condition (4) the matrix is regular. From this follows that rank F = r a n k (C', A,C', . . . . (A')m-Ic'), where ' denotes the transpose, the latter being n according Kalman's well known criterion, [6], which completes the proof. Condition (4) is not very restrictive because of the fact, that for the functions C~k(t) defined in (3) the following Lemma is valid.

Lemma 1 If al . . . . . a,, are m different real numbers and if Tk=akT, k= 1, 2 . . . . . m with TeR +, then the determinant (4) regarded as a function of T has a finite nmnber of zeros on every interval of finite length. Proo[ (idea). The m functions ~k(t) are linearly independent solutions of a system of linear differential equations of first order with constant coefficients, whose characteristic polynomial is the

Sampling with arbitrary choice of the sampling instants minimal polynomial of the matrix A. Furthermore, they fulfill the initial conditions ( 6 i k . . . Kronecker symbol) ~}k)(0)= Oik

i, k =0, 1. . . . .

m- 1

(7)

Remembering further the fact that for a Chebyshevsystem over the interval I the determinant (4) does not vanish for every m-tupel (Tx . . . . , Tm)~.lm with T ~ * T k for i 4 : k , makes the use of this Lemma evident.

what can be shown by direct calculations. They are therefore regular functions of t and the same is true for the determinant a(T) = Det{~,(akT)}

i=0,1 ..... k=l,2 .....

119

THE SIGNIFICANCE OF THE OBSERVABILITYolNDEX

In accordance with Kalman's fundamental criterion, the observability-index is defined by:

m-I m.

Definition 3

Because of this regularity only two cases are possible: Either the zeros of 8(T) have no point of accumulation in the finite or ~ vanishes identically. It is therefore sufficient to show that the latter cannot be true. This is done by differentiating ~(T)N=m(m-1)/2 times according the rules for the differentiation of determinants. Calculating now ~(N)(0) shows that only one determinant in the sum for ~(N)(0) has no parallel rows. This one can--because of the initial conditions (7)--be written in the form of a Vandermonde determinant thus leading to

The number a=

Min {k[rank(C', A ' C ' , . . . . Z_
(A')k-lc')=n}

(11)

is called observability-index of system (1). For observation by means of sampling with constant period, this means, that a samples are sufficient for the calculation of the initial state 9~. This is true also [9], for the case ofp-observability, or formulated more exactly: Theorem 2 Let be Y ' = ( Y x . . . . .

y~) and let n x. . . .

,

n~ be s

integers with

N (m-l)! ! l-[ (aj-a~). ~
8(~0(0)=2!3 ! . . .

nl+n2+

The calculations are somewhat lengthy and can be found in [8]. If the sampling instants shall be chosen according Definition 1 and moreover it should be assured that condition (4) is fulfilled without testing it, then the following Lemma 2, [10], gives a quite useful sufficient condition. It makes use of the concept of a Chebyshev-system, which is well known in the theory of approximation [7]. Lemma

...

02)

+ns=n

and let further cx, e2 c s denote the row vectors of C. If now the vectors . . . . .

c1, c x A , . . . , c l A ' ' - ~ ,

...,

cs, CsA, . . . , CsAn~-x

(13) are n linearly independent row vectors of the observability matrix Q, then the following holds: (1) n~, . . . , n~ can be chosen in a way such that

2

Let ? be a real number and ~=Axz

Max{nk/k=

(8)

be a system of differential equations of order p with constant coefficients. Let co be the maximum eigenfrequency (i.e. the greatest imaginary part of the eigenvalues of Ax), then the following holds: p linearly independent solutions of (8) form a Chebyshev-system on the interval I = (% ? = n/og).

(9)

If all eigenvalues of Ax are real, then

I=(r, oo).

0o)

The functions ctk defined in (3) are linearly independent solutions of a system (8) with p = m .

1, 2, . . . , s} =cr.

(14)

(2))~ can be calculated from the n values yl(T~) .....

yl(T.) .....

y~,(Ta) . . . . .

y~(T. s)

05) provided T 1. . . . . (4).

T, fulfill an inequality similar to

P r o o f (outline). The first assertion follows from equation (11). For the second one, the proof runs similar to that of Theorem 1. The initial state 2~ can be calculated as the solution of the system y i ( T k ) = Ci " e x p ( A T k ) " ;c Ili--1

C)k = l , "z, . . . , ni

= j=o ~ Otg(Tk)'Ci" AJfc. [ i = 1, 2,

,s

120

INGE TROCH

Now let Qt be the matrix with the row vectors (13) and Q2 the matrix with the row vectors q,

clA , . . . , clA m-I,...,%

Example

1

Let the system

c~A. . . . , c , A m 1

1 = , / t X l "~- VX2 ['£2 = - - F N 1 -[-/LV 2

Then there exists a constant matrix R such that

(17)

]l, v~R

y = 2x ~+ .v2 Q2 = R Q l .

As before, the matrix of the coefficients, F~, can be written as product F~ = D i a g { ~ . . . . .

~,}.R.Q~

be given. Let further the output y be observed at Ta >/0 and at T z > T~, denote the observed values by yl and Y2 resp. and let 9~ to be calculated. Then the following equations are to be solved:

(16) 21(2 cos v T i - sin v T i ) e x p ( # T i )

q-3~2(2 sin

vT i

with + c o s v T ~ ) e x p ( # T ; ) = y i, i = 1, 2.

(18)

A measuring error Ay leads to an error A2 and it might be measured, [3], with the function

(I) k

• o(T,~)

•..

%,-t(T,~) e=

k = l , 2. . . .

0-71

,

,s. where 11' ] 1 " " Euclidean Norm.

Reflections similar to those used in the p r o o f of L e m m a 1 show that for T i = a;T, (i= l, 2 . . . . . a) the function

This results for (17) in 5E-

fl(T) = Det{Diag(~,

.....

~.,)

"R

cannot vanish identically. Thus the instants T i can always be chosen such that in (16) the product of the first two matrices is a regular matrix. In that case rank F 1 = r a n k Q1 = n follows, what completes the proof. The somewhat troublesome, but not difficult, calculations can be found in [9].

I+exp(-2#(T2-T1))

- 5 E ( T ~ , T2).

(20)

exp(2#T1)sin 2 v(T2 - T1)

For # 4 0 , this function--regarded as a function of the two independent variables T 1and T2--has no relative minimum over R 2. Let now T 1 be fixed and T2 o r - - w h a t is equivalent--AT= T2 - T1 be varied. For this function of AT exist an infinite number of relative minima at AIT, A2T, ... where AkT is a solution of ~tan

v • AT = -

l -exp(2#

• AT).

(21)

V

The corresponding value of E is given by SOME SIMPLE EXAMPLES At this point a question may be asked: is it possible to make use of the foregoing in a way which assures that measuring errors do not "grow" during calculation. In terms of numerical analysis this means: Observation by means of sampling leads in any case to a system of linear algebraic equations for x(0) and it depends on the condition of its coefficient-matrix how much measuring errors influence the solution. This leads to the idea of using the freedom in the choice of the sampling instants to improve the condition of the coefficient matrix. To make this more clear, two examples shall be discussed, which are chosen to be simple in order to minimize calculation difficulties.

5/~ - [-#2 + v2(1 + exp(2# • AkT)) 2] • exp( - 2#A~T) exp(2# T 1)v2(1 + exp(2#A kT)) = 5/~(T1, A k T ) .

(22)

Inspection of the last expression shows, that in case/~ > 0, /~ is decreasing with increasing AkT, so that the largest meaningful positive solution of (21) is the best one. If # < 0 then the contrary is true: the smallest meaningful solution of (21) is now the best one. These considerations show further, that the optimal AT is independent of the choice of T t and this results in the following strategy: (1) Regard T 1 as fixed (e.g.) and choose AT optimal,

Sampling with arbitrary choice of the sampling instants (2) Take T1 as large (small) as possible for p > 0 (#<0). If /z=0, expression (20) is a minimum for v.AT= (2k + 1)n/2. Consider now

row vectors specified in (13) and (16) whereas the vector q has the components (15). A small variation of the component tlk of/7 leads to a variation of ~, given by

f 2 _ f_.(G_lq)=G_%k '

Example 2

n

(27)

where ek is the k-th unity vector (0 . . . . . 1 , . . To measure these changes the function

, 0).

(Jl~k X1 ~ O~X1

YCz=flxz, az +fl2 +-O

121

k = l , 2. . . . .

(]qk

(23a)

y~ = ax I + bx2

(28) Y2 = CX1+ dx2.

(23b)

Taking again (19) as the measure of the resulting error, leads to 1

E = (ad - bc:2[(a2) + c2)exp( - 2fiT,) + (b 2 + d2)exp(- 2aT1) ] .

(24)

is taken. T 1 . . . . . T o shall be chosen thus, that E as a function of T1 . . . . , To is a minimum. Normally here will be defined a region ~7"___(R +)" out of which the instants of observation can be taken. The minimum required can lie on the border of ~" or can be some relative minimum inside it. In the latter case there are a necessary conditions: II

0 ~ _ OE _

aTj

Note that only one observation of y is needed! (1) For a>~0 and fl>~OE is a minimum, if 7'1 is as large as possible. (2) For a~<0 and fl~<0 --contrary to ( 1 ) it is best to choose T 1 as small as possible: /'1 = 0. (3) For aft < 0 there exists a relative minimum at

0 k_~l(G_ iek)2

0Tj =

=2k~=~{G-lek' ~Tj G-Z ek} where (., .) denotes the scalar product. This is equivalent. 0=trace~'(G-1)'( ~Tj ~ G-l}

T 1 - 2 ( ; _ ajln(

fl(a2+c2)'~ a( b 2 + d2) ]

(25)

and regarding only the numerical aspects--this T 1 would be the optimal observation time, provided the expression (25) is positive. These two examples show, that it is possible to improve the numerical properties by an appropriate choice of the observation time. The matter is rather complicated and involves a lot of mathematical theory. In the following, it shall therefore be shown, how to proceed when treating more extended problems, and strategies for some special cases shall also be given. A NECESSARY CONDITION FOR THE OPTIMAL CHOICE OF THE INSTANTS OF SAMPLING Assume, that the output of system (1) is observed at time T~, 7"2. . . . , T, according Theorem 2. Ths leads to a system of linear equations of the form (26)

whose coefficient-matrix G is regular and has the

o'. (29)

As stated before it is not yet possible to make full use of the equations (28, 29) in this generality, but it is possible to succeed--as in the examples before--when treating a given problem with known constants. One also has to think about numerical methods developed for the problem of parameter optimization--instead of using equations (29). These are of more theoretical interest. There is one case where some more results can be deduced. Assume that .4 can be transformed to a diagonal matrix by means of a similarity transformation and that C is a regular n-matrix. Then a = 1 and

G=C.exp(ATI),

G~ =t/

j = l , 2. . . . .

G-X=exp(-AT1)C -a. (30)

Equation (28) becomes E = trace{C'- 1.exp(- A'T1).exp(- AT1).C- 1}

(31) and the necessary condition (29) trace {C ' - 1.exp( - A "T1).A.exp( - A T 1). C - x} = 0. (32)

122

INGE TROCH

If the real eigenvalues of A a r e 21 . . . . . 2p and the complex ones/tk--iVk, k= 1. . . . . q and if further A has been transformed to the real canonical form Diag(P1

Pq, 21 . . . . . 2p)

. . . . .

(33a)

with

duality [6, 12] is valid for the case of nonequidistant sampling, too. Thus the use of these ideas for tbe solution of the control problem indicated is rather straightforward. Before stating this principle the following definition is useful:

Definition 4 The number p defined by k = 1, 2 . . . . . -

vk

q

(33b)

/~k

p = Min {k/rank(B, AB . . . . .

then (32) becomes with

O=trace(H'AH)= ~ t~k ~, k=l

(h2k-l,j+h~_k,j)

.i=1

+ ~ Zk ~ h~q+k,i.

(34)

q

E = ~ exp(-2/tiT1) k=l

r,,(-1) ~ 2i-l,kl

+ ~ exp(-22fl'0 i=1

Principle of duality' for a-observability and pcontrollability The system

j=l

From (34) follows that (32) can have a solution T t >~0 only if not all the eigenvalues of A have real parts with the same sign. This is a further necessary condition for the existence of a relative minimum in the interior of J-. If all eigenvalues of A have real parts with the same sign, then the minimum is reached at the border of J-. In this case a result can be derived in the following way. Because of (33) equation (31) reads explicitly to:

i=1

(36)

is called controllability-index of system (1). Its meaning is similar to that of the observabilityindex. Roughly speaking it means, that p step functions are sufficient to stir system (1) from any initial state to the equilibrium.

H : = (hik) = exp(AT1). C-1

k=l

Ak-lB)=n}

1 _
1 2 ±T rV.'(c-211),~k2 I

[W2q+i, ~ ( - 1 ) k)2.

~=A'¢ +C'u ~1= B' ¢ + D'u

(37)

is dual to system (l) in that sense that (1) is completely p-controllable (a-observable) with controllability (observability)-index p (a), if and only if (37) is p-observable (a-controllable) with observability (controllability)-index p (a). One way to proof this is to verify that a system which is completely controllable in the sense of Kalman is completely p-controllable with controllability-index p, too. The proofs are similar to those of Theorems 1 and 2. They use the functions

(35)

k=l

Cb-1) is an element of C - 1 . From this last equation it follows that there exists a minimum of E reached at the border of @. If the system (1) is stable, no eigenvalue of A has positive real part, then it follows that it is the best to take T x as small as possible, normally T~ =0. If no eigenvalue of A has a negative real part then it is better to wait as long as possible. This fact can be explained as follows: In the first case--if not all real parts equal 0--the solution has portions that get smaller as time passes by and none that increase, and they can therefore be registered the better the earlier the sample is taken. In the second case the contrary is true.

fig(t)=

~(s)ds

where

m- 1

instead of the functions C~k(t). The values Tk are defined as those instants, where at least one component of the control u (cp. (2)) changes its value. Now the freedom in the choice of the Tk'S can be used to make the control optimal in some sense, i.e. according to a given performanceindex. This again shall be demonstrated with a very simple example:

Example 3 ~1 ~

X2 "1"-b/1

~1~2= - - X 1 -{~b/2

A VIEW AN THE CONTROL PROBLEM As mentioned before, these ideas can be used in the control problem, too. Often piecewise constant controls are to be used which shall be optimal in some sense. But it is not always desirable, or possible, to use directly results from the theory of optimal control. The well-known principle of

k = 0 , 1. . . . .

0

ui(t) =

x I ( O ) = 2( 1 X2(0)=X 2

ui . . . O<~t
0

t> T

i=I,2.

(38a) (38b)

Demanding that E=u~+u~ shall be as small as possible leads from dE/dT = 0 to sin T.(cos T-- I ) = 0

(39)

Sampling with arbitrary choice of the sampling instants under the restriction cos T - 1 4=O. From this follows easily T=(2k+l)rc

k = 0 , 1. . . .

(40)

and further

Ul=--.~2/2, //2=2{1/2, E=(.~12+~22)/4. Now the decision is left, which k in (40) is to choose: Eis independent of k, but for k > 0 x(n) = 0 and x ( t ) 4 : 0 for z < t < 3 n , what means, that the control, if applied longer than necessary, forces the system again out of the equilibrium, so that T = rc is in this case optimal.

123

can be calculated from observations of the output at a finite number of arbitrary distributed instants. There is one not too restrictive condition, which can be formulated explicitly. The freedom in the choice of the observation times can be used to improve the numerical aspects, especially to reduce the propagation of measuring errors on the calculated initial state. The problem is rather comprehensive, yet it has been possible to give strategies for some special cases. The dual problem of control by means of step functions has been treated too and results again in a parameter optimization problem. It has been pointed out that these concepts have a lot of further applications as e.g. in the identification problem. Their use is merely a question of an appropriate choice of the performance criterion which is to be used.

REFERENCES FURTHER ASPECTS OF THE CONCEPT OF NONEQU1DISTANT SAMPLING INTERVALS The concept of p-observability as well as that of p-controllability can be useful in case one or more parameters of the system under investigation are not known exactly. The freedom due to nonequidistant samplings can be used to reduce the influence of the uncertainty of these parameters. It seems that this might be important especially for the control problem. For the applications of these ideas it is just necessary to choose the performance criterion such that it covers the influence of the uncertainty of the parameters. But there are still more applications to think of. In the problem of identification, the initial state might be known in advance. Then measuring the output y may be used to calculate the parameters. Again, the instants of measuring Tk are chosen in a way that measuring errors have as little influence on the calculation as possible. All this is only a question of the choice of the performance criterion. Further, difficulties may arise in the practical implementation of equidistant sampling--in the observation problem as well as in the control problem: The equidistancy might be disturbed. The concepts developed above can be used to pursue the propagation and consequences of this inexactitude. Finally it should be mentioned that at the moment it is not possible to give a complete and explicit solution for this optimization problem. But it is "only" an optimization of parameters and the use of well-known numerical methods will lead to good results in concrete cases. CONCLUSIONS It has been shown that for completely observable systems described by equations (1) the initial state

[ll J. ACKERMANN: Zeitoptimale Mehrfach-Abtastregelsysteme. Vorabdruck der Beitr/ige zum IFACSymposium iiber Mehrgr6Benregelsysteme, Diisseldorf (1968). [2] R. G. BROWN: Not just observable, but how observable? Proc. Natn. Electronics Conf. 22, 709-714 (1966). [3] D. K. FADD~EWand W. N. FAODEJEWA: Numerische Methoden der linearen Algebra. Oldenbourg, Miinchen (1964). [4] J. D. GILCHRIST: n-observability for linear systems. IEEE-Trans. AC-11, 388-395 (1966).

[5] R. E. KALMAN: On the General Theory of Control Systems. Proceedings of the 1st Congress of IFAC, pp. 481-492, Moskau 1930, Butterworth, London, Oldenbourg, Mtinchen (1961). [6] R. E. KALMAN,Y. C. Ho and K. S. NARENDRA: Controllability of linear dynamical systems. Contributions to Differential Equations 1, 189-213 (1963). [7] A. F. TIMAN: Theory of Approximation of Functions of a Real Variable. Pergamon Press, Oxford (1963). [8] I. TROCH: Bemerkungen zur n-Beobachtbarkeit und n-Steurerbarkeit. Z A M M Sl, 255-264 (1971). [9] I. TRocrI: Ober die Bedeutung yon Beobachtbarkeitsund Steuerbarkeitsindex. Z A M M Sl, 265-269 (1971). [10] I. TROCH: Crber die LOsungen linearer autonomer Differentialgleichungssysteme als Tschebyscheff-System. Z A M M 52, 193-194 (1972). [lll I. TROCH: Observing the State by means of Sampling wit.h Arbitrary Distribution of the Instants of Sampling.

Proceedings of the 2nd IFAC-Symposium on Multivariable Control Systems, Vol. 1, North-Holland, Amsterdam, Elsevier, New York (1971). [12] L. A. ZAOEHand C. A. DESOER: Linear System Theory. McGraw-Hill, New York (1968).

R~smn6--Pour les syst6mes de contr61e lin6aires multivariables on discute la question de l'observation de l'6tat grace ~t l'6chantillonnage avec choix arbitraire mais fixe des instants d'6chantillonnage. La libert6 due ~t ce choix arbitraire est utilis6e pour d6dver des crit6res pour un choix des instants d'&:hantillonnage qui sont aussi avantageux que possible en ce qui concerne la propagation des erreurs de mesure. Le principe 6tendu de dualit6 est formul6 et le probl6me dual de contr61e par fonctions en 6tapes est consid66. On fait mention par ailleurs du probl6mer d'identification et de l'influence des incertitudes de param&res.

124

INGE TROCH

Z u s a m m e n f a s s u n g - - E s wird fiir lineare, multiver/inderliche Steuersysteme die Frage der B e o b a c h t u n g des Z u s t a n d e s mittels Representativstatistik mit beliebiger, aber fester W a h l der Statistikmomente besprochen. Die Freiheit zufolge dieser beliebigen W a h l wird benutzt, Kriterien for eine W a h l der M o m e n t e der Statistik abzuleiten, die, was das Mitlaufen der Messfehler anbelangt, so weit wie m6glich von Vorteil ist. Es wird das erweiterte Prinzip der Dualitiit formuliert, u n d das D o p p e l p r o b l e m der Steuerung mittels Schrittfunktionen wird behandelt. Weiter erwiihnte T h e m e n sind z.B. das P r o b l e m der K e n n t l i c h m a c h u n g u n d der Einfluss der U n b e s t i m m t h e i t e n von P a r a m e t e r n .

PeatOMe----PacCMOTpeH ~ c~yqan .nrtne~h~IbiX MHOFOBapHaHTHbIX CHCTeM ynpaBJleHrl~ Bonpoc Ha6YtIo)Ienrin COCTOflHHfl rtyTeM KBaHTOBaHHfl C HpOH3BOJ'IbHblM 1tO FIOCTOgHHblM OT6OpOM MOMeHTOB KBaHTOBaHHfl. CBO60~la ~,aHHa~ 3THM llpOH3BO.rlbHlalM OT6OpOM HCIIO.rlb3OBaHa ~,qa nport3Be~eHn~ rpHTepi,IeB OT6Opa MOMeHTOB KBaHTOBaltl,Lq, BblFO~IHOFO q~M 60-rIblLle BO3MO)KHO tlTO ~:acaeTca pacnpocTpaaerm~ a3MeprlTem, nbix OtUri6OK. CqbopMyaapoBaH gpOTflHyTblfi IIpI,IHIIHII ~BOI~ICTBeHHOCTH H BK.qlOqeHa ~BOfiCTBeHHafl npo6yteMa yr~paBJIeHHfl C IIOMOLI~blO CTyneHbqaTbiX dpyHruHfi. TeMbl pacCMOTpeHHbie B CTaTe BK.rllOqalOT npo6heMy OTO:;,K)IeCTB.rIeHrlg I4 B3mgHrIe Hen3BeCTHOCTe~ napaMeTpoB.