Table Of ContentZEITSCHRIFT FOR STEUERN, REGELN UNO AUTOMATISIEREN
Organ der VDl/VDE-Fachgruppe Regelungstechnik, des Arbeitsausschusses Regelungstechnik
im DNA u. der NAMUR (Normenarbeitsgemeinschaftf. Mel3-u. Regeltechnik i. d. chem. Industrie).
Unter Mitwirkung von Dr.-lng. H. J. Altenhein, Oberingenieur K. Bester, Dr.-lng. O. Grebe,
Dr.-lng. H. Henning, Dr. phil. nat. H. Kaufmann, Prof. Dr.-lng. E. h. K. Kupfmuller, Dr.-lng. R. Oet-
ker, Prof. Dr.-lng. Dr.-lng. E. h. W. Oppelt, Dipl.-lng. H. Schink, Dr. phil. B. Sturm, Dr.-lng.
H. Toeller, Prof. Dr.-lng. E. Wintergerst - Schriftleitung K. F. Fruh - herausgegeben von Prof. Dr.-lng. HANS SARTORIUS
I M V E R LA G R. O L DE N BO U R G, M 0 N CHE N • 14. J AH R GANG 196 6 SE I TE 545-592 H E F T 12
Effective construction of linear state-variable models from input/output functions 1)
Die Konstruktion von linearen Modellen in der Darstellung durch Zustandsvariable
aus den Beziehungen fiir Ein- und AusgangsgroBen 1)
By B. L. H02) and R. E. KALMAN3), Stanford, California (USA)
A new algorithm is given for the ejfective construction of ments on a system. We wish to realize the external description
minimal realizations of a linear, finite-dimensionai dynamical via an equivalent internal description, namely a set of state-
system from its external description, which is displayed via the variable equations. If we are interested in synthesis, these
Markov parameters. The algorithm can a/so be used to prove equations are the first step toward producing the blueprint.
various abstract results in realization theory; we give a proof If we are interested in analyzing experimental data, then these
of the well known fact that any two minimal realizations are equations are an efficient and useful model with which we can
isomorphic. proceed to further analysis and optimization.
The problem of realization for linear, stationary systems was
Es wird ein neuer weitreichender Algorithmus abgeleitet, der zu first stated by Elmer Gilbert [4], who gave an algorithm for
der minimalen Realisierung von linearen dynamischen Systemen computing the map: transfer function matrix-state-variable
endlicher Ordnung fiihrt, wenn man von den Beziehungen aus- differential equations4). A second algorithm for the same
geht, wie sie bei der Beschreibung von Systemen durch die problem was given at the same time by Kaiman [1], using the
Ein- und Ausgangsgroj]e Verwendung finden,· er macht von den theory of controllability .and observability. Both algorithms
sogenannten Markovschen Parametern Gebrauch. Den Algo- require linear-algebra type computations. The deeper theory
rithmus kann man auch zum Beweis von einigen abstrakten of the problem consists of a generalization of the classical
Ergebnissen der ,,Realisierungstheorie" heranziehen. So wird theory of elementary divisors [2], which is today best expressed
z. B. die bekannte Tatsache bewiesen, dajJ zwei beliebige in the language of modules [3]. This theory leads to a third
minimale Realisierungen isomorph sind. algorithm, namely the well-known one which exhibits thc
canonica! form under equivalence of a rectangular polynomial
1. lntroduction matrix [5, Volume I, Chapter VI].
We present here an entirely different method for computing
The basic task of engineering is to transiate performance
a realization, which was evolved from the study of the so-
specifications into a blueprint from which a system can be
called Markov parameters, well known in linear stability
built. We shall be concerned bere with a very special but
theory [6]. The following claims can be made for the new
important idealization of this problem, which is technically
method:
known as the problem of realization [1], [2], [3].
Suppose we are given an external description of a linear a) Only very simple numerica! operations are needed. In fact,
dynamical system, such as an impulse-response matrix or a this is probably the simplest method for computing a realiza-
frequency-response matrix. This data may be a performance tion that will ever be found.
specification or it may be the result of experimental measure- b) Theoretical properties of realizations can be easily proved
via this method. For instance, there is a very simple formula
1) This research was supported in part by the National Aeronautics and
Space Administration under NASA Grant NGR 05-020-073. (Proposition 2) for the dimension of a minimal realization.
The paper is based in part on a dissertation of B. L. Ho at Stanford c) Every realization of a given input/output relation can be
University.
obtained with this method.
2) Engineering Mechanics, Stanford University, Stanford, California,
and IBM ASDD Laboratory, San José, California. 4) The scalar case of this problem is qui te elementary and has long been
3) Engineering Mechanics and Electrical Engineering, Stanford Univer- known. It is not widely appreciated that the generai case is difficult as
sity, Stanford, California. well as important.
546 · Heft 12 (14. Jahrgang 1966) Regel ngstedznih
d) Given experimental input/output data, there is no need to functions Y or T [ W or U] construct con
fix a priori the dimension ( = number of state variables) of the G, H] [or F, G, H] such that the identities given above holct'.
system. lf the dimension is finite, the method will give an exact It is implicitly assumed that we are interested only in fìnite-
realization with the right dimension. dimensional realizations, i.e., where F is an n x n matrix. We
make an important preliminary observation:
The main limitation of the method is that it is of course
restricted to linear systems. Indeed, we know almost nothing Proposition O
today about the nonlinear realization problem.
The problern stated above is equivalent in all f
Our method is so much simpler than its predecessors that we
following: Given a sequence of p X m constant matrices
expect to see many new applications. There are also many
ｻｙｫｽｯｾ＠ k<oo, find a triple {F, G, H} of constant matrices such
well-known problems where conventional techniques of data
that -
analysis have been cumbersome and expensive; here the new
method could bring about major improvements. The follow- (2.3)
ing areas seem to us to be especially promising:
Proof
a) Experimental determination of process dynamics.
a) Relation (2.3) is obviously the same as the first realization
b) Study of the human operator in a control loop.
problern if we set <J> = F and t = k.
c) Biologica! model building.
b) Given the function T, a realization exists only if T is a
d) Conversion of time-domain data to the frequency-domain
meromorphic function which is O at oo. Hence in a suitable
and vice versa.
neighborhood of oo T has a power-series representation
It is only fair to warn the reader that our present results relate
only to the noise-free case. We are well aware that the noisy T ( z) = I0 0 yk - 1 z -k.
case is the really important one. Research on this aspect of
k=l
the problem is continuing. Let us point out also that according
to research experience [7] major advances in statistica! system It is clear that T (z) = H (zl - <J>)-1G if and only if (2.3)
theory depend mainly on a better understanding of linear holds.
deterministic systems. This paper is directed to the latter c) lf Wis given, a realization exists only if Wis a real analytic
problem. function; more precisely, if W has the power series
w I0 0
2. Linear dynamical systems (t)= Yktk/k !
k= o
As is well known (see especially [3]), a finite-dimensionai,
valid for ali t; a realization exists if and only if (2.3) holds.
discrete-time, linear, constant dynamical system has an
internal description given by the state-variable (difference) d) We bave a realization only if U is a meromorphic function
equations and U ( oo) = O, just as in case b). For the same reasons, (2.3)
is necessary and sufficient for a realization.
x(t+l)=<Px(t)+Gu(t), y(t)=Hx(t)
In short, for different reasons in each case, an equivalent
where t = integer and x, u, y are real vectors. The dimension external description of a finite-dimensiona!, linear, constant
of x is the dimension of the system; u and y have dimension dynamical system (be it discrete-time or continuous-time) is
m and p respectively. given by the sequence {Yk}. So the realization problem is
expressible concisely by (2.3). In view of certain classica!
Similarly, the internal desçription of a finite-dimensiona!,
results [5, Volume Il, last chapter; 6], we call {Yk} the
continuous-time, linear, constant dynamical system is given
Markov parameters of the system.
by the state-variable (differential) equations
We say the realization is minimal if the dimension of F
dx/dt = Fx + Gu (t), y (t) = H (t) x (t). satisfying (2.3) is as small as possible [1-3]. The dimension
The external description of a linear system may be stated of the minimal realizations is denoted by n0• lf needed, we
write F = Fmin to indicate the fact that Fbelongs to a minimal
either in the time domain or in the transform domain. See [2].
realization.
For the system (2.1), the time-domain description is given by
An important fact in realization theory [l-3] is the theorem
the (pulse-response) function
that all minimal realizations are equivalent modulo the
t-t Yc= H<PtG Ｈｴ］ｩｮｴ･ｧ･ｲｾｏＩ［＠ choice of basis used to represent the abstract vector x. This
can be proved directly and easily by the present method
while the transform domain-description is given by the (Proposition 3).
(z-transform) transfer function
z-t T (z) = H (zl - <P) - l G (z = complex number) ｾ ｬ＠ (2.1 r) 3. Construction of a minimal realization
1
For the system (2.2), we have analogously the impulse- We shall now describe the new algorithm for the construction
response function of a minimal realization. We assume implicitly that we know
the degree r of an annihilating polynomial of Fmin· For
t-t W (t)=HeFtG (t=real ｮｵｭ｢･ｲｾｏＩ＠ instance, if we are given a transfer-function matrix in rational
form, then r ｾ＠ q, the degree of the least common denomina-
and the (Laplace-transform) transfer function
tor ( = degree of minimal polynomial of Fmin [2]). If the
s-tU(s)=H(sl-F)-1 G (s=complex number). (2.2r) input/output description is given as experimental data, a
1 special procedure is necessary to determine r; this is discussed
The problem of realization is then the following: Given the in the next section.
Regelungstedmih Heft 12 (14. Jahrgang 1966) · 547
Proposition 1 Y1c = HFlcG,
The sequence { Ylc} has a finite-dimensionai realization if and Iq
O=HFJ11J(F)G= fJiHp-i+Jc,
only if there is an integer r and constants LX0, a1, •.• , <Xr--1 ì=O
such that
r so that (3.1) holds with r = q and cxi = -(k In fact, q just
Y,.+i= I aiY,.-i+i for ｡ｬｬｪｾｏＮ＠ (3.1) defined is the minimum value of r for which (3. I) holds.
i= 1 Sufficiency of (3.1) follows from Theorem 1.
The algorithm begins by forming the r x r block matrix
Proof of Proposition 2
("generalized Hankel matrix") built out of the Markov
parameters Take a minimal realization {F, G, H}. Write S = V'W
(' = transpose), where
Yo Y1 . . . Y,. _ 1 ] V=[H' F'H' ... (F'y-1 H'J,
Sr_- Y.1 Y·2 .. · Y,. -_ [ yi +J-2 J W = [G FG ... F'"-1G].
[
Y, _ Yr ·· ·
1 Y2 ,. - 2 Since Fis n0 x n0 it is clear that rank ｖｾ＠ n0 and rank W ｾ＠ n0•
oSifm thilea rtlrya,n wslea twedri tsee 1q:ulSern fcoer {tYhlec +rl }X trh balto ics,k matrix formed out Htthheeen rmec eei nxaiilmsstosa lraiat yrne koa lfSi zn a0ｾ＠t. ionn0• oIff draimnken Ss io<n n<0, nth0•e nT hbiys cTohnetoraredmic t1s
(3.2) Proof of Theorem 1
Proposition 2 Form the "block companion matrix"
If Ylc has a finite-dimensiona! realization, then op IP op op
n =rank S,.. (3.3) op op IP op
0
M=
Theorem 1
op op op op
The following algorithm provides a minimal realization of
arip ar - IP ar - IP ... a IP
{ Ylc} whenever a finite-dimensionai realization exists: i 2 1
a) Form Sr, where r satisfies (3.1). Observe that, by (3.1),
b) Find nonsingular matrices P, Q (which may be taken to be tkS=MkS, k=O, 1, .... (3.7)
lower and upper triangular) such that
Similarly, there is an r x r block companion matrix N (given
ｐｓＬｑ］｛ｾＧ＠ ｾｊ］ｉ］ｶ［ｵＬＮ＠
by the transpose of the block pattern of M but with mx m
(3.4)
blocks) such that
(1 is an s x s unit matrix, s = rank Sr, and J is idempotent, tkS=SNk, k=O, 1, .... (3.8)
8
Us = [18 O]. Note that this is a standard computing problem It is easy to verify that
in linear algebra.)
e) Let Ep be the block matrix [/p, Op, ... , Op]. Then Yk =EP (-rkS) E; ］ｅｐｍ ｫｓｅｾ ＱＬ＠ ｫｾｏＬ＠ (3.9)
11
F= Vs [JP(tSr) QJ] ｕｾＬ＠ so that {M , SEm', Ep} is a realization. In general, it is non-
minimal. It must be therefore reduced so that its dimension
G= Us ｛ｊｐｓｲｅｾ Ｑ｝Ｌ＠ (3.5)
becomes equal to rank S. How this is done is difficult to
H=[EpSrQJ] ｕｾ＠ motivate but easy to demonstrate.
Write S# = QJP. Then S# is a pseudo-inverse of S, that is,
is a realization of {Y7c}. (Proposition 2 shows that this
S = SS# S. With the aid of this identity and the fact that
realization is minimal.)
MkS = SNk (see (3. 7-8)), we rearrange (3.9) successively
Proposition 3 as follows
Any two minimal realizations {Fi, Gi, Hi}, i= 1, 2, of the Y,,=EPMksE:i
1,
same { Yk} are isomorphic; that is, there exists a nonsingular
］ｅｐｓｎｫｅｾＬ＠
matrix T such that
=E SS#SNkE'
p m'
=E ss#MkSE'
p m'
in fact, T is given explicitly by (V;:. V2')- 1 V2 V1' (notation = EPSS # MkSS ＪｓｅｾＧ＠
explained below).
=(EPSQJ)(JPMkSQJ)(JPSE;
Proposition 4 11),
］ＨｅｐｓｑｊＩＨｊｐｍｓｑｊＩｫＨｊｐｓｅｾ ＱＩＮ＠
Every minimal realization may be obtained by suitably
choosing P and Q in the algorithm described under theorem 1. Hence by (3.7) {JP (r:S)QJ, JPSEm', EpSQJ} is a realization.
We shall now present the main ideas of the proofs. A detailed The O blocks in these matrices may be dropped without
demonstration may be found in [8]. affecting the fundamental relation (2.3). Then we bave (3.5).
For simplicity, we assume that r is suitably fixed and drop it Proof of Proposition 3
as a subscript. Each of the four matrices {Vi, Wi}, i= 1, 2, associated with
Necessity of (3.1). Let lJJ (z) = (J0zq + f31zq-l + ... + {Jq, the two minimal realizations have n0 rows, and each of them
{30 = 1, be the minimal polynomial of some Fmin· Then, if has rank n0• So Vi Vi' and Wi W/ are all nonsingular. By the
1•
548 · Heft 12 (14. Jahrgang 1966) Regelungstedznik
definition of a realization V1' W1 = S = V2' W2• Multiplying is satisfied for all O ｾ＠ k ｾ＠ 2 r, that is, for every elernent of
on the left by V2 v.;e get TW1 = W2, with T = ( V2 V2')- 1 V2 V1'. the matrix Sr+i·
T is nonsingular because if U = W1W2' (W2W2')- 1, then Corol/ary
TU = I. Thus V1'U = Vi'T-1 = V2'. But rS = V1'F1W1
= V21TF1T-1W2 = V2'F2W2,whichimpliesthatF2 = TF1T-1• If rank St = constant for all t ｾ＠ t0, then (3.5) with r t 0
The remaining relations may be easily verified. provides a realization of { Y k}.
Proof of Proposition 4 Tbe proofs of these results require subtle rnodifications
proof of Theorern 1 and may be found in [8].
We must exhibit a pair P2, Q2 such that a given minimal
realization {F2, G2, H.J is expressed by formulas (3.5). First We have now a practical procedure for cornputing a realiza-
we take any P = P and Q = Q for which (3.4) holds and tion which does not require looking at infinitely many Yk' s:
1 1
construct the corresponding realization {F1, G1, H1}. Then We examine in succession the rank of S1, S2, ... until rank
we let T = (V2 V2')-1 V2 V1' and Sr = rank Sr+i ·
ｐ Ｑ ］｛ｾ＠ ｾ｝＠ ｛ｾ Ｍ ｩ＠ ｾ｝＠ Then we compute {F, G, H} by (3.5) and set Yk = HFkG.
P., Q2=Q1 The difference sequence formed by the Yk = ｾ＠ - Yk will
+
have O's for at least its first 2 r 1 elements. In view of
Relation (3.4) is preserved. By Proposition 3 we see that the formula (2.1 tr) , if the { Yk} are not identically zero and hav e a
realization corresponding to P2, Q2 is {F2, G2, H2}. finite-dimensionai realization, tbey correspond to a transfer
Two points deserve special emphasis: function rnatrix whicb has sP as a factor in its least common
denominator, with p ｾ＠ 2 r + 1. This observation allows us
1. The intermediate step of constructing the block canonica!
matrix M is a theoretical trick. It can be carried out only if to dispense with the O terms in the { Yk} sequence and consider
the coefficients of an annihilating polynomial are known. As only sequences in which Y0 =F O. By repeating this procedure,
a rule, they cannot be determined directly from { Yk}. If the we eventually obtain a realization if a finite-dimensiona!
initial data is given in rational transfer-function form, then realization exists at ali. Note that it may be possible to extract
the desired polynomial is simply the least common denomi- a finite-dimensiona! part of a sequence which has no finite-
nator. dirnensional realization.
2. The formulas for the minimal realization no longer involve With respect to the corollary, it is interesting to observe that
M because of the relation MS = rS. (rS is known of course if r is the least number such that rank St = constant for t > r,
from {Yk}.) Thus the only piece of information required in then r may be strictly less than the degree q of the minimal
addition to a suitable number of Yk' s is the integer r. This is polynomial of F. For example, take F = n x n diagonal matrix,
an important aspect of the new algorithm. Trying to find the with distinct nonzero diagonal elements, G = In, and H = In.
rceoanlsiztaantitosn < X0w, o.•u. l, d<X ri-n1v, oclovnes tmruuccthin gla Mbo,r awndh itchhe nis readvuocidinegd tbhye sT]h e=n n{ Fa,n Gd , hHen}c er era l=iz els, {wFhki}le mq in=i mna. lly. We see that rank
applying formulas (3.5) directJy. In fact, it is not even neces-
sary to hav e a recurrence relati on such as (3 .1) ( see Theorem 2). Refe rences :
We have here a strong plausibility argument that the algo-
rithm given by Theorem I is maximally efficient. [l] Kaiman, R. E.: Mathematical description of linear dynamical
systems. J. Control (SIAM), 1 (1963), 152-192.
[2] Kaiman, R. E.: Irreducible realizations and the degree of a
4. Determination of r
rational matrix. SIAM J., 13 (1965), 520-544.
[3] Kaiman, R. E.: Algebraic structure of linear dynamical systems.
The procedure described in the preceding section requires
I. The module of ｾＮ＠ Proc. Nat. Acad. Sci. (USA), 53 (1965),
only one abstract assumption, namely knowledge of the
1503-1508.
integer r in the recurrence relation (3.1). How can r be [4] Gilbert, E. G.: Controllability and observability in multivariable
determined? systems. J. Control (SIAM), 1 (1963), 128- 151.
If { Yk} has no finite-dimensiona! realization, then we are [5] Gantmakher, F. R.: Tbe theory of matrices (book). Chelsea,
faced with an approximation problem. We bave no significant 1959.
[6] Jarominek, W.: First and Second IFAC Congress, 1960 and
results as yet in this area.
1963.
If { Yk} has a finite-dimensiona! realization, the following [7] Kaiman, R. E., and Bucy, R. S.: New results in linear filtering
results provi de a method for effectively choosing r: and prediction theory. J. Basic Engr. (Trans. ASME), 83 D
(1961), 95-108.
Theorem 2
[8] Ho, B. L., and Kaiman, R. E.: Markov parameters, the moment
Suppose that rank Sr = rank Sr+1 for some integer r. Then problem, and minimal realizations. J. Contro! (SIAM), 5
if {F, G, H} are given by (3.5) the fundamental relation (2.3) (1967), (to appear).
