Space System Identification. Marc Moonen and JosC Ramos. Abstract-In an earlier paper ([SI), an algorithm has been introduced for identifying multivariable ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 1I, NOVEMBER 1993 M. E. Salukvadze, Vector-Valued Optimization Problems in Control Theory. New York: Academic, 1979. P. M. Makila, “On multiple criteria stationary linear quadratic control,” IEEE Trans. Automat. Contr., vol. 34, pp. 1311-1313, 1989. D. Li and Y. Y. Haimes, “Decomposition technique in multiobjective discrete-time dynamic problems,” in Control and Dynamic Systems, vol. 28, C. T. Leondes, Ed. San Diego, CA: Academic, 1988, pp. 109-180. D. Li, “Multiple objectives and nonseparability in stochastic dynamic programming,” Int. J . Syst. Sci., vol. 21, pp. 933-950, 1990. P. P. Khargonekar and M. A. Rotea, “Multiple objective optimal control of linear systems: The quadratic norm case,” IEEE Trans. Automat. Contr., vol. 36, pp. 14-24, 1991. J. Medanic and M. Andjelic, “On a class of differential games without saddle-point solutions,” J . Optimiz. Appl., vol. 8, pp. 413-430, 1971. J. Medanic and M. Andjelic, “Minimax solution of the multipletarget problem,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 597-604, 1972. J. Medanic, “Minimax Pareto optimal solutions with application to linear-quadratic problems,” in Multicriteria Decision making, G. Leitmann and A. Marzollo, Eds. New York: Springer-Verlag, 1975, pp. 55-124. D. Li, “On the minimax solution of multiple linear-quadratic problems,” IEEE Trans. Automat. Contr., vol. 35, pp. 1153-1156, 1990. -, “A new solution approach to Salukvadze’s problem,” in Proc. 1990 Amen. Contr. Conf., San Diego, CA, May 1990, pp. 409-4 14. N. T. Koussoulas and C. T. Leondes, “The multiple linear quadratic Gaussian problem,” Int. J . Contr., vol. 43, pp. 337-349, 1986. H. T. Toivonen, “A multiobjective linear quadratic Gaussian control problem,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 279-280, 1984. V. Chankong and Y. Y. Haimes, Multiobjectice Decision Making: Theory and Methodology. New York North-Holland, 1983. D. Li and Y. Y. Haimes, “Extension of dynamic programming to nonseparable problems,” Computer Math. Appl., vol. 21, pp. 51-56, 1991. P. L. Yu, “Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives,” J . Optimiz. Theory Appl., vol. 14, pp. 319-377, 1974. D. G. Luenberger, Introduction to Linear and Nonlinear Programming. Reading, MA: Addison-Wesley, 1973. R. W. Reid and S. J. Citron, “On noninferior performance index vector,” J . Optimiz. Theory App/., vol. 7, pp. 11-28, 1971. L. S. Lasdon, Optimization Theory for Large Systems. London: Macmillan Company, 1970. Y. C. Ho, “Comments on a paper by J. Medanic and M. Andjelic,“ J . Optimiz. Theory Appl., vol. 10, pp. 187-189, 1972.
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note, we extend this algorithm by incorporating a balancing step, such that the identified model is always in balanced coordinates. With this modification, one obtains a data driven counterpart for Kung’s realization algorithm.
I. INTRODUCTION During the past decade, a great deal of attention has been given to state space models that have a so-called balanced structure [lo]. This is due to their fundamental importance in system identification [6], [ 2 ] , model reduction [7], [lo], minimum sensitivity analysis [5], and roundoff error quantization in digital filtering [ll]. Presently, a balanced representation can be obtained either from a sequence of Markov parameters (identification approach) or by transforming a given model to balanced coordinates (transformation approach). In the identification approach, it is unfortunate that such a rich theory relies on Markov parameters as a starting point, something rather difficult to measure in practice. If, for instance, the only information available from a system is an input-output record, one would have to apply a deconvolution technique to obtain the impulse response data, then apply Kung’s algorithm [6] to obtain a balanced model. However, the process of extracting a finite impulse response sequence from input-output data is not exact and may introduce additional noise to the identification process. A recent state-space identification algorithm has been introduced in [8], [9], which uses measurable input-output data directly. This algorithm is not directly suited for model reduction because the states of the model are not ordered with respect to a given measure or criterion, i.e., observability and controllability. Therefore, it can only be compared to other balanced realization algorithms if an additional balancing step is included. In this note, we add a balancing procedure up front, such that a balanced state space model is obtained directly. The balancing procedure does not add any computational complexity compared to the original algorithm of [81, [9]. In Section 11, we briefly outline the algorithm of [8], [9], using the same notation for convenience. In Section 111, the balancing step is then added, which resembles a principal component extraction as suggested in [lo], [13], [l]. The resulting algorithm is summarized in Section IV.
IDENTIFICATIONALGORITHM OF [8], [91 11. STATESPACE Consider the linear, discrete, time invariant, multivariable system with state space representation
A Subspace Algorithm for Balanced State Space System Identification Marc Moonen and JosC Ramos Abstract-In an earlier paper ([SI), an algorithm has been introduced for identifying multivariable linear systems directly from input-output data. This new algorithm falls in the class of so-called subspace methods and resembles impulse response based realization algorithms. In this
Manuscript received March 20, 1991; revised June 10, 1992. This work was carried out at the ESAT laboratorium of the Katholieke Universiteit Leuven, in the framework of the Community Research programme ESPIRIT Basic Research Action Nr. 3280, and was partially sponsored by the European Commission. The authors are with ESAT, Katholieke Universiteit Leuven, K. Mercierlaan 94, 3001 Heverlee, Belgium. IEEE Log Number 9212877.
where uk E 9im, y, E %‘,and x k E !H” denote, respectively, the input, output, and state vector at time k . The dimension of x k is the minimal system order, n . Furthermore, A , B, C, and D are unknown system matrices, to be identified-up to a similarity transformation-by means of recorded I/O-sequences. We define input-output matrices as follows
0018-9286/93$03.00 0 1993 IEEE
IEEE TRANSACI-IONS ON AUTOMATIC CONTROL, VOL. 38, NO. 11, NOVEMBER 1993
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where XI
with
u1
...
U,
=
[
4 + 1
...
u,+2
y+3
U1 +/
...
...
U2I+ 1
U21
HT =
1
CAI-zB
where
{XJ
=
span,,
(3)
x,
=
[x,+1
XI
(4)
... ...
x:."]
=
[A'
C'
Yl+l-~
-rtHT.Ul
"'1.
... U ~ + I
X, = -TtHT.U2
+ r'.Y2
r
with T t the n x li pseudo-inverse of r, s.t. rt . = In.Also, by premultiplying (10) by L, the orthogonal complement of s.t. = O l i - n x n , one obtains
r
.r
(5)
X,+,l.
D'
+ rt.Yl
O = -TiHT,Ul +rL.Y,
Thus, any basis for the above intersection can be chosen as a valid state vector sequence Xh, with X ; = TX, for some nonsingular transformation. Once X i is known, the corresponding system matrices A ' = TAT-', B' = TB, C' = CT-' and D' = D are identified by solving
Y,+ I
=
(12)
as follows:
.'.
XI+,
CA'-
From this
U21 +I - I
{HI}n span,, {H2J
X, is a state vector sequences
... D
CAt-3B
(11)
( j B i), and Yl, Y2similarly defined. It is shown in [8] that under certain mild conditions span,,
[ x l x 2 ... x , ] , and H , is a block Toeplitz matrix
with the Markov parameters
.'. ...
U2
=
...
r,
O = -TLHT.U2+ri.Y2. (13)
The derivation below works with the orthogonal complement part in the decomposition (7), which is rewritten as follows:
[U;
1 U&]
=
[TI
I T2 I T3 1 G I .
(14)
The partitioning now corresponds to the input-output parts in
H I and H 2 , respectively. Explicit formulas for the T's can be x:+]-2
derived as follows. First, we know that
U1,I-z
(6) In the original algorithm, the intersection of the row spaces of
H , and H2 is recovered from the SVD of the concatenation:
so that
[ T3
--
2mr
From this,
+ n 211 - n computed as
----,--- (7) 2mr
+n
211 - n
where 7is a certain n X 2fi - n reduction matrix, see [8]. In the sequel, we introduce an alternative computational scheme for S, such that the obtained state vector sequence X; is known beforehand to correspond to a balanced realization, where both the controllability and observability gramians are diagonalized, i.e.,
A ' . A'T = r".
r' = diagonal matrix
[4
- = d . X , = d .[ - r t H ,
I r'l] ;. [
for some 21i - n x n matrix d.From this, together with (131, it follows that
[ T3
Xi may be
1 T4] .
1 T4] =d.[ - r t H ,
I
[
I.']+ B . - r L H T
I r L ](15)
where 9 is an arbitrary 2Zi - n X Zi - n matrix. A similar formula can be derived for TI and T2. First, it is easily verified that
X2=A'.X,
+ A,.U,
(16)
where A R = [ A ' - ' B ... A B B ] is a reversed controllability matrix. By substituting this in
(9) and by making use of (12) and (131, one derives
[ A'
=
[ B ' A'B'
...
1
T2]
=&'. [ A ' T t H T - AR
1 -Airr]
A"-'B'].
111. BALANCED REALIZATION STEP
To fix ideas, we first need the input-output equations from [SI
where E? is again an arbitrary 21i - n X li - n matrix. The key point now is that a block Toeplitz matrix with Markov parameters can be computed as follows
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 11, NOVEMBER 1993
This block Toeplitz matrix is of course important for the balancing step, see below. Equation (18) follows from the formulas for the T’s. It is seen that
-. HT
T4
211 - n x 11
=
3 ) Compute the state vector sequence
-T3
x mi
/I
4) Compute the system matrices
so that
TJ.T,
=
-H,
T2.TJ.T,- TI = & . A R . Finally,
and therefore
If one would now compute the SVD of this matrix
z, --
T;.(T,.T:.T,-T,)=
U,,
.
iixn
. I/dr
(19)
nXmi
n x n
then, from [7], it is well known that a balanced realization is obtained with xb
=
T-X,
where the balancing transformation as given as
T
U .;
= x;1/2.
r.
One easily verifies that with the following choice for F
y= 2; it
1/2.
U T . TJ
is possible to compute Xi directly. Indeed,
1’
y,
T
\
x2 IV. SUMMARY The algorithm is finally summarized as follows:
1) Construct H , and H2 from input-output data, and compute the 21i - n dimensional orthogonal complement
[TI
I
T2
I
T3
2) Compute the SVD of
TJ . ( T 2 .TJ . T3 - T I )
T4] *
[4 =
0.
For additional algorithmic details we refer the reader to [SI, [9], where it is shown how an explicit computation of X i can be avoided, resulting in a considerable computational saving. The balancing procedure has been tested numerically and the results are seen to match those of Kung’s algorithm [6]. The above algorithm yields a balanced state space model, which is directly suitable for balanced model reduction. It should be noted that with Kung’s algorithm, the realization step and the model reduction step may be interchanged. The link with model approximation, which is by now clearly established for Kung’s algorithm [12], [3],[4],is still under investigation for the present algorithm. REFERENCES El] K. S. Arun, D. V. Bhaskar Rao, and S . Y. Kung, “A new predictive efficiency criterion for approximate stochastic realization,” in Proc. 22nd IEEE Con& Decision Contr., San Antonio, TX, 1983, pp.
1353-1355. [2] U. B. Desai, D. Pal, and R. Kirkpatrick, “A realization approach to stochastic model reduction,” Int. J. Contr., vol. 42, no. 4, pp. 821-838, 1985. [3] D. F. Enns, “Model reduction with balanced realization: An error bound and a frequency weighted generalization,” in h o c . 23rd IEEE Con& Decision Contr., Las Vegas, NV, 1984, pp. 127-132. [4] K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their L“ error bounds,” Int. J. Contr., vol. 39, no. 6, pp. 1115-1193, 1984. [5] S. W. Gray and E. I. Verriest, “On the sensitivity of generalized state-space systems,” in Proc. 28th IEEE Con$ Decision Contr., Tampa, FL, 1989. [6] S. Y. Kung, “A new identification and model reduction algorithm via singular value decomposition,” in Proc. 12th Asilomar Conf Circuits, Systems Computers, Pacific Grove, CA, 1978, pp. 705-714. [7] A. J. Laub, M. T. Heath, C. C. Paige, and R. C. Ward, “Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms,” IEEE Trans. Automat. Contr., vol. AC-32, no. 2, pp. 115-122, 1987. [8] M. Moonen, B. De Moor, L. Vandenberghe, and J. Vandewalle, “On- and off-line identification of linear state space models,” Int. J. Contr., vol. 49, no. 1, pp. 219-232, 1989. 191 M. Moonen and J. Vandewalle, “A QSVD approach to on- and off-line state space identification,” Int. J. Contr., vol. 51, no. 5, pp. 1133-1146, 1990. [lo] B. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Automat. Contr., vol. AC-26, no. 1, pp. 17-32, 1981. 1111 C. T. Mullis and R. A. Roberts, “Synthesis of minimum roundoff noise fixed point digital filters,” IEEE Trans. Circuits Syst., vol. CAS-23, no. 9, pp. 551-562, 1976. [12] L. M. Silverman and M. Bettayeb, “Optimal approximations of linear systems,” in Proc. Joint Automat. Contr. Conf, San Francisco, CA, 1980, pp. FA8-A. [13] R. J. Vaccaro, “Deterministic Balancing and Stochastic model reduction,” IEEE Trans. Automat. Contr., vol. AC-30, no. 9, pp. 921-923, 1981.
