State space identification of linear deterministic ...

WATER RESOURCES RESEARCH, VOL. 31, NO.6, PAGES 1519-1531, JUNE 1995

State space identification of linear deterministic rainfall-runoff models Jose Ramos/ Dirk Mallants, and Jan Feyen Institute for Land and Water Management, Katholi.eke Universiteit Leuven, Leuven, Belgium

Abstract. Rainfall· runoff models of the black box type abound in the water resources literature (i.e., transfer function, autoregressive moving average (ARMA), ARMAX, state space, etc.). The corresponding system identification algorithms for such models are known to be numerically efficient and accurate, leading in most cases to good parsimonious representations of the rainfall-runoff process. Alternatively, every model in transfer function, ARMA, and ARMAX form has an equivalent state space representation. However, state space models do not necessarily have simple system identification algorithms, unless the system matrices are restricted to some canonical form. Furthermore, state space system identification algorithms that work with the rainfall/runoff data directly (i.e., covariance free), require initial conditions and are inherently iterative and nonlinear. In this paper we present a state space system identification theory which overcomes these limitations. One advantage of such a theory is that the corresponding algorithms are highly robust to additive noise in the data. They are referred to as "subspace algorithms" due to their ability to separate the signal subspace from the noise subspace. The main advantages of the subspace algorithms are the automatic structure identification (system order), geometrical insights (notions of angle between subspaces), and the fact that they rely on robust numerical procedures (singular value decomposition). In this paper, two algorithms are presented. The first one is a two-step procedure, where the impulse response (unit hydro graph ordinates for the single-input, single-output case) are computed from the input/output data by solving a constrained deconvolution problem. These impulse response ordinates are then used as inputs for identifying the system matrices by means of a Hankel-based realization algorithm. The second approach uses the data directly to identify the system matrices, bypassing the deconvolution step. The algorithms are tested with real data from the Voer catchment in Belgium.

1. Introduction Modeling the rainfall-runoff process is a rather difficult problem due to its complex interaction with other hydrometeorological and geomorphological processes within the hydrologic cycle. The geophysical processes contributing to the hydrologic cycle are generally desc:dbed by nonlinear partial differential equations in time and space (i.e., mass and energy transfer). This gives rise to a number of possible alternatives for modeling the rainfall-runoff components of the hydrologic cycle [Chow, 1964]. On the one hand, physically based hydrologic simulation models, characterized by moisture-accounting elements from the various hydrologic units, are by far the most mathematically detailed descriptions of the surface and groundwater components of the hydrologic cycle. In general, these models can answer questions about the watershed before even physically altering it. Although the physical laws idealized for these models are well suited for small-scale laboratory experiments, they turn out to be of marginal value in the field scale because of the enormous heterogeneity in the data. Furthermore, modeling the space and time dependence of the rainfall-runoff 1

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Copyright 1995 by the American Geophysical Union. Paper number 95WR00234. 0043-1397/95/95WR-00234$05 .00

process may therefore be an insurmountable task, both because of u,ncertainty about the exact description of the processes and because certain properties of the rainfall-rpnoff process which are i'epresented in the model are not accessible to measurement. Hydrodynamic models, on the other hand, represent another class of models commonly used in hydrology.and hydraulics. To this class belong the St. Venant equatiOns; representing consetvation of mass and momentum in open charinels. In the case of surface runoff from natural catchments or flow in open channels, application of the hydrodynamic approach would require a detailed topographic map and_geometric arid hydraulic characteristics of the system, along with initial and boundary conditions. The difficulties in meeting these data requirements, the distributed .nature of the data, and nonlinearities of the processes have led to simplified models which fall in the realm of conceptual models. In the case of flood routing, the conceptual model is described by a lumped form of the cOn.setvation of mass equation, together with a simplified linear conceptual equation representing consetvation of momentum. These types of models have received a great deal of attention due to their compatibility with state space models and realtime forecasting capabilities [Wood and SzOllbsy-Nagy, 1980; Szollosy-Nagy, 1982; Bras and Rodriguez-Iturbe, 1985; Georgakakos et al., 1990]. Adding to the fact that a major hurdle in hydrologic analysis is the heterogeneity in the data, .and that physically based

1519

1520

RAMOS ET AL.: STATE SPACE RAINFALL-RUNOFF MODELS

equations are seldom met in practice, there are compelling arguments that only a handful of parameters cari be identified from rainfall-runoff records [Jakeman et al., 1989]. In view of these limitations, one can then take a simpler route and view the rainfall-runoff process from a transfer function pOint of view. That is, formulate the rainfall-runoff process in terms of an integral property of the Watershed, the unit hydrograph. Here all heterogeneities are lumped by dividing the catchment into smaller subcatchments with approximately uniform areal rainfall. We can ignore the interrial details of the physical mechanisms and treat the subcatchments as a lumped, black box system which converts niinfall excess into direct runoff. One should allow a separate mechanism for converting tOtal rainfall into rainfall exc~ss and total runoff into direct runoff. The total hydrograph is then obtained by routing the different hydrographs through the different tributaries [Chow, 1964]. The problem then is to ideil.tify the causal relationship between rainfall and runoff. This toristitutes a system identification problem, and its solution is the topic of this paper. In the black box domain, one can take a parametric or nonparametric approach. That is, one can postulate a causal, linear, tinie invariant (CLTI) inodel such as

(I) where uk E ffi:', Yk E ffi:"', and ekE ffi"' are, respectively, the input, output, and innovations vectors at time k, .sd(z- 1 ), OO(z- 1), ~(z- 1 ), and 2il(z- 1) are polynomial operators, defined as

.sll(z-')

~

.silo+ .sll,z-1 + .sll,z-2 +

i'M(z- 1)

~

00 0 + OO,z- 1 + oo,z- 2 +

~(z-') ~ ~o

2il(z-')

~

+ ~,z-1 + ~,z-2 +

2ilo + 2il,z-' + 2il,z-' +

and z- 1 is the backward shift operator, that is, z- 1Yk = Yk- 1 • The matrices of .sll(z- 1), ~(z- 1 ), and 2il(z- 1) are (I X I) and the matrices of OO(z- 1 ) are(! X m). The model (1) can be summarized as

(6) 9l 1,

1

where xk E ffin, uk E m:m, Yk E ek E 9l denote, respectively, the state, input, output, and innovations vectors at time k, and [A, B, C, D, E] are parameter inati-ices of appropriate dimensions. These two families of models have been used extensively in hydrology [Cooper and Wood, 1982; Szollosy-Nagy, 1982; Bras and Rodriguez-Iturbe 1985· Delleur 1991] and are good approximations for modeli~g mo~t naturai phenomena, including the rainfall-runoff process. However, the nonlinearities associated with infiltration and subsurface flow processes limit the validity of the above models. Thus it is necessarY to subtract all abstractions from the data and work with effective rainfall/direct runoff data. The problem in the parametric approach is to identify the set of parameter matrices, given a record of rainfall (input)/runo:ff (output) data {u 0 , Uv Uz, ···, uN_ 1 } and {y 0 ,y 1 ,y 2 , ••• •YN----: 1 }, respectively. The basis of impulse response-based models and, in general, the nonparametric or functional approach is to identify the impulse response from the measured rainfall-runoff data. For the single-input, single-output case (SISO), the impulse response corresponds to the unit hydrograph ordinates. This corresponds to a constrained deconvolution problem which for noisy data is not well behaved in practice. Under the assumption of the system's being initially at rest, the nonparametric approach assumes a CLTI relationship between rainfall and runoff, that is,

(7) which is the convolution of the input with the impulse response {hP};=o· For stable and damped systems, which is the case with most hydrologic systems, we can assume that hP < 8, for p > M - 1, where 8 is usually a small number and M is the memory of the system. Furthermore, if one iffiposes the physical constraints for conservativeness and cOpositivity [Boneh and Golan, 1979], we have the following system of equations: M-1

Yk =

2:

hpuk-p

k

=

0, 1, · · ·, N- 1

(8)

P'''o

(2) M-J

and is known in the literature as the Box-Jenkins _(BJ) model [Box and Jenkins, 1976]. An important special case is when H(z- 1) = 11 (11 is an l X l identity matrix), in which case it is known as an output error (OE) model [Jakeman et at., 1989, Ljung and Glad, 1994]; Furthermore, a common variant is to let ~(z- 1 ) ~ .sll(z- 1), in which case (I) becomes

(3) and is known as an _autoregressive moving average with exogenous input (ARMAX) model [Cooper and Wood, 1982]. Finally, we have a special case when ~(z- 1 ) .sll(z- 1) and 1 2il(z- ) ~It> that is,

( 4) which is known as an ARX model. Alternatively, every model of the general form (2) can be represented as a linear state space model such as

(5)

2j

hP ~ I

(9)

p=O

p

~

0, I, · · · , M- I

(10)

The problem of unit hydrograph identification basically amounts to solving the above system of equations. We should point out that parametric models usually lead to the same input/output relation (8), with the exception that the {hP} sequence is related to the model parameters; thus there is no need to directly identify the impulse response as in the nonparametric approach. This is the reason for making the distinction between the parametric and the nonparametric approach. We will come back to this point in a later section. In this paper we will approach the rainfall-runoff model identification problem from a deterministic state space point of view (i.e., ek = 0). We show that the results obtained from the nonparametric approach can be used as inputs to the parametric approach. We further introduce a direct parametric approach, which from a computational point of view iS robust and

' I


efficient. We should point out that correlated input/output data n be modeled within the same framework, but such a model ~:slightly more computationally involved [Moonen and Vande1990]. In section 2 we review some system theory concepts from a ealization theory point of view. In section 3 we present a , ~a-step state space identification procedure. The first step is the constrained deconvolution problem, which we formulate via several optimization techniques. The second step is the realization of the impulse response via a state space model. In Section 4 we introduce a direct subspace identification alga·. ·""~ along the lines of De Moor [1988], Moonen et al. [1989], and Moonen and Ramos [1993], bypassing the deconvolution step. In section 5 we compare the various algorithms, and we draw some conclusions in section 6.

Realization Theory for Linear Dynainic:al Systems Consider the linear, discrete, time invariant, multivariable system with state space representation

Let us now suppose the initial state in the remote. past (i.e., k = -r) is zero, that is, x_r = 0 and that from time k = 0 onward the inputs are also zero, i.e., u[ 011 _ 11 = 0, then from (13) and (14) we have Xo

or (17) = ;1£;,,U[-lf-r]

Thus from (15) and (16) one can see that the initial statex 0 is an interface between past inputs and future outputs [De Moor, 1988; Moonen et al., 1989]. In linear systems theory, the matrix ~'·' is called a block Hankel matrix (a Hankel matrix has constant elements along the opposite or southwest diagonals) and is given by

xk+r = A'xk

+ ~,U[k+r-tfkJ

(13)

CA 2B

CAB CA 2B CA 3B

CA 2B CA 3B CA 4B

CA'- 1B

CA'- 2B

CA;- 3B

[ CAB C' 'iK

where uk E m:m, y k E !}t:i, and xk E m:n denote, respectively, the input, output, and state vectors at time k. The dimension of xk corresponds to the minimal system order (n ). Furthermore,A E ffinxn, B E ffinxm, C E ffilxn, and D E ffilxm are unknown system matrices to be identified (up to a similarity transformation) by means of recorded input/output sequences {u 0 , u 1 , ••• , uN_ 1 } and {y 0 , y 1 , ••• , YN- 1 }. Solving the state equation (11) recursively, starting from time k up to time k + r, we get

(15)

= Cf6,u[-1f-r]

(16)

(11)

(12)

1521

'·'

=

C'd': CA'B CA'+ 1B

CAiL-2B (18)

This matrix has many interesting properties related to the input/output behavior of linear systems [Kailath, 1980]. Furthermore, the singular value decomposition (SVD) (see Appendix A) of this matrix is perhaps the most robust quantitative measure of observability and controllability [Moore, 1981]. This is due to the following properties: rank {'iK,,,} = n rank {0,} =rank {'ll,} = n

where

u,+,-1] 'll, = [BjABj· · ·jA'- B] 1

U[k+r-!lk] =

[

Uk+r-2

~k

Now concatenating future outputs from time k up to time k + i - 1 and using (13), we arrive at [Kailath, 1980; De Moor, 1988]

+ HJU[kfk+i-ll

Y[kfk+t-tJ = f!J;xk

where

Y[kfk+i-tJ =

y,

Yk+t

[

:-

l

U[kfk+i-1]

=

Yk+t-1

u,

uk+l

[ :

(14)

l l

uk+i-1

O I

=

CA c

[ C~i-1

and H r is a block lower triangular Toeplitz matrix of Markov par_ameters (a Toeplitz matrix has constant elements along the mam southeast diagonals), i.e.,

gB Hr=

C~B

[CA;- 2B

D CB

D

CA;- 3B

CA;- 4B

for i, r ;;:::: n. Thus the rank of the Hankel matrix gives us the order of the system. Finally, the entries of the Hankel matrix correspond to the Markov parameters (unit hydrograph ordinates for SISO systems) of a linear system. That is, let the output response of the system be given by k

y, = CA'x 0 +

L;

+ Du,

CAP- 1Bu,_P

(19)

p=l

and for systems initially at rest, i.e., x 0

=

0, we then have

k

yk=

22

hpuk-p

(20)

p=O

where the Markov parameters are given by

pCZ1

p=O

(21)

From this last property one can see that knowledge of the unit hydrograph ordinates allows us to compute the system matrices [A, B, C, D] and vice versa. This can be done by factoring the Hankel matrix using the SVD (see Appendix A).

1522

3.


Indirect System Identification Algorithms

3.1. Constrained Deconvolution Step

In the previous section we showed the connection between unit hydrograph ordinates and the Markov parameters of the system. In order to identify the system matrices [A. B, C, D] via a Hankel matrix approach, we require the unit hydrograph ordinates as inputs. This constitutes a two-step procedure, the first step being the constrained deconvolution problem. We now present several approaches for solving this problem. Here we will assume u k andy k to be scalars, but in the second step we shall make no distinction with their vector dimensions originally introduced in the previous section. The reason is to allow {hP} E m-txm to be general enough for problems in other areas where a matrix impulse response needs to be identified (i.e., multi-input, multioutput systems). In such cases the deconvolution step may have a different form, in which case one should adjust the first step of the algorithm and continue with the second step accordingly. Whenever possible we shall remind the reader of the dimensions so as to avoid confusion. First, let us define the following matrices which will be common to all the algorithms:

l l l h ~ [ h, ho

y ~ [ Yo ~1

h~-1

YN-1

Uo u, Ur~

e

b~m g

~ [~]

where the errors are given by e = a - {3 and 0N is a 1 X N zero row vector. The constrained deconvolution problem is now formulated as the following linear programming (LP) problem:

subject to Kg~

e,

e~-I

uo u,

Uo

U(M-1)

U(M-2)

U(M-3)

Uo

UM

U(M-l)

U(M-2)

u,

U(N-1)

ucN-2)

ucN-3)

U(N-M)

b

(23)

A comparative study between (22) and (23) is reported by Wang and Yu [1986]. 3.1.3. Total least squares. If one relaxes the nonnegativity constraint, a total least squares (TLS) approach can be used for solving the augmented system of equations Fz

~

0

(24)

where

where e k is a scalar error term. Notice that for causal (i.e., nonanticipative) systems, u~i = 0, i ;:::.: 1; therefore Ur corresponds to a Toeplitz matrix. 3.1.1. Quadratic programming. The constrained deconvolution problem can now be stated as the following quadratic programming (QP) problem: Minimize (y- Urh)T(y- Urh)

subject to (22) where 1rM ~

1rM

= [ eo.

u,

ere~

linear one by introducing extra vector variables a and {3 [Hino, 1986; Wang and Yu, 1986]. First, let us define the following vectors and matrices:

F~[ ~] 1 1rM

z

_\]

TLS makes use of the SVD of the coefficient matrix F and is a numerically robust procedure [Van Huffel and Vandewalle, 1991]. However, there exists the possibility of obtaining negative ordinates in the unit hydrograph. Imposing nonnegativity constraints in the 1LS solution leads to a parametric generalized linear complementarity problem [De Moor, 1990]. However, it is not known whether this will lead to any improvement in the solution compared to (22) and (23). Thus it may not warrant the extra computations. Let the SVD ofF (see Appendix A) be given by

[1 1 • •" 1]

(with M elements). The solution can be obtained by any commercially available package. We particularly suggest the optimization toolbox from MATLAB (MATLAB is a registered trademark of the Math Works, Inc.), since it can be used in conjunction with the signal processing and control toolboxes already available. 3.1.2. Linear programming. A more conservative approach is to use linear programming. Here, we are interested in minimizing the sum of the absolute values of the errors, subject to a set of linear constraints. However, it has been shown that this nonlinear problem can be transformed into a

~[

F~ Up"2;pVJ

where UF E ffi(N+l)x(N+l}, lF E ffi(N+ 1 )x(M+l}, and Vp E m