Published in Soil Sci. Soc. Am. J. 68:1437-1444 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-8—NUTRIENT MANAGEMENT & SOIL & PLANT ANALYSIS
State-Space Modeling to Simplify Soil Phosphorus Fractionation
Xiufu Shuai and
Russell S. Yost*
3190 Maile Way, Room 102, Dep. of Tropical Plant and Soil Sci., Univ. of Hawaii, Honolulu, HI 96822
* Corresponding author (rsyost{at}hawaii.edu).
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ABSTRACT
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Soil P can be fractionated by methods such as the Hedley extraction method to help identify and quantify soil P. Measuring the increases in soil P fractions from P fertilizer does not indicate whether external P can control a P fraction because of transfers among P fractions. Similarly, correlations between soil P fractions and plant P uptake do not directly indicate whether a P fraction affects plant P uptake. The objectives of this research were to describe the transformation among fractions, the effect of P fertilizer on P fractions, and the relationship between plant P uptake and P fractions by a state-space modeling approach, and to minimize the dimension of the state-space model by introducing concepts of controllability and observability and by implementing the General Decomposition Theorem. A P fraction was said to be controllable if its size can be controlled directly by the amount of external P fertilizer, or indirectly through transformations among P fractions. A P fraction was considered observable if it contributed to plant P uptake directly, or indirectly through transformations among P fractions. The jointly controllable and observable P fractions identified by the General Decomposition Theorem were minimum to describe the measured soil P dynamics in selected soil-plant systems, while other fractions were redundant in the cases described. Results showed that only strip-P, inorganic NaHCO3–Pi, organic NaHCO3–Po, and inorganic NaOH-Pi of the seven fractions were necessary and sufficient to describe the P dynamics in example Mollisol, Vertisol, Ultisol, and Oxisol.
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INTRODUCTION
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SOIL P is dynamically affected by chemical, physical, and biological processes (Russell, 1977; Stevenson, 1986). Soil P status is frequently studied by sequential extraction methods (Chang and Jackson, 1957; Hedley et al., 1982). The Hedley approach, for example, has been widely used to separate soil P into eight fractions of varying availability for plants (Tiessen and Moir, 1993; Schmidt et al., 1996), and to characterize their transformation (Tiessen et al., 1984; Sharpley et al., 1987; Agbenin and Tiessen, 1994; Cross and Schlesinger, 1995). It is reasonable to assume that the P fractions of the Hedley approach are dynamic and transform among each other. Guo and Yost (1998) compared the dynamics of an individual soil P fraction and cumulative crop P uptake. Their results indicated that fractions in different soils differed in availability and that some fractions were related to plant availability in some soils but not in other soils. Linquist et al. (1997) studied the relationship between soybean P uptake and soil P fractions. In these studies, the transformations and dynamics among the P fractions could not be characterized. Path analysis (Turner and Stevens, 1959) was used to study the interactions among P pools (Tiessen et al., 1984; Beck and Sanchez, 1994). Path analysis, however, is not applicable for a dynamic system far from equilibrium because the path coefficients of the structural equations may vary with time and the relationship of cause and effect among P fractions may also vary with time. A methodology of quantifying the dynamic transformation of soil P fractions still remains an unsolved problem.
The sequential extraction of soil P is a relatively complicated, time-consuming, and expensive procedure. Some fractions have no contribution to crop growth and are not greatly influenced by added P fertilizer, and hence are redundant from the point of view of fertilizer recommendations. It is important, therefore, to identify the kind and minimum number of P fractions necessary to characterize plant–soil P dynamics and to determine whether the same fractions should be quantified in soils that vary in P sorption capacity.
The objectives of the research reported here were (i) model the dynamics of P fractions via a state-space approach, (ii) introduce the concepts of controllability and observability of P fractions, and (iii) identify the jointly controllable and observable P fractions that necessarily and sufficiently describe the dynamics of P status of seven selected soils.
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MATERIALS AND METHODS
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Soils and Greenhouse Experiments
Seven surface soils (0–15 cm) were used for the experiment, and their properties are listed in Table 1 (Guo and Yost, 1998). Four levels of P as Ca(H2PO4)2 were thoroughly mixed with each 5-kg soil portion to adjust the initial P conditions. Maize (Zea mays L.) was grown for 10 crops in the soils in a greenhouse. After each two crop harvests, the soils were sampled and seven soil P fractions were determined by fractionation including strip-P (p1), inorganic NaHCO3–Pi (p2), organic NaHCO3–Po (p3), inorganic NaOH-Pi (p4), organic NaOH-Po (p5), HCl-P (p6), and residual-P (p7) (Hedley et al., 1982; Guo and Yost, 1998). The crop P uptake of each harvest was measured, and the sums of each two harvests within two adjacent soil samplings, denoted as z, were used in the data analysis. The detailed procedures of soil treatment and greenhouse cropping can be found in Guo and Yost (1998). The original sampling sequences of p1 through p7 and z are shown in Fig. 1
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Fig. 1. The dynamic changes of seven P fractions (p1–p7, mg kg–1) and P uptake (z, mg) under different external P application (u, g kg–1 soil).
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State-Space Modeling, Observability, and Controllability
A state-space model is a set of first-order differential or difference equations (state equation) and an observation equation to describe the relationship between the input and output of a dynamic system. The model can be constructed on the basis of physical laws and relationships, rate laws of chemical reactions, or system identification (Ljung and Söderström, 1983; Juang, 1994; Ljung, 1999; Schoukens and Pintelon, 1991). Russell (1977) used two-dimension state-space models to describe and predict the transformation of available and unavailable soil nutrients. Mansell et al. (1977) used four-dimension state-space models to describe the mechanisms of phosphorus transfer among solution, adsorbed, immobilized (chemisorbed), and precipitated phases within a soil. On the basis of state-space modeling, there have already been abundant research results in the area of control system theory that could be widely applied in a thorough study related to soil dynamics. For example, the widely used Kalman-Filter can be used to predict the states and estimate the parameters (Ljung and Söderström, 1983) of the state-space models in spatial and temporal statistics of soils and their vegetation (Nielsen et al., 1994; Wendroth et al., 1999; Stevenson et al., 2001; Wendroth et al., 2001; Timm et al., 2003; Wendroth et al., 2003). One prospective application of system theory in soil science is to design the optimal strategy of fertilizing and irrigation aiming at final maximum of annual harvest/profit and minimum of environmental risk (Li and Yost, 2000).
State-space models are appropriate to describe the dynamic system of soil P fractions when the soil P fractions, the P fertilizer added, and the crop P uptake are regarded as the states, input, and output of the system, respectively. The aim of this study was to develop state-space models to describe the dynamic behavior of soil P fractions. If the states are not controllable and observable, which will be discussed later in this article, there is a flaw in the modeling of the system dynamics, and the Kalman Filter built on this system may not work. Thus, the state-space models need to be reduced to its minimum dimension besides the reasons in the introduction part of this article.
A discrete linear system with single input and signal output can be characterized by
 | [1] |
where k is the sampling sequence, A, b, c are n x n, n x 1, and 1 x n constant matrices, x(k) is n x 1 state vector composed of n state variables, and u(k) and y(k) are scalars termed the input and output. In terms of system theory, because of the soil sampling in discrete time, the discrete state-space model (Eq. [1]) is used to describe dynamics of P fraction in a soil-plant system with the soil sampling sequence as k, with the above-mentioned seven soil P fractions as the state variables in the state vector x = [x1,x2,...,x7]' of the soil-plant system, with the crop P uptake as the system output y, and with the external P fertilizer added as the system input u with
As such, A, b, and c are matrices with dimensions of 7 x 7, 7 x 1, and 1 x 7, respectively. The transformation of the P fractions, the effect of external P fertilizer added, and the availability of P fractions to plant uptake can be described by Eq. [1]. The relationship between soil P fractions and added external P fertilizer cannot sufficiently describe whether external P can control a P fraction if transfers among P fractions are not considered. Similarly, the relationship between soil P fractions and plant P uptake cannot sufficiently describe whether or not a P fraction contributes to plant P uptake if transfers among P fractions are not considered. However, the analysis of transformation among P fractions is complicated since the effects of certain P fractions may be dynamically coupled with each other. Fortunately, developments in modern system and control theories supply powerful tools for such an analysis.
Originally, the concepts of controllability and observability were introduced in system theory to consider the capability of the input to excite all the states and the capacity of all the states to excite the output (D'Angelo, 1970). The System Eq. [1] is said to be state controllable if there exists a control signal u(k) defined across a finite sampling interval 0 
k m such that, starting from any initial state, the state sequence {x(k)} can be made zero for k m (Ogata, 1970). The System Eq. [1] is said to be observable if, given the output y(k) for finite sampling sequence, it is possible to determine the initial state vector x(0) (Ogata, 1970). In the analysis and synthesis of complex multivariable systems, the concepts of controllability and observability provide a firm framework.
As for the soil-plant system, the controllability and observability of P fractions as its state variables may be interpreted as follows: if the size of a certain P fraction xi, i = 1, 2,..., 7, can be controlled directly by u, or indirectly via dynamic transformations among P fractions, then xi is said to be controllable; if xi has contribution to y directly, or indirectly via dynamic transformations among P fractions, then xi is said to be observable. The importance of controllability and observability of a P fraction lies in that they can determine the minimum realization of a soil–plant system model; that is, only those jointly controllable and observable P fractions are minimum to describe the system dynamics, whereas the uncontrollable and/or unobservable P fractions are actually redundant and hence can be eliminated from Eq. [1]. In this manner, the system model can be greatly reduced in many cases.
General Decomposition Theorem
Before the subsystem constituted of jointly controllable and observable states is identified, it is necessary to check whether uncontrollable or unobservable states exist in the system. According to system theory, the controllability and observability of the System Eq. [1] can be judged with the following controllability matrix
 | [2] |
and observability matrix
 | [3] |
The System Eq. [1] is controllable if the rank of MAB is equal to n, and uncontrollable if the rank of MAB is less than n. The System Eq. [1] is observable if the rank of NCA is equal to n and uncontrollable if the rank of NCA is less than n. In case the System Eq. [1] is uncontrollable or unobservable we have the following General Decomposition Theorem (Gilbert, 1963; Kalman, 1963; Kailath, 1980; Chui and Chen, 1989) to obtain the jointly controllable and observable subsystem.
General Decomposition Theorem: There exists an invertible state transformation 
= Tx and the System Eq. [1] can be rewritten into the form of
 | [4] |
where = 
',
and- The subsystem
 | [5] |
is controllable and observable, and it has the same irreducible transfer function as the original System Eq. [4],
where z is the z-transform operator and I is the identity matrix. All the states in this subsystem are jointly controllable and observable.
2. The subsystem 
is controllable, that is, all the states in this subsystem are controllable.
3. The subsystem 
is observable, that is, all the states in this subsystem are observable.
4. The subsystem 
is neither controllable nor observable, that is, all the states in this subsystem are neither controllable nor observable.
The jointly controllable and observable subsystem 
is just a minimum realization of System Eq. [1], which is the ideal objective of our state-space model reduction. The conventional way to find such a minimum realization through the above general decomposition is to compose a similarity transform matrix T by utilizing the linearly independent rows of the controllability matrix MAB or linearly independent columns of the observability matrix NCA combined with other suitably chosen linearly independent rows and columns (Fortmann and Hitz, 1977). This, however, generally leads to appearance of new state variables that are linear combinations of part or all the original ones and hence may no longer possess any physical or chemical meaning. Obviously, this is not desired in practical applications.
Fortunately, the choice of similarity transform for the above decomposition is not unique (Erol and Loparo, 1983), and the matrices A, b, and c encountered in our case usually have a highly sparse feature if a suitable significant level is used to select entries into the model via least squares regression. So, in this paper, the method of state transformation for realizing the similarity transform is restricted in row changes and the corresponding column changes, which is equivalent to the similarity transform with a sparse linear transform matrix with only a single one in each row and column. Thus, the jointly controllable and observable state variables can simply be found from the existing seven P fractions of interest so as to keep their original chemical meaning. The procedure for finding the jointly controllable and observable state variables is as follows.
Step 1: Calculate the rank of NCA, denoted as r. If r < n, {A,c} in Eq. [1] are transformed by row exchanges and the corresponding column exchanges into the formats as follows:
 | [6] |
where the sizes of submatrices Â11, Â21, Â22 are r x r, (n – r) x r, and (n – r) x (n – r), respectively, and 
1 has r columns. The corresponding state variables of pair (Â11, 1) are observable.
Step 2: Calculate the rank of MAB, denoted as s. If s < n, {A,b} in Eq. [1] are transformed by row exchanges and the corresponding column exchanges into the formats as follows:
 | [7] |
where the sizes of submatrices Ã11, Ã12, Ã22 are s x s, s x (n – s), and (n – s) x (n – s), respectively, and 
1 has s rows. The corresponding state variables of pair (Ã11, 1) are controllable.
Step 3: The common state variables shown in Step 1 and Step 2 are controllable and observable, and thus the subsystem constituted of these jointly controllable and observable state variables is just a minimum realization of the System Eq. [1].
Standardization of Soil Phosphorus Fractions and Crop Phosphorus Uptake
The original seven P fractions [pi(k), i = 1, 2, ..., 7] and the P uptake in the above ground biomass by corn [z(k)] were measured at different sampling times (k = 0, 1, 2, ..., 5) and added P fertilizer levels. Data Standardization is necessary in state-space modeling (Nielsen et al., 1999; Wendroth et al., 2001; Nielsen and Wendroth, 2003). Since the intercept is not allowed in Eq. [1] and different scales of P fractions may result in large approximation error in regression in this study, the following method of data standardization was used
 | [8] |
where xi(k) is the normalized ith P fraction at the kth harvest time, 
i is the standard deviation of pi for all harvest times and all levels of external P fertilizer added, and 
i is the ith P fraction at the last harvest time when no external P fertilizer had been added. The standardization is an extension of the concept of minimum value of soil test P (Cox et al., 1981; Cox, 1994) to soil P fraction, and soil P fraction at the last harvest time when no external P fertilizer had been added was assumed to be close to the minimum values. The crop P uptake at the last harvest time when no external P fertilizer had been added, denoted by ß, was assumed to be close to its minimum value when soil P fractions were at their minimum values. Crop P uptake (z) was standardized with the following method:
 | [9] |
where y(k) is the standardized crop P uptake at the kth harvest time and z is the standard deviation of z for all harvest times and all levels of external P fertilizer added.
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RESULTS AND DISCUSSION
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The Lualualei soil was used as an example to illustrate the technique of state-space modeling and model reduction. For the remaining soils, only the basic results were given.
State-Space Model Development for Lualualei Soil Experiment
The dynamic changes of soil P fractions and plant P uptake are shown in Fig. 1. The P fractions and crop P uptake of Lualualei soil were standardized as follows:
The relationships among the soil P fractions xi(k + 1) and x1(k), x2(k),..., x7(k) were estimated by least squares regression via a forward selection method to select those state variables with a significance level of 0.01 as the entries in the model and the intercept term being suppressed. The same regression method was also used to set up the relationship between crop P uptake y(k) and x1(k), x2(k),..., x7(k). The significance level 0.01 ensured that the matrices A, b, and c were sparse enough to reduce the model dimension effectively while still keeping the main effects of P fractions in the model. The software SAS version 8.0 (SAS Institute, 1999) was used for the regression analysis. The regression results among the P fractions were as follows:
 | [10] |
The matrix A in Eq. [1] was constructed from the regression coefficients in Eq. [10].
The linear relationship between the crop P uptake y(k) and x1(k), x2(k),..., x7(k) was
 | [11] |
The matrix c in Eq. [1] was constructed from the coefficient in Eq. [11].
The linear relationship between the initial P fractions xi(0) and added P fertilizer u was
 | [12] |
The interceptions were simply because of the initial condition of the soil, and thus, they were discarded. If a coefficient of u in Eq. [12] was not significantly different from zero at significance level 0.01, it was set to 0. The resulting matrix b in Eq. [1] was
The state-space model of the seven P fractions of the fractionation was thus captured in the form of Eq. [1] with the matrices A, b, and c obtained above:
 | [13] |
Model Reduction for Lualualei Soil
Step 1. The rank of observability matrix [2] was r = 2 < n = 7, thus, Eq. [13] was not observable. It was easy to find that, in this case, the row and column exchange were not necessary. According to the General Decomposition Theorem, the original state vector
and the corresponding matrices A, c in the format of Eq. [6]
directly showed that the two state variables [x1 x2]' were observable.
Step 2. The rank of controllability matrix [3] was s = 5 < n = 7. Thus, Eq. [13] was not controllable. After row and corresponding column exchanges or, equivalently, with the matrix
the state vector was transformed into
and the corresponding matrices A, b in the format of Eq. [7] were
Thus, according to the General Decomposition Theorem, the five state variables [x1 x7 x6 x4 x2]' were controllable.
It is worthy to note here that, when making the row/column exchanges in Step 1 and 2, it does not matter how the state variables in one variable group are ordered. For example, in Step 2 the order of x1, x7, x6, x4, and x2 in group [x1, x7, x6, x4, x2], and the order of x3 and x5 in group [x3, x5] can be arbitrarily changed without affecting the final result. This fact greatly facilitates the choice of row and column exchanges.
Step 3. The states [x1 x2]' were shown in both Steps 1 and 2, and thus were jointly controllable and observable. The reduced state-space equations can be obtained by withdrawing the corresponding equations from Eq. [13]:
 | [14] |
The dimension of state-space model [14] was greatly reduced from seven to two, but the dynamic behavior of Eq. [14] was the same as Eq. [13]:
The nonzero coefficients of x1 in bc,o and cc,o of Eq. [14] indicated that x1 was definitely jointly controllable and observable. Since both the two nondiagonal elements of Matrix A in Eq. [14] were nonzero, the transformations between x1 and x2 existed. Because of this transformation, x2 was still jointly controllable and observable although its coefficients in bc,o and cc,o of Eq. [14] were zeros.
State-Space Modeling and Model Reduction for Other Soils
With the same techniques as used with the Lualualei soil, the state-space models of other soils including Nohili, Wahiawa, Paaloa, Leilehua, Mahana, and Kapaa were developed and reduced. The state-space models with seven P fractions are shown in Table 2 and the reduced state-space models are shown in Table 3.
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Table 3. The reduced matrices Ac,o, bc,o, and cc,o in state-space model Eq. [5] resulting from the application of the General Decomposition Theorem to the full equations of Table 2 for seven soils.
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Controllable and Observable Phosphorus Fractions for All Seven Soils
The controllable and observable P fractions are listed in Table 3. For the Mollisol (Nohili soil) with low P sorption, only the first fraction (strip-P) was controllable and observable and hence the minimum to describe the P dynamics in this particular Mollisol-maize system. For the Vertisol (Lualualei soil) with low P sorption, the first two inorganic fractions (strip-P and inorganic NaHCO3–Pi) were controllable and observable and hence minimum to describe the P dynamics in this particular Vertisol-maize system. For the Oxisol (Wahiawa soil) with medium P sorption, the Oxisol (Kapaa soil) with high P sorption, and the Ultisol (Paaloa soil) with medium P sorption, the first three inorganic fractions (strip-P, inorganic NaHCO3–Pi, and inorganic NaOH-Pi) were controllable and observable and hence minimum to describe the specific Oxisol/Ultisol-maize system. For the Oxisol (Mahana soil) with high P sorption, and the Ultisol (Leilehua soil) with high P sorption, the first four fractions (strip-P, inorganic NaHCO3–Pi, organic NaHCO3–Po, and inorganic NaOH-Pi) were controllable and observable and hence minimum to describe the specific Oxisol/Ultisol-maize system. These results are not necessarily valid for all soils of the indicated orders. Also, these results only appear to represent dynamics of nutrient P in greenhouse growing conditions. We expect that some of the dynamics will be different in field-grown maize and thus suggest that field verification is necessary.
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CONCLUSIONS
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State-space models of P fractionations in soil-plant systems were developed to describe the transformations and dynamics among soil P fractions, the effect of adding external P on the size of certain soil P fractions (controllability), and the contributions of certain soil P fractions to plant P uptake (observability). A scientific evaluation of a soil P fractionation method was obtained via a system analysis of state controllability and observability. In combination with suitable choices of the significance level for selecting the entries in the model via least squares regression and simple matrix row/column exchanges, the General Decomposition Theorem was successfully used to identify and select the jointly controllable and observable soil P fractions that were the minimum to describe the soil P dynamics in the soil-plant systems. The advantage of this approach is that the transformation among soil P fractions is taken into account in the selection. The results show that strip-P could describe P dynamics in one soil that was a Mollisol with low P sorption; strip-P and inorganic NaHCO3–Pi in one soil that was a Vertisol with low P sorption; strip-P, inorganic NaHCO3–Pi, and inorganic NaOH-Pi in two soils that were Oxisols with medium and high P sorption and one soil that was Ultisol with medium sorption; and strip-P, inorganic NaHCO3–Pi, organic NaHCO3–Po, and inorganic NaOH-Pi in one soil that was Oxisol with high P sorption and one soil that was Ultisol with high sorption. As P sorption capacity increases, it appears that more sequentially extracted P fractions were needed to describe the P dynamics in the soil-plant system. The reduced number of P fractions suggests an economic way to interpret and use the information of the Hedley approach. These results also suggest that more than one extractant or combinations of extractants might give a more precise assessment of P nutrient status.
This article is just an initial attempt to apply some existing results in linear system theory to the state-space modeling of soil dynamics and its minimum realization, and it is found that the system theory is a powerful tool for solving difficult dynamical problems in soil science. On the basis of the state-space models thus obtained, further work can be done in the optimal predication of future soil P fractions with the Kalman filtering technique and the optimal fertilizing strategy design aiming at final maximum crop P uptake.
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ACKNOWLEDGMENTS
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We thank Prof. Ning-shou Xu for his insightful help in preparing the manuscript.
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NOTES
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Journal Series No. 4672. Univ. of Hawaii at Manoa, College of Tropical Agriculture and Human Resources.
Received for publication April 11, 2003.
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INTRODUCTION
