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Observability analysis for soil moisture estimation⁎
Proceedings of the Control Conference Africa, 2017 Proceedings of South the Control Conference 2017 Johannesburg, Africa, December 7-8, Johannesburg, South Africa, December Africa, 7-8, 2017 2017 Proceedings of South the Control Conference 2017 Johannesburg, Africa, December Africa, 7-8, 2017 Available online at www.sciencedirect.com Johannesburg, South Africa, December 7-8, 2017
ScienceDirect IFAC PapersOnLine 50-2 (2017) 110–114 Observability analysis for soil moisture Observability analysis for moisture soil Observability estimation analysis for soil moisture estimation estimation ∗∗ Sirish L. Shah ∗∗∗ ∗∗∗ Jannatun Nahar ∗∗ Jinfeng Liu ∗∗
Jannatun Nahar ∗∗ Jinfeng Liu ∗∗ Sirish L. Shah ∗∗∗ Jannatun Nahar Jinfeng Liu ∗∗ Sirish L. Shah ∗∗∗ ∗ ∗∗ Jannatun Nahar Jinfeng Liu (e-mail: [email protected]). L. Shah ∗∗∗ ∗ ∗ University of ALberta, Alberta, Canada University of ALberta, Alberta, Canada (e-mail: [email protected]). ∗ ∗ University of ALberta, Alberta, Canada (e-mail: [email protected]). ∗∗ ∗∗ University of Alberta, Canada (e-mail: [email protected]) ∗ University of Alberta, Alberta, Canada (e-mail: [email protected]) ∗∗ ∗∗ ∗∗∗ University ALberta, Alberta, Canada (e-mail: [email protected]). University of Alberta, (e-mail: ∗∗∗ of Alberta, University of Alberta, Alberta, Canada ,, (e-mail: University Alberta,Canada Alberta, [email protected]) (e-mail: ∗∗ ∗∗∗ ∗∗∗ University University of Alberta,of Canada (e-mail: [email protected]) ofAlberta, Alberta, Alberta, Canada , (e-mail: [email protected]) [email protected]) ∗∗∗ University of [email protected]) Alberta, Alberta, Canada , (e-mail: [email protected]) Abstract: The The knowledge knowledge of of soil soil moisture moisture is is important important in in studying studying climatology, climatology, earth earth science science Abstract: Abstract: The knowledge of soildecision moisture is important inbut studying climatology, earth science and most importantly irrigation support systems, is often hard to determine since and most importantly irrigation decision support systems, but is often hard to determine since Abstract: The knowledge of soildecision moisture is important inmoisture studying climatology, earth science and most importantly irrigation support systems, but is often hard to determine since it is not possible to use critical measurements including sensors all over the entire it is not possible to use critical measurements including moisture sensors all over the entire and most importantly irrigation decision supportincluding systems, but is often hard to to determine since it is not possible to use critical measurements moisture sensors all over the entire agricultural grid sector. As a result, soil moisture at unmeasured region needs be estimated, agricultural grid sector. As a result, soil moistureincluding at unmeasured region needs toover be estimated, it is not possible to use critical measurements moisture sensors all the entire agricultural grid sector. As a result, soil moisture at unmeasured region needs to be estimated, which can can be be done done using using state state estimators estimators such such as as Kalman Kalman based based estimators. estimators. The The model model that that which agricultural sector. Asstate atransfer result, soil moisture unmeasured region needs to bemodel estimated, which can begrid done using estimators such asatKalman based estimators. The that is used to represent water between atmosphere, plant and soil, also known as agrois used to represent water transfer between atmosphere, plant and soil, also known as agrowhich can bemodel, done using state estimators such as Kalman based estimators. The model that is used to represent water transfer between atmosphere, plant and soil, also known as agrohydrological is highly nonlinear. Since ‘strong’ rather than ‘weak’ observability of the hydrological model, iswater highly nonlinear. Sinceatmosphere, ‘strong’ rather than ‘weak’ observability ofagrothe is used to represent transfer between plant and soil, also known as hydrological model, is highly nonlinear. Since ‘strong’ rather than ‘weak’ observability of the system ensures ensures better better performance performance of of Kalman Kalman based based estimators estimators to to develop develop aa reliable reliable soil soil moisture moisture system hydrological model, highly Since ‘strong’ than ‘weak’ observability of this the system ensures betterisperformance of Kalman based estimators to develop a reliable soil moisture estimation algorithm, the mainnonlinear. objective of this this study israther to discuss discuss observability analysis of estimation algorithm, the main objective of study is to observability analysis of this system ensures better performance of Kalman based estimators to develop a reliable soil moisture estimation algorithm, the main objective of this study is to discuss observability analysis of this nonlinear agro-hydrological system. nonlinear agro-hydrological system. estimation algorithm, the main objective of this The study is to discuss observability analysis chosen of this nonlinear agro-hydrological system. The study was performed using synthetic The study was performed using synthetic data. data. The extended extended Kalman Kalman filter filter (EKF) (EKF) was was chosen nonlinear agro-hydrological system. The study was performed using synthetic data. The extended Kalman filter (EKF) was chosen as the the state state estimator. estimator. As As would would be be expected, expected, the the results results show show that that the the EKF EKF performance performance is is as The study was performed using data. Theresults extended Kalman filter (EKF) was chosen as the state estimator. Assystem wouldsynthetic be‘strongly’ expected, the show that the EKF performance is better in cases where the is observable. better in cases where the system is ‘strongly’ observable. as the in state estimator. Assystem would isbe‘strongly’ expected,observable. the results show that the EKF performance is better cases where the © 2017,inIFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. better cases(International where the system is ‘strongly’ observable. Keywords: Observability, Observability, state state estimation, estimation, extended extended Kalman Kalman filter, filter, agriculture, agriculture, nonlinear nonlinear Keywords: Keywords: Observability, state estimation, extended Kalman filter, agriculture, nonlinear systems. systems. Keywords: Observability, state estimation, extended Kalman filter, agriculture, nonlinear systems. systems. 1. INTRODUCTION INTRODUCTION the knowledge knowledge of of moisture moisture content content at at different different depths depths 1. the the knowledge of Due moisture content atnature different depths 1. INTRODUCTION inside soil matrix. to the nonlinear of the agroinside soil matrix. Due to the nonlinear nature of the agro1.the INTRODUCTION the knowledge of Due moisture content atnature different depths inside soil matrix. to the nonlinear of the agroThe basic need of human species for survival is food hydrological model the state estimation is performed using The basic need of the human species for survival is food hydrological model the state estimation is performed using inside soil matrix. Due to the nonlinear nature of the agroThe basic need of the human species for survival is food hydrological model the state estimation is performed using and water, where food production through agriculture in nonlinear estimators such as extended Kalman filter or and water, where food production through agriculture in nonlinear estimators such asestimation extendedisKalman filter or The basic need of the human species for survival is food hydrological model the state performed using and water, where food production through agriculture in nonlinear estimators such as extended Kalman filter or itself requires requires water. water. As As a a result result of of these these basic basic needs needs along along ensemble ensemble Kalman Kalman filter. filter. Several Several such such studies studies have have been been itself and water, where food production through agriculture in nonlinear estimators suchSeveral as extended Kalman filter or ensemble Kalman filter. such studies have been itself requires water. As a result of these basic needs along with all other uses of water, water usage at global level has performed which includes filter design for soil water eswith all other uses of As water, waterofusage at global level has performed which includes filter design for soilhave waterbeen esitself all requires water. a result these basic needs along ensemble Kalman filter. data Several such studies performed which includes filter design for soil water eswith other uses of water, water usage at global level has grown to a rate more than twice the rate of population timation with synthetic (Walker and Kalma, 2001; grown a rate more than water twiceusage the rate of population timation with synthetic data (Walker and Kalma, 2001; with allto uses of water, at global level has performed which includes filter design for soil water esgrown toother a the rate more than twice the rate of population timation with synthetic data (Walker and Kalma, 2001; increase in 20th century (Steduto et al., 2012). In Reichle et al., 2002), remote sensing (Houser et al., 1998) increase in the 20th century (Steduto et al., 2012). In Reichle et al., 2002), remote sensing (Houser et al., 1998) grown to a rate more than twice the rate of population timation with synthetic data (Walker and Kalma, 2001; increase in the 20th century (Steduto et al., 2012). In Reichle et al., 2002), remote sensing (Houser et al., 1998) addition, due due to to adverse adverse climate climate conditions conditions water water scarcity scarcity and and ground ground data data (Lannoy (Lannoy et et al., al., 2008), 2008), but but none none of of these these addition, increase in 20th century (Steduto et water al.,confined 2012). In Reichle et al., 2002), remote (Houser etanalysis al., and ground data (Lannoy et sensing al., 2008), but none of 1998) these addition, duethe to adverse climate conditions scarcity has become a global problem and is no more in studies have taken into account observability of has become a global problem and is no more confined in studies have taken into account observability analysis of addition, duea to adverse climate conditions water scarcity and ground data (Lannoy et al., 2008), but none of study these studies have taken into account observability analysis of has become global problem and is no more confined in arid and semi-arid regions. Agriculture is the sector where the system which ensures reliable estimates. In this arid and semi-arid regions. Agriculture is the sector where the system which ensures reliable estimates. In this study has become a global problem and is no more confined in the studies haveobservability takenensures into account observability analysis of arid and semi-arid regions. Agriculture is the sector where system which reliable estimates. In this study water scarcity has greatest relevance due to the fact that we discuss analysis of the agro-hydrological water scarcity has regions. greatest Agriculture relevance due to the factwhere that we discuss observability analysis of the agro-hydrological arid and semi-arid is the sector the system which ensures reliable estimates. In this study we discuss observability analysis of the agro-hydrological water scarcity has greatest relevance due to the fact that 70 percent percent of of global global fresh fresh water water is is used used for for agriculture agriculture system system and and discuss discuss the the importance importance of of soil soil profile profile discretizadiscretiza70 water scarcity greatest relevance due toforthe factwater that we discuss observability analysis ofofthe system andon discuss theobservability. importance soil agro-hydrological profile discretiza70 percent of has global fresh water is used agriculture (Aquastat, 2012), and among all other agricultural tion based system (Aquastat, 2012), and among all other agricultural water tion based on system observability. 70 percent of global fresh water is used for agriculture andon discuss theobservability. importance of soil profile discretization based system (Aquastat, 2012), among all least other agricultural water system usage irrigation is operated with efficiency. usage irrigation is and operated with (Aquastat, 2012), among all least otherefficiency. agricultural water tion based on system observability. usage irrigation is and operated with least efficiency. 2. DEVELOPING DEVELOPING STATE STATE SPACE SPACE SYSTEM SYSTEM 2. The irrigation practice involves open usageusual irrigation is operated with least efficiency. The usual irrigation practice involves open loop loop control control 2. DEVELOPING STATE SPACE SYSTEM The usual irrigation practice involves open loop control where no feed back is considered and irrigation is mostly where no feed back ispractice considered and irrigation mostly 2. DEVELOPING STATE SPACE SYSTEM The irrigation involves open loopis whereusual no feed is considered and irrigation is control mostly done based on back farmers observation which in terms terms would 2.1 2.1 Agro-hydrological Agro-hydrological model model done based on farmers observation which in would wherebased no feed back is considered and irrigation is mostly done on less farmers observation which in terms would 2.1 Agro-hydrological model cause over or irrigation. The main barrier in implecause over or less irrigation. The main barrier in imple2.1 agro-hydrological Agro-hydrological model mainly considers the system done based on less farmers observation whichbarrier inis terms would cause over or irrigation. main implementing closed loop control in lack of An agro-hydrological model model mainly considers the system menting closed loop control The in irrigation irrigation is the thein lack of An cause over or less irrigation. The main barrier in impleAn agro-hydrological model the system soil water matrix. The inputs and of menting closed loop controlwhich in irrigation isthen the used lack for of with soil moisture measurements could be with soil water matrix. Themainly inputs considers and outputs outputs of this this soil moisture measurements which could be then used for An agro-hydrological model mainly considers system menting closed loop control in irrigation is the lack of with soil water matrix. The inputs and outputs of this soil moisture measurements which could be then used for system includes includes events events such such as as rain, rain, root root water waterthe extraction feedback control. control. Since Since agricultural agricultural areas areas are are vast vast and and system extraction feedback with soil water matrix. The inputs and outputs of this soil moisture measurements which could be are then used the for by plants includes events such as rain, root water and as in 1. feedback control. Since agricultural areas diverse and it to all over by plants and drainage drainage as shown shown in figure figure 1. extraction diverse and it is is impractical impractical to put put sensors sensors all vast over and the system system includes events such as rain, root water extraction feedback control. Since agricultural areas are vast and diverse andtherefore it is impractical putmoisture sensors profile all over the by plants and drainage as shown in figure 1. entire field the overalltosoil soil moisture profile needs entire field the overall needs Theplants dynamics of this this as agro-hydrological system can be be and drainage shown in figure system 1. diverse andtherefore it is impractical tosoil putmoisture sensors profile all over the by The dynamics of agro-hydrological can entire field therefore the overall needs to be estimated. to be estimated. dynamics of this agro-hydrological system can be expressed using Richards equation (Richards, 1931), which entire field therefore the overall soil moisture profile needs The expressed using Richards equation (Richards, 1931), which to be estimated. The dynamics of this agro-hydrological system can be expressed using Richards equation (Richards, 1931), which is shown below : Soil moisture at unmeasured regions can be estimated to bemoisture estimated. shown below :Richards equation (Richards, 1931), which Soil at unmeasured regions can be estimated is expressed using is shown below : Soil moisture at unmeasured regions can be estimated ∂θ using state and model. The ∂h − 1)] − S(t) ∂θ: = ∂ ∂ [K(θ)( ∂h using state estimators estimators and agro-hydrological agro-hydrological The is shown below (1) Soil moisture atofunmeasured regions cansystem bemodel. estimated ∂θ ∂ [K(θ)( ∂h using statestates estimators agro-hydrological model. − 1)] − S(t) (1) = ∂z unknown states thisand agro-hydrological are The the ∂z ∂t unknown of this agro-hydrological system are the (1) [K(θ)( = ∂z − 1)] − S(t) ∂z ∂t ∂h ∂θ ∂ using state estimators and agro-hydrological model. The unknown states of this agro-hydrological system are the water pressure heads at different depths, which gives us 3 3 ∂z − 1)](cm ∂z [K(θ)(content ∂tsoil water pressure heads atagro-hydrological different depths, system which gives us where − S(t) (1) = moisture 3 /cm3 ), K is the θθ is the where is the soil moisture content (cm /cm ), K is the unknown states of this are the 3 ∂z ∂z (cm/h) ∂tsoil moisture water pressure heads at different depths, which gives us where 3 /cm3 3 ), K is the θ is the content (cm hydraulic conductivity which itself is a function Natural Sciences and Engineering Research Council, Canada. hydraulic (cm/h) which(cm itself Natural Sciencesheads and Engineering Research Council, Canada. water pressure at different depths, which gives us where 3 3 a function θ is conductivity the soil moisture content /cmis is the hydraulic conductivity (cm/h) which itself is ),a K function Natural Sciences and Engineering Research Council, Canada. Natural Sciences and Engineering Research Council, Canada. hydraulic conductivity (cm/h) which itself is a function
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where ∆t is the time interval between time instants k and k + 1, ∆zi is the thickness of compartment i and Ki−1/2 (hk ) is the algebraic mean of hydraulic conductivity from compartments i and i + 1. Evaporation
Rain, irrigation
Eq. (4) can be rearranged as: k+1 k k+1 + ai (hk )hi+1 = ei (hk ) − uki di (hk )hk+1 i−1 + bi (h )hi (5) where ai (h) includes the coefficients of hi+1 (k + 1), di (h) includes the coefficients of hi−1 (k + 1), bi (h) includes the coefficients of hi (k +1), ei (h) includes the coefficients from time instant k, and ui (k) includes all source or sink terms shown in Eq. 4.
Root water extraction Surface runoff
Nodes or soil compartments
Soil layers
Eq. (5) represents water flow inside a single compartment i. For N compartments there will be N of these equations which can be expressed with the following matrix. A(hk )hk+1 = E(hk ) + uk (6) h = A(hk )−1 E(hk ) + A(hk )−1 uk where A(h) is a N × N tridiagonal matrix. The boundary conditions of the system are considered as source and sink in the model and are included in u. Eq. (6) is the nonlinear state equation of the system. k+1
Drainage or deep seepage
Fig. 1. The agro-hydrological system with four different soil layers. of soil moisture (Mualem, 1976), z is the vertical distance inside soil matrix (cm), S includes the source and sink terms (cm3 cm−3 h−1 ) and h is the matric potential (cm). Root water extraction is a sink term in the model. The differential soil moisture term in Richards equation can be represented by a differential pressure term using hydraulic capacity, which is shown as follows: dh ∂ ∂h C(h) = [K(h)( − 1)] − S(t) (2) dt ∂z ∂z where C is the hydraulic capacity. The soil moisture content θ can be expressed in terms of matric potential h in the following way (Genuchten, 1980): θ(h) = θres + (θsat − θres )(1 + |αh|η )−m (3) where θres is the residual moisture content (cm3 /cm3 ), θsat is the saturated moisture content (cm3 /cm3 ), η and m are empirical shape factors, and α is the inverse of air entry suction (cm−1 ).
The system output equation is given by: y = DΘ (7) where D is a matrix indicating at which nodes the soil moisture contents are measured and Θ is a N × 1 vector representing the soil moisture using Eq. 3 for N nodes. 2.3 Model validation The non-linear statespace model as shown above is validated against HYDRUS 1d model (Simunek et al., 2005) by simulating soil moisture for a soil profile of 110 cm with four different soil layers. The first 15 cm is represented by layer 1, the next 35 cm is represented by layer 2, then another 40 cm represents layer 3 and lastly the remainder 30 cm represents layer 4. We next consider a constant flux of 0.2 cm/h infiltration at the top boundary. The bottom boundary condition was considered as free drainage. Under no crop condition the soil moisture were simulated for 48
2.2 Model discertization Richards equation is a highly nonlinear partial differential equation, but can be discretized and solved numerically. The discretized Richards equation is developed by implicit backward, finite difference scheme with explicit linearization of hydraulic conductivities following Van Dam et al. (2008) and is shown by Eq. 4. hk+1 − hk+1 1 − hki i + = Ki−1/2 hk 1 i−1 t zi 2 (∆zi−1 + ∆zi ) hk+1 − hk+1 i+1 − Ki+1/2 (hk ) Ki−1/2 (hk ) − Ki+1/2 (hk ) 1 i (∆z + ∆z i i+1 ) 2 −Sik (4) hk+1 Ci (hki ) i
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Fig. 2. The simulated results of soil moisture after 48 hours starting from 0.24 cm/cm using HYDRUS 1d model and developed model.
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hours starting from 0.24 cm/cm and the final results are presented in figure 2. The result clearly shows that the results from the developed model closely matches with the results from HYDRUS 1d model.
Table 1. Parameter values used for simulating the system. Parameter Ksat
3. OBSERVABILITY OF THE SYSTEM When the knowledge of the input and the output over a finite time interval [t0 , tf ] is adequate to uniquely determine the initial state x(t0 ) then the system is said to be observable (Chen, 1995). To determine the observability of the nonlinear system, the agro-hydrological system is linearized at different operating conditions and the local observability of the system is then determined (Zeng et al., 2016). The linearization of the system is performed using the system Jacobians F k and Gk which are obtained numerically and are given by: ∂f |ˆ k|k k (8) ∂h h ,u ∂g |ˆ k+1|k (9) Gk+1 = ∂h h where f = A(hk )−1 E(hk ) + A(hk )−1 uk and g = DΘ representing Eq. (6) and Eq. (7), respectively. Fk =
To study system observability at first the rank test is performed. The system is considered to be observable at time k if the observability matrix Ok has a full column rank, where the observability matrix is given by: Gk Gk F k . Ok = (10) ... Gk (F k )N −1 Since the rank test does not provide any information on how strongly or weakly observable a system is, therefore we further perform degree of observability analysis of an observable system using a deterministic approach as mentioned by Ham (1980).
θres
θsat
α
η
Soil Layer 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Value 1.0 0.250 1.0 0.250 0.05 0.04 0.03 0.03 0.43 0.46 0.45 0.46 0.016 0.016 0.016 0.016 1.37 1.37 1.37 1.37
4.2 Observability analysis To perform the state estimation of this system the soil profile needs to be discretized such that the system remains observable. In this study, the soil profile is discretized into 44, 33, 31, 23 and 15 nodes in order to create the system with 44, 33, 31, 23 and 15 states, respectively. It was found that for 44 states, the rank of Ok is not full at any portion of the time trajectory, therefore the system is unobservable. The cases with 33 states and 31 states is unobservable at some parts of the time trajectory so they are partially observable as shown in Figure (3). But if the number of states are reduced to 23 states the system becomes observable in terms of rank test. In the figure both 15 states and 23 states are observable but to better understand the system we perform further analysis based on degree of observability.
In this deterministic approach the row vectors of Ok are normalized into unit length vectors N and the relative measure of system observability is given by the ratio of the largest and the smallest eigenvalue of the matrix N T N . For highly observable system this ratio denoted by λmax /λmin will be one or closer to one. Due to the time varying property of the agro-hydrological system the observability analysis is performed over the entire time trajectory. 4. RESULTS 4.1 Data description The study was conducted with synthetic data considering a soil profile of 110 cm with four different homogeneous soil layers. The soil parameter values considered are shown in table 1. The synthetic data was generated using the agro-hydrological model and weather data from St.Albert Weather Station, Canada (Government of Canada, 2013). It is considered that only four measurements are available at depths 5 cm, 20 cm, 50 cm and 100 cm. The measurement data were collected on hourly basis. 112
Fig. 3. The rank of the observability matrix of the system with different states. The results for degree of observability is shown in Figure (4). The result shows that as we reduce the number of states λmax /λmin starts to decrease which suggests that the system becomes more observable. Although system
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with 23 states and 15 states both can be considered observable based on rank test, the eigenvalue analysis suggests that the system with 15 states is more observable relative to the system with 23 states. The system with 33 and 31 states which was partially observable performed very poorly in terms of the degree of observability analysis. It was observed that fewer states would ensure better observability, but at the same time reducing the number of states would create issues in the numerical solution of equation 4. In this work the system was not reduced below 15 states because of this reason. In both the rank test (figure 3) and relative observability analysis (figure 4), it is seen that most of the unobservability occurred in the first part of simulation. Though the system with 23 and 15 states were observable in the rank test all over the time trajectory as shown in figure 3 but the relative observability were initially poor as shown in figure 4 due to higher value of λmax /λmin . One possible reason for such behavior is the fact that extreme dry or wet soil condition persisted at those operating regions. The saturated soil moisture for this soil was set between 0.43 and 0.45 as shown in table 1 and the initial soil moisture of the system was considered between 0.35 and 0.4. Though these values do not refer to extreme wet condition but they are relatively near those extreme regions. Linearizing the system in that near saturation region may have introduced some modeling errors.
Fig. 4. The ratio of the largest and the smallest eigen value of matrix N T N for systems with 33, 31, 23 and 15 states at different time trajectory. 4.3 Performance of EKF To investigate the performance of EKF based on observability, an comparative analysis based on system with 31 states, which is partially observable, and system with 15 states, which is fully observable, is performed. The same synthetic data is used but small amount of noise was added to the data. The noise is generated using MATLAB function ’randn’ and only 1% of this noise is added with the measurements. The process noise covariance Qk is taken as 1e5I and the measurement noise covariance is taken as 0.5I where I is the identity matrix with dimension of states. Since we want to compare the EKF performance, 113
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from four measurements only three were chosen for EKF update. They are measurements from depth 5 cm, 50 cm and 100 cm. The soil moisture at depth 20 cm is estimated by the EKF and compared against the actual data. From figure 5, it is clearly observed that the EKF output given by system with 15 states performs better than the system with 31 states. In the estimated node, that is depth 20 cm, the EKF estimates for 15 states converge to actual value but the estimate with 31 states always had a bias. In fact in the case where the measurements are available, that is depth 5 cm, 50 cm and 100 cm, the EKF output from 31 states fails to track the actual data. This result agrees with the system observability analysis. The system with poor observability would not perform well in EKF relative to a system with good observability.
Fig. 5. The extended Kalman filter result at four depths for systems with 31 and 15 states. The measurements from depth 5 cm, 50 cm and 100 cm were used in EKF for updates. The measurement from depth 20 cm was not used inside EKF, this was estimated. 5. CONCLUDING REMARKS The agro-hydrologial system under study is highly nonlinear. To perform state estimation to determine soil moisture, this highly nonlinear model requires linearization at many steps. In such case a rank test itself may not represent the system well. For example the observability of the system with 31 state depends upon the time trajectory. If the time trajectory is not sufficiently long then it would
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be hard to understand whether the system is observable or not. So it is better to perform degree of observability analysis along with the rank test to better understand the system. If the system is observable by the rank test and has fairly low λmax /λmin ratio then it would be reasonable to consider that nonlinear system to be observable over the entire time trajectory. The degree of observability may also be used for determining measurement locations. In that case the eigenvectors corresponding to lowest eigenvalues must be analyzed. An observable system would give much better performance in Kalman based estimation. Moreover when a discretized numerical scheme is used to develop the agro-hydrological model, the discretization or the designing of the soil profile must be done such that the system observability is ensured. This would eventually increase the ease of control application in such system. ACKNOWLEDGEMENTS We would like to thank Agriculture and Agri-Food Canada, Alberta Agriculture and Forestry for providing field and weather data. REFERENCES Aquastat (2012). FAO’s information System on Water and Agriculture. http://www.fao.org/nr/water/aquastat/water use/ index.stm. Chen, C.T. (1995). Linear system theory and design. Oxford University Press, Inc. Genuchten, M.T.V. (1980). A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil science society of America journal, 44(5), 892–898. Government of Canada (2013). Current and historical Alberta weather station data viewer, AgroClimatic Information Service. http://agriculture.alberta.ca/acis/alberta-wea ther-data-viewer.jsp. [Online; accessed August 01,2014]. Ham, F.M. (1980). Determination of the degree of observability in linear control systems. Ph.D. thesis, Iowa State University. Houser, P.R., Shuttleworth, W.J., Famiglietti, J.S., Gupta, H. V.and Syed, K.H., and Goodrich, D.C. (1998). Integration of soil moisture remote sensing and hydrologic modeling using data assimilation. Water Resources Research, 34(12), 3405–3420. Lannoy, G.J.D., Houser, P.R., Pauwels, V., and Verhoest, N.E. (2008). State and bias estimation for soil moisture profiles by an ensemble kalman filter: Effect of assimilation depth and frequency. Water Resources Research, 43(6). Mualem, Y. (1976). A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resources Research, 12(3), 513–522. Reichle, R.H., Walker, J.P., Koster, R.D., and Houser, P.R. (2002). Extended versus ensemble Kalman filtering for land data assimilation. Journal of Hydrometeorology, 3(6), 728–740. 114
Richards, L.A. (1931). Capillary conduction of liquids through porous mediums. Journal of Applied Physics, 1(5), 318–333. Simunek, J., Van Genuchten, M.T., and Sejna, M. (2005). The hydrus-1d software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media. University of California-Riverside Research Reports, 3, 1–240. Steduto, P., Faur`es, J., Hoogeveen, J., Winpenny, J., and Burke, J. (2012). Coping with water scarcity: an action framework for agriculture and food security. Food and Agriculture Organization of the United Nations Rome. Van Dam, J.C., Groenendijk, P., Hendriks, R.F.A., and Jacobs, C.M.J. (2008). SWAP version 3.2: Theory description and user manual. Wageningen UR, Alterra, Postbus 47, 6700 AA Wageningen, The Netherlands, 2 edition. Walker, J. P. andWillgoose, G.R. and Kalma, J.D. (2001). One-dimensional soil moisture profile retrieval by assimilation of near-surface observations: a comparison of retrieval algorithms. Advances in Water Resources, 24(6), 631–650. Zeng, J., Liu, J., Zou, T., and Yuan, D. (2016). Distributed extended kalman filtering for wastewater treatment processes. Industrial & Engineering Chemistry Research, 55(28), 7720–7729.
ScienceDirect IFAC PapersOnLine 50-2 (2017) 110–114 Observability analysis for soil moisture Observability analysis for moisture soil Observability estimation analysis for soil moisture estimation estimation ∗∗ Sirish L. Shah ∗∗∗ ∗∗∗ Jannatun Nahar ∗∗ Jinfeng Liu ∗∗
Jannatun Nahar ∗∗ Jinfeng Liu ∗∗ Sirish L. Shah ∗∗∗ Jannatun Nahar Jinfeng Liu ∗∗ Sirish L. Shah ∗∗∗ ∗ ∗∗ Jannatun Nahar Jinfeng Liu (e-mail: [email protected]). L. Shah ∗∗∗ ∗ ∗ University of ALberta, Alberta, Canada University of ALberta, Alberta, Canada (e-mail: [email protected]). ∗ ∗ University of ALberta, Alberta, Canada (e-mail: [email protected]). ∗∗ ∗∗ University of Alberta, Canada (e-mail: [email protected]) ∗ University of Alberta, Alberta, Canada (e-mail: [email protected]) ∗∗ ∗∗ ∗∗∗ University ALberta, Alberta, Canada (e-mail: [email protected]). University of Alberta, (e-mail: ∗∗∗ of Alberta, University of Alberta, Alberta, Canada ,, (e-mail: University Alberta,Canada Alberta, [email protected]) (e-mail: ∗∗ ∗∗∗ ∗∗∗ University University of Alberta,of Canada (e-mail: [email protected]) ofAlberta, Alberta, Alberta, Canada , (e-mail: [email protected]) [email protected]) ∗∗∗ University of [email protected]) Alberta, Alberta, Canada , (e-mail: [email protected]) Abstract: The The knowledge knowledge of of soil soil moisture moisture is is important important in in studying studying climatology, climatology, earth earth science science Abstract: Abstract: The knowledge of soildecision moisture is important inbut studying climatology, earth science and most importantly irrigation support systems, is often hard to determine since and most importantly irrigation decision support systems, but is often hard to determine since Abstract: The knowledge of soildecision moisture is important inmoisture studying climatology, earth science and most importantly irrigation support systems, but is often hard to determine since it is not possible to use critical measurements including sensors all over the entire it is not possible to use critical measurements including moisture sensors all over the entire and most importantly irrigation decision supportincluding systems, but is often hard to to determine since it is not possible to use critical measurements moisture sensors all over the entire agricultural grid sector. As a result, soil moisture at unmeasured region needs be estimated, agricultural grid sector. As a result, soil moistureincluding at unmeasured region needs toover be estimated, it is not possible to use critical measurements moisture sensors all the entire agricultural grid sector. As a result, soil moisture at unmeasured region needs to be estimated, which can can be be done done using using state state estimators estimators such such as as Kalman Kalman based based estimators. estimators. The The model model that that which agricultural sector. Asstate atransfer result, soil moisture unmeasured region needs to bemodel estimated, which can begrid done using estimators such asatKalman based estimators. The that is used to represent water between atmosphere, plant and soil, also known as agrois used to represent water transfer between atmosphere, plant and soil, also known as agrowhich can bemodel, done using state estimators such as Kalman based estimators. The model that is used to represent water transfer between atmosphere, plant and soil, also known as agrohydrological is highly nonlinear. Since ‘strong’ rather than ‘weak’ observability of the hydrological model, iswater highly nonlinear. Sinceatmosphere, ‘strong’ rather than ‘weak’ observability ofagrothe is used to represent transfer between plant and soil, also known as hydrological model, is highly nonlinear. Since ‘strong’ rather than ‘weak’ observability of the system ensures ensures better better performance performance of of Kalman Kalman based based estimators estimators to to develop develop aa reliable reliable soil soil moisture moisture system hydrological model, highly Since ‘strong’ than ‘weak’ observability of this the system ensures betterisperformance of Kalman based estimators to develop a reliable soil moisture estimation algorithm, the mainnonlinear. objective of this this study israther to discuss discuss observability analysis of estimation algorithm, the main objective of study is to observability analysis of this system ensures better performance of Kalman based estimators to develop a reliable soil moisture estimation algorithm, the main objective of this study is to discuss observability analysis of this nonlinear agro-hydrological system. nonlinear agro-hydrological system. estimation algorithm, the main objective of this The study is to discuss observability analysis chosen of this nonlinear agro-hydrological system. The study was performed using synthetic The study was performed using synthetic data. data. The extended extended Kalman Kalman filter filter (EKF) (EKF) was was chosen nonlinear agro-hydrological system. The study was performed using synthetic data. The extended Kalman filter (EKF) was chosen as the the state state estimator. estimator. As As would would be be expected, expected, the the results results show show that that the the EKF EKF performance performance is is as The study was performed using data. Theresults extended Kalman filter (EKF) was chosen as the state estimator. Assystem wouldsynthetic be‘strongly’ expected, the show that the EKF performance is better in cases where the is observable. better in cases where the system is ‘strongly’ observable. as the in state estimator. Assystem would isbe‘strongly’ expected,observable. the results show that the EKF performance is better cases where the © 2017,inIFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. better cases(International where the system is ‘strongly’ observable. Keywords: Observability, Observability, state state estimation, estimation, extended extended Kalman Kalman filter, filter, agriculture, agriculture, nonlinear nonlinear Keywords: Keywords: Observability, state estimation, extended Kalman filter, agriculture, nonlinear systems. systems. Keywords: Observability, state estimation, extended Kalman filter, agriculture, nonlinear systems. systems. 1. INTRODUCTION INTRODUCTION the knowledge knowledge of of moisture moisture content content at at different different depths depths 1. the the knowledge of Due moisture content atnature different depths 1. INTRODUCTION inside soil matrix. to the nonlinear of the agroinside soil matrix. Due to the nonlinear nature of the agro1.the INTRODUCTION the knowledge of Due moisture content atnature different depths inside soil matrix. to the nonlinear of the agroThe basic need of human species for survival is food hydrological model the state estimation is performed using The basic need of the human species for survival is food hydrological model the state estimation is performed using inside soil matrix. Due to the nonlinear nature of the agroThe basic need of the human species for survival is food hydrological model the state estimation is performed using and water, where food production through agriculture in nonlinear estimators such as extended Kalman filter or and water, where food production through agriculture in nonlinear estimators such asestimation extendedisKalman filter or The basic need of the human species for survival is food hydrological model the state performed using and water, where food production through agriculture in nonlinear estimators such as extended Kalman filter or itself requires requires water. water. As As a a result result of of these these basic basic needs needs along along ensemble ensemble Kalman Kalman filter. filter. Several Several such such studies studies have have been been itself and water, where food production through agriculture in nonlinear estimators suchSeveral as extended Kalman filter or ensemble Kalman filter. such studies have been itself requires water. As a result of these basic needs along with all other uses of water, water usage at global level has performed which includes filter design for soil water eswith all other uses of As water, waterofusage at global level has performed which includes filter design for soilhave waterbeen esitself all requires water. a result these basic needs along ensemble Kalman filter. data Several such studies performed which includes filter design for soil water eswith other uses of water, water usage at global level has grown to a rate more than twice the rate of population timation with synthetic (Walker and Kalma, 2001; grown a rate more than water twiceusage the rate of population timation with synthetic data (Walker and Kalma, 2001; with allto uses of water, at global level has performed which includes filter design for soil water esgrown toother a the rate more than twice the rate of population timation with synthetic data (Walker and Kalma, 2001; increase in 20th century (Steduto et al., 2012). In Reichle et al., 2002), remote sensing (Houser et al., 1998) increase in the 20th century (Steduto et al., 2012). In Reichle et al., 2002), remote sensing (Houser et al., 1998) grown to a rate more than twice the rate of population timation with synthetic data (Walker and Kalma, 2001; increase in the 20th century (Steduto et al., 2012). In Reichle et al., 2002), remote sensing (Houser et al., 1998) addition, due due to to adverse adverse climate climate conditions conditions water water scarcity scarcity and and ground ground data data (Lannoy (Lannoy et et al., al., 2008), 2008), but but none none of of these these addition, increase in 20th century (Steduto et water al.,confined 2012). In Reichle et al., 2002), remote (Houser etanalysis al., and ground data (Lannoy et sensing al., 2008), but none of 1998) these addition, duethe to adverse climate conditions scarcity has become a global problem and is no more in studies have taken into account observability of has become a global problem and is no more confined in studies have taken into account observability analysis of addition, duea to adverse climate conditions water scarcity and ground data (Lannoy et al., 2008), but none of study these studies have taken into account observability analysis of has become global problem and is no more confined in arid and semi-arid regions. Agriculture is the sector where the system which ensures reliable estimates. In this arid and semi-arid regions. Agriculture is the sector where the system which ensures reliable estimates. In this study has become a global problem and is no more confined in the studies haveobservability takenensures into account observability analysis of arid and semi-arid regions. Agriculture is the sector where system which reliable estimates. In this study water scarcity has greatest relevance due to the fact that we discuss analysis of the agro-hydrological water scarcity has regions. greatest Agriculture relevance due to the factwhere that we discuss observability analysis of the agro-hydrological arid and semi-arid is the sector the system which ensures reliable estimates. In this study we discuss observability analysis of the agro-hydrological water scarcity has greatest relevance due to the fact that 70 percent percent of of global global fresh fresh water water is is used used for for agriculture agriculture system system and and discuss discuss the the importance importance of of soil soil profile profile discretizadiscretiza70 water scarcity greatest relevance due toforthe factwater that we discuss observability analysis ofofthe system andon discuss theobservability. importance soil agro-hydrological profile discretiza70 percent of has global fresh water is used agriculture (Aquastat, 2012), and among all other agricultural tion based system (Aquastat, 2012), and among all other agricultural water tion based on system observability. 70 percent of global fresh water is used for agriculture andon discuss theobservability. importance of soil profile discretization based system (Aquastat, 2012), among all least other agricultural water system usage irrigation is operated with efficiency. usage irrigation is and operated with (Aquastat, 2012), among all least otherefficiency. agricultural water tion based on system observability. usage irrigation is and operated with least efficiency. 2. DEVELOPING DEVELOPING STATE STATE SPACE SPACE SYSTEM SYSTEM 2. The irrigation practice involves open usageusual irrigation is operated with least efficiency. The usual irrigation practice involves open loop loop control control 2. DEVELOPING STATE SPACE SYSTEM The usual irrigation practice involves open loop control where no feed back is considered and irrigation is mostly where no feed back ispractice considered and irrigation mostly 2. DEVELOPING STATE SPACE SYSTEM The irrigation involves open loopis whereusual no feed is considered and irrigation is control mostly done based on back farmers observation which in terms terms would 2.1 2.1 Agro-hydrological Agro-hydrological model model done based on farmers observation which in would wherebased no feed back is considered and irrigation is mostly done on less farmers observation which in terms would 2.1 Agro-hydrological model cause over or irrigation. The main barrier in implecause over or less irrigation. The main barrier in imple2.1 agro-hydrological Agro-hydrological model mainly considers the system done based on less farmers observation whichbarrier inis terms would cause over or irrigation. main implementing closed loop control in lack of An agro-hydrological model model mainly considers the system menting closed loop control The in irrigation irrigation is the thein lack of An cause over or less irrigation. The main barrier in impleAn agro-hydrological model the system soil water matrix. The inputs and of menting closed loop controlwhich in irrigation isthen the used lack for of with soil moisture measurements could be with soil water matrix. Themainly inputs considers and outputs outputs of this this soil moisture measurements which could be then used for An agro-hydrological model mainly considers system menting closed loop control in irrigation is the lack of with soil water matrix. The inputs and outputs of this soil moisture measurements which could be then used for system includes includes events events such such as as rain, rain, root root water waterthe extraction feedback control. control. Since Since agricultural agricultural areas areas are are vast vast and and system extraction feedback with soil water matrix. The inputs and outputs of this soil moisture measurements which could be are then used the for by plants includes events such as rain, root water and as in 1. feedback control. Since agricultural areas diverse and it to all over by plants and drainage drainage as shown shown in figure figure 1. extraction diverse and it is is impractical impractical to put put sensors sensors all vast over and the system system includes events such as rain, root water extraction feedback control. Since agricultural areas are vast and diverse andtherefore it is impractical putmoisture sensors profile all over the by plants and drainage as shown in figure 1. entire field the overalltosoil soil moisture profile needs entire field the overall needs Theplants dynamics of this this as agro-hydrological system can be be and drainage shown in figure system 1. diverse andtherefore it is impractical tosoil putmoisture sensors profile all over the by The dynamics of agro-hydrological can entire field therefore the overall needs to be estimated. to be estimated. dynamics of this agro-hydrological system can be expressed using Richards equation (Richards, 1931), which entire field therefore the overall soil moisture profile needs The expressed using Richards equation (Richards, 1931), which to be estimated. The dynamics of this agro-hydrological system can be expressed using Richards equation (Richards, 1931), which is shown below : Soil moisture at unmeasured regions can be estimated to bemoisture estimated. shown below :Richards equation (Richards, 1931), which Soil at unmeasured regions can be estimated is expressed using is shown below : Soil moisture at unmeasured regions can be estimated ∂θ using state and model. The ∂h − 1)] − S(t) ∂θ: = ∂ ∂ [K(θ)( ∂h using state estimators estimators and agro-hydrological agro-hydrological The is shown below (1) Soil moisture atofunmeasured regions cansystem bemodel. estimated ∂θ ∂ [K(θ)( ∂h using statestates estimators agro-hydrological model. − 1)] − S(t) (1) = ∂z unknown states thisand agro-hydrological are The the ∂z ∂t unknown of this agro-hydrological system are the (1) [K(θ)( = ∂z − 1)] − S(t) ∂z ∂t ∂h ∂θ ∂ using state estimators and agro-hydrological model. The unknown states of this agro-hydrological system are the water pressure heads at different depths, which gives us 3 3 ∂z − 1)](cm ∂z [K(θ)(content ∂tsoil water pressure heads atagro-hydrological different depths, system which gives us where − S(t) (1) = moisture 3 /cm3 ), K is the θθ is the where is the soil moisture content (cm /cm ), K is the unknown states of this are the 3 ∂z ∂z (cm/h) ∂tsoil moisture water pressure heads at different depths, which gives us where 3 /cm3 3 ), K is the θ is the content (cm hydraulic conductivity which itself is a function Natural Sciences and Engineering Research Council, Canada. hydraulic (cm/h) which(cm itself Natural Sciencesheads and Engineering Research Council, Canada. water pressure at different depths, which gives us where 3 3 a function θ is conductivity the soil moisture content /cmis is the hydraulic conductivity (cm/h) which itself is ),a K function Natural Sciences and Engineering Research Council, Canada. Natural Sciences and Engineering Research Council, Canada. hydraulic conductivity (cm/h) which itself is a function
Copyright © IFAC 110 Copyright © 2017 2017 110 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 110 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 110 Control. 10.1016/j.ifacol.2017.12.020
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where ∆t is the time interval between time instants k and k + 1, ∆zi is the thickness of compartment i and Ki−1/2 (hk ) is the algebraic mean of hydraulic conductivity from compartments i and i + 1. Evaporation
Rain, irrigation
Eq. (4) can be rearranged as: k+1 k k+1 + ai (hk )hi+1 = ei (hk ) − uki di (hk )hk+1 i−1 + bi (h )hi (5) where ai (h) includes the coefficients of hi+1 (k + 1), di (h) includes the coefficients of hi−1 (k + 1), bi (h) includes the coefficients of hi (k +1), ei (h) includes the coefficients from time instant k, and ui (k) includes all source or sink terms shown in Eq. 4.
Root water extraction Surface runoff
Nodes or soil compartments
Soil layers
Eq. (5) represents water flow inside a single compartment i. For N compartments there will be N of these equations which can be expressed with the following matrix. A(hk )hk+1 = E(hk ) + uk (6) h = A(hk )−1 E(hk ) + A(hk )−1 uk where A(h) is a N × N tridiagonal matrix. The boundary conditions of the system are considered as source and sink in the model and are included in u. Eq. (6) is the nonlinear state equation of the system. k+1
Drainage or deep seepage
Fig. 1. The agro-hydrological system with four different soil layers. of soil moisture (Mualem, 1976), z is the vertical distance inside soil matrix (cm), S includes the source and sink terms (cm3 cm−3 h−1 ) and h is the matric potential (cm). Root water extraction is a sink term in the model. The differential soil moisture term in Richards equation can be represented by a differential pressure term using hydraulic capacity, which is shown as follows: dh ∂ ∂h C(h) = [K(h)( − 1)] − S(t) (2) dt ∂z ∂z where C is the hydraulic capacity. The soil moisture content θ can be expressed in terms of matric potential h in the following way (Genuchten, 1980): θ(h) = θres + (θsat − θres )(1 + |αh|η )−m (3) where θres is the residual moisture content (cm3 /cm3 ), θsat is the saturated moisture content (cm3 /cm3 ), η and m are empirical shape factors, and α is the inverse of air entry suction (cm−1 ).
The system output equation is given by: y = DΘ (7) where D is a matrix indicating at which nodes the soil moisture contents are measured and Θ is a N × 1 vector representing the soil moisture using Eq. 3 for N nodes. 2.3 Model validation The non-linear statespace model as shown above is validated against HYDRUS 1d model (Simunek et al., 2005) by simulating soil moisture for a soil profile of 110 cm with four different soil layers. The first 15 cm is represented by layer 1, the next 35 cm is represented by layer 2, then another 40 cm represents layer 3 and lastly the remainder 30 cm represents layer 4. We next consider a constant flux of 0.2 cm/h infiltration at the top boundary. The bottom boundary condition was considered as free drainage. Under no crop condition the soil moisture were simulated for 48
2.2 Model discertization Richards equation is a highly nonlinear partial differential equation, but can be discretized and solved numerically. The discretized Richards equation is developed by implicit backward, finite difference scheme with explicit linearization of hydraulic conductivities following Van Dam et al. (2008) and is shown by Eq. 4. hk+1 − hk+1 1 − hki i + = Ki−1/2 hk 1 i−1 t zi 2 (∆zi−1 + ∆zi ) hk+1 − hk+1 i+1 − Ki+1/2 (hk ) Ki−1/2 (hk ) − Ki+1/2 (hk ) 1 i (∆z + ∆z i i+1 ) 2 −Sik (4) hk+1 Ci (hki ) i
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Fig. 2. The simulated results of soil moisture after 48 hours starting from 0.24 cm/cm using HYDRUS 1d model and developed model.
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hours starting from 0.24 cm/cm and the final results are presented in figure 2. The result clearly shows that the results from the developed model closely matches with the results from HYDRUS 1d model.
Table 1. Parameter values used for simulating the system. Parameter Ksat
3. OBSERVABILITY OF THE SYSTEM When the knowledge of the input and the output over a finite time interval [t0 , tf ] is adequate to uniquely determine the initial state x(t0 ) then the system is said to be observable (Chen, 1995). To determine the observability of the nonlinear system, the agro-hydrological system is linearized at different operating conditions and the local observability of the system is then determined (Zeng et al., 2016). The linearization of the system is performed using the system Jacobians F k and Gk which are obtained numerically and are given by: ∂f |ˆ k|k k (8) ∂h h ,u ∂g |ˆ k+1|k (9) Gk+1 = ∂h h where f = A(hk )−1 E(hk ) + A(hk )−1 uk and g = DΘ representing Eq. (6) and Eq. (7), respectively. Fk =
To study system observability at first the rank test is performed. The system is considered to be observable at time k if the observability matrix Ok has a full column rank, where the observability matrix is given by: Gk Gk F k . Ok = (10) ... Gk (F k )N −1 Since the rank test does not provide any information on how strongly or weakly observable a system is, therefore we further perform degree of observability analysis of an observable system using a deterministic approach as mentioned by Ham (1980).
θres
θsat
α
η
Soil Layer 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Value 1.0 0.250 1.0 0.250 0.05 0.04 0.03 0.03 0.43 0.46 0.45 0.46 0.016 0.016 0.016 0.016 1.37 1.37 1.37 1.37
4.2 Observability analysis To perform the state estimation of this system the soil profile needs to be discretized such that the system remains observable. In this study, the soil profile is discretized into 44, 33, 31, 23 and 15 nodes in order to create the system with 44, 33, 31, 23 and 15 states, respectively. It was found that for 44 states, the rank of Ok is not full at any portion of the time trajectory, therefore the system is unobservable. The cases with 33 states and 31 states is unobservable at some parts of the time trajectory so they are partially observable as shown in Figure (3). But if the number of states are reduced to 23 states the system becomes observable in terms of rank test. In the figure both 15 states and 23 states are observable but to better understand the system we perform further analysis based on degree of observability.
In this deterministic approach the row vectors of Ok are normalized into unit length vectors N and the relative measure of system observability is given by the ratio of the largest and the smallest eigenvalue of the matrix N T N . For highly observable system this ratio denoted by λmax /λmin will be one or closer to one. Due to the time varying property of the agro-hydrological system the observability analysis is performed over the entire time trajectory. 4. RESULTS 4.1 Data description The study was conducted with synthetic data considering a soil profile of 110 cm with four different homogeneous soil layers. The soil parameter values considered are shown in table 1. The synthetic data was generated using the agro-hydrological model and weather data from St.Albert Weather Station, Canada (Government of Canada, 2013). It is considered that only four measurements are available at depths 5 cm, 20 cm, 50 cm and 100 cm. The measurement data were collected on hourly basis. 112
Fig. 3. The rank of the observability matrix of the system with different states. The results for degree of observability is shown in Figure (4). The result shows that as we reduce the number of states λmax /λmin starts to decrease which suggests that the system becomes more observable. Although system
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with 23 states and 15 states both can be considered observable based on rank test, the eigenvalue analysis suggests that the system with 15 states is more observable relative to the system with 23 states. The system with 33 and 31 states which was partially observable performed very poorly in terms of the degree of observability analysis. It was observed that fewer states would ensure better observability, but at the same time reducing the number of states would create issues in the numerical solution of equation 4. In this work the system was not reduced below 15 states because of this reason. In both the rank test (figure 3) and relative observability analysis (figure 4), it is seen that most of the unobservability occurred in the first part of simulation. Though the system with 23 and 15 states were observable in the rank test all over the time trajectory as shown in figure 3 but the relative observability were initially poor as shown in figure 4 due to higher value of λmax /λmin . One possible reason for such behavior is the fact that extreme dry or wet soil condition persisted at those operating regions. The saturated soil moisture for this soil was set between 0.43 and 0.45 as shown in table 1 and the initial soil moisture of the system was considered between 0.35 and 0.4. Though these values do not refer to extreme wet condition but they are relatively near those extreme regions. Linearizing the system in that near saturation region may have introduced some modeling errors.
Fig. 4. The ratio of the largest and the smallest eigen value of matrix N T N for systems with 33, 31, 23 and 15 states at different time trajectory. 4.3 Performance of EKF To investigate the performance of EKF based on observability, an comparative analysis based on system with 31 states, which is partially observable, and system with 15 states, which is fully observable, is performed. The same synthetic data is used but small amount of noise was added to the data. The noise is generated using MATLAB function ’randn’ and only 1% of this noise is added with the measurements. The process noise covariance Qk is taken as 1e5I and the measurement noise covariance is taken as 0.5I where I is the identity matrix with dimension of states. Since we want to compare the EKF performance, 113
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from four measurements only three were chosen for EKF update. They are measurements from depth 5 cm, 50 cm and 100 cm. The soil moisture at depth 20 cm is estimated by the EKF and compared against the actual data. From figure 5, it is clearly observed that the EKF output given by system with 15 states performs better than the system with 31 states. In the estimated node, that is depth 20 cm, the EKF estimates for 15 states converge to actual value but the estimate with 31 states always had a bias. In fact in the case where the measurements are available, that is depth 5 cm, 50 cm and 100 cm, the EKF output from 31 states fails to track the actual data. This result agrees with the system observability analysis. The system with poor observability would not perform well in EKF relative to a system with good observability.
Fig. 5. The extended Kalman filter result at four depths for systems with 31 and 15 states. The measurements from depth 5 cm, 50 cm and 100 cm were used in EKF for updates. The measurement from depth 20 cm was not used inside EKF, this was estimated. 5. CONCLUDING REMARKS The agro-hydrologial system under study is highly nonlinear. To perform state estimation to determine soil moisture, this highly nonlinear model requires linearization at many steps. In such case a rank test itself may not represent the system well. For example the observability of the system with 31 state depends upon the time trajectory. If the time trajectory is not sufficiently long then it would
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be hard to understand whether the system is observable or not. So it is better to perform degree of observability analysis along with the rank test to better understand the system. If the system is observable by the rank test and has fairly low λmax /λmin ratio then it would be reasonable to consider that nonlinear system to be observable over the entire time trajectory. The degree of observability may also be used for determining measurement locations. In that case the eigenvectors corresponding to lowest eigenvalues must be analyzed. An observable system would give much better performance in Kalman based estimation. Moreover when a discretized numerical scheme is used to develop the agro-hydrological model, the discretization or the designing of the soil profile must be done such that the system observability is ensured. This would eventually increase the ease of control application in such system. ACKNOWLEDGEMENTS We would like to thank Agriculture and Agri-Food Canada, Alberta Agriculture and Forestry for providing field and weather data. REFERENCES Aquastat (2012). FAO’s information System on Water and Agriculture. http://www.fao.org/nr/water/aquastat/water use/ index.stm. Chen, C.T. (1995). Linear system theory and design. Oxford University Press, Inc. Genuchten, M.T.V. (1980). A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil science society of America journal, 44(5), 892–898. Government of Canada (2013). Current and historical Alberta weather station data viewer, AgroClimatic Information Service. http://agriculture.alberta.ca/acis/alberta-wea ther-data-viewer.jsp. [Online; accessed August 01,2014]. Ham, F.M. (1980). Determination of the degree of observability in linear control systems. Ph.D. thesis, Iowa State University. Houser, P.R., Shuttleworth, W.J., Famiglietti, J.S., Gupta, H. V.and Syed, K.H., and Goodrich, D.C. (1998). Integration of soil moisture remote sensing and hydrologic modeling using data assimilation. Water Resources Research, 34(12), 3405–3420. Lannoy, G.J.D., Houser, P.R., Pauwels, V., and Verhoest, N.E. (2008). State and bias estimation for soil moisture profiles by an ensemble kalman filter: Effect of assimilation depth and frequency. Water Resources Research, 43(6). Mualem, Y. (1976). A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resources Research, 12(3), 513–522. Reichle, R.H., Walker, J.P., Koster, R.D., and Houser, P.R. (2002). Extended versus ensemble Kalman filtering for land data assimilation. Journal of Hydrometeorology, 3(6), 728–740. 114
Richards, L.A. (1931). Capillary conduction of liquids through porous mediums. Journal of Applied Physics, 1(5), 318–333. Simunek, J., Van Genuchten, M.T., and Sejna, M. (2005). The hydrus-1d software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media. University of California-Riverside Research Reports, 3, 1–240. Steduto, P., Faur`es, J., Hoogeveen, J., Winpenny, J., and Burke, J. (2012). Coping with water scarcity: an action framework for agriculture and food security. Food and Agriculture Organization of the United Nations Rome. Van Dam, J.C., Groenendijk, P., Hendriks, R.F.A., and Jacobs, C.M.J. (2008). SWAP version 3.2: Theory description and user manual. Wageningen UR, Alterra, Postbus 47, 6700 AA Wageningen, The Netherlands, 2 edition. Walker, J. P. andWillgoose, G.R. and Kalma, J.D. (2001). One-dimensional soil moisture profile retrieval by assimilation of near-surface observations: a comparison of retrieval algorithms. Advances in Water Resources, 24(6), 631–650. Zeng, J., Liu, J., Zou, T., and Yuan, D. (2016). Distributed extended kalman filtering for wastewater treatment processes. Industrial & Engineering Chemistry Research, 55(28), 7720–7729.