Determining Optimal Sensor Locations for State and Parameter

Ind. Eng. Chem. Res. 2005, 44, 5645-5659

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Determining Optimal Sensor Locations for State and Parameter Estimation for Stable Nonlinear Systems Abhay K. Singh and Juergen Hahn* Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122

This paper presents a new approach for sensor location for state and parameter estimation for stable nonlinear systems. The unique feature of this technique is that sensor locations for state estimation and measurement locations for parameter estimation can be determined within the same framework. To compute optimal sensor locations, information derived from observability covariance matrices is combined with already existing measures, which were proposed either for state or parameter estimation, to compute the degree of observability of a nonlinear system over an operating region. The optimal sensor locations then correspond to the configuration that returns the highest value of the measure for the degree of observability of a system. The proposed method is illustrated in case studies where optimal sensor locations for state and parameter estimation for a binary distillation column and a fixed-bed reactor are computed. The results obtained from the presented approach are compared with a technique based upon a linearized system. 1. Introduction Modern chemical plants have to satisfy a variety of criteria, e.g., safety and environmental requirements, which can only be met if a sufficient amount of information about the state of the plant is available. Incidents of failures of sensors to respond to process upsets, due to improper locations, have been reported in the past.1 The failure of sensors to respond accurately can not only create operational difficulties but can also have detrimental effects on the life of process equipment and on the ability to meet environmental requirements. While it is possible to install additional sensors to monitor many aspects of a plant’s operation, this can be expensive both in terms of the initial as well as the maintenance costs. Model-based state and parameter estimation techniques can be used to address this problem: by strategically measuring some key variables of the process, it is possible to reconstruct other variables and parameters by using an observer. However, to gain the largest benefit from this technique, the sensors have to be placed at “optimal” locations. Determining optimal sensor locations in a chemical plant is complicated by the facts that most chemical processes are nonlinear in nature and available tools are mainly confined to linear or linearized systems. Additionally, the current state of the research is that separate approaches are used for determining sensor locations for state estimation and for computing sensor locations for parameter estimation. The technique presented in this paper will address these points since (1) it is applicable to linear as well as nonlinear stable systems and (2) it can determine optimal sensor locations for state estimation as well as sensor locations for parameter estimation within the same framework. Additionally, it can be shown that the technique reduces to already existing approaches for sensor location for state or parameter estimation if the investigated system is linear and if only one of the objectives, i.e., sensor * Corresponding author. Tel.: (979) 845-3568. Fax: (979) 845-6446. E-mail: [email protected].

location for state estimation or measurement location for parameter estimations, has to be met. The main idea of the presented technique is to combine the computation of the observability covariance matrix2 with established measures for locating sensors for linear systems.3-6 This combination of directly using the nonlinear model and applying established measures for linear systems allows more accurate prediction of optimal sensor locations while at the same time resulting in a computationally tractable procedure. Additionally, by augmenting the system with the parameters to be estimated, it can be shown that a submatrix of the observability covariance matrix, i.e., the part of the matrix corresponding to the augmented states, is closely related to the Fisher information matrix (FIM) commonly used for sensor location for parameter estimation. The technique is illustrated in case studies where the optimal sensor locations for state estimation for a binary distillation column and for a fixed-bed reactor are computed. Additionally, the optimal sensor location for on-line estimation of the heat transfer coefficient, for the fixed-bed reactor example, and the relative volatility, for the distillation column model, is determined. The results obtained from the presented approach are compared with a technique based upon the linearized system. 1.1. Previous Work. In the past three decades, there have been a number of contributions on sensor location for state estimation. In these contributions, several measures have been defined to place sensors on a system. Optimality criteria based on the error covariance matrix of Kalman filters have been presented by Omatu et al.7 Similar measures, like the trace and the determinant of error covariance, have been used by Kumar and Seinfeld,8 Colantuoni and Padmanabhan,10 and Harris et al.9 Jorgensen et al.11 base their technique on qualitative knowledge of process variables in addition to the variance of the state prediction error. Morari and O’Dowd12 and Morari and Stephanopoulos13 employ nonstationary noise models and present criteria based on minimizing the estimation error caused by the

10.1021/ie040212v CCC: $30.25 © 2005 American Chemical Society Published on Web 06/18/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005

unobservable subspace. Romagnoli et al.14 and Alvarez et al.15 present a technique that considers variable measurement structures. Another category of measures for sensor location is based on the observability matrix or the observability gramian. Muller and Weber3 use the smallest eigenvalue, the determinant, and the trace of the inverse of the gramian as measures for sensor location. Damak et al.16 and Dochain et al.4 present the condition number of the observability matrix as a criterion for sensor location. Similar measures have been employed by Waldraff et al.,5 in the form of the smallest singular value and the condition number of the observability matrix, for sensor location. van den Berg et al.6 make use of the trace and the spectral norm of the observability gramian as the criteria for sensor location in their work. In addition to the above presented measures, there are contributions that take into account measurement cost and sensor failure in addition to process information. Ali and Narasimhan17 introduced a new concept considering reliability for sensor placement by using the probability of sensor failure in addition to observability and measurement redundancy. Bagajewicz18 and Chmielewski et al.19 present techniques related to the minimization of cost, subject to constraints related to data reconciliation. Muske and Georgakis20 present a sensor location technique that provides the best compromise between measurement cost and process information. While these techniques return good results for the systems to which they were applied, it is important to note that some processes may not be accurately described by linear systems over their operating region. Therefore, it is desirable to have a sensor location technique that captures the nonlinear behavior of the system. However, nonlinear observability analysis21,22 is computationally expensive for sensor location.16 Lopez and Alvarez23 have defined measures, based on an approach similar to the one presented in refs 21 and 22, to compute the degree of estimatability for nonlinear systems. These measures reduce to a criterion based on the linear observability matrix if the system is linear and time invariant. Another alternative for the observability analysis of nonlinear systems is the nonlinear observability gramian presented by Scherpen.24 Georges25 has used these nonlinear observability gramians for sensor location. However, the application of this method is limited to low-order systems because of numerical difficulties. Other significant contributions on sensor location for nonlinear systems include the work by Wouwer et al.26 and Alonso et al.27 for distributed systems. A closely related problem to sensor location for state estimation is sensor location for parameter estimation. However, extension to sensor location for parameter estimation is not straightforward.28 Unlike sensor location for state estimation, there are very few contributions that address optimal sensor location for parameter estimation.26,29 Most of the existing optimality criteria for sensor location for parameter identification are based on scalar measures of the Fisher information matrix,30,31 e.g., the determinant of the Fisher information matrix, which has been used by Qureshi et al.28 and Wouwer et al.26 Using a related approach relying on principal component analysis of the output sensitivity matrix, Li et al.32 computed the best set of parameters to be

estimated if a set of measurement locations is already given. All of these techniques require computation of the parametric-output sensitivity coefficients based on local sensitivity analysis. Therefore, the results obtained by these methods may not capture the nonlinearity and may only be suitable for small changes in the parameters.33 2. Preliminaries 2.1. Observability. Observability refers to the property of a system that allows the reconstruction of the values of the state variables given the outputs.34 For linear time-invariant systems of the form

x3 ) Ax + Bu

(1a)

y ) Cx + Du

(1b)

the observability gramian

WO,linear )

∫0∞eA tCTCeAt dt T

(2)

can be computed in order to determine the observability of the system. If the observability gramian is a matrix of full rank, then the system is observable. However, if the gramian is rank deficient, then the system will not be observable and some of the states (or directions in state space) cannot be reconstructed from the output data. 2.2. Observability Covariance Matrix. While the observability gramian can be used for determining the observability of linear systems, it may not result in sufficient information if the system is nonlinear and has strongly varying operating conditions. Extensive efforts have been undertaken in the last two decades to derive conditions for the observability of nonlinear systems.21,22,24 However, the results derived from these conditions are usually too complex to be interpreted for all but very simple systems. One alternative is to use the relatively new concepts of the observability covariance matrix2 or the empirical observability gramian.35 To present the definition, the following quantities need to be defined:

Tn ) {T1, ..., Tr; Ti ∈ Rn×n, TTi Ti ) I, i ) 1, ..., r} M ) {c1, ..., cs; ci ∈ R, ci > 0, i ) 1, ..., s} En ) {e1, ..., en; standard unit vectors in Rn} with r being the number of matrices for the perturbation directions, s being the number of different perturbation sizes for each direction, and n being the number of states of the system. The observability covariance matrix can be computed for stable nonlinear systems

x3 ) f(x,p,u)

(3a)

y ) h(x,p,u)

(3b)

where p denotes a vector of parameters and is given by r

WO )

s

∑ ∑ l)1m)1

1

∞ TlΨlm(t)TTl dt ∫ 0 2

(4)

(rscm )

where Ψlm(t) ∈ Rn×n corresponds to Ψijlm(t) ) (yilm(t) -

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yss)T(yjlm(t) - yss); yilm(t) is the output of the system corresponding to the initial condition x(0) ) cmTlei + xss; and yss is the steady-state output of the system. The observability covariance matrix defined above and the empirical observability gramian differ from each other in the way Ψlm(t) is computed. The steady-state output, “yss” in Ψlm(t), is the initial steady state while computing the empirical gramians, and it is equal to the final steady state for the observability covariance matrix. If the system is asymptotically stable, then the observability covariance matrix reduces to the empirical observability gramian. The observability covariance matrix or the empirical observability gramian are determined from simulation data, collected within the operating region. The state trajectories generated by simulation are computed only for the nominal value of the inputs. Hence, observability analysis by the covariance matrices or by the empirical gramians is applicable only for nominal values of the inputs. Selection of the T and M matrices is important for computing the observability covariance matrix: The T matrix is usually chosen to be {I, -I}, where the submatrix {I} refers to positive perturbations in the state variables while {-I} corresponds to decreases in the value of the state variables. More complicated choices for T are possible for some nonlinear systems as long as they satisfy the conditions stated above. The matrix M contains the different perturbation sizes. If the system is linear, then any perturbation size will return the same observability covariance matrix.35 However, for nonlinear systems, different perturbation sizes are chosen to capture the nonlinear behavior of the system over an operating region. For example, if during operation the various state variables can undergo 20% perturbation from their steady-state values, then the M matrix can be chosen as M ) {0.01, 0.05, 0.1, 0.15, 0.2}. The largest perturbation size should always be chosen such that system stays in the region of attraction of the nominal operating point. 2.3. Measures for the Degree of Observability. While the rank of the observability gramian can be used in order to determine if a system is observable, this information is not sufficient for determining optimal sensor locations. To address this, a variety of different measures for the degree of observability of linear systems have been introduced, and an overview of several of these is presented below. Muller and Weber3 have outlined three candidates for measuring the degree of observability of a system based upon the linear observability gramian. One measure is

µ1 ) λmin(WO,linear)

(5a)

where the eigenvector corresponding to the smallest eigenvalue λmin refers to the least observable direction of the system and the eigenvalue corresponds to the degree of observability of this worst direction. Higher values of this measure imply a higher degree of observability of the least observable direction. Another measure makes use of the trace of the inverse of the observability gramian

n µ2 ) -1 trace(WO,linear )

of observability. The last measure given by Muller and Weber3 makes use of the determinant of the observability gramian:

µ3 ) [det(WO,linear)]1/n

(5c)

A value of the determinant of zero indicates that the matrix is rank deficient, corresponding to a system that is not observable. Similarly, larger values of the determinant can be used as an indicator of the increased observability of a system. However, it should be pointed out that a single eigenvalue of zero will result in a measure of zero, regardless of the magnitude of all other eigenvalues of the system. Attention has to be paid to these and other factors resulting from representing the state-to-output behavior of a system by a single numerical value. Dochain et al.4 made use of the condition number

CN )

σmax(WO,linear) σmin(WO,linear)

(6)

for the observability analysis of a fixed-bed bioreactor. The σs refer to the smallest and largest singular values of the observability gramian, which are identical to the smallest and largest eigenvalues. A smaller value of the condition number generally implies increased observability of a system. Waldraff et al.5 made use of the smallest singular value of the observability gramian

NS ) σmin(WO,linear)

(7)

as one of the measures for determining sensor locations in tubular reactors. This method is similar to the measure based on the smallest eigenvalue given by Muller and Weber,3 as it serves as an indicator of how far the system is from being unobservable. Higher values of this criterion imply an increased degree of observability. All the above measures are strongly influenced by the smallest singular value or the smallest eigenvalue. While it can be useful to ensure that a minimum degree of observability exists even for the worst directions, this may not always be the best measure for sensor placement. The reason for this is that it is unlikely that every single state of a system needs to be observed; instead, the main focus should be on ensuring that the most important states can be observed easily, e.g., measures which are strongly influenced by the smallest singular values can return misleading information if some states that are not important for plant operation may be unobservable. Some approaches falling into this category are based upon criteria which focus either on the maximal response which can be observed in sensor readings or upon the sum of the observability of all states. van den Berg et al.6 suggested two measures based upon these ideas. The first measure is the spectral radius

F(WO,linear) ) σmax(WO,linear)

(8)

(5b)

where n refers to the number of states of the system. Larger values of the measure indicate a higher degree

which corresponds to the largest eigenvalue (which, in the case of a gramian, is equal to the largest singular value). A large value of the measure indicates that the dominant direction in the observability gramian can be

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3. Optimal Sensor Location for Nonlinear Systems

Table 1. Measures for Degree of Observability smallest eigenvalue

largest eigenvalue

measure

equation

measure

equation

µ1 µ2 µ3 CN NS

eq 5a eq 5b eq 5c eq 6 eq 7

F(WO,linear) trace(WO,linear)

eq 8 eq 9

easily observed, while the trace of the observability gramian n

trace(WO,linear) )

σi(WO,linear) ∑ i)1

(9)

can be interpreted as the sum of the singular values of the matrix. Larger values of the trace correspond to an increase in the overall observability of a system. Summarizing, the presented measures can be put into one of two categories: measures which are mainly (or even exclusively) based upon the least observable direction in state space, and measures which are predominantly influenced by the largest eigenvalue of the observability gramian. These findings are summarized in Table 1. 2.4. Observability Analysis and Sensor Placement Based on Linear Gramians. Sensor locations are usually determined in order to be able to directly measure certain system states or to reconstruct required information about a system. As a minimum requirement, the controlled variables need to be measured for feedback control. However, other considerations, e.g., safety or product quality, may result in a need for additional measurement locations. Observability analysis can be performed in order to maximize the amount of information gained from the available measurements. Thereby, it is possible to determine an a priori estimate for the sensor locations in order to (1) get the most information from a certain number of measurements and (2) use as small a number of measurements as possible to obtain a required amount of information about the system. Perturbations in any of the state variables must be reflected in the sensor readings. This situation will be the case if the gramian has full rank. However, in addition to being able to determine if state perturbations are reflected in the available measurements, it is also important to determine the degree of computational difficulty for extracting this information. If the observability gramian has full rank but is ill-conditioned, then it is most likely not possible to reconstruct all the states or to estimate the process parameters in a realistic setting, i.e., under the influence of plant-model mismatch and measurement noise. The following procedure is usually applied for determining optimal sensor locations for linear systems based upon the observability gramian: observability gramians are computed for a variety of combinations of sensors at different locations. Scalar measures are computed from the gramians in order to compare the degree of observability of various locations for the sensor placement. A sensor configuration corresponding to the largest value of an observability measure indicates a good candidate for optimal sensor location.

3.1. Sensor Location for State Estimation. Since all systems in nature are nonlinear to a certain degree, a gramian of the linearized system will not always result in a good description of the state-to-output behavior if the operating conditions of a process can vary significantly. Because of this, it is possible that observability analysis based upon linear gramians may result in conclusions not in line with the physical knowledge of a system. To address this issue, the methodology presented in this work is based upon observability covariance matrices that are directly computed for the nonlinear system. This has the advantage that it can result in a more accurate description of the state-tooutput behavior than if the gramian of the linearized system is used,2 and at the same time, it is computationally much less expensive than if Lie-algebra-based approaches21,22 or nonlinear observability gramians24 would be used for the observability analysis of nonlinear systems. While the observability covariance matrix is used for observability analysis, some of the measures that have originally been proposed for the quantification of the degree of observability of linear/linearized systems can still be incorporated into this procedure: instead of being used to compute the measures for the observability gramian of the linear system, they are now applied to the observability covariance matrix. Applying the same measures to the covariance matrix instead of the linear gramian is possible because the covariance matrix is an n×n, symmetric, and positive semidefinite matrix, similar to the gramian of a linear system. Since the observability covariance matrix can be viewed as an extension to the gramian of the linear system,2 the information extracted from these measures can be viewed as a measure for the “degree of observability” of the nonlinear system over an operating region. To compute the observability covariance matrices, state trajectories are generated by simulation of the nonlinear model. These state trajectories are generated such that they capture the behavior over the required operating region. The computed covariance matrices are then used for the observability analysis of the nonlinear system, and the degree of observability for a chosen sensor location is determined. The optimal sensor location is computed by maximizing the degree of observability over the entire set of possible measurements. The procedure for computing optimal sensor locations by this approach is summarized in Figure 1. 3.2. Sensor Location for Parameter Estimation. Sensor location for parameter estimation is closely related to sensor location for state estimation. However, a modification of the approach needs to be used to determine the degree of observability of the process parameters. This is achieved by augmenting the original nonlinear system (eq 3a) with equations for the parameters to be estimated:

() (

f(x,p,u) x3 ) p˘ 0 y ) h(x,p,u)

)

(10a) (10b)

The parameters are assumed to be constant or slowly varying with time. Observability analysis and sensor location are then performed for this augmented system.


Figure 1. Sensor location procedure for state estimation.

The covariance matrix of the augmented system can be decomposed into:

(

W W WO ) WO,nn WO,np O,pn O,pp

)

(11)

where the submatrix WO,nn represents the observability covariance matrix of the system (eq 3) before augmentation with the parameters as additional states, WO,pp represents the variance-covariance of the outputs caused by changes in the parameters, and WO,np and WO,pn represent the covariance of the outputs resulting from changes in the state variables and parameters. For small perturbations, WO,pp is closely related to the Fisher information matrix for a linearized system and, therefore, local identifiability of the parameters for the nonlinear system can be established (see Appendix I). However, for this work, it is not required to linearize the system, as these matrices can be directly computed from data generated by the nonlinear model. Therefore, they capture the behavior of the nonlinear system over

the operating region in addition to guaranteeing the local identifiablity of the parameters. To ensure that the parameters are observable, the number of parameters must be less than the number of outputs. However, this is not a just a restriction of the presented method but a general limitation of the observability of parameters for any system (see Appendix II). The degree of observability of the set of parameters is determined by a suitable measure of the submatrix WO,pp. The diagonal elements of this matrix represent the variance that changes in the parameters cause in the outputs, and the other entries are an indicator of the degree of interaction between the parameters. Hence, measures based on the submatrix WO,pp can be used to determine the best sensor location for parameter estimation. Since WO,pp is a square, symmetric, and positive semidefinite matrix, similar measures as defined in Section 2.3 can be used to estimate the degree of observability of the parameters:

ω ) measure(WO,pp)

(12)

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Figure 2. Sensor location procedure for parameter estimation.

The optimal sensor location for parameter estimation is computed by maximizing the measure ω (where the measures to estimate the degree of observability of the parameters are given by eqs 5-9) over the entire set of possible measurement locations. For the case where only a single process parameter needs to be estimated, WO,pp reduces to a scalar that needs to be maximized. The

sensor location technique for parameter estimation is described in Figure 2. 3.3. Sensor Location for State and Parameter Estimation. One important observation is that the approach presented in this paper does not distinguish between sensor locations for state estimation and measurement determination for parameter estimation.


Unlike measures proposed previously in the literature, both of these goals can be accomplished within the same framework. This is achieved by treating the parameters as additional states and computing the observability covariance matrix for the augmented system given by eq 10. Since the observability covariance matrix can be decomposed as shown in eq 11, it is possible to apply the measures for sensor location for state estimation to WO,nn, while measures for sensor location for parameter estimation can be applied to the submatrix WO,pp. This avoids the need for two separate analyses for optimal measurement determination for state and parameter estimation. It is important to point out that WO,nn in the decomposed observability covariance matrix (eq 11) is the observability covariance matrix defined in Section 2.2 for state estimation. If the equilibrium point of the original system is asymptotically stable, this matrix reduces to the empirical observability gramians35 that can be used for sensor location for state estimation for nonlinear systems.36 Further, if the system is linear, WO,nn reduces to the linear observability gramian and the sensor location procedure reduces to existing techniques (e.g., the one by van den Berg et al.6). At the same time, WO,pp from the observability covariance matrix (eq 11) is closely related to the parameteroutput sensitivity matrix given in Wouwer et al.26 and Li et al.32 and to the Fisher information matrix (See Appendix I). Therefore, the presented technique can be viewed as an extension of existing approaches for sensor location, as (1) it can be used for nonlinear systems without resorting to linearizations and (2) it integrates methods for sensor location for state and for parameter estimation within one framework. 3.4. Measuring the Observability of Specific States or Estimating a Secondary Measurement. During process operations, it is often required to have precise information about a particular state for process control or quality control of the product. In these cases, the values of specific states are of greater importance than the overall observability of the process. The observability of a specific state can be determined by analyzing the diagonal entry of the observability covariance matrix corresponding to this state.36 This diagonal entry represents the variance of the output caused by perturbations in this particular state. Hence, the larger the magnitude of this entry, the easier it will be to reconstruct the value of this state from the available measurements. This procedure can also be used to estimate a secondary measurement when the desired state cannot be measured directly by employing a sensor, e.g., if a product concentration needs to be known and it is not feasible to measure concentrations. In such cases, the value is computed from measurements of other properties closely related to the state of interest. The presented sensor location framework serves as an indicator of the properties to be measured and the measurement location for the most accurate information about a specific state. 4. Case Studies This section illustrates the presented technique by performing optimal sensor location for state and parameter estimation for a distillation column model as well as a fixed-bed reactor model. The results from the presented approach are compared to locations determined for a linearized model. Additionally, a nonlinear

observer has been designed for parameter estimation for the distillation column. The performance of the nonlinear observer is compared for optimal and nonoptimal sensor locations. 4.1. Process Models. 4.1.1. Distillation Column. Consider a distillation column model with 30 trays for the separation of a binary mixture of cyclohexane and n-heptane. The column has 32 states and is assumed to have a constant relative volatility of 1.6. The feed is introduced in the middle compartment (17th tray) as a saturated liquid. The feed stream has a composition of xf ) 0.5, and the distillate and bottom product purities are xD ) 0.935 and xB ) 0.065, respectively. The boiling points of cyclohexane and n-heptane are 353 and 371 K, respectively, at a constant column pressure. The reflux ratio is held at a constant value of 3.0. The original column model as given in Hahn and Edgar37 and Benallou et al.38 has been modified based on the derivation relating concentrations and temperature presented in Skogestad39 under the assumption of constant molar flows. The column model is described by a set of 32 nonlinear ordinary differential equations with temperatures as state variables and 33 explicit algebraic equations. 4.1.2. Catalytic Fixed-Bed Reactor. The second model is an important industrial process for the vaporphase oxidation of o-xylene to phthalic anhydride. The reaction is highly exothermic and carried out in a fixedbed reactor. The reactor model is assumed to be onedimensional pseudo-homogeneous. The dynamic version of the steady-state model described in van Welsenaere and Froment40 and Froment and Bischoff41 has been used in this work. The resulting model consists of two partial differential equations, one each for the mass and the energy balance along the length of the reactor. The boundary conditions used for this model are p(t,0) ) 0.015 atm and T(t,0) ) 625 K, corresponding to the inlet reactant partial pressure and inlet reactant fluid temperature. The reactor wall temperature is assumed to be equal to the inlet reactant fluid temperature. The infinite dimensional reactor model is discretized in space using finite differences, converting the original model into a set of 2 × n nonlinear ordinary differential equations, where n is the number of discretization points in space. For this work, 60 discretization points were chosen, resulting in a set of 120 nonlinear ordinary differential equations. 4.2. Sensor Location for State Estimation. The presented case studies are restricted to determining the optimal location if a single sensor is used, to make the results easier to interpret. However, if the influence of placing a sensor on one part of the model has only a minor effect on the observability of other parts of the system, then conclusions can be drawn about subsequently placing several sensors. An extension to the current case, where the numbers as well as the location of sensors need to be determined via the solution of an optimization problem, is currently under investigation. 4.2.1. Distillation Column. To determine the optimal location for a temperature sensor in the column, observability covariance matrices are computed for every possible location, i.e., measuring one state at a time and determining the observability covariance matrix for each case. Scalar measures are computed for the observability covariance matrices, and the location resulting in a scalar measure with the largest value is considered the best location for placing a sensor.

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Figure 3. Values of measures for sensor location for distillation column.

When analyzing the results obtained for sensor location (Figure 3) using the spectral norm (eq 8) and the trace (eq 9) of the observability covariance matrix (measures strongly influenced by the largest eigenvalue), both measures return the 26th state in the stripping section as the best sensor location. If, additionally, a sensor has to be placed in the rectification section, then the 7th state would be the best place for a temperature sensor. These results are inline with expectations from physical insight: the temperature sensitivity at the very top and bottom is at a minimum because the driving force for mass transfer is very small compared to the rest of the column,42 and therefore, the best sensor location should be somewhere in the middle of the rectification and/or stripping sections. Along the same lines, Bequette and Edgar43 observed that the most sensitive trays in a distillation column are located approximately one-fourth of the distance from each end of the column. The computed results are in line with both of the observations from the literature. Another conclusion that can be drawn from the results is that the feed tray is not a good choice for the placement of a sensor. This seems physically justified, as the feed tray is affected by the feed temperature and is least sensitive to perturbations around its nominal point. To corroborate these findings, the location of the feed tray has been varied. Figures 4 and 5 show the measures for a feed located at the 10th and the 25th tray. In both cases, the feed tray is one of the worst locations for placing a sensor. However, it should be pointed out that measuring the feed temperature, but not the temperature on the feed tray, has beneficial effects if feedforward control strategies are to be applied. Observability measures based on the smallest eigenvalue have also been computed. However, these measures do not provide consistent results for placing a sensor (Figure 6). Moreover, the minimum eigenvalue, the determinant, the trace of the inverse of the covariance matrix, and the condition number measures cannot be used if the system is partially observable or has poor observability. In such a case, the minimum eigenvalue will be close to zero and no conclusions can be drawn for sensor location with theses measures.36 4.2.2. Catalytic Fixed-Bed Reactor. Since the reactor contains states corresponding to concentrations

Figure 4. Values of measure for sensor location with the 10th tray as the feed tray.

Figure 5. Values of measures for sensor location with the 25th tray as the feed tray.

and temperatures, unlike the column where each state referred to a temperature, the type of measurement also needs to be considered. From the steady-state temperature and concentration profiles (Figures 7 and 8), it can be concluded that small changes in the partial pressure of the reactant cause significant changes in the temperature around the hot spot of the reactor, indicating a high sensitivity of temperature measurements. To further investigate the choice of temperature or composition sensors, the observability covariance matrix is partitioned as

(

W W WO ) WO,TT WO,TC O,CT O,CC

)

where WO,TT represents the variance-covariance of the temperature variables and WO,CC represents the variance-covariance of the composition variables, whereas WO,TC and WO,CT represent the crossterms. For this case study, the norm of WO is always dominated by WO,TT, i.e., the temperature variables, irrespective of the type of measurement used, even if temperatures and concentrations are scaled. Therefore,


Figure 6. Values of measures based on the smallest eigenvalue for sensor location.

it is appropriate to consider placing a temperature sensor in the reactor. To determine the best sensor location, observability covariance matrices are computed for 60 locations along the reactor. Figure 9 shows the values of two measures (the trace (eq 9) and the spectral radius (eq 8)) for placing a temperature sensor along the length of the reactor. Both measures return the best sensor location at or near the hotspot of the reactor. While the trace determines the optimal sensor location to be exactly at the hotspot of the reactor, the optimal sensor location using the spectral radius shifts slightly toward the inlet of the reactor. These results indicate that the trace may be a more appropriate measure for sensor location than the spectral radius because it takes the observability

Figure 7. Steady-state temperature profile of the reactor.

of all modes of the system, and not just the dominant mode into account. 4.3. Sensor Location for Parameter Estimation. 4.3.1. Catalytic Fixed-Bed Reactor. The temperature profile of the reactor (Figure 7) shows that the reactor temperature exhibits a hotspot. A small variation in the process parameters (inlet temperature, inlet concentration, heat transfer coefficient, etc.) can cause significant changes in the temperature at the hotspot and may lead to a shift in the location of the hotspot. Uncertainty in these parameters has an effect not only on the maximum temperature at the hotspot but also on the location of the hotspot. Additionally, the changes in the hotspot are not symmetrical around the nominal point, so choosing a measurement based upon the nominal value of the parameters may not result in an appropriate sensor location. In this case study, the optimum sensor location for determining the overall heat transfer coefficient in the fixed-bed reactor has been investigated. Observability covariance matrices are computed for the augmented system (eq 10) for 60 spatial locations along the reactor. These matrices are than decomposed as shown in eq 11, and the matrix WO,pp is extracted. In this case, WO,pp is a scalar, since only one parameter is considered. Hence, the measure for the degree of observability of parameters such as the spectral radius (eq 8) and the trace (eq 9) reduces to maximizing WO,pp over the entire set of sensor locations. Figure 10 shows the values of WO,pp along the length of the reactor for the nonlinear model and compares the results to WO,pp computed for a linearized model. Uncertainty of 20% in the overall heat transfer coefficient is considered for determining the sensor location. These results show that the analysis based on the linearized model returns sensor locations that are quite different than those of the presented method based

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Figure 8. Steady-state concentration profile of reactor.

Figure 9. Values of the measure for placing a temperature sensor along the length of the reactor for state estimation.

on nonlinear models. For the nonlinear model, the best location for estimating the parameter is 0.65 m after the inlet, whereas the best location for a linearized model is determined to be 0.55 m from the reactor inlet. The reason for the different sensor location for the linearized and the nonlinear model is that the state trajectories following a perturbation, shown in Figures 11 and 12, are different for the linear and the nonlinear case. The transients in these figures are shown to

emphasize the difference in the movement of the hotspot location for the linear and the nonlinear system. It can be seen from the figures that the actual hotspot location, as given by the nonlinear model, shifts to 0.65 m from the reactor inlet for a 20% perturbation in the parameter. This is in contrast to the location for the linearized model that is at 0.5 m from the reactor inlet. 4.3.2. Distillation Column. Sensor location for estimating the relative volatility of the nonlinear distil-


Figure 10. Values of the measure for placing a temperature sensor along the length of the reactor for parameter estimation.

Figure 11. Transient response for 20% perturbation for nonlinear model.

Figure 12. Transient response for 20% perturbation for linearized model.

lation column model is also investigated. An uncertainty of 10% in the relative volatility has been assumed. Like the reactor, the nonlinear and the linearized distillation models return different sensor locations for parameter estimation. The nonlinear model gives the best location for estimating relative volatility to be the 9th tray in the rectification section, whereas according to the linearized model, the optimal location should be the 6th tray (Figure 13). A nonlinear observer for on-line estimation of relative volatility is designed, and the observer performance is evaluated for optimal and nonoptimal sensor location in the column. A constant gain for the nonlinear observer is computed on the basis of local linearization about the nominal operating point.44 To compute the

observer gain, all the poles of the observer are placed at the same location as the poles of the linearized plant at the nominal point and the pole corresponding to the parameter is placed at -0.05 (faster than the slowest plant mode). It can be observed from Figure 14 that the nonlinear observer with measurement at the optimal location (9th tray) provides faster convergence of the parameter to the actual plant value for a 10% perturbation in relative volatility. The above examples indicate that the presented method for sensor location captures the nonlinearity of the system over the operating region. The sensor location based on the presented technique returns results that are physically intuitive and that provide a sound basis for designing estimators for nonlinear systems.

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Figure 13. Values of the measure for placing a temperature sensor in the distillation column for estimating relative volatility.

provide better results than methods based upon maximizing the observability of the least observable direction, because the magnitude of the smallest singular value is often very close to zero. The presented method for sensor location identifies one sensor location, in the first example, in the stripping section of the distillation column, and one location, in the second example, at the hotspot of the reactor for state estimation. These results are in agreement with physical insight into the process and observations presented in the literature.9,43 Additionally, the optimal sensor location for estimating the heat transfer coefficient in the reactor and the relative volatility in the distillation example has been computed. While this method does not know that there is a hotspot in the reactor, much less that its location shifts, it nevertheless results in the computation of a physically meaningful location for the temperature sensor by taking the uncertainty in the process parameter into account. The advantages that can be gained from using the presented approach over one based on linear models have been illustrated in the examples. Also, a comparison of the estimator performance for optimal and nonoptimal locations for estimating relative volatility for the distillation column has been presented. The performance of a parameter estimator that is based upon the determined sensor location was found to be superior to the one for a nonoptimal location. Acknowledgment The authors gratefully acknowledge partial financial support from the ACS Petroleum Research Fund (Grant PRF# 43229-G9). Appendix I A sufficient condition for the local identifiability of a nonlinear model

Figure 14. Nonlinear observer performance for the temperature measurement at optimal and nonoptimal trays.

x3 ) f(x,p,u)

(Ia)

y ) h(x,p,u)

(Ib)

5. Conclusions This paper presented a methodology for determining optimal sensor locations for stable nonlinear systems for state and parameter estimation. The procedure consists of first computing the observability covariance matrices for the nonlinear system over a prespecified operating region and then determining observability measures based upon the covariance matrices. This method has advantages over other techniques in that it does not resort to linearization of the model, while at the same time, it is computationally tractable. In addition, it is possible to automate the presented procedure enabling the computation of optimal sensor locations by formulating and solving an optimization problem, which is currently under investigation. It has also been shown that the technique can be used to compute optimal sensor locations for state as well as for parameter estimation, while conventional approaches can address only one of these two issues at a time. When analyzing several different observability measures used for sensor location, it can be concluded that, for systems where the number of states far exceeds the number of measurements, as is the case in the presented case studies, methods mainly influenced by the largest singular value of the observability covariance matrix

has been derived by Grewal and Glover.45 Linearizing the nonlinear model for a constant input results in

δx3 ) A δx + A1 δp

(IIa)

δy ) C δx

(IIb)

where

A)

∂f | ∂x xss,u,pss

A1 )

∂f | ∂p xss,u,pss

C)

∂h | ∂x xss,u,pss

and δx, δp, and δy are the state variables, parameters, and outputs given as deviation variables:

δx ) x - xss; δy ) y - yss; δp ) p - pss The change in the outputs for a perturbation δp is given by

∫0tO(t,τ,pss)A1 dτ]δp

δy ) (y - yss) ) [C

where O(t,τ,pss) is the state transition matrix.

(III)


This equation can be rewritten as

the original value of the parameters pss. The output trajectory is then given by

δy ) M(t,pss) δp

(IV)

∫0t O(t,τ,pss)A1(cmTlei) dτ

(yilm(t) - yss(0)) ) C

where

) (C

∫0 O(t,τ,pss)A1 dτ]

M(t,pss) ) [C

t

(V)

Glover45

that the It has been shown by Grewal and parameters of nonlinear systems are locally identifiable if and only if the columns of M(t,pss) are linearly independent or if the matrix

∫0TM(t,pss)MT(t,pss) dt

∫0tO(t,τ,pss)A1 dτ)(cmTlei)

) M(cmTlei)

(X)

Therefore, Ψijlm(t) is given by

Ψijlm(t) ) M(cmTlei)TM(cmTlej)

(VI)

) cm2eiTTlTMTMTlej

(XI)

Ψlm(t) ) cm2TlTMTMTl

(XII)

resulting in has full rank. The relationship between the condition of local identifiability and the rank conditions based upon (i) empirical observability gramians,35 (ii) observability covariance matrices,2 and (iii) the deterministic form of the Fisher information matrix26,28 will be investigated next. This will put these concepts into the context of parameter identifiability and will present the relationship between them. The augmented system obtained by linearization of the nonlinear system (I), with constant parameters as additional state variables, is given by

)

( )[ ]( ) () x˘ A A1 x p˘ 0 0 p

y )[C 0 ]

(VIIa)

x p

(VIIb)

For perturbations of the parameters only, one type of symmetric, positive, semidefinite matrix can be defined r

WO )

s

∑ ∑ l)1m)1

1 2

∫0TTlΨlm(t)TTl dt

(VIII)

(rscm )

where the integration time T is usually chosen to approach ∞ for a stable system. Since it is only of interest to extract the information about the parameters, p, the matrices T, M, and E are redefined as

Tnp ) {T1, ..., Tr; Ti ∈ Rnp×np, TTi Ti ) I, i ) 1, ..., r} M ) {c1, ..., cs; ci ∈ R, ci > 0, i ) 1, ..., s} Enp ) {e1, ..., enp; standard unit vectors in Rnp} where np is the number of parameters. Ψijlm(t) is then given by

Ψijlm(t) ) (yilm(t) - yss)T(yjlm(t) - yss)

Substituting eq XII in eq VIII, the empirical observability gramian comes out to be

WO )

∫0TM(t,pss)TM(t,pss) dt

(XIII)

The rank condition resulting from eq XIII is identical to the one given by eq VI. Case (ii): Observability Covariance Matrix. The computation of the observability covariance matrix is similar to that of the empirical gramian above. However, the difference between the empirical observability gramian and the observability covariance matrix arises from the choice of yss. While the empirical observability gramian uses yss equal to the steady state of the system for the nonperturbed values of the parameters, pss, for the observability covariance matrix, yss is chosen to be equal to the resulting steady state after each parameter perturbation is applied. This has implications insofar as the empirical gramians will not converge for T f ∞46 becasue eq IX will converge to a constant, but usually nonzero, value. Instead, it is required to choose a high, but finite, value for the integration time to perform the rank test for the empirical observability gramian. The observability covariance matrix does not suffer from this drawback, because the value of eq IX will always tend to zero for large times for any case where the nonaugmented system is stable. However, the choices of yss and the final integration time do not have an effect on the rank condition for the local identifiability of the parameters. Case (iii): Fisher Information Matrix. From eq IV, it can be seen that

δy ) M(t,pss) δp

(XIV)

Hence, the empirical gramians can be written as

(IX)

where yilm(t) is the output of the system corresponding to the perturbation only in the parameter given by cmTlei. Case (i): Empirical Observability Gramian. For the empirical observability gramian, yss is chosen to be equal to the steady state of the system and yss(0) is for

WO )

δy δy ∫0T(δp ) (δp) dt T

(XV)

For δp f 0, eq XV reduces to

WO )

∂y ∂y ∫0T(∂p ) (∂p) dt T

(XVI)

which is a special case of the deterministic form of the

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Fisher information matrix in Wouwer et al.26 and Qureshi et al.28 Appendix II In this section it is shown that the number of measurements must be greater than the number of unknown parameters to ensure the identifiability of the parameters. From eq IV, it can be seen that output is related to the unknown parameters:

δy ) M(t,pss) δp

(IV)

This equation can be multiplied on the left by a matrix MT:

MT(t,pss) δy ) MT(t,pss)M(t,pss) δp

(XVI)

The parameters can then be obtained by

δp ) [MT(t,pss)M(t,pss)]-1MT(t,pss) δy

(XVI)

To compute the parameters, the inverse of MT(t,pss)M(t,pss)is required to exist. However, this is only the case if the number of measurements is equal to or greater than the number of unknown parameters.47 Literature Cited (1) Kister, H. Z. Distillation Operation; McGraw-Hill Publishing Company: New York, 1990. (2) Hahn, J.; Edgar, T. F.; Marquardt, W. Controllability and observability covariance matrices for the analysis and order reduction of stable nonlinear systems. J. Process Control 2003, 13, 115. (3) Muller, P. C.; Weber, H. I. Analysis and optimization of certain quantities of controllability and observability for linear dynamic system. Automatica 1972, 8, 237. (4) Dochain, D.; Tali-Mammar, N.; Babary, J. P. On modeling, monitoring and control of fixed bed bioreactors. Comput. Chem. Eng. 1997, 21, 1255. (5) Waldraff, W.; Dochain, D.; Bourrel, S.; Magnus, A. On the use of observability measures for sensor location in tubular reactor. J. Process Control 1998, 8, 497. (6) van den Berg, F. W. J.; Hoefsloot, H. C. J.; Boelens, H. F. M.; Smilde, A. K. Selection of optimal sensor position in a tubular reactor using robust degree of observability criteria. Chem. Eng. Sci. 2000, 55, 827. (7) Omatu, S.; Koide, S.; Soeda, T. Optimal sensor location for a linear distributed parameter system. IEEE Trans. Autom. Control 1978, 23, 665. (8) Kumar, S.; Seinfeld, S. H. Optimal location of measurements in tubular reactors. Chem. Eng. Sci. 1978, 33, 1507. (9) Harris, T. J.; Macgregor, J. F.; Wright, J. D. Optimal sensor location with an application to a packed bed tubular reactor. AIChE J. 1980, 26, 910. (10) Colantuoni, G.; Padmanabhan, L. Optimal sensor location for tubular-flow reactor systems. Chem. Eng. Sci. 1977, 32, 1035. (11) Jorgensen, S. B.; Goldschmidt, L.; Clement, K. A sensor location procedure for chemical processes. Comput. Chem. Eng. 1984, 3, 195. (12) Morari, M.; O’Dowd, M. J. Optimal sensor location in the presence of nonstationary noise. Automatica 1980, 16, 463. (13) Morari, M.; Stephanopoulos, G. Optimal selection of secondary measurements within the framework of state estimation in the presence of persistent unknown disturbances. AIChE J. 1980, 26, 247. (14) Romagnoli, J.; Alvarez, J.; Stephanopolus, G. Variable measurement structures for process control. Int. J. Control 1981, 33, 269. (15) Alvarez, J.; Romangnoli, J. A.; Stephanopoulous, G. Variable measurements structures for the control of a tubular reactor. Chem. Eng. Sci. 1981, 36, 1695.

(16) Damak, T.; Babary, J. P.; Nihtila, M. T. Observer design and sensor location in distributed parameter bioreactors. Proceedings of DYCORD, Maryland, 1992; p 87. (17) Ali, Y.; Narasimhan, S. Sensor network design for maximizing reliability of linear processes. AIChE J. 1993, 39, 821. (18) Bagajewicz, M. J. Design and retrofit of sensor networks in process plants. AIChE J. 1997, 43, 2300. (19) Chmielewski, D. J.; Palmer, T.; Manousiouthakis, V. On the theory of optimal sensor placement. AIChE J. 2002, 48, 1001. (20) Muske, K. R.; Georgakis, C. Optimal measurement system design for chemical processes. AIChE J. 2003, 49, 1488. (21) Isidori, A. Nonlinear Control Systems, 3rd ed.; SpringerVerlag: New York, 1995. (22) Hermann, R.; Kerner, A. J. Nonlinear controllability and observability. IEEE Trans. Autom. Control 1977, 22, 728. (23) Lopez, T.; Alvarez, J. On the effect of the estimation structure in the functioning of a nonlinear copolymer reactor estimator. J. Process Control 2004, 14, 99. (24) Scherpen, J. M. A. Balancing of nonlinear systems. Syst. Control Lett. 1993, 21, 143. (25) Georges, D. The use of observability and controllability gramians or functions for optimal sensor and actuator location in finite-dimensional systems. Proceedings of the 34th Conference on Decision and Control, New Orleans, LA, 1995; p 3319. (26) Wouwer, A. V.; Point, N.; Porteman, S.; Remy, M. An approach to the selection of optimal sensor locations in distributed parameter systems. J. Process Control 2000, 10, 291-300. (27) Alonso, A. A.; Kevrekidis, I. G.; Banga, J. R.; Frouzakis, C. E. Optimal sensor location and reduced order observer design for distributed process system. Comput. Chem. Eng. 2004, 28, 27. (28) Qureshi, Z. H.; Ng, T. S.; Goodwin, G. C. Optimum experimental design for identification of distributed systems. Int. J. Control 1980, 31, 21. (29) Basseville, M.; Benveniste, A.; Moustakides, G. V.; Rougee, A. Optimal sensor location for detecting changes in dynamical behavior. IEEE Trans. Autom. Control 1987, 23, 1067. (30) Mehra, R. K. Optimal input for linear system identification. IEEE Trans. Autom. Control 1974, 19, 192. (31) Walter, E.; Pronzato, L. Qualitative and quantitative experiment design for phenomenological modelssA survey. Automatica 1990, 26, 195. (32) Li, R.; Henson, M. A.; Kurtz, M. J. Selection of model parameters for off-line parameter estimation. IEEE Trans. Control Syst. Technol. 2004, 12, 402. (33) Varma, A.; Morbidelli, M.; Wu, H. Parametric Sensitivity in Chemical Systems; Cambridge University Press: Cambridge, U.K., 1999. (34) Brockettt, R. W. Finite Dimensional Linear System; Wiley: New York, 1970. (35) Lall, S.; Marsden J. E.; Glavaski, S. A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 2002, 12, 519. (36) Singh, A. K.; Hahn, J. Optimal sensor location for nonlinear dynamic systems via empirical gramians. Proceedings DYCOPS 7, Boston, MA, 2004. (37) Hahn, J.; Edgar, T. F. An improved method for nonlinear model reduction using balancing of empirical gramians. Comput. Chem. Eng. 2002, 26, 1379. (38) Benallou, A.; Seborg, D. E.; Mellichamp, D. A. Dynamic compartmental models for separation processes. AIChE J. 1986, 32, 1067. (39) Skogestad, S. Dynamics and control of distillation columnss A critical survey. Model., Identif. Control 1997, 18, 177. (40) van Welsenaere, R. J.; Froment, G. F. Parameter sensitivity and runaway in fixed bed catalytic reactors. Chem. Eng. Sci. 1970, 25, 1503. (41) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd ed.; Wiley: New York, 1990. (42) Luyben, W. L. Practical Distillation Control; Van Nostrand Reinhold: New York, 1992. (43) Bequette, B. W.; Edgar, T. F. Noninteracting control system design methods in distillation. Comput. Chem. Eng. 1989, 13, 641. (44) Wallman, P. H. Reconstruction of unmeasured quantities for nonlinear dynamic processes. Ind. Eng. Chem. Fundam. 1979, 18, 327. (45) Grewal, M. S.; Glover, K. Identifiability of linear and nonlinear dynamical systems. IEEE Trans. Autom. Control 1976, 21, 833.

Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005 5659 (46) Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control; Wiley: New York, 1997.

Received for review August 4, 2004 Accepted May 18, 2005

(47) Crassidis, J. L.; Junkins, J. L. Optimal Estimation of Dynamic Systems; Chapman & Hall/CRC: New York, 2004.

IE040212V

Determining Optimal Sensor Locations for State and Parameter

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