Sensitivity Coefficient Equations

The measurement error, V-, is often assumed to be a normalized, normally distributed random variable with a zero mean and a known variance. The normal probability density function for the measurement error at condition i is 

Maple can be used to determine the sensitivity equations (see Example A.2 of Rawlings and Ekerdt) for a first order, isothermal constant volume reactor model where 

We have two parameters C g (6 = C q and 6 —k) %o will have two sensitivity coefficients and two sensitivity coefficient equations. The first sensitivity coefficient is defined as the rate of change of C (t) with respect to the 

The second sensitivity coefficient is the rate of change of ( ) with respect to 

These sensitivity coefficients were determined by the Maple Jacobian command for this model and evaluated at the experimental times in the Maple worksheet for 

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Because the initial species concentrations are independent of the rate constants, the initial conditions for these sensitivity coefficient equations are that all Ski — 0 at t — 0. Integrating these equations, along with the equations for the species concentrations, provides all the sensitivity coefficients as a function of time. For the case of the rate equations given by equation (35), letting the hrst I parameters be the forward rate constants and the second I parameters be the reverse rate constants, the function fk can be written (using new summation subscripts / and m) as... 

Continuing with the example based on the reaction mechanism of Table 10, the equations for the sensitivity coefficients (equation (50)) were integrated numerically using Matlab. For a mechanism of this size, it was feasible to solve for the species concentrations (15 of them) and all the sensitivity coefficients (39 x 15x2 = 1170 of them) simultaneously. Rather than examining the sensitivity coefficients themselves, it is often more informative to examine scaled or normalized sensitivity coefficients. These are usually defined as... 

The sensitivity coefficient, equation (20), indicates the magnitude and direction of change of the output varible y. caused by deviations of the parameter p. from its nominal value. For a vector of output variables y, the ilatrix of the derivatives of the system responses with respect to the system parameters p is called the sensitivity matrix. 

To obtain the different values of p, it is only hecessary to produce as many independent equations as there are components in the mixture and, if the mixture has n components, to solve a system of n equations having n unknowns. Individual analysis is now possible for mixtures having a few components but even gasoline has more than 200 It soon becomes unrealistic to have ail the sensitivity coefficients necessary for analysis in this case, 200. ... 

These two contributions can be evaluated separately in terms of the sensitivity coefficients of the result to the measured quantities by using the propagation equation of Kline and McClintock (1953) ... 

In summary, at each iteration of the estimation method we compute the model output, y(x kw), and the sensitivity coefficients, G for each data point i=l,...,N which are used to set up matrix A and vector b. Subsequent solution of the linear equation yields Akf f 1 and hence k[Pg.53]

Equations 4.14 and 4.15 are used to evaluate the model response and the sensitivity coefficients that are required for setting up matrix A and vector b at each iteration of the Gauss-Newton method. 

Equation 6.9 is a matrix differential equation and represents a set of nxp ODEs. Once the sensitivity coefficients are obtained by solving numerically the above ODEs, the output vector, y(tl,k l+I ), can be computed. 

Thus, the error in the solution vector is expected to be large for an ill-conditioned problem and small for a well-conditioned one. In parameter estimation, vector b is comprised of a linear combination of the response variables (measurements) which contain the error terms. Matrix A does not depend explicitly on the response variables, it depends only on the parameter sensitivity coefficients which depend only on the independent variables (assumed to be known precisely) and on the estimated parameter vector k which incorporates the uncertainty in the data. As a result, we expect most of the uncertainty in Equation 8.29 to be present in Ab. 

As we have already pointed out in this chapter for systems described by algebraic equations, the introduction of the reduced sensitivity coefficients results in a reduction of cond(A). Therefore, the use of the reduced sensitivity coefficients should also be beneficial to non-stiff systems. 

Finally it is noted that in the above equations we can substitute G(t) with GR(t) and Ak(i+1) with AkRtrM) in case we wish to use the reduced sensitivity coefficient formulation. 

Equations 10.15 to 10.17 define a set of (nxp) partial differential equations for the sensitivity coefficients that need to be solved at each iteration of the Gauss-Newton method together with the n PDEs for the state variables. 

Therefore, efficient computation schemes of the state and sensitivity equations are of paramount importance. One such scheme can be developed based on the sequential integration of the sensitivity coefficients. The idea of decoupling the direct calculation of the sensitivity coefficients from the solution of the model equations was first introduced by Dunker (1984) for stiff chemical mechanisms... 

Essentially this is equivalent to using (Sf/dk kj instead of (<3f/<3k,) for the sensitivity coefficients. By this transformation the sensitivity coefficients are normalized with respect to the parameters and hence, the covariance matrix calculated using Equation 12.4 yields the standard deviation of each parameter as a percentage of its current value. 

Step 2. For each grid point of the operability region, compute the sensitivity coefficients and generate A" given by Equation 12.9. 

Equations (/I) through (74) are absolute sensitivity coefficients. Similarly, we can develop expressions for the sensitivity of D°pt ... 

Finally, we should mention that in addition to solving an optimization problem with the aid of a process simulator, you frequently need to find the sensitivity of the variables and functions at the optimal solution to changes in fixed parameters, such as thermodynamic, transport and kinetic coefficients, and changes in variables such as feed rates, and in costs and prices used in the objective function. Fiacco in 1976 showed how to develop the sensitivity relations based on the Kuhn-Tucker conditions (refer to Chapter 8). For optimization using equation-based simulators, the sensitivity coefficients such as (dhi/dxi) and (dxi/dxj) can be obtained directly from the equations in the process model. For optimization based on modular process simulators, refer to Section 15.3. In general, sensitivity analysis relies on linearization of functions, and the sensitivity coefficients may not be valid for large changes in parameters or variables from the optimal solution. 

The sensitivity analysis of a system of chemical reactions consist of the problem of determining the effect of uncertainties in parameters and initial conditions on the solution of a set of ordinary differential equations [22, 23], Sensitivity analysis procedures may be classified as deterministic or stochastic in nature. The interpretation of system sensitivities in terms of first-order elementary sensitivity coefficients is called a local sensitivity analysis and typifies the deterministic approach to sensitivity analysis. Here, the first-order elementary sensitivity coefficient is defined as the gradient... 

For a local sensitivity analysis, Eq. (2.69) may be differentiated with respect to the parameters a to yield a set of linear coupled equations in terms of the elementary sensitivity coefficients, cXM /daj. 

Solution of the associated sensitivity analysis equations (Fig. 3.8) gives the normalized linear sensitivity coefficients for the CO mass fraction with respect to various rate constants. A rank ordering of the most important reactions in decreasing order is... 

Consider the example of quantitative NMR. Spreadsheet 6.3 gives the standard uncertainties and relative standard uncertainties of the components of the combined uncertainty. It is usual to graph the relative standard uncertainties, the standard uncertainties multiplied by the sensitivity coefficient [dy/dx uc(x)], or the squares of the latter expressed as a percentage contribution to the combined uncertainty (see equation 6.23). A horizontal bar chart for each component in decreasing order is one way of displaying these values (figure 6.9). 

Differentiating the standard-form equation (Eq. 15.58) produces the following differential equation for the sensitivity-coefficient matrix ... 

The first term on the right-hand side is the product of the physical problem s current Jacobian matrix and the sensitivity-coefficient matrix (i.e., the dependent variable). Assuming that the underlying physical problem (i.e., Eq. 15.58) is solved by implicit methods, the Jacobian evaluation is already part of the solution algorithm. The second term, which is the matrix that describes the explicit dependence of f on the parameters, must be evaluated to form the sensitivity equation. Note that all terms on the right-hand side are time dependent, as are the sensitivity coefficients S(t). 

Importantly, recognize that the sensitivity problem is a linear equation for the sensitivity coefficients regardless of whether the original problem is linear or nonlinear. Once the solution to the underlying problem is determined, the sensitivity coefficients can be computed efficiently, exploiting the inherent linearity [57,102,110,232,321], There is recent sensitivity software by Petzold that builds on the DASSL family of codes [258],... 

In solving the underlying model problem, the Jacobian matrix is an iteration matrix used in a modified Newton iteration. Thus it usually doesn t need to be computed too accurately or updated frequently. The Jacobian s role in sensitivity analysis is quite different. Here it is a coefficient in the definition of the sensitivity equations, as is 3f/9a matrix. Thus accurate computation of the sensitivity coefficients depends on accurate evaluation of these coefficient matrices. In general, for chemically reacting flow problems, it is usually difficult and often impractical to derive and program analytic expressions for the derivative matrices. However, advances in automatic-differentiation software are proving valuable for this task [36]. 

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