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Vector output

To set up the problem for a microcomputer or Mathcad, one need only enter the input matrix with a 1.0 as each element of the 0th or leftmost column. Suitable modifications must be made in matrix and vector dimensions to accommodate matrices larger in one dimension than the X matrix of input data (3-56), and output vectors must be modified to contain one more minimization parameter than before, the intercept otq. [Pg.88]

A single experiment consists of the measurement of each of the m response variables for a given set of values of the n independent variables. For each experiment, the measured output vector which can be viewed as a random variable is comprised of the deterministic part calculated by the model (Equation 2.1) and the stochastic part represented by the error term, i.e.,... [Pg.9]

The state variables are the minimal set of dependent variables that are needed in order to describe fully the state of the system. The output vector represents normally a subset of the state variables or combinations of them that are measured. For example, if we consider the dynamics of a distillation column, in order to describe the condition of the column at any point in time we need to know the prevailing temperature and concentrations at each tray (the state variables). On the other hand, typically very few variables are measured, e.g., the concentration at the top and bottom of the column, the temperature in a few trays and in some occasions the concentrations at a particular tray where a side stream is taken. In other words, for this case the observation matrix C will have zeros everywhere except in very few locations where there will be 1 s indicating which state variables are being measured. [Pg.12]

The measurements of the output vector are taken at distinct points in time, t, with i=d,...,N. The initial condition x0, is also chosen by the experimentalist and it is assumed to be precisely known. It represents a very important variable from an experimental design point of view. [Pg.12]

Again, the measured output vector at time t denoted as y, is related to the value calculated by the mathematical model (using the true parameter values) through the error term,... [Pg.13]

The choice of the objective function is very important, as it dictates not only the values of the parameters but also their statistical properties. We may encounter two broad estimation cases. Explicit estimation refers to situations where the output vector is expressed as an explicit function of the input vector and the parameters. Implicit estimation refers to algebraic models in which output and input vector are related through an implicit function. [Pg.14]

Given N measurements of the output vector, the parameters can be obtained by minimizing the Least Squares (LS) objective function which is given below as the weighted sum ofsquares of the residuals, namely,... [Pg.14]

Let us consider first the most general case of the multiresponse linear regression model represented by Equation 3.2. Namely, we assume that we have N measurements of the m-dimensional output vector (response variables), y , M.N. [Pg.27]

Output vector (dependent variables) Model equations ... [Pg.54]

In this case, the unknown parameter vector k is the 2-dimensional vector [rmax, the independent variables are only one, x = [SI and similarly for the output vector, y = [r]. Therefore, the model in our standard notation is... [Pg.60]

Parameter vector Independent variables Output vector Model Equation... [Pg.62]

Experimental data are available as measurements of the output vector as a function of time, i.e., [yj, t ], i=l,...,N where withyj we denote the measurement of the output vector at time t,. These are to be matched to the values calculated by the model at the same time, y(t,), in some optimal fashion. Based on the statistical properties of the experimental error involved in the measurement of the output vector, we determine the weighting matrices Qj (i=l,...,N) that should be used in the objective function to be minimized as mentioned earlier in Chapter 2. The objective function is of the form,... [Pg.85]

Again, let us assume that an estimate kw of the unknown parameters is available at the j,h iteration. Linearization of the output vector around and retaining first order terms yields... [Pg.85]

Assuming a linear relationship between the output vector and the state variables (y = Cx), the above equation becomes... [Pg.86]

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. [Pg.87]

When the output vector (measured variables) are related to the state variables (and possibly to the parameters) through a nonlinear relationship of the form y(t) = h(x(t),k), we need to make some additional minor modifications. The sensitivity of the output vector to the parameters can be obtained by performing the implicit differentiation to yield ... [Pg.92]

As both state variables are measured, the output vector is the same with the state vector, i.e., yi=x, and y2=x2. The feed to the reactor was pure benzene. The equilibrium constants K and K2 were determined from the run at the lowest space velocity to be 0.242 and 0.428, respectively. [Pg.100]

Equations 6,71, 6.75 and 6.76 can be solved simultaneously to yield g(t) and G(t) when the initial state vector x0 and the parameter estimate vector k(J) are given. In order to determine ktl+l) the output vector (given by Equation 6.2) is inserted into the objective function (Equation 6.4) and the stationary condition yields,... [Pg.112]

By taking the last term on the right hand side of Equation 6.83 to the left hand side one obtains Equation 6.11 that is used for the Gauss-Newton method. Hence, when the output vector is linearly related to the state vector (Equation 6.2) then the simplified quasilinearization method is computationally identical to the Gauss-Newton method. [Pg.114]

When the output vector is nonlinearly related to the state vector (Equation 6.3) then substitution of x<,+l> from Equation 6.74 into the Equation 6.3 followed by substitution of the resulting equation into the objective function (Equation 6.4) yields the following equation after application of the stationary condition (Equation 6.78)... [Pg.114]

The above equation represents a set of p nonlinear equations which can be solved to obtain kcomputational algorithm and the Gauss-Newton method yield the same results. [Pg.114]

However, an important question that needs to be answered is "what constitutes a satisfactory polynomial fit " An answer can come from the following simple reasoning. The purpose of the polynomial fit is to smooth the data, namely, to remove only the measurement error (noise) from the data. If the mathematical (ODE) model under consideration is indeed the true model (or simply an adequate one) then the calculated values of the output vector based on the ODE model should correspond to the error-free measurements. Obviously, these model-calculated values should ideally be the same as the smoothed data assuming that the correct amount of data-filtering has taken place. [Pg.117]

The distributed state variables w,(t,z), j=are generally not all measured. Furthermore the measurements could be taken at certain points in space and time or they could be averages over space or time. If we define as y the in-dimensional output vector, each measured variable, y/t), j=l,...,/w, is related to the state vector w(t,z) by any of the following relationships (Seinfeld and Lapidus, 1974) ... [Pg.168]

As usual, it is assumed that the actual measurements of the output vector are related to the model calculated values by... [Pg.169]

Following the same approach as in Chapter 6 for ODE models, we linearize the output vector around the current estimate of the parameter vector kw to yield... [Pg.169]

Next let us turn our attention to models described by a set of ordinary differential equations. We are interested in establishing confidence intervals for each of the response variables y, j=l,...,/w at any time t=to. The linear approximation of the output vector at time to,... [Pg.181]

The main consideration for the choice of the most appropriate time interval is the availability of parameter sensitivity information. Kalogerakis and Luus (1984) proposed to choose the time interval over which the output vector (measured variables) is most sensitive to the parameters. In order to obtain a measure of the available sensitivity information with respect to time, they proposed the use of... [Pg.197]

The procedure for the selection of the most appropriate time interval requires the integration of the state and sensitivity equations and the computation of lj(t), j=l,...,p at each grid point of the operability region. Next by plotting I/t), j=l,...,p versus time (preferably on a log scale) the time interval [t, tNp] where the information indices are excited and become large in magnitude is determined. This is the time period over which measurements of the output vector should be obtained. [Pg.198]

The above results indicate again the need to assign to each grid point a different time interval over which the measurements of the output vector should be made. The Information Index can provide the means to determine this time interval. It is particularly useful when we have completely different time scales as shown by the Information Indices (shown in Figure 12.9) for the grid point (4,... [Pg.209]

The fifteen layers of constant permeability and porosity were taken as the reservoir zones for which these parameters would be estimated. The reservoir pressure is a state variable and hence in this case the relationship between the output vector (observed variables) and the state variables is of the form y(t,)=Cx(t,). [Pg.373]


See other pages where Vector output is mentioned: [Pg.460]    [Pg.862]    [Pg.8]    [Pg.12]    [Pg.12]    [Pg.50]    [Pg.50]    [Pg.57]    [Pg.59]    [Pg.85]    [Pg.91]    [Pg.92]    [Pg.95]    [Pg.117]    [Pg.152]    [Pg.153]    [Pg.193]    [Pg.198]   
See also in sourсe #XX -- [ Pg.98 ]




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