General Estimation Problem

The preceding section discusses the mathematical formulation of the problem under consideration and the general conditions for redundancy and estimability. Now, we are ready to analyze the decomposition of the general estimation problem. The division of linear dynamic systems into their observable and unobservable parts was first suggested by Kalman (1960). The same type of arguments can be extended here to decompose a system considered to be at steady-state conditions. 

When the results of matrix theory are applied to the general estimation problem (see Appendix A), the following can be stated. 

The system of equations (2.12) can be written using the column echelon form of matrix M as follows  

If each row of has only one nonzero element, then physically this means that in the new coordinates Xc = [Xr Xg r], where Xr is a 7-dimensional vector, the subsystem 

If some rows of have more than one nonzero element, there are linear combinations between variables in Xr and variables in Xg r- Thus, the estimable portion of the system is of dimension ob less than 7 (ob 7) and the nonestimable one is of dimension (g — ob). 

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It should be noted here that this formulation of the problem is totally equivalent to that of previous chapter, since the data reconciliation problem is only a special case of the general estimation problem, where we directly measure the process variables. 

The effectiveness of these control schemes will depend to a great extent on how accurately the various state variables and culture parameters can be estimated on-line and under a variety of operating conditions. Several aspects of the general estimation problem have been studied in situations related to chemical reactors (J, 2). With biochemical reactors, however, the estimation problem is considerably more involved because of the growth... 

Tasks that fall within the paradigm of unsupervised learning are in general estimation problems the applications include clustering and estimating statistical distributions, compression, and filtering. 

Generally speaking, for condition numbers less than 10 the parameter estimation problem is well-posed. For condition numbers greater than 1010 the problem is relatively ill-conditioned whereas for condition numbers 10 ° or greater the problem is very ill-conditioned and we may encounter computer overflow problems. 

In this chapter we concentrate on dynamic, distributed systems described by partial differential equations. Under certain conditions, some of these systems, particularly those described by linear PDEs, have analytical solutions. If such a solution does exist and the unknown parameters appear in the solution expression, the estimation problem can often be reduced to that for systems described by algebraic equations. However, most of the time, an analytical solution cannot be found and the PDEs have to be solved numerically. This case is of interest here. Our general approach is to convert the partial differential equations (PDEs) to a set of ordinary differential equations (ODEs) and then employ the techniques presented in Chapter 6 taking into consideration the high dimensionality of the problem. 

Chapter 9 deals with the general problem of joint parameter estimation data reconciliation. Starting from the typical parameter estimation problem, the more general formulation in terms of the error-in-variable methods is described, where measurement errors in all variables are considered. Some solution techniques are also described here. 

In this chapter, the mathematical tools and fundamental concepts utilized in the development and application of modem estimation theory are considered. This includes the mathematical formulation of the problem and the important concepts of redundancy and estimability in particular, their usefulness in the decomposition of the general optimal estimation problem. A brief discussion of the structural aspects of these concepts is included. 

From the results of the previous theorem, we conclude that any system that is estimable and redundant (r > 0) admits a decomposition into its redundant (x0 and nonredundant parts (X2). This conclusion is of paramount importance when applied within the framework of the overall estimation problem. Such a decomposition then allows a new equivalent two-problem formulation of the general least squares problem ... 

The data reconciliation problem can be generally stated as the following constrained weighted least-squares estimation problem ... 

In this section, we first consider that, in addition to the measurement equations model (6.33), we have also the conditions defined by (6.35) imposed on the vector of process variables. Furthermore, we will assume, as a more general formulation, that a priori information enters into the estimation problem. 

In this section the extension of the use of nonlinear programming techniques to solve the dynamic joint data reconciliation and parameter estimation problem is briefly discussed. As shown in Chapter 8, the general nonlinear dynamic data reconciliation (NDDR) formulation can be written as ... 

In this chapter, the general problem of joint parameter estimation and data reconciliation was discussed. First, the typical parameter estimation problem was analyzed, in which the independent variables are error-free, and aspects related to the sequential processing of the information were considered. Later, the more general formulation in terms of the error-in-variable method (EVM), where measurement errors in all variables are considered in the parameter estimation problem, was stated. Alternative solution techniques were briefly discussed. Finally, joint parameter-state estimation in dynamic processes was considered and two different approaches, based on filtering techniques and nonlinear programming techniques, were discussed. 

Knipling R, Wang S. Revised Estimates of the U.S. Drowsy Driver Crash Problem Size Based on the General Estimates System Case Reviews, the 39th Annual Proceedings of the Association for the Advancement of Automotive Medicine, Chicago, IL, October 16-18, 1995. 

Almasy and Sztano [6] and Mah and his coworkers [12] have dealt with this problem and developed structural or probabilistic rules that will determine the location of the gross error. A throuth review of the related problems and the proposed solutions can be found in [15]. When all the measurements are corrected, then they can be used to estimate the value of the variables which are not directly measurable. Such situation entails the solution of a nonlinear estimation problem, in general. 

The classical approach to the fit parameter estimation problem in dielectric spectroscopy is generally formulated in terms of a minimization problem finding values of X which minimize some discrepancy measure S(, s) between the measured values, collected in the matrix s and the fitted values = [/(co,-, x(7 ))] of the complex dielectric permittivity. The choice of S(e,e) depends on noise statistics [132]. 

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