When modeling dynamic processes for several inputs with freely existing data, such as data collected with normal process operations, the ability to accurately model the output response for a given input change is impeded when inputs are cross- (i.e., pair-wised) correlated as this adversely affects accurate estimation of the causative effects of inputs on the response variable. The causative effects of the inputs can be evaluated functionally and analytically via the Jacobian Matrix which is done in this work for NARMAX and Wiener structures that are linear and nonlinear in model parameters. This analysis shows that the Wiener structure with physically-based nonlinear parameterization is superior. This conclusion is also supported in this work by a modeling study on a real distillation column consisting of eight test runs over a period of three years.
In chemical processes, output variables are determined by the values of input variables, which may or may not be measured and some input variables may not even be known. Mathematical modeling is the process of mapping measured variables to output variables using a mathematical formulation. In mathematical modeling a structure must be selected and any unknown coefficients (i.e., parameters) must be estimated. For a general model structure, let its expectation be represented as hi = f(Xi;q), where hi is the expected value of the response (i.e., output) at the ith sampling time, i = 1, . . ., n; Xi is the vector of input values at the ith sampling time; and q is the vector of unknown model parameters with q =[q1 . . . qq]T. Therefore, the element of its Jacobian Matrix, Jnxq, in the ith row and jth column is that is, Moreover, the jth column represents qj and its column vector represents the change in the response space as qj changes for the set of experimental conditions. If two columns, say j and k, are orthogonal, their correlation coefficient is zero, and the information to estimate qj is decoupled (i.e., separate or independent) from the information to estimate qk, and vice versa. The advantage of this is that causative relationships of inputs on the response can be obtained and standard estimation errors of parameters are minimum1. Correlated columns in the Jacobian Matrix arise from pairwise correlation of inputs. Thus, obtaining orthogonal columns for each parameter necessitates setting Xi according to some predetermined statistical experimental design. However, running a statistically designed experiment is not always practical or possible. The ability to accurately model the output response for given inputs is impeded when inputs are cross-correlated. Consequently, model identification often involves the use of experimental data with correlated inputs that result in columns of J that are not orthogonal (i.e., correlated). Notwithstanding, for a given set of experimental data, with pairwise correlated inputs, the best model (in the sense of evaluating causative relationship between inputs and response and extracting scientific knowledge from parameters ) will be the one that produces columns of J with the smallest pair-wise correlation. Therefore, the goal of this work is to evaluate dynamic model structures based on pair-wise correlation of the columns of J when the inputs are pair-wise correlated. This scope is restricted to transfer function models that are linear in the time-dependent process variables. More specifically, this work compares structures that are developed from transfer functions that are applicable to Nonlinear Autoregressive Moving Average with eXogenous variables (NARMAX) and Wiener models.
The basic difference of NARMAX and Wiener transfer functions is the characteristic equation which is the same for each input for NARMAX but can be different for each input for Wiener. With discrete-time modeling, this work will evaluate linear and nonlinear regression transfer function models of NARMAX and Wiener networks. The engineering literature typically defines linear models based on the form of the time-dependent variables in the differential equations of transfer functions. However, this scope is parameter estimation with focus on J which is based on the behavior of the parameters. Hence, this work adopts the statistical definition of linear and nonlinear models in parameter estimation. More specifically, a linear model in this work is one that is linear in parameters and a nonlinear model is one that is nonlinear in parameters. All parameters in this work are estimated with the least squares criterion.
From a search in the process identification literature, we discovered that for NARMAX models and its subclasses, linear forms in parameters are widely, if not exclusively, used. It also revealed that the common practice of Wiener Modeling is the use of transfer functions that are linear in the estimated parameters. The only discrete-time Wiener approach that we found to use nonlinear transfer functions is the one developed by Rollins and co-workers. The nonlinear parameterized NARMAX structure will be developed in this work following this approach.