Feedforward Control with Block-Oriented Modeling with Demonstration to Nonlinear Parametrized Wiener Modeling

            This work proposes a feedforward control (FFC) approach with block-oriented modeling. It is demonstrated using a nonlinear parametrized Wiener Modeling approach in feedback feedforward (FBFF) control. The process used for this demonstration is a simulated continuous stirred tank reactor (CSTR) with four (4) measured inputs. The Wiener model is nonlinear in the physically-based dynamic parameters of the transfer functions and linear in the static parameters of the static gain function. The static gain function has a second order linear regression form with interaction and quadratic terms.  The Wiener model is built under open-loop conditions using a Box-Behnken statistical experimental design consisting of 27 sequential step tests. Under a sequence of multiple input changes, the addition of feedforward control (FFC) reduced the standard deviation of the controlled variable from its set point by 70%.  

Traditional feedback control (FBC) makes adjustments to some manipulated variable after the process has deviated from its desired operating condition or set point (SP) for the controlled variable. In the past few decades, more sophisticated control algorithms have been introduced that use models determined from process data to achieve tighter control. Some of the most well-known model-based control systems include FFC, internal model control (IMC), and model predictive control (MPC).  However, while FFC is the only model-based control system that can totally cancel the effect of a disturbance, theoretically, it appears to have seen limited implementation in real processes as indicated by the very low number of articles in the process control literature. For effective FFC the model must consists of only inputs and the inputs have to map accurately and causatively to changes in the controlled variable which is very difficult to achieve in modeling real data for several reasons. The first one is due to shifts in the output caused by changes in the levels of unmeasured disturbances. Minimizing the number of unmeasured disturbances by measuring them and including them in the set of inputs can help to alleviate this behavior but as the number of inputs in the model increases, the complexity of the model increases and parameter estimation becomes more difficult. Furthermore, as the number of inputs increase, the longer the data collection will need to be and thus, the greater the likelihood of shifts in the output from changes in unmeasured inputs. Secondly, causative input modeling accuracy will suffer when the inputs are pairwise cross-correlated, especially for structures with linear parametrization. Finally, over the input operating space, the inputs typically map nonlinearly to the response space due to interactive and curvilinear relationships between the inputs and the output. As a consequence of these modeling challenges, FFC in practice has often been limited to a few variables, narrow input ranges, and linear static gain functionality.

Some models for nonlinear gain behavior have been proposed for model-based controllers including radial basis functions (RBF), genetic algorithms (GA), Nonlinear Auto Regressive Models And eXogenous inputs (NARMAX) models, and block-oriented models (BOMs). An important advantage of NARMAX and BOMs is that they can use transfer functions, i.e., linear dynamic equations with physically interpretable parameters. However, a limitation of the NARMAX structure is that all of its transfer functions have the same characteristic equation or denominator dynamics. BOMs use the outputs from blocks of dynamic (transfer) functions that are linear (L) differential equations as inputs to functions that can be nonlinear (N) with respect to static gain parameters. The simplest of the BOMs is the Hammerstein network (NL), which has an N block followed by an L block and the Wiener network (LN), which reverses the order of these two blocks. More complicated block-oriented structures include sandwich models such as an LNL network, which has linear dynamic blocks, followed by a nonlinear static block, followed by a second linear dynamic block. When the inputs can have different dynamic behavior, the Wiener network is the preferred choice over Hammerstein and is superior to NARMAX because the inputs can have completely different dynamic structures.

A number of researchers have studied the identification of model parameters for the Hammerstein and Wiener networks. The LNL network has not gotten as much attention, but some have proposed methods for its parameter identification. There has been much progress over the last decade in the identification of BOMs and recently, by taking a nonlinear parametrized approach for estimation of the dynamic parameters, accurate Wiener modeling was demonstrated using nine (9) inputs on a real distillation process with large variation due to unmeasured disturbances and with highly pairwise cross-correlation of the inputs. While there has been progress in the use of BOMs in model based control, progress of FFC using BOMs appears to have been limited to single input models. Thus, the objective of this work is the development of a general FFC framework for multiple-input BOMs in FFC with nonlinear static gain behavior. This framework will be demonstrated using a nonlinear parametrized Wiener modelwith a second-order static gain structure on a simulated CSTR. The Wiener model is built under open-loop conditions using a Box-Behnken statistical experimental design consisting of 27 runs or sequential step tests. Under a sequence of multiple input changes, the addition of feedforward control reduced the standard deviation of the controlled variable from its set point by nearly 70%.