Method for removal of PID dynamics from MPC models

[0001] Not applicable

[0002] Not applicable

[0003] Model Predictive Control (MPC) refers to a class of algorithms that compute a sequence of manipulated variable adjustments in order to optimize the future behavior of complex multivariable processes. Originally developed to meet the needs of petroleum refineries and chemical processes, MPC can now be found in a wide variety of application areas including chemicals, food processing, automotive, aerospace, metallurgy, and pulp and paper. A well-known implementation of MPC in chemical and refinery applications is Dynamic Matrix Control or DMC.

[0004] The MPC Controller employs a software model of the process to predict the effect of past changes of manipulated variable and measurable disturbances on the output variables of interest. The independent variables are computed so as to optimize future system behavior over a time interval known as the prediction horizon. In the general case any desired objective function can be used for the optimization. The system dynamics are described by an explicit process model, which can take, in principle, a number of different mathematical forms. Process input and output constraints are included directly in the problem formulation so that future constraint violations are anticipated and prevented.

[0005] In practice a number of different approaches have been developed and commercialized in implementing MPC Controllers. The most successful implementations have made use of a linear model for the plant dynamics. The linear model is developed in a first step by gathering data on the process by introducing test disturbances on the independent variables and measuring the effects of the disturbances on the dependent variables. This initial step is referred to as Identification and the novel use of this identification data is the essence of this invention.

[0006] U.S. Pat. Nos. 4,349,869 and 4,616,308 describe an implementation of MPC control called Dynamic Matrix Control (DMC). These patents describe the MPC algorithms based on linear models of a plant and describe how process constraints are included in the problem formulation. Initial identification of the MPC controller using process data is also described.

[0007] By way of further background this Identification of process dynamics requires a pre-test in which the independent variables of the process are moved in some pattern to determine the effect on the dependent variables. In a chemical or refinery process the independent variables include the PID (proportional-integral-derivative) controller set points for selected dependent variables, the valve positions of PID controllers in manual, and temperatures, material flows, pressures and compositions that are determined outside the scope of the controller's domain. For any process Identification test, the independent variables are fixed for the analysis of the data. Further the tuning of any of the PID controllers in the domain of the MPC controller is fixed. The MPC controller that is built to use the dynamic process models from the Identification must have exactly the same configuration of independent variables that existed when the Identification was performed. Thus the PID controller configuration that is present during Identification imbeds the PID controller dynamics in the dynamic model.

[0008] This characteristic of current Identification technology represents an unsolved problem that is addressed by this invention. The problem creates a limitation on the use of MPC technology that manifests itself in two different areas.

[0009] The first application area is MPC itself. Because the dynamics of the PID controllers are imbedded in the MPC model, any change in the tuning of a PID controller or changing of the PID state from auto to manual or vice versa changes the dynamic model. To correct this it has been required to retest the process unit with the changed conditions. A well-designed Identification test for a complex multivariable process will be a 2-3 week effort with skilled people, costing two to three hundred thousand dollars.

[0010] The second application area is in the field of Operator Training Simulators. Effective training simulators are important to the chemical process industry. The large investments in new chemical processes and the safety implications of the complex processes require a well-trained operator group. This is important especially for process units that remain on computer control for extended periods of time, since the operators do not have the opportunity to control the unit. MPC models have not been used in creating training simulators because of the aforementioned problem that the PID controller configuration that is present during Identification imbeds the PID controller dynamics in the dynamic model. The result of this is that authentic training is difficult because the operators cannot change the state (auto or manual) of the PID controllers without reducing the fidelity of the model. Current state of the art training simulators are based on approximated first principle engineering equations that have been simplified to such a degree that they do not represent the actual process dynamics. A survey of control rooms in the chemical process industry will reveal that they are rarely used after a start-up as the operating personnel learn that the simulator does not reflect the actual dynamic behavior. A training simulator based on an identification model that has the fidelity to hold a process at constraints, display all temperatures, pressures, flows, and valve positions and allow the operator to switch any PID controller to manual or auto would be a powerful tool for training.

[0011] Numerous unsuccessful attempts have been made by practitioners in the field to address this identification issue. One approach would be to run the identification test with the regulatory control scheme in manual. This of course fails because there are not enough operators available to handle a complex process with 50 to 100 PID controllers on manual. Other attempts have been made to conduct a standard identification test with the regulatory scheme in place but to then set up the model with the valve positions as the independent variables. These approaches always lead to failure with erratic results. It has become recognized that this approach fails because the valve positions are correlated through the PID dynamics via measured and unmeasured disturbances that are always present in a real world identification test and are thus not independent.

[0012] The recognition of this fact and the method of removing the noise and unmeasured disturbances from the data set are the essence of this invention. The method of removing the noise and unmeasured disturbances is a procedural one. The process is first identified with a commercial tool such as AspenTech's DMC-Model. This step breaks the correlation between the unmeasured disturbances, the noise, and the PID controllers. The PID controller dynamics can then be removed from the resulting model without interference from the noise and unmeasured disturbances.

[0013] An object of this invention is to provide a method for removing the dynamics of the PID controllers from the MPC controller used in a multivariable control process. This allows creation of valve-based off-line process simulators.

[0014] It is a further object of this invention to provide such a method that can be used in various implementations of MPC controllers.

[0015] It is a further object of this invention to provide a method to generate new MPC controllers for complex multivariable process control when a change has been made in any PID control configuration or tuning and to do so without having to conduct new identification testing of the process.

[0016] It is a further object of this invention to generate a process simulator based on valve positions with the effect of unmeasured disturbances removed so that a high fidelity process simulator is available for process simulation and training. Such a simulator could be used for simulation in any controller configuration and with various tuning configurations on each individual controller.

[0017] In accordance with this invention there is provided a method used in model predictive control applications for removing PID controller dynamics from a controller model of a process having a plurality of independently controllable, manipulated variables and at least one controlled variable dependent upon the independently controllable, manipulated variables that includes at least the steps of: gathering data about the process by separately introducing a test disturbance in each of the manipulated variables and measuring the effect of the disturbances on the controlled variable; using the effects of the disturbances on the controlled variable to generate a first linearized matrix model relating the at least one controlled variable to the independently controllable, manipulated variables; interchanging selected valve position controlled variables with their corresponding selected independently controllable, manipulated PID controller set point variables in the linearized model using matrix row elimination mathematics to generate a second linearized model that has a new set of independently controllable, manipulated variables, the second model having the dynamics of the selected independently controllable, manipulated PID controller set point variables removed from the model.

[0018] To use this model in a control context the method includes controlling a process having a plurality of independently controllable, manipulated variables and at least one controlled variable dependent upon the independently controllable, manipulated variables including the steps of: gathering data about the process by separately introducing a test disturbance in each of the manipulated variables and measuring the effect of the disturbances on the controlled variable; using the effects of the disturbances on the controlled variable to generate a first linearized dynamic model relating the at least one controlled variable to the independently controllable, manipulated variables; interchanging selected valve position controlled variables with their corresponding selected independently controllable, manipulated PID controller set point variables in the first linearized dynamic model using matrix row elimination mathematics to generate a second linearized dynamic model that has a new set of independently controllable, manipulated variables, the second linearized dynamic model having the dynamics of the selected independently controllable, manipulated PID controller set point variables removed from the second linearized dynamic model; measuring the present value of the variables; calculating for discrete intervals of time from the gathered data about the process, the measured present values and pre-selected operating constraints a set of moves for present and future times for at least the manipulated variables to obtain new values for the manipulated variables and to move the at least one dependent controllable variable towards at least one of the constraints; and changing the process by adjusting the manipulated variables for the set of moves for present and future times to cause the process to move the at least one dependent controllable variable towards at least one of the constraints.

[0019] To use this invention to generate new MPC controllers for complex multivariable process control when a change has been made in any PID control configuration or tuning and to do so without having to conduct new identification testing of the process the following process can be used: by interchanging at least one PID controller set point variable in an original linearized dynamic model with its corresponding valve position controlled variable in the original linearized dynamic model using matrix row elimination mathematics to generate a secondary linearized dynamic model that has the at least one corresponding valve position as a new independently controllable, manipulated variable; then externally emulating new desired PID tuning via mathematical emulator to emulate the effect of the at least one PID controllers new tuning with the secondary linearized dynamic model; then testing the secondary linearized dynamic model with it's emulated PID tuning by stepping each of it's manipulated variables to obtain the new linearized dynamic model that will now contain the dynamics of the at least one PID controllers.

[0020] It should be noted that a regulatory control scheme can be easily emulated external to the process model via a DCS console or console emulator available in modern control packages. This allows the operator to put PID controllers in Manual-mode, break cascades, retune PID controller, or even re-configure the regulatory control scheme.

[0021] To use this invention to generate a process simulator based on valve positions with the effect of unmeasured disturbances removed so that a high fidelity process simulator is available for process simulation and training the following method is used: first gathering data about the process by separately introducing a test disturbance in each of the manipulated variables and measuring the effect of the disturbances on the controlled variable; then using the effects of the disturbances on the controlled variable to generate a first linearized dynamic model relating the at least one controlled variable to the independently controllable, manipulated variables; then interchanging each independently controllable, manipulated PID controller set point variable with its corresponding valve position controlled variable in the first linearized dynamic model using matrix row elimination mathematics to generate a second linearized dynamic model that has the corresponding valve positions as a new set of independently controllable, manipulated variables, the second linearized dynamic model having the dynamics of the selected independently controllable, manipulated PID controller set point variables removed from the second linearized dynamic model; then externally emulating a desired regulatory control schemes via mathematical emulators to emulate PID controllers in either manual, cascade, or automatic modes. As before it should be noted that a regulatory control scheme can be easily emulated external to the process model via a DCS console or console emulator available in modern control packages. This allows the operator to put PID controllers in Manual-mode, break cascades, retune PID controller, or even re-configure the regulatory control scheme.

[0022] The most common method of Identification currently used in oil refining and chemical processes is the Dynamic Matrix Identification (DMI). DMI will be used to illustrate the methodology of this invention, but it should be understood that the invention is not limited to a specific Identification technique.

[0023] FIG. 1 is a schematic demonstrating dependent and independent variables

[0024] FIG. 2 is a more complex flow schematic of a fractionator

[0025] FIG. 3 is a simulation of the fractionator model based on valve positions

[0026] FIG. 4 demonstrates the results from a plant test of the fractionator

[0027] FIG. 5 is a simulation of the fractionator with the PID controllers

[0028] FIG. 6 is a demonstration of the fractionator with the original and recovered values

[0029] The invention is a method used in conjunction with model predictive control for removing the dynamics of PID controllers from MPC controllers.

[0030] An MPC process model is a set of linear equations so it should be mathematically possible to interchange any independent variable with a dependent variable provided a relation exists between the independent and dependent variable. A candidate set for that transformation is the set point (independent) for a PID controller and the associated valve position (dependent) for that PID controller.

[0031] An MPC controller is often based on a linear model of a process system. For example see FIG. 1. Although the invention to be described here has applications in many fields the examples used will be from chemical and refinery process applications.

[0032] There are two types of variables in any system; the independent variables and the dependent variables. The independent variables are inputs to the system. The independent variables are further divided into manipulated and disturbance (feedforward) variables. Manipulated variables are those that can be changed by the human operator, such as valve positions or PID controller set points. The furnace fuel gas flow set point (1) is a manipulated independent variable. Disturbance variables are those independent variables that have an effect on the system, but cannot be changed by the human operator. Variables such as feed composition, feed temperature, and ambient temperature are examples of disturbance variables. The furnace inlet temperature (2) is a disturbance variable.

[0033] Dependent variables are outputs from the system. Dependent variables are affected by changes in the independent variables. The human operator cannot directly change them. The values of dependent variables can be controlled, however, by correctly changing the values of the manipulated variables. Further, as disturbances enter the system, the manipulated variables must be correctly adjusted to counteract the disturbance. The furnace outlet temperature (3) and stack temperature (4) are examples of dependent variables.

[0034] The use of linear models allows the use of matrix mathematics in describing complex and multivariable control. There are several general formulations of MPC models. A general model for control is the step response model: $\begin{array}{c}{\stackrel{\_}{O}}_1=A_{1,1}{\stackrel{\_}{I}}_1++A_{1,1}{\stackrel{\_}{I}}_1++A_{1,\mathrm{nind}}{\stackrel{\_}{I}}_{\mathrm{nind}}\\ \\ {\stackrel{\_}{O}}_i=A_{i,1}{\stackrel{\_}{I}}_1++A_{i,j}{\stackrel{\_}{I}}_j++A_{i,\mathrm{nind}}{\stackrel{\_}{I}}_{\mathrm{nind}}\\ \\ {\stackrel{\_}{O}}_{\mathrm{ndep}}=A_{\mathrm{ndep},1}{\stackrel{\_}{I}}_1++A_{\mathrm{ndep},j}{\stackrel{\_}{I}}_j++A_{\mathrm{ndep},\mathrm{nind}}{\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}$

[0035] where, ${\stackrel{\_}{O}}_i=\left[\begin{array}{c}{O}_{i,1}-O_{i,0}\\ O_{i,2}-O_{i,0}\\ O_{i,3}-O_{i,0}\\ \\ O_{i,\mathrm{ncoef}}-O_{i,0}\end{array}\right],$

[0036] the accumulative change in the i^th dependent variable at each time step, ${\stackrel{\_}{I}}_j=\left[\begin{array}{c}{I}_{j,1}\\ I_{j,2}\\ I_{j,3}\\ \\ I_{j,\mathrm{ncoef}}\end{array}\right],$

[0037] the step change in the j^th independent variable at each time step, and $A_{i,j}=\left[\begin{array}{cccc}{a}_{i,j,1}& & & \\ a_{i,j,2}& a_{i,j,1}& & \\ a_{i,j,3}& a_{i,j,2}& a_{i,j,1}& \\ & & & \\ a_{i,j,\mathrm{ncoef}}& a_{i,j,\left(\mathrm{ncoef}-1\right)}& a_{i,j,\left(\mathrm{ncoef}-2\right)}& a_{i,j,1}\end{array}\right],$

[0038] the Dynamic Matrix.

[0039] An alternate form of this Step Response equation is the Finite Impulse Response (FIR) form. It can be derived from the Step Response form as described below.

[0040] Recalling from the definitions that:

[0041] and that

[0042] we can difference the above system of equations to give: $\mathrm{Equation}2:\mathrm{Finite}\mathrm{Impulse}\mathrm{Response}\mathrm{Equations}\text{-}\mathrm{Block}\mathrm{Matrix}\mathrm{Form}$ $\begin{array}{c}{\stackrel{\_}{O}}_{\underset{}{1}}\underset{}{=}{B}_{\underset{}{1,1}}{\stackrel{\_}{I}}_1+B_{\underset{}{1,j}}{\stackrel{\_}{I}}_j+B_{1,\underset{}{\mathrm{nind}}}{\stackrel{\_}{I}}_{\mathrm{nind}}\\ {\stackrel{\_}{O}}_{\underset{}{i}}\underset{}{=}{B}_{\underset{}{i,1}}{\stackrel{\_}{I}}_1+B_{\underset{}{i,j}}{\stackrel{\_}{I}}_j+B_{i,\underset{}{\mathrm{nind}}}{\stackrel{\_}{I}}_{\mathrm{nind}}\\ {\stackrel{\_}{O}}_{\mathrm{ndep}}=B_{\mathrm{ndep},1}{\stackrel{\_}{I}}_1+B_{\mathrm{ndep},j}{\stackrel{\_}{I}}_j+B_{\mathrm{ndep},\mathrm{nind}}{\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}$

[0043] where ${\stackrel{\_}{O}}_i=\left[\begin{array}{c}{O}_{i,1}-O_{i,0}\\ O_{i,2}-O_{i,1}\\ O_{i,3}-O_{i,2}\\ \\ O_{i,\mathrm{ncoef}}-O_{i,\left(\mathrm{ncoef}-1\right)}\end{array}\right],$

[0044] the change in the i^th dependent variable across each time interval, ${\stackrel{\_}{I}}_j=\left[\begin{array}{c}{I}_{j,1}\\ I_{j,2}\\ I_{j,3}\\ \\ I_{j,\mathrm{ncoef}}\end{array}\right]$

[0045] as above, and $B_{i,j}=\left[\begin{array}{ccccc}{b}_{i,j,1}& & & & \\ b_{i,j,2}& b_{i,j,1}& & & \\ b_{i,j,3}& b_{i,j,2}& b_{i,j,1}& & \\ & & & & \\ b_{i,j,\mathrm{ncoef}}& b_{i,j,\left(\mathrm{ncoef}-1\right)}& b_{i,j,\left(\mathrm{ncoef}-2\right)}& & b_{i,j,1}\end{array}\right],$

[0046] the model matrix of Impulse Coefficients.

[0047] There are five forms of these equations, and we have shown only the first two. While these forms are mathematically equivalent, and while all forms may be used for identification prediction and control, they have very different properties.

[0048] {overscore (O)}A{overscore (I)}Most often used for control calculations.

[0049] {overscore (O)}B{overscore (I)}Used for identification of steady state variables.

[0050] {overscore (O)}B{overscore (I)}Used for identification of ramp variables.

[0051] {overscore (O)}B{overscore (I)}Not commonly used. Old IDCOM control formulation.

[0052] {overscore (O)}A{overscore (I)}Not commonly used.

[0053] C. R. Cutler and C. R. Johnston discuss the properties of these forms of the matrix in a paper, Analysis of the Forms of the Dynamic Matrix, in the Proceedings of the Instrument Society of America ISA 85 Advances in Instrumentation Volume 40, Number 1October 1985.

[0054] The use of these linear modeling techniques, including the identification of the model and the use of the model for control and the use in control with constraints is described in two U.S. Pat. Nos. 4,349,869 and 4,616,308. These patents are incorporated herein by reference.

[0055] We will now derive the algorithm of this invention to demonstrate the removal of the PID dynamics from the controller. The derivation is from the FIR model of equation 2. To derive the algorithm, we assume that the ^th independent variable is the set point of a PID controller and the ^th dependent is the PID valve response to that set point change. We wish to re-constitute the model so that the valve is the independent variable in the process model; that is to say, we wish to remove the dynamics of this PID controller from all affected model responses. This can be accomplished by interchanging the ^th dependent variable with the ^th dependent variable, as follows: $\left[\begin{array}{ccccccc}{B}_{1,1}& & B_{1,\left(\hat{j}-1\right)}& B_{1,\hat{j}}& B_{1,\left(\hat{j}+1\right)}& & B_{1,\mathrm{nind}}\\ & & & & & & \\ B_{\left(\hat{i}-1\right),1}& & B_{\left(\hat{i}-1\right),\left(\hat{j}-1\right)}& B_{\left(\hat{i}-1\right),\hat{j}}& B_{\left(\hat{i}-1\right),\left(\hat{j}+1\right)}& & B_{\left(\hat{i}-1\right),\mathrm{nind}}\\ B_{\hat{i},1}& & B_{\hat{i},\left(\hat{j}-1\right)}& B_{\hat{i},\hat{j}}& B_{\hat{i},\left(\hat{j}+1\right)}& & B_{\hat{i},\mathrm{nind}}\\ B_{\left(i+1\right),1}& & B_{\left(\hat{i}+1\right),\left(\hat{j}-1\right)}& B_{\left(\hat{i}+1\right),\hat{j}}& B_{\left(\hat{i}+1\right),\left(\hat{j}+1\right)}& & B_{\left(\hat{i}+1\right),\mathrm{nind}}\\ & & & & & & \\ B_{\mathrm{ndep},1}& & B_{\mathrm{ndep},\left(\hat{j}-1\right)}& B_{\mathrm{ndep},\hat{j}}& B_{\mathrm{ndep},\left(\hat{j}+1\right)}& & B_{\mathrm{ndep},\mathrm{nind}}\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{I}}_1\\ \\ {\stackrel{\_}{I}}_{\left(\hat{j}-1\right)}\\ {\stackrel{\_}{I}}_{\hat{j}}\\ {\stackrel{\_}{I}}_{\left(\hat{i}+1\right)}\\ \\ {\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}\right]=\left[\begin{array}{ccccccc}I& & 0& 0& 0& & 0\\ & & & & & & \\ 0& & I& 0& 0& & 0\\ 0& & 0& I& 0& & 0\\ 0& & 0& 0& I& & 0\\ & & & & & & \\ 0& & 0& 0& 0& & I\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{O}}_1\\ \\ {\stackrel{\_}{O}}_{\left(\hat{i}-1\right)}\\ {\stackrel{\_}{O}}_{\hat{i}}\\ {\stackrel{\_}{O}}_{\left(\hat{i}+1\right)}\\ \\ {\stackrel{\_}{O}}_{\mathrm{ndep}}\end{array}\right]$

[0056] where $I=\left[\begin{array}{ccc}10& & 00\\ 01& & 00\\ & & \\ 00& & 10\\ 00& & 01\end{array}\right],$

[0057] the Identity Matrix.

[0058] Note that this is nothing more than equation 2 above with an Identity matrix multiplying the O's.

[0059] By performing row elimination operations (pivoting), we get; $\left[\begin{array}{ccccccc}{\hat{B}}_{1,1}& & {\hat{B}}_{1,\left(\hat{j}-1\right)}& 0& {\hat{B}}_{1,\left(\hat{j}+1\right)}& & {\hat{B}}_{1,\mathrm{nind}}\\ & & & & & & \\ {\hat{B}}_{\left(\hat{i}-1\right),1}& & {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}-1\right)}& 0& {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}+1\right)}& & {\hat{B}}_{\left(\hat{i}-1\right),\mathrm{nind}}\\ {\hat{B}}_{i,1}& & {\hat{B}}_{{}_{\hat{i},\left(\hat{j}-1\right)}}& -I& {\hat{B}}_{\hat{i},\left(\hat{j}+1\right)}& & {\hat{B}}_{\hat{i},\mathrm{nind}}\\ {\hat{B}}_{\left(\hat{i}+1\right),1}& & {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}-1\right)}& 0& {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}+1\right)}& & {\hat{B}}_{\left(\hat{i}+1\right),\mathrm{nind}}\\ & & & & & & \\ {\hat{B}}_{\mathrm{ndep1}}& & {\hat{B}}_{\mathrm{ndep}\left(\hat{j}-1\right)}& 0& {\hat{B}}_{\mathrm{ndep}\left(\hat{j}+1\right)}& & {\hat{B}}_{\mathrm{ndep},\mathrm{nind}}\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{I}}_1\\ \\ {\stackrel{\_}{I}}_{\left(\hat{j}-1\right)}\\ {\stackrel{\_}{I}}_{\hat{j}}\\ {\stackrel{\_}{I}}_{\left(\hat{j}+1\right)}\\ \\ {\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}\right]=\left[\begin{array}{ccccccc}I& & 0& -{\hat{B}}_{1,\hat{j}}& 0& & 0\\ & & & & & & \\ 0& & I& -{\hat{B}}_{\left(\hat{i}-1\right),\hat{j}}& 0& & 0\\ 0& & 0& -{\hat{B}}_{\hat{i},\hat{j}}& 0& & 0\\ 0& & 0& -{\hat{B}}_{\left(\hat{i}+1\right),\hat{j}}& I& & 0\\ & & & & & & \\ 0& & 0& -{\hat{B}}_{\mathrm{ndep}\hat{j}}& 0& & I\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{O}}_1\\ \\ {\stackrel{\_}{O}}_{\left(\hat{i}-1\right)}\\ {\stackrel{\_}{O}}_{\hat{i}}\\ {\stackrel{\_}{O}}_{\left(\hat{i}+1\right)}\\ \\ {\stackrel{\_}{O}}_{\mathrm{ndep}}\end{array}\right]$

[0060] Which can be re-written as: $\begin{array}{c}\left[\begin{array}{c}{\hat{B}}_{1,1}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),1}\\ {\hat{B}}_{\hat{i},1}\\ {\hat{B}}_{\left(\hat{i}+1\right),1}\\ \\ {\hat{B}}_{\mathrm{ndep},1}\end{array}\right]{\stackrel{\_}{I}}_1++\left[\begin{array}{c}{\hat{B}}_{1,\left(\hat{j}-1\right)}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}-1\right)}\\ {\hat{B}}_{\hat{i},\left(\hat{j}-1\right)}\\ {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}-1\right)}\\ \\ {\hat{B}}_{\mathrm{ndep},\left(\hat{j}-1\right)}\end{array}\right]{\stackrel{\_}{I}}_{\left(\hat{j}-1\right)}+\left[\begin{array}{c}0\\ \\ 0\\ -I\\ 0\\ \\ 0\end{array}\right]{\stackrel{\_}{I}}_{\hat{j}}+\end{array}\begin{array}{c}\left[\begin{array}{c}{\hat{B}}_{1,\left(\hat{j}+1\right)}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}+1\right)}\\ {\hat{B}}_{\hat{i},\left(\hat{j}+1\right)}\\ {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}+1\right)}\\ \\ {\hat{B}}_{\mathrm{ndep},\left(\hat{j}+1\right)}\end{array}\right]{\stackrel{\_}{I}}_{\left(\hat{j}+1\right)}+\left[\begin{array}{c}{\hat{B}}_{1,\mathrm{nind}}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),\mathrm{nind}}\\ {\hat{B}}_{\hat{i},\mathrm{nind}}\\ {\hat{B}}_{\left(\hat{i}+1\right),\mathrm{nind}}\\ \\ {\hat{B}}_{\mathrm{ndep},\mathrm{nind}}\end{array}\right]{\stackrel{\_}{I}}_{\mathrm{nind}}=\left[\begin{array}{c}I\\ \\ 0\\ 0\\ 0\\ \\ 0\end{array}\right]{\stackrel{\_}{O}}_1++\end{array}\begin{array}{c}\left[\begin{array}{c}0\\ \\ I\\ 0\\ 0\\ \\ 0\end{array}\right]{\stackrel{\_}{O}}_{\left(\hat{i}-1\right)}+\left[\begin{array}{c}-{\hat{B}}_{1,\hat{j}}\\ \\ -{\hat{B}}_{\left(\hat{i}-1\right),\hat{j}}\\ -{\hat{B}}_{\hat{i},\hat{j}}\\ -{\hat{B}}_{\left(\hat{i}+1\right),\hat{j}}\\ \\ -{\hat{B}}_{\mathrm{ndep},\hat{j}}\end{array}\right]{\stackrel{\_}{O}}_{\hat{i}}+\left[\begin{array}{c}0\\ \\ 0\\ 0\\ I\\ \\ 0\end{array}\right]{\stackrel{\_}{O}}_{\left(\hat{i}+1\right)}++\left[\begin{array}{c}0\\ \\ 0\\ 0\\ 0\\ \\ I\end{array}\right]{\stackrel{\_}{O}}_{\mathrm{ndep}}\end{array}$

[0061] Which can be rearranged to; $\begin{array}{c}\left[\begin{array}{c}{\hat{B}}_{1,1}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),1}\\ {\hat{B}}_{\hat{i},1}\\ {\hat{B}}_{\left(\hat{i}+1\right),1}\\ \\ {\hat{B}}_{\mathrm{ndep},1}\end{array}\right]{\stackrel{\_}{I}}_1++\left[\begin{array}{c}{\hat{B}}_{1,\left(\hat{j}-1\right)}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}-1\right)}\\ {\hat{B}}_{\hat{i},\left(\hat{j}-1\right)}\\ {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}-1\right)}\\ \\ {\hat{B}}_{\mathrm{ndep},\left(\hat{j}-1\right)}\end{array}\right]{\stackrel{\_}{I}}_{\left(\hat{j}-1\right)}+\left[\begin{array}{c}{\hat{B}}_{1,\hat{j}}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),\hat{j}}\\ {\hat{B}}_{\hat{i},\hat{j}}\\ {\hat{B}}_{\left(\hat{i}+1\right),\hat{j}}\\ \\ {\hat{B}}_{\mathrm{ndep},\hat{j}}\end{array}\right]{\stackrel{\_}{O}}_{\hat{i}}+\end{array}\begin{array}{c}\left[\begin{array}{c}{\hat{B}}_{1,\left(\hat{j}+1\right)}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}+1\right)}\\ {\hat{B}}_{\hat{i},\left(\hat{j}+1\right)}\\ {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}+1\right)}\\ \\ {\hat{B}}_{\mathrm{ndep},\left(\hat{j}+1\right)}\end{array}\right]{\stackrel{\_}{I}}_{\left(\hat{j}+1\right)}+\left[\begin{array}{c}{\hat{B}}_{1,\mathrm{nind}}\\ \\ {\hat{B}}_{\left(\hat{i}-1\right),\mathrm{nind}}\\ {\hat{B}}_{\hat{i},\mathrm{nind}}\\ {\hat{B}}_{\left(\hat{i}+1\right),\mathrm{nind}}\\ \\ {\hat{B}}_{\mathrm{ndep},\mathrm{nind}}\end{array}\right]{\stackrel{\_}{I}}_{\mathrm{nind}}=\left[\begin{array}{c}I\\ \\ 0\\ 0\\ 0\\ \\ 0\end{array}\right]{\stackrel{\_}{O}}_1++\end{array}\begin{array}{c}\left[\begin{array}{c}0\\ \\ I\\ 0\\ 0\\ \\ 0\end{array}\right]{\stackrel{\_}{O}}_{\left(\hat{i}-1\right)}+\left[\begin{array}{c}0\\ \\ 0\\ I\\ 0\\ \\ 0\end{array}\right]{\stackrel{\_}{I}}_{\hat{j}}+\left[\begin{array}{c}0\\ \\ 0\\ 0\\ I\\ \\ 0\end{array}\right]{\stackrel{\_}{O}}_{\left(\hat{i}+1\right)}++\left[\begin{array}{c}0\\ \\ 0\\ 0\\ 0\\ \\ I\end{array}\right]{\stackrel{\_}{O}}_{\mathrm{ndep}}\end{array}$

[0062] or reassembling the matrix equation we get; $\left[\begin{array}{ccccccc}{\hat{B}}_{1,1}& & {\hat{B}}_{1,\left(\hat{j}-1\right)}& {\hat{B}}_{1,\hat{j}}& {\hat{B}}_{1,\left(\hat{j}+1\right)}& & {\hat{B}}_{1,\mathrm{nind}}\\ & & & & & & \\ {\hat{B}}_{\left(\hat{i}-1\right),1}& & {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}-1\right)}& {\hat{B}}_{\left(\hat{i}-1\right),\hat{j}}& {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}+1\right)}& & {\hat{B}}_{\left(\hat{i}-1\right),\mathrm{nind}}\\ {\hat{B}}_{\hat{i},1}& & {\hat{B}}_{{}_{\hat{i},\left(\hat{j}-1\right)}}& {\hat{B}}_{\hat{i},\hat{j}}& {\hat{B}}_{\hat{i},\left(\hat{j}+1\right)}& & {\hat{B}}_{\hat{i},\mathrm{nind}}\\ {\hat{B}}_{\left(\hat{i}+1\right),1}& & {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}-1\right)}& {\hat{B}}_{\left(\hat{i}+1\right),\hat{j}}& {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}+1\right)}& & {\hat{B}}_{\left(\hat{i}+1\right),\mathrm{nind}}\\ & & & & & & \\ {\hat{B}}_{\mathrm{ndep1}}& & {\hat{B}}_{\mathrm{ndep}\left(\hat{j}-1\right)}& {\hat{B}}_{\mathrm{ndep}\hat{j}}& {\hat{B}}_{\mathrm{ndep}\left(\hat{j}+1\right)}& & {\hat{B}}_{\mathrm{ndep},\mathrm{nind}}\end{array}\right]\begin{array}{c}\left[\begin{array}{c}{\stackrel{\_}{I}}_1\\ \\ {\stackrel{\_}{I}}_{\left(\hat{j}-1\right)}\\ {\stackrel{\_}{O}}_{\hat{i}}\\ {\stackrel{\_}{I}}_{\left(\hat{j}+1\right)}\\ \\ {\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}\right]\end{array}=\left[\begin{array}{ccccccc}I& & 0& 0& 0& & 0\\ & & & & & & \\ 0& & I& 0& 0& & 0\\ 0& & 0& I& 0& & 0\\ 0& & 0& 0& I& & 0\\ & & & & & & \\ 0& & 0& 0& 0& & I\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{O}}_1\\ \\ {\stackrel{\_}{O}}_{\left(\hat{i}-1\right)}\\ {\stackrel{\_}{I}}_{\hat{j}}\\ {\stackrel{\_}{O}}_{\left(\hat{i}+1\right)}\\ \\ {\stackrel{\_}{O}}_{\mathrm{ndep}}\end{array}\right]$

[0063] Note that {overscore (O)} and {overscore (I)} have been interchanged so that the valve position is now an independent variable and the PID set point is now a dependent variable. This illustrates removing the PID dynamics from only one PID controller, but the algorithm is clearly general in that multiple independent/dependent variable pairs can be interchanged to remove the dynamics for multiple controllers.

[0064] By way of further illustration a numerical example problem will now be illustrated to show how this approach is applied to a model predictive controller to remove the dynamics of a particular PID controller.

[0065] Given an FIR model with two (2) independent variables, two (2) dependent variables and four (4) model coefficients, where the second independent variable is the set point of a PID controller and the second dependent variable is the valve position of the PID controller, we wish to re-constitute the model with the PID valve position as an independent variable instead of the PID set point. This requires that the dynamics of the PID controller be removed from all system responses according to the algorithm previously discussed. This example is also valid for the OBI₁, OBI, and OBI forms of the equation.

Dependent Var-1
	Independent Var-1	Independent Var-2
	b1,1,1 1.5	b1,2,1 0.5
	b1,1,2 0.6	b1,2,2 0.4
	b1,1,3 0.2	b1,2,3 0.2
	b1,1,4 0.1	b1,2,4 0.1

[0066]

Dependent Var-2
	Independent Var-1	Independent Var-2
	b2,1,1 0.3	b2,2,1 0.75
	b2,1,2 0.4	b2,2,2 0.25
	b2,1,3 0.1	b2,2,3 0.15
	b2,1,4 0.05	b2,2,4 0.05

[0067] The problem is specified in the matrix below.

	Indicates Pivot Element
1.5	0	0	0	0.5	0	0	0	1	0	0	0	0	0	0	0
0.6	1.5	0	0	0.4	0.5	0	0	0	1	0	0	0	0	0	0
0.2	0.6	1.5	0	0.2	0.4	0.5	0	0	0	1	0	0	0	0	0
0.1	0.2	0.6	1.5	0.1	0.2	0.4	0.5	0	0	0	1	0	0	0	0
0.3	0	0	0	0.75	0	0	0	0	0	0	0	1	0	0	0
0.4	0.3	0	0	0.25	0.75	0	0	0	0	0	0	0	1	0	0
0.1	0.4	0.3	0	0.15	0.25	0.75	0	0	0	0	0	0	0	1	0
0.05	0.1	0.4	0.3	0.05	0.15	0.25	0.75	0	0	0	0	0	0	0	1
Multiply Equation-5 by (1/0.75)
1.5	0	0	0	0.5	0	0	0	1	0	0	0	0	0	0	0
0.6	1.5	0	0	0.4	0.5	0	0	0	1	0	0	0	0	0	0
0.2	0.6	1.50	0	0.2	0.4	0.5	0	0	0	1	0	0	0	0	0
0.1	0.2	0.6	1.5	0.1	0.2	0.4	0.5	0	0	0	1	0	0	0	0
0.4	0	0	0	1	0	0	0	0	0	0	0	1.333	0	0
0.4	0.3	0	0	0.25	0.75	0	0	0	0	0	0	0	1	0	0
0.1	0.4	0.3	0	0.15	0.25	0.75	0	0	0	0	0	0	0	1	0
0.05	0.1	0.4	0.3	0.05	0.15	0.25	0.75	0	0	0	0	0	0	0	1
Multiply Equation-5 by 0.5, add it to Equation-1 and replace Equation-1
Multiply Equation-5 by 0.4, add it to Equation-2 and replace Equation-2
Multiply Equation-S by 0.2, add it to Equation-3 and replace Equation-3
Multiply Equation-5 by 0.1, add it to Equation-4 and replace Equation-4
Multiply Equation-5 by 0.25, add it to Equation-6 and replace Equation-6
Multiply Equation-5 by 0.15, add it to Equation-7 and replace Equation-7
Multiply Equation-5 by 0.05, add it to Equation-8 and replace Equation-8
1.7	0	0	0	0	0	0	0	1	0	0	0	0.667	0	0	0
0.76	1.5	0	0	0	0.5	0	0	0	1	0	0	0.533	0	0	0
0.28	0.6	1.5	0	0	0.4	0.5	0	0	0	1	0	0.267	0	0	0
0.14	0.2	0.6	1.5	0	0.2	0.4	0	0	0	0	1	0.133	0	0	0
0.4	0	0	0	1	0	0	0	0	0	0	0	1.333	0	0	0
0.3	0.3	0	0	0	0.75	0	0	0	0	0	0	0.333	1	0	0
0.04	0.4	0.3	0	0	0.25	0.75	0	0	0	0	0	0.20	0	1	0
0.03	0.1	0.4	0.3	0	0.15	0.25	0.70	0	0	0	0	0.067	0	0	1
Multiply Equation-6 by (1/0.75)
1.7	0	0	0	0	0	0	0	1	0	0	0	0.667	0	0	0
0.76	1.5	0	0	0	0.5	0	0	0	1	0	0	0.533	0	0	0
0.28	0.6	1.5	0	0	0.4	0.5	0	0	0	1	0	0.267	0	0	0
0.14	0.2	0.6	1.5	0	0.2	0.4	0.5	0	0	0	1	0.133	0	0	0
0.4	0	0	0	1	0	0	0	0	0	0	0	1.333	0	0	0
0.4	0.4	0	0	0	1	0	0	0	0	0	0	0.444	1.333	0	0
0.04	0.4	0.3	0	0	0.25	0.75	0	0	0	0	0	0.2	0	1	0
0.03	0.1	0.4	0.3	0	0.15	0.25	0.75	0	0	0	0	0.067	0	0	1
Multiply Equation-5 by 0.5, add it to Equation-2 and replace Equation-2
Multiply Equation-5 by 0.4, add it to Equation-3 and replace Equation-3
Multiply Equation-5 by 0.2, add it to Equation-4 and replace Equation-4
Multiply Equation-5 by 0.25, add it to Equation-7 and replace Equation-7
Multiply Equation-5 by 0.15, add it to Equation-8 and replace Equation-8
1.7	0	0	0	0	0	0	0	1	0	0	0	0.667	0	0	0
0.96	1.7	0	0	0	0	0	0	0	1	0	0	0.311	0.667	0	0
0.44	0.76	1.5	0	0	0	0.5	0	0	0	1	0	0.089	0.533	0	0
0.22	0.28	0.6	1.5	0	0	0.4	0.5	0	0	0	1	0.044	0.267	0	0
0.4	0	0	0	1	0	0	0	0	0	0	0	1.333	0	0	0
0.4	0.4	0	0	0	1	0	0	0	0	0	0	0.444	1.333	0	0
0.06	0.3	0.3	0	0	0	0.75	0	0	0	0	0	0.089	0.333	1	0
0.03	0.04	0.4	0.3	0	0	0.25	0.75	0	0	0	0	0	0.2	0	1
Multiply Equation-7 by (1/0.75)
1.7	0	0	0	0	0	0	0	1	0	0	0	0.667	0	0	0
0.96	1.7	0	0	0	0	0	0	0	1	0	0	0.311	0.667	0	0
0.44	0.76	1.5	0	0	0	0.5	0	0	0	1	0	0.089	0.533	0	0
0.22	0.28	0.6	1.5	0	0	0.4	0.5	0	0	0	1	0.044	0.267	0	0
0.4	0	0	0	1	0	0	0	0	0	0	0	1.333	0	0	0
0.4	0.4	0	0	0	1	0	0	0	0	0	0	.444	1.333	0	0
0.08	0.4	0.4	0	0	0	1	0	0	0	0	0	.119	0.4444	1.333	0
0.03	0.04	0.4	0.3	0	0	0.25	0.75	0	0	0	0	0	0.2	0	1
Multiply Equation-5 by 0.5, add it to Equation-3 and replace Equation-3
Multiply Equation-5 by 0.4, add it to Equation-4 and replace Equation-4
Multiply Equation-5 by 0.25, add it to Equation-8 and replace Equation-8
1.7	0	0	0	0	0	0	0	1	0	0	0	0.667	0	0	0
0.96	1.7	0	0	0	0	0	0	0	1	0	0	0.311	0.667	0	0
0.4	0.96	1.7	0	0	0	0	0	0	0	1	0	0.030	0.311	0.667	0
0.188	0.44	0.76	1.5	0	0	0	0.5	0	0	0	1	0.003	0.089	0.533	0
0.4	0	0	0	1	0	0	0	0	0	0	0	1.333	0	0	0
0.4	0.4	0	0	0	1	0	0	0	0	0	0	0.444	1.333	0	0
0.08	0.4	0.4	0	0	0	1	0	0	0	0	0	0.119	0.444	1.333	0
0.01	0.06	0.3	0.30	0	0	0	0.75	0	0	0	0	0.030	0.089	0.333	1
Multiply Equation-8 by (1/0.75)
1.7	0	0	0	0	0	0	0	1	0	0	0	0.667	0	0	0
0.96	1.7	0	0	0	0	0	0	0	1	0	0	0.311	0.667	0	0
0.4	0.96	1.7	0	0	0	0	0	0	0	1	0	0.030	0.311	0.667	0
0.188	0.44	0.76	1.5	0	0	0	0.5	0	0	0	1	0.003	0.089	0.533	0
0.4	0	0	0	1	0	0	0	0	0	0	0	1.333	0	0	0
0.4	0.4	0	0	0	1	0	0	0	0	0	0	0.444	1.333	0	0
0.08	0.4	0.4	0	0	0	1	0	0	0	0	0	0.119	0.444	1.333	0
0.013	0.08	0.4	0.4	0	0	0	1	0	0	0	0	0.040	0.119	0.444	1.333
Multiply Equation-5 by 0.5, add it to Equation-4 and replace Equation-4
1.7	0	0	0	0	0	0	0	1	0	0	0	0.667	0	0	0
0.96	1.7	0	0	0	0	0	0	0	1	0	0	0.311	0.667	0	0
0.4	0.96	1.7	0	0	0	0	0	0	0	1	0	0.030	0.311	0.667	0
0.181	0.4	0.96	1.7	0	0	0	0	0	0	0	1	0.017	0.030	0.311	0.667
0.4	0	0	0	1	0	0	0	0	0	0	0	1.333	0	0	0
0.4	0.4	0	0	0	1	0	0	0	0	0	0	0.444	1.333	0	0
0.08	0.4	0.4	0	0	0	1	0	0	0	0	0	0.119	0.444	1.333	0
0.013	0.08	0.4	0.4	0	0	0	1	0	0	0	0	0.040	0.119	0.444	1.333
Rearrange Equations
1.7	0	0	0	0.667	0	0	0	1	0	0	0	0	0	0	0
0.96	1.7	0	0	0.311	0.667	0	0	0	1	0	0	0	0	0	0
0.4	0.96	1.7	0	0.030	0.311	0.667	0	0	0	1	0	0	0	0	0
0.181	0.4	0.96	1.7	0.017	0.030	0.311	0.667	0	0	0	1	0	0	0	0
0.4	0	0	0	1.333	0	0	0	0	0	0	0	1	0	0	0
0.4	0.4	0	0	0.444	1.333	0	0	0	0	0	0	0	1	0	0
0.08	0.4	0.4	0	0.119	0.444	1.333	0	0	0	0	0	0	0	1	0

[0068] The new model coefficients with the PID dynamics removed are as follows:

Dependent Var-1
	Independent Var-1	Independent Var-2
	{circumflex over (b)}1,1,1 1.7	{circumflex over (b)}1,2,1 0.667
	{circumflex over (b)}1,1,2 0.96	{circumflex over (b)}1,2,2 0.311
	{circumflex over (b)}1,1,3 0.4	{circumflex over (b)}1,2,3 0.030
	{circumflex over (b)}1,1,4 0.181	{circumflex over (b)}1,2,4 0.017

[0069]

Dependent Var-2
	Independent Var-1	Independent Var-2
	{circumflex over (b)}2,1,1 0.4	{circumflex over (b)}2,2,1 1.333
	{circumflex over (b)}2,1,2 0.4	{circumflex over (b)}2,2,2 0.444
	{circumflex over (b)}2,1,3 0.08	{circumflex over (b)}2,2,3 0.119
	{circumflex over (b)}2,1,4 0.0133	{circumflex over (b)}2,2,4 0.040

[0070] Note that all the coefficient values changed. This new controller now has the dynamics of the second independent variable (a PID set point) removed. This controller can now be used to control the process and the development of this controller was done off line without having to do an additional time consuming expensive identification test on the process.

[0071] In the derivation and example, we discussed the algorithm to remove PID dynamics from an FIR model based on the impulse, or derivative, form of the equations. A similar algorithm can be derived for the Step coefficient form of the model, OAI, as well, as we will now illustrate with a 2 independent, 2 dependent variable example problem. For purposes of this example, we will assume that the second independent and second dependents are to be exchanged. The problem can be written in matrix notation as: $\left[\begin{array}{cc}{A}_{1,1}& A_{1,2}\\ A_{2,1}& A_{2,2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{I}}_1\\ {\stackrel{\_}{I}}_2\end{array}\right]=\left[\begin{array}{c}\mathrm{I0}\\ 0I\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{O}}_1\\ {\stackrel{\_}{O}}_2\end{array}\right]$

[0072] We perform elimination operations (pivoting) to get: $\left[\begin{array}{c}{\hat{A}}_{1,1}0\\ {\hat{B}}_{2,1}-I\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{I}}_1\\ {\stackrel{\_}{I}}_2\end{array}\right]=\left[\begin{array}{c}I-{\hat{B}}_{1,2}\\ 0-{\hat{C}}_{2,2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{O}}_1\\ {\stackrel{\_}{O}}_2\end{array}\right]$

[0073] Rearranging, we get: $\left[\begin{array}{c}{\hat{A}}_{1,1}\\ {\hat{B}}_{2,1}\end{array}\begin{array}{c}{\hat{B}}_{1,2}\\ {\hat{C}}_{2,2}\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{I}}_1\\ {\stackrel{\_}{O}}_2\end{array}\right]=\left[\begin{array}{c}\mathrm{I0}\\ 0I\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{O}}_1\\ {\stackrel{\_}{I}}_2\end{array}\right]$

[0074] Which can be written as:

[0075] Recall that the Impulse coefficients are defined as:

[0076] Likewise, we define the second difference coefficients as:

[0077] Note that: $b_{i,j,m}=\underset{m=1}{\overset{l}{}}\left(c_{i,j,l}\right)$ $a_{i,j,k}=\underset{l=1}{\overset{k}{}}\left(b_{i,j,l}\right)=\underset{l}{\overset{k}{}}\left(\underset{m=1}{\overset{l}{}}\left(c_{i,j,m}\right)\right)$

[0078] Note that the matrix is now a mixed bag of Step response coefficients (A), Impulse coefficients (B), and 2^nd difference coefficients (C). This is due to the fact that our new independent variable is in the accumulative form instead of the delta form and the new dependent variable is in the delta form instead of the accumulative form. To convert this system of equation to the Step form and thereby recover the Step coefficients, two steps must be performed:

[0079] Step-1: Convert new independent variable from accumulative to delta form, {overscore (O)}₂>{overscore (O)}₂.

[0080] Step-2: Convert new dependent variable from delta to accumulative form, {overscore (I)}₂>{overscore (I)}₂.

[0081] Step-1: Convert new independent variable from accumulative to delta form.

[0082] This step requires only a rearrangement of terms in the equations. Note that, {overscore (O)}₂ appears in two sections of the matrix: $\left[\begin{array}{c}{\hat{B}}_{1,2}{\stackrel{\_}{O}}_2\\ {\hat{C}}_{2,2}{\stackrel{\_}{O}}_2\end{array}\right]=\begin{array}{c}[\begin{array}{cccc}{\hat{b}}_{1,2,1}(O_{2,1}-O_{2,0})& & & \\ {\hat{b}}_{1,2,2}(O_{2,1}-O_{2,0})+{\hat{b}}_{1,2,1}& (O_{2,2}-O_{2,0})& & \\ {\hat{b}}_{1,2,3}(O_{2,1}-O_{2,0})+{\hat{b}}_{1,2,2}& (O_{2,2}-O_{2,0})+{\hat{b}}_{1,2,1}& (O_{2,3}-O_{2,0})& \\ & & & \\ {\hat{c}}_{2,2,1}(O_{2,1}-O_{2,0})& & & \\ {\hat{c}}_{2,2,2}(O_{2,1}-O_{2,0})+{\hat{c}}_{2,2,1}& (O_{2,2}-O_{2,0})& & \\ {\hat{c}}_{2,2,3}(O_{2,1}-O_{2,0})+{\hat{c}}_{2,2,2}& (O_{2,2}-O_{2,0})+{\hat{c}}_{2,2,1}& (O_{2,3}-O_{2,0})& \\ & & & \end{array}]\end{array}$