Cutler, Charles R. (San Antonio, TX, US)

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11/212252

11/04/2008

08/26/2005

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Cutler Technology Corporation (San Antonio, TX, US)

700/83

700/30, 700/38, 700/29, 700/31, 700/39

G05B13/02; G05B15/00; G06F17/50

700/83, 700/38, 700/29-31, 703/2, 700/39

4349869	Dynamic matrix control method	September, 1982	Prett et al.	700/39
4616308	Dynamic process control	October, 1986	Morshedi et al.	700/39
5347446	Model predictive control apparatus	September, 1994	Iino et al.
5519605	Model predictive control apparatus and method	May, 1996	Cawlfield	700/31
5566065	Method and apparatus for controlling multivariable nonlinear processes	October, 1996	Hansen et al.
5568378	Variable horizon predictor for controlling dead time dominant processes, multivariable interactive processes, and processes with time variant dynamics	October, 1996	Wojsznis
6056781	Model predictive controller	May, 2000	Wassick et al.
6088630	Automatic control system for unit operation	July, 2000	Cawlfield
6980938	Method for removal of PID dynamics from MPC models	December, 2005	Cutler	703/2
7263473	Method for removal of PID dynamics from MPC models	August, 2007	Cutler	703/2
20030195665	Method for removal of PID dynamics from MPC models	October, 2003	Cutler
20050137721	Use of core process models in model predictive controller	June, 2005	Attarwala	700/30
20050149208	Automated closed loop step testing of process units	July, 2005	Harmse et al.	700/29
20060287741	On-line dynamic advisor from MPC models	December, 2006	Cutler	700/83

JP53125739

November, 1978

Patry, et al. ; Model Predictive Controller Design with Process Constraints and implicit Economic Criteria. IEEE Instr. & Meas. Tech. Conf. May 1997. pp. 559-563.

Decady, Albert

Chang, Sunray R.

M. A. Ervin & Associates
Ervin, Michael A.

I claim:

1. A method for controlling a process having a plurality of independently controllable, manipulated variables and at least one controlled variable dependent upon said independently controllable, manipulated variables comprising the steps of: gathering data about said process by separately introducing test disturbances in each of said manipulated variables and measuring the effect of the disturbances on said controlled variable; using said effects of the disturbances on said controlled variable to generate a first linearized dynamic model relating said at least one controlled variable to said independently controllable, manipulated variables; interchanging selected final control element position controlled variables with their corresponding selected independently controllable, manipulated PID controller set point variables in said 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, said second linearized dynamic model having the dynamics of said selected independently controllable, manipulated PID controller set point variables removed from said second linearized dynamic model; measuring the present value of said variables; pre-setting operating limits for said at least one controlled variable; pre-setting test limits for said at least one controlled variable that fall within said operating limits; using the second linearized dynamic model to calculate a first set of prediction vectors for said at least one controllable variable; placing said second linearized dynamic model in an adaptive test mode wherein all of the PID controllers are switched to manual; perturbing the final control element manipulated variables with random excitation signals; calculating for discrete intervals of time from said gathered data about said process, said measured present values and pre-selected operating constraints a set of moves for present and future times for at least said manipulated variables to obtain new values for said manipulated variables and to move said at least one dependent controllable variable towards at least one of said constraints; and when the prediction vector is outside of test limits for said at least one controlled variable, changing said process by adjusting said manipulated variables for said set of moves for present and future times to cause said process to move said at least one dependent controllable variable towards at least one of said constraints.

2. The method of claim 1, wherein said perturbing step additionally comprises perturbing said test limits for said at least one controlled variable with random excitation signals.

3. The method of claim 2, wherein said process comprises at least one uncontrolled variable that is dependent on said manipulated variables and wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that said at least one uncontrolled variable is limited to a predetermined constraint.

4. The method of claim 3, wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that at least one of said manipulated variables is limited to a predetermined constraint.

5. The method of claim 1, wherein said process comprises at least one uncontrolled variable that is dependent on said manipulated variables and wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that said at least one uncontrolled variable is limited to a predetermined constraint.

6. The method of claim 5, wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that at least one of said manipulated variables is limited to a predetermined constraint.

7. The method of claim 1, wherein said step of calculating said set of moves for present and future times comprises calculating said set of moves employing quadratic programming techniques.

8. The method of claim 7, wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that at least one of said manipulated variables is limited to a predetermined constraint.

9. The method of claim 7, wherein said process comprises at least one uncontrolled variable that is dependent on said manipulated variables and wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that said uncontrolled variable is limited to a predetermined constraint.

10. The method of claim 1, wherein said step of calculating said set of moves for present and future times comprises calculating said set of moves employing linear programming techniques.

11. The method of claim 10, wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that at least one of said manipulated variables is limited to a predetermined constraint.

12. The method of claim 10, wherein said process comprises at least one uncontrolled variable that is dependent on said manipulated variables and wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that said uncontrolled variable is limited to a predetermined constraint.

13. The method of claim 1, wherein said step of calculating said set of moves further comprises calculating said set of moves such that at least one of said manipulated variables is limited to a predetermined constraint.

14. The method of claim 13, wherein said process comprises at least one uncontrolled variable that is dependent on said manipulated variables and wherein said step of calculating said set of moves for present and future times further comprises calculating said set of moves such that said uncontrolled variable is limited to a predetermined constraint.

BACKGROUND OF THE INVENTION

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.

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.

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 (manipulated) variables and measuring the effects of the disturbances on the dependent (controlled) variables. This initial step is referred to as identification.

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.

By way of further background this identification of process dynamics requires a pre-test in which the manipulated variables of the process are moved in some pattern to determine the effect on the dependent (controlled) variables. In a chemical or refinery process the independent variables include the PID (proportional-integral-derivative) controller set points for selected dependent variables, the final control element 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. Because the PID dynamics are a part of the plant behavior there is an inherent correlation of variables that happens as unmeasured disturbances occur in the process. The various PID control loops respond to those unmeasured disturbances and move many of the controlled variables in response. This has historically always prevented practitioners from creating MPC controllers free of the PID dynamics using standard identification tests.

U.S. Pat. No. 6,980,938 by the inventor is incorporated by reference into this application in its entirety. This application addresses the aforementioned issue and describes a methodology for removing the PID dynamics from the dynamic model by use of a novel mathematical matrix algorithm that interchanges selected final control element position (usually valve positions) 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. This second linearized model is an open loop model based on final control element positions only. Because it is an open loop finite impulse response model it has been shown that it can run 50 to 100 times faster than real time. U.S. Pat. No. 6,980,938 describes and claims the use of this type of model in both control and in the development of off line training simulators.

A greatly desired but unmet need in the control of complex multivariable processes such as chemical manufacturing and oil refining is the use of an adaptive controller. In the past, adding an adaptive mechanism to MPC has been approached in a number of ways. Researchers have primarily focused on updating the internal process model used in the control algorithm. Several articles review various adaptive control mechanisms for controlling nonlinear processes (Seborg, Edgar, & Shah, 1986; Bequette, 1991; Di Marco, Semino, & Brambilla, 1997). In addition, Qin and Badgwell (2000) provide a good overview of nonlinear MPC applications that are currently available in industry. As illustrated by these works, the adaptive control mechanisms consider the use of a nonlinear analytical model, combinations of linear empirical models or some mixture of both.

MPC using nonlinear models is likely to become more common as users demand higher performance and new software tools make nonlinear models more readily available. Developing adequate nonlinear empirical models is very challenging, however. There is no model form that is clearly suitable to represent general nonlinear processes. From a theoretical point of view using a nonlinear model changes the control problem from a convex QP to a non-convex Non-Linear Program (NLP), the solution of which is much more difficult. There is no guarantee, for example, that the global optimum can be found. It is important to note that because of these factors none of these non-linear approaches have been successfully implemented commercially on large-scale controllers.

An alternative approach would be to use first-principles models developed from well-known mass, momentum, and energy conservation laws. However, the cost of developing a reasonably accurate first-principles model is likely to be prohibitive until new software tools and validation procedures become available.

A desirable solution though would be an adaptive controller based on linear MPC type models such as dynamic matrix control (DMC) models. This approach would be highly desirable to a control practitioner who is already conversant with the use of DMC type control.

The recognition of this unmet need and a method of addressing the need by use of an open loop finite impulse response model with the PID set points and replaced with final control element positions coupled with an adaptive control methodology is an aspect of this invention.

BRIEF SUMMARY OF THE INVENTION

The inventive concept can be described as follows: First a method is employed to remove the dynamics of the PID controllers from the MPC controller that was created by a plant identification test. This enables the creation of a final control element based open loop FIR model of the process. This model is then used to build an adaptive multivariable controller.

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

It is a further aspect of this invention to provide a method to create such an adaptive controller for complex multivariable processes that can be modified with new regulatory control configurations or new tuning and to do so without having to conduct new identification testing of the process.

It is a further aspect of this invention to provide a method of creating an off-line simulator based on the adaptive model of this invention.

It is a further aspect of this invention to provide a method of creating an on-line operator advisor based on the adaptive model of this invention.

An object of the invention is achieved by a method for controlling a process having a plurality of independently controllable, manipulated variables and at least one controlled variable dependent upon the independently controllable, manipulated variables comprising the steps of: gathering data about the process by separately introducing test disturbances 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 the final control 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; pre-setting operating limits for the at least one controlled variable; pre-setting test limits for the at least one controlled variable that fall within the operating limits; using the second linearized dynamic model to calculate a first set of prediction vectors for the at least one controllable variable; placing the second linearized dynamic model in an adaptive test mode wherein all of the PID controllers are switched to manual; perturbing the final control element manipulated variables with random excitation signals; 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 when the prediction vector is outside of test limits for the at least one controlled variable, 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.

Another aspect of the invention is a method for creating an off-line process simulator for use in process simulation and for training simulators for a process by use of the adaptive multivariable controller model created by the methods just described comprising the steps of: porting the adaptive multivariable controller model to any computer; coupling regulatory control schemes to the adaptive multivariable controller model via mathematical emulators to emulate PID controllers in manual, cascade, or automatic modes; initializing the completed model with initial conditions; and mathematically solving the completed model to predict future variable states.

Another aspect of the invention is a method for creating an on-line operator advisor for use in process simulation and for training simulators for a process by use of the adaptive multivariable controller model created by the methods described above comprising the steps of: coupling regulatory control schemes to the adaptive multivariable controller model via mathematical emulators to emulate PID controllers in manual, cascade, or automatic modes; accessing real time data from the process to initialize the completed model with the state of the manipulated and controlled variables; initializing the configuration and tuning of the regulatory control system; initializing with the state of the open loop final control element prediction vectors; and mathematically solving the completed model to predict the future path of the at least one controlled variable.

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.

DESCRIPTION OF DRAWINGS

FIG. 1 is a flow schematic of a fractionator

FIG. 2 is a simulation of the fractionator model based on valve positions

FIG. 3 demonstrates the results from a plant test of the fractionator

FIG. 4 is a simulation of the fractionator with the PID controllers

FIG. 5 is a demonstration of the fractionator with the original and recovered values

FIG. 6 is a flow sheet representation of the steps of an aspect of the inventive method.

FIG. 7 is a flow sheet representation of the steps of an aspect of the inventive method.

FIG. 8 is a flow sheet representation of the steps of an aspect of the inventive method.

FIG. 9 is a flow sheet representation of the steps of an aspect of the inventive method.

FIG. 10 is a depiction of the adaptive test limits for a controlled variable.

DETAILED DESCRIPTION OF INVENTION

The invention is a method used in conjunction with model predictive control for removing the dynamics of PID controllers from an MPC controller of a process in order to subsequently create an adaptive multivariable controller.

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 final control element position (dependent) for that PID controller.

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

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 final control element positions or PID controller set points. 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.

Dependent variables are outputs from the system. They are also called controlled variables and those two terms are used interchangeably in this discussion. 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 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:

$\begin{array}{ccccccccccc}\delta {\stackrel{\_}{O}}_1& =& A_{1,1}\Delta {\stackrel{\_}{I}}_1& +& \dots & +& A_{1,j}\Delta {\stackrel{\_}{I}}_j& +& \dots & +& A_{1,\mathrm{nind}}\Delta {\stackrel{\_}{I}}_{\mathrm{nind}}\\ & ⋮& & ⋮& & ⋮& & ⋮& & ⋮& \\ \delta {\stackrel{\_}{O}}_i& =& A_{i,1}\Delta {\stackrel{\_}{I}}_1& +& \dots & +& A_{i,j}\Delta {\stackrel{\_}{I}}_j& +& \dots & +& A_{i,\mathrm{nind}}\Delta {\stackrel{\_}{I}}_{\mathrm{nind}}\\ & ⋮& & ⋮& & ⋮& & ⋮& & ⋮& \\ \delta {\stackrel{\_}{O}}_{\mathrm{ndep}}& =& A_{\mathrm{ndep},1}\Delta {\stackrel{\_}{I}}_1& +& \dots & +& A_{\mathrm{ndep},j}\Delta {\stackrel{\_}{I}}_j& +& \dots & +& A_{\mathrm{ndep},\mathrm{nind}}\Delta {\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}$
Equation 1: Step Response Dynamic Matrix, Block Matrix Form
where,

$\delta {\stackrel{\_}{O}}_i=\left[\begin{array}{c}{O}_{i,1}-O_{i,0}\\ O_{i,2}-O_{i,0}\\ O_{i,3}-O_{i,0}\\ ⋮\\ O_{i,\mathrm{ncoef}}-O_{i,0}\end{array}\right],$
the accumulative change in the i ^th dependent variable at each time step,

$\ddot{A}{\stackrel{\_}{I}}_j=\left[\begin{array}{c}\Delta I_{j,1}\\ \Delta I_{j,2}\\ \Delta I_{j,3}\\ ⋮\\ \Delta I_{j,\mathrm{ncoef}}\end{array}\right],$
the step change in the j ^th independent variable at each time step, and

$A_{i,j}=\left[\begin{array}{ccccc}{a}_{i,j,1}& & & & \\ a_{i,j,2}& a_{i,j,1}& & & \\ a_{i,j,3}& a_{i,j,2}& a_{i,j,1}& & \\ ⋮& ⋮& ⋮& ⋰& \\ a_{i,j,\mathrm{ncoef}}& a_{i,j,\left(\mathrm{ncoef}-1\right)}& a_{i,j,\left(\mathrm{ncoef}-2\right)}& \dots & a_{i,j,1}\end{array}\right],$
the Dynamic Matrix.

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.

Recalling from the definitions that:
b _i,j,k =a _i,j,k for k=1,
b _i,j,k =a _i,j,k −a _i,j,(k-1) for k:2→ncoef
and that
Δ O _i,k =O _i,k −O _i,(k-1) for k:1→ncoef
we can difference the above system of equations to give:

$\begin{array}{ccccccc}\ddot{A}{\stackrel{\_}{O}}_1& =& B_{1,1}\ddot{A}{\stackrel{\_}{I}}_1& \dots +& B_{1,j}\ddot{A}{\stackrel{\_}{I}}_j& \dots +& B_{1,\mathrm{nind}}\ddot{A}{\stackrel{\_}{I}}_{\mathrm{nind}}\\ ⋮& ⋮& ⋮& & ⋮& & ⋮\\ \ddot{A}{\stackrel{\_}{O}}_i& =& B_{i,1}\ddot{A}{\stackrel{\_}{I}}_1& \dots +& B_{\mathrm{ij}}\ddot{A}{\stackrel{\_}{I}}_j& \dots +& B_{i,\mathrm{nind}}\ddot{A}{\stackrel{\_}{I}}_{\mathrm{nind}}\\ ⋮& ⋮& ⋮& & ⋮& & ⋮\\ \ddot{A}{\stackrel{\_}{O}}_{\mathrm{ndep}}& & B_{\mathrm{ndep},1}\ddot{A}{\stackrel{\_}{I}}_1& \dots +& B_{\mathrm{ndep},j}\ddot{A}{\stackrel{\_}{I}}_j& \dots +& B_{\mathrm{ndep},\mathrm{nind}}\ddot{A}{\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}$
Equation 2: Finite Impulse Response Equations—Block Matrix Form
where

$\ddot{A}{\stackrel{\_}{O}}_i=\left[\begin{array}{c}{O}_{i,1}-O_{i,0}\\ O_{i,2}-O_{i,1}\\ O_{i,3}-O_{i,2}\\ ⋮\\ O_{i,\mathrm{ncoef}}-O_{i,\left(\mathrm{ncoef}-1\right)}\end{array}\right],$
the change in the i ^th dependent variable across each time interval,

$\ddot{A}{\stackrel{\_}{I}}_j=\left[\begin{array}{c}\Delta I_{j,1}\\ \Delta I_{j,2}\\ \Delta I_{j,3}\\ ⋮\\ \Delta I_{j,\mathrm{ncoef}}\end{array}\right]\mathrm{as}\mathrm{above},\mathrm{and}$ $B_{i,j}=\left[\begin{array}{ccccc}{b}_{i,j,1}& & & & \\ b_{i,j,2}& b_{i,j,1}& & & \\ b_{i,j,3}& b_{i,j,2}& b_{i,j,1}& & \\ ⋮& ⋮& ⋮& ⋰& \\ b_{i,j,\mathrm{ncoef}}& b_{i,j,\left(\mathrm{ncoef}-1\right)}& b_{i,j,\left(\mathrm{ncoef}-2\right)}& \dots & b_{i,j,1}\end{array}\right],$
the model matrix of Impulse Coefficients.

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.

• δ Ō=AΔĪ—Most often used for control calculations.
• ΔŌ=BΔĪ—Used for identification of steady state variables.
• ΔΔŌ=BΔΔĪ—Used for identification of ramp variables.
• δ Ō=Bδ Ī—Not commonly used. Old IDCOM control formulation.
• ΔŌ=AΔΔĪ—Not commonly used.

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 1—October 1985.

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.

We will now derive the algorithm of this invention to demonstrate the removal of the PID (proportional-integral-derivative) dynamics from the controller. The derivation is from the FIR model of equation 2. To derive the algorithm, we assume that the ĵ ^th independent variable is the set point of a PID controller and the î ^th dependent is the PID final control element response to that set point change. The final control element in many processes is a valve but it can be other things such as the speed setting on a motor. The term final control element anticipates any of those elements that are responsive to a PID control loop. We wish to re-constitute the model so that the final control element 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 î ^th dependent variable with the ĵ ^th dependent variable, as follows:

$[\begin{array}{ccccccc}{B}_{1,1}& \dots & B_{1,\left(\hat{j}-1\right)}& B_{1,\hat{j}}& B_{1,\left(\hat{j}+1\right)}& \dots & B_{1,\mathrm{nind}}\\ ⋮& ⋰& ⋮& ⋮& ⋮& ⋰& ⋮\\ B_{\left(\hat{i}-1\right),1}& \dots & B_{\left(\hat{i}-1\right),\left(\hat{j}-1\right)}& B_{\left(\hat{i}-1\right),\hat{j}}& B_{\left(\hat{i}-1\right),\left(\hat{j}+1\right)}& \dots & B_{\left(\hat{i}-1\right),\mathrm{nind}}\\ B_{\hat{i},1}& \dots & B_{\hat{i},\left(\hat{j}-1\right)}& B_{\hat{i},\hat{j}}& B_{\hat{i},\left(\hat{j}+1\right)}& \dots & B_{\hat{i},\mathrm{nind}}\\ B_{\left(\hat{i}+1\right),1}& \dots & B_{\left(\hat{i}+1\right),\left(\hat{j}-1\right)}& B_{\left(\hat{i}+1\right),\hat{j}}& B_{\left(\hat{i}+1\right),\left(\hat{j}+1\right)}& \dots & B_{\left(\hat{i}+1\right),\mathrm{nind}}\\ ⋮& ⋰& ⋮& ⋮& ⋮& ⋰& ⋮\\ B_{\mathrm{ndep},1}& \dots & B_{\mathrm{ndep},\left(\hat{j}-1\right)}& B_{\mathrm{ndep},\hat{j}}& B_{\mathrm{ndep},\left(\hat{j}+1\right)}& \dots & B_{\mathrm{ndep},\mathrm{nind}}\end{array}]\times [\begin{array}{c}\Delta {\stackrel{\_}{I}}_1\\ ⋮\\ \Delta {\stackrel{\_}{I}}_{\left(\hat{j}-1\right)}\\ \Delta {\stackrel{\_}{I}}_{\hat{j}}\\ \Delta {\stackrel{\_}{I}}_{\left(\hat{j}+1\right)}\\ ⋮\\ \Delta {\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}]=\left[\begin{array}{ccccccc}I& \dots & 0& 0& 0& \dots & 0\\ ⋮& ⋰& ⋰& ⋰& ⋰& ⋰& ⋮\\ 0& ⋰& I& 0& 0& ⋰& 0\\ 0& ⋰& 0& I& 0& ⋰& 0\\ 0& ⋰& 0& 0& I& ⋰& 0\\ ⋮& ⋰& ⋰& ⋰& ⋰& ⋰& ⋮\\ 0& \dots & 0& 0& 0& \dots & I\end{array}\right]\times \left[\begin{array}{c}\Delta {\stackrel{\_}{O}}_1\\ ⋮\\ \Delta {\stackrel{\_}{O}}_{\left(\hat{i}-1\right)}\\ \Delta {\stackrel{\_}{O}}_{\hat{i}}\\ \Delta {\stackrel{\_}{O}}_{\left(\hat{i}+1\right)}\\ ⋮\\ \Delta {\stackrel{\_}{O}}_{\mathrm{ndep}}\end{array}\right]$
where

$I=\left[\begin{array}{ccccc}1& 0& \dots & 0& 0\\ 0& 1& ⋰& 0& 0\\ ⋮& & ⋰& & ⋮\\ 0& 0& ⋰& 1& 0\\ 0& 0& \dots & 0& 1\end{array}\right],$
the Identity Matrix.
Note that this is nothing more than equation 2 above with an Identity matrix multiplying the ΔO's.
By performing row elimination operations (pivoting), we get;

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

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

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

$[\begin{array}{ccccccc}{\hat{B}}_{1,1}& \cdots & {\hat{B}}_{1,\left(\hat{j}-1\right)}& {\hat{B}}_{1,\hat{j}}& {\hat{B}}_{1,\left(\hat{j}+1\right)}& \cdots & {\hat{B}}_{1,\mathrm{nind}}\\ ⋮& ⋰& ⋮& ⋮& ⋮& ⋰& ⋮\\ {\hat{B}}_{\left(\hat{i}-1\right),1}& \cdots & {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}-1\right)}& {\hat{B}}_{\left(\hat{i}-1\right),\hat{j}}& {\hat{B}}_{\left(\hat{i}-1\right),\left(\hat{j}+1\right)}& \cdots & {\hat{B}}_{\left(\hat{i}-1\right),\mathrm{nind}}\\ {\hat{B}}_{\hat{i},1}& \cdots & {\hat{B}}_{\hat{i},\left(\hat{j}-1\right)}& {\hat{B}}_{\hat{i},\hat{j}}& {\hat{B}}_{\hat{i},\left(\hat{j}+1\right)}& \cdots & {\hat{B}}_{\hat{i},\mathrm{nind}}\\ {\hat{B}}_{\left(\hat{i}+1\right),1}& \cdots & {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}-1\right)}& {\hat{B}}_{\left(\hat{i}+1\right),\hat{j}}& {\hat{B}}_{\left(\hat{i}+1\right),\left(\hat{j}+1\right)}& \cdots & {\hat{B}}_{\left(\hat{i}+1\right),\mathrm{nind}}\\ ⋮& ⋰& ⋮& ⋮& ⋮& ⋰& ⋮\\ {\hat{B}}_{\mathrm{ndep},1}& \cdots & {\hat{B}}_{\mathrm{ndep},\left(\hat{j}-1\right)}& {\hat{B}}_{\mathrm{ndep},\hat{j}}& {\hat{B}}_{\mathrm{ndep},\left(\hat{j}+1\right)}& \cdots & {\hat{B}}_{\mathrm{ndep},\mathrm{nind}}\end{array}]\times [\begin{array}{c}\Delta {\stackrel{\_}{I}}_1\\ ⋮\\ \Delta {\stackrel{\_}{I}}_{\left(\hat{j}-1\right)}\\ \Delta {\stackrel{\_}{O}}_{\hat{i}}\\ \Delta {\stackrel{\_}{I}}_{\left(\hat{j}+1\right)}\\ ⋮\\ \Delta {\stackrel{\_}{I}}_{\mathrm{nind}}\end{array}]=[\begin{array}{ccccccc}I& \cdots & 0& 0& 0& \cdots & 0\\ ⋮& ⋰& ⋰& ⋰& ⋰& ⋰& ⋮\\ 0& ⋰& I& 0& 0& ⋰& 0\\ 0& ⋰& 0& I& 0& ⋰& 0\\ 0& ⋰& 0& 0& I& ⋰& 0\\ ⋮& ⋰& ⋰& ⋰& ⋰& ⋰& ⋮\\ 0& \cdots & 0& 0& 0& \cdots & I\end{array}]\times [\begin{array}{c}\Delta {\stackrel{\_}{O}}_1\\ ⋮\\ \Delta {\stackrel{\_}{O}}_{\left(\hat{i}-1\right)}\\ \Delta {\stackrel{\_}{I}}_{\hat{j}}\\ \Delta {\stackrel{\_}{O}}_{\left(\hat{i}+1\right)}\\ ⋮\\ \Delta {\stackrel{\_}{O}}_{\mathrm{ndep}}\end{array}]$

Note that ΔŌ _î and ΔĪ _ĵ have been interchanged so that the final control element 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.

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.

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 final control element position of the PID controller, we wish to re-constitute the model with the PID final control element 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 ΔO=BΔI, δ O=Bδ I, and ΔΔO=BΔΔI forms of the equation.

	Independent Var-1	Independent Var-2
Dependent Var-1
	b 1,1,1 = 1.5	b 1,2,1 = 0.5
	b 1,1,2 = 0.6	b 1,2,2 = 0.4
	b 1,1,3 = 0.2	b 1,2,3 = 0.2
	b 1,1,4 = 0.1	b 1,2,4 = 0.1
Dependent Var-2
	b 2,1,1 = −0.3	b 2,2,1 = 0.75
	b 2,1,2 = −0.4	b 2,2,2 = 0.25
	b 2,1,3 = −0.1	b 2,2,3 = 0.15
	b 2,1,4 = −0.05	b 2,2,4 = 0.05

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)