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Model Predictive Control with MATLAB® and Simulink®

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Open access peer-reviewed chapter

By Rainer Dittmar

Reviewed: June 27th 2019Published: December 4th 2019

DOI: 10.5772 / intechopen.88257

Abstract

Model-based predictive controls (Model Predictive Control, MPC) have developed over the last three decades into a powerful approach for solving demanding tasks in multi-variable control with restrictions on the manipulated and controlled variables. They are now used successfully in many areas of industry. The MPC Toolbox of the MATLAB® / Simulink® program system is a tool that is used in practice to prepare for the use of real MPC controls, but is also used for teaching and research at universities and colleges. This book gives an overview of the basic ideas and application advantages of the MPC concept. It shows how MPC controls can be designed, set and simulated with the help of the toolbox in MATLAB® and Simulink®. Selected examples from the field of process engineering demonstrate possible approaches and deepen understanding. The book is aimed at engineers working in industry who want to plan, develop and operate MPC controls with the help of MATLAB® / Simulink®, but also at students from various technical disciplines who want to get into the field of Model Predictive Control.

Keywords

  • Model predictive control
  • Advanced process control
  • MATLAB® / Simulink®
  • System identification
  • Process automation

1 Introduction

It has now been four decades since the first publications on the industrial application of predictive controls appeared [6, 7]. Since then, model-based predictive control (MPC for short) has become the standard technology for solving demanding multivariable control tasks in continuous process engineering [5]. No other modern control method has such a successful history of industrial application, especially in the process industry. There are currently well over 15,000 MPC applications in use worldwide, not only in refinery and petrochemical plants, but also in plants in the chemical and polymer, pulp and paper industries, the foodstuffs industry, the cement industry and the Energy generation [8]. In recent years in particular, MPC regulations have also penetrated other areas that are characterized by significantly faster process dynamics. Over 95% of industrial MPC applications are based on linear process models, which are usually found through active tests in the process systems and subsequent process identification. However, MPC methods are not limited to linear process models; non-linear and hybrid models are also increasingly being used.

The reasons for the widespread use and acceptance of MPC technology include:

  • the systematic consideration of restrictions for the manipulated and controlled variables in the control algorithm,

  • the suitability for multi-variable control with any number of manipulated and controlled variables,

  • the ability to handle multi-variable systems in which the number of available manipulated variables and the controlled variables to be taken into account changes during operation,

  • their suitability for controlled systems with complex dynamics,

  • the simple expandability to include disturbance variable feedforward

  • an integrated function of business optimization,

  • the comprehensibility of the basic MPC concept for the user,

  • the availability of powerful development tools and service providers, and standardized project management.

MPC applications are often the focus of Advanced Process Control (APC) projects in the process industry. The further development of MPC technologies in recent years has resulted in project costs (relative to the other costs of automation) falling while the quality of the applications has increased. For this purpose z. For example, developments such as the use of standardized data interfaces (OPC technology), the development of browser-based visualization tools, the shortening of system tests through the use of advanced identification tools and the development of tools for monitoring and maintaining operational MPC regulations have contributed. Experience shows that APC projects using MPC technology pay for themselves on average in less than a year in the refinery sector. In other areas of the process industry, the average payback period is around two years. Depending on the application, the economic benefit arises from an increase in throughput while respecting the system limitations, by increasing the yield and ensuring consistently high quality of valuable products, by reducing the specific energy consumption or by a combination of several of these factors. MPC regulations are therefore increasingly seen as an attractive option for operating process systems more cost-effectively [9, 10, 11].

MPC applications were initially promoted primarily by process and automation engineers working in industry, before they also attracted greater attention in the academic field. The situation has now completely changed: the number of publications on MPC has risen sharply and is no longer manageable, and there is a mature theory for MPC with linear models.

The focus of this book is on the use of the MPC Toolbox for the design and simulation of MPC controls, but not on the detailed explanation of the MPC theory. For readers who are not familiar with the basics of MPC technology, Table 1 lists selected specialist books that deal with MPC controls with linear process models.

Author (s)titlesourceyear
MaciejowskiPredictive control with constraints[12]2002
Camacho, BordonsModel predictive control[13]2007
Rawlings, MayneModel Predictive Control-Theory and Design[14]2009
Haber, Bars, SchmitzPredictive control in process engineering[15]2011
Borelli, Bemporad, MorariPredictive control for linear and hybrid systems[16]2017
RossiterA first course in predictive control[17]2018

Table 1.

Selected specialist books on MPC controls with linear process models.

Sections on predictive control also contain some control engineering textbooks in English for the training of process engineers. A selection is compiled in Table 2.

Author (s)titlesourceyear
Ogunnaike, RayProcess Dynamics, Modeling and Control[18], chap. 271994
marlinProcess Control[19], chap. 232000
BequetteProcess Control[20], chap. 162003
Seborg, Edgar, Mellichamp, DoyleProcess Dynamics and Control[21], chap. 20th2011
CorriouProcess Control[22], chap. 162017

Table 2.

Control engineering textbooks with sections on Model Predictive Control.

Introductory presentations have been published in German, which are primarily aimed at users in practice (e.g. [8, 23] and Section 6 from [24]). An excellent description of the design and implementation of LMPC projects in the process industry can be found in [25]. The recently published “Handbook of Model Predictive Control” [26] describes both advances in MPC theory and novel practical applications.

In connection with MPC controls, a number of new research directions have emerged over the years, which expand and supplement the classic MPC control (with linear process models) in certain directions. Above all, this includes

  • MPC controls with non-linear process models (for strongly non-linear systems),

  • robust and stochastic MPC controls (for applications with high model uncertainty),

  • explicit MPC regulations (for very fast processes),

  • So-called "Economic MPC regulations" (Economic MPC), in which the tasks of operating point optimization and control are combined,

  • Distributed MPC controls for larger systems and processes (Distributed MPC).

Table 3 lists a monograph on each of these areas of work.

Author (s)titlesourceyear
Greens, PannekNonlinear Model Predictive Control[29]2017
Kouvaritakis, CannonModel Predictive Control: classical, robust and stochastic[30]2016
Grancharova, JohansonExplicit Nonlinear Model Predictive Control[31]2012
Ellis, Christofides, LiuEconomic Model Predictive Control[32]2017
Li, ZhenDistributed Model Predictive Control for plant-wide systems[33]2015

Table 3.

Books on the further development of MPC regulations.

Of the instructional videos available on YouTube on the subject of “Model Predictive Control”, we particularly recommend:

  • a 16-part video series designed by John Anthony Rossiter of the University of Sheffield [27], and

  • a 7-part video series provided by MathWorks Inc. for an introduction to MATLAB® Model Predictive Control ToolboxTM [28].

The first version of MATLAB® Model Predictive Control ToolboxTM was launched in 2004. Since then, current, detailed documentation has been available on the Mathworks website for registered users and of course for MathWorks customers. For the 2018b release used in this book, these are:

  • Model Predictive Control ToolboxTM User’s Guide R2108b [34]

  • Model Predictive Control ToolboxTM Getting Started Guide R2108b [35]

  • Model Predictive Control ToolboxTM Reference R2108b [36]

Authors are Alberto Bemporad (IMT School of Advanced Studies Lucca, Italy), Manfred Morari (University of Pennsylvania, formerly ETH Zurich) and N. Lawrence Ricker (University of Washington at Seattle)

In addition, current “release notes” are published in each case, in which further developments and corrections are described. Videos and webinars are offered on the MathWorks website, which, in addition to introductions to the various tools, also explain application examples from various areas.

As already mentioned, the vast majority of industrial MPC applications are based on linear process models that are obtained from measurement data through system tests and identification. Selected specialist books in which the methods used for system identification are explained are listed in Table 4.

Author (s)titlesourceyear
LjungSystem Identification - Theory for the User[39]1999
ZhuMultivariable System Identification for Process Control[40]2001
Isermann, MünchhofIdentification of Dynamic Systems - an Introduction with Applications[41]2011
TangiralaPrinciples of System Identification[42]2014
Bohn, uncutIdentification of dynamic systems[43]2016

Table 4.

Selected specialist books on the subject of system identification.

Also for MATLAB® Model System Identification ToolboxTM detailed documentation is available on the MathWorks website:

  • System Identification ToolboxTM User’s Guide R2018b [37]

  • System Identification ToolboxTM Getting Started Guide R2018b [38]

The author is Lennart Ljung (Linköping University, Sweden).

As for the MPC Toolbox, the System Identification ToolboxTM Videos and webinars are offered on the MathWorks website. The webinar “Introduction to System Identification” (Lennart Ljung), which is also available on Youtube [44], deserves special mention.

This book is structured as follows:

In First there is an introduction to the field of model-based, predictive control (Model Predictive Control, MPC). This introduces the MPC terminology, and the basic elements of predictive control algorithms are first described in words, then mathematically (formulas). The formulaic description and the notation used follow [34, 35]. This is followed by a brief overview of the industrial application of MPC technologies in the process industry and in other areas. Finally, the components and possible uses of the Model Predictive Control Toolbox are presentedTM explained.

describes in detail the steps involved in the design and simulation of MPC controls with linear process models (LMPC) using the MPC Designer of the MPC Toolbox. The “handling” of the different elements of the toolbox is explained as well as the data and parameters to be entered. The “MPC Designer” used for this is a graphical user interface of the MPC Toolbox, which makes a structured approach to MPC controller design much easier for the user. The possibility of programming at the command line level of MATLAB, which is also available® is not considered. An exception is the design of MPC controllers with non-linear process models (NMPC) - see Chapter 9 - which is currently not supported by the MPC Designer. An important design step is the offline simulation of the MPC control system. The book describes two possibilities for this: (a) the simulation within the MPC Designer App and (b) the simulation with the help of Simulink models, in which software function blocks from the Simulink library are used.

The following chapters are devoted to different application examples for MPC controls. All examples come from the field of process engineering, but different emphases are set.

In, the MPC control of a thermal separation process with three manipulated and three controlled variables as well as two measured disturbance variables is examined. A linear process model in the form of a matrix of transfer functions is given, so it does not have to be developed first. The focus is on the illustration of the work with the MPC Toolbox using a concrete example. In addition, it will be demonstrated what improvement in control quality can be achieved when using a central MPC multivariable control compared to a decentralized PI control.

is dedicated to the MPC control of an evaporator system for which a theoretical process model is available. A model must therefore first be obtained before designing and simulating the LMPC control. In this case, this takes place by means of the simulative generation of measurement data (by applying test signals to the rigorous model) and by subsequent system identification. This path is usually also followed in industrial practice, there of course with tests on the large-scale industrial plant and using real measurement data for empirical modeling.

This takes up the example of a benchmark process that has often been used to examine modern concepts of multivariable control. A non-linear theoretical process model is also available for this example. In contrast to Chapter 5, the generation of a linear model for the LMPC controller is done by linearizing the non-linear model equations with the help of Simulink Control DesignTM. The design of non-square multivariable controls with an unequal number of manipulated and controlled variables is another focus of this section.

The and deal with the application of MPC controllers with process models to the control systems. Chapter 7 introduces the concept of adaptive MPC control, in which a linear model is continuously adapted to the current process behavior using recursive regression methods.In contrast, in Chapter 8 a “Gain Scheduled” MPC control is designed and simulated, in which a “bank” of linear models is developed for different operating points of the process, an LMPC controller is designed for each of these models and one of these controllers is automatically selected during operation and is used. The adaptive MPC control is demonstrated using the example of a continuous stirred tank reactor, the “Gain Scheduled” MPC control using the example of the pH value control of a neutralization reactor. In both cases, special Simulink function blocks from the MPC Toolbox are used to simulate the control system.

occupies a special position. Using the example of the theoretical model of an ethylene oxide reactor, an MPC control (NMPC) is designed. The MPC Designer App cannot be used here for the control design, but must be based on the command line programming in MATLAB® can be used. A “Nonlinear MPC Controller” component, which has been available for the MPC Toolbox since Release 2018b, is used to simulate the NMPC control system.

In order to understand the book, knowledge from the areas of mathematics, modeling and control engineering is required, such as those imparted in the bachelor's and master's degree in technical courses at universities and technical colleges. Basic knowledge of various forms of mathematical models of the dynamic behavior of systems (differential and difference equations, transfer functions, state models) or the design of single-loop PID control loops is therefore not presented again. It is also assumed that the reader has a working knowledge of MATLAB® and Simulink® disposes. If this is not the case, easily accessible sources of information are available to familiarize yourself with these programming environments and tools, including books (e.g. [45, 46] and Sections 16 and 17 of the Pocket Book of Control Engineering [47]), the extensive ones and well-designed teaching materials on the MathWorks website and - especially recommended for beginners - “MATLAB and Simulink Onramp”: a way to step into MATLAB online® and Simulink® without having installed these programs on the computer yourself.

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2. Introduction to Model Predictive Control and MATLAB® Model Predictive Control ToolboxTM

This chapter provides an introduction to the area of ​​model predictive controls and MATLAB® Model Predictive Control ToolboxTM. MPC terminology is explained and the basic elements of predictive control algorithms are described in both verbal and formulaic terms. After a brief overview of the industrial application of MPC technology, the components and functions of the Model Predictive Control ToolboxTM shown.

2.1 Plant model, MPC control loop and MPC terminology

The mathematical model of the dynamic behavior of the multivariable controlled system (hereinafter referred to as “system model”) used in the MPC Toolbox has the structure shown in Figure 1 ([35], p. 2–2).

It consists of the actual process model that shows the relationships between the

  • manipulated variables (manipulated variables)

  • measured disturbance variables

  • input interference not measured

and the

  • measured output variables (controlled variables), as well as the

  • unmeasured output variables

describes. The unmeasured input disturbances, the measured output disturbances (Index… output disturbance) and the measurement noise (Index… noise) are described by further mathematical models that generate these interference signals from white noise signals. The interference signals can therefore be described as “filtered white noise” or “colored noise”, the properties of which are determined by the parameters of the interference models.

Since the controlled system is generally a multiple-input multiple-output or MIMO system for short, all variables are vectors and are identified by bold notation. So called for example

the vector of the manipulated variables (MV) or manipulated variables. The same applies to the other vector quantities. The index in brackets denotes the current point in time in discrete-time notation. In many specialist books and commercial MPC tools, other abbreviations than those used in the MPC Toolbox have become common: the measured disturbance variables are often referred to as DV (disturbance variables) instead of MD (measured disturbances) and the measured output variables instead of MO (measured outputs ) often referred to as CV (control variables).

Figure 2 shows the action plan of a closed control system as it is created in the MPC Toolbox. The disturbance variable models have been omitted for this illustration.

As can be seen, as in every control loop, the controlled variables measured in the process are fed back to the MPC controller, which is also provided with setpoints (alternatively setpoint ranges or lower / upper limit values). In addition, measured disturbance variables can be made known to the MPC controller, which can be used for feedforward compensation. The output variables that are not measured are usually only observed. Alternatively, they can also be estimated and then regulated.

2.2 Basic elements of MPC control algorithms

Regardless of the specific mathematical design of the individual steps, all MPC control algorithms have the following common elements:

  • Estimation of the unmeasured state variables of the process model and the disturbance models,

  • Model-based prediction of the future course of the controlled variables (prediction),

  • Determination of an optimal sequence of future manipulated variable changes,

  • Applying the principle of the receding horizon.

The mode of operation of an MPC control can be described using Figure 3. In order not to make the situation unnecessarily complicated, only one manipulated variable and one controlled variable are considered in the picture (single variable control). In the left half of the figure the courses of the controlled and manipulated variable in the past are shown, the right half of the figure shows their future development. MPC control algorithms work time-discrete. The current sampling time is denoted by, future by, and past by.

2.2.1 Estimation of unmeasured state variables of the process model and the disturbance models

Since linear state models are used in the MPC Toolbox to describe the dynamic behavior (both of the process and the disturbances) and the state variables of the process and disturbance models are generally not measurable, they must be estimated (with the help of the measured manipulated and controlled variables) become. A stationary Kalman filter is used for this. Alternatively, the user can use a self-developed estimation method. The estimated quantities are then inserted into the prediction equations. By using the measured controlled variable, the control loop of the MPC control is closed in this step.

2.2.2 Prediction

With PID controls, a dynamic model of the process is only used in the design phase (to determine the controller parameters). With MPC controls, on the other hand, it is also used in the working phase of the control: At the current point in time, the future behavior of the controlled variable is predicted with the help of a mathematical model for the dynamic behavior of the process. The prediction extends over the “prediction horizon” (or the time with the sampling time). It is made up of two parts, the "free" and the "forced" movement of the system. The “free” movement describes the future course of the controlled variable under the assumption that the manipulated variable will not change in the future. This is shown in dashed lines in Figure 3. One also speaks of a prediction in an open circle or of “unforced prediction”. The “forced” movement results from the consideration of the future course of the manipulated variables over the “tax horizon”. This part is also called “closed-loop prediction” or “forced prediction” and is shown in solid lines in the picture. A future setpoint curve must be known for the controlled variable. A constant setpoint is shown in the figure.

If the setpoint curve is known, the future control differences (setpoint minus prediction value in the future sampling times) can then be predicted. The future course of the manipulated variable is calculated in a further step (see below) by iterative solution of an optimization problem. In each optimization step, a new sequence of future manipulated variable changes is created, with the help of which the forced movement must be recalculated over and over again.

For the prediction of the controlled variable, measurable disturbance variables can be included if dynamic models are known for their relationship with the controlled variable. This procedure corresponds to a feedforward control as it is known from control loops with PID controllers ([24], p. 143 ff.).

2.2.3 Determination of an optimal sequence of future manipulated variable changes (dynamic optimization)

The third element of the MPC algorithm includes the determination of an optimal sequence of future manipulated variable changes over a given “control horizon”, which in practice is usually chosen to be much shorter than the prediction horizon. Note that the tax horizon denotes the number of future changes in the manipulated variable. Since the first change in the manipulated variable is determined for the current point in time, the index runs to, see Figure 3. The notation here means that changes in the manipulated variable are calculated for the points in time on the basis of information that is available up to the current point in time.

The consequence of the future changes in the manipulated variable is determined in real time by solving a quadratic optimization problem. The optimization goal is to minimize the future control differences while making do with the smallest possible manipulated variable changes (control target function). Inequality secondary conditions can be specified for the manipulated variables and their changes between two sampling times, which the MPC controller must always adhere to (so-called “hard” secondary conditions). Either secondary equation conditions (setpoints) or secondary inequality conditions (setpoint ranges or upper / lower limit values) can be specified for the controlled variables. Constraints for the controlled variables may be violated temporarily, so they are understood as “soft” constraints. By specifying weighting factors, a compromise is made between the goals of "lowest possible future control differences" and "lowest possible adjustment effort". The weight factors are the essential element of the controller setting (tuning). Since the independent variables of the optimization problem represent a sequence of manipulated variable changes, this step is also referred to as “optimization”.

2.2.4 Application of the receding horizon principle

Although a whole sequence of future manipulated variable changes was calculated in the previous optimization step, only the first element of this sequence is output to the process or to the actuating device. However, this also means that one does not wait for sampling intervals until the next optimization is carried out, but that the entire procedure of estimation, prediction and optimization is repeated in each sampling interval. Before that, the observed time horizon and with it the data vectors for the control loop variables are shifted forward by one sampling step. This procedure is known as applying the “receding horizon principle”. Its application enables a quick reaction to changes in disturbance variables that have not been measured.

2.2.5 Note: Static operating point optimization

In the case of commercial LMPC program packages (an overview can be found in [24], p. 312 f.), A further function is implemented in addition to the elements of the MPC control algorithm described: the determination of permissible, economically optimal stationary values ​​of the manipulated and controlled variables ( static operating point optimization). The calculation of the optimal control variable sequence does not ensure that the MPC control moves to an economically optimal operating point of the process in the steady state. Therefore, at the same time as the solution of the dynamic optimization task, one can solve a functionally superordinate but program-technically integrated static optimization problem. The aim here is to determine the permissible, optimal values ​​of the control and regulation variables for the steady-state condition of the system and by minimizing a (or derived technological) target function. Mathematically, this task is usually solved in real time with the help of a linear optimization with the constraints already described. The mathematical model for the behavior of the process is required here. In the MPC literature, the static operating point optimization is also referred to as “target selection”, the variables and are then the “targets” or target variables for the dynamic part of the MPC controller. However, the MPC Toolbox has not yet supported the static operating point optimization function.

Figure 4 shows the interaction of the basic elements of an MPC control system without target selection in the form of an action plan.

2.3 MPC math

The explanation of the basic elements of MPC control algorithms given in section 2.2 is specified in the following with the help of mathematical formulas for multi-variable systems. The descriptions refer to the algorithms implemented in the MPC Toolbox. To facilitate understanding, the terminology and notation used in the documentation [34, 35] are used. First, the (state) models used in the MPC Toolbox for the process and the disturbances are described mathematically. The formulas for the essential steps of the MPC control algorithm (state and disturbance variable estimation, prediction of the controlled variables and determination of optimal manipulated variable sequences) are then given.

2.3.1 Models used

2.3.1.1 Process model

In the MPC Toolbox, all calculations (estimation, prediction, optimization) are carried out with a discrete-time, dead-time-free, linear state model of the process with dimensionless input and output variables. If a different process model is specified by the user - e.g. B. a transfer function model or one through the System Identification ToolboxTM model determined from measurement data - then the steps are internally

  • Conversion into a linear state model,

  • temporal discretization,

  • Absorption of existing dead times by introducing additional time-discrete state variables and poles in (MATLAB function), and

  • Conversion of the input and output variables (not the state variables) into dimensionless variables

carried out. The last step is made possible by the user specifying scaling factors in the unit of measurement of the respective input and output variable. If the scaling factors are summarized in diagonal matrices for the input and output variables, then the conversion into the dimensionless form z. B. by

E2

It eventually becomes a state model of the form

E3

generated. Here,, and the vectors of the state variables, the manipulated variables, the measured and the non-measured disturbance variables, denote the vector of the output variables. The index means “process” or “plant”. The matrices and are constant matrices of the state model. The MPC controller forces, i. H. the model of the controlled system cannot jump (the manipulated variables do not have an immediate effect on the controlled variables).

2.3.1.2 Input disturbance model

The linear state model of the input disturbances is

E4

This denotes the vector of the unmeasured, dimensionless input disturbance variables, a vector of dimensionless white noise signals with mean value zero and variance one, the state vector of the model of the input disturbances and and constant matrices. The index means “input disturbance”. The input interference is therefore “filtered white noise”, the properties of which are generated by the structure and parameters of the interference model. This approach can be used to describe many of the interfering signals that occur in practice. If the user does not specify a model of the input interference, a preset model is used. Details are described in [35], pp. 2-4 ff. The choice of the model of the input disturbances influences both the dynamics of the disturbance behavior of the MPC control as well as the static behavior, in particular the avoidance of permanent control differences analogous to the I component in a PID control.

2.3.1.3 Output disturbance model

The linear state model of the (unmeasured) output disturbances is

E5

with the state variables, the dimensionless white noise, the constant matrices and as well as the index (“output disturbance”). The dimensionless output disturbance variables are added to the process output variables (see Fig. 1). The output disturbance model is thus much more general than the model of a step-constant output disturbance originally used for “Dynamic Matrix Control” (one of the earliest MPC control algorithms) (cf. [24], p. 285 ff.). The exclusive use of this model in the early versions of commercial MPC program packages resulted in to an unsatisfactory control quality in the compensation of input disturbances [48, 49]. These disadvantages are avoided by the interference models used in the MPC Toolbox (but also in other modern commercial MPC program packages).

If the user does not define a model of the output disturbances, a preset model is also used here, for details see [35], pp. 2-6 ff.

2.3.1.4 Model of the measurement noise

The linear state model for the measurement noise is

E6

with the state variables, the dimensionless white noise, the constant matrices and as well as the index (“noise”). The dimensionless measurement noise is also added to the output variables of the process, the measured output variables then result in. If the user does not specify a model of the measurement noise, no state variables are used (or their number is equal to zero), empty matrices are preset for and a unit matrix with as many rows and columns as the number of measured output variables. The model of the measurement noise is thus reduced to.

How the interference models can be specified or changed in the user interface of the MPC Toolbox is described below in section 3.5.

2.3.2 Estimate

In order to predict the future course of the controlled variables, the MPC controller needs the current values ​​of the state variables (both of the process model and the disturbance models). Since these generally cannot be measured, they have to be estimated from measurable input and output variables. For this purpose, a state vector of the MPC controller is in the form

E7

Are defined. So it summarizes the state vectors of the process model and the three disturbance models. The index means “current”. The current values ​​of the state variables are obtained via a state observer of the form

E8

determined. The input variables of the observer are as

E9

defined, are made up of the manipulated variables, the measured disturbance variables and the white noise signals of the disturbance models. The constant matrices and are composed of the matrices of the process and disturbance models as follows: