# Model Predictive Control Is Powerful Technique Accounting Essay

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Model Predictive Control (MPC) is a methodology that refers to a class of control algorithms in which a dynamic model of the plant is used to predict and optimize the future behaviour of the process. At each of the control intervals, an open-loop sequence of the manipulated variables is computed in in order to predict future behaviour of the plant.

Model predictive control is powerful technique for optimizing the performance of constrained systems.It was initially developed in the in the process industries in the 60's and 70's in order to deal with the specialized control requirements petroleum refineries and power plants. This technology is now found in a large variety of applications in areas including food processing, automotive, chemicals and aerospace .It can handles multivariable control problems naturally and most importantly has the ability to handle constraints.The MPC has three main elements:

Prediction Model

Objective Function

Obtaining the Control Law

This approach has three important characteristics namely:

A model of the process is used to determine the proper adjustment to the manipulated variable.This is done in order to predict the future behavior of the controlled variable from the values of the manipulated variable.

The difference between the predicted model response and the actual process response gives the important feedback information.The control would be perfect and no further correction would be needed if this difference were zero.

This particular feedback approach can result in the controlled variable approaching its set point after several iterations.

The MPC's main attributes that makes it a successful approach to industrial control design are:

Practicality: It is often the resolution of problems such as satisfying output and control constraints which helps in determining the utility of a controller.

Simplicity: MPC's basic ideas are generally intuitive and do not require complex mathematics.

Demonstrability: The MPC works as shown by several real applications in the industry where it is routinely and profitably employed.

The Model Predictive Control is suitable for several kinds of problems but its main strengths are displayed when applied to problems involving:

Large numbers of controlled and manipulated variables.

Changing control objectives and/or equipment failure.

Constraints that are imposed on both the controlled and manipulated variables.

Figure 1.1 illustrates the basic principles of an MPC controller. Model M(Î¸) parameterized by a set Î¸ which is used to predict the future behavior of the plant. This prediction has two main components namely the free response (fr) and forced response (fo).The free response is the expected behavior of the output assuming zero future control actions while the forced response is the additional component due to the candidate set of future controls(u). [1]

The total prediction for a linear system can be calculated as fo + fr .The error of the future system can be calculated as e=r-(fo + fr),where fo,fr and r are vectors of the appropriate dimensions.

Past Output Controls

M (fr)

Future Reference (r)

M

(fo) Total Response

Optimizer

Future u Controls Future Errors(e)

Objective J

Fig. 1.1 Basic Structure of MPC

## HISTORY OF MPC

Several techniques have been developed since the 1970s for the design of model based control systems for robust multivariable control of industrial processes. Predictive control was simultaneously pioneered by Richalet and Cutler & Ramaker. The first implemented algorithms and successful applications were reported in the referenced papers. Model Predictive Control technology has evolved from a basic multivariable process control technology to a technology that enables operation processes within well-defined operating constraints. The several reasons for the acceptance of this technology by the process industry the following:

Model Predictive Controller is a model based controller design procedure that easily handles processes that involve unstable processes , large time-delays and non-minimum phase.

MPC can handle the limitations that the industrial processes would have in a systematic way during the design and implementation of the controller. These limitations include those in valve capacity and other technological requirements .

The MPC is an easy to tune method which in principle has only three basic parameters to be tuned.

Lastly the Model Predictive Controller can handle structural changes, such as sensor and actuator failures, changes in system parameters and system structure by adapting the control strategy on a sample-by-sample basis.

Like all other controller design methodologies, MPC also has its drawbacks:

A detailed process model is required. This means that either one must have a good insight in the physical behavior of the plant or system identification methods have to be applied to obtain a good model.

The methodology is open, and many variations have led to a large number of MPC methods.

Although, in practice, stability and robustness are easily obtained by accurate tuning, theoretical analyses of stability and robustness properties are difficult to derive.

## PROJECT OBJECTIVE

This lab oriented project would deal with the Model Predictive Controller approach for a temperature process.The initial stages of the project deals with the study of the Model Predictive Control (MPC) and the Internal Model Control (IMC).These concepts would be further implemented using the Matlab software.A simulation study would then be conducted after which its analysis would be done wherein a Matlab code would be implemented to a basic Model Predictive Controller for a temperature process.

## INTRODUCTION TO IMC

The Internal Model Control is based on the Internal Model Principle which states that 'Control can be achieved only if the control system encapsulates,either implicitly or explicitly,some representation of the process to be controlled.If the control scheme has been developed based on an exact model of the process,then perfect control is possible.[3]

The Internal Model Controller consists of the following three parts:[4]

Internal Model : To predict the process response in an attempt to adjust the manipulated variable to achieve control objective.

Filter : To achieve certain robustness in controller design.

Control Algorithm : To calculate the future values of manipulated variable so that the process output is within the desired value.

The exact inverse of a model transfer function is not possible so the IMC approach segregates and eliminates the aspects of this transfer function that make the calculation of a realizable inverse possible.The first step would be to separate the model into the product of the two factors

Gm(s)=Gm+(s) Gm-(s) (2.1)

Where

Gm+(s)=the noninvertible part has ann invferse that is not casual or unstable,The steady state gain of this term must be 1.0.

Gm-(s)= the invertible part has an inverse that is casual and stable,leading to a realizable,stable controller.The steady-state gain of this term is gain of the process model Km.

Consider the system shown in the figure 2.1 below:

Gp(s)

Gc(s) Set Point Output

Fig 2.1 Open Loop Control Strategy

A controller Gc(s) is used to control the process Gp(s).Consider Äžp(s) to be a model of Gp(s).On setting Gc(s) to be the inverse of the model of the process,we get

Gc(s)= Äžp(s)-1 (2.2)

If Gp(s)= Äžp(s)-1, the model would be an exact representation of the process.

It would become clear that the output and setpoint would be equal.This ideal control performance is achieved without feedback.This shows us that we can achieve perfect control if we have complete knowledge about the process.It also shows us that the feedback knowledge would be required only if the knowledge about the process is incomplete or inaccurate.[2]

## THE IMC STRATEGY

The Internal Model Control has the general structure depicted in figure 2.2 below:

d(s)

Process

Gp(s)

Gc(S)

## âˆ‘

E(s) U(s) + Output Y(s)

Process Model

Äžp(s)

## -

ï¤(s)

Fig 2.2 Schematic of the IMC scheme

In the above figure,d(s) is an unknown disturbance affecting the system.U(s) is the manipulated input that is introduced to both the process and its model.The process outout Y(s) is compare witht the output of the model resulting in the signal Äž (s).

ï¤ (s)= [ Gp(s)- Äžp(s)]U(s) + d(s) (2.3)

If d(s) is zero then Äž (s) would be a measure of the difference in behavior of the process and its model.If Gp(s)= Äžp(s) then Äž (s) would be equal to the unknown disturbance.

Thus Äž (s) would now be regarded as the information that is missing in the Äžp(s) model and can be used to improve control.This is done by subtracting Äž (s) from the setpoint R(s).

The control signal resulting from this subtraction is given by

U(s) = [R(s)- Äž (s)] Gc(s) = {R(s)-[ Gp(s)- Äžp(s)]U(s)-d(s)} Gc(s) (2.4)

Thus U(s)= [ R(s)-d(s) ] Gc(s) (2.5)

1+[ Gp(s)- Äžp(s)] Gc(s)

Since Y(s)= Gp(s) U(s) + d(s) (2.6)

The closed loop transfer function for the internal model control scheme would therefore be

Y(s) = [R(s)-d(s)] Gc(s) Gp(s) + d(s) (2.7)

1+[ Gp(s)- Äžp(s)] Gc(s)

From this closed loop expression it is seen that if Gc(s)= Äžp(s)-1 and if Gp(s) = Äžp(s), then the perfect setpoint tracking and disturbance rejection would be achieved.

Theoretically,if Gp(s) â‰  Äžp(s),the perfect rejection can be realized as long as Gc(s)= Äžp(s)-1 .

The Internal Model Controller is usually designed as the inverse of the process model which is in series with a low pass filter where GIMC= Gc(s) Gf(s).The order of the filter would normally be chosen such that the Gc(s) Gf(s) is proper to prevent excessive differential control action.

The closed loop that results would become

Y(s)= GIMC(s)Gp(s)R(s)+[1-GIMC(s) Äžp(s)]d(s) (2.8)

1+[Gp(s)- Äžp(s)]GIMC(S)

## 2.3 PRACTICAL DESIGN OF THE INTERNAL MODEL CONTROLLER[5]

The designing of an Internal Model Controller is relatively easy.If a model of the process Äžp(s) were given,the first step involved is to factor Äžp(s) into invertible and non-invertible components.

Äžp(s)= Äžp+(s) Äžp-(s) (2.9)

Äžp-(s) , the non-invertible component contains terms that if inverted,would lead to realisability and instability problems,e.g. terms containing positive zeros and time delays.

The next step would involve setting Äžc(s)= Äžp+(s)-1 and then GIMC(s)= Gc(s) Gf(s),where

Gf(s) is a low pass function.

## INTRODUCTION

The PT326 Process Trainer is a self contained process and control equipment.It has the basic characteristics of a large plant,enabling distance/velocity lag,transfer lag,system response,proportional and two step control etc.Due to its relatively fast response changes in set value and measured value can be displayed on an oscilloscope.

In this equipment,air drawn from the atmosphere from a centrifugal blower is driven past a heater grid and through the length of tubing to the atmosphere again.The process consists of heating the air flowing in the tube to the desired temperature level and the purpose of the control equipment is to measure the air temperature,compare it with a value set by the operator and generate a control signal which determines the amount of electrical power supplied to the correcting element .

The instrument contains integrated cicuit operational amplifiers and has self-contained power supplies.It can be coupled to the Feedback Process Control Simulator PCS327 for the application of three term control to the process.

Fig 3.1 Process Trainer PT326

## OPEN LOOP OPERATION

With the connections shown in fig 3.1 the process temperature may be controlled manually by the adjustment of 'Set Value'.With this connection the % proportional band adjustment is by-passed ,leaving the controller set to unity gain(100% proportional band).

The response to a step change or ramp input signal enables distance/velocity or transfer lags to be measured and the effect of changing the air flow rate can be observed.With a sinusoidal input signal the frequency response characteristics can be determined.

## EXPERIMENTAL PROCEDURE

The response of the detector to a step change in heater power is affected by two time lags-distance/velocity lag.In any stage of a thermal process where the heat is transferred through thermal resistance to or away from a thermal capacity,the temperature rise following a step change of input is exponential.It reaches 63.2% of its final value Vf after time T which is the exponential lag of the stage.

In this experiment the control loop is open and the controller is set to 100% proportional band.With the process temperature at ta low level a step change in set value is introduced either from an external source or by operation of the switch on the process trainer.The amplified response of the detecting element is displayed on an oscilloscope.

The process trainer connections using the figure 3.1.are made.

The set value is adjusted to the desired value.

A function generator was not used hence the set value disturbance switch on the PT326 was used to provide the step change in set value.

The oscilloscope is triggered from the 'trigger CRO' socket.The output was then recorded to give the graphs in the following pages.