A Report on

DYNAMIC MATRIX CONTROL

OF

JACKETED TANK HEATER

Submitted by: Rishikesh Bagwe (2012A8PS401G)

M Dileep

AUG 2015 – DEC 2015

Submitted in the partial fulfilment of the Advance Process Control course

(APC)

i

Abstract

This report explains the concepts of Model Predictive Control (MPC) and Dynamic Matrix

Control (DMC). This controller technique is applied on a tank heater system. The objective is

to control height and the temperature of the liquid in the tank. The DMC controller is tuned

and the results are simulated in MATLAB.

ii

Table of Contents

Abstract ....................................................................................................................................... i

Table of Contents ....................................................................................................................... ii

Introduction ................................................................................................................................ 1

Model Predictive Control (MPC)............................................................................................... 2

Dynamic Matrix Control (DMC) ............................................................................................... 3

Tank Heater Process .................................................................................................................. 5

Open Loop Graphs ..................................................................................................................... 6

Closed Loop Response ............................................................................................................... 8

Conclusion ................................................................................................................................. 9

1

Introduction

Conventional controllers are the ones which gauge the deviation of the output variable (control

variable) from its set-point and deliver a proportional output on the process input side. PI

controller is the most commonly used conventional controller. These are easy to design but

they have some disadvantages. They attempt to rectifier the error only after it has affected the

output of the process. Industries are costly affair, a percent change in the output performance

or output composition, even for sometime, can lead to big losses.

In order to avoid the loses, the process industries use advanced controllers which either curb

the disturbance in the input variable to contaminate the system or predict the future output and

then give the control action accordingly. Usually the advanced controllers are used along with

the conventional controllers.

There are different types of advanced controllers:

i. Advanced Regulatory Control (ARC)

It includes techniques such feedforward, override or adaptive gain.

ii. Model Predictive control (MPC)

This method identifies important independent and dependent process variables and the

dynamic relationships (models) between them, and uses matrix-math based control and

optimization algorithms, to control multiple variables simultaneously.

iii. Inferential Control

iv. Sequential Control

It refers to dis-continuous time and event based automation sequences that occur within

continuous processes. These may be implemented as a collection of time and logic

function blocks, a custom algorithm, or using a formalized Sequential function

chart methodology.

Manufacturers are being asked to do more with less: improve process performance with fewer

engineers, increase reliability with lower maintenance budgets, and guarantee quality during

changing conditions. Advanced control has proven to be an effective tool in optimizing

operations, reliability, and quality but can be expensive to implementation and maintain than

traditional control systems.

2

Model Predictive Control (MPC)

MPC is a linear algebra method for predicting the result of a sequence of control variable

manipulations. Once the results of specific manipulations (in the past) are predicted, the

controller can then proceed with the sequence that produces the desired result.

MPC is a widely used means to deal with large multivariable constrained control issues in

industry. The main aim of MPC is to minimize a performance criterion in the future that would

possibly be subject to constraints on the manipulated inputs and outputs, where the future

behaviour is computed according to a model of the plant. The model predictive controller uses

the models and current plant measurements to calculate future moves in the independent

variables. The MPC then sends this set of independent variable moves to the corresponding

regulatory controller set-points to be implemented in the process.

MPC uses the mathematical expressions of a process model to predict system behavior. These

predictions are used to optimize the process over a defined time period. An MPC controller can

operate according to the following algorithm.

1. Development of a process model by the control engineers.

2. At time t, previous process inputs and outputs are used, along with the process

model, to predict future process outputs u(f) over a "prediction horizon."

3. Control signals that produce minimum error are found out by the optimizer

4. The control signal is implemented over a pre-defined time interval.

5. Time advances to the next interval, and the procedure is repeated from step 2.

The basic structure of Model Predictive Control:

3

Dynamic Matrix Control (DMC)

Dynamic Matrix Control is a control algorithm designed explicitly to predict the future

response of a plant. This algorithm was first developed by Shell Oil engineers in late 1970’s

and was intended for its use in petroleum refineries. Now-a-days its applications are found in

a wide variety of areas including chemicals, food processing, automotive, and aerospace

applications.

It is a form of control algorithm in which the current control action is obtained by solving a

finite horizon of open loop optimal control problem using the current state of the plant as the

initial state. This process is repeatedly done for each sampling point. The optimization yields

an optimal control sequence and the first control in this sequence is applied to the plantṣ. In

DMC, the models which are used, determine the behaviour of complex dynamical systems.

These models compensate for the effect of nonlinearities present in the variables. Hence the

models are used to predict the behaviour of dependent variables or outputs of the modelled

dynamical system with respect to changes in the process independent variables or inputs.

The plant model used by DMC algorithm is the step response model. This model uses the gi

coefficients that are the output of the lineal system when it is excited using a step. To reduce

the number of coefficients we assume that the system is stable and the output does not change

after some sampling time k. The expression of the output of the system is given by the following

equation:

So the output of the process at any time instant in future (prediction horizon)

Where, G is the Dynamic Matrix, Δu is the control horizon , f is the free response

4

The system has a reference trajectory as an input to the system. The error between this reference

trajectory and the predicted output is the predicted error. The predicted error also depends on

the weightage given to the control horizon. Our objective is to minimize the error. So the

objective function (J) becomes:

Where, w is the reference trajectory λ is the weightage. The objective function is for P future

outputs and M future inputs.

The reference trajectory can be of different profile depending on another parameter α. The

general equation for reference trajectory is

wi = α*wi-1 + (1- α)*ysp

α lies between 0 and 1.

5

Tank Heater Process

The process equations are:

𝑑ℎ

𝑑𝑡=

𝐹𝑖 − 𝐶𝑣√ℎ

𝐴

𝑑𝑇

𝑑𝑡=

𝐹𝑖(𝑇𝑖 − 𝑇)

𝐴. ℎ+

𝑈𝐴(𝑇𝑗𝑜 − 𝑇)

𝐴. ℎ. 𝑅𝑜𝐶𝑝

𝑑𝑇𝑗𝑜

𝑑𝑡=

𝐹𝑗𝑖(𝑇𝑗𝑖 − 𝑇𝑗𝑜)

𝑉𝑗−

𝑈𝐴(𝑇𝑗𝑜 − 𝑇)

𝑉𝑗. 𝑅𝑜𝐶𝑝

The following are the values of the characteristics of the considered system:

Fi = 0.75*10^-3 m3/s

RoCp = 9356.41

Ti = 300 K

A = 0.54 m2 is the cross-sectional area of the tank

Tji = 453 K

Vj = 0.0975 m3 is the volume of the jacket part

UA = 12.63

Fji = 10*10^-4 m3/s

6

Open Loop Graphs

1. t vs h; t vs Fi

2. t vs T; t vs Fji

7

3. Disturbance Ti

4. Past inputs and output

8

Closed Loop Response

The DMC can be tuned by 4 parameters viz P, M, α, λ. This tuning is a Hit and Try process.

First we fix value of P and accordingly smaller value of M. Then we fix our reference trajectory

by fixing α and then try to adjust λ to get the desired response.

Therefore the graphs for different values α are

0.2 - red, 0.5 - green, 0.8 – blue, 1 – black; λ = 8, P = 20, M = 3

The graphs for different values λ are

8 – red, 18 - green, 0.7- blue, 25 - black

9

Conclusion

We were successfully able to tune the DMC controller. We performed the tuning by hit and try

method by changing each variables while other are fixed. P and M were fixed to 20 and 3

respectively. The past inputs were the sample inputs to the process in order to predict the future

outputs based on those inputs. More fine tuning is required and we need add a filter which

restricts the manipulated variable (here Fi) to change in limited steps. Here as soon as the

controller is started there is a sudden jump in the manipulated variable which even though is

within the limits but not at all recommended.