Model Predictive Controller Design for Performance Study ... · Model Predictive Controller Design...

ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -3, 2013

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Model Predictive Controller Design for Performance Study

of a Coupled Tank Process

J. Gireesh Kumar & Veena Sharma

Department of Electrical Engineering, NIT Hamirpur, Hamirpur, Himachal Pradesh, India

Emial : [email protected], [email protected]

Abstract - Model predictive control (MPC) is the class of

advanced control techniques. A primary advantage to this

approach is the explicit handling of constraints. In

addition, the formulation for multivariable systems with

time-delays is straightforward in this control. MPC utilizes

an internal model to predict system dynamic behavior over

a finite horizon. Control decisions are based on optimizing

that predicted response. MPC is a discrete-time form of

control, so inaccuracies in predicted behavior are

corrected at the next control interval. This technique

makes the control of processes to become more efficient

and cost effective. Most of its applications are in the

refining, petrochemical industries and in other chemical

plants. Dynamic Matrix Control is a kind of model

predictive control technique based on step response model

of the process. In this paper, the dynamic matrix control

algorithm is implemented on coupled tank test system and

control quality has been analyzed using a simulation model

with different setting parameters. From the simulation

results it has been observed that dynamic matrix control

algorithm can achieve good results with accuracy even

with cross coupling and disturbance.

Index Terms— Model Predictive Control (MPC), Dynamic

Matrix Control (DMC), Coupled tank

I. INTRODUCTION

Multivariable control techniques have an great

importance in process industries[1]. The common

problem in a process control industry is to control the

process variables like fluid level, temperature and

pressure in storage tanks and chemical reactors [2]. To

solve these control problems we generally use P, PI and

PID controllers. PID controllers are easy to implement

and robust in nature. Although the PID perform well on

wide class of process with robust performance, due to

the feedback nature of these controllers it is difficult to

control MIMO processes and the complexity of

controlling increases for processes with interactions and

disturbances [3]. The PID controllers have three

parameters to be adjusted. Generally this can be done by

trial and error basis or by using tuning algorithms. The

main disadvantages of these controllers are they can’t

handle constraints and tuning of PID controllers is very

difficult task. So we need a control strategy that can

handle constraints and give better controller

performance than PID controllers. One of the solutions

is Model Predictive Control. In this paper to solve the

problem of couple tank dynamic matrix control

algorithm is used.

II. MODEL PREDICTIVE CONTROL

MPC is successful technique used in process

industry for more than 30 years. The main advantage of

MPC is handling of input and output variable constraints

[4]. The term Model Predictive Control does not mean a

specific control strategy, it is a collection of control

methods that makes use of the process model to obtain

the control signal by minimizing the objective function

[5].

The applications of predictive control which are

successful in use are as follows:

Clinical anesthesia.

Robot Manipulators.

Distillation columns.

Cement industry.

Drying towers, etc.

The advantages of MPC over other methods are as

follows:

ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)


71

Compensates measurable disturbances by using

feed forward control in a natural way.

Resulting controller is easier to implement

control law.

Handles constraints.

Very useful when future references are known.

Strategy is easy to understand.

The drawbacks of MPC are:

When constraints are considered, the amount of

computation required is higher.

Greatest drawback is to identify process model

correctly.

A. MPC Strategy

The strategy followed by the controllers belonging

to the MPC family [6] is explained by using the

following figure.

Fig.1. MPC strategy.

The basic idea is to predict the output 𝑌 (𝑘) of

process for p steps and future control moves are selected

such that the predicted response has optimal

characteristics. Here p is prediction horizon and m is

control horizon. The future values of output 𝑌 (𝑘) are

predicted using the process model. This works fine

when there is no model mismatch and disturbances.

When there is model mismatch, the predicted output will

not match actual output. So, only the first instant of the

control action is applied. This control strategy is also

called as “Receding Horizon Control”.

D. Dynamic Matrix Control

Dynamic Matrix Control (DMC) was introduced by

Cutler and Ramaker through their publication in the year

1970 [7]. DMC algorithm is one of the most popular

control algorithms of MPC. DMC is widely accepted in

industries, mainly by petrochemical industries [8]. DMC

uses step response of the model to predict the output.

The control actions are calculated using dynamic matrix

which is formed by using step response coefficients.

1. Cost function

Cost function plays an important role in finding

control action. The cost function is formulated in such a

way that the summation of present and future error is

minimized by using the minimum control action. Due to

process interactions it is not possible to keep all the

outputs close to their set points. So to have a preference

between outputs we include weights to the objective

function.

The cost function of DMC is given as follows:

(1

Where

: future output at k+l instant.

: future setpoint at k+l instant.

: change in control action at k+l instant.

.

Γ𝑙y = Positive definite error weight matrix.

= Positive semi definite controller weight matrix.

The cost function is to be minimized with subject to the

following constraints:

A. Manipulated variable constraints:

The solution DMC contains the current and future

control moves to be implemented. To avoid violations

the constraints on manipulated variable is considered as

(2)

Where

(3)

B. Manipulated Variable Rate Constraints

The limitations of the rate of change in controller

value is considered by adding manipulated variable rate

constraints as

(4)



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C. Output Variable Constraints

The limitations on the output is considered by output

constraints as

(5)

2. DMC Tuning

The tuning parameters of DMC are the prediction

horizon p, control horizon m, sampling time t, weight

matrices Γ𝑙y and Γ𝑙

u . The prediction horizon p is used to

predict the plant response for p future steps and find the

optimal control action such that it minimizes the future

error. The controller gives better performance for a long

prediction horizon but it increases computational burden

[9]. The control horizon m is used to find the optimal

control actions for m steps. Generally control horizon m

is chosen as m<p, long control horizon leads to

unnecessary control action and long computational time

and short control horizon leads to control actions which

are insensitive to modeling errors. The matrix Γ𝑙y

reduces the tracking errors and guides the system to

follow the set point. The matrix Γ𝑙u controls the

aggressiveness of the controller.

III. THE COUPLED – TANK PROCESS

The coupled tank process is a two input two output

process. The inputs to the process are the voltages to the

pumps i,e 𝑢1(𝑡) and 𝑢2(𝑡). The outputs of the process

are water level in tank 1 and tank2 i,e ℎ1(𝑡) and ℎ2(𝑡). The structure of coupled tank is as shown in figure 3.1.

Fig.2. coupled – tank system

To provide interaction between the two tanks, they are

connected through a pipe. It allows water flow between

the tanks. It introduces cross coupling in the system

[10]. The system model can be formulated by ordinary

differential equations using Bernoulli’s equation as

follows:

(6)

(7)

Where

- Cross-sectional area of tank.

- Cross-sectional area of the outlet hole.

– Water level in tank i.

Equilibrium point calculation:

We can calculate the equilibrium points from equations

6 and 7 equating to zero.

(8)

(9)

Solving the above equations 8 and 9 using the

parameters specified in the table 3.1 results:

(10)

(11) Now linearising equations 6 and 7 around equilibrium

points we have

(13)

(14)



73

(15)

Let

(16)

(17)

(18)

Taking Laplace transform on both sides of equation, we have

(19)

(20)

(21)

(22)

By substituting the parameters specified in table 3.1 results in

the following plant transfer function.

(23)

From the above transfer function we can easily derive the

transfer function for a coupled tank system without

interactions as follows

(24)

The following table shows the system parameters which are

used in simulation.

System Parameters Value

Cross sectional area of couple tank

reservoir (A)

0.01389 m2

Cross sectional area of the outlet (ai) 50.265*10-6 m2

Range of input signal (ui) 0 – 5 Volts

Maximum allowable height in tank (hi) 0.3 m

Constant relating control voltage with the

water flow from the pump (ƞ)

0.0024 m/V-sec

Table 3.1: Coupled – Tank system parameters

IV. SIMULATION RESULTS

A. Coupled Tank with interactions

The DMC algorithm is applied on the coupled tank

system with transfer function model with interactions i,e

transfer function specified in equation 23. While applying

interaction the valve𝑅𝑥 is fully open, i,e the gain related to

valve𝑅𝑥 is 1. Here the objective is to control the coupled tank

problem with the following constraints:

Manipulated variable constraint

(25)

Manipulated variable rate constraint

(26)

Output variable constraint

(27

For simulation the prediction horizon p is chosen as 40

and control horizon m as 4.The results are as follows

Fig.3. plant response with interactions



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Fig.4. control signal with interactions

B. Coupled Tank with interaction and disturbance

The DMC algorithm is applied on the coupled tank

system with transfer function model with interactions

and disturbance i,e transfer function specified in

equation 23. The transfer function of the disturbance is

as follows

(28)

The transfer function of disturbance is same as plant

transfer function because disturbance is applied by

opening 𝑞𝑜𝑑1 and 𝑞𝑜𝑑2 which is equivalent to applying a

negative step to an single tank system. Here the

disturbance of amplitude 1 cm is created by opening the

valves R1 and R2.The results are as follows

Fig.5. plant response and control signal with disturbance

V. REFERENCES

[1] Karl Henrik Johansson, “The Quadruple-Tank

Process: A Multivariable Laboratory Process

with an Adjustable Zero,” IEEE Trans. on

Control Systems Technology, vol. 8, pp. 456-

465, May 2000.

[2] W. Grega and A. Maciejczyk, “Digital Control of

a Tank System”, IEEE Trans. on Education, vol.

37, pp. 271-276, Aug. 1994.

I. Kaya, N, Tan and D. P. Atherton, “A Simple

Procedure for Improving the Performance of PID

Controllers”, IEEE Conf. on Control

Applications, vol. 2, pp. 882-885, 2003.

[3] J. H. Lee, “Model Predictive Control in the

Process Industries: Review, Current Status and

Future Outlook,” in Proc. 2nd Asian Conf. on

Control, vol. 2, pp.435-438, 1997.

[4] J. Gireesh Kumar and Veena Sharma, “An

Application of Dynamic Matrix Control to a

Process with Constraints”, in Proc. 2nd Int.

Conf. on Biomedical Engineering & Assistive

Technologies, pp.190-194, dec 6-7, 2012.

[5] M. Morari, J. H. Lee and C. E. García, “Model

Predictive Control”, unpublished, 2002.

[6] C. R. Cutler and B. L. Ramaker, “Dynamic

Matrix Control – a computer control algorithm,”

in Proc. American Conf. on Control, San

Francisco, 1980.

[7] S. J. Qin and T. A. Badgwell, “A survey of

industrial model predictive control technology”,

Control Engineering Practice, vol.11, pp.733-

764, 2003.

[8] S. A. Nirmala, B. Veena Abirami and D.

Manamalli, “Design of Model Predictive

Controller for a Four-Tank Process Using Linear

State Space Model and Performance Study for

Reference Tracking under Disturbances”, in Proc.

Int. Conf. on Process Automation, Control and

Computing, pp.1-5, 2011.

[9] Uma Shankar, “Modeling of Hybrid Dynamical

System”, M.Tech dissertation, NIT Hamirpur,

july 2012.

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