Model Predictive Controller Design for Performance Study of a Coupled Tank Process

(1)

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

70

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:

(2)

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

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.

. Γ_𝑙^y = 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)

(3)

72 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 Γ_𝑙^y and Γ_𝑙^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 Γ_𝑙^y reduces the tracking errors and guides the system to follow the set point. The matrix Γ_𝑙^u 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)

(4)

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 (a_i) 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 valve𝑅_𝑥 is fully open, i,e the gain related to valve𝑅_𝑥 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

(5)

74

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 𝑞𝑜𝑑 1 and 𝑞𝑜𝑑 2 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.

