Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 57 (2015) 1351 – 1358
3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015)
Design of Deadbeat Algorithm for a Nonlinear Conical tank system Marshiana.Da* , Thirusakthimurugan.Pb a
b
Research scholar ,Sathyabama University,Chennai,India Professor,Pondicherry Engineering College,Pondicherry,India
Abstract The control of nonlinear process is the main problem in process industries which repeatedly show its nonlinear activity. In this paper the control action of nonlinear framework using Digital controller with Deadbeat Algorithm was presented. The Nonlinear process which has been considered for its application is conical tank system. System identification of this nonlinear method is made using mathematical modeling and it is approximated to a first order system. Simulation is carried out by the MATLAB software and its servo and regulatory operation is resolved. ©©2015 Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 2015 (ICRTC-2015). (ICRTC-2015) Keywords: :Nonlinear process;Mathematical modeling;Deadbeat Algorithm;MATLAB software;
1. Introduction The control of fluid level in tanks is a essential issues in process industries. The nonlinear system show numerous testing control issues because of their nonlinear vibrant deeds and time changing constraint. The conical tank shows its nonlinearity because of its shape. Design a controller for a nonlinear process is perplexing and excessively hard to implement it. The principle assignment of the controller configuration is to accomplish the preferred working conditions and to design the controller to attain its optimum execution performance. V.R.Ravi et al.[1] proposes that there is a necessitate to control a Level due to the fact that if the level is excessively high may annoy its reaction equilibrium of the entire methodology which may cause harm to equipment, or bring out spillage of profitable or risky material from the process. If the event that the level is excessively low, it may
* Tel:9843195241. E-mail address:
[email protected]
1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015) doi:10.1016/j.procs.2015.07.449
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have terrible results for the sequential operation completed by the process. Henceforth, control of fluid level is an paramount and common chore in the methodology of process industries. Nonlinear models are utilized where precision over a more extensive range of operation is obliged where they can be specifically incorporated into control algorithms. Due to the innate nonlinearity most of the chemical process industries are in need of innovative control techniques. Anandanatarajan.R et al. [2] have done work in nonlinear conical tank system which finds wide application in process industry. S.Nithya et al [3, 4, 5] proposes that the nonlinearity is because of its change in shape. Their shape assures optimal rousing and mixing of ingredients and provides a fast and hygienic cleaning. M.Tham [6] proposes that the flexibility of the digital computer, digital control algorithms need not be restricted to discrete versions of analog designs. In particular, it is possible to formulate controllers that, under ideal conditions, will produce desired closed loop response. Basically, digital controllers are focused based on its process models, which will have very few special cases, the design start with the determination of some desired closed loop properties. Distinctive controllers of diverse complexities will result depending upon the criterion and the form of the process model. Mikulas Alexik [7] proposes the discrete version of time optimal control or DB(n) algorithm is generally known as an elementary control algorithm. Its disadvantage is a great jump of actuating variable usually exceeding actuating variable limitation. “Derived three extensions of DB(n) algorithm where this limitation is respected. First extension is the anti windup extension of classical DB(n) algorithm - eDB algorithm. The second is the state space version of eDB algorithm. The third extension is a version of eDB algorithm, which improves behaviour quality of control processes with time delay. This version is called eDBd algorithm.” The paper is organized as follows: In section II, the mathematical modeling of the conical tank system to determine the transfer function is described. In section III the design of digital control system with deadbeat algorithm.. In section IV , finally the results and conclusion are discussed. 2. Mathematical Modeling The mathematical model of the conical tank is deciding by considering two assumptions (i) by taking level as the control variable and (ii) inflow to the tank as the manipulated variable [8] .This is accomplished by controlling the input flow into the tank. The Figure 1 shows the schematic diagram of the conical tank system. Operating Parameters are F1 -Tank Inflow rate F2 -Tank Outflow rate H - Total height of conical tank. R – Conical Tank Top radius h - Nominal level of tank r - Radius at nominal level of tank
Figure 1. Schematic diagram of the Conical Tank system.
Mass balance Equation is given by ௗ
F1 –F2=A1
ௗ௧
(1)
D. Marshiana and P. Thirusakthimurugan / Procedia Computer Science 57 (2015) 1351 – 1358
F2 = bξ݄ Where b=a√(2g)
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(2)
Where a- area of the tank at the outlet g- acceleration due to gravity By substituting the equation and considering the cross sectional area of the tank at any level h
tanθ=r/h =R/H A=пr2 A=пR2h2/H2 Where r= R2h2/H2
(3)
Transfer function (TF) is given by taking the partial differentiation of the linearised equation and its corresponding Laplace transform [8] = ிଵሺ௦ሻ ఛ௦ାଵ ሺ௦ሻ
Where
߬= K=
ଶξ ଶξ
Specifications H - Total height of the conical tank R - Top radius of the conical tank h- Nominal Level of the tank a- area of the tank at the outlet
=50cm =15cm =30cm =1.1cm
Based on the specification the transfer function is given by ܩሺܵሻ ൌ
ͲǤͷͷͺ ͳͶʹǤʹͳͻܵ ͳ
3. Dead Beat Algorithm The dead-beat controller aim for the finest conceivable to a set point change. Subjectively, this implies that taking after a set-point change, and after a time period equivalent to the system time-delay, which yield ought to be at setpoint and remains there. This necessity is detailed as[9,10,11]:
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ܩ ሺݖሻ ൌ ି ݖ ͳ where k is the system delay, uttered as an integer multiple of the sampling interval Ts. The system delay must be less than one,due to the fact the system delay will incorporate the unit delay because of the utilization of the sampler [12,13,14]. At any specific sampling instant, the current condition of the controlled output will yield is sampled while in the meantime, a control signal is sent to the process. It is just at the next sampling instant that the result of this control action will be experimental. That’s why, with digital control loops, there is dependably a minimum delay of 1 sampling interval. The schematic of a conventional sampled data control system is shown in figure 2.
Figure 2. Schematic diagram of a conventional sampled data control system
where the position of the samplers are depicted clearly. Since it is a standard set up, to remove the samplers. Instead, it can be simplified into a following block diagram which shows conventional sampled data control system
Figure 3. Simplified Block diagram of a conventional sampled data control system
Where W(z)-Set-point of the system, E(z)- Error function U(z)- Controller output value Y(z)-Controlled output Value. D(z) –Transfer function of digital controller, HGp(z) Transfer function of z-transform of zero-order-hold device with the process to be controlled.
The Laplace transform of zero-order-hold ܮሼܼܱܪሺݏሻሽ ൌ
ܩܪ ሺݖሻ ൌ ܼ The expression of closed loop system is
ͳ െ ሺെܶݏ௦ ሻ ݏ
ͳ െ ሺെܶݏ௦ ሻ ܩ ሺݏሻ ݏ
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ܦሺݖሻܩܪ ሺݖሻ ܻሺݖሻ ൌ ܹሺݖሻ ͳ ܦሺݖሻܩܪ ሺݖሻ
The closed loop dynamics of Gm(z). ܻሺݖሻ ൌ ܩ ሺݖሻ ܹሺݖሻ The controller output with closed loop response characteristic which satisfies is ܦሺݖሻ ൌ
ܩ ሺݖሻ ͳ ൨ ൨ ܩܪ ሺݖሻ ͳ െ ܩ ሺݖሻ
Which is known as the synthesis Equation obtained by re-arranging the closed-loop transfer function[15] and replacing Y(z)/W(Z) by Gm(z).The essential controller can be designed for the process, HGp(z). Depending on Gm(z), various methods of controllers will arise. The most commonly used is the Deadbeat controller and Dahlin Controller. The dead-beat controller equation is given by: ܦሺݖሻ ൌ
ͳ ି ݖ ൨ቈ ܩܪ ሺݖሻ ͳ െ ି ݖ
The Response of the deadbeat controller provides various characteristics like zero steady state error, minimun rise time , minimum settling time and very high control signal output 3.1 Deadbeat Controller Design The transfer function of the conical tank system is ܩሺܵሻ ൌ
ͲǤͷͷͺ ͳͶʹǤʹͳͻܵ ͳ
The Discrete transfer function is given by ܩሺܼሻ ൌ ܼሾܩሺሻ ሺሻሿ
ൌ ሾ
ଵǦୣǦ౩
Ǥହହ଼
Ǥ ଵସଶǤଶଵଽୗାଵሿ
ୱ
ሺଵି షబǤబబళబయళ ሻ
ൌ ͲǤͷͷͺሾ
ሺି షబǤబబళబయ ሻ
The Dead beat controller is given by ܦሺܼሻ ൌ
ܦሺܼሻ ൌ
ܼ ିଵ ͳ ܩሺܼሻ ͳ െ ܼ ିଵ
ܼ െ ͲǤͻͻʹ ͲǤͲͲ͵ͻͲͺͻܼ െ ͲǤͲͲ͵ͻͲͺͻ
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4. Dead Beat Algorithm Simulation is performed using MATLAB for the Deadbeat Algorithm to validate the performance Servo and regulatory response is determined for the setpoint of 30cm and 40cm to control the level of the tank using deadbeat controller. The simulation is carried out by taking 60% and 80% as the nominal value . It provides minimum settling time and very high stable output .In figure 4 and 5 gives the closed loop response of a deadbeat controller having the setpoint of 30 cm and 40cm correspondingly of the tank level .Figure 6 and 7 shows the Closed loop response of a deadbeat controller having setpoint 30cm with the setpoint change of 32cm, and Closed loop response of a deadbeat controller having setpoint 40cm with the setpoint change of 42cm, Figure 8 provides Servo and Regulatory response of Closed loop system with deadbeat controller having setpoint 40cm , Figure 8 shows the Servo and Regulatory response of Closed loop system with deadbeat controller having setpoint 40cm. Figure 9and 10 gives he output of Regulatory response of Closed loop system with deadbeat controller having setpoint 40cm and 30 cm respectively.
Figure 4.Closed loop response of a deadbeat controller having setpoint 30cm
Figure 5. Closed loop system response with deadbeat controller having set point 40cm
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Figure 6.Closed loop response of a deadbeat controller having set point 30cm with the set point change of 32cm
Figure 7.Closed loop response of a deadbeat controller having setpoint 40cm with the setpoint change of 42cm
Figure8. Servo and Regulatory response of Closed loop system with deadbeat controller having setpoint 40cm
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Figure 9. Regulatory response of Closed loop system with deadbeat controller having setpoint 40cm
Figure 10. Regulatory response of Closed loop system with deadbeat controller having setpoint 30cm
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