This chapter describes the mold and the basic PI accountant tuning i.e. Ziegler – Nichols tuning for a CSTR procedure in item. Since it is necessary to keep a temperature degree changeless ever, set point trailing is really indispensable. So, in order to do a proper and accurate control action, tuning of different parametric quantities of the accountant should be really of import. A conventional tuning known as Ziegler – Nichols tuning method is described. Dominant pole method is besides developed and it is compared with Z-N reaction curve method.
Mathematical mold is the procedure of building mathematical objects whose behaviours or belongingss correspond in some manner to a peculiar real-world system. A mathematical object could be a system of equations, a stochastic procedure, a geometric or algebraic construction, an algorithm, or even merely a set of Numberss. The term real-world system could mention to a physical system, a fiscal system, a societal system, an ecological system, or basically any other system whose behaviours can be observed.
There are of class many specific grounds for mold, but most are related in some manner to the undermentioned two:
To derive understanding: A mathematical theoretical account accurately reflects some behaviour of a real-world system of involvement which can frequently derive improved apprehension of that system through analysis of the theoretical account. Furthermore, in the procedure of constructing the theoretical account, certain factors are most of import in the system, and how different parts of the system are related.
To foretell or imitate: Very frequently it is necessary to cognize what a real-world system will make in the hereafter, but it is expensive, impractical, or impossible to experiment straight with the system. Examples include atomic reactor design, infinite flight, extinction of species, conditions anticipation, neutralisation procedure, and so on.
2.2.1 The Scheme of Process Modeling
Modeling is of import in procedure industries. There is no definite algorithm to build a mathematical theoretical account that will work in all state of affairss. Modeling is sometimes viewed as an art. It involves taking whatever cognition it has made of mathematics and of the system of involvement and besides utilizing that cognition to make something. Since everyone has a different cognition base and a alone manner of looking at jobs, different people may come up with different theoretical accounts for the same system. There is normally plenty of room for statement about which theoretical account is “ best ” . It is really of import to understand at the beginning that for any existent system, there is no “ perfect “ theoretical account.
One is ever faced with trade-offs between
Increasing the truth of a theoretical account by and large increases cost and decreases flexibleness. The end in making a theoretical account is normally to obtain a sufficiently accurate and flexible theoretical account at a low cost.
Real World Data
Predictions and Explanations
Figure 2.1 Schematic flow of patterning procedure
One of the most utile ways to see mold is as a procedure, as illustrated in Figure 2.1. The get downing point is in the upper left-hand corner, existent universe information. This could stand for quantitative measurings of the system of involvement, general cognition about how it works, or both. In any instance, there is a demand for some information refering to the system. From that information, a theoretical account has been formulated or constructed. Constructing a theoretical account requires:
A clear image of the end of the mold exercising.
A image of the cardinal factors involved in the system and how they relate to each other. This frequently requires taking a greatly simplified position of the system, neglecting factors known to act upon the system, and doing premises which may or may non be right. A theoretical account may dwell of algebraic, differential, or built-in equations, stochastic procedures, geometrical constructions, etc. The undermentioned state of affairss are really common in mold:
Good theoretical accounts already exist for parts of the system. The end is so to piece these “ bomber theoretical accounts ” to stand for the whole system of involvement.
Good theoretical accounts already exist for a different system, which can be translated or modified to use to the system of involvement.
A general theoretical account exists which includes the system of involvement as a particular instance, but it is really hard to calculate with or analyse the general theoretical account. The end is so to simplify or do estimates to the general theoretical account which will still reflect the behaviour of the peculiar system of involvement. The scientific method goes something like this:
1. Make general observations of phenomena
2. Explicate a hypothesis
3. Develop a method to prove hypothesis
4. Obtain informations
5. Test hypothesis against informations
6. Attempt to corroborate or deny hypothesis
2.2.2 Mathematical Modeling of a CSTR procedure
188.8.131.52 Process Description
Chemical reactions in a reactor are either exothermal ( let go of energy ) or endothermal ( necessitate energy input ) and hence require that energy either be removed or added to the reactor for a changeless temperature to be maintained.
Figure 2.2 shows the schematic of the CSTR procedure. In the CSTR procedure theoretical account under treatment, an irreversible exothermal reaction takes topographic point. The heat of the reaction is removed by a coolant medium that flows through a jacket around the reactor. A unstable watercourse A is fed to the reactor. A accelerator is placed inside the reactor. The fluid inside the reactor is absolutely assorted and sent out through the issue valve. The jacket environing the reactor besides has feed and exit watercourses. The jacket is assumed to be absolutely assorted and at a lower temperature than the reactor.
Figure 2.2: CSTR procedure
Table 2.1 The CSTR parametric quantities
Process flow rate
Chemical reaction rate invariable
1 A- 104 K
Inlet coolant temperature
Heat of reaction
I? , I?c
1 A- 103 g/l
Inlet provender concentration
Heat transportation coefficient
7 A- 105 cal
The parametric quantities of the CSTR procedure [ 165 ] used in this work are tabulated in Table 2.1.
2.3 Linearization of the Chemical Reactor Model
The constituent balance for the reactor can be given as:
( 2.1 )
And the energy balance equation by:
( 2.2 )
Equation ( 2.1 ) can be written as:
( 2.3 )
Equation ( 2.1 ) can be linearized as:
( 2.4 )
which can be written as:
( 2.5 )
Rearranging footings and presenting the Laplace operator consequences in:
( 2.6 )
( 2.7 )
The 2nd energy equation of the reactor theoretical account can be rewritten as:
( 2.8 )
This equation can be written as:
which can be rewritten as:
( 2.10 )
Rearranging footings and presenting the Laplace operator consequences in:
( 2.11 )
The response of the alteration in reactor mercantile establishment concentration to a alteration in the reactor throughput F can be obtained by uniting equations ( 2.5 ) and ( 2.10 ) while puting the alterations in and to zero.
( 2.12 )
This equation can be rearranged as:
( 2.13 )
which is a pseudo-first-order equation.
Similarly, can be obtained.
Substitution of the steady province values in the clip changeless and procedure additions Consequences in:
This equation can be rearranged to:
( 2.14 )
( 2.15 )
( 2.16 )
The above two equations are the transportation map of concentration and temperature of the CSTR theoretical accounts.
2.3.1 State Space Model
The additive province infinite theoretical account of a CSTR is given by
Where Ten is the province variable, U is the input variable & A ; Y is the end product variable.
Substituting the steady province solution for the province infinite matrices A, B, C & A ; D are: –
By replacing the parametric quantities of CSTR from Table2.1 in above matrix the province infinite theoretical account is: –
Figure 2.3 shows the energy balance theoretical account of CSTR in MATLAB SIMULINK block. It is developed from the energy balance equation of CSTR given in equation ( 2.2 ) .
Figure 2.3.SIMULINK theoretical account of Energy balance Equation
Figure 2.4.SIMULINK theoretical account of component balance Equation
Equation ( 2.1 ) implies the component balance equation of the CSTR and is modeled in simulink block as shown in Figure.2.4.Combining these two theoretical accounts gives us the complete nonlinear theoretical account of CSTR as the TITO ( Two Input Two Output ) procedure is represented in figure 2.5.
Figure 2.5 Nonlinear theoretical account of CSTR
The unfastened cringle features of CSTR system described above is shown in Figure 2.6 and Figure 2.7. for temperature and Concentration variables. The public presentation step of the system obtained from measure trial is tabulated in Table 2.2.It is observed that the temperature portion of the CSTR procedure implies the under damped response holding peak wave-off of 63.7 % and settling clip of 128.12 seconds, both are unwanted for an chemical industry. Step response.The concentration portion of the procedure outputs an over damped response which is more sulky with 137.66 sec settling clip and approximately0 % wave-off.
Fig2.6 Open loop step response of CSTR procedure for Temperature input
Fig2.7 Open loop step response of CSTR procedure for Concentration input
It is compulsory to plan a accountant which can get by up with both over damped and under damped parts of the same procedure.The wave-off and the settling clip should be eliminated or reduced without impacting the public presentation of the system is the chief undertaking to be achieved in this work.
Table 2.2 Performance indices of unfastened loop measure response of CSTR systems
Rise Time ( Secs )
Settling Time ( Secs )
Peak Time ( Secs )
Peak Overshoot ( % )
PID Controller Design
The PID accountant is besides known as a three manner accountant. In industrial pattern, it is normally known as proportional-plus-reset-plus-rate accountant. The combination of relative, built-in and derivative manner is one of the most powerful but complex accountant operations. This system can be used for virtually any procedure status. The equations of relative manner, built-in manner and derivative manner are combined to an have analytic look for the PID manner,
U ( s ) /E ( s ) = Kp ( 1 + 1/Tis + Tds ) ( 2.17 )
This manner eliminates the beginning of the relative manner and still provides a speedy response. The three accommodation parametric quantities here are the relative addition, the built-in clip and the derivative clip. PID accountant is the most complex of the conventional control manner combinations. The PID accountant can ensue in better control than the one or two accountants. In pattern, control advantage can be hard to accomplish because of the trouble of choosing the proper tuning parametric quantities.
2.4.1 Proportional term
The relative term ( sometimes called addition ) makes a alteration to the end product that is relative to the current mistake value. The relative response can be adjusted by multiplying the mistake by a changeless Kp, called the relative addition.
The relative term is given by:
( 2.18 )
Pout: Proportional term of end product
Kitchen polices: Proportional addition, a tuning parametric quantity
vitamin E: Mistake = Set Point ( SP ) – Procedure Variable ( PV )
T: Time or instantaneous clip ( the nowadays )
A high relative addition consequences in a big alteration in the end product for a given alteration in the mistake. If the relative addition is excessively high, the system can go unstable. On the other manus, a little addition consequences in a little end product response to a big input mistake and hence, a less antiphonal ( or sensitive ) accountant. If the relative addition is excessively low, the control action may be excessively little when reacting to system perturbations.
2.4.2 Built-in term
The part from the built-in term ( sometimes called reset ) is relative to both the magnitude of the mistake and the continuance of the mistake. Summarizing up the instantaneous mistakes over clip ( incorporating the mistake ) , gives the accrued beginning that should hold been corrected antecedently. The accrued mistake is so multiplied by the built-in addition and added to the accountant end product. The magnitude of the part of the built-in term to the overall control action is determined by the built-in addition, Ki.
The built-in term is given by:
( 2.19 )
: Built-in term of end product
: Built-in addition, a tuning parametric quantity
: Mistake = Set Point ( SP ) – Procedure Variable ( PV )
: Time or instantaneous clip ( the nowadays )
The built-in term ( when added to the relative term ) accelerates the motion of the procedure towards the set point and eliminates the residuary steady-state mistake that occurs with a relative merely accountant. However, since the built-in term is reacting to accumulated mistakes from the yesteryear, it can do the present value to overshoot the set point value.
The derivative action is sensitive to measurement noise. Hence, the derivative term is non used in the design of the accountant.
2.5 Loop Tuning
Tuning a control cringle is the accommodation of its control parametric quantities ( gain/proportional set, built-in gain/reset ) to optimum values for the desired control response. Stability ( bounded oscillation ) is a basic demand, but beyond that, different systems behave otherwise and the different applications besides different applications have different demands. Further, some procedures have a grade of non-linearity and so parametric quantities that work good at full-load conditions do n’t work when the procedure is get downing up from no-load ; this can be corrected by addition programming ( utilizing different parametric quantities in different operating parts ) . PI accountants frequently provide acceptable control even in the absence of tuning, but public presentation can by and large be improved by careful tuning.
2.5.1 Tuning methods
There are several methods for tuning a PI cringle. The most effectual methods by and large involve the development of some signifier of procedure theoretical account, and so taking P and I based on the dynamic theoretical account parametric quantities. Manual tuning methods can be comparatively inefficient, peculiarly if the cringles have response times on the order of proceedingss or longer. The pick of method will depend mostly on whether or non the cringle can be taken “ offline ” both for tuning and for the response clip of the system. If the system can be taken offline, the best tuning method frequently involves subjecting the system to a measure alteration in input, mensurating the end product as a map of clip, and utilizing this response to find the control parametric quantities.
Normally PI accountants are used in all procedure industries. Integral accountant adds a pole at the beginning and increases the system type by one. It besides reduces the steady province mistake due to a measure map input to a nothing. The PI accountant transportation map is given by:
( 2.20 )
Figure 2.8 CSTR procedure with feedback control cringle
From the mathematical theoretical account of the CSTR procedure, it is good known that the concentration of the merchandise is affected by the alteration in temperature of the reactor. The chief aim of this work is to keep the temperature of the reactor at peculiar value, say 350K in order to maintain the concentration at changeless rate. The demand of the accountants developed in this work is to keep the temperature by pull stringsing the coolant flow rate.
A PI accountant is used to cut down or extinguish the steady province mistake. The control diagram of the CSTR procedure is shown in figure 2.8. It uses a feedback accountant, which makes the works less sensitive to alterations in the surrounding environment. The feedback accountant tries to extinguish the impact of burden alterations and to maintain the end product to the desired response. The divergence of the procedure end product to the desired set point value is known as the mistake ( E ) . A PI accountant eliminates the mistake by pull stringsing the input ( U ) with regard to the procedure. A PI accountant is capable of accurate control when decently tuned and used. If the PI accountant parametric quantities are chosen falsely, the system can be unstable, i.e. its end product diverges with or without any oscillation.
2.5.2 Ziegler-Nichols Tuning Rule
The Ziegler-Nichols tuning regulation would function as the footing for the PI engineering. The Ziegler-Nichols tuning is non a really limited engineering. It is successful in that function because of its improved public presentation, easiness of usage and low cost. In 1942, Ziegler and Nichols, both employees of Taylor Instruments, described simple mathematical processs ; the first and 2nd methods severally, for tuning PI accountants. These processs are now accepted as standard in control systems pattern. Both techniques make a priori premises on the system theoretical account, but do non necessitate that these theoretical accounts be specifically known. Ziegler-Nichols expression for stipulating the accountants are based on works measure responses. Ziegler and Nichols conducted legion experiments and proposed regulations for finding values of Kp, Ki and Kd based on the transeunt measure response of a works. Two methods are popular for finding the accountant parametric quantities.
184.108.40.206 Z-N Open Loop Method
The first method known as the Ziegler-Nichols Step Response Method ( Using the reaction curve ) is applied to workss with measure responses. It is besides typical of a works made up of a series of first order systems. Figure 2.9 shows the reaction curve for a measure response. The S-shaped reaction curve can be characterized by two invariables, hold clip L and clip changeless T, which are determined by pulling a tangent line at the inflexion point of the curve and happening the intersections of the tangent line with the clip axis and the steady-state degree line. It does non use to workss with neither planimeter nor dominant complex-conjugate poles, whose unit-step response resemble an S-shaped curve with no wave-off
Figure 2.9 Reaction curve for a unit measure response
This S-shaped curve is called the reaction curve. Using the parametric quantities L and T, the values of Kp, Ki and Kd can be set harmonizing to the expression shown in the Table 2.4, below [ 166 ] .
Table 2.3 Controller Parameters utilizing Reaction Curve Method
These parametric quantities will typically give a response with an wave-off of approximately 25 % and a good subsiding clip. Fine-tuning of the accountant utilizing the basic regulations relate each parametric quantity to the response features.
220.127.116.11. Model Reduction
The higher order procedure can be reduced as a First Order procedure with Dead Time ( FODT ) by the half regulation decrease technique. The largest neglected ( denominator ) clip changeless ( slowdown ) is distributed equally to the effectual hold and the smallest maintained clip changeless. The half regulation [ 167 ] is used to come close the procedure as a first or 2nd order theoretical account with effectual hold. For a first-order theoretical account parametric quantities are thousand, I„ and I? .
For illustration, the 2nd order procedure can be approximated as. It is proposed to call off the numerator term ( T0s+1 ) against a ”neighbouring ” denominator term ( I„0s+1 ) ( where both T0 and I„0 are positive and existent ) utilizing the undermentioned estimates.
Here i?± is the ( concluding ) effectual hold, for which the exact value depends on the subsequent estimate of the clip invariables ( half regulation ) , so one may necessitate to think i?± and iterate. If there is more than one positive numerator clip changeless, so one should come close one T0 at a clip, get downing with the largest T0.
Normally I„0 is selected as the closest larger denominator clip changeless ( I„0 & gt ; T0 ) and usage Rules T2 or T3.The exclusion is if there exists no larger I„0, or if there is smaller denominator clip changeless ”close to ” T0, in which instance I„0 is selected as the closest smaller denominator clip changeless ( I„0 & lt ; T0 ) and usage regulations T1, T1a or T1b. To specify ”close to ” more exactly, allow I„0a ( big ) and I„0b ( little ) denote the two neighbouring denominator invariables to I„0.
Then, choice I„0=I„0b ( little ) if T0/I„0b & lt ; I„0a/ T0 and T0/I„0b & lt ; 1.6 ( both conditions must be satisfied ) .
Using the half regulation decrease technique described above, the CSTR procedure can be reduced as a FODT ( First Order procedure with Dead Time ) .
( 2.22 )
From equation 2.22, To=458, I„a=1000, I„b = 250 and K=1
Using half regulation, I?d=125 and I?=375.
Therefore FODT transportation function= ( 2.23 )
The measure response of equation ( 2.23 ) is shown in figure 2.10.The incline can be obtained by pulling tangent on the curve and the procedure addition K is the ratio of maximal end product to input and the value of clip changeless T which is the clip taken to make 63.2 % of concluding end product is 400seconds can besides be determined from the figure2.8. This curve is settled at 1700 seconds which is about four times of the clip changeless obtained.The curve indicates the slowdown L=125 seconds. The parametric quantities of PID accountant Kp= 3.6, Ki = 0.004 and Kd=6 are obtained from Table 2.1.
The closed cringle response shown in figure 2.12 is obtained by permutation of these PID accountant parametric quantities in the block shown in figure 2.11.
Figure 2.10 Step response of CSTR procedure after theoretical account decrease
Figure 2.11 Simulation of PID accountant by reaction curve technique
Figure 2.12 Response of PID accountant tuned by reaction curve technique
2.6. Design of PID Controller Using Dominant Pole Method
PID accountants are likely the most normally used accountant constructions in industry. They do, nevertheless, present some challenges to command and instrumentality applied scientists in the facet of tuning of the additions required for stableness and a good transient public presentation. There are several normative regulations used in PID tuning. An illustration could be what was proposed by Ziegler and Nichols in the 1940 ‘s. These regulations are by and big based on certain false theoretical accounts.
2.6.1. Calculation of Kp, Ki and Kd Values Dominant Pole Method
The concentration portion implies the transportation map whose measure response is in over damped in nature. It is hard to tune a over damped 2nd order procedure with Zero utilizing Z-N closed cringle tuning method because Z-N closed cringle tuning regulation based on sustained oscillation. It is hard to happen ultimate addition in a overdamped procedure since really high addition is required to bring forth bound rhythm oscillation which leads to unstability in system. Dominant pole is one popular manner to plan PID accountant for such a procedure by puting a pole in a safe and stable part such a manner that the consequence of nothing can be compensated. The consequence of Zero in PI accountant is cancelled, ( Astrom ( 2001 ) ) by the inclusion of new pole. The design start with turn uping a pole =0.8 and =2.5, the significance of turn uping a pole in this part will keep stableness and cut down the impact of over damping. The coefficients of PI accountant can be obtained by the following design.The transportation map of the concentration of the chemical procedure can be rearranged as:
( 2.24 )
Put s=sd, Dominant Pole, in the above equation
( 2.25 )
The Sd can be found out by the expression
( 2.26 )
Assume =0.8 and =2.5
Therefore, ( 2.27 )
( 2.28 )
Converting into polar signifier
( 2.29 )
Therefore D=2 and
Utility Equation ( 2.29 ) in ( 2.26 )
( 2.30 )
Solving the equation into polar signifier as
( 2.31 )
Equation is reduced as
( 2.32 )
From the above equation figure ( 2.32 ) Ad and I†d can be calculated as
Ad= 0.0589 and I¦d = -115.33A°
Where Ad is the magnitude and I¦d is the stage.
From these values the derivative changeless Kd value can be found as by the expression
( 2.33 )
( 2.34 )
The relative changeless value Kp can be found out by the expression
( 2.35 )
( 2.36 )
The built-in changeless value Ki can be found out by the expression
( 2.37 )
( 2.38 )
The values obtained for the PID accountant parametric quantities are summarized as below:
Proportional Constant value,
Integral Constant value,
Derivative Constant value,
A proportional-Integral ( PI ) accountant is a generic control loop feedback mechanism that is widely used in industrial control systems. If a accountant starts from a stable province at zero mistake i.e. Process Variable peers to Put Point ( PV = SP ) , so further alterations by the accountant will be in response to alterations in other measured or immeasurable inputs to the procedure that impact on the procedure, and therefore on the PV. Variables that impact on the procedure other than the Manipulated Variable ( MV ) are known as perturbations. By and large, accountants are used to reject perturbations or implement set point alterations. Changes in provender H2O temperature constitute a perturbation to the temperature control procedure.
2.7 Simulation Consequences
The tuning of a PI accountant for the CSTR procedure is carried out by the dominant pole tuning method. Figures 2.13 illustrate the end product response of a PI – tuned CSTR procedure. Table 2.4 shows that the dominant pole tuned method makes the CSTR procedure produce an end product response with a marginally good transient response and a good steady province response.
Figure 2.13 Response of PID accountant tuned by dominant pole technique
Table 2.4 Performance indices of Conventional PID Controller
Delay Time ( Secs )
Rise Time ( Secs )
Peak Time ( Secs )
Settling Time ( Secs )
Peak Overshoot ( % )
Dominant Pole method
In this chapter, the higher order system is reduced into FODT procedure utilizing a novel technique called half regulation and tuning processs are implemented utilizing Reaction Curve Method and Dominant Pole Analysis Method. From the consequences, it is inferred that the Dominant Pole Method implies less overshoot and settling clip compared with Reaction Curve method. This system should be all right tuned and peak shoot should be nullified. Hence, different tuning algorithms have been developed for bettering its public presentation in the undermentioned chapters.