On Robust Control System Design for Plants with Recycle

Muzaffar Iqbal University of Lahore Registration MSE-F16-105

Email Address: [email protected]

Abstract: Robust control system design for plants with a direct procedure for driving the component transfer in plants with recycle when mechanistic is used in the mathematical modeling is demonstrated. Controlling is the basic and most important component of modern age industrial manifesto, Therefore Multi-loop feedback controllers are introduced for the plants on the base of recycle facility; it may be with or without recycle compensator. In general it is observed that controller parameterized using the recycle compensated plant model resulting in close loop system advantages includes satisfactory nominal and robust performance characteristics. When implementation is carried out, It is found that compensated plant model covers better efficiency than when its controller is parameterized using the uncompensated plant model.

1. INTRODUCTION

In recent few decades, the control design has become back bone and integral part of process industries as it plays an important role to make the process more efficient, reliable, optimum and unambiguous. Recycling is an important aspect for process industries as it enhances the efficiency of energy usage thus resulting in greater over all conversion in reactor. Almost in all emerging power plants and production plants recycling is used to enhance the efficiency of the plant and reduce the losses. Recycling process has been studied by many technologists. An organized investigation of feedback system can achieved the robustness in the recycling. There are many organized approaches to overcome the recycling compensations. If we review it is concluded that the implementation of such a control system results in poor dynamic characteristics. To maintain steady state characteristics we wish to achieve while avoiding the poor dynamic characteristics all the major process units on the global are aligned with the recycle facility. When system comes under steady state operation robust designing with recycling is distinctly fascinating because it lead to make the operation more optimum and prime, for example separation process in chemicals industries and more percentage of overall conversion in a reactor. In prospective to keep the steady state edge while avoiding the dynamic penalty, Taiwo 1986 was the scientist who recommended the recycling compensator and controller to study the recycling. This proposal was further adopted by scali and Ferrari (1997). The basic function of recycle compensator is to minimize the belongings of recycle. Taiwo (1985, 1986, 1993, and 1996) suggest recycling compensator whose capability has broadly been endorsed by many researchers, and the penalties of examples are present on the globe. As recycle compensator has ability to make the process coherent , so therefor to access recycle compensator , the transfer are divided into three major paths or functions including forward path, recycle path and finally disturbance transfer function. To calculate recycling compensation the method of transfer function must be broken into onward track, recycle track at the end malfunctioned transfer function But in practical scenario this compensator cannot reduces the effects of recycle process.to minimize this insecurity designing of main controller is important. The doubt in existing control system leads to corrupting the enactment of process industry in air craft engineering the mathematical model of air craft is fabricated after done experiments on air craft body. According to concerns the results obtained from mathematical modeling were not matched to specific values. So available data was not matched actual data.

Section 2 is mainly comprised of specifications or properties of recycle compensator, direct procedure about this disintegrated and controller design will incorporated in section 3 and the appendix contains various types of balances including energy, material and momentum. The concluding remarks will be discussed in chapter 4.

2. PROCESS OVERVIEW AND PROPERTIES OF COMPENSATOR

Block diagram is given below involving a plant with recycle in fig1, where j, k/, n/, k, Jo are controlled, manipulated, disturbance, controller output and measured variables. X(s) is the feedback controller while R(s) is the recycle compensator. Each Wi(s) with perhaps the exception of W2(s) is assumed to be a square matrix. First note that the open loop recycle sub plant output M is given by

M(s) = (1–W?)-1W1k?(s) + (1–W?)-1W2n?(s) (1a)

While the recycle plant output, j is given by

j(s) =W4(1–W?)-1W1k?(s)+W4(1–W?)-1W2n?(s) (1b)

Where W3(s) is the transfer function of recycle process .the compensator F that totally cancels the detrimental effect of recycle is known as the perfect recycle compensator. Such a compensator restores dynamically favorable transfer function of the original process without recycle, that is

M(s) = W1k?(s) + W2n?(s) (2a)

j(s) = W4 W1m?(s) + W4W2n?(s) (2b)

In order to specify the recycle compensator, apply block diagram algebra to inner loop W in fig 1 (Taiwo, 1985, 1986, 1993, Taiwo and Krebs 1996) eqn. (3)

Fig.1: block diagram of a system consisting of a plant with recycle, recycle compensator R and feedback controller X

R(s) = W1 -1 W 3 W 4-1 W 5-1 (s) (3)

R(s) is realizable when equation number three is proper and casual. However, if reliability problems occurs, a compensator which almost the recycle compensator may be employed. Even the use of R (0) alone should resulting addition in process dynamics as it carry over to the removal of raising study state sensitivity connected with plant with recycle.

3. DECOMPOSITION OF RECYCLE PLANTS AND FEEDBACK CONTROLLER DESIGN

3.1 Example 1

The block diagram shown in this example (fig.2a) for this SISO plant is obtained from Del-Muro-Cuellar et al. (2005).Taking block diagram algebra, it is clear that the enclosed block WR in fig 1 is equivalent to fig 2 b which is transformation of Fig2a.

Hence, the recycle compensator (3) is enlisted below

R = wr Given that wf(s) = (x-3) e-1.2x (x-1)2 and wr(s) = e-0.8x (x+2) , the enclosed transfer function W in Fig.1 which is the compensated system is given by R(s) = ym(s) u(s) = wf (s), since in this example W4(s) = W5(s) = 1 whereas, without recycle compensator,

WR(s) = m(s) u (s) = (x+2)(x-3) e-1.2x (x+1)(x+2) e-2x Feedback controller design for the compensated plant

It may be achieved clearly employing IMC parameterization. If controlling of plant is objective then using a proportional plus integral (PI) controller , wf (s) may be simplified to a first order plus dead time model using , for example , Skogestad,s half rule (Seborg et al.2011) , yielding -3e-2.033x(1.5x+1) . The final controller adopting a tuning parameter ?c = 2 is X (s) = -1 8 (1+11.5x) the step response of this process to reference step changes are good as shown in fig .5.Not too much efforts are required to design a controller satisfying a desired robust performance measure for space economy . However, expositions of these techniques are per found in (Taiwo and Kreb, 1996)

Feedback controller design for the uncompensated plant

A simple analysis of the uncompensated plant WR(s) shows that the Nyquist plt marginally avoid the critical point (-1, 0) because it crosses the real negative axis at the frequency = 0.69 with WR (Du) = -0.987. It is therefore not surprising that his open loop response is highly oscillatory as shown in fig 3. This is sharp contrast to the well damped characteristics of the compensated plant (Fig.4).It has not been easy to designing a feedback controller providing a system satisfactory performance. One technique is to decrease close loop efficiency measured such as the integral of the absolute error to a unit step change in reference input, IAE. When it was carried out, the parameter of the PID controller with the metrics of its efficiency is given below. It is proved that the system with recycle compensator outperforms the system without the recycle compensator especially with respect to adequate damping of the close loop response and its acceptable gain margin GM. its phase margin, PM. Is also adequate.

The performance metrics of the compensated system, are IAE = 5.45, GM = 4.56, PM = 50.77. For the uncompensated plant, PID controller used is X (s) = – 0.25 (1 + 11.08x+3.33x ). Performance metrics are IAE = 5.20, GM = 2.135, PM = 121.85

3.2 Two CSTR in series with recycle

In this process, two CSTR reactors are under experiment with recycle facility; see Fig 6 (scali and Ferrari 1999). Following the exposition in the appendix, the state variable model for the process in deviation variable at the given operating point takes the form (note hat for this plant X = I2)

H1H2=-12.2501.051.6 H1 (t)H2(t) +01000 H1 (t)H2(t-2) + 1000.05 u1 tu2t+0.50dt (4)

W1(s) = 1x+12.501.05(x+12.5)(x+1.6)0.05x+1.6

W2(s) = 0.5x+12.50.525(x+12.5)(x+1.6)

W3(s) = 010e-2sS+12.501.05e-2s(S+12.5)(S+1.6)

W4(s) = I2and

W5(s) = diag (e-s, e s)

So equation of recycle compensator is given as

R(s) =010e-x00The model for user compensation feedback controller is given by

j(s) = W5 (s) W 1 (s) k? (s) + W5 (s) W2 (s) n? (s)

Where

W5 (s) W1 (s) = 0.08e-x0.08s+101.05e-x(x+12.5)(x+1.6)0.03125e-x0.625x+1And

W5 (s) W2 (s) = 0.5e-xx+12.50.525e-x(x+12.5)(x+1.6)Whereas the recycle plant transfer function without recycle compensation is given by

Jm (s) = W5 (1-W3)-1 W1 n? (s) + W5 (1-W3)-1 W2 n?(s)

=e-s/D(s) (x+1.6)0.5e-2×1.050.05(x+12.5)k? 0.05(x+1.6)0.525n?) W(s) = x+12.5x+1.6-10.5e-2xFeedback controller design for compensated system

Using the compensated plant model

R (s) =0.08e-x5x+101.05e-x(x+12.5)(x+1.6)0.03125e-x0.625x+1, RGA = 1001This suggests a 1-1/2-2 pairing. Niederlinsk index is 1 affirming, by its non-negativity, that a multi-loop P1 controller should be stable. With the uncertainty weight chosen as

Du = 0.2 0.2x+10.2x+1 and the performance weight as Dp = 0.2 x2.5+0.1x robust control indices computed as summarized in table 1 shown the system is robust.

Table I: controller parameters and robustness indices for the compensated plant

Kp Ki URP UNP URS

Loop1 1.7 6.45 0.97 0.956 0.3

Loop2 12.5 19.9

Feedback controller design for uncompensated system

For the uncompensated plant,

WT(s)=1/W(s) x+1.6e-x0.5e-3×1.05e-x0.05x+12.5

Where W(s) = (x+12.5)(x+1.6)-10.5e-2x

RGA = 2.11-1.11-1.112.11 .

This suggests a 1-1/2-2 pairing. Niederlinski index is 0.4755 affirming by its non-negativity, that a multi loop controller could be stable. Good controller setting obtained are summarizes in table II. with the input weight chosen as Du = 0.2 0.2x+10.2x+1 and its performance weight as Dp = 0.2 x2+0.0075x robust performance indices computed as summarized in table II show that the system is robust . When applying the controller tuned for the compensated plant model has wider gains and delay in its open loops properties quasi polynomial, with this it also consist bigger delay in its (1,2) element. Instability in close loops may happen when ever more realistic model is employed. What is adoptable in this scenario to minimize the controller gains as has been done here where the close loop system was stabilizing multiplying all the controller gains in table I by 0.4. This was assumed to give a robust system also, as shown by the asterisked values in table II

Table II: controller parameters and robustness indices for the uncompensated plant

Kp Ki URP UNP URS

Loop1 4.197 1 0.8578 0.7588 0.3

Loop2 9.678 3.00

Loop1* 0.7 3.3 0.9 0.980 0.3

Loop2* 5.56 7.90

As commented in the last section, the controller parameterized using the compensated plant model result in a faster system when implemented on the uncompensated plant (see figs 7 & 8). It is also robust. The relative sluggishness of the system incorporating the controller parameterized using the uncompensated plant model is a carryover from its open loop relatively larger time constants and other unfavorable characteristics. Utilizing the recycle compensator eliminates such undesired characteristics thus making the recycle compensated system possess the most attractive characteristics as shown in fig 7 & 8.

3.3 Experimental three tank system with recycle

Three tank experiments are under consideration in this example. This system with recycle is shown in fig 9. This is a two input, two output process. The controlled variables h1 and h2 inside tank 1 and 2.the manipulated variable are the VFR q1 and q2 respectively and the water level h3 in tank 3 is found but not controlled

Following (iii) in the appendix, the linearized equations in deviation variables are

?h=A?h + HZ + B? + B?qWhere ?h = ?h1?h2?h3 ,?h1?h1, C = 100010 also C = -0.015300010W1 = M( sI – A)-1B , a third order model which easily simplified to G1r below using element by element simplification using power series expansion:

W1r (s) =0.60120.4x+10.122240.2x+10.122240.2x+10.34484.3x+1,

W3 = M (sI-A)-1 H, which simplifies to W3r below using a similar procedure to the one above:

W3r (s) =03120.2x+100.611240.2x+1,

From (i) in appendix , WR = M( sI –( A+E )-1B, where E is 3 by 3 with zeros apart from e12 which is 0.0435, compute the transfer function matrix for the uncompensated plant model, which, a similar procedure as above , simplified to:

WRr (s) = 1.5×498.3x+12.78567.5x+10.32618.2x+10.344462.4x+1The recycle compensator is calculated as

F= W1-1 W3 = 0500The controller design for the compensated plant

Using upon the recycle compensator the transfer function matrix for the compensated plant is given by

G1r (s) =0.60120.4x+10.122240.2x+10.122240.2x+10.34484.3x+1,

It is very easy to confirm that G(s) is diagonally dominant; hence designing a feedback controller using the diagonal elements would give a stable closed loop system involving G(s)

IMC-PI controller thus obtained as

X(s) = diag (20.07 + 0.17 /x, 30.63+0.36/x (6)

Feedback controller design for uncompensated plant

IMC-PI controller were parameterized for the uncompensated plant using WRT (s) obtained above

The RGA for GRT is given by

RGA = 2.77-1.77-1.772.77Consequently, a multi loop IMC-PI controller

X(s) = diag (40.18 + 0.081 /x , 65.33+0.14/x (7)

The controllers thick were applied on a nonlinear SIMULINK model of three tank system. The simulation results produced in fig 10 shows that the compensated system (line the controllers thick were implemented on a nonlinear SIMULINK model of three tank system. The simulation results represented in fig 10 shows that the compensated system (line) displays a faster set point tracking for both controlled levels h1 and h2 when compared to the uncompensated system (dash lined) being control using controller 7 which has been parameterized using the uncompensated plant model . on applying controller in equation 6 designed for compensated plant model on the uncompensated plant model (dash lined) , a better set point tracking result is obtained than when using controller 7 parameterized by uncompensated plant model as shown in fig 10. The recycle compensated plant using controller 6 however displays a faster set point tracking for the control level h1 than the uncompensated plant. It is noted that the controlled level h2 is indistinguishable for both the compensated and uncompensated plants. This behavior characteristics simple model with adequate open loop damping and without dead time. Otherwise controllers designed for the compensated plant may lead to instability or oscillatory responses when implemented on the uncompensated plant, see example 2 where controller gain reduction was mandatory.

Fig 10 close loop system outputs and inputs in response to step changes in reference 1 and 2 .(Legend: thick, dashed and dotted lines respectively represent controller 6 on compensated plant Controller 7on uncompensated plant and controller 6 on uncompensated plant.

4. DISCUSSION AND CONCLUSION

In this work three impotent examples are discussed about robust control system design for plants with recycle. Each example elaborates the significance of this technique effectively, further illustration comprised below:

In example 1 the recycle compensator may be employed to reestablish proper stability to open loop marginally stable or unstable plants, in this way it is facilitating the design of robust close loop systems. It can be clearly narrate those plants with recycle shows much bigger settling time, when compared to the plant without recycle. Hence parameterizing the feedback controller for the uncompensated plant using the compensated plant model usually results in better close loop systems involving the uncompensated plants parameters. The only caution is this type of controller may have to be modified, considering an example, by controller gain reduction, in order to ensure adequate stability of close loop systems involving the uncompensated plants. This has been shown in example 2.

This work used the IMC to parameterize PI controllers for both SISI and MIMO systems. It was only the case of marginally stable process in example 1 that PID controller parameters of the uncompensated system had to be determined using optimization. Generally it can be clearly narrated that the nominal and robust characteristics of the recycle compensated system are considered better than uncompensated systems.

ACKNOWLEDGMENT

We are greatly grateful to Almighty Allah, the Merciful, the Generous we express gratefulness toward HIM for all his blessings especially on mankind. We place on top and sincerely acknowledge the unremitting backing, precious advising, timely recommendations and inspired supervision presented by Prof…………….., at getting this report to effective completion.

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