chapter 6 closed loop studies -...

180

CHAPTER 6

CLOSED LOOP STUDIES

Improvement of closed-loop performance needs proper tuning of

controller parameters that requires process model structure and the estimation

of respective parameters which are discussed in detail in the previous chapter.

The present chapter discusses the methods for the design of multi-loop control

strategy and tuning of controller parameters. In order to achieve desired

quality, specified output characteristics at the cost of spending optimum

inputs one needs to design a controller and run the plant under closed loop so

that optimal production is achieved under safe operation.

6.1 INTRODUCTION

Controllers for MIMO systems can be either multi-loop

(Controllers are designed only for diagonal elements of process models of

transfer function matrix) or multivariable (Controllers are designed for all the

elements of the MIMO transfer function matrix). Multi-loop control scheme

has an edge over multivariable as the former can work even if a single-loop

fails. In the presence of interactions between input/ output the process needs

to be decoupled and then multi-loop controllers can be designed. When

interaction effects produce a significant deterioration in control system

performance, decoupling control has to be considered. Thus multi-loop or

multivariable controller involves the objective of maintaining several

controlled variables at independent set points.

181

Many researchers have worked on design of multi-loop controllers

for MIMO systems. Luyben (1986) presented a simple practical approach to

the problem of finding reasonable controller settings for N SISO controllers in

Nth order which is a typical industrial multivariable process. Loh et al (1993)

proposed the autotuning of multi-loop PI controller. This procedure is the

combination of sequential loop closing and relay tuning often used for tuning

single PI/ PID controllers. Huang et al (1993) proposed and implemented

PID controllers with the form of no proportional kick. Shen et al (1994)

proposed multivariable automatic tuning that performs the identification-

design procedure in a sequential manner which discusses the concept of

multivariable autotuner and the underlying theory for sequential design and

sequential identification employed in autotuning procedure. Palmor et al

(1995) proposed the automatic tuning of decentralized PID controllers for

TITO processes. Huang et al (2000) derived tuning rule from inverse based

PID controllers. Toh et al (2002) proposed a methodology for autotuning

decentralized proportional-integral-derivative (PID) controllers for

multivariable systems. Lee et al (2004) proposed a tuning method for multi-

loop PID controllers by extending the generalized IMC PID tuning method

for SISO systems. Liu et al (2005) proposed an analytical multi-loop

controller design for industrial and chemical 2-by-2 processes with time

delays. Vu et al (2008) proposed a new analytical method based on direct

synthesis approach for design of multi-loop PID controllers. This method is

aimed to achieve desired closed loop response for MIMO systems with

multiple time delays. Lin et al (2009) proposed a systematic procedure to

design multivariable controllers that have options for selective decoupling of

different structures (e.g. full or partial decoupler). Vu et al (2010) proposed a

novel method for independent design of multi-loop PI/ PID controllers. The

idea of an effective open-loop transfer function (EOTF) is introduced to

decompose multi-loop control system into a set of equivalent independent

single-loops. Veronesi and Visoli (2011) proposed a new automatic tuning

182

technique for multi-loop PID controllers applied to MIMO systems. Jeng et

al (2011) proposed the methods including model identification, controller

monitoring and controller retuning which in turn combined to develop an

intelligent control systems. Rajapandiyan and Chidambaram (2012) proposed

a method for the independent design of PI/ PID controllers based on

equivalent transfer function (ETF) model of individual loops and simplified

decoupler matrix.

6.2 DIFFERENT CONTROL DESIGN STRATEGIES

MIMO systems came into use in chemical industries as the

processes were redesigned to improve efficiency. Multivariable control

involves the objective of maintaining several controlled variables at

independent set points. Interaction between inputs and output cause a

manipulated variable to affect more than one controlled variable. The various

control schemes studied here are decentralized, centralized and decoupled

systems. In decentralized structure diagonal controllers are used hence, result

in system having n controllers whereas in the centralized control systems

having n x n controllers. In decoupled systems the process interactions are

decoupled before they actually reach and affect the processes.

6.2.1 Centralized Structure

Centralized control scheme is a full multivariable controller where

the controller matrix is not a mere diagonal one. The decentralized control

scheme is preferred over the centralized control scheme mainly because the

control system has only n manipulated signals controlling n output variables,

and the operator can easily understand the control loops. However, the design

methods of such decentralized controllers require first pairing of input-output

variables whereas tuning of controllers requires trial and error steps. The

centralized control system requires n x n controllers for controlling n output

183

variables using n manipulated variables. While calculating the control action

using computer, the problem of requiring n x n controllers does not exist. The

advantage of centralized controller is easy to tune even with the knowledge of

steady state gain matrix alone thereby multivariable PI controllers can be

easily designed.

For the centralized structure, Internal model control-proportional

integral tuning is adopted based on the studies and recommendations of

Reddy et al (1997) on tuning of centralized PI controllers for a Multi-stage

flash desalination plant using Davison, Maciejowski and Tanttu-Lieslehto

methods.

The IMC-PID tuning relations are used in tuning the controller.

When a first order system is in the formp

p

p 1

D sk es

, the PI controller settings are

as follows:

C

p

Kk l

(6.1)

pI (6.2)

where p pmax 1/ 0.7 ,0.2l D

These tuning relations are derived by comparing IMC control with

the conventional PID controller and thereby formulating the equations to

determine the proportional gain and integral time.

184

6.2.2 Decentralized Structure

In spite of developments in advanced controller synthesis for

multivariable controllers, decentralized controller remain popular in industries

because of the following reasons:

1. Decentralized controllers are easy to implement.

2. They are easy for operators to understand.

3. The operators can easily retune the controllers for change in

process conditions.

4. Some manipulated variables may fail. Tolerances of such

failures are easily incorporated into the design of

decentralized controllers than the full controllers.

5. The control system can be bought gradually into service

during process start up and taken gradually out of service

during shut down.

The design of decentralized control system consists of two main

steps:

Step 1 control structure selection

Step 2 design of SISO controller for each loop.

In decentralized control of multivariable systems, the system is

decomposed into number of subsystems and individual controllers are

designed for each subsystem.

185

For tuning of controller, Biggest Log Modulus Tuning (BLT)

method is used (Luyben, 1986), which is an extension of Multivariable

Nyquist Criterion and gives satisfactory response. Detuning factor F (typical

values are said to vary between 2 and 5) is chosen so that closed-loop log

modulus Lcmmax >= 2n,

cmw20log

1+wL = (6.3)

( )p cw 1 det I G G= - + + (6.4)

where GC is n x n diagonal matrix of PI controller transfer functions, Gp is n x

n matrix containing process transfer functions relating n controlled variables

to n manipulated variables.

Now the PI controller parameters are given as:

CiZ-NCi

KK F= (6.5)

Ii IiZ-NF (6.6)

where i stands for individual transfer function component, CiZ-NK and IiZ-N are

Ziegler-Nichols tuning parameters. These parameters are calculated from the

system perturbed in closed loop by relay of amplitude h, reaches a limit cycle

whose amplitude a and period of oscillation Pu. These parameters are

correlated with the ultimate gain (Ku) and frequency ( u) by the following

relationships:

4u

hKa

(6.7)

186

uu

2P

(6.8)

Detuning factor F determines the stability of each loop. The larger

the value of F more stable the system is but the set point and load responses is

sluggish. This method yield settings give a reasonable compromise between

stability and performance in multivariable systems.

The decentralized scheme is more advantageous in fact that the

system remains stable even when one controller goes down and easier to tune

because of less number of tuning parameters. However, pairing analysis

needs to be done as n! pairings between input/ output.

6.2.3 Decoupled Structure

This structure has additional elements called decouplers to

compensate for interaction phenomenon. When RGA shows strong interaction

then a decoupler is designed. However, decouplers are designed only for

order less than 3 as the design procedure becomes more complex as the order

increases.

The BLT (Luyben, 1986) procedure of tuning the decentralized

structure follows the generalized way for all n x n systems as mentioned

above. The centralized controllers are tuned using IMC-PI tuning relations

which are appropriately selected for first order and second order systems.

The decoupled structure adopts the various methods like partial,

static and dynamic decoupling to produce the best results. The design

equations for general decoupler for n x n systems are conveniently

summarized using matrix notations, defined as follows:

187

Transfer function matrix, 11 1n

n1 nn

G s G sG

G s G s

Decoupler matrix, 11 1n

n1 nn

D s D sD

D s D s

Diagonal matrix of decoupler, sH

sHsH

H

33

22

11

...0......0...

1

n

...u

uu

; 1

n

...M

MM

; 1

n

...y

yy

Manipulated variable (new) Manipulated variable (old) Output variable

For a decoupled multivariable system output can be written as:

GMy (6.9)

M=Du (6.10)

Equation (6.10) becomes:

GDy u (6.11)

Equation (6.11) becomes:

Hy u (6.12)

where,

GD=H (6.13)

188

(or) -1D=G H (6.14)

which defines the decoupler.

For 2-by-2 system, equations are derived for decouplers by taking

that loop as well as other interacting loops into account.

The scope of the discussion is the decentralized PI controller for

MIMO systems and the closed loop responses for set point tracking using

BLT tuning and IMC-PID Laurent series. The illustrations of decentralized

PI controllers are given in the following section using BLT tuning.

6.3 BIGGEST LOG MODULUS TUNING

Luyben proposed BLT tuning involves the following four steps:

Step 1 : Calculate the Ziegler-Nichols settings for each individual

loop. The ultimate gain and ultimate frequency of each

diagonal transfer function Gjj(s) are calculated in the classical

way. To do numerically, a value of frequency, is guessed.

The phase angle is calculated and the frequency is varied to

find the point where the Nyquist plot of Gjj crosses the

negative real axis (the phase angle is -180 degrees). The

frequency where it occurs is u. The reciprocal of the real part

of Gjj is the ultimate gain.

Step 2 : Detuning factor F is assumed which is always greater than 1.

Typical values are between 1.5 and 4. The gain of all

feedback controllers KCi are calculated by assuming Ziegler-

Nichols gain KZNi by the factor F.

ZNiCi F

KK

189

where, uiZNi 2.2KK

Then all feedback controller reset times Ii are calculated by

multiplying the Z-N reset times ZNi by the same factor F.

Ii ZNiF

where uiZNi 1.2P

The F factor can be considered as detuning factor which is

applied to all loops. Larger the value of F more stable the

system will be but more sluggish will be the set point and load

responses. The method yield settings that give a reasonable

compromise between stability and performance in

multivariable systems.

Step 3 : Using the guessed value of F and the resulting controller

settings, a multivariable nyquist plot of scalar function

Mi 1 det iW I G B is made. The point (-1, 0)

is closer to this contour then the system is closer to instability.

Therefore the quantity w1+w will be similar to closed loop

servo transfer function for SISO loop MM

B1+ B

GG . Based

on intuition and empirical grounds, multivariable closed loop

log modulus Lcm is defined.

cm 20log1

wLw

The peak in the plot of Lcm over the entire frequency range is

the biggest log modulus Lcmmax.

190

Step 4 : The F factor is varied until Lcmmax is equal to 2N. Where N is

the order of the system. For N=1 in SISO case, the familiar

+2dB max closed loop log modulus criterion is obtained. For

a 2-by-2 system, a +4dB value of Lcmmax is used and for 3-by-3

system a +6dB; hence forth.

An example given here is 2-by-2 MIMO system (WB column) with

transfer function matrix as in Equation (2.1).

Step 1: The ultimate gain and ultimate frequency for loop 1 and loop 2 are

Ku1 = 2.112, Pu1 = 3.9 and Ku2 = -0.418, Pu2 = 11.04. Using Ziegler-

Nichols settings the following controllers are obtained for loop 1 and

loop 2 that are KC1,ZN 11,ZN = 3.25 and KC2,ZN = 0.19,

12,ZN = 9.2.

Step 2: Assuming, F = 2.55.

Step 3: The F factor is varied until Lcmmax is equal to 4 for 2-by-2 MIMO

systems.

This empirically determined BLT criterion is tested on ten

multivariable distillation columns, varying from 2-by-2 system upto 4-by-4

MIMO systems (Luyben, 1986).

6.4 IMC PID LAURENT TUNING

IMC-PI controller parameters are derived by equating the closed

loop response to desired closed-loop response involving user defined tuning

in process transfer function. Controller synthesis procedure using desired

closed loop response involves synthesis of coefficient terms s0, s-1 and s1 in

191

PID parallel structure are discussed briefly by Panda (2009) in section 4.3.1.

Thus, the IMC-PI controller parameters for FOPDT processes are computed

using the following expressions:

2

pII p

p p p

;C

DK

k D D

6.5 RESULTS AND DISCUSSION

The parameters of multi-loop controllers using IMC and BLT

design for MIMO processes are listed in the following Table 6.1.

192

Table 6.1 Parameters of multi-loop PI controllers for MIMO processes

S.no Process BLT design IMC design CK I CK I

1 WB 0.375 -0.075

8.29 23.6

0.3102 -0.0975

17.2327 15.3156

2 VL -1.07 1.97

7.1 2.58

-1.2391 1.0044

7.3602 9.458

3 WW 27.4 -13.3

41.4 52.9

27.5573 -14.4122

62.5 37.3565

4 TS -16.6 70.6

20.6 80.1

-4.0008 8.5810

6.5389 57.233

5 OR 1.28 -0.251 2.24

19.3 21.1 7.78

1.5862 -0.2972 3.3905

7.349 5.6806 6.9280

6 T1 -14.3 0.602 -0.21

5.6 6.98 173

-2.4507 0.3303 -0.2243

68.3547 2.4088

39.1936 7 T4 -7.04

-2.2 -0.114

11.3 23.2 24.2

-2.4507 -2.4040 -0.2834

68.3547 4.0055

11.9384

8 DL -0.084 -5.16 0.305 0.529

33 15.5 17

11.2

-0.118 -7.26 0.429 0.743

23.5 11

12.1 7.94

9 A1 0.945 1.22

0.491 0.837

174 18

17.8 66.9

0.3896 1.2820 0.8924 0.1059

47.8713 32.8688 16.7250 34.3824

10 A2 0.393 0.495 0.31

0.927

145 31 31

93.8

0.1461 0.6651 0.8765 0.1194

28.7163 45.7664 19.0727 35.6131

193

To evaluate the output control performance, it is considered a unit

setpoint change of all control loops one by one and the integral square error

ISE ( i i ie y r ) used to evaluate the control performance.

2i

0

ISE e dt (6.15)

The simulation results and ISE values are given in Figures 6.1 to

6.10. The results show that IMC design provides better performance than

BLT design.

Figure 6.1 Step response and ISE values of multi-loop PI controllers

for WB column (solid line: IMC design; dashed line: BLT

design)

194


for VL column (solid line: IMC design; dashed line: BLT

design)

195


for WW column (solid line: IMC design; dashed line: BLT

design)

196


for TS column (solid line: IMC design; dashed line: BLT

design)

19

Figure 6.5 Step response and ISE values of multi-loop PI controllers for OR column (solid line: IMC design; dashed

line: BLT design)

Figure 6.6 Step response and ISE values of multi-loop PI controllers for T1 column (solid line: IMC design; dashed

line: BLT design)

Figure 6.7 Step response and ISE values of multi-loop PI controllers for T4 column (solid line: IMC design; dashed

line: BLT design)

Figure 6.8 Step response and ISE values of multi-loop PI controllers for DL column (solid line: I

line: BLT design)

Figure 6.9 Step response and ISE values of multi-loop PI controllers for A1 column (solid line: IMC design; dashed

line: BLT design)

Figure 6.10 Step response and ISE values of multi-loop PI controllers for A2 column (solid line: IMC design; dashed

line: BLT design)

203

6.6 OPTIMAL CONTROL DESIGN

The main objective of the present work is to capture the disturbance

dynamics thereby measure the interaction in terms of area under the closed

loop undesired response. The goal is to minimize the interaction by using

medium-scale algorithm with termination tolerances for step and objective

function in the order of 0.001. The optimization gives the solution for

proportional and integral gains (KC, KI) of the controller after 73 function

evaluations.

6.6.1 Parameter Optimization

The performance of closed-loop system is measured with single

scalar quantity performance index. Performance of each loop needs to be

better or improved at the cost of spending optimal/ minimum inputs. Thus the

problem is to formulate performance criteria that will lead to find optimal

solution of manipulated inputs. It needs to determine control configuration

selection and the free parameters of controller that optimizes the performance

index. A linear 2-by-2 MIMO process is considered to formulate

performance criteria. The optimal controller parameters found from the

solution will be used to retune the loops so as to minimize loss through

undesirable response and unnecessary disturbance through undesired loops.

A commonly used performance criterion is the area under the

regulatory response which is given by:

20

J y t dt (6.16)

This criterion has good mathematical track ability properties which

is acceptable in practice as a measure of system performance. Smaller the

204

value J results in small overshoot in the system. Since, the integration is

carried out ov all error lasting for

long time and thus results in small settling time. Also, a finite J implies that

the steady state error is zero.

Therefore, a more realistic performance index is of the form:

20

J y t dt subject to the following constraint on control signalu t , max

u t M for some constant M M determined by the

linear range of plant.

2u t is a measure of instantaneous rate of energy expenditure. To

minimize energy expenditure: 2

0

u t dt .

To replace the performance criterion the following quadratic

performance index as:

2 2

0

J e t u t dt (6.17)

To allow greater generality, a real positive constant can be

inserted to obtain the performance criterion J.

2 2

0

J e t u t dt (6.18)

By adjusting the weighting factor , one can weight the relative

importance of the system error and the expenditure in energy. Increasing the

by giving sufficient weight to control effort, the amplitude of the control

205

signal which minimizes the overall performance index which may be kept

within practical bounds although at the expenses of increased system error.

In this work, the main intention is to show the amount of interaction

obtainable in area calculation of closed loop undesirable/ regulatory

responses. For a general MIMO systems cost is minimized by changing the

input control signal, u. The control pairing which needs the least area to

fulfill its control targets will be the most efficient control pairing.

The design approach based on parameter optimization consists of

the following steps:

Step 0 : Compute the performance index (J) as a function of free

parameters k1, k2 kn, of the system with fixed configuration.

1 2 nJ J , ...k k k (6.19)

Step 1 : Determine the solution set ki of the equations

i

J 0; 1,2...nk

i (6.20)

Equation (6.17) gives the necessary conditions for J to be

minimum. From the solution set of these equations, find the

subset that satisfies the sufficient conditions which requires

that hessian matrix given is positive definite.

2 2 2

21 2 1 n1

2 2 2

22 1 2 n2

2 2 2

2n 1 n 2 n

J J J....

J J J....

. . . .

J J J....

k k k kk

H k k k kk

k k k k k

206

Since, 2 2

i j j i

J Jk k k k

the matrix H is always symmetric.

Step 2 : If there are two or more sets of ki satisfying the necessary and

sufficient conditions for minimization of J, then compute the

corresponding J for each set. The set which gives the smallest

J is optimum set.

In this work, the performance of the control system can be

adequately specified in terms of settling time, overshoot and steady state

error. Thus, the performance index can be chosen as

J k1 (settling time) + k2 (overshoot) + k3 (steady state error) (6.21)

In this work, the performance index which includes the undesirable

system characteristics and in addition good mathematical track ability are

presented. The performance indexes often involve integrating closed loop

regulatory response when the system is subjected to a standard disturbance

such as a step. The system whose design minimizes the selected performance

index on controller configuration is the optimal.

6.6.2 Monitoring of Closed Loop Undesired Responses and Redesign

of Controllers

The optimization toolbox routine offers a choice of algorithms. For

constrained minimization, minimax, goal attainment and semi-infinite

optimization, variations of sequential quadratic programming are used.

Nonlinear least squares problem uses the gauss-newton and levenberg-

marquardt methods.

To optimize the control parameters in simulink model optsim.mdl

(This model can be found in the optimization toolbox optim directory) which

207

includes nonlinear process plant (Equation 2.11) modeled as a simulink block

diagram. The problem is to design a feedback control law that tracks unit step

input to the system. One way to solve this problem is to minimize the error

between the output and the input signal. The variables are parameters of PI

controllers. The routine lsqnonlin is used to perform a least squares fit.

The function tracklsq must run the simulation. The simulation can be run

either in the base workspace or in current workspace. To run the simulation

in optsim, the variables KP, KI, a1 and a2 (a1 and a2 are variables in the plant

block) must be defined. KP and KI are the variables that are optimizing here.

The simulation is performed using fixed-step fifth-order method to 100

seconds.

The controller settings before and after optimization are tabulated

in Table 6.2:

Table 6.2 Controller parameters and area under undesired response for

coupled tanks system

Controller parameters

Before optimization

After optimization

ck 28.9520 41.9656

I 1.4476 0.0215

Area Before optimization

After optimization

2793.2 2655.2

With these controller settings, the closed loop undesired responses

before and after optimization is shown in Figure 6.11.

208

Figure 6.11 Closed loop undesired responses

6.6.3 Performance measure

Control effort calculated using this objective function is optimum

that saved energy. Thus, for the coupled tanks system gain optimum is

calculated as:

Gain optimum = 2793.2 2655.2 *100 4.95%2793.2

Optimum gain in this coupled tanks system is 5% when the

disturbance is 10%, thus the control effort is calculated using this objective

function which is found to be optimum that saved energy. Hence, the

optimized control signal has saved 5% of utility in this particular coupled

tanks system.

209

The performance is measured using standard deviation and

variances are shown in Table 6.3.

Table 6.3 Performance measure

6.7 SUMMARY

In this chapter multi-loop PI controllers are designed and tuned for

all 10 distillation columns for the cited examples. The control effort and

performance of closed-loop for all cases are evaluated here. It is found that

IMC-PI control tuning rule produces lower IAE with smoother responses.

Thus, it can be concluded that IMC-PI works better than BLT method.

Performance measure

Before optimization

After optimization

Standard deviation

0.2397 0.1368

Variance 0.0574 0.0187

chapter 6 closed loop studies -...

Documents