A filter based approach for stochastic performance monitoring of feedback control systems

TABLE OF CONTENTS Acknowledgements i Table of contents iii Summary vi Nomenclature viii List of Figures x List of Tables xi Chapter 1 Introduction 1 Chapter 2 Review of Control L

ACKNOWLEDGEMENTS

I would like to express my sincere thanks and gratefulness to Dr Lakshminarayanan Samavedham for supervising my research activities His consistent encouragement, especially during my unproductive times, coupled with valuable practical ideas motivated me throughout my Masters’ program Under his friendly guidance, I have gained technical knowledge, self-confidence and maturity, which I feel is an important enhancement for me and keep on guiding me in my future endeavors Further, I am also thankful to him for providing me an opportunity to be a tutor for PDC (Process Dynamics and Control) course, which was truly a fantastic learning experience for me In addition to the technical stuff, I enjoyed non-technical discussions with him, filled with humor, in canteen, in cricket field, in departmental corridors and in group parties I am indebted to him for improving me so much

I am also thankful to all the DACS group members for inspiring me in some way or the other Kyaw’s passion for programming was inspiring enough to make me learn basics of GUI programming Prabhat’s technical knowledge always helped me - in fact, ideas generated during this research work are influenced by his thesis work Madhukar deserves special thanks, for helping me during my initial days in singapore and later for teaching me basics of Matlab I always wondered how somebody can be

so consistent and hardworking - I would like to be as steady as him in my future I am also thankful to Dharmesh for giving me technical as well as non-technical advices Ramprasad and May Su were wonderful batch mates - both of them helped me a lot Finally, I like to thank our new group members Rohit and Balaji for their cooperation Balaji’s company, during my thesis writing, has given me a tremendous amount of enthusiasm

Pavan deserves special gratitude as he guided me throughout from the application stage to the joining stage to NUS I am very much thankful to Murthy, Anand, Ganesh and Ramkrishna for providing me wonderful memories, especially tennis playing and swimming, as my flat mates I feel lucky to have friends like Murthy and others I will always relish the friendliness and affection I received from Arul, Mohan, Lalitha, Suresh, Raj, Biswajit, Shanthi, Ravi, Ashwin, Khare, Anju, Atreyyee, Pavan, Naveen, Manish, Manoj, Bhupendra, Vipul, Nirantar, Saurabh, Rahul, Sudhakar, Srinivas, Reddy and many more

I like to express a deep gratitude to my family, especially to my father, my mother and

my brother, without their love and efforts I would have never made up to this point

My wife, Ritika, deserved special thanks for her love and support, help in thesis writing and patience during the many hours I spent in front of the computer

Finally, I like to acknowledge National University of Singapore for providing me financial support and excellent facilities during this study

Far better it is to dare mighty things,

even though chequered by failure,

than to dwell in that perpetual twilight

that knows not victory or defeat

……… …… Theodore Roosevelt

TABLE OF CONTENTS

Acknowledgements i

Table of contents iii

Summary vi

Nomenclature viii

List of Figures x

List of Tables xi

Chapter 1 Introduction 1

Chapter 2 Review of Control Loop Performance Monitoring 6

2.1 Introduction 6

2.2 History of Control Loop Performance Monitoring 7

2.3 Minimum Variance Benchmark 11

2.3.1 Minimum Variance Estimation: SISO Case 11

2.3.2 Minimum Variance Estimation: MIMO Case 15

2.4 Limitations of MVC Benchmark 22

2.5 Control Loop Performance Monitoring: Present Status and

Challenges 23

Chapter 3 Calculation of the PI Achievable Performance 25

3.1 Introduction 25

3.2 Existing methods for PI achievable performance calculation 26

3.2.1 ASDR technique 26

3.2.2 PI achievable performance calculation from closed loop data 28

3.3 A New filter based method for PI achievable performance calculation 29

3.3.1 Central idea 29

3.3.2 Theoretical development 30

3.4 Case Studies 32

3.5 Conclusions 38

Chapter 4 PI Achievable Performance Calculation: Multiloop Case 39

4.1 Introduction 39

4.2 PI achievable performance calculation for multiloop control

systems 40

4.2.1 Multiloop control systems 40

4.2.2 Theoretical development 41

4.3 PI achievable performance calculation with alternate pairing 43

4.3.1 Pairing concerns in multiloop control systems 43

4.3.2 Methodology 46

4.4 PI achievable performance calculation with decouplers 47

4.4.1 Decouplers 47

4.4.2 Methodology 48

4.5 PI achievable performance calculation with a centralized control scheme 50

4.5.1 Centralized control 50

4.5.2 Theory 51

4.6 Case studies 52

4.7 Conclusions 66

Chapter 5 Variance Tradeoff Issues in PI Achievable Performance Assessment 67

5.1 Introduction 67

5.2 LQG Benchmark 68

5.3 PI achievable performance assessment: Variance Tradeoff Curve 69

5.3.1 Theoretical development: SISO case 70 5.3.2 Theoretical development: MIMO case 70

SUMMARY

The chemical industry has long recognized process control as a key enabling technology to improve its safety record and ensure its economic viability In today’s quality conscious market environment, product variability must be kept to as minimum levels as possible in order to generate sustained economic benefits for the shareholders Chemical process industries rely on instrumentation and control technologies to meet these needs Consequently, there is a demand for good performance of the control loops in a plant (typically consisting of several thousand control loops) on a continuous basis Over the past decade, academia and industry have focused on the topic of control loop benchmarking and monitoring Great strides have been made in the assessment of control loop performance using the minimum variance controller as the benchmark In a scenario where more than 95% of process controllers belong to the PID family, this yardstick is unrealistic The performance index of an industrial controller must really be based on the best performance obtainable from this class of controllers (controllers of reduced/restricted complexity) This work deals with the performance assessment and enhancement of regulatory level PI controllers for SISO as well as MIMO (multiloop) control configurations The benchmark employed is the best performance achievable with PI controllers

For realistic benchmarking of PI control loops, a filter based method is introduced for the calculation of best achievable performance by a PI controller In this filter based framework, the variance tradeoff between output variance and the variance of the manipulated variable is also considered The benchmark standard and the

consideration of variance tradeoffs make this work very relevant for the chemical process industry

Industrial process control systems are configured as multiloop feedback controllers Therefore, the scope of our work was extended to the domain of multiloop control systems A methodology that facilitates the calculation of best PI achievable performance and the optimal multiloop PI settings for stochastic disturbance rejection

is provided in this thesis Often, multiloop performance is not acceptable even with the properly tuned controllers due to interactions To enhance multiloop performance, advanced control strategies are often employed The most important questions that comes to mind, before any kind of restructuring of control system, is whether the new control strategy will offer significant benefits than the current strategy and if the extent of performance enhancement can be quantified beforehand (i.e before making the change in the actual system) The filter based approach proposed here has been extended to answer questions such as: (a) performance enhancement possible with the alternate pairing scheme (b) benefits that will accrue through the employment of decouplers and (c) the performance achievable with the use of multivariable controller (as opposed to a multiloop controller) Further, tradeoff curve between output variance and control effort is generated for all the above stated control strategies The methods proposed here permit a critical analysis of the current system and leads to the selection of the best control strategy for the process

The developed theory is validated with realistic simulation examples

NOMENCLATURE

t

a :White noise sequence

d :Time delay (SISO); Delay order (MIMO)

D :Interactor matrix

F :First (d-1) parameters of closed loop impulse response sequence

FCOR :Filtering and Correlation analysis

G :Closed loop servo process transfer function (matrix for MIMO systems)

*

G :Decoupled closed loop servo transfer function

J :Index of linear quadratic objective function (cost function)

LQG :Linear Quadratic Gaussian

MIMO :Multiple input multiple output

MVC :Minimum variance control

N :Open loop disturbance transfer function

Q :Controller transfer function (matrix for MIMO systems)

*

Q :New controller, Optimal controller (matrix for MIMO systems)

R :Input weighting matrix

SISO :Single input single output

T :Open loop process transfer function (matrix for MIMO systems)

*

T :Decoupled open loop transfer function

T~ :Delay free process transfer function (matrix for MIMO systems)

LIST OF FIGURES

Figure 2.1: White noise driven closed loop system 12

Figure 3.1: Experimental data for example 1 33

Figure 3.2: Results with optimal controller for example 1 33

Figure 3.3: Experimental data for example 2 35

Figure 3.4: Results with optimal controller for example 2 35

Figure 3.5: Experimental data for example 3 37 Figure 3.6: Results with optimal controller for example 3 37

Figure 4.1: Stochastic disturbance rejection with various control strategies for example 1 57

Figure 4.2: Set point tracking with various control strategies for example 1 59 Figure 4.3: Closed loop experimental data for example 2 60

Figure 4.4: Stochastic disturbance rejection with various control strategies for example 2 64 Figure 4.5: Set point tracking with various control strategies for example 2 65 Figure 5.1: An example of a variance tradeoff curve 68 Figure 5.2: Variance tradeoff curve for example 1 72

Figure 5.3: Comparison of tradeoff curve obtained from exact G and identified G for example 1 73 Figure 5.4: Variance tradeoff curve for example 2 74

Figure 5.5: Comparison of tradeoff curve obtained from exact G and identified G for example 2 75 Figure 5.6: Variance tradeoff curve for example 3 76

Figure 5.7: Comparison of tradeoff curve obtained from exact G and identified G for example 3 77 Figure 5.8: Comparison of tradeoff curve for various control strategies for example 4, obtained with exact G 78

LIST OF TABLES

Table 4.1: Comparison of results with exact G and Identified G for

example 1 56 Table 4.2: Comparison of results with exact G and Identified G for

example 2 61

CHAPTER 1

INTRODUCTION

In order to sell a product in a competitive market, it is necessary for a company that

its product must meet the demands of the consumer in a cost effective way One of

these demands is that product variability must be kept to a minimum and it has to

conform to consumer specifications Reducing variability will also enable process

economic optimization by operating closer to the constraints (Latour, 1986) The

constraint might be the optimum operating point that results in the lowest energy

consumption or lowest product giveaway, but beyond which the process should not be

operated for other reasons such as safety Thus, if variability is within acceptable

limits, it adds to the profits of the company (Shunta, 1995) by avoiding:

1 Storing off-quality product, which increases inventories and cycle time

2 Blending various off-spec products to meet consumer specifications

3 Selling at a reduced price

4 Disposing of off-grade product and other wastes

5 Reworking product, which reduces plant capacity, throughput, and energy

efficiency

There are many sources of variability within a manufacturing process Examples are

small, random variations in steam supply pressure, vibrations and turbulence, and

slight randomness in the composition of raw material Some of these sources can be

eliminated through proper equipment maintenance Some sources of variability such

as the physical and chemical properties of raw materials, which vary due to their

source and storage conditions, are unavoidable Once this variability is present, one of

the roles of the downstream process is to reduce this variability as much as possible

The extent of attenuation that can be achieved depends on two main factors:

1 Process design

2 The selection of control strategy, controller structure and controller tuning

Traditionally, process design is carried out based on steady state considerations only

without bothering about the dynamic controllability aspects of the plant Hence,

designs based on steady state aspects alone may result in a plant with unfavorable

static and dynamic characteristics These plants may be poorly controllable or even

uncontrollable Further, dead time dominant processes and processes with poles in

right half plane are examples of deficiencies in the process design which not only

adds to the product variability but also limits the performance of the control system

Hence, the process design is a more fundamental factor in determining product quality

than controller tuning For instance, a process design that unnecessarily contributes to

final product variability may represent a loss of disturbance attenuation that cannot be

recovered by even the most sophisticated control strategy Hence, it is essential to

consider the controllability aspects of a process during its design phase itself Lewin

(1996), Luyben (1993), Tseng et al. (1999) and many other researchers have worked

on the integration of design and control to achieve low variability designs

Control strategy, controller structure and controller tuning, collectively known as

control system design, will certainly have an impact on the attenuation characteristics

of the process and is the primary factor considered by process engineers to solve

variability problems once the plant is constructed and is in operation Different

processes and performance specifications require different control strategies,

controller structure and controller tuning parameters Feedback control is the most

common and widely used control strategy employed in the chemical industry

Feedforward control, cascade control, decoupling control, multivariable control,

model predictive control (MPC) are some of the advanced control strategies employed

by the control engineers as and when the situation demands higher “muscle power”

As far as controller structure is considered, PI or PID controllers are still the first

choice of industry because of their simple structure, easy implementation, low

maintenance requirements and high reliability as compared to other higher order

controllers Once the control strategy and controller structure is fixed for a process,

there are hundreds of methods to design the PI/PID controller i.e tuning the controller

parameters Discussions about various control strategies, controller structures and

methods for controller tuning can be found in many standard control textbooks e.g

Marlin (1995), Seborg et al (1999) etc Even though process design is a more

fundamental factor to be considered for achieving low variability, for an existing

plant, process engineers typically avoid process redesign / process modification owing

to the higher costs involved Their option is likely to be proper control system design

(controller structure, variable pairing and controller parameters)

In a typical process plant, where thousands of control loops are present, continuous

good performance of existing controllers is mandatory to maintain the variability

within acceptable limits, so as to meet the quality standards and to generate sustained

economic benefits This indicates the need of assessing control loop performance and

it has received a lot of attention from industry as well as academia This study is an

attempt towards the development of tools for performance evaluation and

performance enhancement of existing control loops As we understand, control loop

performance assessment is simply the evaluation of controller capability with

reference to some pre-defined benchmark in meeting the objectives defined for a

control loop Careful examination of the above definition reveals two keywords,

benchmark and controller objectives So far, research in this area has been governed

by these two keywords Chapter 2 provides an overview of the research

accomplishments in this field It discusses various performance assessment

approaches based on different benchmarks and different controller objectives At the

end of this chapter, a suitable benchmark, PI achievable performance, is identified

after comparing advantages and disadvantages of each benchmark, keeping in view a

particular controller objective In this study, stochastic disturbance rejection is

considered as primary controller objective i.e maintaining variability within

acceptable limits in presence of random disturbances, which is in accordance with the

control objectives in industry Chapter 3 deals extensively with the PI achievable

performance calculation for the single loop case It starts with a detailed explanation

of existing methods for PI achievable performance calculation After a summary of

existing methods, a new methodology is proposed for the PI achievable performance

calculation and is compared with the existing methods Various case studies are

shown to prove the utility of the proposed methodology Chapter 4 looks at the

multiloop PI control performance monitoring issues It begins with the theoretical

development for the extension of the proposed method to solve the PI achievable

problem for multiloop systems Later in the chapter, some of the key issues in the

field of multiloop performance monitoring are addressed These include:

(a) what would be the PI achievable performance with alternate pairing

arrangement?

(b) what performance improvement will accrue through the use of decouplers in

multiloop control systems?

(c) what will be the improvement in the performance with the use of a

multivariable controller?

All the above mentioned questions are answered based on the proposed method and

examples are presented to demonstrate the efficacy of the developed theory Chapter 5

introduces a new benchmark, LQG benchmark with PI structure limitation, which

considers both controller structure limitation as well as control effort in assessing a

controller’s performance In the last chapter, conclusions are drawn based on this

study A brief description of the problems evolved during this study is given, which

can be taken as recommendations for the future work in this area

CHAPTER 2

REVIEW OF CONTROL LOOP PERFORMANCE MONITORING

2.1 Introduction

Most modern industrial plants have hundreds and even thousands of automatic control

loops These are usually Proportional-Integral (PI) controllers, but can also include

more sophisticated model based linear or non-linear controllers The performance of a

process controller often changes during plant operation An initially well tuned

controller may become undesirably sluggish or aggressive due to many reasons, such

as changes in process gain, process dynamics, valve stiction or constraints,

sensor/actuator failure, equipment fouling, feed variability and seasonal influences A

controller with poor performance increases manufacturing costs, lowers product

quality and even risks process safety Therefore, monitoring controller performance is

important, and has become a routine task for process control engineers In practice,

many poorly performing controllers in plants go unnoticed for quite a long time

before being detected and corrected It has been reported that as many as 60% of all

industrial controllers have performance problems (Ender 1993; Bialkowski 1993)

The last decade has witnessed a plethora of techniques put forth for control loop

performance assessment Also, a lot of research efforts, academic as well as industrial,

are underway in this area owing to its direct relevance to practical industrial needs

The next section (section 2.2) gives an overview about the history of performance

assessment techniques Later in section 2.3, performance assessment with MVC

(minimum variance controller as benchmark) is covered in fair detail Both SISO as

well as MIMO control loop performance assessment is considered Section 2.4

presents the limitations of the MVC benchmark Finally, section 2.5 discusses the

industrial status as well as the research/implementation challenges in this area

2.2 History of Control Loop Performance Monitoring

Traditionally, the problem of assessing control loop performance is dealt with design

only when complete information about loop is available Mostly, step response data is

used for evaluating performance of control system i.e response to a step change in the

set point Various performance criteria such as rise time, settling time, overshoot,

decay ratio and offset, calculated from the step response data, are used for evaluating

a particular control system The rise time along with the settling time is a measure of

the speed of the response, whereas overshoot, decay ratio and offset are related to the

quality of response Further, any new controller, before implementation, is compared

with the existing controller in terms of its response to set point changes or load

disturbance changes Apart from these criteria, various statistical measures are also

used to monitor the performance Some of them are mean and variance of the process

output, number of constraint violations, percentage of time the controller is on closed

loop, number of operator interventions, frequency of alarms etc Another performance

index used relates the actual product variability to the specifications issued by the

consumer

The solution to the problem of performance assessment without any design

information about the loop was initiated by Astrom (1970), where he employed

autocorrelation plots from closed loop output data for performance monitoring A

poor controller can be identified if the autocorrelation is significant after the dead

time Later, Devries and Wu (1978) used spectral dispersion and spectral methods for

MIMO performance assessment Recent interest in this area was sparked when Harris

(1989) employed simple time series analysis to extract the controller invariant part of

variance from the routine operating data and used this minimum variance as a

benchmark for control loop performance assessment This contribution by Harris was

significant as it marked a new direction in this area and initiated considerable

academic research and industrial applications especially in the pulp, paper, chemical

and petrochemical industries

Stanfelj et al. (1993) used cross correlation analysis for feed forward plus feedback

control systems to diagnose the root cause of a poor performance Later Desborough

and Harris (1993), and Vishnubhotla et al. (1997) used analysis of variance

(ANOVA) for performance assessment of feed forward plus feedback control systems

of SISO processes MVC based performance monitoring was extended to

multivariable processes by Huang et al. (1997a) They proposed a filtering and

correlation (FCOR) algorithm for MIMO performance assessment Harris et al.

(1996) also extended MVC for MIMO performance assessment based on the

estimation of interactor matrix The issue of estimation of time delay from closed loop

operating data, which is mandatory for MVC based assessment, was addressed by

Lynch and Dumont (1996) Estimation of the interactor matrix (generalized delay

structure for MIMO process) using closed loop data was demonstrated by Huang et al

(1997b) A detailed explanation of MVC benchmark and its estimation from routine

data for both SISO and MIMO cases is presented in section 2.3 of this thesis

Several performance measures are proposed in literature based on the MVC

benchmark Some of them are:

(1) Ratio of output variance to the theoretical minimum variance

2

mv

2 y

) d (

1 ) d (

σ

σ

η = − Eqn – 2.2.2

(3) The extended horizon performance index, utilized by Desborough and Harris

(1993), Harris et al. (1996), Kozub (1997), Thornhill et al. (1999) etc

F F

F 1

F

F

F 1 1 ) h d

d 2 1 d 2

1

2 1 h d 2

1 d 2

1

++++

+

+++++

−

=+

−

− +

d

σ

σ

η = Eqn – 2.2.4

In this study, the CLPI will be employed as the performance metric A value of CLPI

close to zero implies poor controller performance while a value close to 1 indicates

the best possible controller performance (best refers to the lowest possible output

variance)

Eriksson and Isaksson (1994) pointed out that the MVC based performance index is

insufficient for deterministic performance monitoring Also, they initiated a need for

benchmark that takes PI structure of controller into consideration, as MVC based

variance is not achievable by PI/PID controllers when delay is significant or if the

disturbance is non-stationary Later, a more realistic performance measure called the

PI achievable performance was introduced by Ko and Edgar (1998) and an

industrially relevant methodology to estimate this performance index was developed

by Agrawal and Lakshminarayanan (2003) The PI achievable performance is covered

extensively in chapter 3

All the work discussed above is concerned with the stochastic performance

monitoring, as they deal with the assessment of control loop performance under

unmeasured, stochastic disturbances Few researchers have also addressed the issue of

deterministic performance assessment Shinskey (1994) proposed the use of time

delay to determine the best achievable deterministic performance Swanda and Seborg

(1999) used normalized settling time to evaluate the deterministic performance of PID

controllers based on set point response data The tradeoff between stochastic and

deterministic performance is well known i.e the two types of performances cannot be

achieved simultaneously There are certain advantages and disadvantages associated

with both approaches In this study, our focus is on stochastic performance monitoring

as we are primarily concerned with the control loops in the regulatory layer

Apart from the performance assessment based on the MVC benchmark, few other

approaches are also reported in literature Kendra and Cinar (1997) discussed the use

of frequency domain techniques for performance monitoring Tyler and Morari (1995)

developed a performance monitoring method based on likelihood ratios Rengaswamy

diagnosing different kind of oscillations in control loops MVC based benchmark is

useful as a first check in performance monitoring For higher level performance

assessment (even for the computation of PI achievable performance), a model of the

process is typically required The LQG benchmark is one such benchmark and is

relevant when both the input and output variances are of concern Patwardhan (1999)

used a comparison of the designed performance and the achieved performance to rate

the performance of model predictive controllers (MPC)

An excellent overview of the research in the area of control loop performance

monitoring based on MVC can be found in Harris et al (1999), Hoo et al (2003) and

Qin (1998) Huang and Shah (1999) provide a through treatment of this area

2.3 Minimum Variance Benchmark

The most fundamental limitation to the controller’s performance is delay, which

characterizes the control invariant part of output variance This control invariant part

of output variance is referred as minimum variance that would result if we employ a

minimum variance controller The minimum variance is the global lower bound on

the output variance, hence it can be used as a benchmark to assess any controller’s

performance by comparing the present variance with the minimum variance Harris

(1989) showed that the minimum variance bound can be estimated using time series

modeling Only routine closed loop operating data and knowledge of process delay is

required to estimate the minimum variance The estimation of minimum variance

using routine output data is presented in the next section

2.3.1 Minimum Variance Estimation: SISO Case

Consider a closed loop system driven by white noise as shown in Figure 2.1

Here, the process transfer function Tis separated into a pure delay part z-d and a delay

free transfer function T such that T = z−d T ~ The gaussian white noise signal a t is driving the closed loop system

Now, for the white noise driven closed loop system, the process output y t can written

as

Q

T ~ z 1

N a

Q T 1

N y

( )

t d

d d

d

Q

T ~ z 1

Q

T ~ z F R z Q

T ~ z 1 F y

d

Q

T ~ z 1

Q

T ~ F R z F y

−

Eqn – 2.3.4

Finally, the output expression can be separated into control variant and control

invariant part as shown in the following equation

d t t d t d

t

Q

T ~ z 1

Q T F R a

−+

As the polynomial F is independent of the controllerQ, it is termed as the controller

invariant part ofy t The polynomial L (in powers of z-1) represents the control variant

part of the output variance It is easy to note that if

F

T ~

R

Q= then Lequals to zero, in

which casey t = Fa t This choice of Q gives us the Minimum Variance Controller (MVC) If the MVC is used to regulate the process, the process output has the

minimum possible variance The output variance under MVC can be written as

a 2 1 d 2

1 2 0 MVC , t 2

Now, any other choice of Q will result in a higher output variance than minimum

variance as we have L≠0 and outputy tis equal to

y t =F a t+L a t−d ; L≠0 Eqn – 2.3.7

Under this situation, expression for output variance is given by

a 2

1 2 0 2 1 d 2

1 2 0 t 2

Any controller other than the MVC will result in an output variance σ which is 2y

greater than or equal to the minimum variance 2

mv

σ The minimum variance controller thus fits well as a theoretically sound benchmark for the performance assessment of linear time-invariant feedback controllers The closed loop performance index η, using MVC as benchmark can be given as

( ) 2

y

2 mv

L L F

F F

F

F F )

d

1 2 0 2 1 d 2

1 2 0

1 d 1

0

+++++

+

++

With this definition of the control loop performance index (CLPI), it is evident that a value of η( d ) close to 1 will indicate that the current controller is performing as well

as the minimum variance controller It is impossible to obtain better performance (in terms of reducing the output variance) via loop retuning Further, a value of η( d )

close to 0 will indicate that there is a possibility for performance enhancement by retuning the existing linear feedback controller If loop retuning does not result in improved performance, extra attention must be paid to that particular loop - one must investigate various issues such as control system restructuring, sensor and actuator relocation, maintenance of hardware or even a process modification/ process redesign

2.3.2 Minimum Variance Estimation: MIMO Case

This section presents the multivariate control loop performance assessment using minimum variance as benchmark Again the key idea is the estimation of control invariant part from the routine data using multivariate time series analysis We have seen that controller performance evaluation, based on the minimum variance benchmark (MVC), requires knowledge of the delay free part of the process transfer

function For a SISO system, the transfer function T may be represented as the ratio of polynomial A and B (in z-1) and the delay component z-d

1 d

1 1

z ) z ( B

) z ( A ) z (

−

− = Eqn – 2.3.11

The delay free part of the transfer function T ~ can be obtained by multiplying the

transfer function T with scalar D= z di.e

B

A DT

T~= = Eqn – 2.3.12

In the MIMO case, this D is no longer a scalar, but takes a matrix form, known as the

“interactor matrix” Multiplication of transfer function matrix T by the interactor matrix D , removes the infinite zeros from the original transfer function matrix T and yields the delay free transfer function matrixT ~ The interactor matrix has several applications in control theory, for example, it is an important entity in solving the multivariable minimum variance control problem Also, it is an important prerequisite for the evaluation of control loop performance for MIMO systems as it facilitates the evaluation of control invariant part

The interactor matrix, proposed by Wolovich and Falb (1976), had a lower triangular form With this form of interactor matrix, the minimum variance law (Goodwin and

Sin 1984; Dugard et al 1984) is not unique but output order dependent Later Shah et

al. (1987) pointed out that interactor matrix can take an upper triangular form or a full

matrix form Rogozinski et al (1987) proposed an algorithm for the calculation of the

nilpotent interactor matrix from the matrix of coefficients of the numerator of right matrix fraction (RMF) description of the system Peng and Kinnaert (1992) found unitary interactor matrix as a special case of nilpotent interactor matrix The unitary

interactor matrix is found very useful for MIMO performance assessment using minimum variance control as benchmark, as factorization of interactor matrix doesn’t change the spectral property of the system Further, minimum variance law is found to

be unique, when derived with the unitary interactor matrix Huang et al (1997b) have suggested factoring the unitary interactor matrix directly from the first d markov

parameter matrices of the process using closed loop data with dither excitation

For any m × proper, rational transfer function matrix T , the interactor matrix D is n

where K is a full rank matrix i.e rank ( K )= min( m , n ) The integer r is the number

of infinite zeros of T The interactor matrix D can be written as

D=D 0 z d +D 1 z d−1 +D d−1 z Eqn – 2.3.14

where d is the delay order for a given transfer function matrix It is worth mentioning

that for a given transfer function matrix T , the interactor matrix D can be

non-unique but the delay order d is always unique

The interactor matrix can be of following types:

• Simple Interactor - If D is of the form D=q d I (where I is the identity matrix), then the transfer function matrix T is regarded as having a simple

interactor matrix

• Diagonal Interactor – If D is diagonal, i.e D= diag ( z d 1 , z d 2 , , z n ), then

the transfer function matrix T is regarded as having a diagonal interactor

matrix

• General Interactor – If D does not take any of the above forms, then it is

known as general interactor matrix It can be a lower triangular or an upper

triangular or a full matrix

One realization of general interactor matrix is the unitary interactor matrix, which

satisfies the following property

D T ( z−1 ) * D ( z )=I Eqn – 2.3.15

Now, the feedback control invariant part of the output response can be estimated with

the routine output data as follows Consider the simplest form of a multivariable

process - a square process transfer function matrix with a simple interactor matrix If

the multivariable process is represented by the following equation

J = Eqn – 2.3.17

As we are considering the process with simple interactor matrix, the transfer function

matrix can be written as

T =z−d T ~ Eqn – 2.3.18

Here T ~ is the delay-free transfer function matrix Substituting equation 2.3.18

into equation 2.3.17 yields

Y t = z−d T ~ U t +Na t

Eqn – 2.3.19

Consider the feedback control law given byU t =−QY t The closed-loop transfer function is then determined by

1

F

F = + − +⋅ ⋅+ − − − and R is the remaining proper and rational

transfer function matrix Substituting equation 2.3.21 into 2.3.20 gives

( ) ( ) t

d d

1 d

d d

d d

t z I z T ~ I z Q T ~ Q z z F z R a

1 d d

t d t 1 d d

t d

1 d 1 d 1

1 0 t mv

E ≥ Eqn – 2.3.26

The equality holds under minimum variance control i.e whenL= The minimum 0

variance control law is therefore obtained by simply settingL= The resulting 0

controller transfer function, U t =−QY t is determined by

NR I z

Therefore, if a closed-loop response under feedback control is modeled by a multivariate moving-average process as

44

44

14444

4444

1

iant var control

d t d t iant

var in control

d t d t

y mv t

T t

min t t

trace

trace ]

Y Y [ E

] Y Y [ E iance var actual

iance var imum min ) d

Now, consider the process with general unitary interactor matrix

Y t =TU t+Na t = D−1 T ~ U t +Na t Eqn – 2.3.30

Multiplying both sides of above equation by z−d D yields the following equation

t d t d t

and performance measure for can be calculated for interactor filtered outputY ~ t Huang and Shah (1999) showed that if D is a unitary interactor matrix, then the minimum variance control law, which minimizes the objective function of the interactor filtered output Y ~ t

[ t]

T

t Y ~

Y ~ E

effect completely just after the process delay has elapsed Such large moves are undesirable because of their detrimental effect on actuators Hence, due to its excessive moves and poor robustness properties, MVC is usually not employed in the industry Further, as most of the industrial controllers are of PI type, the benchmark to

be used for industrial controller’s performance assessment must take into account this controller structure limitation As MVC doesn’t account for controller structure limitation as well as variation in the input, it is not considered as a pragmatic benchmark

Considering the fact that 95% of the industrial control loops belong to the PID family, Eriksson and Isaksson (1994) recommended use of PI achievable performance as a benchmark Ko and Edgar (1998) developed ASDR technique to determine PI achievable performance calculation using known open loop process model and routine data Later, Agrawal and Lakshminarayanan (2003) proposed a method to obtain PI achievable performance using closed loop experimental data The PI achievable performance benchmark is discussed in depth in the next chapter

2.5 Control Loop Performance Monitoring: Present Status and Challenges

Today, there is a strong interest in assessing control loop performance as industries are forced to push the limits of their performance further and further Commercial software such as ProcessDoc (1997; developed by Matrikon consulting Inc.), LoopScout by Honeywell Hi spec solutions (Minneapolis, MN) etc are able to provide automated control loop performance monitoring and diagnostics All these tools are useful for continuous monitoring of loops, diagnosing and resolving

regulatory control problems and re-tuning loops for optimal performance Apart from these commercial softwares, some chemical companies like Eastman Chemicals and DuPont have developed in-house monitoring tools (reported in Hoo et al. (2003)) In these applications, MVC is the most widely used performance benchmark Various researchers (Kozub 1997; Thornhill et al 1999; Haarsma and Nikolau 2000) also reported successful industrial case studies based on the MVC benchmark

There has been a significant development in theory as well as in practice in the field

of control loop performance monitoring over the last decade Even after the successful implementation of various industrial applications, there are a lot of issues to be explored in this area Some of the areas that deserve attention include root cause analysis of the low performance loops, detection and isolation of plantwide oscillations, extension of MVC to nonlinear and time variant systems and benefit analysis with control system redesign/restructuring Also, optimization and identification strategies need to be reinforced to make performance monitoring more fruitful

in the industry Desborough and Miller (2001) surveyed the status of controllers employed in the chemical industry and concluded that a typical chemical plant has 98% PID type controllers and a vast majority of these controllers are PI controllers There are many reasons why industry prefers PI controllers over high complexity controllers These include familiarity, ease of design and maintenance etc However, due to the inherent controller structure limitation, no matter how much tuning effort is applied to the PI/PID controllers, their performance cannot match with the performance given by minimum variance controller when the process is delay dominant or if the process is subject to non-stationary disturbance dynamics Moreover, very little percentage of industrial PI controllers provide performance equivalent to that of the minimum variance controller Hence, using MVC benchmark for performance monitoring of PI controllers is not only inappropriate but also misleading as it may lead to inaccurate diagnosis of the factors causing low performance For instance, it may happen that a PI controller is performing up to its potential i.e giving maximum performance that can be achieved with a PI controller,

1

A shorter version of this chapter was presented as “A filter based approach for estimation of PI

but at the same time, when adjudged with reference to MVC, its performance may appear low The above mentioned situation can lead to a situation where the control engineer keeps tuning the loop without any success Therefore, it will be better if the

PI controllers are assessed against PI achievable performance rather than the MVC performance

PI achievable performance, in terms of CLPI (performance measure defined in section 2.3), can be defined as maximum value of CLPI that can be obtained by restricting controller structure to PI type With this definition, it is evident that the controller structure limitation is taken into account in assessing the controller performance - this makes it a more pragmatic benchmark

The organization of this chapter is as follows: In section 3.2, a brief overview of existing methods for PI achievable performance calculation is presented Section 3.3 contains theoretical development of a new methodology for PI achievable calculation The proposed methodology is also compared with the existing methods Various simulation examples are presented in the section 3.4 to showcase the usefulness of the proposed method followed by the conclusions in section 3.5

a PID type controller In ASDR method, assuming open loop model is available, routine data is employed to calculate the PI achievable performance A brief explanation of ASDR method is given below

Assuming linear time invariant process (T ) and noise dynamics ( N ), the output can

be represented as

yt = Tut + Nat Eqn – 3.2.1

wherey t, u tand a t are respectively the process output, input and white noise sequences When there is no set point change, the closed loop output is given by the following equation

as follows:

N = 1( +TQ)H Eqn – 3.2.3

Once, the process and noise models are known, Ko and Edgar (1998) employed a numerical optimization procedure to estimate the highest control performance index reachable by restricting the feedback controller Q to a PI or PID form Usually, open loop models are not available; in such cases, the method presented in next section requires no open loop model of the plant will be useful for practical applications

3.2.2 PI Achievable Performance Calculation From Closed Loop Data

Process models are rarely available in the chemical industry Desborough and Miller (2001) estimate that dynamic process models are available only for about 1% of chemical process In such a scenario, the demands placed by the method of Ko and Edgar (1998) are too much to be of real use in the industry Agrawal and Lakshminarayanan (2003) proposed an alternate way of determining the PI achievable performance from closed loop experimental data (set point excited data) Their method uses identified closed loop process and disturbance models The relationship between the controlled variable and the set point under closed loop is given by the following equation

TQ 1

N y

TQ 1

TQ

+

++

Here, G is the closed loop servo response model From equation 3.2.4, we can write

Q ) G 1 (

G T

-= Eqn – 3.2.5

) G 1 (

H N

-= Eqn – 3.2.6

Assuming time invariant process T and noise dynamics N, for a new controller Q *

the closed loop impulse response H * is given by

=

−+

−

=+

=

1 Q

Q G 1

H Q

) G 1 (

G Q

1

) G 1 ( H

T Q 1

N H

Recall that the CLPI (control loop performance index) can be obtained from the

estimated closed loop impulse response H if the process delay dis known Equation

3.2.7 implies that with the knowledge of the current closed loop impulse response H ,

closed loop servo transfer function G and the current controller Q, it is possible to estimate the closed loop impulse response H * for any given controllerQ * Given that

the process delay d remains constant, it is possible to determine the optimal PI type

controller Q * that maximizes the performance Hence, the PI achievable control loop performance can be computed from the knowledge of the current controller and current closed loop servo and disturbance transfer functions Agrawal and Lakshminarayanan (2003) demonstrated the workability of the above scheme using several examples They also ensured that deterministic control loop performance measures like the normalized integral absolute error, gain and phase margins are also within acceptable limits

3.3 A New Filter Based Method For PI Achievable Performance Calculation

3.3.1 Central Idea

It has been shown that CLPI may be calculated by extracting the closed loop impulse coefficients from routine closed loop data via time-series modeling Now, the PI achievable performance calculation is essentially an optimization problem i.e starting from CLPI calculation for any initial control settings, its calculation requires continuous estimation of CLPI for various controller settings until we reach the optimal PI settings which maximizes the CLPI In the proposed method, the key idea

is to derive a filter which can provide routine data corresponding to any new controllerQ * - this will facilitate calculation of the CLPI for any controllerQ * Hence, incorporation of this idea within an optimization routine enables us to calculate PI achievable performance Further, derivation of the filter requires knowledge of