Helsinki University of Technology Control Engineering Laboratory Espoo 2000 Report 121

FUZZY GAIN SCHEDULING OF MULTIVARIABLE PROCESSES Matteo Cavazzutti

TEKNILLINEN KORKEAKOULU TEKNISKA HÖGSKOLAN HELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D´HELSINKI

Helsinki University of Technology Control Engineering Laboratory Espoo September 2000 Report 121

FUZZY GAIN SCHEDULING OF MULTIVARIABLE PROCESSES 1 Matteo Cavazzutti "La Sapienza" University of Rome Engineering Faculty Electronic Engineering Department Abstract: The intense and increasing interest of industry on Automation Technologies claimed

deep researches in this field. Emerging techniques like Fuzzy Logic has been studied and

successful applied to process control.

Traditional control methods, e.g. PID-based, have poor performances when applied to modern

processes whose models are strongly non-linear and multivariable-based. Fuzzy logic controllers

might provide better results through advanced methodologies like gain scheduling, for example.

In this thesis, multivariable non-linear systems have been considered. To perform the control of

the process, multivariable PI controller has been used. A fuzzy gain scheduler has been

developed to allow the controller producing the optimal control action at different operating

points. An algorithm has been used to execute the off-line tuning in each of operating point, and

a fuzzy sub-system performs a fine tuning.

In that way the system works as follow: as changes in operating points are detected, the

scheduler, which behaves as a supervisor, sets new parameters for the controller. Then, the

fine-tuning provides the performances to achieve the desired results. The parameters of the gain

scheduler are stored in a fuzzy system, based on linearization at different work points of the

process. Thus a mathematical model of the process is not required, and necessary data for

process study could come simply from on-field experiments.

An example has been considered and the proposed method has been applied on it.

Keywords: fuzzy gain scheduling, multivariable control, non-linear control, PI controller, fuzzy

logic

Helsinki University of Technology

Department of Automation and Systems Technology

Control Engineering Laboratory

1 A re-edition of a Master’s Thesis work, September 2000.

Distribution:

Helsinki University of Technology

Control Engineering Laboratory

P.O. Box 5400

FIN-02015 HUT, Finland

Tel. +358-9-451 5201

Fax. +358-9-451 5208

E-mail: control.engineering@hut.fi

ISBN 951-22-5281-3

ISSN 0356-0872

Picaset Oy

Helsinki 2000

Acknowledgements The thesis has been entirely developed in the Control Engineering Laboratory of

the Helsinki University of Technology. I want to express my personal gratitude to Professor Heikki Koivo for his proficient

support during my work. His marvelous leadership, oriented to build cooperation and friendship within the Laboratory Staff, has represented always an excellent reference in my improvements.

I also wish to thank all the Control Engineering Laboratory Staff for their

acquaintance and for the motivating work environment in which I found myself working.

I am deeply thankful to Professor Alessandro De Carli from the Computer and

System Science Department of “La Sapienza” University of Rome, who gave me the possibility to experience this work.

Finally, but not at least, I want to thank my family for their extraordinary

encouragement and patience in bearing me. I cannot avoid spending a few words about the country in which I have lived for six

months: Finland. Looking on the globe somebody might see it’s not so far from my own country, Italy. But between those nations people and habits, as it appeared to me, are in general quite dissimilar. I had an unforgettable life experience because just the different culture I discovered here made me growing-up. The day I will leave (“I left” to the reader) Finland will be (“was”) sad and pleased at the same time: “Sad” because I will leave friends, habits that I successfully enjoyed, and my own life I built day by day. “Glad” because I will keep the memory of all the life teachings I learnt here.

Helsinki, September 2000 Matteo Cavazzutti

Contents

Chapter 1 _________________________________________________________ 1

Introduction ____________________________________________________ 1 1.1 – Historical Background ______________________________________ 1 1.2 – Gain Scheduling and multivariable processes_____________________ 2 1.3 – Thesis organization _________________________________________ 4

Chapter 2 _________________________________________________________ 5

Basics on Fuzzy Logic ____________________________________________ 5 2.1 – Looking over fuzzy world____________________________________ 5 2.2 – Fuzzy sets ________________________________________________ 6 2.3 – Fuzzy operations and relations ________________________________ 7 2.4 – Fuzzy rules _______________________________________________ 9 2.5 – An example: washing machine fuzzy control system ______________ 12

Chapter 3 ________________________________________________________ 16

PID control ____________________________________________________ 16 3.1 – General aspects of PID controller _____________________________ 16 3.2 – Implementation ___________________________________________ 18

Chapter 4 ________________________________________________________ 20

Fuzzy controller design __________________________________________ 20 4.1 – Overview ________________________________________________ 20 4.2 – Parameters design _________________________________________ 21 4.3 – Reliability _______________________________________________ 25

Chapter 5 ________________________________________________________ 26

Multivariable feedback control____________________________________ 26 5.1 – Generalities ______________________________________________ 26 5.2 – Problem definition_________________________________________ 27 5.3 – Pettinen-Koivo tuning method _______________________________ 28

Chapter 6 ________________________________________________________ 31

Gain Scheduling ________________________________________________ 31 6.1 – Overview ________________________________________________ 31 6.2 – Problem definition_________________________________________ 33

Chapter 7 ________________________________________________________ 36

Case study _____________________________________________________ 36 7.1 – Presentation ______________________________________________ 36 7.2 – The process ______________________________________________ 38 7.3 – The fuzzy gain scheduler ___________________________________ 40 7.4 – The fuzzy fine-tuning ______________________________________ 41 7.5 – Trial and error: manual adjustments ___________________________ 46 7.6 – Simulation and results______________________________________ 47

Chapter 8 ________________________________________________________ 51

Conclusions ____________________________________________________ 51

Bibliografy _______________________________________________________ 52

1

Chapter 1

Introduction

1.1 – Historical Background Although advanced controllers are currently available in process control, most

industrial methods utilize well-known proportional, integral, derivative (PID) controllers. It has been estimated [33], that in 1996 at least 90% of controllers employed in the industry are PIDs or PID-based. The success of PID controllers is due to the following simple reasons:

Its straightforward structure makes relations between its parameters and the system

response to be clear. This means that the plant operators acquire a deep knowledge about how to tune controller parameters.

The tuning task is nowadays fairly straightforward and deterministic. Popular tuning algorithms are widely used, e.g. Ziegler-Nichols or Internal Model Control (IMC) method.

The overall performance obtained is, generally speaking, considered satisfactory with respect to the relative quickness and low cost of implementation.

In recent years new alternative methods have been explored, like, for example,

genetic algorithms, fuzzy logic and neural networks. They have been introduced successfully into control theory in many different ways. For example, a large field is oriented in projecting the so-called PID-like controllers. They can be considered implementations of standard PID, using those different mathematical methods. Other studies investigated supervising PID-controlled systems, introducing techniques like self-tuning and scheduling. The aim is to continue using simple PID- based controllers, since they are demanded by industry, but supporting them with more advanced control strategies.

All those investigations set sights on improving controllers where PID shows lack of acceptable performance. For example, when one has to deal with strong process non-linearities, or when switching between several operating points, as this could lead to the necessity of controller tuning at each step. Moreover, in some industrial processes it is not possible to use conventional control methods, because of the scarcity of a sufficient number of data regarding the input-output plant relations. In

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multivariable system, additionally, experience shows that PID controllers, by themselves, are not sufficient to perform the control action.

In 1965 Zadeh introduced the concept of fuzzy logic for the first time [47]. Nine

years later the first Fuzzy Logic Controller (FLC) was introduced by Mamdami [19]. From that time FLC technique, that might be defined as “rule-based control theory able to emulate the human expert’s knowledge” [27], has been used in a significant amount of industrial applications, like for instance, in paper machine control [42], water quality control [6], transmission control [6], elevator control [8]. Studies and researches in fuzzy control have been growing up quickly, as the interest of researchers and scientists have been turned more and more on that subject.

The success of FLC is connected to the undeniable advantage of providing a way of converting a linguistic control strategy, based on human expert skill, into an automatic control strategy.

In fact fuzzy logic ability is in connecting human nature and mathematical models. In that sense, the core of a FLC is a set of linguistic rules, built with the simple consequence if-then logic connected by implications and inferences. Linguistic rules arise from the practice, the knowledge and the know-how of expert operators and designers, who do not form mathematical models for decision-making and control.

Long lasting experiences show that the FLC frequently yields better results than those gained by conventional controllers. At not last, personnel supposed to handle with FLC, is usually not required to have high educational background.

On the other hand, a designer has to take into consideration that there are not yet

systematic and straightforward procedures to manage the development of a FLC. The most commonly used ways to proceed are still founded on trial-and-error methods. Furthermore, usually fuzzy systems contain redundancies, that is, a FLC has more parameters than the conventional equivalent; and this could introduce more trouble in managing all of them than in the other case. Since the number of rules grows quickly with the number of inputs, inputs variables have to be restricted, say to two or three.

The performance of fuzzy systems is strictly related to the ability and experience of the designer. As a result, stability and robustness of FLC are not achieved automatically and they need usually an ad hoc investigation.

1.2 – Gain Scheduling and multivariable processes Since realistic models of engineering systems are nonlinear, the dynamic behavior

of the system to be controlled changes with the operating point. A common approach to this problem is to apply the notion of gain scheduling.

Gain scheduling was introduced during the 1950s with particular reference to flight control systems [23]. It can be considered a kind of adaptive control technique: the original idea was to evaluate auxiliary variables (other than the plant outputs) that are strongly correlated with the changes in process dynamics [34]. In such a way there is no need of system parameter estimation; thus control variables can be changed as soon as changes in system dynamics are observed by the auxiliary measurement. This means

3

that change of the controller parameters depends on how quickly the operating conditions are measured.

The conventional scheduling procedure is first to linearize the system model about different operating points. Then linear design methods are employed to the linearized models in order to compute local controller parameters, as they are supposed to work just within linear regions. The last step is the development of the real gain scheduler: it has to deal with changes in operating conditions. At its simplest level it means, that it behaves like a switch between the operating points, forcing new control parameters as it detects modifications in the set point. This implies that the gain scheduler together with linear controllers handles non-linearity. In the area between two linearized regions, the control system has to compute kind of interpolation between the control parameters related to the close regions. The principle is shown in Fig. 1.1.

Input Output

Plant

Gain Scheduler

Controller

Figure 1.1: Example of Gain Scheduling. In this scheme the input variables

to the scheduler are the current operating point and the reference signal. Moreover, if the linearization task is not sufficiently accurate, inducted errors can

be big enough to produce instability, and therefore the failure of the control action. In such a context, it is easily understood that fuzzy logic has been widely applied to

gain scheduling, at least for its ability to smooth sharpness, and the intuitive approach to non-linearity.

Multivariable plants models are nowadays getting more and more studied because many industrial processes need to be modeled with more than one variable. Fuzzy gain scheduling techniques haven’t yet applied to multivariable systems. In this thesis the problem of control of a 2 2× process is considered. A multivariable fuzzy gain scheduler has been built to control the process.

4

1.3 – Thesis organization The thesis is organized as follows: In Chapter 2 the basics on Fuzzy Logic are presented. This is required to give the

reader a summary of the main notions about Fuzzy Logic. An example is also explained to familiarize with the subject.

Chapter 3 deals with PID controllers. General aspects about PID are explained and the problem of implementation is shortly considered.

In Chapter 4 the design of fuzzy controllers is taken into account. When designing a fuzzy system many parameters are available and different procedures exist to manage them. This chapter highlights the main concepts and suggestions to keep on mind during designing.

An approach to multivariable systems control is the subject of Chapter 5. These systems are very difficult to be controlled because the knowledge on how parameters affect the system is hardly understandable. Automatic tuning methods are therefore needed. The Pettinen-Koivo tuning algorithm is described. This algorithm is then utilized in the case study, in Chapter 7.

Gain scheduling techniques are illustrated in Chapter 6. The evolution of this methods is presented and advantages and disadvantages of the gain scheduling are investigated.

Conclusions and suggestions for future works are reported in Chapter 8.

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Chapter 2

Basics on Fuzzy Logic

2.1 – Looking over fuzzy world In 1961 Lofti A. Zadeh felt that new kinds of mathematic theories were required to

support recent research developments [46]:

“We need a radical different kind of mathematics, the mathematic of fuzzy or cloudy quantities which are not described in terms of probability distributions [..]”

A few years later, with the first publication on fuzzy logic, the connection between

shadowy nature and computers world was built. Since that moment fuzzy logic has been studied, developed and applied to many different fields; nowadays it is possible to find a huge amount of literature dealing with fuzzy logic. Nevertheless, if someone asks what is fuzzy logic, it’s not simple and straightforward to answer, at least because a rigorous definition of fuzzy logic doesn’t exist. It is commonly accepted that there are two way of meaning fuzzy logic: the first one is motivated by the necessity to overtake difficulties encountered in analyzing or developing analytically complex systems. The second comes from the observation that human knowledge and concepts have not sharp boundaries.

According to the former approach every mathematical field can be fuzzified changing its native set by a fuzzy set; whereas the latter requires fuzzy logic to work with vague concepts and shaded theories.

Both of these meanings make fuzzy logic to be a natural bridge between the quantitative and the qualitative world.

In fact fuzzy logic essence stays in its ability to develop a connection between human nature and mathematical models. Human language is not made of digits or formulas, but it’s… fuzzy: it’s built on ideas, opinions and, why not, fantasy. Depending on them, words like “a bit”, “a few”, “a lot”, “many” etc. are definitely not numbers, but everyone can get the idea of what his interlocutor means using those words. A computer can’t, if it doesn’t have a rigorous definition of each word. Thus fuzzy logic can be thought as an interface between approximate and inexact nature of real world on one side, and the planet of numbers and bits on the other.

6

With the same mentioned view points it’s clearly understandable the double reasonings that made fuzzy logic widely used in control theory: on one side there is the approach to manage, for example, plants models (e.g. with their non-linearities), on the other the ability to transfer human know-how in the controllers, and to make them easily usable by operators.

The following paragraphs do not pretend to be a handbook on fuzzy logic: the development of such a work would be as hard as unnecessary. Fuzzy logic theory is mainly based on some central concepts, e.g. fuzzy sets, fuzzy rules and inference. In this contest, we only want to draw reader’s attention on some principles of fuzzy logic theory.

2.2 – Fuzzy sets Classical set theory define the membership of an element x to a generic set A as

follows:

A

0( )

1

xx

x

⇔ ∉ Αµ = ⇔ ∈ Α

(2.1)

The nature of the universe in which such a definition acts is black and white type,

that is every element can only belongs or not to a set. On the other hand a fuzzy set element can partially belong to a set, and the border of its belonging is shaded. With this guideline we could give the following [44]:

Definition 2.1: (membership function). Let X be a nonempty set (e.g.

nX ⊆ � ), and be called the universe of discourse. A fuzzy set A X⊆ is characterized by the membership function:

[ ]A : X 0,1µ → (2.2)

Membership functions are in general related to fuzzy subsets. Such a definition represents a generalization of the concept of membership, since it

includes the classical set theory as a special case, in fact: { } [ ]0,1 0,1∈ .

For instance, let’s consider, as universe of discourse X, the set of real number included in the interval [ ]0,1 . The following figures show how two membership

functions, Aµ and Βµ , of sets ,A B X⊂ could be represented geometrically in the classical and fuzzy cases:

7

µΑ

0 x1 x2 x3 x4 x5 x6 x7 x8 1 x 0

1

µΒ

0 x1 x2 x3 x4 x5 x6 x7 x8 x9 1 x 0

1

Figure 2.1.a Figure 2.1.b

In classical set theory (Figure 2.1.a) elements 1 2 5, ,x x x , for example, belongs to the

set A. In fuzzy set theory (Figure 2.1.b) the same elements belongs to the set B as well, but with a certain rate, whose value is given by the corresponding value of membership function. This means that any element partially belonging to a fuzzy set is also a partial member of its complement. Note that the membership function of Figure 2.1.a can be even thought as a particular fuzzy membership function, coherently with the fact that fuzzy set theory is an extension of classic theory.

The most common properties of traditional set theory, like commutativity, associativity, distributivity, identity etc. can be directly applied to fuzzy theory, but

one should note that it is not true, in general, that, given a set A X⊂ , then Α ∩ Α = ∅ (law of contradiction) and XΑ ∪ Α = (principle of excluded middle). This is actually another way to see how the essence of fuzziness lies in the lack of distinction between Α and Α .

Nevertheless, the last two mentioned principles keep their validity when fuzzy sets are crisp.

Lastly, we called nX ⊆ � universe of discourse. Such a name seems to fit to fuzzy logic better than, for example, a domain, if we mean fuzzy logic as an approach to generic problems instead of a mathematical discipline.

2.3 – Fuzzy operations and relations The notions of intersection and union are extended and generalized in fuzzy set

theory with the concepts of T-norm and S-norm operators. In particular:

Definition 2.2: (generic norm operators). Let’s consider a generic function :[0,1] [0,1] [0,1]F × → with the following properties

( , ) ( , )F x y F y x= (Symmetry) (2.3) ( ( , ), ) ( , ( , ))F F x y z F x F x y= (Associativity) (2.4)

8

if x z≤ and y v≤ then ( , ) ( , )F x y F z v≤ (Monotonocity) (2.5)

An operator T having these properties, and, also, such that:

( ,1)T x x= (One identity) (2.6) is called T-norm operator.

In the same way, an operator S is called S-norm operator if, additionally to properties (2.3-5), it is also:

( ,0)S x x= (Zero identity) (2.7)

The functions Min and Max are examples of T-norm and S-norm operators,

respectively, and they can be used to easily characterize how fuzzy set operations affect membership functions, as proposed by Zadeh:

Definition 2.3: (fuzzy set operations).

: ( ) 1 ( )AAx X x x∀ ∈ µ = − µ (Complement) (2.8)

: ( ) max( ( ), ( ))A B A Bx X x x x∪∀ ∈ µ = µ µ (Union) (2.9)

: ( ) min( ( ), ( ))A B A Bx X x x x∩∀ ∈ µ = µ µ (Intersection) (2.10)

Figure 2.2.a: intersection membership function Figure 2.2.b: union membership function

Fuzzy relations characterize how fuzzy sets are combined with classic operations.

They extend the conventional notion of relation into a matter of degree: in general a membership function obtained combining fuzzy sets belonging to iX ( 1,..,i n= )

universes of discourse is defined as: 1: .. [0,1]R nX Xµ × × → . In classical theory the

same definition would have had, as co-domain, the set {0,1}. Fuzzy relations are simply kinds of fuzzy subset (i.e. membership functions) over a base set [13].

Thus generalizing the definition of Cartesian product it is possible to directly obtain fuzzy relations from membership functions:

9

Definition 2.4: (Cartesian product). Given i iA X⊂ ( 1,..,i n= ) fuzzy

subsets, the Cartesian product is:

1 .. 11,..,

( ,.., ) min( ( ))n iA A n A i

i nx x x× × =

µ = µ (2.11)

The next two definitions are needed to provide fuzzy logic with operators: in other

words, given the space in which fuzzy logic lives, it’s now missing the ability to process fuzzy information, that is, for example, how to perform arithmetic operations between fuzzy numbers.

Definition 2.5: (composition). The composition of two relations R S� is defined as a membership function in X V× :

( , ) sup ( ( , ), ( , ))R S v R Sx y T x v v yµ = µ µ

o

(2.12)

where: , : [0,1]R S X V× → , x X∈ , y Y∈ ,and T is a T-norm. Definition 2.6: (extension principle). Given a function

1: .. n Cf X X X× × → operating between crisp sets ,i CX X ( 1,..,i n= ), let

i iA X⊂ to be fuzzy sets, the fuzzy extension is a membership function defined

as:

111

1.. ( )

( ) sup min[ ( ),.., ( )]n

n

F A A nu u f v

u u−∈

µ ν = µ µ (2.13)

The importance of the last definition is that it allows extending every punctual

operation to the fuzzy equivalent, with the fundamental property that the classical operation is a particular case of the fuzzy correspondent. Formally Fµ is simply an extended mapping of f .

2.4 – Fuzzy rules In fuzzy logic variables are called linguistic variables. In order to assume fuzzy

values they are characterized by the parameters listed below. As an example, within parenthesis there are values for the variable “velocity”:

a symbolic name (velocity); the set of linguistic values they can take (slow, medium, fast); the numeric domain in which the linguistic values take numerical values syntactic (and the semantic equivalents) rules which defines the values.

10

0 40 100 160 200 0

degr

ee o

f mem

bers

hip

slow medium fast

Velocity

Figure 2.3: Example for linguistic variable “Velocity”

Fuzzy rules associate a certain linguistic input variables with an output. In our

contest the input is connected to a process state, whereas the output represents a control action. A set of parallel fuzzy rules defines a fuzzy rule base; it can be seen as a function that maps the output space as from rules.

In order to give consistency to the fuzzy system, the rule base has to be complete (i.e. any combination of inputs results in an appropriate output) and consistent (i.e. there can’t be two or more rules with the same antecedent and different consequences).

The structure of the rules is:

IF ( 1x is 1jA AND .. AND nx is j

nA ) THEN ( y is jB ) (2.14)

Variables ix are the inputs to the fuzzy system, y is the output. j

iA and jB are

fuzzy sets. jB can be also a singleton. Note that, in general, membership functions of an input variable should cover the entire input space, which is the universe of discourse, whereas the output membership functions have not such a kind of restriction.

When a new input value enters in a fuzzy system, it has to be evaluated by the fuzzy system to produce the appropriate output, with respect to the involved rule. This task is called fuzzy inference.

First, for each input the degree of membership to the correspondent fuzzy set has to be estimated. For example, with reference to Figure 2.3 a velocity of 60 Km/h is said to be (approximately) 80% “slow” and 20% “medium”. This step is often called matching step, e.g. in [46].

With multiple input conditions rules (e.g. rules with OR and AND conditions) a fuzzy conjunction operator has to be used to combine the matching degree of each condition. Commonly used operators are, for example, Min and the product: the former works cutting off (clipping) the top of membership function whose value is higher than the matching degree. The latter performs scaling the membership function proportionally to the matching degree.

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Then output membership functions are evaluated with respect to the calculated matching degrees. This will produce the conclusion that is the weighed consequent of each rule.

Usually rules have partially overlapping conditions: a particular input involves more than one fuzzy set to have a non-zero matching degree. This means that the conclusions have to be combined (aggregation). A frequently involved fuzzy operator is, for that purpose, the max function.

11

1( )A

xµ

( )B yµ

21( )

Axµ

1x 2x

1 11 2

1 2min( ( ), ( ))A A

x xµ µ 1 2max( , )B B

µ µ

22( )

Axµ

12( )

Axµ

1 ( )B

yµ

2 ( )B

yµ

Figure 2.4: Fuzzy inference. This is an example on how inputs and rules affect the output membership function. The numbering introduced in (2.11-12) is used, and the min() and max() operators are applied aggregate rules and clip fuzzy sets. On the last “shape” on the right a defuzzification method, like center of gravity for instance, has to be applied. Finally, the output of a fuzzy system is very often requested to be a number. That is

to say that there is need to translate fuzzy sets in crisp sets. The procedure that converts the fuzzy output in a numeric format is called defuzzification. There are many methods to perform this step; probably the best known is the center of gravity technique. It works evaluating the center of gravity of a mass whose shape comes from the clipped or scaled output membership functions. The definition of center of gravity is the following:

( )( )

( )

BY

BY

y ydycog B

y dy

µ=

µ∫∫

(center of gravity) (2.15)

Note that the mentioned property of completeness of rule base finds here its

analytical interpretation since an output not included in any membership function will produce an indetermination in calculating (2.15).

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Observe, at last, that the defuzzification is requested even if the output membership functions are singletons. In that case the center of gravity coincides simply with the arithmetic mean.

2.5 – An example: washing machine fuzzy control system The control problem in this example is about the automatic selection of cycles for a

washing machine. The reasons for choosing such a kind of weird problem, at least from a theorist point of view, are mainly two. The first is that the involved human knowledge to be fuzzified is easily understandable due to the fact that it is a day-life problem. The second is that it is a “historical” application, since in 1990 such a type of fuzzy washing machine was actually developed and put on the market. It was one of the first examples of using fuzzy logic to design consumer-oriented products.

Problem statement: We want to build a fuzzy system that optimizes automatically

the laundry washing, by a decision-making task of washing cycles. A housewife could observe that the temperature of the water is another important output to be considered in optimizing washing quality, but the example achieves its target easily with only one output.

Proposed solution: In order to reach the desired result we will proceed through the

following steps: Linguistic variables are identified and fuzzified. This means that also membership

functions, fuzzy sets with their ranges are chosen; Fuzzy rule base is built; Outputs are aggregated; Finally, output is defuzzified to produce the conventional numeric format output. Step 1: As it is demanded, the output of the fuzzy system is the intensity of the washing

cycle. In order to choose the optimal value for the output let’s assume, as input variables, the laundry quantity and the laundry softness. To describe the inputs by fuzzy sets, we can simply use the words we would probably use from the actual know-how of daily experience. In that way, for example, we can associate to the laundry quantity the fuzzy sets: “small”, “medium” and “large”. With respect to the laundry softness it’s reasonable to relate it with the words “hard”, “normal” and “soft”. A more precise description (that is: to add others fuzzy sets) might be done, for instance, calling the intermediate fuzzy sets “hard normal” and “hard soft”. This is actually a designer choice. It’s, in that sense, a trivial example, but it shows how design skills and experience are combined.

For the output variable, the washing cycle, let’s consider four fuzzy sets: “delicate”, “light”, “normal” and “strong”.

13

Note that numeric values for these variables make sense only when the project is physically built, and they come from the technology involved.

Let’s consider the gaussian membership functions for input variables, and triangular for the output. The real variables ranges, which are the universes of discourse, come from the specific implementation that one wants to use. For fuzzy ranges let’s use, through an adequate normalization, the interval [ ]0,1 .

0 0.2 0.5 0.8 1

0

1

Quantity

Deg

ree

of m

embe

rshi

p

Small Medium Large

0 0.2 0.5 0.8 1

0

1

Softness

Deg

ree

of m

embe

rshi

p

Soft NormalSoft NormalHard Hard

Figure 2.5: Membership functions for input variables

0 1 0

Cycle

Deg

ree

of m

embe

rshi

p

Delicate Light Normal Strong

Figure 2.6: Membership function for the output variable “washing cycles”

Step 2: As it will be mentioned in Chapter 4 with reference to Fuzzy Controllers, the

construction of the rule base is an essential task and it might present some difficulties. Concerning our example the whole rule base is shown in Table 2.1. In every box there is the result (the output) of the corresponding inputs. Two rules are listed below, as examples:

IF the Laundry Quantity is Large AND the Laundry Softness is Hard THEN

Washing Cycles is Strong;

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IF the Laundry Quantity is Normal AND the Laundry Softness is Normal Hard THEN Washing Cycles is Normal;

Once more the human experience is widely employed. Heuristic conjectures take

place and the rule base construction “derive benefit” from them.

Laundry Quantity Laundry Softness

SMALL MEDIUM LARGE

SOFT DELICATE LIGHT NORMAL

NORMAL SOFT LIGHT NORMAL NORMAL

NORMAL HARD LIGHT NORMAL STRONG

HARD LIGHT NORMAL STRONG

Table 2.1: Fuzzy rules Step 3: Operators min-max are used to aggregate rules. An example of the inference task is

shown in Figure 2.4. Step 4: The defuzzification is based on center of gravity method, thus it follows Equation

2.15. In Figure 2.7 a surface graph is drawn: for each set of input values it gives the final defuzzified output values according to the designing choices.

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0

0.6 1

0

0.5

1

0.7

SoftnesQuantit

Cyc

le

Quantity Softness

Figure 2.7: With The MATLAB Fuzzy Toolbox it is possible to display the relation

between output and input variables. For instance, let’s suppose to have, as it is shown, a Softness fuzzy value of 0.183,

and a Quantity of 0.809. These are values that “live” in the normalized universe of discourse. The reasoning produces the output value equal to 0.529.

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Chapter 3

PID control

3.1 – General aspects of PID controller As already mentioned, PID controller is the most commonly used control algorithm.

Generally speaking it has some important functions: it can be used in feedback systems, it has the ability to eliminate steady state error through integral action and it can anticipate the future with the derivative action (e.g. [49]). The “theoretical” equation that summarizes its behavior is, with reference to Figure 4.1:

0

1 ( )( ) ( ) ( )

t

di

de tu t K e t e d T

T dt

= + τ τ +

∫ (3.1)

where: ( )u t is the control variable and ( )e t is the control error given by:

( ) ( ) ( )spe t y t y t= − .

C ontroller P lant

cG (s) pG (s)u(t)

e(t)

( )s py t( )y t

F igu re 3.1: A gen eric feedback controlled system and its no menclature

The different components of PID controller can immediately be recognized in the

correspondent order: the P-term (proportional to the error), the I-term (proportional to the integral of the error), and the D-term (proportional to the derivative of the error). The terms K , iT and dT are gain parameters. That means that the I-term, for example

is proportional to the integral of error through the term / iK T . Sometimes [31], in order

17

to emphasize the values of parameters that have to be set in tuning the controller, the Equation (3.1) is presented in the form:

0

( )( ) ( ) ( )

t

p i d

de tu t K e t K e d K

dt= + τ τ +∫ (3.2)

so that, for example d dK K T= ⋅ . In the Laplace domain the corresponding equation

is given by:

1( ) ( ) ( ) ( )d

i

U s K E s E s sT E ssT

= + +

(3.3)

where ( )U s and ( )E s denote the L-transformations of control variable and error.

pK ( )spy t

y(t) spy e 1n 2n

pK

Controller

Kπ

Plant u

Figure 3.2: Simplified closed loop system for static considerations

The idea of PID control is that integral and derivative actions compensate lack of

performances of the simple proportional action. In order to get this point by simple static remarks, let’s consider the model in Figure 3.2. Both the controller and the plant are assumed to be modeled by a pure proportional action, that is:

( )p sp bu K y y u= − + (3.4)

1( )y K u nπ π= + (3.5)

The parameter bu represents a bias or a reset value. When the control error is zero,

the control variable assumes the value ( ) bu t u= . Inputs 1n and 2n are disturbances.

It can be easily figured out that the relation between the plant output and inputs is:

2 1( ) ( )1 1

psp b

p p

K K Ky y n n u

K K K Kπ π

ππ π

= − + ++ +

(3.6)

If we assume 2 0n = and 0bu = then the loop gain (i.e. pK Kπ ) should be high in

order to make spy yπ ≈ as close as possible. In that way the system is also insensitive

18

to load noises 1n . On the other hand, if 2 0n ≠ then 2n influences the system as much as the set point does. To avoid this effect the loop gain should be not too high. Moreover the bias 0bu ≠ affects the system as 1n does. Therefore the only

proportional action can’t satisfy all these requests simultaneously. This feature can be interpreted also in terms of available freedom degrees of a generic system in order to satisfy a certain number of conditions: each freedom degree usually can satisfy only one request. Moreover we argued based only a static model: dynamics also sets limits (upper bound) on how large a loop gain can be . These questions show that the P-action is not sufficient

An integrative action is usually thought to reduce the steady state error, and this is easy to understand: if the control error is, for instance, positive, it will produce an increasing of the control signal, whereas a negative error will generate a decreasing control signal.

The derivative action is used to improve closed loop performances in term of stability, that is to say: to satisfy phase margin requests. In fact instability can occur because of process dynamics: a control action change can’t be immediately seen in the process output because of propagation times. As a result, the control action is always late in correcting errors. The derivative action performs (in “co-operation” with the proportional action) being proportional to the predicted process output, where the prediction is made by approximating the error at the next step with its Taylor series expansion stopped at the first order:

( )

( ) ( )d d

de te t T e t T

dt+ ≈ + (3.7)

Thus the control signal is proportional to an estimation of the control error at time

dT . In that sense the D-action is said to predict the process output.

3.2 – Implementation The standard algorithm shown in Equation (3.1) rarely occurs in practical

implementations because many variations with better performances have been developed. Finally, a question arises about physical feasibility of that equation.

The so-called characteristic equation of a system as shown in Figure 3.1 is defined as:

c p1 G (s)G (s) 0+ = (3.8)

This equation, which corresponds to the denominator of the closed-loop system

response in the Laplace domain, allows determining stability characteristics of the system and also the transient response of the control system. Design methods usually are based on this equation.

19

In the derivative path of a PID a problem occurs: the gain of this part increase as frequency increases. This means that a fast change in the input signal can make the D-action being very strong: for instance, a step signal has an infinite derivative. A real step signal can’t have, of course, an infinite slope; nevertheless it can generate values too high for the system stability, in fact the measurement noise would benefit by all the available gain.

Thus very often the derivative transfer function (i.e. in Laplace domain) is approximated, in order to reduce the gain [16], by:

1

dd

d

sTsT

sT

N

≈+

(3.9)

where N is a parameter to be chosen. At low frequencies the approximation can fit

well depending on the tuning of N ; at high frequencies the gain is limited to N . From a control theory point of view in the second member of Equation (3.9) we simply added the pole / dN Tω = to limit the gain, in the Bode magnitude diagram, at high

frequencies. In designing of industrial processes, derivative action is frequently not used. As a

general rule one can assume that: if the step response of the process seems to belong to a first-order system, than PI control is enough. This takes place, being more precisely, when the Nyquist curve lies only in the first and fourth quadrant. Nevertheless, even if the process doesn’t look like a first-order system, it is commonly accepted that if the process does not require tight control a PI controller is still adequate: in these cases the main targets are related only to steady-state error (I-action) and transient response (P-action). Note that a PI controller cannot obtain good results when it is applied to a non-linear system and set point changes are involved. In fact it behaves in the same way trough the different operating conditions even if the controller is non-linear (PI-like). In this case other tools that recognize set point changes are required.

20

Chapter 4

Fuzzy controller design

4.1 – Overview We assume a Fuzzy Logic Controller to be a system that is used to perform a

generic control on a target process. By “generic control” we mean that a fuzzy controller can be thought not only as a traditional cascade-connected controller, but also as a supervisor in feedback systems.

In design a FLC, first, input and output variables have to be selected. They of course depend on which kind of FLC is going to be implemented. The next step is the choice of the rule base; finally the shape of membership functions is decided and a tuning of the controller is done “moving” and shaping membership functions within the variables range, which represent the universe of the discourse, as seen in Chapter 2.

A general way to represent a FLC is as follow:

( ) ( ( ), ( 1),.. ( ), ( 1), ( 2), ( ))u k F e k e k e k u k u k u k= − − ς − − − ς (4.1) where F is the fuzzy control law. This equation has not to be considered as a

difference equation: it indicates only that the FLC can use information from previous steps over the last one. This means that information about rate of change in variables is available, since we can write, referring to ( )e k , for instance:

( ) ( ) ( 1)e k e k e k∆ = − − (4.2)

When input variables are chosen it has to be considered that every new input will

add anything but a negligible contribution to increment number of rules. In fact the number of rules grows exponentially with number of inputs. For instance, if we assume to use two membership functions for inputs, then covering all the occurrences will cause rules growing at least proportionally to 2 xn , where xn is the number of input

fuzzy sets. This is a rough evaluation, but practice reveals that in real applications the number of inputs is seldom greater than three or four.

The choice of variables to use is demanded by the class of fuzzy controller. Since the target of a control system is to keep the error ( )e k small, usually the change of error, defined in Equation (4.2), is considered, besides the real error, obviously. It

21

gives information about the derivative of error. Therefore it can be used to figure out how quickly the error is changing, and if it is decreasing or not. Additionally, input (i.e. operating point) or output to the system ought to be employed in order to notify the controller about changing in operating point. In fact one should note that providing the controller with information only about error and its changes, it will make the controller behaving in the same way at different operating points, as the error is interpreted to be the same (in a fuzzy set sense), no matter if the plant is, as always it is, non-linear.

If the fuzzy controller is designed as a cascade controller, then its output is a control variable that is forcing the plant. Otherwise, if the fuzzy controller is a supervisor then its output is a general parameter (eventually a set of parameters) that will drive the real controller.

4.2 – Parameters design Input and output variables of the fuzzy controller must be expected to belong to

certain intervals. These intervals have to be specified in the fuzzy controller. In fact, with regards to input variables: every possible input value should be fuzzified, thus fuzzy sets have to cover all the range of varying. Given the error ( ) ( ) ( )spe k y k y k= − ,

its range is defined by the maximal and minimal values of variables that identify it. It is reasonable to assume that set point and the measurement fluctuate in the same interval. Hence, if min max( ), ( ) [ , ]spy k y k y y∈ then min max( ) [ , ]e k e e∈ , where:

min min maxe y y= − (4.3)

max max mine y y= − (4.4)

In the same way, the change of error ( )e k∆ belongs to the range min max[ , ]e e∆ ∆ ,

with:

min min maxe e e∆ = − (4.5)

max max mine e e∆ = − (4.6)

It is common procedure to normalize intervals (that are the universe of the

discourse), introducing scaling factors. These factors are basically functions that map the real measured values into the new intervals.

A mapping function is *:f I I→ where min max[ , ]I i i= is the true interval, and * * *

min max[ , ]I i i= is the new, desired, interval. A general linear mapping is, thus:

( )* *

* *max minmin min

max min

( )i i

x f x x i ii i

−= = − +−

(4.7)

22

where *x represents the new normalized variable. If the ranges are taken symmetric with respect to the origin: (*) (*) (*)

max mini i i= − = then

Equation (4.7) becomes: *

* ix x

i= . Therefore we define:

*

i

ik

i= (4.8)

as a general scaling factor. The reasoning mechanism in the fuzzy controller applies

to the normalized values. The scaling factors are applied also to output variables. It’s obvious that the simplest shape of the scaling factor given by Equation (4.8) can be used if the input variable interval is (almost) symmetric. Otherwise, if this interval is non-negative (or non-positive) then Equation (4.7) has to be considered. However the normalized interval is chosen by the designer, thus it can always be taken symmetric.

The choice of the number of fuzzy subsets is another important task: when a number has to be evaluated, it can be said to be only small, medium or large (with respect to certain implicit estimation criteria), but a more accurate assessment could be done adding intermediate sets like quite small and quite large. Thus it’s plain that accuracy might be improved unlimitedly: the course of the fuzzified variable could be classified in several fuzzy sets and a very thorough fuzzy description will be achieved. An upper bound to the amount of fuzzy sets is given by the growth of the number of rules: having a large amount of sets will increase the number of rules, in order to keep completeness of the fuzzy system. To clarify this aspect if we consider, for instance, a fuzzy system with two inputs, then a new fuzzy set added in one of those inputs, will make rules number rising, by completeness, as much as the number of sets of the other variable.

A good remark is that it is not necessary to have the same shape and dimension (i.e. areas) for fuzzy sets. Therefore it follows that different dimensions of sets can be selected. This technique is useful to deal with non-linearities: where strong non-linear behaviors are handled, a local thick set of membership functions could increase performances. On the other hand, linear-like might need less sets, thus rules number can be kept locally small. However, this is not a straightforward way to proceed, because a fuzzy system introduces by itself non-linearities.

The rule base choice depends, as already pointed out, on input and output variables and fuzzy subsets; but its roots are, above all, in the designer approach, skill and know-how. It is surely the most difficult and crucial aspect in outlining a fuzzy controller.

The construction of the rule base can follow basically two methodologies [44]. The former is based on the concept of template: when a fuzzy controller is designed starting from a template it indicates that common and well-known knowledge coming from designers is applied. The templates should be meant as general guidelines in developing the rules. A very general template rule base is for example the Mac Vicar-Whelan’s. It can be used to build problem-oriented rule bases for PID-like fuzzy controllers, by modifying, add or excluding its rules. Such an approach is recommendable, for example, when an a priori knowledge of the rules is not available: from a template some rules can be extracted to provide a reasonable starting point.

The latter method is based on intuitive knowledge and experience. The fuzzy controller is, in that sense, thought as an expert system. The rule base source can be expert operator’s experience and a good knowledge of the problem to solve. This

23

method probably is more strictly related to the original interpretation of fuzzy logic than the previous one. In the first method, fuzzy logic is a mathematical approach to implement handbook rules. The second provide the connection between human experience and heuristic strategies on one side, and the controller on the other one.

The combination of both of these ways can be also an interesting alternative: first a controller could be designed as a “handbook prototype”, following principles of a template. Then, test and incoming experience can improve and personalize the effectiveness of the rule base.

Considerations on the kind of rules should be done: the rule type shown in Equation (2.14) is said to be a Mamdami rule type. Its main characteristic is that the consequence is a fuzzy proposition, as the antecedents are. Another kind of rule is commonly adopted, the Sugeno rule type. The general form of this kind of rule has, as a consequence, a function of the controller inputs.

A set of Sugeno rules can be seen as a set of local controllers, each one with its own set of controller parameters. Thus different controller parameters for different input combinations are defined. It acts as a sort of gain scheduler, if the consequences functions are linear. Although the ideas in Mamdami’s and Suogeno’s rules come from different reasoning, it can be shown that in practical implementations rules can be equivalent. This occurs when in Sugeno reasoning the output functions (i.e. the functions adopted as rules consequences) are constant.

A practical consideration takes place in choosing which of kind of rules to apply. In fact, actual implementations of standard PID controllers are very different from the textbook version and, above all, more complicated. This means that Sugeno type rule results to be more problematic because every time the fuzzy system has to produce an output, there is a function to be evaluated in each rule, possibly also non linear, with different parameters. Therefore when gain scheduling technique is invoked Mamdami rules are simpler and more efficient to apply.

Figure 4.1: Fuzzy sets can be chosen thicker in regions of strong non-linearity.

0 0.3 0.5 0.7 1

1 input1

Deg

ree

of m

embe

rshi

p

24

Once the rule base has been written down reasonably, a tuning step of the fuzzy controller is required. In fact rule base is made trying to lead to an expected control action, and this task is only the framework of the controller.

It’s worthwhile to emphasize that for our purpose the tuning task in this Chapter is referred to the fuzzy controller meant as a generic fuzzy system. Therefore it deals with general aspects of the controller that are not connected with the controlled process. Tuning with respect to the process is discussed in Chapter 5.

The tuning of a fuzzy controller is a significantly sophisticated phase, especially if we compare it with the tuning of a conventional controller. In fact a fuzzy controller, being a fuzzy system, is extremely flexible; its behavior is determined by a copious number of parameters. These parameters come from, for example, defining membership functions, fuzzy subsets and the inference procedure.

A few organized tuning methods have been presented in literature, but they mostly are concerned about specific problems (see, e.g. [30]). This lack of available methods makes the tuning to be performed usually by trial and error methods.

Tuning by trial and error can be executed by introducing scaling factors, varying the position, within the borders, and the shape of membership functions. “Shape” is here meant as a general varying of the area, keeping the same kind of membership function: for instance if we are handling a triangular membership function, then moving the peaks (making them sliding) of the figure is a “shaping” actions, and the nature of the membership function is not modified. A graphic example of this concept is, for example, Figure 4.1.

The scaling factors are an essential aspect of tuning. They change the normalized interval of input and output; therefore they operate a transformation of the space state, as inputs are error and change of error. In general, high values of the scaling factor

related to error, *

e

ek

e= , improve responsiveness of the system but they increase the

overshoot and thus the stability might be compromised. Moreover a fast convergence

is bounded by high values of both ek and *

e

ek

e∆∆=∆

that is the scaling factor of the

change of error. When insufficient results are obtained locally, that is: the system does not reach the

expected performance into certain subintervals of the universes of the discourse of input variables, then fuzzy subsets are “moved” into the prescribed range as already mentioned above. The same tuning can be done for output variables, but if the output membership functions are singletons then this task is easier, because a change in the value of a singleton will affect only the support of the corresponding rules.

The choice of the kind of membership function (e.g. triangle, trapezoid, gaussian etc.) might be also a designing task. But, at least in control theory, it is seldom taken into consideration because suitable results are usually achieved using very simple membership functions.

25

4.3 – Reliability When building a controller the designer always encounters problems at least

because designing is supposed not to be related to trivial tasks. Obviously the project of a fuzzy controller meets difficulties, as well. It is not true that overall performances of a fuzzy controller are much better than those achieved by traditional controllers.

As it has been shown above, the design of a fuzzy controller has many parameters to take into consideration, and managing all of them is not a negligible work. Non-linearities are involved, and the connection between the adjustment of one parameter and the generated consequence is often not clearly understandable.

Stability and robustness of the system are not easily achievable and predictable, and there are not systematic procedures to lead to them. They are still deeply studied, and even if big efforts have been put on finding connections between non-linear control theory and fuzzy control theory, much needs to be done.

What we want to highlight is that the project of a controller by means of fuzzy logic is only a different way to tackle the control problem. Designing difficulties threaten whatever path the designer chooses to follow. The power of fuzzy logic in control theory takes place mainly in two different aspects: the first is the likelihood to develop advanced control strategies without a specific and deep knowledge of complex theories. The second is that a fuzzy controller can be understood, and thus maintained, by personnel whose education level is not as high as it is required for complex ordinary controllers.

26

Chapter 5

Multivariable feedback control

5.1 – Generalities The task of tuning a multivariable controller with respect to the involved process is

considered. In a multivariable process the design of the controller must take into consideration interactions between process variables. This is not a trivial problem.

The simplest way to proceed is to have scalar controllers, each one of these is forcing a control action only on one channel. However, if the interactions are enough strong so that a scalar approach is not sufficient, then a multivariable controller has to be used. If we focalize our attention on PID controllers, then for a generic n n× process the controller has 3n n× × parameters to tune. Geometrically, tuning such a controller corresponds to search a point or a set of points in the 3n n× × space.

A difficulty in that case is that in multivariable control the relations between modifications in control parameters and response of the system are not straightforward to be figured out. Practically it means that parameters that in the scalar case have a predictable action on the process, in the multivariable case they simply have not. Therefore desired performances and robustness characteristics of the system are not easily achievable. This means that it’s almost impossible to adjust parameters for a multivariable controller by trial and error. Automatic tuning methods are required. In literature many tuning methods have been presented and tested. The choice of the best method is usually related to the specific purpose.

Moreover if using a fuzzy system, the number of parameters increases quickly and is even greater than in the traditional approach. While a big number of parameters gives more degrees of freedom in designing, it becomes harder to keep a general outlook of the system.

A first alternative is to consider a PI controller, thus the derivative action is not used. A smaller number of parameters is therefore involved. The feasibility of this choice has been considered in Chapter 3. Several methods have been proposed to facilitate the tuning of multivariable PI controllers [30].

27

5.2 – Problem definition Let’s consider a linear system model in a transfer function matrix form:

Y(s)=G(s)U(s) (5.1) where Y(s) and U(s) are, respectively, the vectors of transfer functions of output

and input signals. In this context we assume the number of elements of each vectors to be two. Thus the matrix G(s) is 2×2 .

A conventional MIMO PI controller is described by a transfer function matrix:

p i

1U(s)= K +K E(s)

s

(5.2)

where E(s) is the vector of the transfer functions of measured errors and:

i i11 12

i i i21 22

k kK =

k k

p p

11 12p p p

21 22

k kK =

k k

(5.3)

are the gain matrixes of the proportional (P) and integral (I) parts of the

multivariable controller. Terms on the main diagonals of the matrices correspond to the direct action of one channel on itself. While terms on the secondary diagonals represent the cross control actions. Therefore control scalar variables are, in the Laplace domain:

1 1 2

1 1( ) ( ) ( ) ( ) ( )p i p i

11 11 12 12u s k k e s k k e ss s

= + + +

2 1 2

1 1( ) ( ) ( ) ( ) ( )p i p i

21 21 22 22u s k k e s k k e ss s

= + + +

With this representation we highlight how each channel interferes the other one. A

PI scalar controller has two parameters to set. With a simple 2 2× PI controller (that is actually the simplest multivariable controller) the number of parameters has become eight. With a 3 3× PI controller eighteen parameters would have to be adjusted. When vector approach is used, and interferences are taken into account, number of parameters grows very quickly.

Another thing to be considered is that when a step occurs in one of the input variables, there is a fast and big change in the system conditions. At the beginning the output is far from the set point because the controller has still to act. This means that the control parameters are changing, as it is expected. Therefore if we consider, for instance, the channel “number one”, a change in 1( )spy t will affect, immediately, 1( )e t

and 1( )e t∆ . The new inputs to the controller will force changes in control parameters.

It can be seen, by Equations (5.4) and (5.5) that even if 2 ( )e t and 2 ( )e t∆ are not changing (or at least if they are changing because of previous adjustments, without any

28

regard about the last step change) the control variable on this channel, 2 ( )u t , changes.

And it is not exaggerated to say that probably the change is important. In fact 1( )e t is

big and its coefficients p21k and i

21k come from the previous evaluation. Hence everything changes.

One could think to set equal to zero coefficients of secondary diagonals in pK and

iK matrices. Actually this is a simpler way to develop the controller, and it

corresponds to the scalar case. The number of control parameters is half of the previous case. However, in this case interferences inside the process cannot be taken into consideration because the controller parameters on each channel vary only depending on the correspondent variable. Moreover fewer parameters are involved and this could give less autonomy in designing.

U1

U2

Y1

Y2

PI

PI controller 4

PI

PI controller 3

PI

PI controller 2

PI

PI controller 1

Figure 5.1: Structure of a 2x2 PI controller.

5.3 – Pettinen-Koivo tuning method The present tuning method has been developed for multivariable PI-controllers and

can be seen as an improved extension of the Davison’s method. Davison proposed to take the tuning matrices both for proportional and integrative part. They differ only for a fine tuning parameters. Davidson tuning matrices are:

1

pK G (0)−= δ 1

iK G (0)−= ε

This method aims to decouple the system at low frequencies. Pettinen and Koivo

modified this method in order to achieve decoupling at high frequencies as well.

29

It has been found out in [20] that their method can achieve good performances, even if no general rules exist to state if one method is better than others for every application.

Given the state space representation of the system model:

x=Ax+bu

y=Cx

� (5.4)

then the transfer function matrix G(s) can be expanded in series:

...2

2 3

CB CAB CA BG(s)=

s s s+ + + (5.5)

The method suggests to use the first term of the series expansion as a rough tuning

matrix for the proportional part of the controller:

( ) 1

pK CB−= δ (5.6)

The reason for such a kind of choice originates from evaluating the limit of the

transfer function with the Laplace variable. In fact, since:

CB G(s)

s→ as |s| → ∞ (5.7)

then

p

I G(s)K

s→ δ as |s| → ∞ (5.8)

This selection of pK aims to decouple the system at high frequencies.

Following the same reasoning, for the integral part of the controller, we remark that the steady state gain matrix is:

1(0) ( )G C A B−= − (5.9)

therefore its inverse is used as a rough tuning matrix for the integral part iK :

1 1( ( ) )iK C A B− −= ε − (5.10)

The parameters δ and ε are intended to be for fine-tuning. Often they can be

selected to be equal. Usually tight controller tuning can be achieved by selecting large values for these parameters, whereas smaller values produce looser tunings.

30

A problem arises when the matrix CB is not invertible. In that case [29] the tuning matrix should assumed to be:

( ) 1

pK CB CAB−= δ + (5.11)

The matrix CAB is taken from the second term of the series expansion of Equation

(5.5), so that the essence of the method is kept. If one element of G(s) is such that the degree of the element’s denominator is two

or more greater than the degree of its numerator, then the corresponding element in matrix CB is zero. Thus the matrix CB will be singular if all elements of G(s) have this property. An approach to that problem is to reduce the model order before computing the tuning matrices. Therefore in this case, firstly all elements of G(s) are reduced into first order transfer functions. Then the method is applied to the reduced model.

This method has been originally developed for tuning of multivariable PI controllers. If an extension to the PID case is needed, one possibility might be to assume for the derivative part the same tuning matrix as for the proportional part.

Note that the method, as long as it has been presented based on a linear system (see Equations: (5.4), (5.6), (5.10)), would presume a linear model of the process to be available. However Equations (5.6) and (5.10) can be computed even from step response experiments, without any assumption of linearity on the process.

31

Chapter 6

Gain Scheduling

6.1 – Overview Realistic models of engineering systems often are non linear. One consequence is

that the dynamic behavior of a system to be controlled changes with the operating region. The notion of gain scheduling is widely applied to such a kind of situation. A definition has been proposed in [12]:

Gain Scheduling: a linear parameter varying feedback regulator whose parameters are changed as a function of operating conditions.

Historically the roots of gain scheduling come after the Second World War, when

aviation technology started dealing with jets and supersonic airplanes. Gain scheduling was applied to autopilot systems. Other fields of applications are automotive engine control, for example: gains scheduling came as a response to de demand of fuel economy and exhaust reductions. In power electronics gain scheduling has been applied for instance to control systems for switch-mode AC/DC converters [5] or for Load Frequency Control [36].

How to apply gain scheduling? The first step is to linearize the model about some operating points. The linearization can be based mainly on two different approaches: the evaluation of the Jacobian is probably the most common and well-known way. Another method is the quasi-linear parameter varying scheduling (or simply quasi-LPV scheduling). It is based on the idea of rewriting a process model in which non-linear terms have been replaced with time-varying terms that are then included in the scheduling variable.

Then linear design methods are applied to the linearized model at each operating point, in order to arrive at a set of linear feedback control laws that perform satisfactorily when the closed-loop system is operating near the respective operating points. What one should have is, therefore, a set of linear controllers corresponding to the different linearizations.

The gain scheduling is then intended to handle non linear aspects of the design problem. The basic idea involves interpolating the linear control law designs at intermediate operating conditions. That is: a scheme is devised for changing

32

(scheduling) the parameters (gains) in the linear control law structure based on the monitored operating condition.

Many types of gain scheduling methods have been experimented. Usually the model is arranged so that the operating condition is specified by the values of one or more variables of the process, and the gains are scheduled according to the instantaneous values of these scheduling variables.

As probably it is already clear to the reader, the main advantage of the gain scheduling approach is that linear design methods are applied to the linearized system at each operating point. The wealth of linear control methods, performances measures, design intuition and computational tools can be brought to support control design for general, multivariable non linear processes. For example frequency domain notions can be used in specifying performances, while this tool is much less understood in an explicit non linear process. Moreover, methods of robust control for linear systems can be applied to counter uncertainty in the plant parameters in order to obtain robustness in the final gain scheduled system. Another advantage is related to the fact that a gain scheduled control system has the ability to rapidly respond to changing in operating conditions. This means also that it is crucial to choose the scheduling variables that reflect the process dynamic so that changing in the operating conditions are immediately notified.

X

U

Figure 6.1: Gain scheduling based on a series of local approximations: the global controller is

obtained by patching the overall model by a series of local controllers associated with the linearization of the model.

Gain scheduling does not require strict assumptions on the plant model, thus it can

be used also without an analytical model of the process. This is indeed an interesting feature because industrial applications need very often to deal with databases, i.e. inexact models.

On the other side some difficulties can slightly compensate this advantages. One difficulty is the selection of appropriate scheduling variables. General proposals have not yet reached and rules of thumb mainly show the way to the designers. Practice shows that scheduling variables selection usually is based on the “physics” of the situation, and on particular characteristics of the process.

Perhaps the major difficulty in gain scheduling design is the selection of the scheduling procedure, which is the control law that varies gains changing as a function

33

of the scheduling variables. This aspect is actually seldom discussed in literature, and it appears that scheduling is currently an art, and simple curves-fitting approaches are used. On the other hand it is also true that as complexity increases, scheduling become much more difficult to be managed satisfactory with not general methods.

Another aspect of gain scheduling is that the overall performances of the control system must be checked by deep simulation studies. This is a consequence of the local use of linear design methods. However, one should note that in real application most of non linear methods are local as well.

When stability has to be studied it is difficult to perform an overall analysis. Usually stability can be guaranteed only locally and with slow time variations. However gain scheduling has become a popular reality in many applications. Rugh wrote [32]: “Machines that walk, swim, or fly are gain scheduled”.

Gain scheduling has been studied also on multivariable systems [16]. However, up to now, none of these control design studies have led to implementation of such multivariable control laws to operational and running processes. One of the difficulties with implementing the gain scheduled multivariable control laws is the complexity of these control laws, due to the large number of controller parameters that need to be scheduled.

With the evolution and increase of studies on gain scheduling many ideas have been investigated. For a long time gain scheduling appeared mainly to be a point of view to approach control problem instead of a branch of control theory. Only in the last ten years gain scheduling has been much more considered a real design methodology for non–linear control, e.g. in [1] and [3]. Nevertheless literature doesn’t exhibit yet uniform analytical treatments.

6.2 – Problem definition We shortly show a formalism proposed by Rough [32], as it appears the best one

that summarizes exhaustively many years of studies gain scheduling oriented. Given the non-linear plant:

1

2

( , , , )

( , , , )

( , , )

x a x u w v

z c x u w v

y c x w v

= = =

�

(6.1)

where: x is the state, u is the input, z denotes an error signal to be controlled, and

y is a measured output available to the controller. Both v and w denote exogenous inputs to the plant. They may be easily confused, but v represents external signals like references, disturbances etc. The linearization is computed with respect to v , while w represents the parametric dependence of the plant on exogenous variables.

A plant equilibrium point ( , , , )e e ex u u w is a point such that:

( , , , ) 0e e e ea x u w v = (6.2)

34

Let’s consider σ a scheduling variable. It is function of w and y . We then affirm that:

Definition 6.1 (equilibrium family): the functions ( )ex σ , ( )eu σ , ( )ew σ

and ( )ev σ define an equilibrium family on the set S for the process described

by Equation (6.1) if:

( ( ), ( ), ( ), ( )) 0e e e ea x u w vσ σ σ σ = Sσ ∈ (6.3)

Associated with this equilibrium family there is the error equilibrium family:

1( ) ( ( ), ( ), ( ), ( ))e e e e ez c x u w vσ σ σ σ σ= Sσ ∈ (6.4)

and the measured equilibrium family:

2( ) ( ( ), ( ), ( ))e e e ey c x w vσ σ σ σ= Sσ ∈ (6.5)

For example, if the target is that ( )z t tracks a constant reference input r with zero

steady state error, then we should have:

( ( ), ( ), ( ), ( )) 0e e e ea x u w vσ σ σ σ =

( ) 0ez σ =

and σ should include r . Note that ( ( ), ( ))g y t w tσ = . The variable σ is treated as

a parameter throughout the process design, and then it becomes an input signal to the gain scheduler due to the mentioned dependence: ( ( ), ( ))g y t w tσ = . Here stays the main trick of the gain scheduling techniques. It is clear that the definition of equilibrium points and the selection of scheduling signals depend on the particular problem to be solved.

Corresponding to an equilibrium family there is a plant linearization:

1 2

1 11 12

2 21

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

x xA B B

z C D D z

C Dy y

δ δ

δ δ

δ δ

σ σ σσ σ σσ σ

=

�

��

Sσ ∈ (6.6)

with, for example:

1( ) ( ( ), ( ), ( ), ( ))e e e e

cA x u w v

xσ σ σ σ σ∂=

∂ (6.7)

112 ( ) ( ( ), ( ), ( ), ( ))e e e e

cD x u w v

uσ σ σ σ σ∂=

∂ (6.8)

and with the deviation variables defined as:

35

( ) ( ) ( )ex t x t xδ σ= − , ( ) ( ) ( )eu t u t uδ σ= − . (6.9)

Thus, for each fixed σ , Equation (6.6), that is the linearization, describes the local

behavior of the non-linear model of Equation (6.1), about the corresponding equilibrium.

Given a set of equilibrium values 1,.., Kσ σ of the equilibrium family Sσ ∈ , let’s assume to have evaluated a linear controller for each value according to the specifications. Thus we describe a linear controller family as:

( )c ck

k kc c

x xF G

H Eu u

δ δ

δ δ

σ =

�

� 1,..,k K= (6.10)

then, the linear parameter-varying controller is based on the set of controllers of

Equation (6.10). Usually interpolation between indexed controllers is done, with respect to σ .

Based on Equation (6.10), we need a controller of the form:

( , , )c cx f x y w=� , ( , , )cu h x y w= . (6.11)

At the equilibrium, Equations (6.1), (6.2) and (6.10) show that there must be a function ( )cx σ� so that:

( ( ), ( ), ( )) 0c

e e ef x y wσ σ σ = (6.12)

( ) ( ( ), ( ), ( ))ce e e eu h x y wσ σ σ σ= (6.13)

these equations require, for example, that:

( )[ ( )] ( )[ ( )] ( )c ce eu H x x E y y uσ σ σ σ σ= − + − + ,

( ( ), ( ), ( )) ( )ce e eh x y w E

yσ σ σ σ∂ =

∂ Sσ ∈ (6.14)

The most direct way to satisfy these equations is to consider the following

controller: ( )[ ( )] ( )[ ( )]c c c

ex F x x G y yσ σ σ σ= − + −� ,

( )[ ( )] ( )[ ( )] ( )c ce eu H x x E y y uσ σ σ σ σ= − + − + . (6.15)

A real implementation of Equation (6.15) is simply achieved by replacing

( ( ), ( ))g y t w tσ = . Thus we obtain the gain scheduled controller:

( ( , ))[ ( ( , ))] ( ( , ))[ ( ( , ))]c c cex F g y w x x g y w G g y w y y g y w= − + −� , (6.16)

( ( , ))[ ( ( , ))] ( ( , ))[ ( ( , ))] ( ( , ))c ce eu H g y w x x g y w E g y w y y g y w u g y w= − + − + .

36

Chapter 7

Case study

7.1 – Presentation A multivariable, non-linear model of a plant has been studied. A proposed control

scheme, based on fuzzy gain scheduling is presented. Let’s consider Figure 7.1. The overall control system is supposed to work as follow.

From the system model different linearizations have been calculated, in correspondence to a certain number of points. These points are, hence, work points.

We need to specify that we use the term “work point” to indicate a point on which the linearization is computed, while with “set point” we refer to the operating conditions of the system.

The choice of work points has been led by the non-linearities of the process. This means that characteristics at steady state and relationships between input and output have been evaluated from a generic linear approximation of the model. Since the linearizations are supposed to approximate the non-linear behavior, where strong non-linearities have been found the layout of work point has been chosen to be thicker.

Therefore every work point has a range of “validity” in which the difference between the real characteristic and its approximation is small, with respect to the involved values. The bigger is the number of these intervals (i.e. more work points are chosen), the more the approximations match the real model.

The number of linear pieces cannot be expanded indefinitely as one could think to do in order to gain an even better approximation, in fact a problem arises in the conjunction between intervals, as it is explained below.

For those ranges of linearity control parameters have been calculated using a tuning algorithm. In the roughest way to mean gain scheduling technique, we can state that: as long as the set point stays within a certain range, control parameters are kept the same.

When a set-point change is detected, the scheduler switches from one work point to another, in the way that the new set point belongs to the range of the new work point. These “switching action” are strongly non linear and they are sources of troubles.

In fact two events occur simultaneously: one is the change in the set point, and the other is the subsequent change of the “linear” control parameters, due to the fact that the new set point resides in the range of another work point.

37

2

Out2

1 Out1

Process PI Controller

Fuzzy Tuner

Fuzzy Scheduler

2

In2

1 In1

pKiK

εδ

Figure 7.1: General control scheme overview.

These control parameters are optimized for the work point, which reasonably

corresponds to the middle point of each interval. Thus, when the set point moves from one range to a neighborhood range, the system is in the worst operating condition (that is the distance between the work point and the real set point is the largest, relatively to the interval length).

Another consideration that one should keep on mind, is that the process we are handling is not linear. Therefore better or, more realistically, even worse occurrences can take place, depending strictly on the local non-linearity in the process.

Fuzzy logic is used in the scheduler to deal with these non-linearities. The idea is that best control parameters cannot be obviously the same for the whole interval in which they are defined. Moreover the conjunctions between intervals might need some specific evaluations. In fact, the error of the linearization, i.e. the difference between it and the real value can be relatively high.

Nevertheless note that the different derivatives that two linearizations have in the border point between two intervals do not affect the possible hitch that can be found in that point. In fact the parameters concern only the work points. Therefore the fuzzy scheduler works modifying in a fuzzy way parameters between intervals, so that the ability of fuzzy logic to fit non-linear functions is widely involved.

Thus, when the input signal changes, that is a new set point is defined, the fuzzy scheduler receives a new input. This input is then processed in the fuzzy system so that the output is proposed.

The controller chosen is a multivariable PI controller. Its control parameters are the two gain matrices, pK and iK .

As those matrices are defined (Pettinen-Koivo method), parameters for fine-tuning have to be calculated. Experimentally it can be seen that usually a stronger control action is achieved by choosing high values of fine-tuning parameters, while small values trigger weaker actions.

38

In fact, the gain scheduling, even being fuzzy, needs an additional tuning, because of the linearization, whose implications we already pointed out. Furthermore, the presence of two different interfering channels makes the control even harder. In the proposed solutions those fine-tuning parameters have been made variable with the set points through a fuzzy system. They are supposed to take care of the real fine-tuning of the system. They have been chosen to be equal, because it was figured out that different values could engender oscillations and instability.

A fuzzy system has been developed for optimizing the choice of their values depending on the requested control actions.

The whole development of the control system is based on linearization of the involved process. Therefore a mathematical model of the process is not actually demanded. The data source can be only experimental information, and from them a linear model can be developed. This feature is remarkable because it shows the way to use multivariable fuzzy gain scheduling techniques in industrial applications.

7.2 – The process The considered process is governed by the following equations system:

1 2 1

2 1 2 2 1 1

1 1

2 2

( ) ( ) ( )

( ) [4.75 4.5 ( )] ( ) 0.7[ ( ) ( )] 0.25 ( )

( ) ( )

( ) ( )

x t x t u t

x t x t u t x t u t x t

y t x t

y t x t

= − = − − − − = =

�

� (7.1)

where ( )iu t are inputs, ( )iy t are outputs and ( )ix t are state variables. A feasible

block-type representation is shown in Figure 7.2. The generic linearization, is:

1 1 1

20 102 2 2

0 1 1 0

4.5 0.25 0.7 0.7 4.75 4.5

x x u

u xx x u

− = + − − − −

��

1 1

2 2

1 0

0 1

y x

y x

=

(7.2)

0ix and 0iu are points in which the linearization is computed.

At steady state, i.e. 1 2 0x x= =� � , it comes out that:

2 1

101 2

20

4.75 4.5

4.5 0.25

x u

xx u

u

= − = −

(7.3)

39

x2

x1 4.75-4.5x1

(4.75-4.5x1)u2

.25x1 .7(x2-u1)

x1’ Input 1

Input 2

Output 1

Output 2

s 1

I

s 1

.25

.7

4.5

4.75

Figure 7.2: Studied process, represented in Matlab.

The linearization of 1 2( )x f u= has been achieved through five linear pieces, as it is

shown in Figure 7.3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.2

0.4

0.6

0.8

1

1.2

1.4

u2

x1

Figure 7.3: Proposed linarization for 1 2( )x f u=

Not all elements of gain matrices are varying with the work points, but this is

strictly related to the process. In Table 7.1 are represented the values assumed by gain matrices elements in different intervals of linearity.

40

Zone 1

Zone 2

Zone 3

Zone 5

Zone 5

p11k 1 1 1 1 1 p

12k 0.2662 0.3965 0.6691 1.2471 2.429 p

21k 0.1996 0.3418 0.5711 0.9922 1.9582 p

22k 0.28513 0.4883 0.8159 1.4174 2.7975 i11k 0 0 0 0 0 i12k 1 1 1 1 1 i21k 0.10015 0.2869 0.8208 2.4368 9.7758 i22k 0.0651 0.1224 0.4155 1.3409 5.0175

Table 7.1: Values of gain matrices elements for each interval (“Zone”) of linearity

7.3 – The fuzzy gain scheduler The gain scheduler detects the input signal, i.e. the set point, and it produces, as

output, the values for elements of gain matrices of the PI multivariable controller. Therefore it has, as input signals, the set points of the system. As output, it has the

mentioned matrices pK and iK . A representative scheme is shown in Figure 7.4.

1

1

ki11 ki12 ki21 ki22

kp11 kp12 kp21 kp22

Set Point 1

Set Point 2

Mux

Fuzzy Logic Gain Scheduler

Demux

In1 Out1

DeNormalization 2

In1 Out1

DeNormalization 1

Figure 7.4: Fuzzy gain scheduler

The wires between logical boxes are vector-cables. In that figure there are normalization blocks. This is due to a design choice. In fact,

for each matrix, the parameters are calculated in the same way. Thus the normalized interval [0,1] has been chosen as the fuzzified universe of discourse. Then a de-normalization step is required to obtain the real values. Consequently Equation (4.7) has been applied to each parameter.

41

Thus: since control parameters come from a linear model, which is the generic approximation of the system, and since the tuning equations are linear, it follows that the relationships between set point and parameters are linear.

Therefore normalizations blocks contain the implementation of Equation (4.7), with [0,1] being the normalized interval, and with maximum and minimum of output values corresponding to the equivalent values taken into consideration in the set point.

Note that a fuzzy system is non-linear, but it doesn’t affect the linear mapping from an interval to another one.

Membership functions of the scheduler are shown in Figure 7.5. The rule base is simply a correspondence between intervals (fuzzy sets) and control parameters.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

Y s p

Deg

ree

of m

embe

rshi

p

m f1 mf2 mf3 mf4 mf5

Figure 7.5: Membership function for input variable in the fuzzy scheduler

7.4 – The fuzzy fine-tuning The control parameters turned out by the scheduler are supposed to present their

best effects during set point changes, like steps signals. The choice of the parameters comes from the Pettinen-Koivo algorithm. For this algorithm the more the frequency of the signals is high, the more one could expect good performances. Therefore during input step signals the fuzzy scheduler is expected to work properly. However, due to approximations from the non-linear model, the parameters cannot achieve optimal results. The scheduler, as it has been concerned in this work, takes as inputs only the set points. Therefore the presence of a fine-tuning system is compulsory.

The fine-tuning parameters δ and ε are the choice as output of the fuzzy tuner. As inputs, set points and measured errors are considered.

42

e2

de2

e1

de1

1

d,e

Fuzzy Logic Tuner

1 1

Discrete Filter

1 1

Discrete Filter

1-z -1

1

Discrete Filter

1-z -1

1

Discrete

4

Y2

3

Ysp2

2

Y1

1

Ysp1

Figure 7.6: Fuzzy tuner

A rule base has been developed and successfully applied to the systems.

Experimentally it was discovered that δ and ε performed better if taken of the same value. Thus a common rule base had to be studied. Difficulties in choosing rule-base were related mostly to the fact that contrasting actions can be needed sometimes. This means, for instance, that the tuner could have to manage opposite control situations. A problem, in this case, arises in selecting which action the controller should prefer.

If the knowledge on the system is not enough deep, there is no way to be sure that the action the controller is acting will be, in the best case, the right one. The risk is to generate, with a strong and incorrect action, oscillations and possibly the failure of the control.

An alternative is what we might call a “do nothing” procedure. The idea starts from the hardest case, which is when the two channels need opposite kinds of action. The central consideration is that the system is already going, or at least we presume it, to a stable situation, due to the scheduling. “Do nothing” means: “wait, and see how the overall system behave”. In fact the tuning algorithm, by itself is supposed to be enough.

When something changes (i.e. the derivative of one error is going to zero or, even better, it changes sign, that is the correspondent error is decreasing), and the worst situation is running out, the controller may start “doing something” that is: changing the parameters.

When the two channels still require different actions, but one of them needs a softer action, it has been chosen the control action to prefer more the demand of the channel that needs more control.

43

When measured errors are going in the same “direction”, that is they have similar relative bigness, and derivative, obviously there is no doubt in choosing the control action, because there is not ambiguity.

Therefore let’s summarize the inspiring guidelines for the tuner: If the errors need strongly opposite control actions, then “do nothing”; If the errors need opposite actions but not heavily, then perform a soft action that is

a mid point between the requests; If the errors are coherent, then perform the action, as it is demanded. Note that even for coherent control actions there could be a problem with the

requested power. We apply the same rule as for “weak” contradictories actions, That is: evaluate a medium action, softly moderated, to avoid instability warning.

For the fuzzy system, as we already mentioned, inputs are errors and their changes. The output is a normalized gain. After the normalization it represents the values of δ and ε .

The fuzzy rule-base has been practically developed as follow: firstly the need of every channel has been fuzzified individually. Then the two rule-tables have been combined following the principles shown above, obtaining one 7 7× table. Therefore 39 rules have been imposed.

Abbreviations:

BB=”big-big” BM=”big-medium”

BS=”big-small” U=”unitary”

SB=”small-big” SM=”small-medium”

SS=”small-small” DC=”Do nothing”

( )e t∆

( )e t

Positive

(P)

Zero (Z)

Negative

(N)

Positive (P)

BB

BM

BS

Zero (Z)

U

U

U

Negative

(N)

SB

SM

SS

Table 7.1: Rule base of each single variable

44

Table 7.2: Combination of two rule-bases in one table.

SS SM SB U BS BM BB

1 0 Figure 7.7: Correspondences between linguistic set and numerical values.

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1

0

0 . 2

0 . 4

0 . 6

0 . 8

1

e r r o r 1

Deg

ree

of m

embe

rshi

p

n e g a t i v e z e r o

p o s i t i v e

Figure 7.8: Membership function for 1( )e t in the fuzzy tuner

SS SM SB U BS BM BB

SS SS SM SM SM SM SB DC

SM SM SM SB SB SB DC BS

SB SM SB SB DC DC BS BM

U SM SB DC DC DC BS BM

BS SM SB DC DC BS BS BM

BM SB DC BS BS BS BM BM

BB DC BS BM BM BM BM BB

45

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

derr1

Deg

ree

of m

embe

rshi

p negative

zero positive

Figure 7.9: Membership function for 1( )e t∆ in the fuzzy tuner

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

err2

Deg

ree

of m

embe

rshi

p

negative zero

positive

Figure 7.10: Membership function for 2 ( )e t in the fuzzy tuner

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

derr2

Deg

ree

of m

embe

rshi

p

negative zero

positive

Figure 7.11: Membership function for 2 ( )e t∆ in the fuzzy tuner

46

7.5 – Trial and error: manual adjustments During simulation good performances haven’t been reached immediately. We

would certainly have been disappointed if the contrary happened. Fuzzy systems parameters needed a not negligible adjustments phase. For the fine-

tuner our main attention was pointed on the fuzzy sets that characterize the values of errors, changes in errors and on the normalization of the output fine parameters, as in an ordinary procedure.

Nevertheless, additionally manual settings have been done: each element of the matrices pK and iK have been inspected and varied to find out their influence on the

closed loop system. This means that, given the closed loop system CLS , the following

dependence has been investigated:

( )xCL CL ijS S k= (7.4)

where x

ijk (with ,x p i= ) denotes the generic element of matrices pK and iK .

For this purpose, a sensitivity function:

CLxij

Sk

∂∂

(7.5)

can be considered. This task is strictly related to the particular process under control

and no general rules are applicable. Therefore analytical evaluations could be given for example in terms of step response, i.e. rise time, overshoot, settle time etc.

Obviously, the fact that the process is multivariable and non linear is an enormous obstacle in performing such kinds of analysis, because the influence of an element can be extreme in some operating points and negligible in others. Therefore the uncertainty in obtaining useful results doesn’t justify an analytical investigation. For that reason trial and error is still the most reasonable procedure.

What might be found out in this way is that the overall system performs better if some elements of gain matrices pK and iK are multiplied by constant values, which

can be greater or smaller than the unit. With regard to the gain scheduler, what is mainly important is that the linearization

is enough detailed, that is: the maximal error between process and its approximation is small. If the fine-tuning doesn’t make available satisfactory results it is requested to revisit the linearization. We would highlight that this task is anything but easy: interferences between variables are often unpredictable.

47

7.6 – Simulation and results In order to test the system performances, random sources has been put as inputs on

both of the variables. Those sources vary within the whole range of scheduling. Moreover no limits have been put in the height of the steps. In that way they could

be very affecting the system, so that the system had to work in all possible and hard operating conditions.

It’s worth to put our attention on Figure 7.13. Sometimes the output seems to have a spikes type of behavior. As it is possible to see comparing Figures 7.13 and 7.14, those spikes take place when a big change, that is what sort of “long” step, in the value of the other variable, occurs. Thus they are only consequences related to the non-linearity of the process. In fact in these singletons the rise time (or the slope time, if the singleton points downwards) is just about zero, while the slope time (rise time in the second case) is greater and it corresponds to the system reaction.

In order to have a comparison for the present results with other controllers we developed a fuzzy gain scheduler based on a simpler and less accurate linearization. In this case, in fact, only three linear pieces have been used. Thus the fuzzy scheduler has three work points to manage. Performances decrease, due to the fact that the control parameters are less accurate. Figures 7.15 and 7.16 show the same input sequences applied to the simplified scheduled system.

Another comparison has been made excluding the fuzzy fine tuner from the control system. The results, as it is viewable in Figures 7.17 and 7.18, are very clear: the fine tuning parameters are fundamental in achieving reasonable performances. Without any fine-tuning the step response keeps oscillations for a longer time, and those oscillations are much more wide than in the proposed solution.

0 0.2 0.4 0.6 0.8 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u2

x1

Figure 7.12: Less accurate linearization

48

0 50 100 150 200 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

time

y1

Figure 7.13: Random step signal response for input 1( )y t

0 50 100 150 200 0

0.2

0.4

0.6

0.8

1

1.2

time

y2

Figure 7.14: Random step signal response for input 2 ( )y t

49

0 50 100 150 200 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

time

y1

Figure 7.15: Output 1( )y t : the same input sequence applied to a simplified fuzzy gain scheduling

scheme.

0 50 100 150 200 0

0.2

0.4

0.6

0.8

1

1.2

time

y2

Figure 7.16: Output 2( )y t : the same input sequence applied to a simplified fuzzy gain scheduling

scheme.

50

0 50 100 150 200 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time

y1

Figure 7.17: Performances obtained excluding the fuzzy fine-tuner ( 1( )y t )

0 50 100 150 200 0

0.2

0.4

0.6

0.8

1

1.2

time

y2

Figure 7.18: Performances obtained excluding the fuzzy fine-tuner ( 2( )y t )

51

Chapter 8

Conclusions An innovative fuzzy gain scheduler has been designed for a multivariable 2 2×

non-linear system. In order to control a non-linear multivariable process, a conventional scalar controller is not sufficient because it doesn’t know interactions in the process.

Interferences between the two variables have to be taken into consideration, therefore an advanced control strategy is required, and the scheduler needs a tuning algorithm. In the proposed solution, a rough tuning is built off-line in scheduler, and a fine-tuning is performed by another fuzzy system. The combination of those fuzzy systems allows the overall system to work properly.

Experience and intuition of the designer are strongly involved for at least two reasons: the former is that fuzzy logic essence recalls the human know-how; the latter is that this branch of control theory is being developed in these years, and systematic procedures don’t exist yet. Future works in this field are, nowadays, limited only by the brainpower of designers.

The final target of those systems is an implementation in industrial processes. In fact, the use of linearized model of the original process for building the controller, suits sublimely with the lack of rigorous industrial process models, especially when those industrial processes are strongly non-linear. Super visioning the control systems therefore usually guarantees stability and performances. Fuzzy logic helps in this task because it gives an intuitive interface between human operators and the controllers. This means also that possibilities for future works, besides improving gain scheduling techniques, can be oriented in investigating thoroughly stability analysis.

The results, as it has been found out in this work, are encouraging.

52

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HELSINKI UNIVERSITY OF TECHNOLOGY CONTROL ENGINEERING LABORATORY Editor: H. Koivo Report 108 Ojala, P., Design and Control of the Piezo Actuated Micro Manipulator. December 1997. Report 109 Niemi, A. J., Berndtson, J., Karine, S., Automatic Control of Paper Machine by Dry Line Measurement. December 1997. Report 110 Ylén, J-P, Nissinen, A. S., Sumean logiikan sovelluksen kehistysprosessi ja sen soveltaminen fuzzyTECH-ohjelmiston evaluointiin.

December 1997. Report 111 Hyötyniemi, H., Mental Imagery: Unified Framework for Associative Representations. August 1998. Report 112 Hyötyniemi, H., Koivo, H. (eds.), Multivariate Statistical Methods in Systems Engineering. December 1998. Report 113 Robyr, S., FEM Modelling of a Bellows and a Bellows-Based Micromanipulator. February 1999. Report 114 Hasu, V., Design of Experiments in Analysis of Flotation Froth Appearance. April 1999. Report 115 Nissinen, A. S., Hyötyniemi, H., Analysis Of Evolutionary Self-Organizing Map. September 1999. Report 116 Hätönen, J., Image Analysis in Mineral Flotation. September 1999. Report 117 Hyötyniemi, H., GGHA Toolbox for Matlab. November 1999. Report 118 Nissinen, A. S. Neural and Evolutionary Computing in Modeling of Complex Systems. November 1999. Report 119 Gadoura, I. A. Design of Intelligent Controllers for Switching-Mode Power Supplies. November 1999. Report 120 Ylöstalo, T., Salonen, K., Siika-aho, M., Suni, S., Hyötyniemi, H., Rauhala, H., Koivo, H. Paperikoneen kiertovesien konsentroitumisen vaikutus mikrobien kasvuun. September 2000. Report 121 Cavazzutti, M. Fuzzy Gain Scheduling of Multivariable Processes. September 2000. ISBN 951-22-5281-3

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