stabilization of fuzzy control systems via a common quadratic Lyapunov function have been presented ..... Countable or discrete universe X [Zimmermann, 1985]. n n. Ã. 2. 2. Ã ..... symbolic descriptions of what are in general time-varying quantities) will be used to describe fuzzy ...... Katsuhiko Ogata, N. J. Prentice-. Hall, third ...
Ministry of Higher Education and Scientific Research, Al-Nahrain University, College of Science
A Thesis Submitted to the Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, as a Partial Fulfillment of the Requirements for the Degree of Master of Science in Applied Mathematics
By
Sarmad Abdulhasan Jamil Salman (B.Sc. 2001)
Supervised By
Dr. Radhi Ali Zboon
August 2004
Jumady Althany 1425
ﺑِﺴ ِﻢ ِ اﷲ اﻟ ﱠﺮ ْﺣﻤ ِﻦ اﻟ ﱠﺮِﺣ ِﻴﻢ ِ◌ ْ
َﲰَﺎء َﻫ ُـﺆﻻء ﺿ ُﻬ ْﻢ َﻋﻠَﻰ اﻟْ َﻤﻼَﺋِ َﻜ ِﺔ ﻓَـ َﻘ َﺎل أَﻧﺒِﺌُ ِﻮﱐ ﺑِﺄ ْ آد َم اﻷ ْ َﲰَﺎء ُﻛﻠﱠ َﻬﺎ ﰒُﱠ َﻋَﺮ َ َو َﻋﻠﱠ َﻢ َ إِن ُﻛﻨﺘﻢ ﺻ ِﺎدﻗِﲔ ﴿ ﴾٣١ﻗَﺎﻟُﻮاْ ﺳﺒﺤﺎﻧَﻚ ﻻَ ِﻋ ْﻠﻢ ﻟَﻨﺎ إِﻻﱠ ﻣﺎ ﻋﻠﱠﻤﺘـﻨﺎ إِﻧﱠﻚ أَﻧﺖ اﻟْﻌﻠِ ﻴﻢ ُْ َ َ ُْ َ َ َ َ َ َ ْ ََ َ َ َ ُ اﳊ ِ ﻴﻢ ﻜ َﲰَﺂﺋِ ِﻬ ْﻢ ﻗَ َﺎل أَ َﱂْ أَﻗُﻞ ﻟﱠ ُﻜ ْﻢ ﺎ ﻳ ﺎل ﻗ ﴾ ٣٢ ﴿ َ َ َﲰَﺂﺋِ ِﻬ ْﻢ ﻓَـﻠَ ﱠﻤﺎ أَﻧﺒَﺄ َُﻫ ْﻢ ﺑِﺄ ْ آد ُم أَﻧﺒِْﺌـ ُﻬﻢ ﺑِﺄ ْ َ َ َْ ُ إِ ﱢﱐ أَﻋﻠَﻢ َﻏﻴﺐ اﻟ ﱠﺴﻤﺎو ِ ات َواﻷ َْر ِ ض َوأ َْﻋﻠَ ُﻢ َﻣﺎ ﺗُـْﺒ ُﺪو َن َوَﻣﺎ ُﻛﻨﺘُ ْﻢ ﺗَ ْﻜﺘُ ُﻤﻮ َن ْ ُ َْ ََ
﴿﴾٣٣
ﺴـورة اﻟﺒﻘرة اﯿﺔ ٣٣ - ٣١
Dedication
To… My dear father… Dr. AbdAlHassan Jameel Salman … The only person … I would give anything to have him beside me, at this moment, here, watching my success … I wish … He is seeing me now from heaven…
Supervisors Certification We certify that this thesis was prepared under our supervision at the department of mathematics and computer applications, College of Science, Al-Nahrian University as a partial fulfillment of the requirements for the degree of master in applied mathematics.
Signature: Name: Dr. Radhi Ali Zboon Date: /10/2004
In view of the available recommendations; I forward this thesis for debate by the examining committee.
Signature: Name: Asist. Prof. Dr. Akram M. Al-Abood Head of the department mathematics and computer applications Date: /10/2004
Acknowledgments Praise is to Allah, the cherisher and sustainer of the worlds. I would like to express my deep appreciation to my supervisor Dr. Radhi Ali Zboon for his patience and appreciable advice. I hope I will give him what he disserves from gratitude. Thanks are extended to the college of science of Al-Nahrain University for giving me the chance to complete my postgraduate study. Also I am very thankful for all the staff members of the department of mathematics and computer applications. Finally, I would like to thank my mother for her care and my family for giving me the aim I fought for.
Sarmad 25/8/2004 Baghdad
Abstract In this thesis, a new approach of fuzzy control for continuous nonlinear dynamical systems is proposed, based on the framework of Takagi-Sugeno fuzzy model and a common controller for all the local models generated by fuzzifying the nonlinear system. Some theorems giving sufficient conditions guarantee the simultaneous stabilization of fuzzy control systems via a common quadratic Lyapunov function have been presented and developed; their theoretical aspects have also been proved and discussed. Design algorithms, illustrative examples and graphs have been presented to show effectiveness of the approach.
Contents Acknowledgments. Abstract. Contents. Introduction.
Chapter One:
Basic Mathematical Concepts.
1.1
Introduction.
1
1.2
Some Basic Concepts of Fuzzy Set Theory.
2
1.2.1
Fuzzy Set.
2
1.2.2
Representation of Fuzzy Set.
2
1.2.3
Support of Fuzzy Set.
10
1.2.4
Height of Fuzzy Set.
10
1.2.5
Complement of Fuzzy Set.
10
1.2.6
Union and Intersection of Fuzzy Set.
10
1.2.7
Inclusion.
10
1.2.8
α-Cuts.
11
1.3
Linguistic Variables, Values, and Rules.
11
1.3.1
Universes of Discourse.
11
1.3.2
Linguistic Variables.
12
1.3.3
Linguistic Values.
12
1.3.4
Linguistic Rules.
13
1.4
Structure of Fuzzy Controller.
14
1.4.1
Preprocessing.
14
1.4.2
Fuzzification.
15
1.4.3
Rule Base.
16
1.4.4
Inference Engine.
16
1.4.5
Defuzzification.
16
1.4.6
Postprocessing.
20
1.5
Mamdani and Sugeno Controller.
21
1.6
Design of a Class of Fuzzy Control Systems.
23
Chapter Two:
Multi-Input Systems Design Problem.
2.1
Introduction.
25
2.2
Lyapunov Stability Analysis.
26
2.2.1
Lyapunov Direct Method.
27
2.2.2
Lyapunov Indirect Method.
28
2.3
Stability Analysis of Takagi-Sugeno Fuzzy Systems.
28
2.4
Block Controllable Companion Form.
31
2.5
Design Algorithm.
37
Chapter Three:
Uncertain Multi-Input Systems Design Problem.
3.1
Introduction.
56
3.2
Coordinates Transformation.
56
3.3
Matching Conditions.
57
3.4
Design Algorithm.
62
Conclusions and Future Studies.
81
References.
82
Introduction A fuzzy control system uses many ideas from the standard control methodology except in fuzzy control it is often said that a formal mathematical model is assumed unavailable so that mathematical analysis is impossible. While it is often the case that it is difficult, impossible, or cost-prohibitive to develop an accurate mathematical model for many processes, it is almost always possible for the control engineer to specify some type of approximate model of the process (after all, we do know what physical object we are trying to control). Indeed, it has been our experience that most often the control engineer developing a fuzzy control system does have a mathematical model available. While it may not be used directly in controller design, it is often used in simulation to evaluate the performance of the fuzzy controller before it is implemented. Certainly there are some applications where one can design a fuzzy controller and evaluate its performance directly via an implementation. In such applications one is not overly concerned with the failure of the control system (e.g., for some commercial products such as washing machines or a shaver) and therefore there may be no need for a mathematical
model
for
conducting
simulation-based
evaluations
before
implementation. In other applications there is the need for a high level of confidence in the reliability of the fuzzy control system before it is implemented. In addition to simulation-based studies, one approach to enhancing our confidence in the reliability of fuzzy control systems is to use the mathematical model of the plant and nonlinear analysis for (i) verification of stability and performance specifications and (ii) possible redesign of the fuzzy controller. Some may be confident that a true expert would (i) never need anything more than intuitive knowledge for rulebase design, and (ii) never design a faulty fuzzy controller. However, a true expert will certainly use all available information to ensure the reliable operation of a control i
system including approximate mathematical models, simulation, nonlinear analysis, and experimentation. We emphasize, however, that mathematical analysis cannot alone provide the definitive answers about the reliability of the fuzzy control system since such analysis proves properties about the model of the process, not the actual physical process. It can be argued that a mathematical model is never a perfect representation of a physical process; hence, while nonlinear analysis may appear to provide definitive statements about control system reliability, it is understood that such statements are only accurate to the extent that the mathematical model is accurate. Nonlinear analysis does not replace the use of common sense and evaluation via simulations and experimentation; it simply assists in providing a rigorous engineering evaluation of a fuzzy control system before it is implemented. It is important to note that the advantages of fuzzy control often become most apparent for very complex problems where we have an intuitive idea about how to achieve high performance control. In such control applications an accurate mathematical model is so complex (i.e., high order, nonlinear, stochastic, with many inputs and outputs) that it is sometimes not very useful for the analysis and design of conventional control systems (since assumptions needed to apply conventional control are often violated). The conventional control engineering approach to this problem is to use an approximate mathematical model that is accurate enough to characterize the essential plant behavior, yet simple enough so that the necessary assumptions to apply the analysis and design techniques are satisfied. However, due to the inaccuracy of the model, upon implementation the developed controllers often need to be tuned via the “expertise” of the control engineer. The fuzzy control approach, where explicit characterization and utilization of control expertise is used earlier in the design process, largely avoids the problems with ii
model complexity that are related to design (i.e., for the most part fuzzy control system design does not depend on a mathematical model. "Fuzzy control system design can depend on a mathematical model if one needs it to perform simulations to gain insight into how to choose the rule-base and membership functions"). However, the problems with model complexity that are related to analysis have not been solved (i.e., analysis of fuzzy control systems critically depends on the form of the mathematical model); hence, it is often difficult to apply nonlinear analysis techniques to the applications where the advantages of fuzzy control are most apparent. For instance, existing results for stability analysis of fuzzy control systems typically requires that the plant model be deterministic, satisfy some continuity constraints, and sometimes require the plant to be linear or “linear-analytic”. The current status of the field, as characterized by these limitations, coupled with the importance of nonlinear analysis of fuzzy control systems, make it an open area for research where an introductory survey can help establish the necessary foundations for a bridge between the fuzzy control and nonlinear analysis communities. The work in this thesis is divided to three chapters; the first chapter entitled “Basic Mathematical Concepts” gives the introductory material that is necessary to understand the next two chapters. In the second chapter theorem 2.1 has been developed, which describes a fuzzy controller that simultaneously quadratically stabilize the multi-input systems whose i th local model is (A i , B) via a common Lyapunov function V(x ) = x T W −1x . Afterwards a step by step design algorithm has been suggested. A practical example and illustration graphs for the local systems are presented. The third chapter can be considered as an extension to the work of the second chapter. The aim of this chapter is to discus the simultaneous quadratic stabilizability of the uncertain multi-input fuzzy dynamical systems whose iii
i th
local model
(A + A i , B + Bi ) ,
for that theorem 3.1 has been developed, which describes a fuzzy
controller that simultaneously quadratically stabilize the above uncertain multi-input fuzzy dynamical systems via a common Lyapunov function V(x ) = (Tx ) T W −1 (T x ) , using T-S fuzzy model and suitable coordinates transformation. A suggested step by step algorithm to design a fuzzy controller and an illustrative example and graphs are shown also. For the graphs and the calculations, we used the software “MathCad 2001 professional” on a “Pentium IV computer” in the laboratories of “Mathematics and Computers Applications Department” in the “College of Science” at “Al-Nahrain University”.
iv
Chapter one
Basic Mathematical Concepts
1.1 Introduction: Most of our traditional tools for formal modeling, reasoning and computing are crisp, deterministic and precise in character. By crisp we mean dichotomous, which is, yes-or-no type rather than more-or-less type. In conventional dual logic, for instance, a statement can be true or false and nothing in between. In set theory an element can either belong to a set or not, and in optimizing a solution is either feasible or not. Precision assumes that the parameters of a model represent exactly either our perception of the phenomenon modeled or the features of the real system that has been modeled [Zimmermann, 1985]. In the real world we distinguish three kinds of inexactness: Generality, which a concept applies to a variety of situations; Ambiguity, that it describes more than one distinguishable sub concept; and Vagueness, that precise boundaries are not defined. Generality occurs when the universe is not just one point; Ambiguity occurs when there is more than one local maximum of a membership function; and vagueness occurs when the function takes values other than just 0 and 1. All three types of inexactness are represented by a fuzzy set [Kandel, 1986]. The development in this field were along two lines: the first was as a formal theory which when maturing , became more sophisticated and specified and which was enlarged by original ideas and concepts as well as by "embracing" classical mathematical areas such as algebra, topology, and soon by generating (fuzzifying) them. The second was as a very powerful modeling language, which can cope with a large fraction of uncertainties of real life situations. Because of its generality, it can be well adapted to different circumstances and context. In many cases this will mean, however, the context-dependent modification and specification of the original concepts of the formal fuzzy set theory [Zimmermann, 1985]. This chapter will give the introduction to some basic concepts in both fuzzy set theory and fuzzy control theory which are necessary to understand the work in the following two chapters. 1
Chapter one
Basic Mathematical Concepts
1.2 Some Basic Concepts of Fuzzy Set Theory: In this section some basic definitions of fuzzy set theory will be presented for the sake of giving a mathematical background to deal with fuzzy control theory: 1.2.1 Fuzzy Set: ~ If X is a collection of objects denoted by x then a fuzzy set A in X is a set of
ordered pairs:
{
}
~ A = (x , µ A~ ( x ) ) x ∈ X
(1.1)
~ where µ A~ ( x ) is called the membership function or grade of membership of x in A
which maps X to the membership space M. (when M contains only the two points 0 ~ and 1, A is nonfuzzy and µ A~ ( x ) is identical to the characteristic function of nonfuzzy set.) the range of membership function is a subset of the nonnegative real numbers whose supremum is finite. Elements with a zero degree of membership are normally not listed [Zimmermann, 1985].
1.2.2 Representation of a Fuzzy Set: In the literature the reader can find different ways of denoting a fuzzy set. In what follows three of these ways shall be presented: The first way is by an ordered set of pairs, the first element of which denotes the object and the second the degree of membership [Zimmermann, 1985]. Example 1.1: The fuzzy set "Comfortable apartment for three persons" can be described as:
{
}
~ A = (x , µ A~ ( x ) ) x is the number of bedrooms
= {(1, 0.4 ), (2, 0.7 ), (3,1), (4,0.7 ), (5, 0.4 ), (6, 0.1)} 2
Chapter one
Basic Mathematical Concepts 1.2
Comfortablity
1
0.8
0.6
0.4
0.2
0
0
1
2
3 4 5 The Number of Bedrooms
6
7
Figure (1.1): Comfortable apartment for three persons.
The second way is represented solely by stating its membership function [Zimmermann, 1985]. Example 1.2: The fuzzy set "Real numbers larger than 5" can be described as follows: 0 1 µ A~ ( x ) = 1 + ( x − 5) − 2
x≤5 x >5
1.2
Degree of Membership
1
0.8
0.6
0.4
0.2
0
0
5
10 Real Number
15
Figure (1.2): Real numbers larger than 5 3
20
Chapter one
Basic Mathematical Concepts
The third way is by the following: a. Countable or discrete universe X [Zimmermann, 1985]. ~ µ ~ (x ) A= A 1 n
=∑
x1
µ A~ ( x i )
i =1
Where + satisfies
a
+
µ A~ ( x 2 )
x2
+ +
µ A~ ( x n )
xn
(1.2.a)
xi u
+ b u = max(a , b ) u , i.e., if the same element has two different
degrees of membership then its membership degree becomes the largest of them [Dimiter, 1996]. The symbol
∑
here stands for successive + operation.
Example 1.3: The fuzzy set "Integers close to 4" can be described as follows: ~ A = 0.1 + 0.4 + 0.8 + 1 + 0.8 + 0.4 + 0.1 1 2 3 4 5 6 7
1.2
Degree of Membership
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4 Integers
5
6
Figure (1.3): Integers close to 4
4
7
8
Chapter one
Basic Mathematical Concepts
b. Uncountable or Continuous universe X [Zimmermann, 1985]. ~ µ ~ (x ) A= A 1
=∫
x1
µ A~ ( x )
x
+
µ A~ ( x 2 )
x2
++
µ A~ ( x n )
xn
(1.2.b)
x
where the symbol ∫ here stands for successive + operation.
Example 1.4: The fuzzy set "Real numbers close to 4" can be described as follows:
(
~ A = ∫ 1 + (x − 4 )2
)
−1
x
R
1.2
Degree of Membership
1
0.8
0.6
0.4
0.2
0
10
6.5
3
0.5
4 7.5 Real Number
11
Figure (1.4): Real numbers close to 4
5
14.5
18
Chapter one
Basic Mathematical Concepts
Remarks 1.1: 1. The membership function is not limited to values between 0 and 1. If ~ ~ sup x µ A~ (x ) = 1 the fuzzy set A is called normal. A non-empty fuzzy set A can always be normalized by dividing µ A~ ( x ) by sup x µ A~ (x ) [Zimmermann, 1985].
2. There is a variety in the choice of the membership function depending on the mathematical background, mathematical model of the plant and designer expertise. In what follows we shall list some of this types: a. The increasing membership functions with straight lines we will call Γ − function , because of the similarity of these functions with this character. This function is a function of one variable and two parameters defined as follow (figure 1.5) [Dimiter, 1996]: Γ − function : The function Γ : U → [0,1] is a function with two parameters
defined as [Dimiter, 1996]:
0 Γ(u; α, β) = (u − α ) (β − α ) 1
uβ
1 ½ 0
α
β Figure (1.5): Γ-Function
6
(1.3)
Chapter one
Basic Mathematical Concepts
Zadeh’s S-function [Dimiter, 1996], define as: x≤α 0 2 2 x − α α< x ≤β γ−α S( x; α, β, γ ) = 2 x−γ 1 − 2 γ − α β < x ≤ γ 1 x>γ
(1.4)
Where β = (α + γ ) / 2 , can be considered as a more fluent variant of the Γ − functions (figure 1.6). It is frequently used in fuzzy logic, but only seldomly
in fuzzy control [Dimiter, 1996].
1 ½ 0
α
γ
β
Figure (1.6): Zadeh's S-function
b. The decreasing membership functions with straight lines we will call L − functions ; they are defined as follows (figure 1.7) [Dimiter, 1996]: L − function : The function L : U → [0,1] is a function with two parameters defined as [Dimiter, 1996]:
u < α, 1 L(u; α, β) = (β − u ) /(β − α) α ≤ u ≤ β 0 u >β
7
(1.5)
Chapter one
Basic Mathematical Concepts 1 ½ 0
α
β
Figure (1.7): L-function
c. Bell-shaped membership functions with straight lines we will call Λ − functions or triangular functions; they are defined as follow (figure1.8) [Dimiter, 1996]: Λ − functions : The function Λ : U → [0,1] is a function with three parameters
defined as [Dimiter, 1996]: x≤α 0 (u − α ) (β − α ) α < u ≤ β Λ (u; α, β, γ ) = (γ − u ) (γ − β) β < x ≤ γ 0 x>γ
(1.6)
1 ½ 0
α
β
γ
Figure (1.8): Λ-function
Zadeh's bell - shaped π − function [Dimiter, 1996] defined as: x≥γ S( x; γ − β, γ − β 2 , γ ) π( x; β, γ ) = 1 − S( x; γ, γ + β 2 , γ + β) x ≤ γ
(1.7)
can be considered as a more fluent variant of the Λ − function . Like the S − function given above, it is of low practical use in fuzzy control [Dimiter,
1996]. 8
Chapter one
Basic Mathematical Concepts
d. Approximating membership functions with straight lines, where the top is not one point but an interval, we will call Π − functions ; these are defined as follows (figure 1.9) [Dimiter, 1996]: Π − function : The function Π : U → [0,1] is a function with four parameters
defined as [Dimiter, 1996]:
0 (u − α ) (β − α ) Π (u; α, β, γ, δ ) = 1 (δ − u ) (δ − γ ) 0
u
