`Advances in Fuzzy SystemsVolume 2008 (2008), Article ID 421525, 15 pageshttp://dx.doi.org/10.1155/2008/421525`
Research Article

## On Controllability and Observability of Fuzzy Dynamical Matrix Lyapunov Systems

Department of Applied Mathematics, Acharya Nagarjuna University, Nuzvid Campus, Nuzvid 521 201, Andhra Pradesh, India

Received 27 September 2007; Accepted 1 January 2008

Copyright © 2008 M. S. N. Murty and G. Suresh Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We provide a way to combine matrix Lyapunov systems with fuzzy rules to form a new fuzzy system called fuzzy dynamical matrix Lyapunov system, which can be regarded as a new approach to intelligent control. First, we study the controllability property of the fuzzy dynamical matrix Lyapunov system and provide a sufficient condition for its controllability with the use of fuzzy rule base. The significance of our result is that given a deterministic system and a fuzzy state with rule base, we can determine the rule base for the control. Further, we discuss the concept of observability and give a sufficient condition for the system to be observable. The advantage of our result is that we can determine the rule base for the initial value without solving the system.

#### 1. Introduction

The importance of control theory in applied mathematics and its occurrence in several problems such as mechanics, electromagnetic theory, thermodynamics, and artificial satellites are well known. In general, fuzzy systems are mainly classified into three categories, namely pure fuzzy systems, T-S fuzzy systems, and fuzzy logic systems, using fuzzifiers and defuzzifiers. In this paper, we use fuzzy matrix Lyapunov system to describe fuzzy logic system. The purpose of this paper is to provide sufficient conditions for controllability and observability of first-order fuzzy matrix Lyapunov system modeled by where is an fuzzy input matrix called fuzzy control and is an fuzzy output matrix. Here , and are matrices of order , whose elements are continuous functions of on .

The problem of controllability and observability for a system of ordinary differential equations was studied by Barnett and Cameron [1] and for matrix Lyapunov systems by Murty et al.[2]. Fuzzy control usually decomposes a complex system into several subsystems according to the human expert's understanding of the system and uses a simple control law to emulate the human control strategy.There exist two major types of fuzzy controllers, namely Mamdani fuzzy controllers and Takagi-Sugeno (TS) fuzzy controllers. They mainly differ in the consequence of fuzzy rules: the former uses fuzzy sets whereas the latter employs (linear) functions. Takagi and Sugeno [3, 4] propose a type of fuzzy model in which the consequent part of the rules is defined not by the membership function but by a crisp analytical function. More and more interest appears to shift towards TS fuzzy controllers in recent years, as evidenced by the increasing number of papers in this direction and due to their applications in real world problems (e.g., [512]).

Recently, the controllability and observability criteria for fuzzy dynamical control systems were discussed by Ding and Kandel [13, 14]. In this paper, by converting the fuzzy matrix Lyapunov system into a Kronecker product system we obtain sufficient conditions for controllability and observability of the system (1) satisfying (2).

The paper is well organized as follows. In Section 2, we present some basic definitions and results relating to fuzzy sets [13] and Kronecker product of matrices. Further, we obtain a unique solution of the system (1), when is a crisp continuous matrix. In Section 3, we generate a fuzzy dynamical Lyapunov system, and also obtain its solution set. In Section 4, we present a sufficient condition for the controllability of the system and illustrate the results by suitable examples. In Section 5, we obtain a sufficient condition for the observability of the fuzzy dynamical Lyapunov system, and the theorem is highlighted by a suitable example. Finally, in Section 6, we present some conclusions and future works.

This paper extends some of the results of Ding and Kandel [13, 14] developed for system of fuzzy differential equations to fuzzy matrix Lyapunov systems and includes their results as a particular case, when , and are column vectors of order .

#### 2. Preliminaries

In this section, we present some definitions and results relating to fuzzy sets [13] and Kronecker product of matrices.

Let denote the family of all nonempty compact convex subsets of . Define the addition and scalar multiplication in as usual. Radstrom [15] states that is a commutative semigroup under addition, which satisfies the cancellation law. Moreover, if and , then and if , then . The distance between and is defined by the Hausdorff metric where

Definition 1. A set-valued function is said to be measurable if it satisfies any one of the following equivalent conditions: (1)for all , is measurable,(2), where are Borel -field of and , respectively (Graph measurability),(3)there exists a sequence of measurable functions such that , for all (Castaing's representation).

We denote by the set of all selections of that belong to the Lebesgue Bochner space , that is, We present the Aumann's integral as follows:

We say that is integrably bounded if it is measurable and there exists a function , such that , . From [16], we know that if is a closed valued measurable multifunction, then is convex in . Furthermore, if is integrably bounded, then is compact in .

Let where (i) is normal, that is, there exists an such that ;(ii) is fuzzy convex, that is, for and , (iii) is upper semicontinuous;(iv) is compact. For , the -level set is denoted and defined by . Then, from (i)–(iv) it follows that for all .

Define by where is the Hausdorff metric defined in . It is easy to show that is a metric in and using results of [15], we see that is a complete metric space, but not locally compact. Moreover, the distance verifies that We note that is not a vector space. But it can be imbedded isomorphically as a cone in a Banach space [15].

Regarding fundamentals of differentiability and integrability of fuzzy functions, we refer to Kaleva [17] and Lakshmikantham and Mohapatra [18].

In the sequel, we need the following representation theorem.

Theorem 1 (see [19]). If , then (1) , for all ; (2) , for all ; (3) if is a nondecreasing sequence converging to , then . Conversely, if is a family of subsets of satisfying (1)–(3), then there exists a such that for and .

A fuzzy set-valued mapping is called fuzzy integrably bounded if is integrably bounded.

Definition 2. Let be a fuzzy integrably bounded mapping. The fuzzy integral of over denoted by is defined level-set-wise by
Let ,and consider the fuzzy differential equation

Definition 3. A mapping is a fuzzy weak solution to (13) if it is continuous and satisfies the integral equation If is continuous, then this weak solution also satisfies (13) and we call it fuzzy strong solution to (13).

Now, we present some properties and rules for Kronecker products and basic results related to matrix Lyapunov systems.

Definition 4 (see [2]). Let and .Then the Kronecker product of and written is defined to be the partitioned matrix which is an matrix and is in .

Definition 5 (see [2]). Let ; one denotes The Kronecker product has the following properties and rules [2].
(1) ( denotes transpose of ).(2).(3)The mixed product rule(. This rule holds, provided the dimension of the matrices is such that the various expressions exist. (4).(5)If and are matrices, then.(6).(7)If and are matrices both of order , then (i),(ii).

Now by applying the Vec operator to the matrix Lyapunov system (1) satisfying (2) and using the above properties, we have where is an matrix and , are column matrices of order .

The corresponding linear homogeneous system of (17) is

Lemma 1. Let and be the fundamental matrices for the systems respectively. Then the matrix is a fundamental matrix of (19) and the solution of (19) is .

Proof. Consider Also .
Hence, is a fundamental matrix of (19). Clearly, is a solution of (19).

Theorem 2. Let and be the fundamental matrices for the systems (20) and (21). Then the unique solution of the initial value problem (17) is given by

Proof. First we show that the solution of (17) is of the form , where is a particular solution of (17) and is given by
Let be any other solution of (17), write , then satisfies (19), hence , .
Consider the vector , where is an arbitrary vector to be determined so as to satisfy (17), Hence, the desired expression follows immediately by noting the fact that and .

#### 3. Formation of Fuzzy Dynamical Lyapunov Systems

Let , , , and define where is the -level set of . From the above definition of and Theorem 1, it can be easily seen that .

Now by using the fuzzy control , we show that the following system determines a fuzzy system.

Assume that is continuous in . The set is a convex and compact set in . For any positive number , consider the following differential inclusions: Let be the solution of (29) satisfying (30).

Claim (i). , for every , .
First, we prove that is nonempty, compact, and convex in . Since has measurable selection, we have that is nonempty.
Let , , , .
If for any , then there is a selection such that Then Thus is bounded.
For any , Therefore Since and are both uniformly continuous on , is equicontinuous. Thus, is relatively compact. If is closed, then it is compact.
Let and . For each , there is a such that Since is closed, then there exists a subsequence of converging weakly to . From Mazur's theorem [20], there exists a sequence of numbers , such that converges strongly to .
Thus, from (35) we have From Fatou's lemma, taking the limit as on both sides of (36), we have Thus, , and hence is closed.
Let , , then there exist such that Let , , then Since is convex, , we have that is . Thus is convex. Therefore, is nonempty, compact, and convex in . Thus, from Arzela-Ascoli theorem, we know that is compact in for every . Also it is obvious that is convex in . Thus, we have , for every . Hence the claim.

Claim (ii). , for all .
Let . Since , we have Thus, we have the selection inclusion and the following inclusion: Consider the differential inclusions Let and be the solution sets of (43) and (44), respectively. Clearly, the solution of (43) satisfies the following inclusion: Thus , and hence . Hence the claim.

Claim (iii). If is a nondecreasing sequence converging to , then .
Let , and consider the inclusions Let and be the solution sets of (46) and (47), respectively. Since is a fuzzy set and from Theorem 1, we have Consider and then . Therefore Thus, we have , , which implies that Let be the solution set to the inclusion Then, It follows that This implies that . Therefore, From (51) and (55), we have and hence, From Claims (i)(iii) and applying Theorem 1, there exists on such that is a solution set to the differential inclusions (29) and (30). Hence, the system (27), (28) is a fuzzy dynamical Lyapunov system, and it can be expressed as The solution set of the fuzzy dynamical system (58), (59) is given by

Remark 1. Consider a special case. If the input is in the form where , are crisp numbers, then the th component of the solution set of (27) is a fuzzy set in .

Proof. The proof follows along similar lines as in the above discussion.

#### 4. Controllability of Fuzzy Dynamical Lyapunov Systems

In this section, we discuss the concept of controllability of the fuzzy system (58) satisfying (59).

Definition 6. The fuzzy system (58), (59) is said to be completely controllable if for any initial state and any given final state there exists a finite time and a control , , such that .

Lemma 2. If is a fuzzy set, then .

Proof. Let be the -level set of . Since From the definition of fuzzy set, we have .

Lemma 3. Let be two fuzzy sets and let be a nonzero continuous function on , satisfying then .

Proof. For each -level, we have Suppose that , then for some , we have . Without loss of generality, we assume that . Let and . Then, we have either (i) or (ii) holds.
If (i) holds, then
If (ii) holds, then Thus, in both cases (i) and (ii), we have This implies that which is a contradiction to (64). Hence .

Definition 7 (see [20]). Let , , and let be the -level set of . One defines the sum of and by the difference between and by and the scalar product by

Definition 8 (see [20]). Let and , . If , then which is defined by If , then which is defined by

Definition 9 (see [20]). Let be an matrix, , let , , be a fuzzy set in , and let be -level sets of . Define the product of and as

All these definitions yield the following lemma.

Lemma 4. is a fuzzy set in .

Proof. The proof is similar to proof of Lemma 3.1 [13].

Theorem 3. The fuzzy system (58),(59) is completely controllable if the symmetric controllability matrix is nonsingular. Furthermore,the fuzzy control which transfers the state of the system from to a fuzzy state can be determined by the following fuzzy rule base: where

Proof. Suppose that the symmetric controllability matrix is nonsingular. Therefore exists. Multiplying on both sides of (76), we have Now our problem is to find the control such that Since is fuzzy and from Lemma 4, must be fuzzy, otherwise the fuzzy left side of (80) cannot be equal to the crisp right side. By Lemma 2, can be written as From (80) and (81), we have Combining (79) and (82), we have It follows that By using Lemma 3, we get Now we have two special cases for (85). First, let be a crisp point, then we will get a corresponding control , satisfying (85).
Second, let , then the corresponding control will take the form in wich the th component of is a fuzzy set in . Obviously, is in , the grade of the membership can be determined by , the grade of the membership of in . Thus, based on the above discussion, we have a fuzzy rule base for the control ,and is given by (77) and (78).

Remark 2. The nonsingularity of the symmetric controllability matrix in Theorem 3 is only a sufficient condition but not necessary because the fuzzy rule base cannot guarantee the nonsingularity of the controllability matrix.

Example 1. Consider the fuzzy dynamical matrix Lyapunov system (1) satisfying (2) with Let , where We select the points , , , and which are in , , , and with grades of the membership are 0.5, 0.75, 0.5, and 0.75, respectively. The fundamental matrices of (20), (21) are Now the fundamental matrix of (19) is Consider where . Therefore, Clearly, it is nonsingular.
Thus, from Theorem 3, the input can be chosen by the following -level sets: Hence, the -level sets of fuzzy control are given by and the corresponding control function to the point is

Example 2. Consider the fuzzy dynamical matrix Lyapunov system (1) satisfying (2) with Let , where We select the points , , , and which are in , , , and with grades of the membership being 0.5, 0.75, 0.25, and 0.5, respectively. The fundamental matrices of (20), (21) are