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Difference of application of fuzzy rough sets and probability random on target control

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Qinge Wu, Tuo Wang, Yongxuan Huang and Jisheng Li

Qinge Wu, Tuo Wang, Yongxuan Huang and Jisheng Li
Systems Engineering Institute School of Electronic and Information Engineering Xian Jiao Tong University, Xian, Shaanxi, 710049, P.R.China

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    Difference of application of fuzzy rough sets and probability random on target control Difference of application of fuzzy rough sets and probability random on target control Presentation Transcript

    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ Difference of Application of Fuzzy Rough Sets and Probability Random on Target Control QingE Wu 1, Tuo Wang 1, YongXuan Huang1 and JiSheng Li1 1. Xi’an Jiaotong University, P.R.China Systems Engineering Institute School of Electronic & Information Engineering 2006-9-11 p. 1 QingE Wu, Ph.D. Candidate, E-MAIL: wqe969699@163.com
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ Abstract With a brand-new theory, this paper not only provides the differences between fuzzy rough sets and probability random attribute in concept, formula expression and function type, but also introduces their differences in algorithm on target control for better evolving efficiency of the two. At the same time, it will be discussed how the two choose respectively the optimum radar to combine so that radars can better control target by an instance. The simulation results indicate that stochastic algorithm is more effectual in multi-target control than single fuzzy rough algorithm. Finally some problems of combination of fuzzy rough sets and probability randomness to be solved and development trends are discussed. Thus, we can know what kind of field they are suitable for applying, respectively, in order to accomplish better task by the investigation. p. 2 Abstract
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ 1. Introduction Probability randomness (PR) divides mostly the probability space for non-deterministic information based on probability, expectation mean theory and Bayes’ rule. PR main strengths include: they are rigorous, systematic, and particularly suitable for random sequential processing. However, fuzzy rough (FR) approaches offer a variety of alternatives with distinctive flexibilities that are valuable for handling many real- world problems based on fuzzy functions and membership degree. Like FR approaches, however, it does not provide a way of fusing old knowledge and new information as the Bayes’ rule does in the probabilistic setting. A main weakness of the FR methods stems from their lack of solid, systematic weight update, the FR relies on heuristic principles and expert systems for weight update. p. 3 Introduction
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ Research frame Previously, concept of FR and stochastic thing believed were identical, which stochastic phenomenon was fuzzy or rough. However, FRS and random events are different in nature and their differences are inherent. Two easy confusion knowledge points Research mainly The difference of FR and PR Collect information Difference of application Difference of theories p. 4 Research frame
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ Difference of application Signal processing Study Fuzzy signal processing General signal processing Using Using General Algebra, probability FS RS FRS Combination Relation on control? p. 5 Research frame
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ Relation on control? Study first the relation of Difference of theories FRS and probability on control. Study the differences of the This paper studies the question, two in concept, formula and parallels to compare the two expression and function in theories, and discusses mainly type their differences in algorithm on target control. p. 6 Research frame
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ 2. Mathematics base of FR control Definition 2.1 Let {U , R} be a given approximate space. Assume A ∈ FR (U ) and let L=[0,1]. N(A, B) is called a FR norm of A and B if the mapping N : FR (U ) × FR (U ) → L satisfies the following conditions: N ( A, B ) = N ( B, A) ⇔ N ( AL , BL ) = N ( BL , AL ) and N ( AU , BU ) = N ( BU , AU ) 0 ≤ N ( A, B ) ≤ 1 ⇔ 0 ≤ N ( AL , BL ) ≤ 1 and 0 ≤ N ( AU , BU ) ≤ 1 N (U , φ ) = 0, where U is a universe set, φ is an empty set p. 7 Mathematics base of FR control
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ A ⊆ B ⊆ C then If N ( A, C ) ≤ N ( A, B ) ∧ N (B, C ) ⇔ N ( AL , C L ) ≤ N ( AL , BL ) ∧ N (BL , C L ) N ( AU , CU ) ≤ N ( AU , BU ) ∧ N (BU , CU ) and U = {u1 , u 2 , , un } Theorem 2.1 If , then ∆1n N ( A, B ) = 1 − ∑ A ( ui ) − B ( ui ) n i =1 is a FR norm of FRS A and B , where A = AL , AU B = BL , BU ∈ FR (U ) ui ∈ U . p. 8 Mathematics base of FR control
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ Definition 2.2 A selected-near principle: If there is a pair of i0 and j0 {( } ( ) )( ) , N ( AL , BL ) to let N Ai0 , B = max N A1 , B , N A2 , B , n L L L L L L { } ( ) N AUj0 , BU = max N ( AU , BU ) , N ( AU , BU ) , , N ( AU , BU ) 1 2 n Definition 2.3 Let C is a conditional attribute set, D is a decision attribute set. Let Xi and Yj be each equivalence class of U/C and U/D respectively, where U/C and U/D signify respectively all equivalence : X i → Yj class of C and D. The decision rule is defined by: rij ρ ( X i ,Yj ) = Yj ∩ X i X i ∩ Yj ≠ φ Xi The decision degree is: p. 9 Mathematics base of FR control
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ 3. Mathematics algorithm of PR control 3.2 procedure of PR algorithm on target control: (1) Controller-conditioned reinitialization: Mixing estimate: ∆ (i ) = E[ xi (k − 1) mk , z k −1 ] = (i ) x k −1 k −1 ∑ E{xi (k − 1) m k −1 m k , z k −1 ) (i ) ( j) ( j) , z }P (m k −1 k −1 j (2) Controller-conditioned interaction: predicted state, control gain, Updated state: Λ ( i ) K k = Pk k −1 ( H k )′( S k ) −1 (i ) (i ) x k k −1 = Fk −1 x k −1 k −1 (i ) (i ) (i ) (i ) Λ (i ) ∆ Λ (i ) x k k = x k k −1 + K k(i ) zki ) ( p. 10 Mathematics algorithm of PR control
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ (3) Controller probability update: Based on Bayes formula, the ∆ , where µ 0 = µ = P(m (i ) (i ) (i ) k probability of a controller is: z) k P ( m0 = m ( i ) ) is known. (4) Overall output: The mixing overall estimate and covariance matrix calculated by the state estimate and covariance of the current controllers can be taken as the final output with this circulation. Λ Λ Pk k = E[( xi (k ) − x k k )( xi (k ) − x k k )′ z ] k Λ (i ) Λ x k k = ∑ x k k µk(i ) Λ (i ) Λ (i ) Λ Λ = ∑ [Pk(k) + E ( x k k − x k k )( x k k − x k k )′]µk(i ) i i i p. 11 Mathematics algorithm of PR control
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ 4. FR decision control and Matlab simulation Proposition 4.1: Here, the control on targets is discussed with three types radar tracking 4 targets as an example. It is shown in the Table 1. The universe is the type of target 1,2,3,4 here. The condition attribute set C position, speed, acceleration , the decision attributes set D { very good, better, not so good, not good } for controlling, then, what is the decision rule? Here, use FR 1-norm formula and the selected-near principle, obtain the decision-making rule and decision-making degree are shown in Table 1.: p. 12 FR decision control and Matlab simulation
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ Table 1: Control decisions of some type radars Let "the control of radars is better" act as a criterion whether a radar or a group of radars is selected. According to the genetic algorithm, the "better" and" very good" radar are carried on the genetic algorithm combination. p. 13 FR decision control and Matlab simulation
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ 5. Simulation results and discussion Fig.1 target orbit and FR tracking curve Fig.2 target orbit and PR tracking curve The tracking curve of FR and PR algorithm are shown in Fig. 1 and Fig. 2. From the Fig. 1 and Fig. 2 known, PR algorithm is better relatively than FR algorithm on the multiple-target control. p. 14 Simulation results and discussion
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ 6. Conclusions The theory of FRS is unique and effective soft scientific method, and it can analyze and deal with some fuzzy and incomplete information. The processing technology of FRS to fuzzy uncertain information, and it combines with PR, which will be enormous potentiality in its application. Because the theories and applications of combining FRS with PR are even in embryonic state, it will require a lot of scholars that can further probe into the combination of the two. These problems to be solved will be directly impulse the development of FRS and PR, and it will show the further and spacious prospect in wide applications. p. 15 Conclusions
    • Systems Engineering Institute, School of Electronic & Information Engineering Xi’an Jiao Tong University, P.R.China Xi’ References [2] W.S.Chaer, R.H.Bishop, and J.Ghosh. Hierarchical Adaptive Kalman Filter for Interplanetary Orbit Determination. IEEE Trans. Aerospace and Electronic Systems. 34 (1998), 883-895. [3] W.N. Liu, Y. JingTao, Y. Yiyu, Rough Approximations under Level Fuzzy Sets. In: Springer-Verlag, Rough Sets and Current Trends in Computing, Fourth International Conference, RSCTC 2004. Tsumoto, S. S., Roman, K, J., et al. 1 (2004), 78-83. p. 16 End