Fuzzy control technique based on continuous \(T\)-norm and \(S\)-norm (Q2766031)

The author studies the following discrete fuzzy relation NEWLINE\[NEWLINEx(k+ 1)= (x(k)\times u(k))\circ P,\quad u(k)= x(k)\circ Q,\quad k\in\mathbb{N}= \{1,2,\dots\},\tag{1}NEWLINE\]NEWLINE with \(P= \bigvee^l_{i=1} P_i\), \(P_i= (A_i, B_i)\to C_i\), \(Q= \bigvee^s_{j=1} Q_j\), \(Q_j= D_j- E_j\), where ``\(\circ\)'' is the Zadeh max-min composition operator, \(x(k)\), \(u(k)\) denote the state variable and the control language variable respectively, \(A_i,B_i,C_j\in X= \{a_1, a_2,\dots, a_n\}\) and \(B_i,E_j\in U= \{b_1, b_2,\dots, b_m\}\) are fuzzy sets.NEWLINENEWLINENEWLINEThe key step of the stability analysis of system (1) is to reduce it to the next form NEWLINE\[NEWLINEx(k+ 1)= x(k)\circ [Q\circ P],\quad k\in\mathbb{N}.\tag{2}NEWLINE\]NEWLINE A necessary and sufficient condition so that (1) can be reduced to a new system of the form (2) is given in terms of fuzzy relation matrices.NEWLINENEWLINENEWLINEAn algorithm on the convergence of the sequence \(\{[Q\circ P]^k\mid k\in\mathbb{N}\}\) is also obtained by using semigroup theory.