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Relaxed design conditions using Takagi-Sugeno fuzzy models and Lyapunov-Floquet transformation for controlling time-varying periodic systems

“…Then, from equations (14) and (15)Adding and subtracting B 0 F 0 q ( t ) into the equation (17), we obtainAs the matrix ( R + B 0 F 0 ) in equation (18) can present all eigenvalues with negative real part (selecting an adequate matrix F 0 ), the systems in equations (14) and (15) can be considered equivalent ifIn general, it is not possible to obtain a control law u ( t ) that exactly satisfy the equation (19). Boghiu et al (1998) proposed to consider an approximate control law u ( t ) given bywhere B*(t)=(B(t)′B(t))−1B(t)′ is the generalized inverse of the matrix B ( t ). In order to obtain the control law as a function of the vector states x ( t ), it suffices to observe that q ( t ) = Q ( t ) −1 x ( t ) and therefore…”

“…In this case, since an approximate solution for u ( t ) is used, the asymptotic stability of closed-loop system is not guaranteed. However, many examples in the literature show that even with the approximate control law it is possible to find a gain F 0 such that the closed-loop system given in equation (22) is asymptotically stable (Boghiu et al, 1998; Peruzzi et al, 2016; Sherrill et al, 2015).…”