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A reprint of this paintings, first released in 1980. the cloth coated maintains to supply a foundation for study within the arithmetic of dynamical structures. It addresses: similar platforms; integration of vector fields; linear structures, linearization and reliable manifolds; reliable platforms; and extra.
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2) to compute local Lyapunov functions which are valid for all the dynamics that are active within a region. e. only the ith rule and the corresponding dynamic is active, and interpolation regions in between them where several dynamics are present (a single Pi matrix common to all of them must be found). The objective will be to compute a matrix Pi for every region, valid for all the dynamics in that region, and in such a way that the resulting Lyapunov function be continuous across regions. Let {Xi }i∈I ⊆ IRn be a partition of the state space into cells, and I the index set of the cells.
Some difficulties appear because the controller is designed to work well at the operating point, but for nonlinear systems this stability is only guaranteed around this point. The system can be locally stable around the operating point but not globally stable, so some initial conditions or large enough perturbations can make it go out of control. However, knowing how global stability may be lost, the problem can be if not avoided at least minimized. This justifies an increased interest in the global analysis of fuzzy control systems.
A matrix of convenient dimensions (a matrix in the MIMO case and a scalar in the SISO case). 9 can be seen as a feedback system composed by two blocks, F and N , so that: F = G(I + CG)−1 , N = H − C. 5 Input/Output Methods 27 Fig. 9. 1. 55) 1 . 56) r where C is called the centre of the cone and r is called the radius of the cone. g(F ) = g(G(I + CG)−1 ) ≤ The conicity criterium also has a geometrical interpretation. , C scalar. 10). 28 2. Fuzzy Systems and Stability Analysis Fig. 10. 56). The problem is somehow equivalent to the search for a candidate Lyapunov function, since it is a sufficient but not necessary stability condition.