This chapter addresses some miscellaneous problems that can be addressed with the framework described in the previous chapters. First, the problem of estimating the DA via union of a continuous family of estimates is considered for polynomial and non-polynomial systems. The family of estimates corresponds to the LEDA provided by a continuous family of LFs, and hence the problem amounts to computing a parameter-dependent LEDA. The problem can be solved by exploiting the results introduced in Chapters 5 and 6 and by constructing a suitable parameter-dependent LF. The second problem addressed in this chapter considers the estimation of the set of admissible equilibrium points for an uncertain nonlinear system, either polynomial or non-polynomial. Specifically, it is shown how outer estimates with fixed shape can be computed by solving an SDP by exploiting SOS polynomials. A sufficient condition for establishing whether the found candidate is tight is hence provided by looking for power vectors in linear subspaces, which turns out to be also necessary in the case of uncertain polynomial systems. Then, the methodology is exteded to address the computation of the minimum volume outer estimate and the construction of the smallest convex outer estimate. The third problem addressed in this chapter considers the computation and the minimization via constrained control design of the extremes of the trajectories of a polynomial or non-polynomial system over a given set of initial conditions. It is shown how upper bounds of the sought extremal values as well as candidates of the sought controllers can be computed by solving optimization problems with BMI constraints by looking for suitable invariant sets through SOS polynomials. Moreover, a necessary and sufficient condition is proposed to establish the tightness of the found upper bound by exploiting a reverse trajectory system. Lastly, the chapter provides some remarks on the estimation of the DA and construction of global LFs for degenerate polynomial systems, in particular polynomial systems whose linearized system at the equilibrium point is marginally stable and polynomial systems where the global LF or its temporal derivative have not positive definite highest degree forms.
KeywordsEquilibrium Point Uncertain System Polynomial System Continuous Family Outer Estimate
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