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Additional resources for Solid state physics, advances in research and applications
5 Summary In this chapter we have presented the global and local variants of the uniform output regulation problem. In all variants of the problem there are three basic requirements: regularity, uniform convergence, and asymptotic output zeroing. The regularity requirement guarantees existence and uniqueness of solutions of the closed-loop system. The uniform convergence requirement guarantees that solutions of the closed-loop system “forget” their initial conditions and converge to some steady-state solution determined only by the input.
In the context of the output regulation problem, we will consider several classes of exosystems. 4) and for any compact set K0 ⊂ W there exists a compact set Kw ⊂ W such that any solution w(t) starting in w(0) ∈ K0 satisﬁes w(t) ∈ Kw for all t ∈ R. For the case of W = Rm , Assumption A1 can be reformulated in the following equivalent way: for any a > 0 there exists b > 0 such that |w(0)| ≤ a implies |w(t)| ≤ b for all t ∈ (−∞, +∞). 4) can be called equi-boundedness of solutions in forward and backward time.
The global output regulation problem for nonlinear systems, which remained a tough and quite unaddressed problem for a long time, has received attention in a number of recent publications [11, 17, 40, 58, 79]. In these papers the authors use problem settings for the global output regulation problem with various internal stability requirements. For example, in  all solutions of the closed-loop system must be bounded; in , in addition to the requirement of boundedness of solutions of the closed-loop system, for w(t) ≡ 0 the closed-loop system must have a globally asymptotically stable equilibrium at the origin.