By Ivan Tyukin
Within the context of this ebook, version is taken to intend a characteristic of a process geared toward reaching the very best functionality, while mathematical versions of our surroundings and the process itself will not be totally on hand. This has purposes starting from theories of visible conception and the processing of data, to the extra technical difficulties of friction reimbursement and adaptive class of indications in fixed-weight recurrent neural networks. principally dedicated to the issues of adaptive legislation, monitoring and id, this booklet provides a unifying system-theoretic view at the challenge of model in dynamical structures. distinctive consciousness is given to structures with nonlinearly parameterized versions of uncertainty. innovations, equipment and algorithms given within the textual content could be effectively hired in wider parts of technological know-how and know-how. The certain examples and history info make this e-book compatible for a variety of researchers and graduates in cybernetics, mathematical modelling and neuroscience.
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Extra resources for Adaptation in Dynamical Systems
The question, however, is how should one do this? This is a typical example of the non-dominating adaptation problem, of which a more formal statement is provided at the end of this section. 2 Example: adaptive tuning to bifurcations In the previous case the set to which the system solutions are to converge was a priori known. There are systems for which information of this kind is not explicitly available. Their goal is not to reach a given state in the system’s state space but rather to maintain adaptively a certain functional property of the system.
The answer to this question is partially provided by Barbalat’s lemma. In order to state the lemma let us recall the property of uniform continuity of a function of a real variable. 1 A function h : R → R is called uniformly continuous iff for every ε > 0, ε ∈ R there exists δ > 0, δ ∈ R such that for all t, τ ∈ R the following inequality holds: |t − τ | < δ ⇒ |h(t) − h(τ )| < ε. 5) The lemma now can be formulated as follows. 1 Let h : R → R be a uniformly continuous function and suppose that the following limit exists: t lim t→∞ t 0 h(τ )dτ = a, t0 ∈ R, a ∈ R.
The question, however, is whether these properties characterize the preferred state with minimal ambiguity. To some degree, thanks to the requirement of invariance in the deﬁnitions, this issue is already taken into account. 4 is replaced with forward-invariance. 1. 4 then the equilibrium of this system will still be weakly attracting. One can easily see that in this case the equilibrium will not be the only attracting set in the state space. In fact, if we were to replace invariance with mere forward-invariance, the bottom half of every disk centered at the point (0, 0) would be a weakly attracting set too.