By Xing-Gang Yan, Sarah K. Spurgeon, Christopher Edwards
This publication systematizes fresh study paintings on variable-structure keep watch over. it's self-contained, featuring precious mathematical preliminaries in order that the theoretical advancements could be simply understood by means of a huge readership.
The textual content starts off with an advent to the basic principles of variable-structure keep an eye on pertinent to their program in advanced nonlinear structures. within the center of the e-book, the authors lay out an process, appropriate for a wide classification of platforms, that bargains with approach uncertainties with nonlinear bounds. Its remedy of advanced platforms during which constrained size info is out there makes the implications constructed handy to enforce. quite a few case-study purposes are defined, from aerospace, via strength structures to river pollutants regulate with helping simulations to help the transition from mathematical idea to engineering practicalities.
The e-book addresses platforms with nonlinearities, time delays and interconnections and considers concerns similar to stabilization, observer layout, and fault detection and isolation. It makes wide use of numerical and functional examples to render its rules extra without difficulty absorbed.
Variable-Structure keep watch over of complicated Systems could be of curiosity to educational researchers learning regulate concept and its software in nonlinear, time-delayed an modular large-scale structures; the robustness of its strategy can also be beautiful to manage engineers operating in industries go together with aerospace, electric and mechanical engineering.
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Additional resources for Variable Structure Control of Complex Systems: Analysis and Design
Example text
Then there exists a continuous function φ2 (·) in [0, a) such that φ1 (s) = φ2 (s)s, s ∈ [0, a) Proof (i) Suppose that β(x, s) : Rn × R + → R+ is a class K I function. 2 Comparison Functions 31 (ii) Since the function φ1 (·) is a C 1 function in [0, a), its derivative continuous in [0, a). 4) From the definition of φ2 (·), it is clear to see that (1) (2) if s = 0, then φ1 (s) = φ2 (s)s; if s = 0, then from φ1 (0) = 0, φ1 (s) = φ2 (s)s. Therefore, the expression φ1 (s) = φ2 (s)s holds for s ∈ [0, a).
A Lipschitz function may not be differentiable and a simple example is the scalar function f (x) = |x| at the origin x = 0 in x ∈ R. 2) is not Lipschitz in the compact set x ∈ [0, 1] for any constant α satisfying 1 < α < 2. 2) is not bounded in the interval [0, 1]. 1 ([91]) Consider a function f (x) : Rn → Rm which is differentiable in the domain Ω. 1) holds. 2 Generalised Lipschitz Condition The well-known Lipschitz condition in Sect. 1 will be extended to a more general case which will be used later in the analysis.
If the uncertainty or disturbance acts in the input/control channel or the effects are equivalent to an uncertainty acting in the input channel, it is called matched uncertainty. Otherwise it is called mismatched uncertainty. 7) experiences uncertainties φ(t, x) and ψ(t, x) described by x˙ = F(t, x) + G(t, x)(u + φ(t, x)) + ψ(t, x). 23) Then, the term φ(t, x) is called matched uncertainty. In addition, if the uncertainty ψ(t, x) can be modelled as ψ(t, x) = G(t, x)χ (t, x) where χ (·) represents the uncertainty, it is clear to see that the uncertainty of the term ψ(·) is reflected by the uncertainty χ (·) which is exactly acting in the input channel.