By Bruce A. Francis, Malcolm C. Smith, Jan C. Willems
This Festschrift encompasses a number of articles via acquaintances, co-authors, colleagues, and previous Ph.D. scholars of Keith Glover, Professor of Engineering on the collage of Cambridge, at the celebration of his 60th birthday. Professor Glover's medical paintings spans a wide selection of issues, the most issues being procedure id, version relief and approximation, strong controller synthesis, and keep an eye on of airplane and engines. The articles during this quantity are a tribute to Professor Glover's seminal paintings in those components.
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Additional resources for Control of uncertain systems--modelling, approximation, and design: a workshop on the occasion of Keith Glover's 60th birthday
Since the relative degree r = 1, under the above assumptions, the zero dynamics can be written as: η(t) ˙ = Hη + g1 ys (t) g2 ys (t) if ys (t) ≤ 0, if ys (t) ≥ 0, (25) with η ∈ Rn−1 . It can be proved that the problem (25) is equivalent to the following (sign constrained) switching problem with systems (H, g1 ) and (H, −g2 ) with a nonnegative input u, see . , the Lie-algebra generated by Hη + [g1 − g2 ][u u]T , needs to be deﬁned. Denote this set by R(H, [g1 , −g2 ]). , the pair (H, [g1 , −g2 ]) has to be controllable.
Lg1 Lrf1 −1 h1 (x) .. , rp −1 rp −1 Lgm Lf hp (x), . . , Lgm Lf hp (x) AIA has rank p at x0 . (14) Control of High-Speed Underwater Vehicles 33 For linear time invariant (LTI) systems given by (A, B, C), we have that Lgj Lfri −1 hi (x) = ci Ari −1 bj and if p = m then the vector relative degree is deﬁned if rankALT I = n where n is the state dimension. The concept of relative degree can be extended to time delay systems, too. Usually this is deﬁned for a discrete time equivalent of the continuous time systems by introducing the discrete time shift operator Δ as Δxt = xt−τ with τ denoting the given time delay.
By this strategy, eventually, a large polytope symmetric around λ = 0 for which the LMIs are satisﬁed will be found, allowing for correspondingly large parameter variations. ¯ a bisection algorithm may be To ﬁnd a good trade-oﬀ between σ and λ, run for diﬀerent values of σ, where the quasi-convex optimization strategy outlined above can be carried out for each chosen value of σ. One has to employ a bisection algorithm (or similar technique) to optimise σ because σ enters the expressions for Φ(s), F (s), F ⊥ (s), G(0, s), .