By Geir E. Dullerud
My major aim in penning this monograph is to supply an in depth therapy of uncertainty research for sampled-data platforms within the context of sys tems regulate concept. right here, sampled-data procedure refers back to the hybrid sys tem shaped while non-stop time and discrete time platforms are intercon nected; via uncertainty research I suggest achieveable functionality within the pres ence of worst -case uncertainty and disturbances. the focal point of the e-book is sampled-data platforms; but the technique offered is acceptable to either typical and sampled-data platforms. The previous few years has visible a wide surge in examine job established round developing systematic tools for sampled-data layout. the purpose of this task has been to deepen and expand the, by means of now, refined perspective built for layout of simply non-stop time or discrete time structures (e.g. J{oo or -I!l optimum synthesis, J1 thought) in order that it may be ap plied to the layout of sampled-data platforms. This learn attempt has been mostly winning, generating either fascinating new mathematical instruments for keep an eye on conception, and new methodologies for useful engineering layout. research of based uncertainty is a vital goal up to the mark layout, since it is a versatile and non-conservative method of studying sys tem functionality, that is compatible in lots of engineering layout scenarios.
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Extra info for Control of Uncertain Sampled-Data Systems
Example text
3). Note that the definition of this mapping cannot be generalized to the case where the F has a direct feedthrough term; F must be low-pass. 3). e2 . We now move on to notions of time invariance, which playa central role in our framework. Define the delay operator DT on if, for each T > 0, to be the map that takes the function U (t) to the shifted function which is u(t - T) for t ~ T, and zero for ° ~ t ~ T. c (if) > 0. ef to be the map ud[k] >-+ ud[k - 1], with (UUd)[O] = 0. ef) is LTI if MdU = UMd.
2 purely in terms of operators on,c ('£2), and convert it to an equivalent frequency domain condition on functions in j\. £2 with transfer functions in j\. We have: Given the previous definitions and suppose ~ E ,cj<\. )-1 exists and is bounded if and only if for each z in the closed unit disk Ifil, the operator I - M(z)Li(z) is nonsingular. )-1 exists and is bounded. ) -1 is L11. olD> so that I - M(z')Li(z') is not invertible. 2. DESTABILIZING PERTURBATIONS II (I - M(z)Li(z) )-111 - dicting that (I - MLi)-1 z - z', see for example [3, pp.
L-C:o. For later reference we will denote the space of operators on L2 that can be identified with the transfer functions JIl by "cj\. J{oo and JIl are complete. l-C:o is complete is standard; see [27]. l-C:o, it is sufficient to show that every Cauchy sequence converges to a continuous function on Ii)). Suppose Tk is a Cauchy sequence in JIl. : :; II Tk - Ttl I which tends to zero as k and l tend to infinity. • 00, CHAPTER 2. PREUMINARIES 26 Analogous to these two spaces we define their half-plane matrix valued versions.
