Distributed parameter control of nonlinear flexible structures with linear finite-dimensional controllers
type: Item
instance of: scholarly article
title: Distributed parameter control of nonlinear flexible structures with linear finite-dimensional controllers
zbMATH Open document ID: 0591.93031
DOI: 10.1016/0022-247X(85)90043-5
author: Mark J. Balas
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: The flexible structures under consideration are governed by a generalized wave equation \((1)\quad u_{tt}+2\epsilon u_ t+A_ 0u=F\) for the displacement \(u=u(x,t)\) of a structure \(\Omega\) from its equilibrium position under the influence of a distributed force \(F=F_ C+F_ D\) where the control force \(F_ C\) is of the form \(F_ C(x,t)=B_ 0f=\sum^{\infty}_{i=1}b_ i(t) f_ i(t)\) and the disturbance force \(F_ D\) is given by \(F_ D(x,t)=F_ 0(t,u,u_ t,f)\) with \(F_ 0\) being continuous in t, uniformly Lipschitz continuous in \((u,u_ t,f)\) and satisfying \(F_ 0(t,0,0,0)=0\). \(A_ 0\) is assumed to be a linear symmetric, time-invariant differential operator whose domain \(D(A_ 0)\) is dense in \(H_ 0=L^ 2(\Omega)\) and which is positive definite. Measurements are made by p sensors with outputs \((2)\quad y=(y_ 1,...,y_ p)^ T=C_ 0u+D_ 0u_ t\) where \(y_ j(t)=<c_ j,u>+<d_ j,u_ t>\) and \(c_ j,d_ j\in H_ 0\) for \(j=1,...,p\). The state equation (1) and the measurement output equation (2) are rewritten in the form \((3)\quad v_ t=Av+Bf+h(t,v,f),\) \(y=Cv\) as a system for \(v=(u,u_ t)^ T\) with A, B, h, and C suitably defined. As a first result it is then proved that, for every \(\epsilon >0\), A generates a group \(U_ A(t)\) with \(\| U_ A(t)\| \leq Me^{-\sigma t},\) \(t\in {\mathbb{R}}\), for some position constants \(M>0\), \(\sigma >0.\) The linear controller f in (3) is assumed to satisfy \((4)\quad f=H_{11}y+H_{12}z,\) \(\dot z=Fz+Hy+Df,\quad z(0)=0.\) As a second result it is then proved that there exists a bounded linear operator T: D(A)\(\to {\mathbb{R}}^ S\) (s being the dimension of z) with \(FT- TA+HC=0\), if the spectrum of F lies to the left of \((-\sigma_ F,0)\) for some \(\sigma_ F>\sigma\). On introducing \(e=z-Tv\) for v in D(A) one obtains (on using (3)) \(\dot e=Fe-Th\) which can be combined with (3) to the system \((5)\quad w_ t-A_ cw+h_ c(t,w)\) for \(w=(v,e)^ T\) with \(A_ c\) and \(h_ c\) suitably defined. As a third result it is proved that the…
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 74P99
zbMATH DE Number: 3950314
zbMATH Keywords: nonlinear flexible structures
zbMATH Keywords: distributed parameter systems
zbMATH Keywords: direct velocity feedback
zbMATH Keywords: compensator feedback
zbMATH Keywords: exponential stability
zbMATH Keywords: mild solution
reviewed by: Werner Krabs
MaRDI profile type: Publication
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full work available at URL: 0022-247x(85)90043-5
OpenAlex ID: W2076413069
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description: scientific article; zbMATH DE number 3950314