Distributed parameter control of nonlinear flexible structures with linear finite-dimensional controllers
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 system (5) is well-posed and that its ''mild'' solution satisfies a certain integral equation. The fourth result of the paper states the exponential stability of this mild solution provided the linearized system (5) with \(h_ c\equiv 0\) is exponentially stable and some additional condition is satisfied. The last section (before the conclusion) then is concerned with the question how the parameters of the linear controller in (4) have to be chosen as to satisfy some performance criterion for \(w_ t=A_ cw\) and to maintain the exponential stability of the group \(U_ c(t)\) generated by \(A_ c\).
scientific article; zbMATH DE number 3950314
scientific article