## (C,A,B)-pairs in infinite dimensions.(English)Zbl 0553.93037

This paper considers infinite-dimensional systems on a real separable Hilbert space X of the form $$\dot x=Ax+Bu$$, $$y=Cx$$ and introduces a (C,A)-invariant, a T(C,A)-invariant, an (A,B)-invariant and a T(A,b)- invariant subspace where A is the infinitesimal generator of a strongly continuous semigroup T(t) on X and $$B\in L[R^ m,X]$$, $$C\in L[X,R^ k]$$. Consider the feedback processor on the Hilbert space W of the form $$\dot w=Nw+My$$, $$u=Lw+Ky$$, where M, L, K are bounded linear operators and N is the infinitesimal generator of a strongly continuous semigroup on W. Defining $$x^ e=(x,w)$$, we obtain the closed-loop operator $A^ e=\left[ \begin{matrix} A+BKC\\ M\end{matrix} \begin{matrix} BL\\ N\end{matrix} \right].$ $$A^ e$$ is also an infinitesimal generator of a semigroup $$T^ e(t)$$. If (S,V) is a trajectory-invariant (C,A,B)-pair and $$S\subset D(A)$$, then there exist an extension space $$W\cong V/S$$, bounded operators M, L, K, a linear operator N on W and a closed subspace H of $$X^ e=X\oplus W$$ which is $$A^ e$$-invariant. Further, N is an infinitesimal generator, the feedback processor is well defined and H is invariant under the semigroup generated by $$A^ e$$. Consider the system on X $$\dot x=Ax+Bu+Eq$$, $$y=Cx$$, $$z=Dx$$, where $$E\in L[R^ r,X]$$, $$D\in L[X,R^ l]$$, q is the disturbance and z is the output to be decoupled. DDPM is to design a feedback processor such that the output z is decoupled from the disturbance q, i.e. $$D^ e\int^{t}_{0}T^ e(t-s)E^ eq(s)ds=0$$, $$t\geq 0$$, where $$T^ e(t)$$ is the semigroup generated by the closed-loop system operator $$A^ e$$. Let $$S^*(L)$$ and $$V^*(K)$$ denote the infimal T(C,A)-invariant subspace containing a closed subspace L and the supremal T(C,A)-invariant subspace contained in a closed subspace K respectively. Assume that $$S^*(Im E)$$ and $$V^*(Ker D)$$ exist and satisfy $$S^*(Im E)\subset D(A)$$. Then, DDPM is solvable if $$S^*(Im E)\subset V^*(Ker D)$$.
Reviewer: M.Kono

### MSC:

 93C25 Control/observation systems in abstract spaces 47A15 Invariant subspaces of linear operators 47D03 Groups and semigroups of linear operators 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems 46C99 Inner product spaces and their generalizations, Hilbert spaces
Full Text:

### References:

 [1] Curtain, R. F.; Pritchard, A. J., Infinite Dimensional Linear Systems Theory, (Lecture Notes in Control and Information Sciences, 8 (1978), Springer: Springer Berlin) · Zbl 0352.49003 [2] R.F. Curtain, Invariance concepts in infinite dimensions, SIAM J. Control Optim.; R.F. Curtain, Invariance concepts in infinite dimensions, SIAM J. Control Optim. · Zbl 0602.93037 [3] Schumacher, J. M., Compensator synthesis using (C, A, B)-pairs, IEEE Trans. Automat. Control, 25, 1133-1137 (1980) · Zbl 0483.93035 [4] Schumacher, J. M., Dynamic Feedback in Finite and Infinite-Dimensional Linear Systems, (M.C. Tracts No. 143 (1982), Mathematisch Centrum: Mathematisch Centrum Amsterdam) · Zbl 0474.93003 [5] Willems, J. C.; Commault, C., Disturbance decoupling by measurement feedback with stability or pole placement, SIAM J. Control Optim., 19, 490-504 (1981) · Zbl 0467.93036 [6] Wonham, W. M., Linear Multivariable Control, A Geometric Approach (1979), Springer: Springer New York · Zbl 0393.93024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.