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Notions of stabilizability and controllability for a class of uncertain linear systems. (English) Zbl 0625.93010
This paper develops a number of results concerning the stabilizability of uncertain linear systems. The uncertain systems under consideration are described by linear state equations which contain norm-bounded time- varying uncertain parameters. The notion of stabilizability considered is that of quadratic stabilizability. That is, it is desired to construct a state feedback control such that the resulting closed-loop uncertain system is stable with a single quadratic Lyapunov function. Furthermore, a notion of complete stabilizability is defined in the paper. This refers to the case of an uncertain linear system which is quadratically stabilizable no matter how large the bound on the uncertain parameters. In order to investigate the property of quadratic stabilizability, the paper establishes a new version of the small gain theorem. This shows that quadratic stability is equivalent to a certain frequency domain condition. This result enables the comlete stabilizability of an uncertain linear system to be investigated using the geometric theory of almost A,B-invariant subspaces.
Apart from considering stabilizability, this paper also considers a number of notions of controllability for uncertain linear systems. The first such notion is the analog of modal controllability for a linear time-invariant system. This notion of modal controllability for an uncertain linear system is referred to as complete stabilizability with an arbitrary degree of stability. The paper presents a geometric condition which is necessary and sufficient for a given uncertain linear system to be completely stabilizable with an arbitrary degree of stability.
The paper presents another notion of controllability referred to as controllability invariance. An uncertain linear system is controllability invariant if for each fixed value of the uncertain parameters, the resulting linear time-invariant system is controllable in the usual sense. The paper shows that under certain conditions, the notions of controllability invariance and stabilizability with an arbitrary degree of stability are equivalent.

##### MSC:
 93B05 Controllability 93C05 Linear systems in control theory 93D15 Stabilization of systems by feedback 47A15 Invariant subspaces of linear operators 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory
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##### References:
 [1] DOI: 10.1007/BF00939145 · Zbl 0549.93045 [2] BROCKETT R. W., Finite Dimensional Linear Systems (1970) [3] HOLLOT , C. V. , 1984 , Construction of quadratic Lyapunov functions for a class of uncertain linear systems . Ph.D. dissertation, Department of Electrical Engineering , University of Rochester , Rochester , N.Y. , U.S.A . [4] KAILATH T., Linear Systems (1980) [5] MEILAKHS A. M., Automat, Telemekk, 10 pp 16– (1978) [6] MOLANDER , P. , 1979 , Stabilization of uncertain systems . Report LUTFD2/(TRFT–1020)/1–111/(1979) , Lund Institute of Technology , Lund , Sweden . [7] NARENDRA K. S., Frequency Domain Criteria for Absolute Stability (1975) [8] DOI: 10.1137/0323020 · Zbl 0563.93054 [9] DOI: 10.1016/0005-1098(86)90045-2 · Zbl 0602.93055 [10] WILLEMS J. C., l.E.E.E. Trans, autom. Control 26 (1980) [11] DOI: 10.1137/0319029 · Zbl 0467.93036 [12] DOI: 10.1137/0321020 · Zbl 0519.93055 [13] WONHAM W. M., Linear Multivariate Control a Geometrical Approach (1979) · Zbl 0424.93001
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