Zero Dynamics and Stabilization for Analytic Linear Systems

被引:12
作者
Berger, Thomas [1 ]
Ilchmann, Achim [2 ]
Wirth, Fabian [3 ]
机构
[1] Univ Hamburg, Fachbereich Math, D-20146 Hamburg, Germany
[2] Tech Univ Ilmenau, Inst Math, D-98693 Ilmenau, Germany
[3] Univ Passau, Dept Comp Sci & Math, D-94030 Passau, Germany
关键词
Time-varying linear systems; Feedback stabilization; Zero dynamics; Strict relative degree; Byrnes-Isidori form; Geometric control theory; Algebraic systems theory; RELATIVE DEGREE; NONLINEAR-SYSTEMS; NORMAL-FORM; INVERSE; MATRIX;
D O I
10.1007/s10440-014-9956-2
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We combine different system theoretic concepts to solve the feedback stabilization problem for linear time-varying systems with real analytic coefficients. The algebraic concept of the skew polynomial ring with meromorphic coefficients and the geometric concept of (A,B)-invariant time-varying subspaces are invoked. They are exploited for a description of the zero dynamics and to derive the zero dynamics form. The latter is essential for stabilization by state feedback: the subsystem describing the zero dynamics are decoupled from the remaining system which is controllable and observable. The zero dynamics form requires an assumption close to autonomous zero dynamics; this in some sense resembles the Byrnes-Isidori form for systems with strict relative degree. Some aspects of the latter are also proved. Finally, using the zero dynamics form, we show for square systems with autonomous zero dynamics that there exists a linear state feedback such that the Lyapunov exponent of the closed-loop system equals the Lyapunov exponent of the zero dynamics; some boundedness conditions are required, too. If the zero dynamics are exponentially stable this implies that the system can be exponentially stabilized. These results are to some extent also new for time-invariant systems.
引用
收藏
页码:17 / 57
页数:41
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