Authors: Thomas Berger Achim Ilchmann Fabian Wirth
Publish Date: 2014/07/16
Volume: 138, Issue: 1, Pages: 17-57
Abstract
We combine different system theoretic concepts to solve the feedback stabilization problem for linear timevarying systems with real analytic coefficients The algebraic concept of the skew polynomial ring with meromorphic coefficients and the geometric concept of ABinvariant timevarying 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 ByrnesIsidori 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 closedloop 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 timeinvariant systemsWe have chosen the multiplication rule 12 for the skew polynomial ring mathcalM D This rule is a consequence of the associative rule Dfh=Dfh for all differentiable functions fh which yields Df h = frac mathrm dmathrm dtf cdot h + f cdot frac mathrm dmathrm dth= frac mathrm dmathrm dtf + fD h In contrast to the commutative ring mathbb R D used in the timeinvariant case mathcalM D is noncommutative It is obvious that mathcal MD does not have any zero divisors allows a right and left division algorithm and hence is a right and left Euclidean domain and even a principal ideal domainMatrices over this ring may be viewed as RD = sumn i=0 R i Di inmathcalM Dgtimes q cong mathcalMgtimes q D The left row rank right column rank of a matrix RDin mathcal MDgtimes q is defined as the rank of the free left right mathcal MDmodule of the rows columns of RD resp As a consequence of Theorem A1 the row and column rank coincide and hence we denote the rank of RD by mathrm rk mathcal MDRDAnother canonical form is the so called Hermite form see eg 12 Theorems 24 and 61 where also a nice overview of various forms for timevarying systems is given If instead of mathcalM D the commutative ring mathbb Rs is considered then the Hermite form over mathbb Rs is well known cf 22 Theorem 2514 Since mathbb Rs is embedded in mathcal MD the Hermite form of any Rsin mathbb R s gtimes q over mathbb Rs and the Hermite form of RD over mathcal MD coincide In particular this yields the following corollary
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