By Günter Ludyk
During this monograph a few balance houses of linear, time-variant, discrete-time structures are summarized, the place a few homes are popular, a few are little-known proof, and some might be new. types for this treatise an the asymp totical behaviour of options of distinction equations are the generally identified very good books of CESARI  and CONTI . within the tables of bankruptcy 1 the definitions and the essen tial statements an balance of discrete-time platforms are summarized, such that bankruptcy 2 to five should be considered as explaining appendices for those tables. i'm thankful to Paul Ludyk, who typed and corrected the manuscript with nice ability and persistence, and Alois Ludyk, who drew the figures with nice creative ability. Gunter Ludyk Bremen, January 1985 Contents Notations 1 1. advent and precis four 2. Mathematical Description of Discrete-Time platforms sixteen 2. 1 country Equations sixteen 2. 2 homes of the Transition Matrix 19 2. three LAGRANGE-Identity and GREEN's formulation for distinction Equations 20 2. four Estimations for the Norm of the Transition Matrix 21 three. balance of loose Discrete-Time structures 34 three. 1 LJAPUNOW- and LAGRANGE-Stability 34 three. 2 couple of minutes Boundedness forty three. three UniformStability forty five three. four Asymptotic balance sixty three three. five P-stability 70 three. 6 Exponential and Uniform Asymptotic balance seventy five three. 7 family among the soundness Glasses eighty four four. balance of pressured Discrete-Time structures 86 four. 1 initial effects 86 four. 2 Input-State balance ninety three four.
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Additional resources for Stability of Time-Variant Discrete-Time Systems
38 If only the distance of the solution x from the trivial equilibrium state xe=o is of interest, then one can use the following definitions. 13) is valid. 3-11 DEFINITION: The system x(k+l)=A(k)x(k) is said to be LAGRANGE-stable, if every solution of the system is bounded. A connection between stability and LAGRANGE-stability is established in the following theorem. , if the system is stable. PROOF: (•) If A(k)ES the system is naturally LAGRANGE~stable according to Definition 3-2 and Definition 3-10.
33) The following lemma can be derived by the same arguments which were needed for the derivation of Lemma 3-5 and Theorem 3-7. 3-22 LEMMA: If a solution of x(k+l)=A(k)x (k) is uniformly stable, then all solutions are uniformly stable. 48 THEOREM: The system x(k+l)=A(k)x(k) is uniformly stable if and only if A(k) E U$ . For this theorem the following definition was used: 3-24 DEFINITION: The system x(k+l)=A(k)x(k) uni[ormZy stabZ~, is said to be if every solution is uniformly stable. If the system matrix A is time-invariant and the system x(k+l) = A x(k) is stable,it is uniformly stable tool Because, if A E S, then for k ~ 0 = a there exists a constant c > 0 such that But since ~(k,k 0 ) = Ak-ko = ~(k-k 0 ,o), it is also for k <: k0 hence A E US.
I + Az- 1 + A2 z - 2 + ••• ) z L Aiz-(i+1) 00 i=o 25 converge for lzl>l~maxl. By assistance of the CAUCHY integral formula k A can be computed. e. • Additionally bounds for the norm of the transition matrix can be obtained in the following way. 41) for all k>k 0 ,k 1 EK. In the case of k0
Stability of Time-Variant Discrete-Time Systems by Günter Ludyk