By Aristide Halanay, Vlad Ionescu
Discrete-time structures come up as a question in fact in modelling organic or monetary procedures. For structures and keep watch over conception they're of significant significance, quite in reference to electronic keep an eye on functions. If sampling is played in an effort to keep watch over periodic techniques, virtually periodic structures are bought. it is a robust motivation to enquire the discrete-time structures with time-varying coefficients. This examine monograph incorporates a research of discrete-time nodes, the discrete counterpart of the speculation elaborated by means of Bart, Gohberg and Kaashoek for the continual case, discrete-time Lyapunov and Riccati equations, discrete-time Hamiltonian platforms in reference to input-output operators and linked Hankel and Toeplitz operators. these kind of instruments goal to unravel the issues of stabilization and attenuation of disturbances within the framework of H2- and H-control idea. The ebook is the 1st of its type to be dedicated to those themes and is composed ordinarily of unique, lately acquired results.
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Additional info for Time-Varying Discrete Linear Systems: Input-Output Operators. Riccati Equations. Disturbance Attenuation
If the Liapunov equation X=A*aXA+C*C has a bounded on Z positive semidefinite solution X (0 ::5 X tially stable evolution. (1) ::5 It I) then A defines an exponen- Proof. Choose Ci ,;) E Z x X arbitrary and let xk+l = Akxk, xi = ;, k ~ i. ~> ~jJ. 1I~1I2 (2) Let K be such that A + K C defines an exponentially stable evolution. , +j=i LJ;)k '+1 K. x,1I 2 1 -l 1 - q j=i J J }:c1-j - 1 00 k=j+i Notes and References 43 Hence 00 L II Ski~ 11 2 ~P-O II ~ 11 2 k=i or equivalently I ~ P. + 1 A. 5. Dualizing Theorem lone obtains 0 Theorem 1'.
Basic operations 1*c ~ D* + B* a(l - 49 A* a)-IC* (15) Proof. According to (15), the right-hand side of (14) is defined by w=A*aw +C*Qy u=QB*aw+QD*Qy Premultiplying the first equation by Q one obtains Q w = (QA*)Q aw + (Q C*)y U = (QB*)Qaw + (QD*)y Let x = Q a w = a -1(Q w). Hence a x = Q w and we obtain finally ax=A*x+c#y U = B* x and the conclusion follows. From Definition 6 we have automatically that + D* Y o (rt:l = Tc' The anticausal version of Proposition 7 is left to the reader as an exercise.
1. 2. For 2. it suffices to evaluate the spectral radius of a-I ALe. p(a- 1 A) = lim sup II (a- 1 Ai 1I 11i. 10) we get ;-+00 1 II «a- Aiw)k II = II st,k-i wk _ i II ~ pill W k _ i II for some W E lZ(Z ,X). Hence II (a- 1 Aiw II; ~pZq2i11 w II;and consequently II (a- 1 Ai II 1Ii ~plliq-+q < 1 as i -+ 00. 3' is Proposition 4'. Assume that A = (Ak)k EZ defines an anticausal exponentially stable evolu- tion. Then 1. a A is well defined and bounded on p(Z , X ). 2. 18) holds on F(Z , X). 0 Let us now assume thatA in (2) defines an exponentially dichotomic evolution and assume also that v E F(Z , X ).
Time-Varying Discrete Linear Systems: Input-Output Operators. Riccati Equations. Disturbance Attenuation by Aristide Halanay, Vlad Ionescu
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