Stability of a Time Varying System

Consider the linear time varying equation

e(t) = /1 u*(t )e(t),

(H.1)

with

0 < umin < u*(t) < umax V t ◦ 0

(H.2)

and such that A = – A/L is a stability matrix with П a negative number greater than the largest real part of its eigenvalues. Use the Gronwall-Bellman inequality Rugh (1996) to conclude that

t

\e(t)|| < \e(0)Wexp{J ||Au%(x)\dx}. 0

(H.3)

Considering (H.2), this implies

\e(t)|| < ||e(0)||enumint.

(H.4)

Since П < 0, this establishes asymptotic stability of (H.1).

References

Goodwin GC, Sin KS (1984) Adaptive filtering prediction and control. Prentice-Hall, New York Ibragimov NH (1999) Elementary Lie group analysis and ordinary differential equations. Wiley, New York

MoscaE, Zappa G, Lemos JM (1989) Robustness of multipredictor adaptive regulators: MUSMAR. Automatica 25(4):521-529

Rugh WJ (1996) Linear system theory, 2nd edn. Prentice – Hall, New York

[1] — 1

y (k + i) = Cx(k + i) = CAjx (k) + C^jAi — j—1 Bu(k + j). (5.39)

j =0

The minimization of (5.33) is made under the assumption that both the control action u and the inlet fluid temperature d are constant along the control horizon, meaning that

[2] LEGO® is a registered trademark of the LEGO Group.

[3] T

T sT(k) [e [(pi – pi)(pT – pT)J + asT(k)E [(pi-1 – pi-1)(PT – Pf-1^] s(k)

[5] —0 i —0

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