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Material Type: Notes; Class: SYSTEMS ANALYSIS TECHNIQUES; Subject: Electrical & Comp. Sys. Engr.; University: Rensselaer Polytechnic Institute; Term: Fall 2004;
Typology: Study notes
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. (^) find the transfer function of the
controllability and observability.
show that sampling may destroy controllability.
Ref: Sections 7.5-7.7. Fall^2004
A Plant with an Observer and Controller
What is the transfer function of the combined
plant -observer-controller?
Does the observer affect the stability of the overall system? u(t) (^) x(t) =Ax(t) + Bu(t)
i(t) =Ai(t) + Bu(t) + L(y(t) - y(t» y(t) = Ci(t)
. FIG. 7.5-
Find Y(s)/V(s),
system:
observer:
Use the following matrix identity,
[I + C(sl - A)-1 B]-1 = 1 - C(sl - A + BC)-1 B
Use this on the 1st term on the RHS of (4),
Use the identity again (right to left this time) on I - Bk T(s I - A + LC +Bk t) -1I
Rewriting BV(s),
BV(s) =
and soX(s) = [sl - A + BkT]-1 BV(s)
and Y(s)/V(s) = C [sl - A + BkT]-1 B
What is the transfer function when all the states are measurable?
subtract row 1 from row 2,
= det(sI - A + BkT) det(sI - A + LC)
. (^) the modes of the overall system can be made stable.
. with feedback, u(t)=-Kx(t) + Fv(t)
v
FIG. 7.7-
D
A
-K
effect on controllability (cont):
Test matrix for the closed-loop system is,
E>e= [BF (A-BK)BF (A-BK)2BF.. ... (A -BK)n-l BF]
and for the open-loop system,
ee = [B AB A2B ... An-lB]
Use column operations to show rank (e c) :
Sampling may destroy controllability.
Theorem: If (A,B) is completely controllable then the equivalent discrete system (F,G) is completely controllable if
Im[Aj(A)-Aj(A)]=F21tn/T;n=xl,x2... (*)
Xl (t) I 1 1 Xl (t) I 0
Show that T == n 7r destroys complete contro liability.
