1

 Consider again the general framework with
 Suppose (A,B_{2}) is stabilizable and (C_{2},A) is
detectable.
 Youla parameterization: all controllers K that internally stabilize G.
 Suppose GÎ RH_{¥}_{ }.
Then K=Q(I+G_{22}Q)^{1}, Q Î RH_{¥}_{ }and
I+G_{22}Q(¥ )
nonsingular.
 Proof: K stabilizes a stable plant G_{22} iff K(IG_{22}K)^{1}
is stable. So the conclusion follows by letting Q = K(IG_{22}K)^{1}.

2

 General Case: Let F and L be such that A+LC_{2} and A+B_{2}F
are stable. Then K = F_{l}(J,Q):
 with any QÎ RH_{¥}_{ }and I+D_{22}Q(¥ ) nonsingular.

3

 Closedloop Matrix:
 F_{l}(G, K)= F_{l}(G, F_{l}(J, Q)) = F_{l}(T,
Q)
 = {T_{11}+T_{12}QT_{21}: Q Î RH_{¥}_{ }and
I+G_{22}Q(¥ )
nonsingular}.
 where T (T_{22}=0) is given by

4

 Coprime Factorization Approach: Let be rcf
and lcf of G_{22} over RH_{¥} respectively. And let them satisfy the Bezout
identity:
