In the context of control theory,


The algebraic design tradeoff by Freudenberg and Looze is a constraint that relates the sensitivity function $\sigma$, and the complementary sensitivity function $\tau : \sigma + \tau = 1$.


However, does this constraint exist in a feedback system designed using an observer and state feedback?



Algebraic design tradeoff by Freudenberg and Looze and State Feedback with Observer
Of course, why would be interested otherwise :-)?


The constraints on sensitivity and complementary sensitivity function are formulated in the frequency domain. When you are talking about feedback systems designed using an observer and state feedback then you are talking about objects defined in the time domain. So the answer comes from relating frequency domain to time domain via the Laplace transform. 


I will assume that you are talking about single-input single-output systems because otherwise the constraints are a bit trickier anyway. So to paraphase your question. Let talk about the frequency domain. I.e. $s$ is a complex variable. We have a "plant" i.e. map relating input signals $u(s)$ to output signals $y(s)$:
$$ y(s) = G(s) u(s),$$
a controller, i.e. a map relating the error signal $e(s)$ to the input signal,
$$ u(s) = C(s) e(s),$$
and a reference signal $w(s)$. The open loop map is $y(s) = G(s) C(s) e(s)$. And we close the loop by letting the error $e(s) = w(s) - y(s)$. If we compute the map from $w$ to $y$ then we obtain
$$ y(s) = \tau(s) w(s) := \frac{G(s) C(s)}{1+G(s)C(s)} w(s). $$
So the complementary sensitivity is the transfer function of the closed loop system. The sensitivity is 
$$ \sigma(s) = 1 - \tau(s) = \frac{1}{1+G(s)C(s)}, $$
and the constraint you mention clearly holds.


Note that this applies to "a controller" $C(s)$. So anything that you write down in the time domain which takes the output $y$ of the system and produces an input $u$ of the system and which can be written as a linear time-invariant system will have a transfer function describing its behaviour in the frequency domain and the calculations apply to that transfer function.


So if we start in the time-domain with a system
$$ \dot x = Ax + Bu ,\quad y = Cx$$
where $A,B,C$ are (real or complex) matrices of appropriate size. Then assuming controllability and observability, we may choose a feedback $F$ such that $A+BF$ is Hurwitz and an observer gain $L$ such that $A+LC$ is Hurwitz. This yields the stabilizing controller
$$ \dot z = (A+BF +LC)z + L(-y),\quad u = Fz.$$
(I am currently ignoring the reference signal $w$. If you want it in there, write $L(w-y)$.)


The relation to the frequency domain is then that the transfer function of the original system is
$$ G(s) = C(sI-A)^{-1}B.$$
But also the controller has a transfer function as 
$$ C(s) = F(sI-(A+BF+LC))^{-1}L.$$
With this $G(s),C(s)$ we are precisely in the situation described above.


The only thing to notice is that in the time domain there may be unobservable or uncontrollable subspaces with dynamics which do not show up in the transfer function. But of course this does not affect the Freudenberg-Looze result.


