State Space Controllability & Observability with Variable Terms in a Transfer Functio

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Homework Statement


The following two gain equations are given (G is the forward-path gain; H is presumably the feedback gain):

G(s) = [itex]\frac{s + 3}{s^{3} + 2s^{2} + as + 1}[/itex] ;

H(s) = [itex]\frac{s + b}{s + 5}[/itex]

The question asks that the range of values for a and b be determined such that the system is completely controllable and observable.


2. Homework Equations /theory

Closed-loop transfer function for no controller, M(s) = [itex]\frac{G(s)}{1 + G(s)H(s)}[/itex]

State space vectors:
[itex]\dot{x}[/itex] = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)

If for these state vectors, the matrix S has full rank, the system is controllable; if V has full rank then the system is observable. S and V are defined as:

S = [B A*B A*A*B ...]
V = Transpose([C C*A C*A*A ...]) (just more convenient to write the transpose)

The Attempt at a Solution



I calculated the closed-loop transfer function as:

M(s) = [itex]\frac{s^{2} + 8s + 5}{s^{4} + 7s^{3} + (11 + a)s^{2} + (5a + b + 4)s + (3b + 5)}[/itex]

From this I multiplied out the highest order of s and transcribed the coefficients into the OCF and CCF forms. However, the S for CCF is always non-singular and a similar case occurs with the matrix V for OCF. This is because those matrices are always triangular (as far as I can tell). As such, it appears as though the variables a and b have no effect on the system, which obviously seems erroneous.

Now perhaps I should calculate the S vector for the OCF form - I haven't done this as it seems absolutely messy and I know for a fact that this question is supposed to be done under time pressure (it is in preparation for an exam). On the other hand, I saw at least one question where the forward-path gain was considered alone, and the feedback treated separately - but in that case it was some unrecognised vector form.

Note: my bad if this is in the wrong forum space - first post and seemed most relevant to engineering homework.
 
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