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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) +

**B**u(t)

**y**(t) =

**Cx**(t) +

**D**u(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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