Determine Range of a and b for Controllability and Observability

[gand4lf]

Homework Statement

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

G(s) = $\frac{s + 3}{s^{3} + 2s^{2} + as + 1}$ ;

H(s) = $\frac{s + b}{s + 5}$

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) = $\frac{G(s)}{1 + G(s)H(s)}$

State space vectors:
$\dot{x}$ = 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) = $\frac{s^{2} + 8s + 5}{s^{4} + 7s^{3} + (11 + a)s^{2} + (5a + b + 4)s + (3b + 5)}$

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.

 

[KataKoniK]

Thank you for your post. It seems like you have made a good attempt at solving the problem. However, there are a few things that you may want to consider in order to fully answer the question.

Firstly, when solving for the range of values for a and b, it is important to keep in mind that the system must be completely controllable and observable. This means that the S and V matrices must have full rank in order for the system to be fully controllable and observable. Therefore, when calculating the closed-loop transfer function, it is important to consider the values of a and b that will result in a non-singular S and V matrix.

Secondly, when calculating the S and V matrices, it is important to consider the order of the system. In this case, the system is a fourth-order system, so the S and V matrices will have four columns. It may be helpful to write out the matrices in full and then try to find a pattern in the coefficients that will result in a non-singular matrix.

Lastly, it is important to note that the feedback gain, H(s), is a constant and does not change with the value of s. Therefore, when calculating the closed-loop transfer function, it may be helpful to treat the feedback gain separately and then substitute it into the overall transfer function.

I hope this helps in your attempt to solve the problem. Good luck on your exam!