RHP zeros in control systems

Question

In the slides of the following link is expressed the concept I am trying to understand : slides

I am studying control systems. I have seen that if I have a zero in the RHP, by using decoupling I can move the zero from one input/output channel to another, so we can choose where it create less problems.

I have that this concept is not really clear to me. With a decoupler I can transform the transfer matrix into a diagonal matrix, so that the first input influences just the first output and the same for the other channels, but what does it means that I can move the RHP zero?

For example, consider the transfer matrix:

\$G(s)=\begin{pmatrix} \frac{2}{s+1} & \frac{3}{s+2}\\ \frac{1}{s+1} & \frac{1}{s+1} \end{pmatrix}\$

So, I am considering a system with two inputs and two outputs.

I use a decoupler of the type:

\$\begin{pmatrix} -1 & 1\\ 1 & \frac{-2(s+2)}{3(s+1)} \end{pmatrix}\$

then the transfer matrix after applying the decoupler will be:

\$\begin{pmatrix} \frac{s-1}{s^{2}+3s+2} & 0\\ 0 & \frac{0.3s^{2}-0.3}{s^{3}+3s^{2}+3s+1} \end{pmatrix}\$

It can be seen that the original transfer matrix has a zero in the right half plane, so with positive real part, at \$s=1\$ since \$det(G(s))=0\$ for \$s=1\$ .If I apply the decoupler I have that in both channels I have a zero at \$s=1\$ , so with positive real part, which imposes limitations in the achivable bandwidth in the frequency response.

So, what does it means that with the decoupling it is possible to move the unstable zeros from one channel to the other?

Answer 1

When you decouple a MIMO system you turn it into a series of SISO systems. So you transform your system from a single 2 inputs 2 outputs system into two independent SISO systems by having a diagonal matrix. Moving the unstable zero to one of the inputs you can control which output is affected. The result is that only one of the outputs remains decoupled since you transform the diagonal matrix into a triangular matrix.

Using this method you can move the drawbacks around the system. Maybe one of the outputs moves slowly. Hence you don’t need good performance on said output. However you need a fast response on the other output.